← #31011055
Subgroups — #31011055
Order 17152.
520 conjugacy classes of subgroups.
The subgroup lattice has 302 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | #22570868 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #22562417 | C2 | 1 | 2 | yes | maximal |
| 3 | #22562373 | C2 | 1 | 2 | yes | maximal, complemented |
| 4 | #22562259 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | #22562143 | C2 | 1 | 2 | yes | maximal |
| 6 | #22562078 | C2 | 1 | 2 | yes | maximal, complemented |
| 7 | C4×#14523989 | C2 | 1 | 2 | yes | maximal, complemented |
| 8 | #22562016 | C2 | 1 | 1 | yes | maximal, characteristic |
| 9 | C67×#2950 | C2 | 1 | 1 | yes | maximal, characteristic |
| 10 | #16864008 | — | 2 | 1 | no | — |
| 11 | C2×#14523989 | C22 | 1 | 2 | yes | — |
| 12 | C2×#14523989 | C22 | 1 | 2 | yes | — |
| 13 | #16863798 | C22 | 1 | 1 | yes | characteristic |
| 14 | #16863776 | C22 | 1 | 1 | yes | characteristic |
| 15 | #16863765 | — | 2 | 1 | no | — |
| 16 | #16863654 | — | 2 | 1 | no | — |
| 17 | #16863637 | C22 | 1 | 2 | yes | — |
| 18 | #16863504 | — | 2 | 1 | no | — |
| 19 | C4×Dic268 | C22 | 1 | 2 | yes | — |
| 20 | #16863223 | C22 | 1 | 2 | yes | — |
| 21 | #16863221 | C22 | 1 | 2 | yes | — |
| 22 | C4⋊(C67⋊C16) | C22 | 1 | 2 | yes | — |
| 23 | #16863200 | C22 | 1 | 2 | yes | — |
| 24 | #16863200 | C22 | 1 | 2 | yes | — |
| 25 | #16863199 | C22 | 1 | 2 | yes | — |
| 26 | #16863199 | C22 | 1 | 2 | yes | — |
| 27 | C22⋊(C67⋊C16) | C22 | 1 | 2 | yes | — |
| 28 | #16863172 | C22 | 1 | 1 | yes | characteristic |
| 29 | C4×C67⋊C16 | C22 | 1 | 1 | yes | characteristic |
| 30 | C22×Q16×C67 | C22 | 1 | 1 | yes | characteristic |
| 31 | #519×C67 | — | 2 | 1 | no | — |
| 32 | #441×C67 | — | 2 | 1 | no | — |
| 33 | #441×C67 | — | 2 | 1 | no | — |
| 34 | C4×Q16×C67 | C22 | 1 | 2 | yes | — |
| 35 | C4×Q16×C67 | C22 | 1 | 2 | yes | — |
| 36 | #427×C67 | C22 | 1 | 1 | yes | characteristic |
| 37 | #419×C67 | — | 2 | 1 | no | — |
| 38 | #405×C67 | C22 | 1 | 1 | yes | characteristic |
| 39 | C2×Dic268 | — | 2 | 1 | no | — |
| 40 | C2×Dic268 | C23 | 1 | 2 | yes | — |
| 41 | #14524043 | — | 2 | 1 | no | — |
| 42 | #14524036 | — | 4 | 2 | no | — |
| 43 | #14524035 | — | 2 | 1 | no | — |
| 44 | #14524033 | — | 4 | 2 | no | — |
| 45 | #14524022 | — | 2 | 2 | no | — |
| 46 | C4×Dic134 | — | 2 | 2 | no | — |
| 47 | #14523989 | — | 2 | 4 | no | — |
| 48 | #14523989 | — | 2 | 4 | no | — |
| 49 | C2×C67⋊C16 | C23 | 1 | 2 | yes | — |
| 50 | C2×C67⋊C16 | C23 | 1 | 2 | yes | — |
| 51 | #14523977 | C23 | 1 | 2 | yes | — |
| 52 | #14523975 | — | 2 | 1 | no | — |
| 53 | #14523968 | — | 2 | 2 | no | — |
| 54 | #14523968 | — | 2 | 2 | no | — |
| 55 | C22×Q8×C67 | — | 2 | 1 | no | — |
| 56 | C2×Q16×C67 | — | 2 | 4 | no | — |
| 57 | C2×Q16×C67 | — | 2 | 1 | no | — |
| 58 | C2×Q16×C67 | C23 | 1 | 2 | yes | — |
| 59 | C2×M16×C67 | D4 | 1 | 1 | yes | characteristic |
| 60 | C22×C8×C67 | C23 | 1 | 1 | yes | characteristic |
| 61 | C4×Q8×C67 | — | 2 | 2 | no | — |
| 62 | C4×Q8×C67 | — | 2 | 2 | no | — |
| 63 | C4×Q8×C67 | — | 4 | 2 | no | — |
| 64 | #117×C67 | — | 2 | 1 | no | — |
| 65 | #117×C67 | — | 4 | 2 | no | — |
| 66 | #117×C67 | D4 | 1 | 1 | yes | characteristic |
| 67 | #107×C67 | C23 | 1 | 2 | yes | — |
| 68 | #106×C67 | — | 2 | 1 | no | — |
| 69 | #103×C67 | — | 2 | 2 | no | — |
| 70 | #103×C67 | — | 2 | 2 | no | — |
| 71 | #97×C67 | D4 | 1 | 1 | yes | characteristic |
| 72 | #97×C67 | D4 | 1 | 1 | yes | characteristic |
| 73 | C4×C8×C67 | C23 | 1 | 1 | yes | characteristic |
| 74 | C4×C8×C67 | C23 | 1 | 1 | yes | characteristic |
| 75 | C2×Dic134 | — | 2 | 2 | no | — |
| 76 | Dic268 | — | 2 | 2 | no | — |
| 77 | Dic268 | — | 2 | 2 | no | — |
| 78 | Dic268 | — | 4 | 2 | no | — |
| 79 | C4⋊Dic67 | — | 2 | 2 | no | — |
| 80 | C4⋊Dic67 | — | 4 | 2 | no | — |
| 81 | Dic67⋊C4 | — | 4 | 2 | no | — |
| 82 | Dic67⋊C4 | — | 4 | 2 | no | — |
| 83 | Dic67⋊C4 | — | 4 | 2 | no | — |
| 84 | C22⋊Dic67 | — | 4 | 2 | no | — |
| 85 | C4×Dic67 | — | 4 | 2 | no | — |
| 86 | C67⋊C16 | — | 2 | 2 | no | — |
| 87 | C67⋊C16 | — | 2 | 2 | no | — |
| 88 | C2×Q8×C67 | — | 2 | 4 | no | — |
| 89 | C2×Q8×C67 | — | 2 | 2 | no | — |
| 90 | C2×Q8×C67 | — | 4 | 2 | no | — |
| 91 | C2×Q8×C67 | — | 4 | 1 | no | — |
| 92 | C22×C4×C67 | — | 4 | 1 | no | — |
| 93 | C22×C4×C67 | C2×D4 | 1 | 1 | yes | characteristic |
| 94 | Q16×C67 | — | 2 | 4 | no | — |
| 95 | Q16×C67 | — | 2 | 2 | no | — |
| 96 | Q16×C67 | #41 | 1 | 4 | yes | — |
| 97 | M16×C67 | — | 4 | 1 | no | — |
| 98 | C2×C8×C67 | — | 2 | 2 | no | — |
| 99 | C2×C8×C67 | C2×D4 | 1 | 1 | yes | characteristic |
| 100 | C2×C8×C67 | C2×D4 | 1 | 1 | yes | characteristic |
| 101 | C2×C8×C67 | C2×D4 | 1 | 1 | yes | characteristic |
| 102 | C2×C8×C67 | C24 | 1 | 1 | yes | characteristic |
| 103 | C4⋊C4×C67 | — | 2 | 2 | no | — |
| 104 | C4⋊C4×C67 | — | 4 | 2 | no | — |
| 105 | C4⋊C4×C67 | — | 4 | 2 | no | — |
| 106 | C4⋊C4×C67 | D8 | 1 | 2 | yes | — |
| 107 | C22⋊C4×C67 | — | 4 | 1 | no | — |
| 108 | C22⋊C4×C67 | D8 | 1 | 2 | yes | — |
| 109 | C42×C67 | — | 4 | 1 | no | — |
| 110 | C42×C67 | — | 4 | 2 | no | — |
| 111 | C42×C67 | — | 4 | 2 | no | — |
| 112 | C42×C67 | C2×D4 | 1 | 1 | yes | characteristic |
| 113 | C42×C67 | C2×D4 | 1 | 1 | yes | characteristic |
| 114 | C23×C67 | C2×D8 | 1 | 1 | yes | characteristic |
| 115 | Q8×C67 | — | 2 | 4 | no | — |
| 116 | Q8×C67 | — | 4 | 4 | no | — |
| 117 | Q8×C67 | — | 4 | 2 | no | — |
| 118 | C2×C4×C67 | — | 2 | 1 | no | — |
| 119 | C2×C4×C67 | — | 2 | 1 | no | — |
| 120 | C2×C4×C67 | — | 4 | 4 | no | — |
| 121 | C2×C4×C67 | — | 4 | 2 | no | — |
| 122 | C2×C4×C67 | — | 4 | 2 | no | — |
| 123 | C2×C4×C67 | C2×D8 | 1 | 2 | yes | — |
| 124 | C2×C4×C67 | C2×D8 | 1 | 2 | yes | — |
| 125 | C2×C4×C67 | C2×D8 | 1 | 1 | yes | characteristic |
| 126 | C2×C4×C67 | C22×D4 | 1 | 1 | yes | characteristic |
| 127 | C2×Dic67 | — | 4 | 2 | no | — |
| 128 | C2×Dic67 | — | 4 | 2 | no | — |
| 129 | Dic134 | — | 4 | 2 | no | — |
| 130 | Dic134 | — | 4 | 2 | no | — |
| 131 | C8×C67 | — | 2 | 1 | no | — |
| 132 | C8×C67 | — | 2 | 1 | no | — |
| 133 | C8×C67 | — | 2 | 1 | no | — |
| 134 | C8×C67 | C2×#41 | 1 | 2 | yes | — |
| 135 | C22×C67 | — | 2 | 1 | no | — |
| 136 | C22×C67 | — | 2 | 2 | no | — |
| 137 | C22×C67 | C22×D8 | 1 | 1 | yes | characteristic |
| 138 | C4×C67 | — | 2 | 1 | no | — |
| 139 | C4×C67 | — | 2 | 2 | no | — |
| 140 | C4×C67 | — | 2 | 2 | no | — |
| 141 | C4×C67 | — | 4 | 4 | no | — |
| 142 | C4×C67 | — | 8 | 2 | no | — |
| 143 | C4×C67 | #525 | 1 | 1 | yes | characteristic |
| 144 | C4×C67 | #575 | 1 | 1 | yes | characteristic |
| 145 | Dic67 | — | 8 | 2 | no | — |
| 146 | Dic67 | — | 8 | 2 | no | — |
| 147 | #33687 | — | 67 | 1 | no | maximal, Sylow |
| 148 | C2×C67 | — | 4 | 1 | no | — |
| 149 | C2×C67 | #3100 | 1 | 1 | yes | characteristic |
| 150 | C2×C67 | #3417 | 1 | 2 | yes | — |
| 151 | #3177 | — | 67 | 1 | no | — |
| 152 | #2950 | D67 | 1 | 1 | yes | characteristic |
| 153 | #2237 | — | 67 | 2 | no | — |
| 154 | #2229 | — | 67 | 2 | no | — |
| 155 | #2209 | — | 67 | 2 | no | — |
| 156 | #2188 | — | 67 | 2 | no | — |
| 157 | #2176 | — | 67 | 2 | no | — |
| 158 | C4×Q32 | — | 67 | 2 | no | — |
| 159 | #2164 | — | 67 | 1 | no | — |
| 160 | C67 | #33687 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 161 | C22×Q16 | C2×D67 | 1 | 1 | yes | characteristic |
| 162 | #541 | — | 134 | 1 | no | — |
| 163 | #519 | — | 2 | 1 | no | — |
| 164 | C2×Q32 | — | 67 | 2 | no | — |
| 165 | C2×Q32 | — | 67 | 2 | no | — |
| 166 | #499 | — | 67 | 1 | no | — |
| 167 | #494 | — | 67 | 1 | no | — |
| 168 | #491 | — | 134 | 1 | no | — |
| 169 | #470 | — | 134 | 1 | no | — |
| 170 | #467 | — | 67 | 2 | no | — |
| 171 | #441 | — | 2 | 1 | no | — |
| 172 | #441 | — | 2 | 1 | no | — |
| 173 | #441 | — | 134 | 1 | no | — |
| 174 | C4×Q16 | — | 67 | 2 | no | — |
| 175 | C4×Q16 | C2×D67 | 1 | 2 | yes | — |
| 176 | C4×Q16 | C2×D67 | 1 | 2 | yes | — |
| 177 | #427 | C2×D67 | 1 | 1 | yes | characteristic |
| 178 | #419 | — | 2 | 1 | no | — |
| 179 | #405 | C2×D67 | 1 | 1 | yes | characteristic |
| 180 | #367 | — | 67 | 2 | no | — |
| 181 | #366 | — | 67 | 2 | no | — |
| 182 | C4⋊C16 | — | 67 | 2 | no | — |
| 183 | #358 | — | 67 | 2 | no | — |
| 184 | #358 | — | 67 | 2 | no | — |
| 185 | #358 | — | 67 | 2 | no | — |
| 186 | #358 | — | 67 | 2 | no | — |
| 187 | C22⋊C16 | — | 67 | 2 | no | — |
| 188 | #346 | — | 67 | 1 | no | — |
| 189 | C4×C16 | — | 67 | 1 | no | — |
| 190 | C22×Q8 | — | 2 | 1 | no | — |
| 191 | C2×Q16 | — | 2 | 4 | no | — |
| 192 | C2×Q16 | — | 2 | 1 | no | — |
| 193 | C2×Q16 | — | 67 | 2 | no | — |
| 194 | C2×Q16 | — | 134 | 1 | no | — |
| 195 | C2×Q16 | C22×D67 | 1 | 2 | yes | — |
| 196 | C2×M16 | C22⋊D67 | 1 | 1 | yes | characteristic |
| 197 | C22×C8 | C22×D67 | 1 | 1 | yes | characteristic |
| 198 | C4⋊Q8 | — | 134 | 1 | no | — |
| 199 | #126 | — | 268 | 2 | no | — |
| 200 | #125 | — | 268 | 2 | no | — |
| 201 | #125 | — | 134 | 1 | no | — |
| 202 | C22⋊Q8 | — | 134 | 2 | no | — |
| 203 | C4×Q8 | — | 2 | 2 | no | — |
| 204 | C4×Q8 | — | 2 | 2 | no | — |
| 205 | C4×Q8 | — | 4 | 2 | no | — |
| 206 | C4×Q8 | — | 134 | 2 | no | — |
| 207 | #117 | — | 2 | 1 | no | — |
| 208 | #117 | — | 4 | 2 | no | — |
| 209 | #117 | C22⋊D67 | 1 | 1 | yes | characteristic |
| 210 | Q32 | — | 134 | 4 | no | — |
| 211 | Q32 | — | 134 | 4 | no | — |
| 212 | C2×C16 | — | 67 | 2 | no | — |
| 213 | C2×C16 | — | 67 | 2 | no | — |
| 214 | #107 | — | 67 | 2 | no | — |
| 215 | #107 | C22×D67 | 1 | 2 | yes | — |
| 216 | #106 | — | 2 | 1 | no | — |
| 217 | #106 | — | 134 | 1 | no | — |
| 218 | #103 | — | 2 | 2 | no | — |
| 219 | #103 | — | 2 | 2 | no | — |
| 220 | #103 | — | 134 | 2 | no | — |
| 221 | #103 | — | 134 | 2 | no | — |
| 222 | #97 | C22⋊D67 | 1 | 1 | yes | characteristic |
| 223 | #97 | C22⋊D67 | 1 | 1 | yes | characteristic |
| 224 | C4×C8 | C22×D67 | 1 | 1 | yes | characteristic |
| 225 | C4×C8 | C22×D67 | 1 | 1 | yes | characteristic |
| 226 | C2×Q8 | — | 2 | 4 | no | — |
| 227 | C2×Q8 | — | 2 | 2 | no | — |
| 228 | C2×Q8 | — | 4 | 2 | no | — |
| 229 | C2×Q8 | — | 4 | 1 | no | — |
| 230 | C2×Q8 | — | 134 | 2 | no | — |
| 231 | C22×C4 | — | 4 | 1 | no | — |
| 232 | C22×C4 | C2×C22⋊D67 | 1 | 1 | yes | characteristic |
| 233 | Q16 | — | 2 | 4 | no | — |
| 234 | Q16 | — | 2 | 2 | no | — |
| 235 | Q16 | — | 268 | 2 | no | — |
| 236 | Q16 | — | 134 | 2 | no | — |
| 237 | Q16 | — | 134 | 2 | no | — |
| 238 | Q16 | #11761628 | 1 | 4 | yes | — |
| 239 | M16 | — | 4 | 1 | no | — |
| 240 | C2×C8 | — | 2 | 2 | no | — |
| 241 | C2×C8 | C2×C22⋊D67 | 1 | 1 | yes | characteristic |
| 242 | C2×C8 | C2×C22⋊D67 | 1 | 1 | yes | characteristic |
| 243 | C2×C8 | C2×C22⋊D67 | 1 | 1 | yes | characteristic |
| 244 | C2×C8 | C23×D67 | 1 | 1 | yes | characteristic |
| 245 | C4⋊C4 | — | 2 | 2 | no | — |
| 246 | C4⋊C4 | — | 4 | 2 | no | — |
| 247 | C4⋊C4 | — | 4 | 2 | no | — |
| 248 | C4⋊C4 | — | 268 | 2 | no | — |
| 249 | C4⋊C4 | — | 268 | 2 | no | — |
| 250 | C4⋊C4 | — | 268 | 2 | no | — |
| 251 | C4⋊C4 | — | 268 | 2 | no | — |
| 252 | C4⋊C4 | — | 134 | 2 | no | — |
| 253 | C4⋊C4 | #11761613 | 1 | 2 | yes | — |
| 254 | C22⋊C4 | — | 4 | 1 | no | — |
| 255 | C22⋊C4 | — | 268 | 2 | no | — |
| 256 | C22⋊C4 | #11761613 | 1 | 2 | yes | — |
| 257 | C42 | — | 4 | 1 | no | — |
| 258 | C42 | — | 4 | 2 | no | — |
| 259 | C42 | — | 4 | 2 | no | — |
| 260 | C42 | — | 268 | 2 | no | — |
| 261 | C42 | C2×C22⋊D67 | 1 | 1 | yes | characteristic |
| 262 | C42 | C2×C22⋊D67 | 1 | 1 | yes | characteristic |
| 263 | C16 | — | 134 | 2 | no | — |
| 264 | C16 | — | 134 | 2 | no | — |
| 265 | C23 | C2×#11761613 | 1 | 1 | yes | characteristic |
| 266 | Q8 | — | 2 | 4 | no | — |
| 267 | Q8 | — | 4 | 4 | no | — |
| 268 | Q8 | — | 4 | 2 | no | — |
| 269 | Q8 | — | 268 | 2 | no | — |
| 270 | Q8 | — | 268 | 2 | no | — |
| 271 | C2×C4 | — | 2 | 1 | no | — |
| 272 | C2×C4 | — | 2 | 1 | no | — |
| 273 | C2×C4 | — | 4 | 4 | no | — |
| 274 | C2×C4 | — | 4 | 2 | no | — |
| 275 | C2×C4 | — | 4 | 2 | no | — |
| 276 | C2×C4 | — | 268 | 2 | no | — |
| 277 | C2×C4 | — | 268 | 2 | no | — |
| 278 | C2×C4 | C2×#11761613 | 1 | 2 | yes | — |
| 279 | C2×C4 | C2×#11761613 | 1 | 2 | yes | — |
| 280 | C2×C4 | C2×#11761613 | 1 | 1 | yes | characteristic |
| 281 | C2×C4 | C22×C22⋊D67 | 1 | 1 | yes | characteristic |
| 282 | C8 | — | 2 | 1 | no | — |
| 283 | C8 | — | 2 | 1 | no | — |
| 284 | C8 | — | 2 | 1 | no | — |
| 285 | C8 | C2×#11761628 | 1 | 2 | yes | — |
| 286 | C22 | — | 2 | 2 | no | — |
| 287 | C22 | — | 2 | 1 | no | — |
| 288 | C22 | C22×#11761613 | 1 | 1 | yes | characteristic |
| 289 | C4 | — | 2 | 2 | no | — |
| 290 | C4 | — | 2 | 2 | no | — |
| 291 | C4 | — | 2 | 1 | no | — |
| 292 | C4 | — | 4 | 4 | no | — |
| 293 | C4 | — | 8 | 2 | no | — |
| 294 | C4 | — | 536 | 2 | no | — |
| 295 | C4 | — | 536 | 2 | no | — |
| 296 | C4 | #16863907 | 1 | 1 | yes | characteristic |
| 297 | C4 | #16864194 | 1 | 1 | yes | characteristic |
| 298 | C2 | — | 4 | 1 | no | — |
| 299 | C2 | #22570209 | 1 | 1 | yes | characteristic |
| 300 | C2 | #22573360 | 1 | 2 | yes | — |
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