← #27537819
Subgroups — #27537819
Order 13568.
484 conjugacy classes of subgroups.
The subgroup lattice has 326 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | #20283690 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #20276622 | C2 | 1 | 2 | yes | maximal, complemented |
| 3 | #20276545 | C2 | 1 | 2 | yes | maximal, complemented |
| 4 | #20276539 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | #20276538 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | #20276521 | C2 | 1 | 2 | yes | maximal, complemented |
| 7 | #20276518 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 8 | #20276472 | C2 | 1 | 2 | yes | maximal, complemented |
| 9 | C53×#2150 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 10 | #16811157 | C4 | 1 | 1 | yes | characteristic |
| 11 | C2×#13133156 | C22 | 1 | 2 | yes | — |
| 12 | C2×C16×D53 | C22 | 1 | 2 | yes | — |
| 13 | #16810753 | C4 | 1 | 1 | yes | characteristic |
| 14 | C8×Dic106 | C22 | 1 | 1 | yes | characteristic |
| 15 | #16810702 | C4 | 1 | 1 | yes | characteristic |
| 16 | #16810694 | C4 | 1 | 2 | yes | — |
| 17 | C8×D212 | C22 | 1 | 1 | yes | characteristic |
| 18 | C8×C22⋊D53 | C22 | 1 | 2 | yes | — |
| 19 | #16810682 | C4 | 1 | 2 | yes | — |
| 20 | #16810680 | C4 | 1 | 1 | yes | characteristic |
| 21 | #16810577 | C22 | 1 | 2 | yes | — |
| 22 | C4⋊#11719908 | C22 | 1 | 1 | yes | characteristic |
| 23 | C4⋊#11719908 | C22 | 1 | 1 | yes | characteristic |
| 24 | #16810441 | C22 | 1 | 2 | yes | — |
| 25 | #16810441 | C22 | 1 | 2 | yes | — |
| 26 | C22⋊#11719908 | C22 | 1 | 2 | yes | — |
| 27 | #16810402 | C22 | 1 | 2 | yes | — |
| 28 | #16810402 | C22 | 1 | 2 | yes | — |
| 29 | #16810399 | C22 | 1 | 2 | yes | — |
| 30 | #16810398 | C22 | 1 | 2 | yes | — |
| 31 | C16×Dic53 | C22 | 1 | 2 | yes | — |
| 32 | C2×M32×C53 | C22 | 1 | 1 | yes | characteristic |
| 33 | C2×M32×C53 | C22 | 1 | 1 | yes | characteristic |
| 34 | C2×C4×C8×C53 | C22 | 1 | 1 | yes | characteristic |
| 35 | C53×C4⋊C16 | C22 | 1 | 2 | yes | — |
| 36 | C53×C4⋊C16 | C22 | 1 | 2 | yes | — |
| 37 | C2×#11719912 | C2×C4 | 1 | 2 | yes | — |
| 38 | C2×C8×D53 | C23 | 1 | 2 | yes | — |
| 39 | #13133211 | C8 | 1 | 2 | yes | — |
| 40 | #13133209 | C8 | 1 | 1 | yes | characteristic |
| 41 | #13133205 | C8 | 1 | 1 | yes | characteristic |
| 42 | #13133200 | C8 | 1 | 2 | yes | — |
| 43 | #13133196 | C8 | 1 | 1 | yes | characteristic |
| 44 | #13133191 | C8 | 1 | 1 | yes | characteristic |
| 45 | C4×Dic106 | C2×C4 | 1 | 1 | yes | characteristic |
| 46 | C4×D212 | C2×C4 | 1 | 1 | yes | characteristic |
| 47 | C4×C22⋊D53 | C2×C4 | 1 | 2 | yes | — |
| 48 | #13133174 | C2×C4 | 1 | 2 | yes | — |
| 49 | #13133156 | — | 4 | 2 | no | — |
| 50 | C2×#11719908 | C23 | 1 | 2 | yes | — |
| 51 | C2×#11719908 | C23 | 1 | 2 | yes | — |
| 52 | C16×D53 | — | 4 | 2 | no | — |
| 53 | C4⋊#88322 | C2×C4 | 1 | 1 | yes | characteristic |
| 54 | C4⋊#88322 | C2×C4 | 1 | 1 | yes | characteristic |
| 55 | #13133146 | C2×C4 | 1 | 2 | yes | — |
| 56 | #13133146 | C2×C4 | 1 | 2 | yes | — |
| 57 | C22⋊#88322 | C2×C4 | 1 | 2 | yes | — |
| 58 | #13133129 | C2×C4 | 1 | 2 | yes | — |
| 59 | #13133129 | C2×C4 | 1 | 2 | yes | — |
| 60 | #13133128 | C2×C4 | 1 | 2 | yes | — |
| 61 | #13133127 | C2×C4 | 1 | 2 | yes | — |
| 62 | C8×Dic53 | C23 | 1 | 2 | yes | — |
| 63 | C22×C8×C53 | C2×C4 | 1 | 1 | yes | characteristic |
| 64 | C22×C8×C53 | C23 | 1 | 1 | yes | characteristic |
| 65 | C2×C42×C53 | C2×C4 | 1 | 1 | yes | characteristic |
| 66 | M32×C53 | — | 4 | 1 | no | — |
| 67 | M32×C53 | — | 4 | 1 | no | — |
| 68 | C2×C16×C53 | C23 | 1 | 2 | yes | — |
| 69 | C2×C16×C53 | C23 | 1 | 2 | yes | — |
| 70 | C4×C8×C53 | C2×C4 | 1 | 1 | yes | characteristic |
| 71 | C4×C8×C53 | C2×C4 | 1 | 1 | yes | characteristic |
| 72 | C4×C8×C53 | C23 | 1 | 1 | yes | characteristic |
| 73 | C4×C8×C53 | C23 | 1 | 1 | yes | characteristic |
| 74 | C22×C4×C53 | C2×C8 | 1 | 1 | yes | characteristic |
| 75 | C22×C4×C53 | C2×C8 | 1 | 1 | yes | characteristic |
| 76 | C22×C4×C53 | C22×C4 | 1 | 1 | yes | characteristic |
| 77 | C2×C22⋊D53 | C2×C8 | 1 | 2 | yes | — |
| 78 | C2×D212 | C2×C8 | 1 | 1 | yes | characteristic |
| 79 | C2×C4×D53 | C22×C4 | 1 | 2 | yes | — |
| 80 | C2×Dic106 | C2×C8 | 1 | 1 | yes | characteristic |
| 81 | C2×C8×C53 | — | 2 | 1 | no | — |
| 82 | C2×C8×C53 | — | 2 | 1 | no | — |
| 83 | C2×C8×C53 | — | 4 | 1 | no | — |
| 84 | C2×C8×C53 | C22×C4 | 1 | 1 | yes | characteristic |
| 85 | C2×C8×C53 | C22×C4 | 1 | 1 | yes | characteristic |
| 86 | C2×C8×C53 | C22×C4 | 1 | 1 | yes | characteristic |
| 87 | C2×C8×C53 | C24 | 1 | 1 | yes | characteristic |
| 88 | C42×C53 | C2×C8 | 1 | 1 | yes | characteristic |
| 89 | C42×C53 | C2×C8 | 1 | 1 | yes | characteristic |
| 90 | C42×C53 | C22×C4 | 1 | 1 | yes | characteristic |
| 91 | C42×C53 | C22×C4 | 1 | 1 | yes | characteristic |
| 92 | C22⋊Dic53 | C2×C8 | 1 | 2 | yes | — |
| 93 | #11719921 | C2×C8 | 1 | 2 | yes | — |
| 94 | #11719921 | C2×C8 | 1 | 2 | yes | — |
| 95 | #11719921 | C2×C8 | 1 | 2 | yes | — |
| 96 | #11719921 | C2×C8 | 1 | 2 | yes | — |
| 97 | C4⋊Dic53 | C2×C8 | 1 | 1 | yes | characteristic |
| 98 | C4⋊Dic53 | C2×C8 | 1 | 1 | yes | characteristic |
| 99 | #11719919 | C2×C8 | 1 | 2 | yes | — |
| 100 | #11719919 | C2×C8 | 1 | 2 | yes | — |
| 101 | #11719919 | C2×C8 | 1 | 2 | yes | — |
| 102 | #11719919 | C2×C8 | 1 | 2 | yes | — |
| 103 | C4×Dic53 | C22×C4 | 1 | 2 | yes | — |
| 104 | C2×#88322 | C22×C4 | 1 | 2 | yes | — |
| 105 | C2×#88322 | C22×C4 | 1 | 2 | yes | — |
| 106 | #11719912 | — | 4 | 2 | no | — |
| 107 | C8×D53 | — | 2 | 2 | no | — |
| 108 | C8×D53 | — | 2 | 2 | no | — |
| 109 | C16×C53 | — | 2 | 2 | no | — |
| 110 | C16×C53 | — | 2 | 2 | no | — |
| 111 | #11719908 | — | 2 | 2 | no | — |
| 112 | #11719908 | — | 2 | 2 | no | — |
| 113 | C23×C53 | C22×C8 | 1 | 1 | yes | characteristic |
| 114 | C22×D53 | C22×C8 | 1 | 2 | yes | — |
| 115 | C2×C4×C53 | — | 2 | 1 | no | — |
| 116 | C2×C4×C53 | — | 2 | 1 | no | — |
| 117 | C2×C4×C53 | — | 4 | 1 | no | — |
| 118 | C2×C4×C53 | — | 4 | 1 | no | — |
| 119 | C2×C4×C53 | C22×C8 | 1 | 1 | yes | characteristic |
| 120 | C2×C4×C53 | C22×C8 | 1 | 1 | yes | characteristic |
| 121 | C2×C4×C53 | C22×C8 | 1 | 1 | yes | characteristic |
| 122 | C2×C4×C53 | C22×C8 | 1 | 1 | yes | characteristic |
| 123 | C2×C4×C53 | C22×C8 | 1 | 1 | yes | characteristic |
| 124 | C2×C4×C53 | C23×C4 | 1 | 1 | yes | characteristic |
| 125 | C22⋊D53 | — | 4 | 2 | no | — |
| 126 | C2×Dic53 | C22×C8 | 1 | 2 | yes | — |
| 127 | C2×Dic53 | C22×C8 | 1 | 2 | yes | — |
| 128 | C2×Dic53 | C22×C8 | 1 | 2 | yes | — |
| 129 | D212 | — | 4 | 1 | no | — |
| 130 | C4×D53 | — | 2 | 2 | no | — |
| 131 | C4×D53 | — | 2 | 2 | no | — |
| 132 | Dic106 | — | 4 | 1 | no | — |
| 133 | C8×C53 | — | 2 | 1 | no | — |
| 134 | C8×C53 | — | 2 | 1 | no | — |
| 135 | C8×C53 | — | 2 | 1 | no | — |
| 136 | C8×C53 | H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 137 | C8×C53 | 2−1+4 | 1 | 1 | yes | characteristic |
| 138 | #88322 | — | 2 | 2 | no | — |
| 139 | #88322 | — | 2 | 2 | no | — |
| 140 | #33509 | — | 53 | 1 | no | maximal, Sylow |
| 141 | C22×C53 | — | 2 | 1 | no | — |
| 142 | C22×C53 | — | 2 | 1 | no | — |
| 143 | C22×C53 | — | 2 | 1 | no | — |
| 144 | C22×C53 | C23×C8 | 1 | 1 | yes | characteristic |
| 145 | C2×D53 | — | 2 | 2 | no | — |
| 146 | C2×D53 | — | 2 | 2 | no | — |
| 147 | C2×D53 | — | 2 | 2 | no | — |
| 148 | C4×C53 | — | 2 | 1 | no | — |
| 149 | C4×C53 | — | 2 | 1 | no | — |
| 150 | C4×C53 | — | 2 | 1 | no | — |
| 151 | C4×C53 | — | 2 | 1 | no | — |
| 152 | C4×C53 | — | 2 | 1 | no | — |
| 153 | C4×C53 | #520 | 1 | 1 | yes | characteristic |
| 154 | C4×C53 | #568 | 1 | 1 | yes | characteristic |
| 155 | Dic53 | — | 2 | 2 | no | — |
| 156 | Dic53 | — | 2 | 2 | no | — |
| 157 | Dic53 | — | 2 | 2 | no | — |
| 158 | #2966 | — | 53 | 1 | no | — |
| 159 | #2183 | — | 53 | 2 | no | — |
| 160 | #2169 | — | 53 | 2 | no | — |
| 161 | #2168 | — | 53 | 2 | no | — |
| 162 | #2168 | — | 53 | 2 | no | — |
| 163 | #2164 | — | 53 | 2 | no | — |
| 164 | #2163 | — | 53 | 1 | no | — |
| 165 | #2151 | — | 53 | 2 | no | — |
| 166 | #2150 | D53 | 1 | 1 | yes | characteristic, complemented |
| 167 | C2×C53 | — | 4 | 1 | no | — |
| 168 | C2×C53 | #2988 | 1 | 1 | yes | characteristic |
| 169 | C2×C53 | #3407 | 1 | 1 | yes | characteristic |
| 170 | C2×C53 | #3407 | 1 | 1 | yes | characteristic |
| 171 | D53 | — | 4 | 2 | no | — |
| 172 | #529 | — | 53 | 1 | no | — |
| 173 | C2×M32 | — | 53 | 2 | no | — |
| 174 | C2×M32 | C2×D53 | 1 | 1 | yes | characteristic |
| 175 | C2×M32 | C2×D53 | 1 | 1 | yes | characteristic |
| 176 | C22×C16 | — | 53 | 2 | no | — |
| 177 | #446 | — | 53 | 1 | no | — |
| 178 | C8×Q8 | — | 53 | 1 | no | — |
| 179 | #436 | — | 53 | 1 | no | — |
| 180 | #435 | — | 53 | 2 | no | — |
| 181 | C8×D4 | — | 53 | 2 | no | — |
| 182 | C8×D4 | — | 53 | 1 | no | — |
| 183 | #433 | — | 53 | 2 | no | — |
| 184 | #432 | — | 53 | 1 | no | — |
| 185 | #405 | — | 53 | 2 | no | — |
| 186 | C2×C4×C8 | C2×D53 | 1 | 1 | yes | characteristic |
| 187 | C4⋊C16 | — | 53 | 2 | no | — |
| 188 | C4⋊C16 | — | 53 | 2 | no | — |
| 189 | C4⋊C16 | — | 53 | 1 | no | — |
| 190 | C4⋊C16 | — | 53 | 1 | no | — |
| 191 | C4⋊C16 | C2×D53 | 1 | 2 | yes | — |
| 192 | C4⋊C16 | C2×D53 | 1 | 2 | yes | — |
| 193 | C22⋊C16 | — | 53 | 2 | no | — |
| 194 | C22⋊C16 | — | 53 | 2 | no | — |
| 195 | C22⋊C16 | — | 53 | 2 | no | — |
| 196 | #346 | — | 53 | 2 | no | — |
| 197 | #346 | — | 53 | 2 | no | — |
| 198 | C4×C16 | — | 53 | 2 | no | — |
| 199 | C53 | #33509 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 200 | C2×M16 | — | 53 | 2 | no | — |
| 201 | C22×C8 | — | 53 | 2 | no | — |
| 202 | C22×C8 | C4×D53 | 1 | 1 | yes | characteristic |
| 203 | C22×C8 | C22×D53 | 1 | 1 | yes | characteristic |
| 204 | #126 | — | 53 | 2 | no | — |
| 205 | #125 | — | 53 | 1 | no | — |
| 206 | #124 | — | 53 | 1 | no | — |
| 207 | #123 | — | 53 | 2 | no | — |
| 208 | C22⋊Q8 | — | 53 | 1 | no | — |
| 209 | #121 | — | 53 | 1 | no | — |
| 210 | C4×Q8 | — | 53 | 1 | no | — |
| 211 | C4×D4 | — | 53 | 2 | no | — |
| 212 | C4×D4 | — | 53 | 1 | no | — |
| 213 | #117 | — | 53 | 2 | no | — |
| 214 | C2×C42 | C4×D53 | 1 | 1 | yes | characteristic |
| 215 | M32 | — | 4 | 1 | no | — |
| 216 | M32 | — | 4 | 1 | no | — |
| 217 | M32 | — | 212 | 2 | no | — |
| 218 | C2×C16 | — | 53 | 2 | no | — |
| 219 | C2×C16 | — | 53 | 2 | no | — |
| 220 | C2×C16 | — | 212 | 2 | no | — |
| 221 | C2×C16 | C22×D53 | 1 | 2 | yes | — |
| 222 | C2×C16 | C22×D53 | 1 | 2 | yes | — |
| 223 | C4⋊C8 | — | 53 | 2 | no | — |
| 224 | C4⋊C8 | — | 53 | 2 | no | — |
| 225 | C4⋊C8 | — | 53 | 1 | no | — |
| 226 | C4⋊C8 | — | 53 | 1 | no | — |
| 227 | C22⋊C8 | — | 53 | 2 | no | — |
| 228 | C22⋊C8 | — | 53 | 2 | no | — |
| 229 | C22⋊C8 | — | 53 | 2 | no | — |
| 230 | #97 | — | 53 | 2 | no | — |
| 231 | #97 | — | 53 | 2 | no | — |
| 232 | C4×C8 | — | 53 | 2 | no | — |
| 233 | C4×C8 | C4×D53 | 1 | 1 | yes | characteristic |
| 234 | C4×C8 | C4×D53 | 1 | 1 | yes | characteristic |
| 235 | C4×C8 | C22×D53 | 1 | 1 | yes | characteristic |
| 236 | C4×C8 | C22×D53 | 1 | 1 | yes | characteristic |
| 237 | C2×Q8 | — | 53 | 1 | no | — |
| 238 | C2×D4 | — | 53 | 2 | no | — |
| 239 | C2×D4 | — | 53 | 1 | no | — |
| 240 | C22×C4 | — | 53 | 2 | no | — |
| 241 | C22×C4 | C8×D53 | 1 | 1 | yes | characteristic |
| 242 | C22×C4 | C8×D53 | 1 | 1 | yes | characteristic |
| 243 | C22×C4 | C2×C4×D53 | 1 | 1 | yes | characteristic |
| 244 | M16 | — | 212 | 2 | no | — |
| 245 | C2×C8 | — | 2 | 1 | no | — |
| 246 | C2×C8 | — | 2 | 1 | no | — |
| 247 | C2×C8 | — | 4 | 1 | no | — |
| 248 | C2×C8 | — | 53 | 2 | no | — |
| 249 | C2×C8 | — | 53 | 2 | no | — |
| 250 | C2×C8 | — | 106 | 2 | no | — |
| 251 | C2×C8 | — | 106 | 2 | no | — |
| 252 | C2×C8 | C2×C4×D53 | 1 | 1 | yes | characteristic |
| 253 | C2×C8 | C2×C4×D53 | 1 | 1 | yes | characteristic |
| 254 | C2×C8 | C2×C4×D53 | 1 | 1 | yes | characteristic |
| 255 | C2×C8 | C23×D53 | 1 | 1 | yes | characteristic |
| 256 | C4⋊C4 | — | 53 | 2 | no | — |
| 257 | C4⋊C4 | — | 53 | 2 | no | — |
| 258 | C4⋊C4 | — | 53 | 1 | no | — |
| 259 | C4⋊C4 | — | 53 | 2 | no | — |
| 260 | C4⋊C4 | — | 53 | 1 | no | — |
| 261 | C4⋊C4 | — | 53 | 2 | no | — |
| 262 | C22⋊C4 | — | 53 | 2 | no | — |
| 263 | C22⋊C4 | — | 53 | 2 | no | — |
| 264 | C22⋊C4 | — | 53 | 2 | no | — |
| 265 | C22⋊C4 | — | 53 | 2 | no | — |
| 266 | C22⋊C4 | — | 53 | 2 | no | — |
| 267 | C42 | — | 53 | 2 | no | — |
| 268 | C42 | C8×D53 | 1 | 1 | yes | characteristic |
| 269 | C42 | C8×D53 | 1 | 1 | yes | characteristic |
| 270 | C42 | C2×C4×D53 | 1 | 1 | yes | characteristic |
| 271 | C42 | C2×C4×D53 | 1 | 1 | yes | characteristic |
| 272 | C16 | — | 2 | 2 | no | — |
| 273 | C16 | — | 2 | 2 | no | — |
| 274 | C16 | — | 106 | 2 | no | — |
| 275 | C16 | — | 106 | 2 | no | — |
| 276 | C23 | — | 53 | 2 | no | — |
| 277 | C23 | C2×C8×D53 | 1 | 1 | yes | characteristic |
| 278 | Q8 | — | 212 | 1 | no | — |
| 279 | D4 | — | 212 | 1 | no | — |
| 280 | D4 | — | 212 | 2 | no | — |
| 281 | C2×C4 | — | 2 | 1 | no | — |
| 282 | C2×C4 | — | 2 | 1 | no | — |
| 283 | C2×C4 | — | 4 | 1 | no | — |
| 284 | C2×C4 | — | 4 | 1 | no | — |
| 285 | C2×C4 | — | 53 | 2 | no | — |
| 286 | C2×C4 | — | 53 | 2 | no | — |
| 287 | C2×C4 | — | 53 | 2 | no | — |
| 288 | C2×C4 | — | 106 | 2 | no | — |
| 289 | C2×C4 | — | 106 | 2 | no | — |
| 290 | C2×C4 | C2×C8×D53 | 1 | 1 | yes | characteristic |
| 291 | C2×C4 | C2×C8×D53 | 1 | 1 | yes | characteristic |
| 292 | C2×C4 | C2×C8×D53 | 1 | 1 | yes | characteristic |
| 293 | C2×C4 | C2×C8×D53 | 1 | 1 | yes | characteristic |
| 294 | C2×C4 | C2×C8×D53 | 1 | 1 | yes | characteristic |
| 295 | C2×C4 | C22×C4×D53 | 1 | 1 | yes | characteristic |
| 296 | C8 | — | 2 | 1 | no | — |
| 297 | C8 | — | 2 | 1 | no | — |
| 298 | C8 | — | 2 | 1 | no | — |
| 299 | C8 | — | 106 | 2 | no | — |
| 300 | C8 | — | 106 | 2 | no | — |
| 301 | C8 | #13133262 | 1 | 1 | yes | characteristic |
| 302 | C8 | #13133265 | 1 | 1 | yes | characteristic |
| 303 | C22 | — | 2 | 1 | no | — |
| 304 | C22 | — | 2 | 1 | no | — |
| 305 | C22 | — | 2 | 1 | no | — |
| 306 | C22 | — | 106 | 2 | no | — |
| 307 | C22 | — | 106 | 2 | no | — |
| 308 | C22 | — | 106 | 2 | no | — |
| 309 | C22 | C22×C8×D53 | 1 | 1 | yes | characteristic |
| 310 | C4 | — | 2 | 1 | no | — |
| 311 | C4 | — | 2 | 1 | no | — |
| 312 | C4 | — | 2 | 1 | no | — |
| 313 | C4 | — | 2 | 1 | no | — |
| 314 | C4 | — | 2 | 1 | no | — |
| 315 | C4 | — | 106 | 2 | no | — |
| 316 | C4 | — | 106 | 2 | no | — |
| 317 | C4 | — | 106 | 2 | no | — |
| 318 | C4 | #16811104 | 1 | 1 | yes | characteristic |
| 319 | C4 | #16811389 | 1 | 1 | yes | characteristic |
| 320 | C2 | — | 4 | 1 | no | — |
| 321 | C2 | — | 212 | 2 | no | — |
| 322 | C2 | #20283824 | 1 | 1 | yes | characteristic |
| 323 | C2 | #20287817 | 1 | 1 | yes | characteristic |
| 324 | C2 | #20287817 | 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