Finite Group DatabasebetaFinite Group DatabasebetaOrder 1792.
806 conjugacy classes of subgroups; only the 155 normal subgroups are shown.
This group has 806 conjugacy classes of subgroups — too many to draw. Only its 155 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #11743338 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #11740780 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | #11736764 | C2 | 1 | 2 | yes | maximal |
| 4 | C7×#1537 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | #11728567 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | #11726131 | C2 | 1 | 4 | yes | maximal, complemented |
| 7 | #11726129 | C2 | 1 | 4 | yes | maximal, complemented |
| 8 | C7×#533 | C22 | 1 | 1 | yes | characteristic |
| 9 | #90412 | — | 2 | 2 | no | — |
| 10 | C7⋊#564 | C22 | 1 | 4 | yes | — |
| 11 | #90316 | — | 2 | 4 | no | — |
| 12 | #90261 | — | 2 | 2 | no | — |
| 13 | #90258 | — | 2 | 2 | no | — |
| 14 | #90209 | — | 2 | 2 | no | — |
| 15 | #90168 | C22 | 1 | 1 | yes | characteristic |
| 16 | #90161 | C22 | 1 | 2 | yes | — |
| 17 | C7×#432 | C22 | 1 | 2 | yes | — |
| 18 | #89861 | — | 2 | 2 | no | — |
| 19 | #89841 | C22 | 1 | 2 | yes | — |
| 20 | #89839 | C22 | 1 | 2 | yes | — |
| 21 | #89774 | — | 2 | 2 | no | — |
| 22 | C2×#6377 | C22 | 1 | 1 | yes | characteristic |
| 23 | #89618 | — | 2 | 2 | no | — |
| 24 | #89614 | — | 2 | 2 | no | — |
| 25 | #89509 | — | 2 | 4 | no | — |
| 26 | #89485 | C22 | 1 | 4 | yes | — |
| 27 | #89474 | C22 | 1 | 4 | yes | — |
| 28 | C7×#330 | C22 | 1 | 4 | yes | — |
| 29 | #89260 | C22 | 1 | 8 | yes | — |
| 30 | #19063 | — | 7 | 1 | no | maximal, Sylow |
| 31 | C2×#1157 | — | 2 | 1 | no | — |
| 32 | C7×C4⋊Q8 | D4 | 1 | 2 | yes | — |
| 33 | C7×#125 | C23 | 1 | 2 | yes | — |
| 34 | C7×C22⋊Q8 | — | 2 | 2 | no | — |
| 35 | C4×C7×Q8 | — | 2 | 2 | no | — |
| 36 | C7×#117 | D4 | 1 | 2 | yes | — |
| 37 | C2×C42×C7 | C23 | 1 | 1 | yes | characteristic |
| 38 | Q8×Dic7 | — | 2 | 4 | no | — |
| 39 | #6507 | — | 2 | 4 | no | — |
| 40 | #6493 | — | 2 | 2 | no | — |
| 41 | #6489 | D4 | 1 | 4 | yes | — |
| 42 | #6487 | — | 2 | 2 | no | — |
| 43 | #6487 | — | 2 | 2 | no | — |
| 44 | C22×C7⋊C8 | — | 2 | 1 | no | — |
| 45 | #6453 | — | 2 | 4 | no | — |
| 46 | #6453 | — | 2 | 8 | no | — |
| 47 | #6452 | — | 2 | 8 | no | — |
| 48 | #6452 | — | 2 | 8 | no | — |
| 49 | #6450 | — | 2 | 4 | no | — |
| 50 | #6442 | — | 2 | 8 | no | — |
| 51 | #6438 | — | 2 | 1 | no | — |
| 52 | #6437 | C23 | 1 | 1 | yes | characteristic |
| 53 | C4×D28 | — | 2 | 2 | no | — |
| 54 | #6433 | — | 2 | 2 | no | — |
| 55 | #6433 | C23 | 1 | 4 | yes | — |
| 56 | #6432 | — | 2 | 2 | no | — |
| 57 | #6432 | D4 | 1 | 2 | yes | — |
| 58 | #6432 | D4 | 1 | 2 | yes | — |
| 59 | #6432 | C23 | 1 | 1 | yes | characteristic |
| 60 | C4×Dic14 | — | 2 | 4 | no | — |
| 61 | C4×Dic14 | — | 2 | 2 | no | — |
| 62 | C4×Dic14 | — | 2 | 2 | no | — |
| 63 | C7×C4⋊C8 | — | 2 | 2 | no | — |
| 64 | C7×C22⋊C8 | — | 2 | 2 | no | — |
| 65 | C7×#97 | C23 | 1 | 4 | yes | — |
| 66 | #6395 | — | 2 | 4 | no | — |
| 67 | #6395 | — | 2 | 4 | no | — |
| 68 | #6392 | — | 2 | 4 | no | — |
| 69 | #6391 | — | 2 | 4 | no | — |
| 70 | #6384 | — | 2 | 4 | no | — |
| 71 | #6383 | — | 2 | 4 | no | — |
| 72 | #6377 | — | 2 | 1 | no | — |
| 73 | #6377 | C23 | 1 | 2 | yes | — |
| 74 | #3556 | — | 7 | 1 | no | — |
| 75 | #3235 | — | 7 | 2 | no | — |
| 76 | #3135 | — | 7 | 1 | no | — |
| 77 | #1663 | — | 7 | 4 | no | — |
| 78 | #1658 | — | 7 | 4 | no | — |
| 79 | #1537 | D7 | 1 | 1 | yes | characteristic, complemented |
| 80 | #1531 | — | 7 | 2 | no | — |
| 81 | C2×C7×Q8 | — | 2 | 2 | no | — |
| 82 | C22×C4×C7 | C2×D4 | 1 | 1 | yes | characteristic |
| 83 | C22×C4×C7 | C2×D4 | 1 | 2 | yes | — |
| 84 | C2×C22⋊D7 | — | 4 | 1 | no | — |
| 85 | #1157 | — | 4 | 4 | no | — |
| 86 | C2×D28 | — | 2 | 2 | no | — |
| 87 | C2×D28 | — | 2 | 1 | no | — |
| 88 | C2×C4×D7 | — | 4 | 1 | no | — |
| 89 | C2×Dic14 | — | 2 | 2 | no | — |
| 90 | C2×Dic14 | — | 2 | 2 | no | — |
| 91 | C2×Dic14 | — | 2 | 1 | no | — |
| 92 | C2×C7×C8 | — | 2 | 4 | no | — |
| 93 | C7×C4⋊C4 | — | 2 | 2 | no | — |
| 94 | C7×C4⋊C4 | — | 2 | 4 | no | — |
| 95 | C7×C4⋊C4 | C2×D4 | 1 | 4 | yes | — |
| 96 | C7×C22⋊C4 | — | 2 | 2 | no | — |
| 97 | C42×C7 | — | 2 | 2 | no | — |
| 98 | C42×C7 | C2×D4 | 1 | 1 | yes | characteristic |
| 99 | C42×C7 | C2×D4 | 1 | 2 | yes | — |
| 100 | C42×C7 | C24 | 1 | 1 | yes | characteristic |
| 101 | C22⋊Dic7 | — | 2 | 4 | no | — |
| 102 | (C2×D7)⋊C4 | — | 4 | 2 | no | — |
| 103 | C4⋊Dic7 | — | 2 | 8 | no | — |
| 104 | C4⋊Dic7 | — | 2 | 2 | no | — |
| 105 | C4⋊Dic7 | — | 2 | 2 | no | — |
| 106 | C4⋊Dic7 | — | 2 | 2 | no | — |
| 107 | C4⋊Dic7 | C2×D4 | 1 | 4 | yes | — |
| 108 | C4⋊Dic7 | C2×D4 | 1 | 4 | yes | — |
| 109 | Dic7⋊C4 | — | 2 | 8 | no | — |
| 110 | Dic7⋊C4 | — | 2 | 8 | no | — |
| 111 | Dic7⋊C4 | — | 2 | 4 | no | — |
| 112 | Dic7⋊C4 | — | 2 | 4 | no | — |
| 113 | C4×Dic7 | — | 2 | 8 | no | — |
| 114 | C4×Dic7 | — | 2 | 4 | no | — |
| 115 | C2×C7⋊C8 | — | 2 | 2 | no | — |
| 116 | C2×C7⋊C8 | — | 4 | 2 | no | — |
| 117 | #564 | — | 7 | 4 | no | — |
| 118 | #557 | — | 14 | 2 | no | — |
| 119 | #557 | — | 14 | 2 | no | — |
| 120 | #557 | — | 14 | 2 | no | — |
| 121 | #541 | — | 14 | 4 | no | — |
| 122 | #541 | — | 14 | 2 | no | — |
| 123 | #533 | — | 7 | 2 | no | — |
| 124 | #533 | C2×D7 | 1 | 1 | yes | characteristic |
| 125 | #532 | — | 7 | 1 | no | — |
| 126 | #501 | — | 7 | 4 | no | — |
| 127 | #491 | — | 7 | 2 | no | — |
| 128 | #490 | — | 7 | 2 | no | — |
| 129 | #490 | — | 7 | 4 | no | — |
| 130 | #489 | — | 14 | 2 | no | — |
| 131 | #482 | — | 14 | 4 | no | — |
| 132 | #476 | — | 14 | 2 | no | — |
| 133 | #474 | — | 14 | 2 | no | — |
| 134 | #432 | C2×D7 | 1 | 2 | yes | — |
| 135 | #416 | — | 14 | 2 | no | — |
| 136 | C2×#97 | — | 7 | 1 | no | — |
| 137 | #330 | — | 7 | 8 | no | — |
| 138 | #330 | C2×D7 | 1 | 4 | yes | — |
| 139 | C23×C7 | #120 | 1 | 1 | yes | characteristic |
| 140 | C22×D7 | — | 4 | 1 | no | — |
| 141 | C7×Q8 | — | 4 | 4 | no | — |
| 142 | C2×C4×C7 | — | 2 | 2 | no | — |
| 143 | C2×C4×C7 | — | 2 | 4 | no | — |
| 144 | C2×C4×C7 | — | 4 | 2 | no | — |
| 145 | C2×C4×C7 | #120 | 1 | 2 | yes | — |
| 146 | C2×C4×C7 | #120 | 1 | 1 | yes | characteristic |
| 147 | C2×C4×C7 | C22×D4 | 1 | 1 | yes | characteristic |
| 148 | C2×C4×C7 | C22×D4 | 1 | 2 | yes | — |
| 149 | C22⋊D7 | — | 8 | 2 | no | — |
| 150 | C2×Dic7 | — | 2 | 4 | no | — |
| 151 | C2×Dic7 | — | 2 | 4 | no | — |
| 152 | C2×Dic7 | — | 4 | 1 | no | — |
| 153 | D28 | — | 4 | 4 | no | — |
| 154 | D28 | — | 4 | 2 | no | — |
| 155 | C4×D7 | — | 8 | 2 | no | — |
| 156 | Dic14 | — | 4 | 4 | no | — |
| 157 | Dic14 | — | 4 | 4 | no | — |
| 158 | Dic14 | — | 4 | 2 | no | — |
| 159 | C7×C8 | — | 4 | 4 | no | — |
| 160 | C7⋊C8 | — | 4 | 2 | no | — |
| 161 | C2×#41 | — | 14 | 1 | no | — |
| 162 | C22×C8 | — | 14 | 1 | no | — |
| 163 | C4⋊Q8 | — | 7 | 2 | no | — |
| 164 | C4⋊Q8 | — | 7 | 2 | no | — |
| 165 | C4⋊Q8 | — | 7 | 1 | no | — |
| 166 | C4⋊Q8 | — | 14 | 2 | no | — |
| 167 | C4⋊Q8 | — | 14 | 4 | no | — |
| 168 | C4⋊Q8 | D28 | 1 | 2 | yes | — |
| 169 | #127 | — | 7 | 1 | no | — |
| 170 | #126 | — | 14 | 8 | no | — |
| 171 | #125 | — | 7 | 4 | no | — |
| 172 | #125 | — | 14 | 2 | no | — |
| 173 | #125 | — | 14 | 8 | no | — |
| 174 | #125 | — | 14 | 8 | no | — |
| 175 | #125 | — | 14 | 4 | no | — |
| 176 | #125 | — | 14 | 8 | no | — |
| 177 | #125 | C22×D7 | 1 | 2 | yes | — |
| 178 | #124 | — | 14 | 1 | no | — |
| 179 | C22⋊Q8 | — | 2 | 2 | no | — |
| 180 | C22⋊Q8 | — | 14 | 2 | no | — |
| 181 | C22⋊Q8 | — | 14 | 2 | no | — |
| 182 | #121 | — | 14 | 2 | no | — |
| 183 | C4×Q8 | — | 2 | 2 | no | — |
| 184 | C4×Q8 | — | 14 | 4 | no | — |
| 185 | C4×Q8 | — | 14 | 4 | no | — |
| 186 | C4×Q8 | — | 14 | 4 | no | — |
| 187 | C4×Q8 | — | 14 | 2 | no | — |
| 188 | C4×Q8 | — | 14 | 2 | no | — |
| 189 | C4×D4 | — | 14 | 2 | no | — |
| 190 | #117 | — | 7 | 4 | no | — |
| 191 | #117 | D28 | 1 | 2 | yes | — |
| 192 | C2×C42 | C22×D7 | 1 | 1 | yes | characteristic |
| 193 | #107 | — | 14 | 4 | no | — |
| 194 | #106 | — | 14 | 4 | no | — |
| 195 | C4⋊C8 | — | 2 | 2 | no | — |
| 196 | #103 | — | 14 | 4 | no | — |
| 197 | #102 | — | 14 | 4 | no | — |
| 198 | #102 | — | 14 | 4 | no | — |
| 199 | #102 | — | 14 | 4 | no | — |
| 200 | C22⋊C8 | — | 2 | 2 | no | — |
| 201 | #97 | — | 7 | 2 | no | — |
| 202 | #97 | — | 14 | 1 | no | — |
| 203 | #97 | C22×D7 | 1 | 4 | yes | — |
| 204 | C22×C7 | — | 2 | 1 | no | — |
| 205 | C22×C7 | — | 2 | 2 | no | — |
| 206 | C22×C7 | C2×#120 | 1 | 1 | yes | characteristic |
| 207 | C2×D7 | — | 8 | 1 | no | — |
| 208 | C2×D7 | — | 8 | 2 | no | — |
| 209 | C4×C7 | — | 2 | 1 | no | — |
| 210 | C4×C7 | — | 2 | 2 | no | — |
| 211 | C4×C7 | — | 2 | 2 | no | — |
| 212 | C4×C7 | — | 4 | 4 | no | — |
| 213 | C4×C7 | #454 | 1 | 2 | yes | — |
| 214 | Dic7 | — | 4 | 4 | no | — |
| 215 | Dic7 | — | 4 | 4 | no | — |
| 216 | Dic7 | — | 8 | 1 | no | — |
| 217 | #41 | — | 28 | 4 | no | — |
| 218 | C2×Q8 | — | 2 | 2 | no | — |
| 219 | C2×Q8 | — | 14 | 2 | no | — |
| 220 | C2×Q8 | — | 14 | 2 | no | — |
| 221 | C2×Q8 | — | 14 | 1 | no | — |
| 222 | C2×D4 | — | 14 | 1 | no | — |
| 223 | C2×D4 | — | 14 | 2 | no | — |
| 224 | C2×D4 | — | 28 | 1 | no | — |
| 225 | C22×C4 | — | 28 | 1 | no | — |
| 226 | C22×C4 | C2×D28 | 1 | 1 | yes | characteristic |
| 227 | C22×C4 | D4×D7 | 1 | 2 | yes | — |
| 228 | C2×C8 | — | 2 | 4 | no | — |
| 229 | C2×C8 | — | 14 | 2 | no | — |
| 230 | C2×C8 | — | 28 | 2 | no | — |
| 231 | C4⋊C4 | — | 2 | 2 | no | — |
| 232 | C4⋊C4 | — | 2 | 4 | no | — |
| 233 | C4⋊C4 | — | 7 | 4 | no | — |
| 234 | C4⋊C4 | — | 7 | 4 | no | — |
| 235 | C4⋊C4 | — | 14 | 8 | no | — |
| 236 | C4⋊C4 | — | 14 | 8 | no | — |
| 237 | C4⋊C4 | — | 14 | 8 | no | — |
| 238 | C4⋊C4 | — | 14 | 2 | no | — |
| 239 | C4⋊C4 | — | 14 | 2 | no | — |
| 240 | C4⋊C4 | — | 14 | 4 | no | — |
| 241 | C4⋊C4 | — | 14 | 4 | no | — |
| 242 | C4⋊C4 | — | 14 | 2 | no | — |
| 243 | C4⋊C4 | C2×D28 | 1 | 4 | yes | — |
| 244 | C22⋊C4 | — | 2 | 2 | no | — |
| 245 | C22⋊C4 | — | 14 | 4 | no | — |
| 246 | C22⋊C4 | — | 28 | 2 | no | — |
| 247 | C42 | — | 2 | 2 | no | — |
| 248 | C42 | — | 14 | 8 | no | — |
| 249 | C42 | — | 14 | 4 | no | — |
| 250 | C42 | C2×D28 | 1 | 1 | yes | characteristic |
| 251 | C42 | D4×D7 | 1 | 2 | yes | — |
| 252 | C42 | C23×D7 | 1 | 1 | yes | characteristic |
| 253 | C2×C7 | — | 4 | 1 | no | — |
| 254 | C2×C7 | C2×#454 | 1 | 1 | yes | characteristic |
| 255 | C2×C7 | #3020 | 1 | 2 | yes | — |
| 256 | D7 | — | 16 | 1 | no | — |
| 257 | C23 | — | 28 | 1 | no | — |
| 258 | C23 | #6445 | 1 | 1 | yes | characteristic |
| 259 | Q8 | — | 4 | 4 | no | — |
| 260 | Q8 | — | 28 | 4 | no | — |
| 261 | Q8 | — | 28 | 2 | no | — |
| 262 | Q8 | — | 28 | 4 | no | — |
| 263 | D4 | — | 28 | 4 | no | — |
| 264 | D4 | — | 28 | 2 | no | — |
| 265 | D4 | — | 56 | 2 | no | — |
| 266 | C2×C4 | — | 2 | 2 | no | — |
| 267 | C2×C4 | — | 2 | 4 | no | — |
| 268 | C2×C4 | — | 4 | 2 | no | — |
| 269 | C2×C4 | — | 14 | 4 | no | — |
| 270 | C2×C4 | — | 14 | 4 | no | — |
| 271 | C2×C4 | — | 28 | 1 | no | — |
| 272 | C2×C4 | — | 56 | 2 | no | — |
| 273 | C2×C4 | #6445 | 1 | 2 | yes | — |
| 274 | C2×C4 | #6445 | 1 | 1 | yes | characteristic |
| 275 | C2×C4 | C22×D28 | 1 | 1 | yes | characteristic |
| 276 | C2×C4 | C2×D4×D7 | 1 | 2 | yes | — |
| 277 | C8 | — | 4 | 4 | no | — |
| 278 | C8 | — | 28 | 2 | no | — |
| 279 | C7 | #19063 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 280 | C22 | — | 2 | 1 | no | — |
| 281 | C22 | — | 2 | 2 | no | — |
| 282 | C22 | — | 56 | 1 | no | — |
| 283 | C22 | — | 56 | 2 | no | — |
| 284 | C22 | C2×#6445 | 1 | 1 | yes | characteristic |
| 285 | C4 | — | 2 | 1 | no | — |
| 286 | C4 | — | 2 | 2 | no | — |
| 287 | C4 | — | 2 | 2 | no | — |
| 288 | C4 | — | 4 | 4 | no | — |
| 289 | C4 | — | 28 | 4 | no | — |
| 290 | C4 | — | 28 | 4 | no | — |
| 291 | C4 | — | 56 | 1 | no | — |
| 292 | C4 | #89525 | 1 | 2 | yes | — |
| 293 | C2 | — | 4 | 1 | no | — |
| 294 | C2 | — | 112 | 1 | no | — |
| 295 | C2 | C2×#89525 | 1 | 1 | yes | characteristic |
| 296 | C2 | #11736441 | 1 | 2 | yes | — |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #11743338 | C2 | semidirect |
| 4. C7×#1537 | C2 | semidirect |
| 5. #11728567 | C2 | semidirect |
| 6. #11726131 | C2 | semidirect |
| 7. #11726129 | C2 | semidirect |
| 79. #1537 | D7 | semidirect |
| 279. C7 | #19063 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #11743338 | 2 |
| #11740780 | 2 |
| #11736764 | 2 |
| C7×#1537 | 2 |
| #11728567 | 2 |
| #11726131 | 2 |
| #11726129 | 2 |
| #19063 | 7 |