← #31719208
Subgroups — #31719208
Order 17152.
1634 conjugacy classes of subgroups.
The subgroup lattice has 290 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | #22567159 | C2 | 1 | 3 | yes | maximal, complemented |
| 2 | #22565601 | C2 | 1 | 6 | yes | maximal, complemented |
| 3 | #22564429 | C2 | 1 | 6 | yes | maximal |
| 4 | #22564373 | C2 | 1 | 6 | yes | maximal |
| 5 | #22563300 | C2 | 1 | 6 | yes | maximal, complemented |
| 6 | #22562690 | C2 | 1 | 3 | yes | maximal |
| 7 | C67×#2810 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 8 | C2×#14524037 | C22 | 1 | 2 | yes | — |
| 9 | C2×#14524033 | C22 | 1 | 6 | yes | — |
| 10 | C2×#14524001 | C22 | 1 | 3 | yes | — |
| 11 | C2×#14524001 | C22 | 1 | 3 | yes | — |
| 12 | C2×C42×D67 | C22 | 1 | 1 | yes | characteristic |
| 13 | #16863330 | C22 | 1 | 6 | yes | — |
| 14 | #16863329 | C22 | 1 | 6 | yes | — |
| 15 | #16863327 | C22 | 1 | 6 | yes | — |
| 16 | #16863325 | C22 | 1 | 6 | yes | — |
| 17 | #16863320 | C22 | 1 | 2 | yes | — |
| 18 | #16863295 | C22 | 1 | 3 | yes | — |
| 19 | #16863294 | C22 | 1 | 3 | yes | — |
| 20 | #16863293 | C22 | 1 | 6 | yes | — |
| 21 | #16863293 | C22 | 1 | 6 | yes | — |
| 22 | #16863292 | C22 | 1 | 3 | yes | — |
| 23 | #16863291 | C22 | 1 | 6 | yes | — |
| 24 | #16863285 | C22 | 1 | 6 | yes | — |
| 25 | #16863281 | C22 | 1 | 6 | yes | — |
| 26 | #16863279 | C22 | 1 | 6 | yes | — |
| 27 | #16863279 | C22 | 1 | 6 | yes | — |
| 28 | #16863278 | C22 | 1 | 6 | yes | — |
| 29 | #16863278 | C22 | 1 | 6 | yes | — |
| 30 | #16863270 | C22 | 1 | 6 | yes | — |
| 31 | #16863267 | C22 | 1 | 6 | yes | — |
| 32 | C4⋊C4×Dic67 | C22 | 1 | 6 | yes | — |
| 33 | C4×Dic67⋊C4 | C22 | 1 | 6 | yes | — |
| 34 | #16863247 | C22 | 1 | 6 | yes | — |
| 35 | #16863247 | C22 | 1 | 6 | yes | — |
| 36 | #400×C67 | C22 | 1 | 6 | yes | — |
| 37 | #389×C67 | C22 | 1 | 3 | yes | — |
| 38 | #387×C67 | C22 | 1 | 6 | yes | — |
| 39 | C22×C4×D67 | C23 | 1 | 3 | yes | — |
| 40 | #14524037 | — | 4 | 4 | no | — |
| 41 | #14524033 | — | 2 | 24 | no | — |
| 42 | #14524001 | — | 2 | 12 | no | — |
| 43 | #14524001 | — | 4 | 6 | no | — |
| 44 | C2×C4⋊Dic67 | C23 | 1 | 6 | yes | — |
| 45 | C2×C4⋊Dic67 | C23 | 1 | 3 | yes | — |
| 46 | C2×C4⋊Dic67 | C23 | 1 | 3 | yes | — |
| 47 | C2×Dic67⋊C4 | C23 | 1 | 6 | yes | — |
| 48 | C2×Dic67⋊C4 | C23 | 1 | 6 | yes | — |
| 49 | C2×Dic67⋊C4 | C23 | 1 | 6 | yes | — |
| 50 | C2×Dic67⋊C4 | C23 | 1 | 6 | yes | — |
| 51 | C2×Dic67⋊C4 | C23 | 1 | 6 | yes | — |
| 52 | C2×Dic67⋊C4 | C23 | 1 | 6 | yes | — |
| 53 | C2×(C2×D67)⋊C4 | C23 | 1 | 6 | yes | — |
| 54 | C2×(C2×D67)⋊C4 | C23 | 1 | 6 | yes | — |
| 55 | C2×C4×Dic67 | C23 | 1 | 6 | yes | — |
| 56 | C2×C4×Dic67 | C23 | 1 | 6 | yes | — |
| 57 | C2×C4×Dic67 | C23 | 1 | 3 | yes | — |
| 58 | C2×C4×Dic67 | C23 | 1 | 6 | yes | — |
| 59 | C42×D67 | — | 4 | 2 | no | — |
| 60 | C67⋊#95 | C23 | 1 | 6 | yes | — |
| 61 | C67⋊#95 | C23 | 1 | 6 | yes | — |
| 62 | C67⋊#95 | C23 | 1 | 6 | yes | — |
| 63 | C67⋊#95 | C23 | 1 | 6 | yes | — |
| 64 | C67⋊#95 | C23 | 1 | 6 | yes | — |
| 65 | C67⋊#95 | C23 | 1 | 6 | yes | — |
| 66 | C2×C4⋊C4×C67 | C23 | 1 | 6 | yes | — |
| 67 | C2×C4⋊C4×C67 | C23 | 1 | 3 | yes | — |
| 68 | C2×C4⋊C4×C67 | C23 | 1 | 3 | yes | — |
| 69 | C2×C42×C67 | C23 | 1 | 2 | yes | — |
| 70 | C2×C42×C67 | C23 | 1 | 1 | yes | characteristic |
| 71 | #95×C67 | C23 | 1 | 6 | yes | — |
| 72 | #95×C67 | C23 | 1 | 6 | yes | — |
| 73 | #95×C67 | C23 | 1 | 6 | yes | — |
| 74 | #95×C67 | C23 | 1 | 2 | yes | — |
| 75 | C23×D67 | C24 | 1 | 1 | yes | characteristic |
| 76 | C22×Dic67 | C24 | 1 | 6 | yes | — |
| 77 | C22×Dic67 | C24 | 1 | 6 | yes | — |
| 78 | C22×Dic67 | C24 | 1 | 3 | yes | — |
| 79 | C2×C4×D67 | — | 2 | 6 | no | — |
| 80 | C2×C4×D67 | — | 2 | 6 | no | — |
| 81 | C2×C4×D67 | — | 2 | 6 | no | — |
| 82 | C4⋊Dic67 | — | 2 | 12 | no | — |
| 83 | C4⋊Dic67 | — | 4 | 3 | no | — |
| 84 | C4⋊Dic67 | #41 | 1 | 12 | yes | — |
| 85 | Dic67⋊C4 | — | 2 | 12 | no | — |
| 86 | Dic67⋊C4 | — | 2 | 12 | no | — |
| 87 | Dic67⋊C4 | — | 2 | 12 | no | — |
| 88 | Dic67⋊C4 | — | 2 | 12 | no | — |
| 89 | Dic67⋊C4 | — | 2 | 12 | no | — |
| 90 | Dic67⋊C4 | — | 4 | 6 | no | — |
| 91 | (C2×D67)⋊C4 | — | 2 | 12 | no | — |
| 92 | (C2×D67)⋊C4 | — | 2 | 12 | no | — |
| 93 | C4×Dic67 | — | 2 | 12 | no | — |
| 94 | C4×Dic67 | — | 2 | 12 | no | — |
| 95 | C4×Dic67 | — | 4 | 3 | no | — |
| 96 | C4×Dic67 | — | 4 | 6 | no | — |
| 97 | C22×C4×C67 | C24 | 1 | 6 | yes | — |
| 98 | C22×C4×C67 | C24 | 1 | 6 | yes | — |
| 99 | C22×C4×C67 | C24 | 1 | 3 | yes | — |
| 100 | C4⋊C4×C67 | — | 2 | 6 | no | — |
| 101 | C4⋊C4×C67 | — | 2 | 12 | no | — |
| 102 | C4⋊C4×C67 | — | 4 | 3 | no | — |
| 103 | C42×C67 | — | 4 | 1 | no | — |
| 104 | C42×C67 | — | 4 | 2 | no | — |
| 105 | C23×C67 | C25 | 1 | 1 | yes | characteristic |
| 106 | C22×D67 | — | 2 | 3 | no | — |
| 107 | C22×D67 | — | 2 | 3 | no | — |
| 108 | C22×D67 | H(5,𝔽2) | 1 | 2 | yes | — |
| 109 | C2×C4×C67 | — | 2 | 3 | no | — |
| 110 | C2×C4×C67 | — | 2 | 3 | no | — |
| 111 | C2×C4×C67 | — | 2 | 6 | no | — |
| 112 | C2×C4×C67 | — | 2 | 6 | no | — |
| 113 | C2×C4×C67 | — | 2 | 6 | no | — |
| 114 | C2×C4×C67 | — | 2 | 6 | no | — |
| 115 | C2×C4×C67 | — | 2 | 6 | no | — |
| 116 | C2×C4×C67 | — | 2 | 6 | no | — |
| 117 | C2×C4×C67 | C2×#41 | 1 | 6 | yes | — |
| 118 | C2×Dic67 | — | 2 | 3 | no | — |
| 119 | C2×Dic67 | — | 2 | 6 | no | — |
| 120 | C2×Dic67 | — | 2 | 3 | no | — |
| 121 | C2×Dic67 | — | 2 | 6 | no | — |
| 122 | C2×Dic67 | — | 2 | 6 | no | — |
| 123 | C2×Dic67 | — | 2 | 6 | no | — |
| 124 | C2×Dic67 | — | 2 | 6 | no | — |
| 125 | C2×Dic67 | C2×#41 | 1 | 12 | yes | — |
| 126 | C2×Dic67 | 2−1+4 | 1 | 6 | yes | — |
| 127 | C4×D67 | — | 2 | 24 | no | — |
| 128 | C22×C67 | C22×#41 | 1 | 3 | yes | — |
| 129 | C22×C67 | C2×H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 130 | C22×C67 | C2×2−1+4 | 1 | 3 | yes | — |
| 131 | C2×D67 | — | 2 | 6 | no | — |
| 132 | C2×D67 | — | 4 | 3 | no | — |
| 133 | C2×D67 | — | 4 | 1 | no | — |
| 134 | C4×C67 | — | 2 | 6 | no | — |
| 135 | C4×C67 | — | 4 | 6 | no | — |
| 136 | C4×C67 | — | 4 | 6 | no | — |
| 137 | Dic67 | — | 2 | 12 | no | — |
| 138 | Dic67 | — | 2 | 6 | no | — |
| 139 | Dic67 | — | 4 | 6 | no | — |
| 140 | #60049 | — | 67 | 1 | no | maximal, Sylow |
| 141 | C2×C67 | #3482 | 1 | 3 | yes | — |
| 142 | C2×C67 | #3492 | 1 | 3 | yes | — |
| 143 | C2×C67 | #3556 | 1 | 1 | yes | characteristic |
| 144 | D67 | — | 4 | 2 | no | — |
| 145 | #2810 | D67 | 1 | 1 | yes | characteristic, complemented |
| 146 | #2726 | — | 67 | 3 | no | — |
| 147 | #2585 | — | 67 | 6 | no | — |
| 148 | #2478 | — | 67 | 6 | no | — |
| 149 | #2475 | — | 67 | 6 | no | — |
| 150 | #2370 | — | 67 | 6 | no | — |
| 151 | #2296 | — | 67 | 3 | no | — |
| 152 | C67 | #60049 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 153 | C2×#126 | — | 67 | 2 | no | — |
| 154 | C2×#125 | — | 67 | 6 | no | — |
| 155 | C2×#117 | — | 67 | 3 | no | — |
| 156 | C2×#117 | — | 67 | 3 | no | — |
| 157 | C22×C42 | — | 67 | 1 | no | — |
| 158 | #400 | C2×D67 | 1 | 6 | yes | — |
| 159 | #398 | — | 67 | 6 | no | — |
| 160 | #398 | — | 67 | 6 | no | — |
| 161 | #398 | — | 67 | 6 | no | — |
| 162 | #397 | — | 67 | 6 | no | — |
| 163 | #396 | — | 67 | 2 | no | — |
| 164 | #389 | — | 67 | 3 | no | — |
| 165 | #389 | — | 67 | 6 | no | — |
| 166 | #389 | — | 67 | 3 | no | — |
| 167 | #389 | — | 67 | 3 | no | — |
| 168 | #389 | — | 67 | 6 | no | — |
| 169 | #389 | — | 67 | 6 | no | — |
| 170 | #389 | C2×D67 | 1 | 3 | yes | — |
| 171 | #388 | — | 67 | 6 | no | — |
| 172 | #387 | — | 67 | 6 | no | — |
| 173 | #387 | — | 67 | 6 | no | — |
| 174 | #387 | — | 67 | 6 | no | — |
| 175 | #387 | — | 67 | 6 | no | — |
| 176 | #387 | — | 67 | 6 | no | — |
| 177 | #387 | C2×D67 | 1 | 6 | yes | — |
| 178 | #385 | — | 67 | 6 | no | — |
| 179 | #385 | — | 67 | 6 | no | — |
| 180 | C4×C4⋊C4 | — | 67 | 6 | no | — |
| 181 | C4×C4⋊C4 | — | 67 | 6 | no | — |
| 182 | C4×C4⋊C4 | — | 67 | 6 | no | — |
| 183 | C4×C4⋊C4 | — | 67 | 6 | no | — |
| 184 | C23×C4 | — | 67 | 3 | no | — |
| 185 | #126 | — | 268 | 4 | no | — |
| 186 | #125 | — | 134 | 24 | no | — |
| 187 | #117 | — | 268 | 6 | no | — |
| 188 | #117 | — | 134 | 12 | no | — |
| 189 | C2×C4⋊C4 | — | 67 | 3 | no | — |
| 190 | C2×C4⋊C4 | — | 67 | 6 | no | — |
| 191 | C2×C4⋊C4 | — | 67 | 3 | no | — |
| 192 | C2×C4⋊C4 | — | 67 | 6 | no | — |
| 193 | C2×C4⋊C4 | — | 67 | 6 | no | — |
| 194 | C2×C4⋊C4 | — | 67 | 6 | no | — |
| 195 | C2×C4⋊C4 | — | 67 | 6 | no | — |
| 196 | C2×C4⋊C4 | — | 67 | 6 | no | — |
| 197 | C2×C4⋊C4 | — | 67 | 6 | no | — |
| 198 | C2×C4⋊C4 | C22×D67 | 1 | 3 | yes | — |
| 199 | C2×C4⋊C4 | C22×D67 | 1 | 6 | yes | — |
| 200 | C2×C4⋊C4 | C22×D67 | 1 | 3 | yes | — |
| 201 | C2×C22⋊C4 | — | 67 | 6 | no | — |
| 202 | C2×C22⋊C4 | — | 67 | 6 | no | — |
| 203 | C2×C42 | — | 268 | 2 | no | — |
| 204 | C2×C42 | — | 67 | 6 | no | — |
| 205 | C2×C42 | — | 67 | 3 | no | — |
| 206 | C2×C42 | — | 67 | 6 | no | — |
| 207 | C2×C42 | — | 67 | 6 | no | — |
| 208 | C2×C42 | C22×D67 | 1 | 2 | yes | — |
| 209 | C2×C42 | C22×D67 | 1 | 1 | yes | characteristic |
| 210 | #95 | — | 67 | 6 | no | — |
| 211 | #95 | — | 67 | 6 | no | — |
| 212 | #95 | — | 67 | 6 | no | — |
| 213 | #95 | — | 67 | 6 | no | — |
| 214 | #95 | — | 67 | 6 | no | — |
| 215 | #95 | — | 67 | 6 | no | — |
| 216 | #95 | C22×D67 | 1 | 6 | yes | — |
| 217 | #95 | C22×D67 | 1 | 6 | yes | — |
| 218 | #95 | C22×D67 | 1 | 2 | yes | — |
| 219 | #95 | C22×D67 | 1 | 6 | yes | — |
| 220 | C24 | — | 67 | 1 | no | — |
| 221 | C22×C4 | — | 67 | 3 | no | — |
| 222 | C22×C4 | — | 67 | 6 | no | — |
| 223 | C22×C4 | — | 67 | 6 | no | — |
| 224 | C22×C4 | — | 134 | 6 | no | — |
| 225 | C22×C4 | — | 134 | 6 | no | — |
| 226 | C22×C4 | — | 134 | 6 | no | — |
| 227 | C22×C4 | C23×D67 | 1 | 6 | yes | — |
| 228 | C22×C4 | C23×D67 | 1 | 6 | yes | — |
| 229 | C22×C4 | C23×D67 | 1 | 3 | yes | — |
| 230 | C4⋊C4 | — | 2 | 6 | no | — |
| 231 | C4⋊C4 | — | 2 | 12 | no | — |
| 232 | C4⋊C4 | — | 4 | 3 | no | — |
| 233 | C4⋊C4 | — | 268 | 3 | no | — |
| 234 | C4⋊C4 | — | 268 | 6 | no | — |
| 235 | C4⋊C4 | — | 67 | 12 | no | — |
| 236 | C4⋊C4 | — | 134 | 12 | no | — |
| 237 | C4⋊C4 | — | 134 | 12 | no | — |
| 238 | C4⋊C4 | — | 134 | 12 | no | — |
| 239 | C4⋊C4 | — | 134 | 12 | no | — |
| 240 | C4⋊C4 | — | 134 | 12 | no | — |
| 241 | C4⋊C4 | — | 134 | 12 | no | — |
| 242 | C22⋊C4 | — | 134 | 12 | no | — |
| 243 | C22⋊C4 | — | 134 | 12 | no | — |
| 244 | C42 | — | 4 | 1 | no | — |
| 245 | C42 | — | 4 | 2 | no | — |
| 246 | C42 | — | 268 | 3 | no | — |
| 247 | C42 | — | 268 | 6 | no | — |
| 248 | C42 | — | 134 | 12 | no | — |
| 249 | C42 | — | 134 | 12 | no | — |
| 250 | C23 | — | 67 | 2 | no | — |
| 251 | C23 | — | 134 | 3 | no | — |
| 252 | C23 | — | 134 | 3 | no | — |
| 253 | C23 | C24×D67 | 1 | 1 | yes | characteristic |
| 254 | C2×C4 | — | 2 | 3 | no | — |
| 255 | C2×C4 | — | 2 | 3 | no | — |
| 256 | C2×C4 | — | 2 | 6 | no | — |
| 257 | C2×C4 | — | 2 | 6 | no | — |
| 258 | C2×C4 | — | 2 | 6 | no | — |
| 259 | C2×C4 | — | 2 | 6 | no | — |
| 260 | C2×C4 | — | 2 | 6 | no | — |
| 261 | C2×C4 | — | 2 | 6 | no | — |
| 262 | C2×C4 | — | 67 | 12 | no | — |
| 263 | C2×C4 | — | 67 | 6 | no | — |
| 264 | C2×C4 | — | 134 | 24 | no | — |
| 265 | C2×C4 | — | 134 | 6 | no | — |
| 266 | C2×C4 | — | 134 | 6 | no | — |
| 267 | C2×C4 | — | 134 | 6 | no | — |
| 268 | C2×C4 | — | 134 | 3 | no | — |
| 269 | C2×C4 | — | 134 | 6 | no | — |
| 270 | C2×C4 | — | 134 | 3 | no | — |
| 271 | C2×C4 | — | 134 | 6 | no | — |
| 272 | C2×C4 | #41×D67 | 1 | 6 | yes | — |
| 273 | C22 | — | 268 | 3 | no | — |
| 274 | C22 | — | 268 | 1 | no | — |
| 275 | C22 | — | 134 | 6 | no | — |
| 276 | C22 | C2×#41×D67 | 1 | 3 | yes | — |
| 277 | C22 | H(5,𝔽2)×D67 | 1 | 1 | yes | characteristic |
| 278 | C22 | 2−1+4×D67 | 1 | 3 | yes | — |
| 279 | C4 | — | 2 | 6 | no | — |
| 280 | C4 | — | 4 | 6 | no | — |
| 281 | C4 | — | 4 | 6 | no | — |
| 282 | C4 | — | 268 | 6 | no | — |
| 283 | C4 | — | 134 | 12 | no | — |
| 284 | C4 | — | 134 | 6 | no | — |
| 285 | C2 | — | 268 | 2 | no | — |
| 286 | C2 | #22573828 | 1 | 3 | yes | — |
| 287 | C2 | #22573914 | 1 | 3 | yes | — |
| 288 | C2 | #22574703 | 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