← #11743335
Subgroups — #11743335
Order 896.
1002 conjugacy classes of subgroups.
The subgroup lattice has 316 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | C7×#533 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #90423 | C2 | 1 | 2 | yes | maximal, complemented |
| 3 | #90418 | C2 | 1 | 2 | yes | maximal, complemented |
| 4 | #90390 | C2 | 1 | 4 | yes | maximal, complemented |
| 5 | #90334 | C2 | 1 | 4 | yes | maximal, complemented |
| 6 | #90319 | C2 | 1 | 4 | yes | maximal, complemented |
| 7 | #90276 | C2 | 1 | 2 | yes | maximal, complemented |
| 8 | #90275 | C2 | 1 | 2 | yes | maximal, complemented |
| 9 | Q8×D28 | C2 | 1 | 2 | yes | maximal, complemented |
| 10 | Q8×Dic14 | C2 | 1 | 2 | yes | maximal, complemented |
| 11 | #90225 | C2 | 1 | 4 | yes | maximal, complemented |
| 12 | #90168 | C2 | 1 | 1 | yes | maximal, characteristic |
| 13 | #90161 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 14 | C2×Q8×D7 | C22 | 1 | 4 | yes | — |
| 15 | C2×#1157 | C22 | 1 | 2 | yes | — |
| 16 | C7×C4⋊Q8 | C22 | 1 | 1 | yes | characteristic |
| 17 | C7×C4⋊Q8 | C22 | 1 | 1 | yes | characteristic |
| 18 | C7×#125 | C22 | 1 | 2 | yes | — |
| 19 | C7×C22⋊Q8 | C22 | 1 | 2 | yes | — |
| 20 | C7×C22⋊Q8 | C22 | 1 | 2 | yes | — |
| 21 | C4×C7×Q8 | C22 | 1 | 2 | yes | — |
| 22 | C4×C7×Q8 | C22 | 1 | 2 | yes | — |
| 23 | C7×#117 | C22 | 1 | 2 | yes | — |
| 24 | C2×C42×C7 | C22 | 1 | 1 | yes | characteristic |
| 25 | #6510 | C22 | 1 | 4 | yes | — |
| 26 | #6509 | C22 | 1 | 4 | yes | — |
| 27 | Q8×Dic7 | C22 | 1 | 4 | yes | — |
| 28 | #6493 | C22 | 1 | 2 | yes | — |
| 29 | #6493 | C22 | 1 | 2 | yes | — |
| 30 | #6489 | C22 | 1 | 2 | yes | — |
| 31 | #6487 | C22 | 1 | 2 | yes | — |
| 32 | #6487 | C22 | 1 | 2 | yes | — |
| 33 | #6461 | C22 | 1 | 8 | yes | — |
| 34 | #6461 | C22 | 1 | 8 | yes | — |
| 35 | #6460 | C22 | 1 | 8 | yes | — |
| 36 | #6460 | C22 | 1 | 4 | yes | — |
| 37 | #6459 | C22 | 1 | 8 | yes | — |
| 38 | #6457 | C22 | 1 | 8 | yes | — |
| 39 | #6456 | C22 | 1 | 4 | yes | — |
| 40 | #6455 | C22 | 1 | 4 | yes | — |
| 41 | #6451 | C22 | 1 | 8 | yes | — |
| 42 | #6451 | C22 | 1 | 4 | yes | — |
| 43 | #6448 | C22 | 1 | 8 | yes | — |
| 44 | #6446 | C22 | 1 | 8 | yes | — |
| 45 | #6438 | C22 | 1 | 2 | yes | — |
| 46 | #6438 | C22 | 1 | 4 | yes | — |
| 47 | #6437 | C22 | 1 | 1 | yes | characteristic |
| 48 | C4×D28 | C22 | 1 | 2 | yes | — |
| 49 | C4×D28 | C22 | 1 | 2 | yes | — |
| 50 | C4×D28 | C22 | 1 | 4 | yes | — |
| 51 | #6433 | C22 | 1 | 2 | yes | — |
| 52 | #6432 | C22 | 1 | 1 | yes | characteristic |
| 53 | #6432 | C22 | 1 | 1 | yes | characteristic |
| 54 | #6432 | C22 | 1 | 1 | yes | characteristic |
| 55 | #6432 | C22 | 1 | 4 | yes | — |
| 56 | C4×Dic14 | C22 | 1 | 2 | yes | — |
| 57 | C4×Dic14 | C22 | 1 | 2 | yes | — |
| 58 | C4×Dic14 | C22 | 1 | 4 | yes | — |
| 59 | #3550 | — | 7 | 1 | no | maximal, Sylow |
| 60 | C2×C7×Q8 | C23 | 1 | 2 | yes | — |
| 61 | C2×C7×Q8 | C23 | 1 | 2 | yes | — |
| 62 | C22×C4×C7 | C23 | 1 | 1 | yes | characteristic |
| 63 | C22×C4×C7 | C23 | 1 | 1 | yes | characteristic |
| 64 | C22×C4×C7 | C23 | 1 | 1 | yes | characteristic |
| 65 | C2×C22⋊D7 | C23 | 1 | 4 | yes | — |
| 66 | Q8×D7 | — | 4 | 8 | no | — |
| 67 | #1157 | — | 4 | 4 | no | — |
| 68 | C2×D28 | C23 | 1 | 2 | yes | — |
| 69 | C2×D28 | C23 | 1 | 2 | yes | — |
| 70 | C2×D28 | C23 | 1 | 2 | yes | — |
| 71 | C2×C4×D7 | C23 | 1 | 4 | yes | — |
| 72 | C2×C4×D7 | C23 | 1 | 8 | yes | — |
| 73 | C2×Dic14 | C23 | 1 | 2 | yes | — |
| 74 | C2×Dic14 | C23 | 1 | 2 | yes | — |
| 75 | C2×Dic14 | C23 | 1 | 2 | yes | — |
| 76 | C2×Dic14 | C23 | 1 | 8 | yes | — |
| 77 | C7×C4⋊C4 | C23 | 1 | 2 | yes | — |
| 78 | C7×C4⋊C4 | C23 | 1 | 4 | yes | — |
| 79 | C7×C4⋊C4 | C23 | 1 | 2 | yes | — |
| 80 | C7×C4⋊C4 | C23 | 1 | 2 | yes | — |
| 81 | C7×C4⋊C4 | C23 | 1 | 4 | yes | — |
| 82 | C7×C4⋊C4 | C23 | 1 | 2 | yes | — |
| 83 | C7×C22⋊C4 | C23 | 1 | 4 | yes | — |
| 84 | C42×C7 | C23 | 1 | 1 | yes | characteristic |
| 85 | C42×C7 | C23 | 1 | 1 | yes | characteristic |
| 86 | C42×C7 | C23 | 1 | 4 | yes | — |
| 87 | C42×C7 | C23 | 1 | 1 | yes | characteristic |
| 88 | C42×C7 | C23 | 1 | 1 | yes | characteristic |
| 89 | C22⋊Dic7 | C23 | 1 | 4 | yes | — |
| 90 | (C2×D7)⋊C4 | C23 | 1 | 4 | yes | — |
| 91 | (C2×D7)⋊C4 | C23 | 1 | 4 | yes | — |
| 92 | (C2×D7)⋊C4 | C23 | 1 | 8 | yes | — |
| 93 | (C2×D7)⋊C4 | C23 | 1 | 8 | yes | — |
| 94 | (C2×D7)⋊C4 | C23 | 1 | 8 | yes | — |
| 95 | C4⋊Dic7 | C23 | 1 | 2 | yes | — |
| 96 | C4⋊Dic7 | C23 | 1 | 2 | yes | — |
| 97 | C4⋊Dic7 | C23 | 1 | 2 | yes | — |
| 98 | C4⋊Dic7 | C23 | 1 | 2 | yes | — |
| 99 | C4⋊Dic7 | C23 | 1 | 2 | yes | — |
| 100 | C4⋊Dic7 | C23 | 1 | 2 | yes | — |
| 101 | C4⋊Dic7 | C23 | 1 | 8 | yes | — |
| 102 | Dic7⋊C4 | C23 | 1 | 4 | yes | — |
| 103 | Dic7⋊C4 | C23 | 1 | 4 | yes | — |
| 104 | Dic7⋊C4 | C23 | 1 | 8 | yes | — |
| 105 | C4×Dic7 | C23 | 1 | 4 | yes | — |
| 106 | C4×Dic7 | C23 | 1 | 8 | yes | — |
| 107 | #563 | — | 7 | 4 | no | — |
| 108 | #559 | — | 7 | 2 | no | — |
| 109 | Q82 | — | 7 | 2 | no | — |
| 110 | #556 | — | 7 | 2 | no | — |
| 111 | #556 | — | 7 | 2 | no | — |
| 112 | D4×Q8 | — | 7 | 2 | no | — |
| 113 | D4×Q8 | — | 7 | 2 | no | — |
| 114 | #542 | — | 7 | 4 | no | — |
| 115 | #542 | — | 7 | 4 | no | — |
| 116 | #536 | — | 7 | 4 | no | — |
| 117 | #533 | — | 7 | 1 | no | — |
| 118 | #533 | D7 | 1 | 1 | yes | characteristic, complemented |
| 119 | #532 | — | 7 | 1 | no | — |
| 120 | C23×C7 | C24 | 1 | 1 | yes | characteristic |
| 121 | C22×D7 | C24 | 1 | 4 | yes | — |
| 122 | C7×Q8 | — | 2 | 4 | no | — |
| 123 | C7×Q8 | — | 2 | 4 | no | — |
| 124 | C2×C4×C7 | — | 2 | 2 | no | — |
| 125 | C2×C4×C7 | — | 2 | 2 | no | — |
| 126 | C2×C4×C7 | — | 2 | 2 | no | — |
| 127 | C2×C4×C7 | C24 | 1 | 4 | yes | — |
| 128 | C2×C4×C7 | C24 | 1 | 1 | yes | characteristic |
| 129 | C2×C4×C7 | C24 | 1 | 1 | yes | characteristic |
| 130 | C2×C4×C7 | C24 | 1 | 4 | yes | — |
| 131 | C2×C4×C7 | C24 | 1 | 1 | yes | characteristic |
| 132 | C2×C4×C7 | C24 | 1 | 1 | yes | characteristic |
| 133 | C2×C4×C7 | C24 | 1 | 1 | yes | characteristic |
| 134 | C2×C4×C7 | C24 | 1 | 1 | yes | characteristic |
| 135 | C22⋊D7 | — | 4 | 4 | no | — |
| 136 | C2×Dic7 | C24 | 1 | 4 | yes | — |
| 137 | C2×Dic7 | C24 | 1 | 4 | yes | — |
| 138 | C2×Dic7 | C24 | 1 | 4 | yes | — |
| 139 | D28 | — | 2 | 4 | no | — |
| 140 | D28 | — | 2 | 4 | no | — |
| 141 | D28 | — | 4 | 2 | no | — |
| 142 | C4×D7 | — | 2 | 8 | no | — |
| 143 | C4×D7 | — | 4 | 8 | no | — |
| 144 | Dic14 | — | 2 | 4 | no | — |
| 145 | Dic14 | — | 2 | 4 | no | — |
| 146 | Dic14 | — | 4 | 2 | no | — |
| 147 | Dic14 | — | 4 | 8 | no | — |
| 148 | C2×#41 | — | 7 | 2 | no | — |
| 149 | C22×Q8 | — | 7 | 4 | no | — |
| 150 | C4⋊Q8 | — | 7 | 8 | no | — |
| 151 | C4⋊Q8 | — | 7 | 4 | no | — |
| 152 | C4⋊Q8 | — | 7 | 1 | no | — |
| 153 | C4⋊Q8 | — | 7 | 1 | no | — |
| 154 | C4⋊Q8 | — | 7 | 1 | no | — |
| 155 | C4⋊Q8 | — | 7 | 4 | no | — |
| 156 | C4⋊Q8 | C2×D7 | 1 | 1 | yes | characteristic |
| 157 | C4⋊Q8 | C2×D7 | 1 | 1 | yes | characteristic |
| 158 | #127 | — | 7 | 1 | no | — |
| 159 | #126 | — | 7 | 8 | no | — |
| 160 | #126 | — | 7 | 8 | no | — |
| 161 | #125 | — | 7 | 2 | no | — |
| 162 | #125 | C2×D7 | 1 | 2 | yes | — |
| 163 | #124 | — | 7 | 4 | no | — |
| 164 | #124 | — | 7 | 8 | no | — |
| 165 | #124 | — | 7 | 2 | no | — |
| 166 | #124 | — | 7 | 4 | no | — |
| 167 | #123 | — | 7 | 8 | no | — |
| 168 | #123 | — | 7 | 8 | no | — |
| 169 | C22⋊Q8 | — | 7 | 4 | no | — |
| 170 | C22⋊Q8 | — | 7 | 2 | no | — |
| 171 | C22⋊Q8 | — | 7 | 2 | no | — |
| 172 | C22⋊Q8 | — | 7 | 4 | no | — |
| 173 | C22⋊Q8 | — | 7 | 8 | no | — |
| 174 | C22⋊Q8 | — | 7 | 8 | no | — |
| 175 | C22⋊Q8 | C2×D7 | 1 | 2 | yes | — |
| 176 | C22⋊Q8 | C2×D7 | 1 | 2 | yes | — |
| 177 | #121 | — | 7 | 2 | no | — |
| 178 | #121 | — | 7 | 2 | no | — |
| 179 | C4×Q8 | — | 7 | 4 | no | — |
| 180 | C4×Q8 | — | 7 | 2 | no | — |
| 181 | C4×Q8 | — | 7 | 2 | no | — |
| 182 | C4×Q8 | — | 7 | 4 | no | — |
| 183 | C4×Q8 | C2×D7 | 1 | 2 | yes | — |
| 184 | C4×Q8 | C2×D7 | 1 | 2 | yes | — |
| 185 | C4×D4 | — | 7 | 4 | no | — |
| 186 | C4×D4 | — | 7 | 2 | no | — |
| 187 | C4×D4 | — | 7 | 2 | no | — |
| 188 | C4×D4 | — | 7 | 4 | no | — |
| 189 | #117 | — | 7 | 2 | no | — |
| 190 | #117 | — | 7 | 4 | no | — |
| 191 | #117 | C2×D7 | 1 | 2 | yes | — |
| 192 | C2×C42 | C2×D7 | 1 | 1 | yes | characteristic |
| 193 | C22×C7 | — | 2 | 1 | no | — |
| 194 | C22×C7 | — | 2 | 1 | no | — |
| 195 | C22×C7 | — | 2 | 1 | no | — |
| 196 | C22×C7 | C25 | 1 | 1 | yes | characteristic |
| 197 | C2×D7 | — | 2 | 4 | no | — |
| 198 | C2×D7 | — | 2 | 4 | no | — |
| 199 | C2×D7 | — | 2 | 4 | no | — |
| 200 | C4×C7 | — | 2 | 4 | no | — |
| 201 | C4×C7 | — | 2 | 1 | no | — |
| 202 | C4×C7 | — | 2 | 1 | no | — |
| 203 | C4×C7 | — | 2 | 1 | no | — |
| 204 | C4×C7 | — | 2 | 4 | no | — |
| 205 | C4×C7 | H(5,𝔽2) | 1 | 2 | yes | — |
| 206 | C4×C7 | 2−1+4 | 1 | 2 | yes | — |
| 207 | C4×C7 | 2−1+4 | 1 | 2 | yes | — |
| 208 | Dic7 | — | 2 | 4 | no | — |
| 209 | Dic7 | — | 2 | 4 | no | — |
| 210 | Dic7 | — | 2 | 4 | no | — |
| 211 | #41 | — | 28 | 4 | no | — |
| 212 | C2×Q8 | — | 7 | 2 | no | — |
| 213 | C2×Q8 | — | 7 | 2 | no | — |
| 214 | C2×Q8 | — | 7 | 8 | no | — |
| 215 | C2×Q8 | — | 7 | 2 | no | — |
| 216 | C2×Q8 | — | 28 | 8 | no | — |
| 217 | C2×Q8 | C22×D7 | 1 | 2 | yes | — |
| 218 | C2×Q8 | C22×D7 | 1 | 2 | yes | — |
| 219 | C2×D4 | — | 7 | 4 | no | — |
| 220 | C2×D4 | — | 7 | 2 | no | — |
| 221 | C2×D4 | — | 7 | 2 | no | — |
| 222 | C2×D4 | — | 7 | 2 | no | — |
| 223 | C22×C4 | — | 7 | 4 | no | — |
| 224 | C22×C4 | — | 7 | 8 | no | — |
| 225 | C22×C4 | C22×D7 | 1 | 1 | yes | characteristic |
| 226 | C22×C4 | C22×D7 | 1 | 1 | yes | characteristic |
| 227 | C22×C4 | C22×D7 | 1 | 1 | yes | characteristic |
| 228 | C4⋊C4 | — | 7 | 4 | no | — |
| 229 | C4⋊C4 | — | 7 | 8 | no | — |
| 230 | C4⋊C4 | — | 7 | 4 | no | — |
| 231 | C4⋊C4 | — | 7 | 2 | no | — |
| 232 | C4⋊C4 | — | 7 | 2 | no | — |
| 233 | C4⋊C4 | — | 7 | 2 | no | — |
| 234 | C4⋊C4 | — | 7 | 2 | no | — |
| 235 | C4⋊C4 | — | 7 | 2 | no | — |
| 236 | C4⋊C4 | — | 7 | 2 | no | — |
| 237 | C4⋊C4 | — | 7 | 8 | no | — |
| 238 | C4⋊C4 | C22×D7 | 1 | 2 | yes | — |
| 239 | C4⋊C4 | C22×D7 | 1 | 2 | yes | — |
| 240 | C4⋊C4 | C22×D7 | 1 | 2 | yes | — |
| 241 | C4⋊C4 | C22×D7 | 1 | 2 | yes | — |
| 242 | C4⋊C4 | C22×D7 | 1 | 4 | yes | — |
| 243 | C4⋊C4 | C22×D7 | 1 | 4 | yes | — |
| 244 | C22⋊C4 | — | 7 | 4 | no | — |
| 245 | C22⋊C4 | — | 7 | 8 | no | — |
| 246 | C22⋊C4 | — | 7 | 4 | no | — |
| 247 | C22⋊C4 | — | 7 | 4 | no | — |
| 248 | C22⋊C4 | — | 7 | 8 | no | — |
| 249 | C22⋊C4 | — | 7 | 8 | no | — |
| 250 | C22⋊C4 | C22×D7 | 1 | 4 | yes | — |
| 251 | C42 | — | 7 | 4 | no | — |
| 252 | C42 | — | 7 | 8 | no | — |
| 253 | C42 | C22×D7 | 1 | 1 | yes | characteristic |
| 254 | C42 | C22×D7 | 1 | 1 | yes | characteristic |
| 255 | C42 | C22×D7 | 1 | 1 | yes | characteristic |
| 256 | C42 | C22×D7 | 1 | 1 | yes | characteristic |
| 257 | C42 | C22×D7 | 1 | 4 | yes | — |
| 258 | C2×C7 | — | 4 | 1 | no | — |
| 259 | C2×C7 | C2×H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 260 | C2×C7 | C2×2−1+4 | 1 | 1 | yes | characteristic |
| 261 | C2×C7 | C2×2−1+4 | 1 | 1 | yes | characteristic |
| 262 | D7 | — | 4 | 4 | no | — |
| 263 | C23 | — | 7 | 4 | no | — |
| 264 | C23 | C23×D7 | 1 | 1 | yes | characteristic |
| 265 | Q8 | — | 2 | 4 | no | — |
| 266 | Q8 | — | 2 | 4 | no | — |
| 267 | Q8 | — | 14 | 4 | no | — |
| 268 | Q8 | — | 14 | 4 | no | — |
| 269 | Q8 | — | 28 | 8 | no | — |
| 270 | Q8 | — | 28 | 2 | no | — |
| 271 | D4 | — | 14 | 4 | no | — |
| 272 | D4 | — | 14 | 4 | no | — |
| 273 | D4 | — | 28 | 2 | no | — |
| 274 | D4 | — | 28 | 4 | no | — |
| 275 | C2×C4 | — | 2 | 2 | no | — |
| 276 | C2×C4 | — | 2 | 2 | no | — |
| 277 | C2×C4 | — | 2 | 2 | no | — |
| 278 | C2×C4 | — | 7 | 4 | no | — |
| 279 | C2×C4 | — | 7 | 4 | no | — |
| 280 | C2×C4 | — | 7 | 4 | no | — |
| 281 | C2×C4 | — | 14 | 8 | no | — |
| 282 | C2×C4 | — | 28 | 8 | no | — |
| 283 | C2×C4 | C23×D7 | 1 | 4 | yes | — |
| 284 | C2×C4 | C23×D7 | 1 | 1 | yes | characteristic |
| 285 | C2×C4 | C23×D7 | 1 | 1 | yes | characteristic |
| 286 | C2×C4 | C23×D7 | 1 | 1 | yes | characteristic |
| 287 | C2×C4 | C23×D7 | 1 | 4 | yes | — |
| 288 | C2×C4 | C23×D7 | 1 | 1 | yes | characteristic |
| 289 | C2×C4 | C23×D7 | 1 | 1 | yes | characteristic |
| 290 | C2×C4 | C23×D7 | 1 | 1 | yes | characteristic |
| 291 | C7 | #3550 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 292 | C22 | — | 2 | 1 | no | — |
| 293 | C22 | — | 2 | 1 | no | — |
| 294 | C22 | — | 2 | 1 | no | — |
| 295 | C22 | — | 14 | 4 | no | — |
| 296 | C22 | — | 14 | 4 | no | — |
| 297 | C22 | — | 14 | 4 | no | — |
| 298 | C22 | C24×D7 | 1 | 1 | yes | characteristic |
| 299 | C4 | — | 2 | 1 | no | — |
| 300 | C4 | — | 2 | 1 | no | — |
| 301 | C4 | — | 2 | 4 | no | — |
| 302 | C4 | — | 2 | 1 | no | — |
| 303 | C4 | — | 2 | 4 | no | — |
| 304 | C4 | — | 14 | 4 | no | — |
| 305 | C4 | — | 14 | 4 | no | — |
| 306 | C4 | — | 14 | 4 | no | — |
| 307 | C4 | #6551 | 1 | 2 | yes | — |
| 308 | C4 | #6553 | 1 | 2 | yes | — |
| 309 | C4 | #6554 | 1 | 2 | yes | — |
| 310 | C2 | — | 4 | 1 | no | — |
| 311 | C2 | — | 28 | 4 | no | — |
| 312 | C2 | C2×#6551 | 1 | 1 | yes | characteristic |
| 313 | C2 | C2×#6553 | 1 | 1 | yes | characteristic |
| 314 | C2 | C2×#6554 | 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