← #27354773
Subgroups — #27354773
Order 13568.
560 conjugacy classes of subgroups.
The subgroup lattice has 286 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | #20285301 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #20283809 | C2 | 1 | 2 | yes | maximal |
| 3 | #20273625 | C2 | 1 | 2 | yes | maximal, complemented |
| 4 | #20273613 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | #20273535 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | C8×Dic212 | C2 | 1 | 2 | yes | maximal, complemented |
| 7 | #20273478 | C2 | 1 | 1 | yes | maximal, characteristic |
| 8 | #20273189 | C2 | 1 | 2 | yes | maximal |
| 9 | C53×#1450 | C2 | 1 | 1 | yes | maximal, characteristic |
| 10 | #16811180 | C22 | 1 | 2 | yes | — |
| 11 | #16811028 | C4 | 1 | 1 | yes | characteristic |
| 12 | #16811003 | C4 | 1 | 1 | yes | characteristic |
| 13 | #16810974 | C4 | 1 | 2 | yes | — |
| 14 | #16810864 | C4 | 1 | 2 | yes | — |
| 15 | #16810864 | C4 | 1 | 2 | yes | — |
| 16 | #16810753 | C22 | 1 | 4 | yes | — |
| 17 | #16810752 | — | 2 | 4 | no | — |
| 18 | C8×Dic106 | C22 | 1 | 4 | yes | — |
| 19 | C4×Dic212 | C22 | 1 | 2 | yes | — |
| 20 | C4×Dic212 | C22 | 1 | 2 | yes | — |
| 21 | #16810681 | — | 2 | 4 | no | — |
| 22 | #16810680 | C22 | 1 | 2 | yes | — |
| 23 | #16810376 | C22 | 1 | 2 | yes | — |
| 24 | #16810374 | C22 | 1 | 2 | yes | — |
| 25 | #16810355 | C22 | 1 | 4 | yes | — |
| 26 | #16810355 | C22 | 1 | 4 | yes | — |
| 27 | C53×#431 | C22 | 1 | 1 | yes | characteristic |
| 28 | C53×#431 | C22 | 1 | 1 | yes | characteristic |
| 29 | C2×C4×C8×C53 | C22 | 1 | 1 | yes | characteristic |
| 30 | C53×#322 | C22 | 1 | 1 | yes | characteristic |
| 31 | C53×#322 | C22 | 1 | 1 | yes | characteristic |
| 32 | C82×C53 | C22 | 1 | 1 | yes | characteristic |
| 33 | C82×C53 | C22 | 1 | 1 | yes | characteristic |
| 34 | C2×Dic212 | C2×C4 | 1 | 2 | yes | — |
| 35 | C2×Dic212 | C2×C4 | 1 | 2 | yes | — |
| 36 | #13133217 | C2×C4 | 1 | 2 | yes | — |
| 37 | #13133217 | C2×C4 | 1 | 2 | yes | — |
| 38 | #13133209 | — | 2 | 2 | no | — |
| 39 | #13133196 | — | 2 | 2 | no | — |
| 40 | #13133196 | C2×C4 | 1 | 4 | yes | — |
| 41 | C4×Dic106 | — | 2 | 2 | no | — |
| 42 | C4×Dic106 | C23 | 1 | 4 | yes | — |
| 43 | #13133176 | — | 2 | 2 | no | — |
| 44 | #13133151 | C2×C4 | 1 | 2 | yes | — |
| 45 | #13133151 | C2×C4 | 1 | 2 | yes | — |
| 46 | C4⋊#88322 | C23 | 1 | 4 | yes | — |
| 47 | #13133146 | — | 2 | 4 | no | — |
| 48 | #13133146 | — | 2 | 4 | no | — |
| 49 | #13133142 | C2×C4 | 1 | 4 | yes | — |
| 50 | #13133142 | C2×C4 | 1 | 4 | yes | — |
| 51 | C22⋊#88322 | — | 2 | 2 | no | — |
| 52 | #13133128 | — | 2 | 2 | no | — |
| 53 | #13133127 | — | 2 | 4 | no | — |
| 54 | C8×Dic53 | — | 2 | 4 | no | — |
| 55 | C22×C8×C53 | C2×C4 | 1 | 1 | yes | characteristic |
| 56 | C22×C8×C53 | C2×C4 | 1 | 1 | yes | characteristic |
| 57 | C2×C42×C53 | C23 | 1 | 1 | yes | characteristic |
| 58 | C4⋊C8×C53 | — | 2 | 1 | no | — |
| 59 | C4⋊C8×C53 | D4 | 1 | 2 | yes | — |
| 60 | C22⋊C8×C53 | — | 2 | 1 | no | — |
| 61 | C22⋊C8×C53 | D4 | 1 | 2 | yes | — |
| 62 | C4×C8×C53 | C2×C4 | 1 | 1 | yes | characteristic |
| 63 | C4×C8×C53 | C2×C4 | 1 | 1 | yes | characteristic |
| 64 | C4×C8×C53 | C23 | 1 | 1 | yes | characteristic |
| 65 | C4×C8×C53 | C23 | 1 | 1 | yes | characteristic |
| 66 | C4×C8×C53 | C23 | 1 | 1 | yes | characteristic |
| 67 | C4×C8×C53 | C23 | 1 | 1 | yes | characteristic |
| 68 | C4×C8×C53 | C23 | 1 | 1 | yes | characteristic |
| 69 | C4×C8×C53 | C23 | 1 | 1 | yes | characteristic |
| 70 | C22×C4×C53 | C22×C4 | 1 | 1 | yes | characteristic |
| 71 | C22×C4×C53 | C2×D4 | 1 | 1 | yes | characteristic |
| 72 | C22×C4×C53 | #41 | 1 | 1 | yes | characteristic |
| 73 | C2×Dic106 | — | 2 | 2 | no | — |
| 74 | C2×Dic106 | C22×C4 | 1 | 4 | yes | — |
| 75 | C2×C8×C53 | — | 2 | 2 | no | — |
| 76 | C2×C8×C53 | — | 2 | 2 | no | — |
| 77 | C2×C8×C53 | — | 2 | 1 | no | — |
| 78 | C2×C8×C53 | — | 2 | 1 | no | — |
| 79 | C2×C8×C53 | C22×C4 | 1 | 1 | yes | characteristic |
| 80 | C2×C8×C53 | C22×C4 | 1 | 1 | yes | characteristic |
| 81 | C2×C8×C53 | C22×C4 | 1 | 1 | yes | characteristic |
| 82 | C2×C8×C53 | C22×C4 | 1 | 1 | yes | characteristic |
| 83 | C2×C8×C53 | C2×D4 | 1 | 2 | yes | — |
| 84 | C2×C8×C53 | C2×D4 | 1 | 2 | yes | — |
| 85 | C42×C53 | C22×C4 | 1 | 1 | yes | characteristic |
| 86 | C42×C53 | C2×D4 | 1 | 1 | yes | characteristic |
| 87 | C42×C53 | #41 | 1 | 1 | yes | characteristic |
| 88 | C42×C53 | C24 | 1 | 1 | yes | characteristic |
| 89 | C22⋊Dic53 | — | 2 | 4 | no | — |
| 90 | C4⋊Dic53 | — | 2 | 2 | no | — |
| 91 | C4⋊Dic53 | — | 2 | 2 | no | — |
| 92 | C4⋊Dic53 | — | 2 | 2 | no | — |
| 93 | C4⋊Dic53 | C22×C4 | 1 | 4 | yes | — |
| 94 | #11719919 | — | 2 | 4 | no | — |
| 95 | #11719919 | — | 2 | 4 | no | — |
| 96 | C4×Dic53 | — | 2 | 4 | no | — |
| 97 | C2×#88322 | — | 2 | 4 | no | — |
| 98 | Dic212 | — | 2 | 4 | no | — |
| 99 | Dic212 | — | 2 | 4 | no | — |
| 100 | C23×C53 | C4×D4 | 1 | 1 | yes | characteristic |
| 101 | C2×C4×C53 | — | 2 | 1 | no | — |
| 102 | C2×C4×C53 | — | 2 | 1 | no | — |
| 103 | C2×C4×C53 | — | 2 | 2 | no | — |
| 104 | C2×C4×C53 | — | 4 | 1 | no | — |
| 105 | C2×C4×C53 | C4×D4 | 1 | 1 | yes | characteristic |
| 106 | C2×C4×C53 | C4×D4 | 1 | 1 | yes | characteristic |
| 107 | C2×C4×C53 | C4×D4 | 1 | 1 | yes | characteristic |
| 108 | C2×C4×C53 | C23×C4 | 1 | 1 | yes | characteristic |
| 109 | C2×C4×C53 | C22×D4 | 1 | 1 | yes | characteristic |
| 110 | C2×C4×C53 | C2×#41 | 1 | 1 | yes | characteristic |
| 111 | C2×Dic53 | — | 2 | 4 | no | — |
| 112 | C2×Dic53 | — | 2 | 4 | no | — |
| 113 | Dic106 | — | 2 | 4 | no | — |
| 114 | Dic106 | — | 2 | 4 | no | — |
| 115 | Dic106 | — | 8 | 2 | no | — |
| 116 | C8×C53 | — | 2 | 1 | no | — |
| 117 | C8×C53 | — | 2 | 1 | no | — |
| 118 | C8×C53 | — | 2 | 2 | no | — |
| 119 | C8×C53 | — | 2 | 2 | no | — |
| 120 | C8×C53 | — | 4 | 1 | no | — |
| 121 | C8×C53 | — | 4 | 1 | no | — |
| 122 | C8×C53 | #131 | 1 | 2 | yes | — |
| 123 | C8×C53 | #131 | 1 | 2 | yes | — |
| 124 | #88322 | — | 4 | 4 | no | — |
| 125 | #17972 | — | 53 | 1 | no | maximal, Sylow |
| 126 | C22×C53 | — | 2 | 1 | no | — |
| 127 | C22×C53 | — | 2 | 1 | no | — |
| 128 | C22×C53 | — | 2 | 1 | no | — |
| 129 | C22×C53 | C2×C4×D4 | 1 | 1 | yes | characteristic |
| 130 | C4×C53 | — | 2 | 1 | no | — |
| 131 | C4×C53 | — | 2 | 1 | no | — |
| 132 | C4×C53 | — | 2 | 1 | no | — |
| 133 | C4×C53 | — | 2 | 1 | no | — |
| 134 | C4×C53 | C2×#131 | 1 | 1 | yes | characteristic |
| 135 | C4×C53 | C2×#131 | 1 | 1 | yes | characteristic |
| 136 | C4×C53 | #578 | 1 | 2 | yes | — |
| 137 | Dic53 | — | 4 | 4 | no | — |
| 138 | Dic53 | — | 4 | 4 | no | — |
| 139 | #3170 | — | 53 | 1 | no | — |
| 140 | #2985 | — | 53 | 2 | no | — |
| 141 | #1593 | — | 53 | 2 | no | — |
| 142 | #1591 | — | 53 | 2 | no | — |
| 143 | #1579 | — | 53 | 2 | no | — |
| 144 | C8×Q16 | — | 53 | 2 | no | — |
| 145 | #1568 | — | 53 | 1 | no | — |
| 146 | #1497 | — | 53 | 2 | no | — |
| 147 | #1450 | D53 | 1 | 1 | yes | characteristic |
| 148 | C2×C53 | — | 4 | 1 | no | — |
| 149 | C2×C53 | #2931 | 1 | 1 | yes | characteristic |
| 150 | C2×C53 | #2950 | 1 | 1 | yes | characteristic |
| 151 | C2×C53 | #2957 | 1 | 1 | yes | characteristic |
| 152 | #533 | — | 53 | 2 | no | — |
| 153 | #500 | — | 53 | 1 | no | — |
| 154 | #494 | — | 53 | 1 | no | — |
| 155 | #487 | — | 53 | 2 | no | — |
| 156 | #467 | — | 53 | 2 | no | — |
| 157 | #467 | — | 53 | 2 | no | — |
| 158 | #446 | — | 53 | 4 | no | — |
| 159 | #446 | — | 106 | 4 | no | — |
| 160 | C8×Q8 | — | 53 | 4 | no | — |
| 161 | C4×Q16 | — | 53 | 2 | no | — |
| 162 | C4×Q16 | — | 53 | 2 | no | — |
| 163 | #433 | — | 106 | 4 | no | — |
| 164 | #432 | — | 53 | 2 | no | — |
| 165 | #431 | C2×D53 | 1 | 1 | yes | characteristic |
| 166 | #431 | C2×D53 | 1 | 1 | yes | characteristic |
| 167 | C2×C4×C8 | C2×D53 | 1 | 1 | yes | characteristic |
| 168 | #335 | — | 53 | 2 | no | — |
| 169 | #334 | — | 53 | 2 | no | — |
| 170 | Q8⋊C8 | — | 53 | 4 | no | — |
| 171 | Q8⋊C8 | — | 53 | 4 | no | — |
| 172 | #322 | C2×D53 | 1 | 1 | yes | characteristic |
| 173 | #322 | C2×D53 | 1 | 1 | yes | characteristic |
| 174 | C82 | C2×D53 | 1 | 1 | yes | characteristic |
| 175 | C82 | C2×D53 | 1 | 1 | yes | characteristic |
| 176 | C53 | #17972 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 177 | C2×Q16 | — | 53 | 2 | no | — |
| 178 | C2×Q16 | — | 53 | 2 | no | — |
| 179 | C22×C8 | C4×D53 | 1 | 1 | yes | characteristic |
| 180 | C22×C8 | C4×D53 | 1 | 1 | yes | characteristic |
| 181 | C4⋊Q8 | — | 53 | 2 | no | — |
| 182 | C4⋊Q8 | — | 53 | 2 | no | — |
| 183 | #125 | — | 106 | 2 | no | — |
| 184 | C22⋊Q8 | — | 53 | 4 | no | — |
| 185 | C22⋊Q8 | — | 106 | 2 | no | — |
| 186 | C4×Q8 | — | 53 | 4 | no | — |
| 187 | C4×Q8 | — | 106 | 2 | no | — |
| 188 | #117 | — | 106 | 2 | no | — |
| 189 | C2×C42 | C22×D53 | 1 | 1 | yes | characteristic |
| 190 | #107 | — | 53 | 2 | no | — |
| 191 | #107 | — | 53 | 2 | no | — |
| 192 | C4⋊C8 | — | 2 | 1 | no | — |
| 193 | C4⋊C8 | — | 53 | 4 | no | — |
| 194 | C4⋊C8 | — | 106 | 4 | no | — |
| 195 | C4⋊C8 | — | 106 | 4 | no | — |
| 196 | C4⋊C8 | D212 | 1 | 2 | yes | — |
| 197 | #103 | — | 53 | 4 | no | — |
| 198 | #103 | — | 53 | 4 | no | — |
| 199 | C22⋊C8 | — | 2 | 1 | no | — |
| 200 | C22⋊C8 | — | 106 | 2 | no | — |
| 201 | C22⋊C8 | D212 | 1 | 2 | yes | — |
| 202 | #97 | — | 106 | 4 | no | — |
| 203 | #97 | — | 106 | 2 | no | — |
| 204 | C4×C8 | — | 106 | 4 | no | — |
| 205 | C4×C8 | C4×D53 | 1 | 1 | yes | characteristic |
| 206 | C4×C8 | C4×D53 | 1 | 1 | yes | characteristic |
| 207 | C4×C8 | C22×D53 | 1 | 1 | yes | characteristic |
| 208 | C4×C8 | C22×D53 | 1 | 1 | yes | characteristic |
| 209 | C4×C8 | C22×D53 | 1 | 1 | yes | characteristic |
| 210 | C4×C8 | C22×D53 | 1 | 1 | yes | characteristic |
| 211 | C4×C8 | C22×D53 | 1 | 1 | yes | characteristic |
| 212 | C4×C8 | C22×D53 | 1 | 1 | yes | characteristic |
| 213 | C2×Q8 | — | 53 | 4 | no | — |
| 214 | C2×Q8 | — | 106 | 2 | no | — |
| 215 | C22×C4 | C2×C4×D53 | 1 | 1 | yes | characteristic |
| 216 | C22×C4 | C2×D212 | 1 | 1 | yes | characteristic |
| 217 | C22×C4 | #11719945 | 1 | 1 | yes | characteristic |
| 218 | Q16 | — | 106 | 4 | no | — |
| 219 | Q16 | — | 106 | 4 | no | — |
| 220 | C2×C8 | — | 2 | 2 | no | — |
| 221 | C2×C8 | — | 2 | 2 | no | — |
| 222 | C2×C8 | — | 2 | 1 | no | — |
| 223 | C2×C8 | — | 2 | 1 | no | — |
| 224 | C2×C8 | — | 106 | 4 | no | — |
| 225 | C2×C8 | C2×C4×D53 | 1 | 1 | yes | characteristic |
| 226 | C2×C8 | C2×C4×D53 | 1 | 1 | yes | characteristic |
| 227 | C2×C8 | C2×C4×D53 | 1 | 1 | yes | characteristic |
| 228 | C2×C8 | C2×C4×D53 | 1 | 1 | yes | characteristic |
| 229 | C2×C8 | C2×D212 | 1 | 2 | yes | — |
| 230 | C2×C8 | C2×D212 | 1 | 2 | yes | — |
| 231 | C4⋊C4 | — | 53 | 4 | no | — |
| 232 | C4⋊C4 | — | 106 | 2 | no | — |
| 233 | C4⋊C4 | — | 106 | 2 | no | — |
| 234 | C4⋊C4 | — | 106 | 4 | no | — |
| 235 | C4⋊C4 | — | 106 | 2 | no | — |
| 236 | C4⋊C4 | — | 106 | 4 | no | — |
| 237 | C22⋊C4 | — | 106 | 4 | no | — |
| 238 | C42 | — | 106 | 4 | no | — |
| 239 | C42 | C2×C4×D53 | 1 | 1 | yes | characteristic |
| 240 | C42 | C2×D212 | 1 | 1 | yes | characteristic |
| 241 | C42 | #11719945 | 1 | 1 | yes | characteristic |
| 242 | C42 | C23×D53 | 1 | 1 | yes | characteristic |
| 243 | C23 | C4×D212 | 1 | 1 | yes | characteristic |
| 244 | Q8 | — | 106 | 4 | no | — |
| 245 | Q8 | — | 106 | 4 | no | — |
| 246 | Q8 | — | 424 | 2 | no | — |
| 247 | C2×C4 | — | 2 | 1 | no | — |
| 248 | C2×C4 | — | 2 | 1 | no | — |
| 249 | C2×C4 | — | 2 | 2 | no | — |
| 250 | C2×C4 | — | 4 | 1 | no | — |
| 251 | C2×C4 | — | 106 | 4 | no | — |
| 252 | C2×C4 | — | 106 | 4 | no | — |
| 253 | C2×C4 | C4×D212 | 1 | 1 | yes | characteristic |
| 254 | C2×C4 | C4×D212 | 1 | 1 | yes | characteristic |
| 255 | C2×C4 | C4×D212 | 1 | 1 | yes | characteristic |
| 256 | C2×C4 | C22×C4×D53 | 1 | 1 | yes | characteristic |
| 257 | C2×C4 | C22×D212 | 1 | 1 | yes | characteristic |
| 258 | C2×C4 | C2×#11719945 | 1 | 1 | yes | characteristic |
| 259 | C8 | — | 2 | 1 | no | — |
| 260 | C8 | — | 2 | 1 | no | — |
| 261 | C8 | — | 2 | 2 | no | — |
| 262 | C8 | — | 2 | 2 | no | — |
| 263 | C8 | — | 4 | 1 | no | — |
| 264 | C8 | — | 4 | 1 | no | — |
| 265 | C8 | — | 212 | 4 | no | — |
| 266 | C8 | #13133223 | 1 | 2 | yes | — |
| 267 | C8 | #13133223 | 1 | 2 | yes | — |
| 268 | C22 | — | 2 | 1 | no | — |
| 269 | C22 | — | 2 | 1 | no | — |
| 270 | C22 | — | 2 | 1 | no | — |
| 271 | C22 | C2×C4×D212 | 1 | 1 | yes | characteristic |
| 272 | C4 | — | 2 | 1 | no | — |
| 273 | C4 | — | 2 | 1 | no | — |
| 274 | C4 | — | 2 | 1 | no | — |
| 275 | C4 | — | 2 | 1 | no | — |
| 276 | C4 | — | 212 | 4 | no | — |
| 277 | C4 | — | 212 | 4 | no | — |
| 278 | C4 | C2×#13133223 | 1 | 1 | yes | characteristic |
| 279 | C4 | C2×#13133223 | 1 | 1 | yes | characteristic |
| 280 | C4 | #16811440 | 1 | 2 | yes | — |
| 281 | C2 | — | 4 | 1 | no | — |
| 282 | C2 | #20283428 | 1 | 1 | yes | characteristic |
| 283 | C2 | #20283584 | 1 | 1 | yes | characteristic |
| 284 | C2 | #20283633 | 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