← #27393262
Subgroups — #27393262
Order 13568.
1270 conjugacy classes of subgroups.
The subgroup lattice has 322 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | #20283194 | C2 | 1 | 2 | yes | maximal, complemented |
| 2 | #20275268 | C2 | 1 | 4 | yes | maximal, complemented |
| 3 | #20275134 | C2 | 1 | 4 | yes | maximal, complemented |
| 4 | #20274826 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | #20274643 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | C53×#3485 | C2 | 1 | 1 | yes | maximal, characteristic |
| 7 | #16811390 | C22 | 1 | 4 | yes | — |
| 8 | #16810607 | — | 2 | 4 | no | — |
| 9 | #16810607 | C22 | 1 | 4 | yes | — |
| 10 | C2×#13133136 | C22 | 1 | 4 | yes | complemented |
| 11 | C2×#13133134 | — | 2 | 4 | no | — |
| 12 | C2×#13133134 | C22 | 1 | 4 | yes | — |
| 13 | #16810587 | C22 | 1 | 4 | yes | complemented |
| 14 | #16810393 | C22 | 1 | 8 | yes | complemented |
| 15 | C2×H(5,𝔽2)×C53 | C22 | 1 | 1 | yes | characteristic |
| 16 | C22×#41×C53 | C22 | 1 | 1 | yes | characteristic |
| 17 | C53×#547 | — | 2 | 4 | no | — |
| 18 | C53×#546 | — | 2 | 4 | no | — |
| 19 | C53×#536 | C22 | 1 | 2 | yes | — |
| 20 | C53×#534 | C4 | 1 | 4 | yes | — |
| 21 | C53×#525 | C22 | 1 | 2 | yes | — |
| 22 | C2×#123×C53 | C4 | 1 | 4 | yes | — |
| 23 | C53×#518 | C22 | 1 | 1 | yes | characteristic |
| 24 | #13133224 | — | 4 | 8 | no | — |
| 25 | C2×#11719917 | — | 2 | 8 | no | — |
| 26 | C2×#11719917 | C23 | 1 | 8 | yes | — |
| 27 | #13133136 | — | 2 | 4 | no | — |
| 28 | #13133136 | — | 4 | 4 | no | — |
| 29 | #13133134 | — | 2 | 4 | no | — |
| 30 | #13133134 | — | 2 | 8 | no | — |
| 31 | C22⋊#88322 | — | 4 | 4 | no | — |
| 32 | H(5,𝔽2)×C53 | — | 4 | 4 | no | — |
| 33 | C2×#41×C53 | — | 2 | 4 | no | — |
| 34 | C2×#41×C53 | — | 4 | 4 | no | — |
| 35 | C2×#41×C53 | D4 | 1 | 4 | yes | — |
| 36 | C2×#41×C53 | C23 | 1 | 2 | yes | — |
| 37 | C22×Q8×C53 | C23 | 1 | 1 | yes | characteristic |
| 38 | C22×D4×C53 | C2×C4 | 1 | 4 | yes | — |
| 39 | C22×D4×C53 | C2×C4 | 1 | 2 | yes | — |
| 40 | C22×D4×C53 | D4 | 1 | 4 | yes | — |
| 41 | C22×D4×C53 | C23 | 1 | 1 | yes | characteristic |
| 42 | C23×C4×C53 | C2×C4 | 1 | 2 | yes | — |
| 43 | C23×C4×C53 | C23 | 1 | 1 | yes | characteristic |
| 44 | C4⋊Q8×C53 | — | 2 | 2 | no | — |
| 45 | #124×C53 | — | 2 | 2 | no | — |
| 46 | #123×C53 | — | 2 | 8 | no | — |
| 47 | #123×C53 | — | 2 | 4 | no | — |
| 48 | #123×C53 | C2×C4 | 1 | 4 | yes | — |
| 49 | #123×C53 | C2×C4 | 1 | 8 | yes | — |
| 50 | C22⋊Q8×C53 | — | 2 | 2 | no | — |
| 51 | C22⋊Q8×C53 | — | 2 | 4 | no | — |
| 52 | #121×C53 | — | 2 | 8 | no | — |
| 53 | #121×C53 | — | 2 | 2 | no | — |
| 54 | #120×C53 | — | 2 | 4 | no | — |
| 55 | #120×C53 | C2×C4 | 1 | 4 | yes | — |
| 56 | C4×D4×C53 | — | 2 | 4 | no | — |
| 57 | C4×D4×C53 | — | 2 | 4 | no | — |
| 58 | #117×C53 | C23 | 1 | 2 | yes | — |
| 59 | C2×C4⋊C4×C53 | — | 2 | 4 | no | — |
| 60 | C2×C22⋊C4×C53 | — | 2 | 4 | no | — |
| 61 | C2×C22⋊C4×C53 | C2×C4 | 1 | 4 | yes | — |
| 62 | C24×C53 | C22⋊C4 | 1 | 4 | yes | — |
| 63 | C24×C53 | C22×C4 | 1 | 1 | yes | characteristic |
| 64 | C24×C53 | C22×C4 | 1 | 2 | yes | — |
| 65 | #41×C53 | — | 2 | 8 | no | — |
| 66 | #41×C53 | — | 4 | 8 | no | — |
| 67 | #41×C53 | — | 4 | 16 | no | — |
| 68 | C2×Q8×C53 | — | 2 | 2 | no | — |
| 69 | C2×Q8×C53 | — | 2 | 4 | no | — |
| 70 | C2×Q8×C53 | C2×D4 | 1 | 2 | yes | — |
| 71 | C2×D4×C53 | — | 2 | 4 | no | — |
| 72 | C2×D4×C53 | — | 2 | 4 | no | — |
| 73 | C2×D4×C53 | — | 2 | 4 | no | — |
| 74 | C2×D4×C53 | — | 4 | 8 | no | — |
| 75 | C2×D4×C53 | — | 4 | 4 | no | — |
| 76 | C2×D4×C53 | — | 4 | 8 | no | — |
| 77 | C2×D4×C53 | C22⋊C4 | 1 | 8 | yes | — |
| 78 | C2×D4×C53 | C22×C4 | 1 | 4 | yes | — |
| 79 | C2×D4×C53 | C2×D4 | 1 | 4 | yes | — |
| 80 | C2×D4×C53 | C2×D4 | 1 | 2 | yes | — |
| 81 | C22×C4×C53 | — | 2 | 4 | no | — |
| 82 | C22×C4×C53 | — | 2 | 4 | no | — |
| 83 | C22×C4×C53 | — | 2 | 2 | no | — |
| 84 | C22×C4×C53 | — | 2 | 4 | no | — |
| 85 | C22×C4×C53 | — | 4 | 4 | no | — |
| 86 | C22×C4×C53 | C22⋊C4 | 1 | 4 | yes | — |
| 87 | C22×C4×C53 | C22×C4 | 1 | 1 | yes | characteristic |
| 88 | C22×C4×C53 | C22×C4 | 1 | 2 | yes | — |
| 89 | C22×C4×C53 | C2×D4 | 1 | 4 | yes | — |
| 90 | C22×C4×C53 | C24 | 1 | 1 | yes | characteristic |
| 91 | C4⋊C4×C53 | — | 2 | 4 | no | — |
| 92 | C4⋊C4×C53 | — | 2 | 8 | no | — |
| 93 | C4⋊C4×C53 | — | 2 | 2 | no | — |
| 94 | C22⋊C4×C53 | — | 2 | 4 | no | — |
| 95 | C22⋊C4×C53 | — | 2 | 8 | no | — |
| 96 | C22⋊C4×C53 | — | 2 | 4 | no | — |
| 97 | C22⋊C4×C53 | — | 2 | 4 | no | — |
| 98 | C22⋊C4×C53 | C22×C4 | 1 | 4 | yes | — |
| 99 | C42×C53 | — | 2 | 2 | no | — |
| 100 | #11719917 | — | 2 | 8 | no | — |
| 101 | #11719917 | — | 2 | 8 | no | — |
| 102 | #11719917 | — | 4 | 8 | no | — |
| 103 | C2×#88322 | — | 2 | 8 | no | — |
| 104 | C2×#88322 | — | 4 | 8 | no | — |
| 105 | C23×C53 | — | 2 | 4 | no | — |
| 106 | C23×C53 | — | 2 | 1 | no | — |
| 107 | C23×C53 | — | 2 | 2 | no | — |
| 108 | C23×C53 | — | 4 | 4 | no | — |
| 109 | C23×C53 | — | 8 | 1 | no | — |
| 110 | C23×C53 | — | 8 | 2 | no | — |
| 111 | C23×C53 | — | 8 | 4 | no | — |
| 112 | C23×C53 | C2×C22⋊C4 | 1 | 4 | yes | — |
| 113 | C23×C53 | C2×C22⋊C4 | 1 | 4 | yes | — |
| 114 | C23×C53 | C23×C4 | 1 | 1 | yes | characteristic |
| 115 | Q8×C53 | — | 2 | 4 | no | — |
| 116 | Q8×C53 | — | 4 | 4 | no | — |
| 117 | D4×C53 | — | 2 | 8 | no | — |
| 118 | D4×C53 | — | 2 | 8 | no | — |
| 119 | D4×C53 | — | 2 | 4 | no | — |
| 120 | D4×C53 | — | 4 | 8 | no | — |
| 121 | D4×C53 | — | 4 | 8 | no | — |
| 122 | D4×C53 | — | 8 | 4 | no | — |
| 123 | C2×C4×C53 | — | 2 | 8 | no | — |
| 124 | C2×C4×C53 | — | 2 | 1 | no | — |
| 125 | C2×C4×C53 | — | 2 | 2 | no | — |
| 126 | C2×C4×C53 | — | 2 | 4 | no | — |
| 127 | C2×C4×C53 | — | 2 | 2 | no | — |
| 128 | C2×C4×C53 | — | 2 | 4 | no | — |
| 129 | C2×C4×C53 | — | 4 | 4 | no | — |
| 130 | C2×C4×C53 | — | 4 | 8 | no | — |
| 131 | C2×C4×C53 | — | 4 | 4 | no | — |
| 132 | C2×C4×C53 | — | 8 | 4 | no | — |
| 133 | C2×C4×C53 | C2×C22⋊C4 | 1 | 4 | yes | — |
| 134 | C2×C4×C53 | C22×D4 | 1 | 2 | yes | — |
| 135 | C2×C4×C53 | H(5,𝔽2) | 1 | 4 | yes | — |
| 136 | #88322 | — | 4 | 8 | no | — |
| 137 | #21779 | — | 53 | 1 | no | maximal, Sylow |
| 138 | C22×C53 | — | 2 | 4 | no | — |
| 139 | C22×C53 | — | 2 | 4 | no | — |
| 140 | C22×C53 | — | 4 | 1 | no | — |
| 141 | C22×C53 | — | 4 | 4 | no | — |
| 142 | C22×C53 | — | 4 | 4 | no | — |
| 143 | C22×C53 | — | 4 | 2 | no | — |
| 144 | C22×C53 | — | 4 | 4 | no | — |
| 145 | C22×C53 | — | 4 | 1 | no | — |
| 146 | C22×C53 | — | 8 | 4 | no | — |
| 147 | C22×C53 | — | 16 | 4 | no | — |
| 148 | C22×C53 | C22×C22⋊C4 | 1 | 1 | yes | characteristic |
| 149 | C22×C53 | #518 | 1 | 2 | yes | — |
| 150 | C4×C53 | — | 2 | 4 | no | — |
| 151 | C4×C53 | — | 4 | 4 | no | — |
| 152 | C4×C53 | — | 4 | 1 | no | — |
| 153 | C4×C53 | — | 4 | 2 | no | — |
| 154 | C4×C53 | #534 | 1 | 4 | yes | — |
| 155 | #3485 | D53 | 1 | 1 | yes | characteristic |
| 156 | #2888 | — | 53 | 2 | no | — |
| 157 | #1915 | — | 53 | 4 | no | — |
| 158 | #1891 | — | 53 | 4 | no | — |
| 159 | #1829 | — | 53 | 2 | no | — |
| 160 | #1785 | — | 53 | 2 | no | — |
| 161 | C2×C53 | — | 2 | 1 | no | — |
| 162 | C2×C53 | — | 2 | 2 | no | — |
| 163 | C2×C53 | — | 4 | 4 | no | — |
| 164 | C2×C53 | — | 4 | 4 | no | — |
| 165 | C2×C53 | — | 8 | 1 | no | — |
| 166 | C2×C53 | #2308 | 1 | 1 | yes | characteristic |
| 167 | C2×H(5,𝔽2) | C2×D53 | 1 | 1 | yes | characteristic |
| 168 | C22×#41 | C2×D53 | 1 | 1 | yes | characteristic |
| 169 | #568 | — | 53 | 4 | no | — |
| 170 | #547 | — | 2 | 4 | no | — |
| 171 | #546 | — | 2 | 4 | no | — |
| 172 | #536 | C2×D53 | 1 | 2 | yes | — |
| 173 | #534 | Dic53 | 1 | 4 | yes | — |
| 174 | #525 | C2×D53 | 1 | 2 | yes | — |
| 175 | C2×#123 | Dic53 | 1 | 4 | yes | — |
| 176 | #518 | C2×D53 | 1 | 1 | yes | characteristic |
| 177 | #413 | — | 53 | 4 | no | — |
| 178 | #413 | — | 106 | 4 | no | — |
| 179 | C2×#101 | — | 53 | 4 | no | — |
| 180 | C2×#100 | — | 53 | 4 | no | — |
| 181 | C2×#100 | — | 106 | 4 | no | — |
| 182 | #407 | — | 53 | 4 | no | — |
| 183 | #344 | — | 53 | 8 | no | — |
| 184 | C53 | #21779 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 185 | H(5,𝔽2) | — | 4 | 4 | no | — |
| 186 | C2×#41 | — | 2 | 4 | no | — |
| 187 | C2×#41 | — | 4 | 4 | no | — |
| 188 | C2×#41 | C22⋊D53 | 1 | 4 | yes | — |
| 189 | C2×#41 | C22×D53 | 1 | 2 | yes | — |
| 190 | C22×Q8 | C22×D53 | 1 | 1 | yes | characteristic |
| 191 | C22×D4 | C2×Dic53 | 1 | 2 | yes | — |
| 192 | C22×D4 | C2×Dic53 | 1 | 4 | yes | — |
| 193 | C22×D4 | C22⋊D53 | 1 | 4 | yes | — |
| 194 | C22×D4 | C22×D53 | 1 | 1 | yes | characteristic |
| 195 | C23×C4 | C2×Dic53 | 1 | 2 | yes | — |
| 196 | C23×C4 | C22×D53 | 1 | 1 | yes | characteristic |
| 197 | #131 | — | 212 | 8 | no | — |
| 198 | C2×M16 | — | 53 | 8 | no | — |
| 199 | C2×M16 | — | 106 | 8 | no | — |
| 200 | C4⋊Q8 | — | 2 | 2 | no | — |
| 201 | #124 | — | 2 | 2 | no | — |
| 202 | #123 | — | 2 | 8 | no | — |
| 203 | #123 | — | 2 | 4 | no | — |
| 204 | #123 | C2×Dic53 | 1 | 4 | yes | — |
| 205 | #123 | C2×Dic53 | 1 | 8 | yes | — |
| 206 | C22⋊Q8 | — | 2 | 2 | no | — |
| 207 | C22⋊Q8 | — | 2 | 4 | no | — |
| 208 | #121 | — | 2 | 2 | no | — |
| 209 | #121 | — | 2 | 8 | no | — |
| 210 | #120 | — | 2 | 4 | no | — |
| 211 | #120 | C2×Dic53 | 1 | 4 | yes | — |
| 212 | C4×D4 | — | 2 | 4 | no | — |
| 213 | C4×D4 | — | 2 | 4 | no | — |
| 214 | #117 | C22×D53 | 1 | 2 | yes | — |
| 215 | C2×C4⋊C4 | — | 2 | 4 | no | — |
| 216 | C2×C22⋊C4 | — | 2 | 4 | no | — |
| 217 | C2×C22⋊C4 | C2×Dic53 | 1 | 4 | yes | — |
| 218 | #101 | — | 106 | 4 | no | — |
| 219 | #101 | — | 212 | 4 | no | — |
| 220 | #100 | — | 106 | 8 | no | — |
| 221 | #100 | — | 106 | 4 | no | — |
| 222 | C22⋊C8 | — | 212 | 4 | no | — |
| 223 | C24 | C22⋊Dic53 | 1 | 4 | yes | — |
| 224 | C24 | C22×Dic53 | 1 | 1 | yes | characteristic |
| 225 | C24 | C22×Dic53 | 1 | 2 | yes | — |
| 226 | #41 | — | 2 | 8 | no | — |
| 227 | #41 | — | 4 | 8 | no | — |
| 228 | #41 | — | 4 | 16 | no | — |
| 229 | C2×Q8 | — | 2 | 2 | no | — |
| 230 | C2×Q8 | — | 2 | 4 | no | — |
| 231 | C2×Q8 | C2×C22⋊D53 | 1 | 2 | yes | — |
| 232 | C2×D4 | — | 2 | 4 | no | — |
| 233 | C2×D4 | — | 2 | 4 | no | — |
| 234 | C2×D4 | — | 2 | 4 | no | — |
| 235 | C2×D4 | — | 4 | 8 | no | — |
| 236 | C2×D4 | — | 4 | 4 | no | — |
| 237 | C2×D4 | — | 4 | 8 | no | — |
| 238 | C2×D4 | C22⋊Dic53 | 1 | 8 | yes | — |
| 239 | C2×D4 | C22×Dic53 | 1 | 4 | yes | — |
| 240 | C2×D4 | C2×C22⋊D53 | 1 | 4 | yes | — |
| 241 | C2×D4 | C2×C22⋊D53 | 1 | 2 | yes | — |
| 242 | C22×C4 | — | 2 | 4 | no | — |
| 243 | C22×C4 | — | 2 | 4 | no | — |
| 244 | C22×C4 | — | 2 | 2 | no | — |
| 245 | C22×C4 | — | 2 | 4 | no | — |
| 246 | C22×C4 | — | 4 | 4 | no | — |
| 247 | C22×C4 | C22⋊Dic53 | 1 | 4 | yes | — |
| 248 | C22×C4 | C22×Dic53 | 1 | 1 | yes | characteristic |
| 249 | C22×C4 | C22×Dic53 | 1 | 2 | yes | — |
| 250 | C22×C4 | C2×C22⋊D53 | 1 | 4 | yes | — |
| 251 | C22×C4 | C23×D53 | 1 | 1 | yes | characteristic |
| 252 | M16 | — | 106 | 8 | no | — |
| 253 | M16 | — | 106 | 8 | no | — |
| 254 | M16 | — | 212 | 8 | no | — |
| 255 | C2×C8 | — | 106 | 8 | no | — |
| 256 | C2×C8 | — | 212 | 8 | no | — |
| 257 | C4⋊C4 | — | 2 | 4 | no | — |
| 258 | C4⋊C4 | — | 2 | 8 | no | — |
| 259 | C4⋊C4 | — | 2 | 2 | no | — |
| 260 | C22⋊C4 | — | 2 | 4 | no | — |
| 261 | C22⋊C4 | — | 2 | 8 | no | — |
| 262 | C22⋊C4 | — | 2 | 4 | no | — |
| 263 | C22⋊C4 | — | 2 | 4 | no | — |
| 264 | C22⋊C4 | C22×Dic53 | 1 | 4 | yes | — |
| 265 | C42 | — | 2 | 2 | no | — |
| 266 | C23 | — | 2 | 4 | no | — |
| 267 | C23 | — | 2 | 1 | no | — |
| 268 | C23 | — | 2 | 2 | no | — |
| 269 | C23 | — | 4 | 4 | no | — |
| 270 | C23 | — | 8 | 1 | no | — |
| 271 | C23 | — | 8 | 2 | no | — |
| 272 | C23 | — | 8 | 4 | no | — |
| 273 | C23 | C2×C22⋊Dic53 | 1 | 4 | yes | — |
| 274 | C23 | C2×C22⋊Dic53 | 1 | 4 | yes | — |
| 275 | C23 | C23×Dic53 | 1 | 1 | yes | characteristic |
| 276 | Q8 | — | 2 | 4 | no | — |
| 277 | Q8 | — | 4 | 4 | no | — |
| 278 | D4 | — | 2 | 8 | no | — |
| 279 | D4 | — | 2 | 8 | no | — |
| 280 | D4 | — | 2 | 4 | no | — |
| 281 | D4 | — | 4 | 8 | no | — |
| 282 | D4 | — | 4 | 8 | no | — |
| 283 | D4 | — | 8 | 4 | no | — |
| 284 | C2×C4 | — | 2 | 8 | no | — |
| 285 | C2×C4 | — | 2 | 4 | no | — |
| 286 | C2×C4 | — | 2 | 1 | no | — |
| 287 | C2×C4 | — | 2 | 2 | no | — |
| 288 | C2×C4 | — | 2 | 2 | no | — |
| 289 | C2×C4 | — | 2 | 4 | no | — |
| 290 | C2×C4 | — | 4 | 4 | no | — |
| 291 | C2×C4 | — | 4 | 8 | no | — |
| 292 | C2×C4 | — | 4 | 4 | no | — |
| 293 | C2×C4 | — | 8 | 4 | no | — |
| 294 | C2×C4 | C2×C22⋊Dic53 | 1 | 4 | yes | — |
| 295 | C2×C4 | C22×C22⋊D53 | 1 | 2 | yes | — |
| 296 | C2×C4 | #13133262 | 1 | 4 | yes | — |
| 297 | C8 | — | 212 | 8 | no | — |
| 298 | C22 | — | 2 | 4 | no | — |
| 299 | C22 | — | 2 | 4 | no | — |
| 300 | C22 | — | 4 | 1 | no | — |
| 301 | C22 | — | 4 | 4 | no | — |
| 302 | C22 | — | 4 | 4 | no | — |
| 303 | C22 | — | 4 | 2 | no | — |
| 304 | C22 | — | 4 | 4 | no | — |
| 305 | C22 | — | 4 | 1 | no | — |
| 306 | C22 | — | 8 | 4 | no | — |
| 307 | C22 | — | 16 | 4 | no | — |
| 308 | C22 | C22×C22⋊Dic53 | 1 | 1 | yes | characteristic |
| 309 | C22 | #16811098 | 1 | 2 | yes | — |
| 310 | C4 | — | 2 | 4 | no | — |
| 311 | C4 | — | 4 | 4 | no | — |
| 312 | C4 | — | 4 | 1 | no | — |
| 313 | C4 | — | 4 | 2 | no | — |
| 314 | C4 | #16811182 | 1 | 4 | yes | — |
| 315 | C2 | — | 2 | 1 | no | — |
| 316 | C2 | — | 2 | 2 | no | — |
| 317 | C2 | — | 4 | 4 | no | — |
| 318 | C2 | — | 4 | 4 | no | — |
| 319 | C2 | — | 8 | 1 | no | — |
| 320 | C2 | #20277293 | 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