← #27300546
Subgroups — #27300546
Order 13568.
266 conjugacy classes of subgroups.
The subgroup lattice has 232 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | C2×#16810452 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #20276497 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | #20276489 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | #20276487 | C2 | 1 | 1 | yes | maximal, characteristic |
| 5 | #20276312 | C2 | 1 | 1 | yes | maximal, characteristic |
| 6 | #20274839 | C2 | 1 | 1 | yes | maximal, characteristic |
| 7 | C53×#1833 | C2 | 1 | 1 | yes | maximal, characteristic |
| 8 | C2×#13133157 | C22 | 1 | 1 | yes | characteristic |
| 9 | C22×#11719908 | C22 | 1 | 1 | yes | characteristic |
| 10 | C2×#13133153 | C22 | 1 | 1 | yes | characteristic |
| 11 | #16810661 | C22 | 1 | 1 | yes | characteristic |
| 12 | #16810452 | — | 2 | 1 | no | — |
| 13 | #16810452 | — | 2 | 1 | no | — |
| 14 | #16810450 | — | 2 | 1 | no | — |
| 15 | #16810448 | — | 2 | 1 | no | — |
| 16 | #16810447 | — | 2 | 2 | no | — |
| 17 | #16810445 | — | 2 | 2 | no | — |
| 18 | #16810399 | C4 | 1 | 2 | yes | complemented |
| 19 | C4×#11719908 | C4 | 1 | 2 | yes | complemented |
| 20 | #16810394 | — | 2 | 1 | no | — |
| 21 | #16810379 | — | 2 | 1 | no | — |
| 22 | C2×#108×C53 | C22 | 1 | 1 | yes | characteristic |
| 23 | C53×#427 | C22 | 1 | 1 | yes | characteristic |
| 24 | C53×#405 | C22 | 1 | 1 | yes | characteristic |
| 25 | C53×#344 | — | 2 | 1 | no | — |
| 26 | C53×H(3,ℤ/4) | — | 2 | 1 | no | — |
| 27 | C2×#11719917 | — | 2 | 1 | no | — |
| 28 | #13133176 | — | 2 | 1 | no | — |
| 29 | #13133157 | D4 | 1 | 2 | yes | — |
| 30 | #13133157 | Q8 | 1 | 2 | yes | — |
| 31 | C2×#11719908 | C2×C4 | 1 | 1 | yes | characteristic |
| 32 | C2×#11719908 | C2×C4 | 1 | 1 | yes | characteristic |
| 33 | C2×#11719908 | C2×C4 | 1 | 2 | yes | — |
| 34 | C2×#11719908 | D4 | 1 | 1 | yes | characteristic |
| 35 | C2×#11719908 | D4 | 1 | 1 | yes | characteristic |
| 36 | C2×#11719908 | Q8 | 1 | 1 | yes | characteristic |
| 37 | C2×#11719908 | Q8 | 1 | 1 | yes | characteristic |
| 38 | #13133153 | — | 2 | 1 | no | — |
| 39 | #13133153 | — | 2 | 1 | no | — |
| 40 | #13133151 | — | 2 | 1 | no | — |
| 41 | #13133149 | — | 2 | 1 | no | — |
| 42 | C2×M16×C53 | — | 2 | 1 | no | — |
| 43 | C2×M16×C53 | D4 | 1 | 1 | yes | characteristic |
| 44 | C22×C8×C53 | C23 | 1 | 1 | yes | characteristic |
| 45 | #117×C53 | — | 2 | 1 | no | — |
| 46 | #117×C53 | D4 | 1 | 1 | yes | characteristic |
| 47 | #108×C53 | — | 2 | 1 | no | — |
| 48 | #108×C53 | — | 2 | 1 | no | — |
| 49 | #107×C53 | — | 2 | 1 | no | — |
| 50 | #106×C53 | — | 2 | 1 | no | — |
| 51 | #97×C53 | Q8 | 1 | 2 | yes | — |
| 52 | C4×C8×C53 | C2×C4 | 1 | 2 | yes | — |
| 53 | C22×C4×C53 | C2×D4 | 1 | 1 | yes | characteristic |
| 54 | M16×C53 | — | 2 | 1 | no | — |
| 55 | M16×C53 | — | 2 | 1 | no | — |
| 56 | M16×C53 | — | 4 | 1 | no | — |
| 57 | M16×C53 | — | 4 | 1 | no | — |
| 58 | C2×C8×C53 | — | 4 | 1 | no | — |
| 59 | C2×C8×C53 | C4⋊C4 | 1 | 2 | yes | — |
| 60 | C2×C8×C53 | C22×C4 | 1 | 1 | yes | characteristic |
| 61 | C2×C8×C53 | C2×D4 | 1 | 1 | yes | characteristic |
| 62 | C2×C8×C53 | C2×D4 | 1 | 1 | yes | characteristic |
| 63 | C2×C8×C53 | C2×Q8 | 1 | 1 | yes | characteristic |
| 64 | C2×C8×C53 | C2×Q8 | 1 | 1 | yes | characteristic |
| 65 | C2×C8×C53 | C2×Q8 | 1 | 1 | yes | characteristic |
| 66 | C4⋊C4×C53 | — | 2 | 1 | no | — |
| 67 | C4⋊C4×C53 | — | 4 | 1 | no | — |
| 68 | C22⋊C4×C53 | — | 2 | 1 | no | — |
| 69 | C22⋊C4×C53 | — | 4 | 1 | no | — |
| 70 | C42×C53 | — | 4 | 1 | no | — |
| 71 | C42×C53 | C4⋊C4 | 1 | 2 | yes | — |
| 72 | C22⋊Dic53 | — | 4 | 1 | no | — |
| 73 | C4⋊Dic53 | — | 4 | 1 | no | — |
| 74 | C4×Dic53 | — | 4 | 1 | no | — |
| 75 | #11719917 | — | 4 | 1 | no | — |
| 76 | #11719917 | — | 4 | 1 | no | — |
| 77 | C2×#88322 | — | 4 | 1 | no | — |
| 78 | #11719908 | C4⋊C4 | 1 | 1 | yes | characteristic |
| 79 | #11719908 | C4⋊C4 | 1 | 1 | yes | characteristic |
| 80 | #11719908 | C4⋊C4 | 1 | 1 | yes | characteristic |
| 81 | #11719908 | C4⋊C4 | 1 | 1 | yes | characteristic |
| 82 | #11719908 | C4⋊C4 | 1 | 4 | yes | — |
| 83 | C23×C53 | #135 | 1 | 1 | yes | characteristic |
| 84 | C2×C4×C53 | — | 2 | 2 | no | — |
| 85 | C2×C4×C53 | — | 8 | 1 | no | — |
| 86 | C2×C4×C53 | C2×C4⋊C4 | 1 | 1 | yes | characteristic |
| 87 | C2×C4×C53 | #127 | 1 | 1 | yes | characteristic |
| 88 | C2×C4×C53 | C4⋊Q8 | 1 | 1 | yes | characteristic |
| 89 | C2×C4×C53 | #135 | 1 | 1 | yes | characteristic |
| 90 | C2×C4×C53 | #135 | 1 | 1 | yes | characteristic |
| 91 | C2×C4×C53 | #135 | 1 | 1 | yes | characteristic |
| 92 | C2×Dic53 | — | 8 | 1 | no | — |
| 93 | C8×C53 | — | 2 | 2 | no | — |
| 94 | C8×C53 | — | 8 | 1 | no | — |
| 95 | C8×C53 | C2×C4⋊C4 | 1 | 1 | yes | characteristic |
| 96 | C8×C53 | C2×C4⋊C4 | 1 | 1 | yes | characteristic |
| 97 | C8×C53 | C4⋊Q8 | 1 | 1 | yes | characteristic |
| 98 | C8×C53 | C4⋊Q8 | 1 | 1 | yes | characteristic |
| 99 | #88322 | — | 8 | 1 | no | — |
| 100 | #12911 | — | 53 | 1 | no | maximal, Sylow |
| 101 | C22×C53 | — | 4 | 1 | no | — |
| 102 | C22×C53 | #427 | 1 | 1 | yes | characteristic |
| 103 | C22×C53 | #495 | 1 | 1 | yes | characteristic |
| 104 | C22×C53 | #499 | 1 | 1 | yes | characteristic |
| 105 | C4×C53 | — | 4 | 2 | no | — |
| 106 | C4×C53 | — | 16 | 1 | no | — |
| 107 | C4×C53 | #384 | 1 | 1 | yes | characteristic |
| 108 | C4×C53 | #427 | 1 | 1 | yes | characteristic |
| 109 | C4×C53 | #495 | 1 | 1 | yes | characteristic |
| 110 | C4×C53 | #499 | 1 | 1 | yes | characteristic |
| 111 | Dic53 | — | 16 | 1 | no | — |
| 112 | C2×#368 | — | 53 | 1 | no | — |
| 113 | #2158 | — | 53 | 1 | no | — |
| 114 | #2155 | — | 53 | 1 | no | — |
| 115 | #2155 | — | 53 | 1 | no | — |
| 116 | #2108 | — | 53 | 1 | no | — |
| 117 | #1833 | — | 53 | 1 | no | — |
| 118 | #1833 | D53 | 1 | 1 | yes | characteristic |
| 119 | C2×C53 | — | 2 | 1 | no | — |
| 120 | C2×C53 | — | 2 | 1 | no | — |
| 121 | C2×C53 | — | 2 | 1 | no | — |
| 122 | C2×C53 | #1846 | 1 | 1 | yes | characteristic |
| 123 | C2×M32 | — | 53 | 1 | no | — |
| 124 | C22×C16 | — | 53 | 1 | no | — |
| 125 | C2×#108 | — | 53 | 1 | no | — |
| 126 | C2×#108 | C2×D53 | 1 | 1 | yes | characteristic |
| 127 | #427 | — | 53 | 1 | no | — |
| 128 | #427 | C2×D53 | 1 | 1 | yes | characteristic |
| 129 | #405 | C2×D53 | 1 | 1 | yes | characteristic |
| 130 | #368 | — | 106 | 1 | no | — |
| 131 | #368 | — | 106 | 1 | no | — |
| 132 | #367 | — | 106 | 1 | no | — |
| 133 | #366 | — | 106 | 1 | no | — |
| 134 | #365 | — | 106 | 2 | no | — |
| 135 | #365 | — | 106 | 2 | no | — |
| 136 | #346 | — | 53 | 2 | no | — |
| 137 | C4×C16 | — | 53 | 2 | no | — |
| 138 | #344 | — | 2 | 1 | no | — |
| 139 | #344 | — | 106 | 1 | no | — |
| 140 | H(3,ℤ/4) | — | 2 | 1 | no | — |
| 141 | H(3,ℤ/4) | — | 106 | 1 | no | — |
| 142 | C53 | #12911 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 143 | C2×M16 | — | 2 | 1 | no | — |
| 144 | C2×M16 | — | 106 | 1 | no | — |
| 145 | C2×M16 | C22⋊D53 | 1 | 1 | yes | characteristic |
| 146 | C22×C8 | C22×D53 | 1 | 1 | yes | characteristic |
| 147 | #117 | — | 2 | 1 | no | — |
| 148 | #117 | — | 106 | 1 | no | — |
| 149 | #117 | C22⋊D53 | 1 | 1 | yes | characteristic |
| 150 | M32 | — | 53 | 2 | no | — |
| 151 | M32 | — | 53 | 2 | no | — |
| 152 | C2×C16 | — | 53 | 1 | no | — |
| 153 | C2×C16 | — | 53 | 2 | no | — |
| 154 | C2×C16 | — | 53 | 1 | no | — |
| 155 | C2×C16 | — | 53 | 1 | no | — |
| 156 | C2×C16 | — | 53 | 1 | no | — |
| 157 | C2×C16 | — | 53 | 1 | no | — |
| 158 | C2×C16 | — | 53 | 1 | no | — |
| 159 | #108 | — | 2 | 1 | no | — |
| 160 | #108 | — | 2 | 1 | no | — |
| 161 | #108 | — | 106 | 1 | no | — |
| 162 | #108 | — | 106 | 1 | no | — |
| 163 | #107 | — | 2 | 1 | no | — |
| 164 | #107 | — | 106 | 1 | no | — |
| 165 | #106 | — | 2 | 1 | no | — |
| 166 | #106 | — | 106 | 1 | no | — |
| 167 | #97 | Dic106 | 1 | 2 | yes | — |
| 168 | C4×C8 | C4×D53 | 1 | 2 | yes | — |
| 169 | C22×C4 | C2×C22⋊D53 | 1 | 1 | yes | characteristic |
| 170 | M16 | — | 2 | 1 | no | — |
| 171 | M16 | — | 2 | 1 | no | — |
| 172 | M16 | — | 4 | 1 | no | — |
| 173 | M16 | — | 4 | 1 | no | — |
| 174 | M16 | — | 212 | 1 | no | — |
| 175 | M16 | — | 212 | 1 | no | — |
| 176 | C2×C8 | — | 4 | 1 | no | — |
| 177 | C2×C8 | — | 212 | 1 | no | — |
| 178 | C2×C8 | #11719919 | 1 | 2 | yes | — |
| 179 | C2×C8 | C2×Dic106 | 1 | 1 | yes | characteristic |
| 180 | C2×C8 | C2×C4×D53 | 1 | 1 | yes | characteristic |
| 181 | C2×C8 | D4×D53 | 1 | 1 | yes | characteristic |
| 182 | C2×C8 | D4×D53 | 1 | 1 | yes | characteristic |
| 183 | C2×C8 | Q8×D53 | 1 | 1 | yes | characteristic |
| 184 | C2×C8 | Q8×D53 | 1 | 1 | yes | characteristic |
| 185 | C4⋊C4 | — | 2 | 1 | no | — |
| 186 | C4⋊C4 | — | 4 | 1 | no | — |
| 187 | C4⋊C4 | — | 212 | 1 | no | — |
| 188 | C22⋊C4 | — | 2 | 1 | no | — |
| 189 | C22⋊C4 | — | 4 | 1 | no | — |
| 190 | C22⋊C4 | — | 212 | 1 | no | — |
| 191 | C42 | — | 4 | 1 | no | — |
| 192 | C42 | — | 212 | 1 | no | — |
| 193 | C42 | #11719919 | 1 | 2 | yes | — |
| 194 | C16 | — | 53 | 1 | no | — |
| 195 | C16 | — | 53 | 1 | no | — |
| 196 | C16 | — | 53 | 1 | no | — |
| 197 | C16 | — | 53 | 1 | no | — |
| 198 | C16 | — | 53 | 4 | no | — |
| 199 | C23 | #13133237 | 1 | 1 | yes | characteristic |
| 200 | C2×C4 | — | 2 | 2 | no | — |
| 201 | C2×C4 | — | 8 | 1 | no | — |
| 202 | C2×C4 | — | 424 | 1 | no | — |
| 203 | C2×C4 | C2×#11719919 | 1 | 1 | yes | characteristic |
| 204 | C2×C4 | #13133213 | 1 | 1 | yes | characteristic |
| 205 | C2×C4 | #13133215 | 1 | 1 | yes | characteristic |
| 206 | C2×C4 | #13133237 | 1 | 1 | yes | characteristic |
| 207 | C2×C4 | #13133237 | 1 | 1 | yes | characteristic |
| 208 | C2×C4 | #13133237 | 1 | 1 | yes | characteristic |
| 209 | C8 | — | 2 | 2 | no | — |
| 210 | C8 | — | 8 | 1 | no | — |
| 211 | C8 | — | 424 | 1 | no | — |
| 212 | C8 | C4⋊C4×D53 | 1 | 1 | yes | characteristic |
| 213 | C8 | C4⋊C4×D53 | 1 | 1 | yes | characteristic |
| 214 | C8 | #13133216 | 1 | 1 | yes | characteristic |
| 215 | C8 | #13133216 | 1 | 1 | yes | characteristic |
| 216 | C22 | — | 4 | 1 | no | — |
| 217 | C22 | #16810659 | 1 | 1 | yes | characteristic |
| 218 | C22 | #16811005 | 1 | 1 | yes | characteristic |
| 219 | C22 | #16811022 | 1 | 1 | yes | characteristic |
| 220 | C4 | — | 4 | 2 | no | — |
| 221 | C4 | — | 16 | 1 | no | — |
| 222 | C4 | — | 848 | 1 | no | — |
| 223 | C4 | #16810492 | 1 | 1 | yes | characteristic |
| 224 | C4 | #16810659 | 1 | 1 | yes | characteristic |
| 225 | C4 | #16811005 | 1 | 1 | yes | characteristic |
| 226 | C4 | #16811022 | 1 | 1 | yes | characteristic |
| 227 | C2 | — | 2 | 1 | no | — |
| 228 | C2 | — | 2 | 1 | no | — |
| 229 | C2 | — | 2 | 1 | no | — |
| 230 | C2 | #20274877 | 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