← #30843537
Subgroups — #30843537
Order 17152.
706 conjugacy classes of subgroups.
The subgroup lattice has 256 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | #22570796 | C2 | 1 | 2 | yes | maximal |
| 2 | #22570793 | C2 | 1 | 2 | yes | maximal |
| 3 | #22570720 | C2 | 1 | 2 | yes | maximal |
| 4 | #22559827 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | #22559818 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | #22559807 | C2 | 1 | 2 | yes | maximal, complemented |
| 7 | #22559803 | C2 | 1 | 2 | yes | maximal, complemented |
| 8 | C82×D67 | C2 | 1 | 1 | yes | maximal, characteristic |
| 9 | C67×#1713 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 10 | #16863951 | C22 | 1 | 4 | yes | — |
| 11 | #16863801 | C22 | 1 | 2 | yes | — |
| 12 | #16863801 | C22 | 1 | 2 | yes | — |
| 13 | #16863797 | — | 2 | 4 | no | — |
| 14 | #16863792 | C22 | 1 | 4 | yes | — |
| 15 | #16863776 | C22 | 1 | 2 | yes | — |
| 16 | #16863748 | C22 | 1 | 4 | yes | — |
| 17 | #16863746 | C22 | 1 | 4 | yes | — |
| 18 | #16863638 | — | 2 | 4 | no | — |
| 19 | #16863634 | — | 2 | 4 | no | — |
| 20 | Q16×Dic67 | — | 2 | 4 | no | — |
| 21 | #16863490 | — | 2 | 4 | no | — |
| 22 | #16863433 | — | 2 | 4 | no | — |
| 23 | C4×C8×D67 | C22 | 1 | 1 | yes | characteristic |
| 24 | C4×C8×D67 | C22 | 1 | 1 | yes | characteristic |
| 25 | C4×C8×D67 | C22 | 1 | 1 | yes | characteristic |
| 26 | C8×C67⋊C8 | C22 | 1 | 1 | yes | characteristic |
| 27 | C8×C67⋊C8 | C22 | 1 | 1 | yes | characteristic |
| 28 | C8×C67⋊C8 | C22 | 1 | 1 | yes | characteristic |
| 29 | #500×C67 | C22 | 1 | 2 | yes | — |
| 30 | #500×C67 | C22 | 1 | 2 | yes | — |
| 31 | #494×C67 | C22 | 1 | 2 | yes | — |
| 32 | C82×C67 | C22 | 1 | 1 | yes | characteristic |
| 33 | C2×Dic268 | — | 2 | 4 | no | — |
| 34 | C2×C8×D67 | D4 | 1 | 2 | yes | — |
| 35 | C2×C8×D67 | D4 | 1 | 2 | yes | — |
| 36 | C2×C8×D67 | D4 | 1 | 2 | yes | — |
| 37 | #14524043 | C23 | 1 | 4 | yes | — |
| 38 | #14524034 | — | 2 | 4 | no | — |
| 39 | #14524023 | — | 2 | 4 | no | — |
| 40 | #14524019 | — | 2 | 4 | no | — |
| 41 | Q8×Dic67 | — | 2 | 4 | no | — |
| 42 | #14524008 | — | 2 | 4 | no | — |
| 43 | #14524001 | — | 2 | 4 | no | — |
| 44 | C42×D67 | C23 | 1 | 1 | yes | characteristic |
| 45 | #14523977 | — | 2 | 4 | no | — |
| 46 | #14523977 | — | 2 | 4 | no | — |
| 47 | #14523974 | — | 2 | 8 | no | — |
| 48 | #14523967 | — | 2 | 8 | no | — |
| 49 | #14523966 | — | 2 | 8 | no | — |
| 50 | C4×C67⋊C8 | C23 | 1 | 1 | yes | characteristic |
| 51 | C4×C67⋊C8 | C23 | 1 | 1 | yes | characteristic |
| 52 | C4×C67⋊C8 | C23 | 1 | 1 | yes | characteristic |
| 53 | C8×Dic67 | D4 | 1 | 2 | yes | — |
| 54 | C8×Dic67 | D4 | 1 | 2 | yes | — |
| 55 | C8×Dic67 | D4 | 1 | 2 | yes | — |
| 56 | C2×Q16×C67 | — | 2 | 4 | no | — |
| 57 | C4⋊Q8×C67 | C23 | 1 | 4 | yes | — |
| 58 | #107×C67 | — | 2 | 4 | no | — |
| 59 | #107×C67 | — | 2 | 4 | no | — |
| 60 | C4×C8×C67 | C23 | 1 | 1 | yes | characteristic |
| 61 | C4×C8×C67 | C23 | 1 | 1 | yes | characteristic |
| 62 | C4×C8×C67 | C23 | 1 | 1 | yes | characteristic |
| 63 | C2×Dic134 | — | 2 | 4 | no | — |
| 64 | C2×C4×D67 | C2×D4 | 1 | 1 | yes | characteristic |
| 65 | C2×C4×D67 | C2×D4 | 1 | 1 | yes | characteristic |
| 66 | C2×C4×D67 | C2×D4 | 1 | 1 | yes | characteristic |
| 67 | Dic268 | — | 4 | 8 | no | — |
| 68 | C2×C67⋊C8 | C2×D4 | 1 | 2 | yes | — |
| 69 | C2×C67⋊C8 | C2×D4 | 1 | 2 | yes | — |
| 70 | C2×C67⋊C8 | C2×D4 | 1 | 2 | yes | — |
| 71 | C8×D67 | — | 2 | 4 | no | — |
| 72 | C8×D67 | — | 2 | 4 | no | — |
| 73 | C8×D67 | Q16 | 1 | 8 | yes | complemented |
| 74 | C4⋊Dic67 | — | 2 | 4 | no | — |
| 75 | C4⋊Dic67 | — | 2 | 4 | no | — |
| 76 | C4⋊Dic67 | — | 4 | 4 | no | — |
| 77 | Dic67⋊C4 | — | 4 | 4 | no | — |
| 78 | (C2×D67)⋊C4 | — | 4 | 4 | no | — |
| 79 | C4×Dic67 | — | 4 | 4 | no | — |
| 80 | C4×Dic67 | C2×D4 | 1 | 1 | yes | characteristic |
| 81 | C4×Dic67 | C2×D4 | 1 | 1 | yes | characteristic |
| 82 | C4×Dic67 | C2×D4 | 1 | 1 | yes | characteristic |
| 83 | C2×Q8×C67 | — | 2 | 4 | no | — |
| 84 | Q16×C67 | — | 4 | 8 | no | — |
| 85 | C2×C8×C67 | C2×D4 | 1 | 2 | yes | — |
| 86 | C2×C8×C67 | C2×D4 | 1 | 2 | yes | — |
| 87 | C2×C8×C67 | C2×D4 | 1 | 2 | yes | — |
| 88 | C4⋊C4×C67 | — | 2 | 4 | no | — |
| 89 | C4⋊C4×C67 | — | 2 | 4 | no | — |
| 90 | C42×C67 | C24 | 1 | 1 | yes | characteristic |
| 91 | C22×D67 | #127 | 1 | 1 | yes | characteristic |
| 92 | Q8×C67 | — | 4 | 8 | no | — |
| 93 | C2×C4×C67 | — | 4 | 4 | no | — |
| 94 | C2×C4×C67 | C22×D4 | 1 | 1 | yes | characteristic |
| 95 | C2×C4×C67 | C22×D4 | 1 | 1 | yes | characteristic |
| 96 | C2×C4×C67 | C22×D4 | 1 | 1 | yes | characteristic |
| 97 | C2×Dic67 | — | 4 | 4 | no | — |
| 98 | C2×Dic67 | #127 | 1 | 1 | yes | characteristic |
| 99 | C2×Dic67 | #127 | 1 | 1 | yes | characteristic |
| 100 | C2×Dic67 | #127 | 1 | 1 | yes | characteristic |
| 101 | C4×D67 | — | 2 | 2 | no | — |
| 102 | C4×D67 | — | 2 | 2 | no | — |
| 103 | C4×D67 | C2×Q16 | 1 | 4 | yes | — |
| 104 | Dic134 | — | 4 | 8 | no | — |
| 105 | C8×C67 | C2×Q16 | 1 | 4 | yes | — |
| 106 | C8×C67 | #135 | 1 | 4 | yes | — |
| 107 | C8×C67 | #135 | 1 | 4 | yes | — |
| 108 | C67⋊C8 | — | 2 | 2 | no | — |
| 109 | C67⋊C8 | — | 2 | 2 | no | — |
| 110 | C67⋊C8 | C2×Q16 | 1 | 4 | yes | — |
| 111 | C22×C67 | C2×#127 | 1 | 1 | yes | characteristic |
| 112 | C2×D67 | — | 2 | 1 | no | — |
| 113 | C2×D67 | — | 2 | 1 | no | — |
| 114 | C2×D67 | #494 | 1 | 2 | yes | — |
| 115 | C4×C67 | — | 8 | 4 | no | — |
| 116 | C4×C67 | C22×Q16 | 1 | 2 | yes | — |
| 117 | C4×C67 | C2×#135 | 1 | 2 | yes | — |
| 118 | C4×C67 | C2×#135 | 1 | 2 | yes | — |
| 119 | Dic67 | — | 2 | 1 | no | — |
| 120 | Dic67 | — | 2 | 1 | no | — |
| 121 | Dic67 | — | 8 | 4 | no | — |
| 122 | Dic67 | #494 | 1 | 2 | yes | — |
| 123 | #19604 | — | 67 | 1 | no | maximal, Sylow |
| 124 | C2×C67 | C2×#494 | 1 | 1 | yes | characteristic |
| 125 | C2×C67 | #3147 | 1 | 1 | yes | characteristic |
| 126 | C2×C67 | #3147 | 1 | 1 | yes | characteristic |
| 127 | D67 | — | 2 | 2 | no | — |
| 128 | #3170 | — | 67 | 2 | no | — |
| 129 | #3170 | — | 67 | 2 | no | — |
| 130 | #3160 | — | 67 | 2 | no | — |
| 131 | #1713 | — | 67 | 1 | no | — |
| 132 | #1713 | D67 | 1 | 1 | yes | characteristic, complemented |
| 133 | #1710 | — | 67 | 2 | no | — |
| 134 | #1707 | — | 67 | 2 | no | — |
| 135 | #1707 | — | 67 | 2 | no | — |
| 136 | C2×C82 | — | 67 | 1 | no | — |
| 137 | C67 | #19604 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 138 | #533 | — | 67 | 4 | no | — |
| 139 | #500 | — | 67 | 2 | no | — |
| 140 | #500 | — | 67 | 2 | no | — |
| 141 | #500 | C2×D67 | 1 | 2 | yes | — |
| 142 | #500 | C2×D67 | 1 | 2 | yes | — |
| 143 | #499 | — | 134 | 4 | no | — |
| 144 | #498 | — | 67 | 4 | no | — |
| 145 | #494 | — | 67 | 2 | no | — |
| 146 | #494 | C2×D67 | 1 | 2 | yes | — |
| 147 | #487 | — | 67 | 4 | no | — |
| 148 | #487 | — | 67 | 4 | no | — |
| 149 | #467 | — | 134 | 4 | no | — |
| 150 | #467 | — | 134 | 4 | no | — |
| 151 | C4×Q16 | — | 134 | 4 | no | — |
| 152 | C4×Q16 | — | 134 | 4 | no | — |
| 153 | #427 | — | 134 | 4 | no | — |
| 154 | C2×C4×C8 | — | 67 | 1 | no | — |
| 155 | C2×C4×C8 | — | 67 | 1 | no | — |
| 156 | C2×C4×C8 | — | 67 | 1 | no | — |
| 157 | C82 | — | 67 | 1 | no | — |
| 158 | C82 | — | 67 | 1 | no | — |
| 159 | C82 | — | 67 | 1 | no | — |
| 160 | C82 | C2×D67 | 1 | 1 | yes | characteristic |
| 161 | C2×Q16 | — | 2 | 4 | no | — |
| 162 | C2×Q16 | — | 134 | 4 | no | — |
| 163 | C22×C8 | — | 67 | 2 | no | — |
| 164 | C22×C8 | — | 67 | 2 | no | — |
| 165 | C22×C8 | — | 67 | 2 | no | — |
| 166 | C4⋊Q8 | — | 67 | 4 | no | — |
| 167 | C4⋊Q8 | C22×D67 | 1 | 4 | yes | — |
| 168 | #125 | — | 134 | 4 | no | — |
| 169 | C22⋊Q8 | — | 134 | 4 | no | — |
| 170 | C22⋊Q8 | — | 134 | 4 | no | — |
| 171 | C4×Q8 | — | 134 | 4 | no | — |
| 172 | C4×Q8 | — | 134 | 4 | no | — |
| 173 | #117 | — | 134 | 4 | no | — |
| 174 | C2×C42 | — | 67 | 1 | no | — |
| 175 | #107 | — | 2 | 4 | no | — |
| 176 | #107 | — | 2 | 4 | no | — |
| 177 | #107 | — | 134 | 4 | no | — |
| 178 | #107 | — | 134 | 4 | no | — |
| 179 | #106 | — | 134 | 8 | no | — |
| 180 | #103 | — | 134 | 8 | no | — |
| 181 | #103 | — | 134 | 8 | no | — |
| 182 | C4×C8 | — | 67 | 2 | no | — |
| 183 | C4×C8 | — | 67 | 2 | no | — |
| 184 | C4×C8 | — | 67 | 2 | no | — |
| 185 | C4×C8 | — | 67 | 1 | no | — |
| 186 | C4×C8 | — | 67 | 1 | no | — |
| 187 | C4×C8 | — | 67 | 1 | no | — |
| 188 | C4×C8 | C22×D67 | 1 | 1 | yes | characteristic |
| 189 | C4×C8 | C22×D67 | 1 | 1 | yes | characteristic |
| 190 | C4×C8 | C22×D67 | 1 | 1 | yes | characteristic |
| 191 | C2×Q8 | — | 2 | 4 | no | — |
| 192 | C2×Q8 | — | 134 | 4 | no | — |
| 193 | C22×C4 | — | 67 | 1 | no | — |
| 194 | C22×C4 | — | 67 | 1 | no | — |
| 195 | C22×C4 | — | 67 | 1 | no | — |
| 196 | Q16 | — | 4 | 8 | no | — |
| 197 | Q16 | — | 268 | 8 | no | — |
| 198 | C2×C8 | — | 67 | 2 | no | — |
| 199 | C2×C8 | — | 67 | 8 | no | — |
| 200 | C2×C8 | — | 67 | 2 | no | — |
| 201 | C2×C8 | — | 67 | 2 | no | — |
| 202 | C2×C8 | — | 134 | 4 | no | — |
| 203 | C2×C8 | — | 134 | 4 | no | — |
| 204 | C2×C8 | D4×D67 | 1 | 2 | yes | — |
| 205 | C2×C8 | D4×D67 | 1 | 2 | yes | — |
| 206 | C2×C8 | D4×D67 | 1 | 2 | yes | — |
| 207 | C4⋊C4 | — | 2 | 4 | no | — |
| 208 | C4⋊C4 | — | 2 | 4 | no | — |
| 209 | C4⋊C4 | — | 268 | 4 | no | — |
| 210 | C4⋊C4 | — | 268 | 4 | no | — |
| 211 | C4⋊C4 | — | 134 | 4 | no | — |
| 212 | C4⋊C4 | — | 134 | 4 | no | — |
| 213 | C22⋊C4 | — | 268 | 4 | no | — |
| 214 | C42 | — | 268 | 4 | no | — |
| 215 | C42 | — | 67 | 1 | no | — |
| 216 | C42 | — | 67 | 1 | no | — |
| 217 | C42 | — | 67 | 1 | no | — |
| 218 | C42 | C23×D67 | 1 | 1 | yes | characteristic |
| 219 | C23 | — | 67 | 1 | no | — |
| 220 | Q8 | — | 4 | 8 | no | — |
| 221 | Q8 | — | 268 | 8 | no | — |
| 222 | C2×C4 | — | 4 | 4 | no | — |
| 223 | C2×C4 | — | 268 | 4 | no | — |
| 224 | C2×C4 | — | 67 | 1 | no | — |
| 225 | C2×C4 | — | 67 | 4 | no | — |
| 226 | C2×C4 | — | 67 | 1 | no | — |
| 227 | C2×C4 | — | 67 | 1 | no | — |
| 228 | C2×C4 | — | 134 | 2 | no | — |
| 229 | C2×C4 | — | 134 | 2 | no | — |
| 230 | C2×C4 | C2×D4×D67 | 1 | 1 | yes | characteristic |
| 231 | C2×C4 | C2×D4×D67 | 1 | 1 | yes | characteristic |
| 232 | C2×C4 | C2×D4×D67 | 1 | 1 | yes | characteristic |
| 233 | C8 | — | 67 | 4 | no | — |
| 234 | C8 | — | 134 | 2 | no | — |
| 235 | C8 | — | 134 | 2 | no | — |
| 236 | C8 | Q16×D67 | 1 | 4 | yes | — |
| 237 | C8 | #14524062 | 1 | 4 | yes | — |
| 238 | C8 | #14524066 | 1 | 4 | yes | — |
| 239 | C22 | — | 67 | 2 | no | — |
| 240 | C22 | — | 134 | 1 | no | — |
| 241 | C22 | — | 134 | 1 | no | — |
| 242 | C22 | #127×D67 | 1 | 1 | yes | characteristic |
| 243 | C4 | — | 8 | 4 | no | — |
| 244 | C4 | — | 536 | 4 | no | — |
| 245 | C4 | — | 67 | 2 | no | — |
| 246 | C4 | — | 134 | 1 | no | — |
| 247 | C4 | — | 134 | 1 | no | — |
| 248 | C4 | C2×Q16×D67 | 1 | 2 | yes | — |
| 249 | C4 | C2×#14524062 | 1 | 2 | yes | — |
| 250 | C4 | C2×#14524066 | 1 | 2 | yes | — |
| 251 | C2 | — | 134 | 2 | no | — |
| 252 | C2 | #494×D67 | 1 | 1 | yes | characteristic |
| 253 | C2 | #22570636 | 1 | 1 | yes | characteristic |
| 254 | C2 | #22570638 | 1 | 1 | yes | characteristic |
Complements
Splittings G = N ⋊ H with N normal; H ≅ G/N.
Maximal subgroups
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
Finite Group Databasebeta