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The database currently contains $512{,}613$ transitive subgroups of $S_n$, including all subgroups (up to conjugacy) for $n \le 47$ and $n \ne 32$. Among the $2{,}801{,}324$ groups in degree $32$, all those with order less than $512$ or greater than $40{,}000{,}000{,}000$ are included.

Distribution of degrees and orders

order
1-15 16-31 32-63 64-127 128-255 256-511 512-1023 1024-2047 2048-65535 65536-40000000000 40000000000- Total
degree
1-3 4 4
100.00% 0.00%
4-7 12 7 6 3 1 1 1 2 33
36.36% 21.21% 18.18% 9.09% 3.03% 3.03% 3.03% 6.06% 0.01%
8 5 9 11 9 7 3 2 2 2 50
10.00% 18.00% 22.00% 18.00% 14.00% 6.00% 4.00% 4.00% 4.00% 0.01%
9-11 5 8 10 12 14 10 10 6 6 6 87
5.75% 9.20% 11.49% 13.79% 16.09% 11.49% 11.49% 6.90% 6.90% 6.90% 0.02%
12 5 10 18 43 48 33 37 40 59 8 301
1.66% 3.32% 5.98% 14.29% 15.95% 10.96% 12.29% 13.29% 19.60% 2.66% 0.06%
13-15 4 5 9 7 10 18 17 17 55 30 4 176
2.27% 2.84% 5.11% 3.98% 5.68% 10.23% 9.66% 9.66% 31.25% 17.05% 2.27% 0.03%
16 14 52 130 251 330 304 248 532 91 2 1954
0.72% 2.66% 6.65% 12.85% 16.89% 15.56% 12.69% 27.23% 4.66% 0.10% 0.38%
17-23 17 39 64 92 136 149 158 921 732 40 2348
0.72% 1.66% 2.73% 3.92% 5.79% 6.35% 6.73% 39.22% 31.18% 1.70% 0.46%
24 15 44 144 375 778 1315 2315 11075 8868 71 25000
0.06% 0.18% 0.58% 1.50% 3.11% 5.26% 9.26% 44.30% 35.47% 0.28% 4.88%
25-31 19 34 64 145 201 249 365 2277 6383 548 10285
0.18% 0.33% 0.62% 1.41% 1.95% 2.42% 3.55% 22.14% 62.06% 5.33% 2.01%
32 51 369 1809 7322 7 8 1 2001 11568
0.44% 3.19% 15.64% 63.30% 0.06% 0.07% 0.01% 17.30% 2.26%
33-35 4 9 10 11 11 8 107 384 140 684
0.58% 1.32% 1.46% 1.61% 1.61% 1.17% 15.64% 56.14% 20.47% 0.13%
36 14 85 210 403 886 1425 14354 96392 7510 121279
0.01% 0.07% 0.17% 0.33% 0.73% 1.17% 11.84% 79.48% 6.19% 23.66%
37-39 5 10 10 11 9 11 87 187 63 393
1.27% 2.54% 2.54% 2.80% 2.29% 2.80% 22.14% 47.58% 16.03% 0.08%
40 14 51 130 246 544 779 17508 281759 14811 315842
0.00% 0.02% 0.04% 0.08% 0.17% 0.25% 5.54% 89.21% 4.69% 61.61%
41-47 17 35 57 114 156 185 1748 13777 6520 22609
0.08% 0.15% 0.25% 0.50% 0.69% 0.82% 7.73% 60.94% 28.84% 4.41%
Total 35 104 328 1035 3169 9617 3697 5567 48734 408617 31710 512613
0.01% 0.02% 0.06% 0.20% 0.62% 1.88% 0.72% 1.09% 9.51% 79.71% 6.19%

Distribution of solvability and degrees

degree
1-3 4-7 8 9-11 12 13-15 16 17-23 24 25-31 32 33-35 36 37-39 40 41-47 Total
solvable
no 9 5 29 36 70 49 542 807 2688 162 331 3252 72 24244 6517 38813
27.27% 10.00% 33.33% 11.96% 39.77% 2.51% 23.08% 3.23% 26.14% 1.40% 48.39% 2.68% 18.32% 7.68% 28.82% 7.57%
yes 4 24 45 58 265 106 1905 1806 24193 7597 11406 353 118027 321 291598 16092 473800
100.00% 72.73% 90.00% 66.67% 88.04% 60.23% 97.49% 76.92% 96.77% 73.86% 98.60% 51.61% 97.32% 81.68% 92.32% 71.18% 92.43%
Total 4 33 50 87 301 176 1954 2348 25000 10285 11568 684 121279 393 315842 22609 512613
0.00% 0.01% 0.01% 0.02% 0.06% 0.03% 0.38% 0.46% 4.88% 2.01% 2.26% 0.13% 23.66% 0.08% 61.61% 4.41%

Distribution of primitivity and degrees

degree
1-3 4-7 8 9-11 12 13-15 16 17-23 24 25-31 32 33-35 36 37-39 40 41-47 Total
primitive
no 15 43 59 295 157 1932 2302 24995 10197 11566 672 121257 376 315834 22564 512264
45.45% 86.00% 67.82% 98.01% 89.20% 98.87% 98.04% 99.98% 99.14% 99.98% 98.25% 99.98% 95.67% 100.00% 99.80% 99.93%
yes 4 18 7 28 6 19 22 46 5 88 2 12 22 17 8 45 349
100.00% 54.55% 14.00% 32.18% 1.99% 10.80% 1.13% 1.96% 0.02% 0.86% 0.02% 1.75% 0.02% 4.33% 0.00% 0.20% 0.07%
Total 4 33 50 87 301 176 1954 2348 25000 10285 11568 684 121279 393 315842 22609 512613
0.00% 0.01% 0.01% 0.02% 0.06% 0.03% 0.38% 0.46% 4.88% 2.01% 2.26% 0.13% 23.66% 0.08% 61.61% 4.41%

Distribution of number of arithmetic equivalent siblings and degrees

degree
1-3 4-7 8 9-11 12 13-15 16 17-23 24 25-31 32 33-35 36 37-39 40 41-47 Total
arithmetic equivalent count
not computed 6 76412 235842 22527 334787
0.06% 63.01% 74.67% 99.64% 65.31%
0 4 32 48 86 295 167 1784 2317 23882 10186 9254 670 43736 388 80000 82 172931
100.00% 96.97% 96.00% 98.85% 98.01% 94.89% 91.30% 98.68% 95.53% 99.04% 80.00% 97.95% 36.06% 98.73% 25.33% 0.36% 33.74%
1 1 2 1 6 9 166 31 1106 81 2020 14 1103 5 4545
3.03% 4.00% 1.15% 1.99% 5.11% 8.50% 1.32% 4.42% 0.79% 17.46% 2.05% 0.91% 1.27% 0.89%
2 11 11 6 22 50
0.04% 0.11% 0.05% 0.02% 0.01%
3 4 270 6 280
0.20% 2.33% 0.00% 0.05%
5 1 1 2
0.00% 0.01% 0.00%
7 18 18
0.16% 0.00%
Total 4 33 50 87 301 176 1954 2348 25000 10285 11568 684 121279 393 315842 22609 512613
0.00% 0.01% 0.01% 0.02% 0.06% 0.03% 0.38% 0.46% 4.88% 2.01% 2.26% 0.13% 23.66% 0.08% 61.61% 4.41%

Distribution of degrees and nilpotency classes

nilpotency class
-1 0 1 2 3 4 5 6 7 8 9 Total
degree
1-3 1 1 2 4
25.00% 25.00% 50.00% 0.00%
4-7 27 5 1 33
81.82% 15.15% 3.03% 0.01%
8 24 3 8 11 4 50
48.00% 6.00% 16.00% 22.00% 8.00% 0.01%
9-11 80 4 2 1 87
91.95% 4.60% 2.30% 1.15% 0.02%
12 298 2 1 301
99.00% 0.66% 0.33% 0.06%
13-15 173 3 176
98.30% 1.70% 0.03%
16 527 5 55 301 420 268 181 114 83 1954
26.97% 0.26% 2.81% 15.40% 21.49% 13.72% 9.26% 5.83% 4.25% 0.38%
17-23 2335 9 3 1 2348
99.45% 0.38% 0.13% 0.04% 0.46%
24 24974 3 8 11 4 25000
99.90% 0.01% 0.03% 0.04% 0.02% 4.88%
25-31 10040 11 14 48 54 52 16 24 16 10 10285
97.62% 0.11% 0.14% 0.47% 0.53% 0.51% 0.16% 0.23% 0.16% 0.10% 2.01%
32 2390 7 499 5048 2967 563 94 11568
20.66% 0.06% 4.31% 43.64% 25.65% 4.87% 0.81% 2.26%
33-35 681 3 684
99.56% 0.44% 0.13%
36 121264 4 8 3 121279
99.99% 0.00% 0.01% 0.00% 23.66%
37-39 390 3 393
99.24% 0.76% 0.08%
40 315816 3 8 11 4 315842
99.99% 0.00% 0.00% 0.00% 0.00% 61.61%
41-47 22596 9 3 1 22609
99.94% 0.04% 0.01% 0.00% 4.41%
Total 501616 1 76 610 5436 3453 883 291 138 99 10 512613
97.85% 0.00% 0.01% 0.12% 1.06% 0.67% 0.17% 0.06% 0.03% 0.02% 0.00%