The first table shows all symmetric positions, counts and depths. These numbers all agree with Radu and Kociemba.
Summary in HTM
dist             pos          modM      modM+inv mod48
---- --------------- ------------- ------------- -----
   0               1             1             1     1
   1              18             2             2    18
   2              51             5             5     3
   3             312            14             8    24
   4           1,335            61            42    39
   5           4,380           191           115    12
   6          17,782           784           461    22
   7          70,188         2,974         1,631    12
   8         229,336         9,729         5,294    40
   9         851,139        35,615        18,604     3
  10       2,989,204       125,103        65,125     4
  11       9,732,164       406,615       208,603    20
  12      35,024,904     1,461,865       746,313    24
  13     122,054,340     5,089,766     2,581,241    36
  14     436,197,214    18,183,156     9,195,333    46
  15   1,763,452,505    73,491,944    37,037,332    41
  16   8,035,307,127   334,845,106   168,373,488    39
  17  37,542,012,922 1,564,371,587   785,249,161    10
  18  95,387,902,305 3,974,821,633 1,994,938,045    33
  19  21,267,102,443   886,299,819   446,607,275    11
  20       1,091,994        46,514        32,625    42
---- --------------- ------------- ------------- -----
 tot 164,604,041,664 6,859,192,484 3,445,060,704     0
The remaining tables give counts for all the individual symmetry subgroups. In the tables below, the "pos" column gives all positions in the subgroup (including those that happen to have more symmetry); the "modM" and "modM+inv" columns however are for only positions with *exactly* that symmetry.
Summary of C_2_(a) in HTM
dist            pos          modM    modM+inv
---- -------------- ------------- -----------
   0              1             0           0
   1              6             0           0
   2             15             0           0
   3             72             8           4
   4            343            34          22
   5          1,060           123          69
   6          4,570           507         277
   7         17,960         2,182       1,143
   8         60,136         7,210       3,793
   9        226,199        28,017      14,361
  10        805,260        99,641      50,974
  11      2,602,404       323,655     164,112
  12      9,295,556     1,158,307     585,163
  13     30,949,920     3,863,416   1,945,189
  14    101,692,042    12,702,003   6,387,506
  15    363,402,817    45,411,240  22,803,625
  16  1,333,918,995   166,705,822  83,639,845
  17  4,550,677,966   568,759,399 285,178,698
  18  7,774,262,733   971,648,345 487,396,809
  19  1,120,268,931   139,966,952  70,596,762
  20         51,094         5,763       4,464
---- -------------- ------------- -----------
 tot 15,288,238,080 1,910,682,624 958,772,816

Summary of C_2_(b) in HTM
dist           pos        modM    modM+inv
---- ------------- ----------- -----------
   0             1           0           0
   2             3           0           0
   4             1           0           0
   6            44           2           1
   7           180          33          23
   8           460          78          49
   9         1,587         365         225
  10         8,072       1,861       1,122
  11        28,146       6,713       3,826
  12       119,054      29,113      16,120
  13       473,702     117,581      63,201
  14     2,168,270     540,239     282,525
  15    12,440,909   3,107,127   1,592,911
  16    84,769,773  21,185,220  10,730,009
  17   548,185,762 137,030,323  69,003,014
  18 1,583,371,387 395,812,880 199,227,132
  19   316,458,701  79,102,176  40,162,141
  20        13,628       3,297       2,725
---- ------------- ----------- -----------
 tot 2,548,039,680 636,937,008 321,085,024

Summary of C_2h_(a) in HTM
dist     pos    modM modM+inv
---- ------- ------- --------
   0       1       0        0
   2       5       0        0
   4      23       2        1
   5       8       2        1
   6     102       5        4
   7      34       7        7
   8     322      42       35
   9     137      33       29
  10     500     115       89
  11   1,768     423      327
  12   2,916     622      521
  13   6,002   1,285    1,043
  14  12,426   2,489    1,997
  15  19,795   4,636    3,566
  16  46,029  11,153    8,057
  17 108,816  26,636   17,388
  18 239,765  58,874   35,586
  19 149,539  36,878   21,920
  20   1,636     350      293
---- ------- ------- --------
 tot 589,824 143,552   90,864

Summary of C_2h_(b) in HTM
dist    pos   modM modM+inv
---- ------ ------ --------
   0      1      0        0
   2      1      0        0
   4      1      0        0
   6      6      1        1
   7      6      1        1
   8     28      7        7
   9     23     10       10
  10     90     33       33
  11    200     72       72
  12    474    192      191
  13    786    329      323
  14  1,470    671      634
  15  3,585  1,720    1,471
  16  8,951  4,360    3,496
  17 21,560 10,513    7,953
  18 42,015 20,776   15,207
  19 18,953  9,360    7,384
  20    154     71       69
---- ------ ------ --------
 tot 98,304 48,116   36,852

Summary of C_2v_(a1) in HTM
dist    pos   modM modM+inv
---- ------ ------ --------
   0      1      0        0
   1      2      0        0
   2      1      0        0
   4      1      0        0
   5      4      0        0
   6      4      0        0
   7     10      1        1
   8     26      1        1
   9      9      1        1
  10    146     22       20
  11    462     92       73
  12    756    134      107
  13    832    130      102
  14  3,112    704      554
  15  5,249  1,249      993
  16  9,769  2,347    1,824
  17 15,690  3,735    2,933
  18 22,009  5,340    4,174
  19  7,285  1,757    1,530
  20    168     39       39
---- ------ ------ --------
 tot 65,536 15,552   12,352

Summary of C_2v_(a2) in HTM
dist       pos    modM modM+inv
---- --------- ------- --------
   0         1       0        0
   1         2       0        0
   2         3       0        0
   3         8       2        1
   4        15       0        0
   5        36       8        6
   6        92       4        3
   7       140      33       21
   8       230      17       13
   9       167      34       26
  10       494     111       86
  11     1,184     280      203
  12     4,284     960      715
  13     9,252   2,080    1,551
  14    20,858   4,549    3,313
  15    32,193   7,712    5,542
  16    89,617  22,012   14,838
  17   223,444  55,305   34,551
  18   505,801 125,363   72,962
  19   289,047  71,764   41,100
  20     2,780     646      445
---- --------- ------- --------
 tot 1,179,648 290,880  175,376

Summary of C_2v_(b) in HTM
dist    pos   modM modM+inv
---- ------ ------ --------
   0      1      0        0
   2      1      0        0
   4      1      0        0
   6     10      3        3
   7      6      1        1
   8     20      3        3
   9     39     18       18
  10     92     34       34
  11    172     58       58
  12    456    183      180
  13    714    293      289
  14  1,468    671      622
  15  3,127  1,491    1,292
  16  8,993  4,381    3,625
  17 21,186 10,329    7,840
  18 42,201 20,873   15,168
  19 19,617  9,696    7,638
  20    200     94       93
---- ------ ------ --------
 tot 98,304 48,128   36,864

Summary of C_3 in HTM
dist       pos    modM modM+inv
---- --------- ------- --------
   0         1       0        0
   6         1       0        0
   8         4       0        0
   9        18       3        3
  10        28       6        5
  11        26       5        3
  12       162      32       28
  13       288      71       46
  14     1,711     418      260
  15     8,924   2,216    1,238
  16    67,176  16,705    9,105
  17   481,063 120,050   62,806
  18 2,083,458 520,076  271,003
  19 1,133,447 282,475  152,809
  20     2,829     659      576
---- --------- ------- --------
 tot 3,779,136 942,716  497,882

Summary of C_3v in HTM
dist pos modM modM+inv
---- --- ---- --------
   0   1    0        0
   6   1    0        0
  13   4    2        2
  14   3    1        1
  15   6    3        3
  16   4    2        2
  17   7    2        2
  18  12    4        4
  19   9    2        2
  20   1    0        0
---- --- ---- --------
 tot  48   16       16

Summary of C_4 in HTM
dist     pos   modM modM+inv
---- ------- ------ --------
   0       1      0        0
   1       6      1        1
   2       9      1        1
   4       1      0        0
   5      12      2        1
   6      50      8        5
   7      42      7        5
   8      56      7        5
   9      33      7        7
  10     188     42       41
  11     758    177      131
  12   1,720    381      267
  13   2,604    583      389
  14   4,040    958      674
  15   9,401  2,302    1,507
  16  21,803  5,362    3,148
  17  46,296 11,461    6,475
  18  48,801 12,066    7,118
  19  11,413  2,756    1,852
  20     222     39       37
---- ------- ------ --------
 tot 147,456 36,160   21,664

Summary of C_4h in HTM
dist   pos modM modM+inv
---- ----- ---- --------
   0     1    0        0
   2     3    1        1
   4     1    0        0
   6    10    3        3
   7     2    1        1
   8    12    3        3
   9     1    0        0
  10     2    0        0
  11    20   10       10
  12    62   28       27
  13    84   35       35
  14    58   20       15
  15    71   34       29
  16   193   87       76
  17   210   95       86
  18   387  184      162
  19   353  171      132
  20    66   32       28
---- ----- ---- --------
 tot 1,536  704      608

Summary of C_4v in HTM
dist   pos modM modM+inv
---- ----- ---- --------
   0     1    0        0
   1     2    1        1
   2     1    0        0
   4     1    0        0
   5     4    2        1
   6     4    0        0
   7     2    1        1
   8    12    3        2
   9     3    1        1
  10    14    6        6
  11    20   10       10
  12    80   37       32
  13   154   70       57
  14   152   67       63
  15   101   49       39
  16   141   61       57
  17   186   83       75
  18   125   53       51
  19    19    4        4
  20     2    0        0
---- ----- ---- --------
 tot 1,024  448      400

Summary of C_i in HTM
dist            pos          modM    modM+inv
---- -------------- ------------- -----------
   0              1             0           0
   2              9             0           0
   4             63             1           1
   5            120             4           4
   6            694            22          18
   7          2,562           102          66
   8         11,338           440         262
   9         44,853         1,846         998
  10        194,740         8,036       4,360
  11        788,992        32,617      16,884
  12      3,315,172       137,692      71,004
  13     13,445,694       559,379     284,571
  14     54,961,138     2,288,348   1,160,146
  15    271,122,047    11,293,497   5,689,550
  16  1,585,605,201    66,058,983  33,189,184
  17  9,134,397,172   380,581,022 190,837,230
  18 27,801,243,853 1,158,344,598 580,705,104
  19  6,999,321,491   291,614,579 146,586,998
  20        259,100        10,540       7,188
---- -------------- ------------- -----------
 tot 45,864,714,240 1,910,931,706 958,553,568

Summary of C_s_(a) in HTM
dist            pos          modM      modM+inv
---- -------------- ------------- -------------
   0              1             0             0
   1              4             0             0
   2             13             1             1
   3             48             4             3
   4            159            17            12
   5            464            48            32
   6          1,554           176           104
   7          4,758           555           324
   8         13,810         1,662           931
   9         40,353         4,977         2,719
  10        114,084        14,062         7,601
  11        324,496        40,016        21,300
  12      1,019,512       125,984        66,943
  13      4,128,898       513,072       266,473
  14     20,060,486     2,501,038     1,280,752
  15    103,107,447    12,877,423     6,527,551
  16    607,357,293    75,889,571    38,240,126
  17  3,601,456,840   450,107,871   226,052,001
  18 11,082,306,785 1,385,122,357   695,129,013
  19  2,925,751,099   365,623,453   184,049,962
  20        197,592        23,793        15,592
---- -------------- ------------- -------------
 tot 18,345,885,696 2,292,846,080 1,151,661,440

Summary of C_s_(b) in HTM
dist         pos        modM   modM+inv
---- ----------- ----------- ----------
   0           1           0          0
   1           2           0          0
   2           1           0          0
   4           1           0          0
   5           4           0          0
   6          24           3          2
   7          50           9          5
   8         126          20         12
   9         533         117         83
  10       1,956         419        267
  11       6,478       1,439        883
  12      25,130       5,906      3,422
  13     114,700      28,154     15,734
  14     546,558     135,189     72,285
  15   3,102,247     772,641    402,389
  16  19,703,503   4,919,061  2,517,679
  17 110,503,448  27,611,515 14,012,771
  18 252,385,203  63,069,968 32,023,935
  19  38,279,025   9,558,403  4,965,270
  20       4,290         948        863
---- ----------- ----------- ----------
 tot 424,673,280 106,103,792 54,015,600

Summary of D_2_(edge) in HTM
dist    pos   modM modM+inv
---- ------ ------ --------
   0      1      0        0
   2      3      0        0
   4      1      0        0
   6     28      4        4
   7     44      6        6
   8    128     22       18
   9     71     10        9
  10    494    111       91
  11  1,030    230      172
  12  1,844    403      304
  13  2,134    461      373
  14  4,604  1,064      861
  15  5,959  1,396    1,111
  16 11,355  2,667    2,051
  17 22,676  5,397    3,980
  18 36,221  8,736    5,922
  19 11,617  2,713    1,807
  20     94     12       11
---- ------ ------ --------
 tot 98,304 23,232   16,720

Summary of D_2_(face) in HTM
dist     pos   modM modM+inv
---- ------- ------ --------
   0       1      0        0
   2       9      0        0
   4      63      4        3
   5      24      2        1
   6     424     27       18
   7     246     18       10
   8   2,000    152       98
   9   1,593    132       79
  10   6,282    515      321
  11   7,776    641      387
  12  17,780  1,428      881
  13  20,250  1,608      897
  14  28,698  2,185    1,257
  15  23,343  1,850    1,135
  16  45,731  3,708    2,207
  17  66,418  5,370    3,104
  18  57,353  4,456    2,601
  19  16,711  1,256      749
  20     210      4        4
---- ------- ------ --------
 tot 294,912 23,356   13,752

Summary of D_2d_(edge) in HTM
dist   pos  modM modM+inv
---- ----- ----- --------
   0     1     0        0
   2     1     0        0
   4     1     0        0
   6     4     0        0
   7     6     3        3
   8    22     8        8
   9    27    13       11
  10    22    10        6
  11    44    22       18
  12    88    41       39
  13   128    57       52
  14   224   103       98
  15   209   103       93
  16   435   208      161
  17   504   242      192
  18   797   389      288
  19   523   256      172
  20    36    17       11
---- ----- ----- --------
 tot 3,072 1,472    1,152

Summary of D_2d_(face) in HTM
dist pos modM modM+inv
---- --- ---- --------
   0   1    0        0
   2   1    0        0
   4   1    0        0
   6   4    0        0
   8   8    1        1
   9   1    0        0
  10  24   11        9
  11  18    9        9
  12  62   28       27
  13  58   22       19
  14  54   18       16
  15  13    5        5
  16  47   14       12
  17  76   28       24
  18 107   44       42
  19  35   12       12
  20   2    0        0
---- --- ---- --------
 tot 512  192      176

Summary of D_2h_(edge) in HTM
dist   pos modM modM+inv
---- ----- ---- --------
   0     1    0        0
   2     1    0        0
   4     1    0        0
   6     4    0        0
   7     4    2        2
   8    14    4        4
   9     3    1        1
  10    24   11       11
  11    56   28       28
  12    90   42       42
  13   128   57       57
  14   126   54       54
  15   145   71       71
  16   231  106      106
  17   528  254      254
  18   455  218      218
  19   225  107      107
  20    12    5        5
---- ----- ---- --------
 tot 2,048  960      960

Summary of D_2h_(face) in HTM
dist    pos  modM modM+inv
---- ------ ----- --------
   0      1     0        0
   2      3     0        0
   4     15     2        2
   6     76    11       11
   8    140    20       20
   9      3     0        0
  10     18     2        2
  12    288    45       45
  13    678   106      106
  14  2,322   378      375
  15  1,041   172      164
  16  1,031   161      137
  17  1,574   252      244
  18  3,465   568      406
  19  1,471   240      143
  20    162    25       17
---- ------ ----- --------
 tot 12,288 1,982    1,672

Summary of D_3 in HTM
dist pos modM modM+inv
---- --- ---- --------
   0   1    0        0
   6   1    0        0
  11   2    1        1
  12   4    2        2
  14  13    6        5
  15  10    5        5
  16  28   14       11
  17  55   26       18
  18 174   85       62
  19 135   65       52
  20   9    4        4
---- --- ---- --------
 tot 432  208      160

Summary of D_3d in HTM
dist pos modM modM+inv
---- --- ---- --------
   0   1    0        0
   6   1    0        0
  14   1    1        1
  17   3    3        3
  18   4    4        4
  19   5    4        4
  20   1    0        0
---- --- ---- --------
 tot  16   12       12

Summary of D_4 in HTM
dist pos modM modM+inv
---- --- ---- --------
   0   1    0        0
   2   3    1        1
   4   1    0        0
   6  12    4        3
   7  10    5        4
   8  16    5        5
   9   3    1        1
  10   8    3        3
  11  10    5        5
  12  66   30       27
  13  62   24       21
  14  34    8        8
  15  27   12       12
  16  59   20       16
  17  96   38       34
  18  63   22       22
  19  39   14       14
  20   2    0        0
---- --- ---- --------
 tot 512  192      176

Summary of D_4h in HTM
dist pos modM modM+inv
---- --- ---- --------
   0   1    0        0
   2   1    1        1
   4   1    1        1
   6   4    3        3
   8   6    6        6
   9   1    1        1
  10   2    2        2
  12   6    6        6
  13  14   14       14
  14  18   18       18
  15   3    3        3
  16  19   19       19
  17  20   20       20
  18  19   19       19
  19  11   10       10
  20   2    1        1
---- --- ---- --------
 tot 128  124      124

Summary of O_h in HTM
dist pos modM modM+inv
---- --- ---- --------
   0   1    1        1
   6   1    1        1
  19   1    1        1
  20   1    1        1
---- --- ---- --------
 tot   4    4        4

Summary of S_4 in HTM
dist     pos    modM modM+inv
---- ------- ------- --------
   0       1       0        0
   2       3       0        0
   4       1       0        0
   6      10       0        0
   7      36       7        7
   8      94      16       16
   9     127      25       19
  10     236      48       40
  11     522     110       90
  12   1,224     256      210
  13   3,990     937      697
  14   8,076   1,944    1,283
  15  20,147   4,965    3,011
  16  52,237  12,900    7,588
  17 125,622  31,218   17,201
  18 176,669  43,854   24,049
  19  52,945  13,014    7,554
  20     428      82       75
---- ------- ------- --------
 tot 442,368 109,376   61,840

Summary of S_6 in HTM
dist   pos  modM modM+inv
---- ----- ----- --------
   0     1     0        0
   6     1     0        0
   8     4     1        1
   9     6     3        2
  10     4     2        2
  11     4     2        1
  12    22    11        8
  14    25    12       10
  15    44    22       16
  16   324   158      106
  17   799   397      261
  18 2,956 1,476      979
  19 3,401 1,698    1,143
  20   185    88       81
---- ----- ----- --------
 tot 7,776 3,870    2,610

Summary of T in HTM
dist pos modM modM+inv
---- --- ---- --------
   0   1    0        0
   6   1    0        0
   8   2    0        0
  12   8    2        2
  16   8    0        0
  17  10    2        2
  18  20    5        5
  19  13    3        3
  20   9    0        0
---- --- ---- --------
 tot  72   12       12

Summary of T_h in HTM
dist pos modM modM+inv
---- --- ---- --------
   0   1    0        0
   6   1    0        0
   8   2    1        1
  16   8    4        3
  17   2    1        1
  19   1    0        0
  20   9    4        3
---- --- ---- --------
 tot  24   10        8

Summary of hard positions in HTM
       symm       pos   modM modM+inv
----------- --------- ------ --------
        O_h         1      1        1
       D_4h         3      1        1
        T_h         8      4        3
D_2h_(edge)        30      5        5
        D_3        32      4        4
 D_2_(face)        48      4        4
D_2d_(edge)       102     17       11
 D_2_(edge)       144     12       11
D_2h_(face)       150     25       17
       C_4h       192     32       28
  C_2v_(a1)       468     39       39
        C_4       468     39       37
        S_6       704     88       81
   C_2h_(b)       852     71       69
        S_4       984     82       75
   C_2v_(b)     1,128     94       93
   C_2h_(a)     4,200    350      293
  C_2v_(a2)     7,752    646      445
        C_3    10,544    659      576
    C_s_(b)    22,752    948      863
    C_2_(b)    79,128  3,297    2,725
    C_2_(a)   138,312  5,763    4,464
        C_i   252,960 10,540    7,188
    C_s_(a)   571,032 23,793   15,592
----------- --------- ------ --------
        tot 1,091,994 46,514   32,625