← #27531001
Subgroups — #27531001
Order 13568.
536 conjugacy classes of subgroups.
The subgroup lattice has 348 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | #20276293 | C2 | 1 | 2 | yes | maximal, complemented |
| 2 | #20275888 | C2 | 1 | 2 | yes | maximal, complemented |
| 3 | #20275884 | C2 | 1 | 2 | yes | maximal, complemented |
| 4 | #20275696 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | #20275547 | C2 | 1 | 1 | yes | maximal, characteristic |
| 6 | #20275097 | C2 | 1 | 2 | yes | maximal |
| 7 | #20274956 | C2 | 1 | 2 | yes | maximal, complemented |
| 8 | #20274954 | C2 | 1 | 2 | yes | maximal, complemented |
| 9 | C53×#1995 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 10 | #16811389 | C22 | 1 | 2 | yes | — |
| 11 | #16811180 | C22 | 1 | 1 | yes | characteristic |
| 12 | #16811173 | C22 | 1 | 1 | yes | characteristic |
| 13 | #16810671 | C22 | 1 | 2 | yes | — |
| 14 | #16810647 | C22 | 1 | 1 | yes | characteristic |
| 15 | #16810647 | C22 | 1 | 1 | yes | characteristic |
| 16 | #16810641 | C22 | 1 | 2 | yes | — |
| 17 | #16810639 | C22 | 1 | 2 | yes | — |
| 18 | C2×#13133145 | C22 | 1 | 2 | yes | — |
| 19 | C2×#13133145 | C22 | 1 | 2 | yes | — |
| 20 | #16810596 | C22 | 1 | 2 | yes | — |
| 21 | #16810394 | C22 | 1 | 2 | yes | — |
| 22 | #16810393 | C22 | 1 | 2 | yes | — |
| 23 | #16810383 | C22 | 1 | 2 | yes | — |
| 24 | #16810383 | C22 | 1 | 2 | yes | — |
| 25 | #16810379 | C22 | 1 | 2 | yes | — |
| 26 | C53×#533 | C22 | 1 | 1 | yes | characteristic |
| 27 | C53×#423 | C22 | 1 | 1 | yes | characteristic |
| 28 | C53×#376 | C22 | 1 | 1 | yes | characteristic |
| 29 | C53×#339 | C22 | 1 | 2 | yes | — |
| 30 | C53×#339 | C22 | 1 | 2 | yes | — |
| 31 | C2×#11719945 | C23 | 1 | 2 | yes | — |
| 32 | #13133225 | — | 4 | 2 | no | — |
| 33 | #13133223 | — | 4 | 2 | no | — |
| 34 | C2×#11719917 | C23 | 1 | 2 | yes | — |
| 35 | C2×#11719917 | C23 | 1 | 2 | yes | — |
| 36 | M16×D53 | — | 4 | 2 | no | — |
| 37 | #13133217 | — | 2 | 1 | no | — |
| 38 | #13133217 | Q8 | 1 | 1 | yes | characteristic, complemented |
| 39 | #13133214 | Q8 | 1 | 1 | yes | characteristic, complemented |
| 40 | #13133209 | — | 2 | 1 | no | — |
| 41 | #13133205 | Q8 | 1 | 2 | yes | complemented |
| 42 | #13133196 | — | 4 | 1 | no | — |
| 43 | #13133191 | — | 4 | 1 | no | — |
| 44 | C4×Dic106 | — | 4 | 1 | no | — |
| 45 | C4×D212 | — | 4 | 1 | no | — |
| 46 | #13133176 | C23 | 1 | 2 | yes | — |
| 47 | #13133152 | — | 4 | 2 | no | — |
| 48 | C4⋊#88322 | — | 4 | 1 | no | — |
| 49 | C4⋊#88322 | — | 4 | 1 | no | — |
| 50 | #13133145 | — | 2 | 2 | no | — |
| 51 | #13133145 | — | 2 | 2 | no | — |
| 52 | #13133145 | — | 2 | 2 | no | — |
| 53 | #13133145 | — | 2 | 2 | no | — |
| 54 | #13133145 | — | 4 | 2 | no | — |
| 55 | #13133143 | — | 4 | 2 | no | — |
| 56 | #13133131 | — | 4 | 2 | no | — |
| 57 | C2×M16×C53 | C23 | 1 | 2 | yes | — |
| 58 | C4⋊Q8×C53 | — | 2 | 1 | no | — |
| 59 | #125×C53 | — | 2 | 1 | no | — |
| 60 | C22⋊Q8×C53 | — | 4 | 1 | no | — |
| 61 | C4×Q8×C53 | — | 4 | 1 | no | — |
| 62 | #117×C53 | C23 | 1 | 2 | yes | — |
| 63 | C2×C42×C53 | C23 | 1 | 1 | yes | characteristic |
| 64 | C2×C42×C53 | C23 | 1 | 1 | yes | characteristic |
| 65 | C2×C42×C53 | C23 | 1 | 1 | yes | characteristic |
| 66 | C4⋊C8×C53 | — | 4 | 1 | no | — |
| 67 | #95×C53 | — | 4 | 1 | no | — |
| 68 | C2×Q8×C53 | — | 4 | 1 | no | — |
| 69 | C22×C4×C53 | — | 2 | 1 | no | — |
| 70 | C22×C4×C53 | — | 2 | 1 | no | — |
| 71 | C22×C4×C53 | — | 2 | 1 | no | — |
| 72 | C22×C4×C53 | C24 | 1 | 1 | yes | characteristic |
| 73 | C2×C22⋊D53 | — | 2 | 2 | no | — |
| 74 | #11719945 | — | 2 | 2 | no | — |
| 75 | #11719945 | — | 2 | 2 | no | — |
| 76 | #11719945 | — | 2 | 2 | no | — |
| 77 | #11719945 | — | 2 | 2 | no | — |
| 78 | C2×D212 | — | 4 | 1 | no | — |
| 79 | C2×D212 | C2×Q8 | 1 | 2 | yes | — |
| 80 | C2×C4×D53 | — | 2 | 2 | no | — |
| 81 | C2×Dic106 | — | 4 | 1 | no | — |
| 82 | C2×Dic106 | C2×Q8 | 1 | 2 | yes | — |
| 83 | M16×C53 | — | 2 | 2 | no | — |
| 84 | M16×C53 | — | 2 | 2 | no | — |
| 85 | C2×C8×C53 | — | 2 | 2 | no | — |
| 86 | C4⋊C4×C53 | — | 2 | 2 | no | — |
| 87 | C4⋊C4×C53 | — | 4 | 2 | no | — |
| 88 | C4⋊C4×C53 | — | 4 | 1 | no | — |
| 89 | C22⋊C4×C53 | — | 2 | 2 | no | — |
| 90 | C42×C53 | — | 2 | 1 | no | — |
| 91 | C42×C53 | — | 2 | 1 | no | — |
| 92 | C42×C53 | — | 2 | 1 | no | — |
| 93 | C42×C53 | — | 2 | 2 | no | — |
| 94 | C42×C53 | C2×Q8 | 1 | 1 | yes | characteristic |
| 95 | C42×C53 | C2×Q8 | 1 | 1 | yes | characteristic |
| 96 | C42×C53 | #41 | 1 | 1 | yes | characteristic |
| 97 | C42×C53 | #41 | 1 | 1 | yes | characteristic |
| 98 | C42×C53 | #41 | 1 | 1 | yes | characteristic |
| 99 | C42×C53 | #41 | 1 | 1 | yes | characteristic |
| 100 | C22⋊Dic53 | — | 2 | 2 | no | — |
| 101 | #11719921 | — | 4 | 2 | no | — |
| 102 | C4⋊Dic53 | — | 2 | 2 | no | — |
| 103 | C4⋊Dic53 | — | 4 | 1 | no | — |
| 104 | C4⋊Dic53 | — | 4 | 1 | no | — |
| 105 | #11719919 | — | 4 | 2 | no | — |
| 106 | C4×Dic53 | — | 2 | 2 | no | — |
| 107 | #11719917 | — | 2 | 2 | no | — |
| 108 | #11719917 | — | 2 | 2 | no | — |
| 109 | #11719917 | — | 2 | 2 | no | — |
| 110 | #11719917 | — | 2 | 2 | no | — |
| 111 | C2×#88322 | — | 2 | 2 | no | — |
| 112 | C2×#88322 | — | 2 | 2 | no | — |
| 113 | #11719912 | — | 4 | 4 | no | — |
| 114 | C8×D53 | — | 4 | 4 | no | — |
| 115 | C23×C53 | H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 116 | C22×D53 | — | 2 | 2 | no | — |
| 117 | Q8×C53 | — | 8 | 2 | no | — |
| 118 | C2×C4×C53 | — | 2 | 1 | no | — |
| 119 | C2×C4×C53 | — | 2 | 1 | no | — |
| 120 | C2×C4×C53 | — | 2 | 1 | no | — |
| 121 | C2×C4×C53 | — | 2 | 1 | no | — |
| 122 | C2×C4×C53 | — | 2 | 1 | no | — |
| 123 | C2×C4×C53 | — | 2 | 1 | no | — |
| 124 | C2×C4×C53 | — | 4 | 2 | no | — |
| 125 | C2×C4×C53 | — | 8 | 1 | no | — |
| 126 | C2×C4×C53 | — | 8 | 1 | no | — |
| 127 | C2×C4×C53 | — | 8 | 1 | no | — |
| 128 | C2×C4×C53 | C22×Q8 | 1 | 1 | yes | characteristic |
| 129 | C2×C4×C53 | C2×#41 | 1 | 1 | yes | characteristic |
| 130 | C2×C4×C53 | C2×#41 | 1 | 1 | yes | characteristic |
| 131 | C2×C4×C53 | H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 132 | C2×C4×C53 | H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 133 | C2×C4×C53 | H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 134 | C22⋊D53 | — | 4 | 2 | no | — |
| 135 | C22⋊D53 | — | 4 | 2 | no | — |
| 136 | C2×Dic53 | — | 2 | 2 | no | — |
| 137 | C2×Dic53 | — | 4 | 2 | no | — |
| 138 | D212 | — | 2 | 2 | no | — |
| 139 | D212 | — | 2 | 2 | no | — |
| 140 | D212 | — | 8 | 2 | no | — |
| 141 | C4×D53 | — | 4 | 2 | no | — |
| 142 | C4×D53 | — | 4 | 2 | no | — |
| 143 | Dic106 | — | 2 | 2 | no | — |
| 144 | Dic106 | — | 2 | 2 | no | — |
| 145 | Dic106 | — | 8 | 2 | no | — |
| 146 | C8×C53 | — | 4 | 2 | no | — |
| 147 | #88322 | — | 4 | 2 | no | — |
| 148 | #88322 | — | 4 | 2 | no | — |
| 149 | #32705 | — | 53 | 1 | no | maximal, Sylow |
| 150 | C22×C53 | — | 4 | 1 | no | — |
| 151 | C22×C53 | #538 | 1 | 1 | yes | characteristic |
| 152 | C22×C53 | #538 | 1 | 1 | yes | characteristic |
| 153 | C22×C53 | #543 | 1 | 1 | yes | characteristic |
| 154 | C2×D53 | — | 4 | 2 | no | — |
| 155 | C2×D53 | — | 8 | 2 | no | — |
| 156 | C4×C53 | — | 4 | 1 | no | — |
| 157 | C4×C53 | — | 4 | 1 | no | — |
| 158 | C4×C53 | — | 4 | 1 | no | — |
| 159 | C4×C53 | — | 4 | 1 | no | — |
| 160 | C4×C53 | — | 4 | 1 | no | — |
| 161 | C4×C53 | — | 4 | 1 | no | — |
| 162 | C4×C53 | — | 8 | 2 | no | — |
| 163 | C4×C53 | #554 | 1 | 1 | yes | characteristic |
| 164 | C4×C53 | #554 | 1 | 1 | yes | characteristic |
| 165 | C4×C53 | #559 | 1 | 1 | yes | characteristic |
| 166 | C4×C53 | H(3,𝔽4) | 1 | 1 | yes | characteristic |
| 167 | Dic53 | — | 4 | 2 | no | — |
| 168 | Dic53 | — | 8 | 2 | no | — |
| 169 | #2102 | — | 53 | 2 | no | — |
| 170 | #2031 | — | 53 | 2 | no | — |
| 171 | #2031 | — | 53 | 2 | no | — |
| 172 | #1995 | — | 53 | 1 | no | — |
| 173 | #1995 | D53 | 1 | 1 | yes | characteristic, complemented |
| 174 | #1963 | — | 53 | 1 | no | — |
| 175 | #1884 | — | 53 | 2 | no | — |
| 176 | #1861 | — | 53 | 2 | no | — |
| 177 | #1861 | — | 53 | 2 | no | — |
| 178 | C2×C53 | — | 2 | 1 | no | — |
| 179 | C2×C53 | — | 2 | 1 | no | — |
| 180 | C2×C53 | — | 2 | 1 | no | — |
| 181 | C2×C53 | #2804 | 1 | 1 | yes | characteristic |
| 182 | D53 | — | 8 | 2 | no | — |
| 183 | #568 | — | 53 | 2 | no | — |
| 184 | #533 | — | 53 | 1 | no | — |
| 185 | #533 | C2×D53 | 1 | 1 | yes | characteristic |
| 186 | #532 | — | 53 | 1 | no | — |
| 187 | #430 | — | 53 | 2 | no | — |
| 188 | #423 | — | 53 | 1 | no | — |
| 189 | #423 | — | 53 | 1 | no | — |
| 190 | #423 | C2×D53 | 1 | 1 | yes | characteristic |
| 191 | #421 | — | 53 | 2 | no | — |
| 192 | #421 | — | 53 | 2 | no | — |
| 193 | C2×#104 | — | 53 | 2 | no | — |
| 194 | C2×#104 | — | 53 | 2 | no | — |
| 195 | #410 | — | 53 | 2 | no | — |
| 196 | #376 | C2×D53 | 1 | 1 | yes | characteristic |
| 197 | #344 | — | 53 | 2 | no | — |
| 198 | #344 | — | 53 | 2 | no | — |
| 199 | #339 | — | 53 | 2 | no | — |
| 200 | #339 | — | 53 | 2 | no | — |
| 201 | #339 | C2×D53 | 1 | 2 | yes | — |
| 202 | #339 | C2×D53 | 1 | 2 | yes | — |
| 203 | H(3,ℤ/4) | — | 53 | 2 | no | — |
| 204 | C53 | #32705 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 205 | C2×#41 | — | 53 | 2 | no | — |
| 206 | #131 | — | 212 | 2 | no | — |
| 207 | #131 | — | 212 | 2 | no | — |
| 208 | C2×M16 | — | 53 | 2 | no | — |
| 209 | C2×M16 | — | 53 | 2 | no | — |
| 210 | C2×M16 | — | 212 | 2 | no | — |
| 211 | C2×M16 | C22×D53 | 1 | 2 | yes | — |
| 212 | C4⋊Q8 | — | 2 | 1 | no | — |
| 213 | C4⋊Q8 | — | 53 | 1 | no | — |
| 214 | C4⋊Q8 | — | 106 | 1 | no | — |
| 215 | #127 | — | 53 | 1 | no | — |
| 216 | #125 | — | 2 | 1 | no | — |
| 217 | #125 | — | 106 | 1 | no | — |
| 218 | #124 | — | 53 | 2 | no | — |
| 219 | C22⋊Q8 | — | 4 | 1 | no | — |
| 220 | C22⋊Q8 | — | 212 | 1 | no | — |
| 221 | #121 | — | 212 | 1 | no | — |
| 222 | C4×Q8 | — | 4 | 1 | no | — |
| 223 | C4×Q8 | — | 212 | 1 | no | — |
| 224 | C4×D4 | — | 212 | 1 | no | — |
| 225 | #117 | — | 53 | 2 | no | — |
| 226 | #117 | C22×D53 | 1 | 2 | yes | — |
| 227 | C2×C42 | C22×D53 | 1 | 1 | yes | characteristic |
| 228 | C2×C42 | C22×D53 | 1 | 1 | yes | characteristic |
| 229 | C2×C42 | C22×D53 | 1 | 1 | yes | characteristic |
| 230 | #108 | — | 212 | 2 | no | — |
| 231 | C4⋊C8 | — | 4 | 1 | no | — |
| 232 | C4⋊C8 | — | 212 | 1 | no | — |
| 233 | C4⋊C8 | — | 212 | 1 | no | — |
| 234 | #104 | — | 106 | 2 | no | — |
| 235 | #104 | — | 106 | 2 | no | — |
| 236 | #104 | — | 106 | 2 | no | — |
| 237 | #104 | — | 106 | 2 | no | — |
| 238 | #104 | — | 212 | 2 | no | — |
| 239 | #104 | — | 212 | 2 | no | — |
| 240 | #99 | — | 212 | 2 | no | — |
| 241 | #95 | — | 4 | 1 | no | — |
| 242 | #41 | — | 106 | 2 | no | — |
| 243 | #41 | — | 106 | 2 | no | — |
| 244 | #41 | — | 106 | 2 | no | — |
| 245 | #41 | — | 106 | 2 | no | — |
| 246 | C2×Q8 | — | 4 | 1 | no | — |
| 247 | C2×Q8 | — | 53 | 2 | no | — |
| 248 | C2×Q8 | — | 212 | 1 | no | — |
| 249 | C2×D4 | — | 53 | 2 | no | — |
| 250 | C2×D4 | — | 106 | 2 | no | — |
| 251 | C2×D4 | — | 212 | 1 | no | — |
| 252 | C22×C4 | — | 2 | 1 | no | — |
| 253 | C22×C4 | — | 2 | 1 | no | — |
| 254 | C22×C4 | — | 2 | 1 | no | — |
| 255 | C22×C4 | — | 106 | 2 | no | — |
| 256 | C22×C4 | C23×D53 | 1 | 1 | yes | characteristic |
| 257 | M16 | — | 2 | 2 | no | — |
| 258 | M16 | — | 2 | 2 | no | — |
| 259 | M16 | — | 106 | 2 | no | — |
| 260 | M16 | — | 106 | 2 | no | — |
| 261 | M16 | — | 106 | 2 | no | — |
| 262 | M16 | — | 106 | 2 | no | — |
| 263 | M16 | — | 212 | 4 | no | — |
| 264 | C2×C8 | — | 2 | 2 | no | — |
| 265 | C2×C8 | — | 106 | 2 | no | — |
| 266 | C2×C8 | — | 106 | 2 | no | — |
| 267 | C2×C8 | — | 212 | 4 | no | — |
| 268 | C4⋊C4 | — | 2 | 2 | no | — |
| 269 | C4⋊C4 | — | 4 | 2 | no | — |
| 270 | C4⋊C4 | — | 4 | 1 | no | — |
| 271 | C4⋊C4 | — | 106 | 2 | no | — |
| 272 | C4⋊C4 | — | 212 | 1 | no | — |
| 273 | C4⋊C4 | — | 212 | 2 | no | — |
| 274 | C4⋊C4 | — | 212 | 1 | no | — |
| 275 | C22⋊C4 | — | 2 | 2 | no | — |
| 276 | C22⋊C4 | — | 106 | 2 | no | — |
| 277 | C22⋊C4 | — | 212 | 2 | no | — |
| 278 | C42 | — | 2 | 1 | no | — |
| 279 | C42 | — | 2 | 1 | no | — |
| 280 | C42 | — | 2 | 1 | no | — |
| 281 | C42 | — | 2 | 2 | no | — |
| 282 | C42 | — | 106 | 2 | no | — |
| 283 | C42 | #11719945 | 1 | 1 | yes | characteristic |
| 284 | C42 | #11719945 | 1 | 1 | yes | characteristic |
| 285 | C42 | Q8×D53 | 1 | 1 | yes | characteristic |
| 286 | C42 | Q8×D53 | 1 | 1 | yes | characteristic |
| 287 | C42 | #11719949 | 1 | 1 | yes | characteristic |
| 288 | C42 | #11719949 | 1 | 1 | yes | characteristic |
| 289 | C23 | — | 106 | 2 | no | — |
| 290 | C23 | #13133262 | 1 | 1 | yes | characteristic |
| 291 | Q8 | — | 8 | 2 | no | — |
| 292 | Q8 | — | 106 | 2 | no | — |
| 293 | Q8 | — | 106 | 2 | no | — |
| 294 | Q8 | — | 424 | 2 | no | — |
| 295 | D4 | — | 106 | 2 | no | — |
| 296 | D4 | — | 106 | 2 | no | — |
| 297 | D4 | — | 424 | 2 | no | — |
| 298 | D4 | — | 212 | 2 | no | — |
| 299 | D4 | — | 212 | 2 | no | — |
| 300 | C2×C4 | — | 2 | 1 | no | — |
| 301 | C2×C4 | — | 2 | 1 | no | — |
| 302 | C2×C4 | — | 2 | 1 | no | — |
| 303 | C2×C4 | — | 2 | 1 | no | — |
| 304 | C2×C4 | — | 2 | 1 | no | — |
| 305 | C2×C4 | — | 2 | 1 | no | — |
| 306 | C2×C4 | — | 4 | 2 | no | — |
| 307 | C2×C4 | — | 8 | 1 | no | — |
| 308 | C2×C4 | — | 8 | 1 | no | — |
| 309 | C2×C4 | — | 8 | 1 | no | — |
| 310 | C2×C4 | — | 106 | 2 | no | — |
| 311 | C2×C4 | — | 212 | 2 | no | — |
| 312 | C2×C4 | — | 212 | 2 | no | — |
| 313 | C2×C4 | — | 212 | 2 | no | — |
| 314 | C2×C4 | C2×Q8×D53 | 1 | 1 | yes | characteristic |
| 315 | C2×C4 | C2×#11719945 | 1 | 1 | yes | characteristic |
| 316 | C2×C4 | C2×#11719949 | 1 | 1 | yes | characteristic |
| 317 | C2×C4 | #13133262 | 1 | 1 | yes | characteristic |
| 318 | C2×C4 | #13133263 | 1 | 1 | yes | characteristic |
| 319 | C2×C4 | #13133263 | 1 | 1 | yes | characteristic |
| 320 | C8 | — | 4 | 2 | no | — |
| 321 | C8 | — | 212 | 2 | no | — |
| 322 | C8 | — | 212 | 2 | no | — |
| 323 | C22 | — | 4 | 1 | no | — |
| 324 | C22 | — | 424 | 2 | no | — |
| 325 | C22 | — | 212 | 2 | no | — |
| 326 | C22 | #16811211 | 1 | 1 | yes | characteristic |
| 327 | C22 | #16811213 | 1 | 1 | yes | characteristic |
| 328 | C22 | #16811248 | 1 | 1 | yes | characteristic |
| 329 | C4 | — | 4 | 1 | no | — |
| 330 | C4 | — | 4 | 1 | no | — |
| 331 | C4 | — | 4 | 1 | no | — |
| 332 | C4 | — | 4 | 1 | no | — |
| 333 | C4 | — | 4 | 1 | no | — |
| 334 | C4 | — | 4 | 1 | no | — |
| 335 | C4 | — | 8 | 2 | no | — |
| 336 | C4 | — | 424 | 2 | no | — |
| 337 | C4 | — | 212 | 2 | no | — |
| 338 | C4 | #16811333 | 1 | 1 | yes | characteristic |
| 339 | C4 | #16811334 | 1 | 1 | yes | characteristic |
| 340 | C4 | #16811358 | 1 | 1 | yes | characteristic |
| 341 | C4 | #16811365 | 1 | 1 | yes | characteristic |
| 342 | C2 | — | 2 | 1 | no | — |
| 343 | C2 | — | 2 | 1 | no | — |
| 344 | C2 | — | 2 | 1 | no | — |
| 345 | C2 | — | 424 | 2 | no | — |
| 346 | C2 | #20282659 | 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