← #13510224
Subgroups — #13510224
Order 1792.
464 conjugacy classes of subgroups.
The subgroup lattice has 296 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | C7×#3177 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #11736022 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | #11729111 | C2 | 1 | 2 | yes | maximal, complemented |
| 4 | #11729070 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | #11728837 | C2 | 1 | 2 | yes | maximal |
| 6 | #11728835 | C2 | 1 | 2 | yes | maximal |
| 7 | #11728670 | C2 | 1 | 2 | yes | maximal, complemented |
| 8 | #11728666 | C2 | 1 | 2 | yes | maximal, complemented |
| 9 | #11727636 | C2 | 1 | 1 | yes | maximal, characteristic |
| 10 | C7×#541 | — | 2 | 1 | no | — |
| 11 | #90209 | — | 2 | 1 | no | — |
| 12 | C7×#499 | C22 | 1 | 1 | yes | characteristic |
| 13 | C7×#494 | C22 | 1 | 1 | yes | characteristic |
| 14 | C7×#491 | — | 2 | 1 | no | — |
| 15 | C7×#470 | — | 2 | 1 | no | — |
| 16 | C7×#467 | C22 | 1 | 2 | yes | — |
| 17 | C7×#441 | — | 2 | 1 | no | — |
| 18 | C4×C7×Q16 | C22 | 1 | 2 | yes | — |
| 19 | C7×#405 | C22 | 1 | 1 | yes | characteristic |
| 20 | C2×#6403 | C22 | 1 | 2 | yes | — |
| 21 | C2×#6403 | C22 | 1 | 2 | yes | — |
| 22 | #89909 | — | 2 | 1 | no | — |
| 23 | #89888 | C22 | 1 | 2 | yes | — |
| 24 | C2×#6386 | C22 | 1 | 1 | yes | characteristic |
| 25 | C22×C7⋊C16 | C22 | 1 | 1 | yes | characteristic |
| 26 | #89488 | — | 2 | 1 | no | — |
| 27 | #89475 | — | 2 | 1 | no | — |
| 28 | #89471 | C22 | 1 | 1 | yes | characteristic |
| 29 | C4×Dic28 | C22 | 1 | 2 | yes | — |
| 30 | #89455 | C22 | 1 | 1 | yes | characteristic |
| 31 | #89359 | C22 | 1 | 2 | yes | — |
| 32 | #89359 | C22 | 1 | 2 | yes | — |
| 33 | #89287 | C22 | 1 | 2 | yes | — |
| 34 | #89287 | C22 | 1 | 2 | yes | — |
| 35 | #89285 | C22 | 1 | 2 | yes | — |
| 36 | #89284 | C22 | 1 | 2 | yes | — |
| 37 | C4⋊(C7⋊C16) | C22 | 1 | 2 | yes | — |
| 38 | C4⋊(C7⋊C16) | C22 | 1 | 2 | yes | — |
| 39 | #33827 | — | 7 | 1 | no | maximal, Sylow |
| 40 | C2×C7×Q16 | — | 2 | 1 | no | — |
| 41 | C2×C7×Q16 | C23 | 1 | 2 | yes | — |
| 42 | C2×C7×M16 | D4 | 1 | 1 | yes | characteristic |
| 43 | C22×C7×C8 | C23 | 1 | 1 | yes | characteristic |
| 44 | C7×C4⋊Q8 | — | 2 | 1 | no | — |
| 45 | C7×#126 | — | 4 | 2 | no | — |
| 46 | C7×#125 | — | 2 | 1 | no | — |
| 47 | C7×#125 | — | 4 | 2 | no | — |
| 48 | C7×C22⋊Q8 | — | 2 | 2 | no | — |
| 49 | C4×C7×Q8 | — | 2 | 2 | no | — |
| 50 | C7×#117 | D4 | 1 | 1 | yes | characteristic |
| 51 | #6487 | — | 2 | 2 | no | — |
| 52 | C2×Dic28 | — | 2 | 1 | no | — |
| 53 | C2×Dic28 | C23 | 1 | 2 | yes | — |
| 54 | #6452 | — | 4 | 2 | no | — |
| 55 | #6442 | — | 4 | 2 | no | — |
| 56 | #6433 | — | 2 | 1 | no | — |
| 57 | #6432 | — | 2 | 1 | no | — |
| 58 | C4×Dic14 | — | 2 | 2 | no | — |
| 59 | C7×#107 | C23 | 1 | 2 | yes | — |
| 60 | C7×#106 | — | 2 | 1 | no | — |
| 61 | C7×#103 | — | 2 | 2 | no | — |
| 62 | C7×#103 | — | 2 | 2 | no | — |
| 63 | C7×#97 | D4 | 1 | 1 | yes | characteristic |
| 64 | C7×#97 | D4 | 1 | 1 | yes | characteristic |
| 65 | C4×C7×C8 | C23 | 1 | 1 | yes | characteristic |
| 66 | C4×C7×C8 | C23 | 1 | 1 | yes | characteristic |
| 67 | #6403 | — | 4 | 2 | no | — |
| 68 | #6403 | — | 4 | 2 | no | — |
| 69 | #6392 | C23 | 1 | 2 | yes | — |
| 70 | #6391 | — | 2 | 1 | no | — |
| 71 | #6390 | — | 2 | 2 | no | — |
| 72 | #6390 | — | 2 | 2 | no | — |
| 73 | #6386 | — | 4 | 1 | no | — |
| 74 | C2×C7⋊C16 | — | 4 | 1 | no | — |
| 75 | C2×C7⋊C16 | C23 | 1 | 2 | yes | — |
| 76 | C2×C7⋊C16 | C23 | 1 | 2 | yes | — |
| 77 | #3177 | — | 7 | 1 | no | — |
| 78 | #3177 | D7 | 1 | 1 | yes | characteristic |
| 79 | #2229 | — | 7 | 2 | no | — |
| 80 | #2229 | — | 7 | 2 | no | — |
| 81 | #2219 | — | 7 | 2 | no | — |
| 82 | #2216 | — | 7 | 2 | no | — |
| 83 | #2209 | — | 7 | 2 | no | — |
| 84 | #2209 | — | 7 | 2 | no | — |
| 85 | #2151 | — | 7 | 1 | no | — |
| 86 | C2×C7×Q8 | — | 2 | 2 | no | — |
| 87 | C22×C4×C7 | C2×D4 | 1 | 1 | yes | characteristic |
| 88 | C2×Dic14 | — | 2 | 2 | no | — |
| 89 | C7×Q16 | — | 2 | 2 | no | — |
| 90 | C7×Q16 | — | 2 | 2 | no | — |
| 91 | C7×Q16 | — | 4 | 2 | no | — |
| 92 | C7×M16 | — | 4 | 1 | no | — |
| 93 | C2×C7×C8 | — | 2 | 1 | no | — |
| 94 | C2×C7×C8 | — | 2 | 1 | no | — |
| 95 | C2×C7×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 96 | C2×C7×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 97 | C2×C7×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 98 | C2×C7×C8 | C24 | 1 | 1 | yes | characteristic |
| 99 | C7×C4⋊C4 | — | 2 | 2 | no | — |
| 100 | C7×C4⋊C4 | — | 4 | 2 | no | — |
| 101 | C7×C4⋊C4 | — | 4 | 2 | no | — |
| 102 | C7×C4⋊C4 | — | 4 | 2 | no | — |
| 103 | C7×C4⋊C4 | — | 4 | 2 | no | — |
| 104 | C7×C4⋊C4 | D8 | 1 | 2 | yes | — |
| 105 | C7×C22⋊C4 | — | 4 | 2 | no | — |
| 106 | C7×C22⋊C4 | D8 | 1 | 2 | yes | — |
| 107 | C42×C7 | — | 4 | 2 | no | — |
| 108 | C42×C7 | C2×D4 | 1 | 1 | yes | characteristic |
| 109 | C42×C7 | C2×D4 | 1 | 1 | yes | characteristic |
| 110 | C22⋊Dic7 | — | 4 | 2 | no | — |
| 111 | C4⋊Dic7 | — | 2 | 2 | no | — |
| 112 | C4⋊Dic7 | — | 4 | 2 | no | — |
| 113 | Dic7⋊C4 | — | 4 | 2 | no | — |
| 114 | Dic7⋊C4 | — | 4 | 2 | no | — |
| 115 | Dic7⋊C4 | — | 4 | 2 | no | — |
| 116 | C4×Dic7 | — | 4 | 2 | no | — |
| 117 | Dic28 | — | 2 | 2 | no | — |
| 118 | Dic28 | — | 2 | 2 | no | — |
| 119 | Dic28 | — | 4 | 2 | no | — |
| 120 | C7⋊C16 | — | 2 | 2 | no | — |
| 121 | C7⋊C16 | — | 2 | 2 | no | — |
| 122 | #541 | — | 2 | 1 | no | — |
| 123 | #541 | — | 14 | 1 | no | — |
| 124 | C2×Q32 | — | 7 | 2 | no | — |
| 125 | C2×Q32 | — | 7 | 2 | no | — |
| 126 | C2×M32 | — | 7 | 1 | no | — |
| 127 | C22×C16 | — | 7 | 1 | no | — |
| 128 | #499 | — | 7 | 1 | no | — |
| 129 | #499 | C2×D7 | 1 | 1 | yes | characteristic |
| 130 | #494 | — | 7 | 1 | no | — |
| 131 | #494 | C2×D7 | 1 | 1 | yes | characteristic |
| 132 | #491 | — | 2 | 1 | no | — |
| 133 | #491 | — | 14 | 1 | no | — |
| 134 | #470 | — | 2 | 1 | no | — |
| 135 | #470 | — | 14 | 1 | no | — |
| 136 | #467 | — | 7 | 2 | no | — |
| 137 | #467 | C2×D7 | 1 | 2 | yes | — |
| 138 | #441 | — | 2 | 1 | no | — |
| 139 | #441 | — | 14 | 1 | no | — |
| 140 | C4×Q16 | — | 7 | 2 | no | — |
| 141 | C4×Q16 | C2×D7 | 1 | 2 | yes | — |
| 142 | #405 | C2×D7 | 1 | 1 | yes | characteristic |
| 143 | #367 | — | 7 | 2 | no | — |
| 144 | #366 | — | 7 | 2 | no | — |
| 145 | C4⋊C16 | — | 7 | 2 | no | — |
| 146 | C4⋊C16 | — | 7 | 2 | no | — |
| 147 | #358 | — | 7 | 2 | no | — |
| 148 | #358 | — | 7 | 2 | no | — |
| 149 | #358 | — | 7 | 2 | no | — |
| 150 | #358 | — | 7 | 2 | no | — |
| 151 | C23×C7 | C2×D8 | 1 | 1 | yes | characteristic |
| 152 | C7×Q8 | — | 4 | 2 | no | — |
| 153 | C7×Q8 | — | 4 | 2 | no | — |
| 154 | C2×C4×C7 | — | 2 | 1 | no | — |
| 155 | C2×C4×C7 | — | 2 | 1 | no | — |
| 156 | C2×C4×C7 | — | 4 | 2 | no | — |
| 157 | C2×C4×C7 | — | 4 | 2 | no | — |
| 158 | C2×C4×C7 | C2×D8 | 1 | 2 | yes | — |
| 159 | C2×C4×C7 | C2×D8 | 1 | 1 | yes | characteristic |
| 160 | C2×C4×C7 | C2×D8 | 1 | 2 | yes | — |
| 161 | C2×C4×C7 | C22×D4 | 1 | 1 | yes | characteristic |
| 162 | C2×Dic7 | — | 4 | 2 | no | — |
| 163 | C2×Dic7 | — | 4 | 2 | no | — |
| 164 | Dic14 | — | 4 | 2 | no | — |
| 165 | Dic14 | — | 4 | 2 | no | — |
| 166 | C7×C8 | — | 2 | 1 | no | — |
| 167 | C7×C8 | — | 2 | 1 | no | — |
| 168 | C7×C8 | — | 2 | 1 | no | — |
| 169 | C7×C8 | H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 170 | C7×C8 | 2−1+4 | 1 | 1 | yes | characteristic |
| 171 | C2×Q16 | — | 2 | 1 | no | — |
| 172 | C2×Q16 | — | 7 | 2 | no | — |
| 173 | C2×Q16 | — | 14 | 1 | no | — |
| 174 | C2×Q16 | C22×D7 | 1 | 2 | yes | — |
| 175 | C2×M16 | C22⋊D7 | 1 | 1 | yes | characteristic |
| 176 | C22×C8 | C22×D7 | 1 | 1 | yes | characteristic |
| 177 | C4⋊Q8 | — | 2 | 1 | no | — |
| 178 | C4⋊Q8 | — | 14 | 1 | no | — |
| 179 | #126 | — | 4 | 2 | no | — |
| 180 | #126 | — | 28 | 2 | no | — |
| 181 | #125 | — | 2 | 1 | no | — |
| 182 | #125 | — | 4 | 2 | no | — |
| 183 | #125 | — | 14 | 1 | no | — |
| 184 | #125 | — | 28 | 2 | no | — |
| 185 | C22⋊Q8 | — | 2 | 2 | no | — |
| 186 | C22⋊Q8 | — | 14 | 2 | no | — |
| 187 | C4×Q8 | — | 2 | 2 | no | — |
| 188 | C4×Q8 | — | 14 | 2 | no | — |
| 189 | #117 | C22⋊D7 | 1 | 1 | yes | characteristic |
| 190 | Q32 | — | 28 | 2 | no | — |
| 191 | Q32 | — | 28 | 2 | no | — |
| 192 | M32 | — | 28 | 1 | no | — |
| 193 | C2×C16 | — | 7 | 2 | no | — |
| 194 | C2×C16 | — | 7 | 2 | no | — |
| 195 | C2×C16 | — | 28 | 1 | no | — |
| 196 | #107 | — | 7 | 2 | no | — |
| 197 | #107 | C22×D7 | 1 | 2 | yes | — |
| 198 | #106 | — | 2 | 1 | no | — |
| 199 | #106 | — | 14 | 1 | no | — |
| 200 | #103 | — | 2 | 2 | no | — |
| 201 | #103 | — | 2 | 2 | no | — |
| 202 | #103 | — | 14 | 2 | no | — |
| 203 | #103 | — | 14 | 2 | no | — |
| 204 | #97 | C22⋊D7 | 1 | 1 | yes | characteristic |
| 205 | #97 | C22⋊D7 | 1 | 1 | yes | characteristic |
| 206 | C4×C8 | C22×D7 | 1 | 1 | yes | characteristic |
| 207 | C4×C8 | C22×D7 | 1 | 1 | yes | characteristic |
| 208 | C22×C7 | — | 2 | 2 | no | — |
| 209 | C22×C7 | — | 2 | 1 | no | — |
| 210 | C22×C7 | C22×D8 | 1 | 1 | yes | characteristic |
| 211 | C4×C7 | — | 2 | 2 | no | — |
| 212 | C4×C7 | — | 2 | 2 | no | — |
| 213 | C4×C7 | — | 2 | 1 | no | — |
| 214 | C4×C7 | — | 8 | 2 | no | — |
| 215 | C4×C7 | — | 8 | 2 | no | — |
| 216 | C4×C7 | #537 | 1 | 1 | yes | characteristic |
| 217 | C4×C7 | #575 | 1 | 1 | yes | characteristic |
| 218 | Dic7 | — | 8 | 2 | no | — |
| 219 | Dic7 | — | 8 | 2 | no | — |
| 220 | C2×Q8 | — | 2 | 2 | no | — |
| 221 | C2×Q8 | — | 14 | 2 | no | — |
| 222 | C22×C4 | C2×C22⋊D7 | 1 | 1 | yes | characteristic |
| 223 | Q16 | — | 2 | 2 | no | — |
| 224 | Q16 | — | 2 | 2 | no | — |
| 225 | Q16 | — | 4 | 2 | no | — |
| 226 | Q16 | — | 14 | 2 | no | — |
| 227 | Q16 | — | 14 | 2 | no | — |
| 228 | Q16 | — | 28 | 2 | no | — |
| 229 | M16 | — | 4 | 1 | no | — |
| 230 | C2×C8 | — | 2 | 1 | no | — |
| 231 | C2×C8 | — | 2 | 1 | no | — |
| 232 | C2×C8 | C2×C22⋊D7 | 1 | 1 | yes | characteristic |
| 233 | C2×C8 | C2×C22⋊D7 | 1 | 1 | yes | characteristic |
| 234 | C2×C8 | C2×C22⋊D7 | 1 | 1 | yes | characteristic |
| 235 | C2×C8 | C23×D7 | 1 | 1 | yes | characteristic |
| 236 | C4⋊C4 | — | 2 | 2 | no | — |
| 237 | C4⋊C4 | — | 4 | 2 | no | — |
| 238 | C4⋊C4 | — | 4 | 2 | no | — |
| 239 | C4⋊C4 | — | 4 | 2 | no | — |
| 240 | C4⋊C4 | — | 4 | 2 | no | — |
| 241 | C4⋊C4 | — | 14 | 2 | no | — |
| 242 | C4⋊C4 | — | 28 | 2 | no | — |
| 243 | C4⋊C4 | — | 28 | 2 | no | — |
| 244 | C4⋊C4 | — | 28 | 2 | no | — |
| 245 | C4⋊C4 | — | 28 | 2 | no | — |
| 246 | C4⋊C4 | #1141 | 1 | 2 | yes | — |
| 247 | C22⋊C4 | — | 4 | 2 | no | — |
| 248 | C22⋊C4 | — | 28 | 2 | no | — |
| 249 | C22⋊C4 | #1141 | 1 | 2 | yes | — |
| 250 | C42 | — | 4 | 2 | no | — |
| 251 | C42 | — | 28 | 2 | no | — |
| 252 | C42 | C2×C22⋊D7 | 1 | 1 | yes | characteristic |
| 253 | C42 | C2×C22⋊D7 | 1 | 1 | yes | characteristic |
| 254 | C16 | — | 14 | 2 | no | — |
| 255 | C16 | — | 14 | 2 | no | — |
| 256 | C2×C7 | — | 4 | 1 | no | — |
| 257 | C2×C7 | #3245 | 1 | 1 | yes | characteristic |
| 258 | C2×C7 | #3417 | 1 | 2 | yes | — |
| 259 | C23 | C2×#1141 | 1 | 1 | yes | characteristic |
| 260 | Q8 | — | 4 | 2 | no | — |
| 261 | Q8 | — | 4 | 2 | no | — |
| 262 | Q8 | — | 28 | 2 | no | — |
| 263 | Q8 | — | 28 | 2 | no | — |
| 264 | C2×C4 | — | 2 | 1 | no | — |
| 265 | C2×C4 | — | 2 | 1 | no | — |
| 266 | C2×C4 | — | 4 | 2 | no | — |
| 267 | C2×C4 | — | 4 | 2 | no | — |
| 268 | C2×C4 | — | 28 | 2 | no | — |
| 269 | C2×C4 | — | 28 | 2 | no | — |
| 270 | C2×C4 | C2×#1141 | 1 | 2 | yes | — |
| 271 | C2×C4 | C2×#1141 | 1 | 2 | yes | — |
| 272 | C2×C4 | C2×#1141 | 1 | 1 | yes | characteristic |
| 273 | C2×C4 | C22×C22⋊D7 | 1 | 1 | yes | characteristic |
| 274 | C8 | — | 2 | 1 | no | — |
| 275 | C8 | — | 2 | 1 | no | — |
| 276 | C8 | — | 2 | 1 | no | — |
| 277 | C8 | #6553 | 1 | 1 | yes | characteristic |
| 278 | C8 | #6554 | 1 | 1 | yes | characteristic |
| 279 | C7 | #33827 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 280 | C22 | — | 2 | 2 | no | — |
| 281 | C22 | — | 2 | 1 | no | — |
| 282 | C22 | C22×#1141 | 1 | 1 | yes | characteristic |
| 283 | C4 | — | 2 | 2 | no | — |
| 284 | C4 | — | 2 | 2 | no | — |
| 285 | C4 | — | 2 | 1 | no | — |
| 286 | C4 | — | 8 | 2 | no | — |
| 287 | C4 | — | 8 | 2 | no | — |
| 288 | C4 | — | 56 | 2 | no | — |
| 289 | C4 | — | 56 | 2 | no | — |
| 290 | C4 | #90513 | 1 | 1 | yes | characteristic |
| 291 | C4 | #90524 | 1 | 1 | yes | characteristic |
| 292 | C2 | — | 4 | 1 | no | — |
| 293 | C2 | #11740755 | 1 | 1 | yes | characteristic |
| 294 | C2 | #11741462 | 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