← #27537762
Subgroups — #27537762
Order 13568.
570 conjugacy classes of subgroups.
The subgroup lattice has 206 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | C8×#11719949 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | Q8×#11719908 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | C16×D212 | C2 | 1 | 3 | yes | maximal, complemented |
| 4 | #20276534 | C2 | 1 | 3 | yes | maximal, complemented |
| 5 | #20276512 | C2 | 1 | 3 | yes | maximal, complemented |
| 6 | C4⋊C16×D53 | C2 | 1 | 3 | yes | maximal, complemented |
| 7 | Q8×C16×C53 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 8 | C4×#11719949 | C4 | 1 | 1 | yes | characteristic |
| 9 | C2×C16×D53 | C22 | 1 | 6 | yes | — |
| 10 | Q8×#88322 | C4 | 1 | 1 | yes | characteristic |
| 11 | C8×D212 | C22 | 1 | 3 | yes | — |
| 12 | #16810688 | C4 | 1 | 3 | yes | — |
| 13 | #16810675 | C4 | 1 | 3 | yes | — |
| 14 | C4×C8×D53 | C22 | 1 | 3 | yes | — |
| 15 | C4⋊#11719908 | C22 | 1 | 3 | yes | — |
| 16 | #16810441 | C22 | 1 | 1 | yes | characteristic |
| 17 | #16810441 | C22 | 1 | 3 | yes | — |
| 18 | #16810402 | C22 | 1 | 3 | yes | — |
| 19 | #16810402 | C22 | 1 | 3 | yes | — |
| 20 | C4×#11719908 | C22 | 1 | 3 | yes | — |
| 21 | C8×Q8×C53 | C22 | 1 | 1 | yes | characteristic |
| 22 | C53×C4⋊C16 | C22 | 1 | 3 | yes | — |
| 23 | C4×C16×C53 | C22 | 1 | 3 | yes | — |
| 24 | C2×#11719949 | C8 | 1 | 1 | yes | characteristic |
| 25 | C2×C8×D53 | C2×C4 | 1 | 3 | yes | — |
| 26 | C2×C8×D53 | C23 | 1 | 3 | yes | — |
| 27 | Q8×Dic53 | C8 | 1 | 1 | yes | characteristic |
| 28 | C4×D212 | C2×C4 | 1 | 3 | yes | — |
| 29 | #13133179 | C8 | 1 | 3 | yes | — |
| 30 | #13133175 | C8 | 1 | 3 | yes | — |
| 31 | C42×D53 | C2×C4 | 1 | 3 | yes | — |
| 32 | C2×#11719908 | C23 | 1 | 3 | yes | — |
| 33 | C2×#11719908 | C23 | 1 | 1 | yes | characteristic |
| 34 | C16×D53 | — | 4 | 6 | no | — |
| 35 | C4⋊#88322 | C2×C4 | 1 | 3 | yes | — |
| 36 | #13133129 | C2×C4 | 1 | 6 | yes | — |
| 37 | C4×#88322 | C2×C4 | 1 | 3 | yes | — |
| 38 | C8×Dic53 | C2×C4 | 1 | 3 | yes | — |
| 39 | C8×Dic53 | C23 | 1 | 1 | yes | characteristic |
| 40 | C4×Q8×C53 | C2×C4 | 1 | 1 | yes | characteristic |
| 41 | C2×C16×C53 | C23 | 1 | 3 | yes | — |
| 42 | C2×C16×C53 | C23 | 1 | 1 | yes | characteristic |
| 43 | C4⋊C8×C53 | C2×C4 | 1 | 3 | yes | — |
| 44 | C4×C8×C53 | C23 | 1 | 3 | yes | — |
| 45 | C2×Q8×C53 | C2×C8 | 1 | 1 | yes | characteristic |
| 46 | #11719949 | C16 | 1 | 8 | yes | complemented |
| 47 | C2×D212 | C2×C8 | 1 | 3 | yes | — |
| 48 | C2×C4×D53 | C2×C8 | 1 | 3 | yes | — |
| 49 | C2×C4×D53 | C2×C8 | 1 | 3 | yes | — |
| 50 | C2×C4×D53 | C22×C4 | 1 | 3 | yes | — |
| 51 | C2×C8×C53 | C22×C4 | 1 | 3 | yes | — |
| 52 | C2×C8×C53 | C24 | 1 | 1 | yes | characteristic |
| 53 | C4⋊C4×C53 | C2×C8 | 1 | 3 | yes | — |
| 54 | C42×C53 | C22×C4 | 1 | 3 | yes | — |
| 55 | #11719921 | C2×C8 | 1 | 6 | yes | — |
| 56 | C4⋊Dic53 | C2×C8 | 1 | 3 | yes | — |
| 57 | C4×Dic53 | C2×C8 | 1 | 3 | yes | — |
| 58 | C4×Dic53 | C2×C8 | 1 | 3 | yes | — |
| 59 | C4×Dic53 | C22×C4 | 1 | 1 | yes | characteristic |
| 60 | C2×#88322 | C22×C4 | 1 | 3 | yes | — |
| 61 | C2×#88322 | C22×C4 | 1 | 1 | yes | characteristic |
| 62 | C8×D53 | — | 2 | 3 | no | — |
| 63 | C8×D53 | — | 2 | 3 | no | — |
| 64 | C8×D53 | — | 4 | 3 | no | — |
| 65 | C16×C53 | — | 2 | 3 | no | — |
| 66 | C16×C53 | — | 2 | 1 | no | — |
| 67 | #11719908 | — | 2 | 3 | no | — |
| 68 | #11719908 | — | 2 | 1 | no | — |
| 69 | C22×D53 | C22×C8 | 1 | 3 | yes | — |
| 70 | Q8×C53 | C2×C16 | 1 | 4 | yes | complemented |
| 71 | C2×C4×C53 | C22×C8 | 1 | 3 | yes | — |
| 72 | C2×C4×C53 | C22×C8 | 1 | 3 | yes | — |
| 73 | C2×C4×C53 | C23×C4 | 1 | 1 | yes | characteristic |
| 74 | C2×Dic53 | C22×C8 | 1 | 1 | yes | characteristic |
| 75 | C2×Dic53 | C22×C8 | 1 | 3 | yes | — |
| 76 | C2×Dic53 | C22×C8 | 1 | 1 | yes | characteristic |
| 77 | D212 | C2×C16 | 1 | 12 | yes | complemented |
| 78 | C4×D53 | — | 2 | 3 | no | — |
| 79 | C4×D53 | — | 2 | 3 | no | — |
| 80 | C4×D53 | — | 4 | 3 | no | — |
| 81 | C4×D53 | C2×C16 | 1 | 12 | yes | complemented |
| 82 | C8×C53 | — | 2 | 3 | no | — |
| 83 | C8×C53 | H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 84 | C8×C53 | 2−1+4 | 1 | 1 | yes | characteristic |
| 85 | #88322 | — | 2 | 3 | no | — |
| 86 | #88322 | — | 2 | 1 | no | — |
| 87 | #33502 | — | 53 | 1 | no | maximal, Sylow |
| 88 | C22×C53 | C23×C8 | 1 | 1 | yes | characteristic |
| 89 | C2×D53 | — | 2 | 3 | no | — |
| 90 | C2×D53 | — | 2 | 3 | no | — |
| 91 | C2×D53 | C22×C16 | 1 | 6 | yes | — |
| 92 | C4×C53 | — | 2 | 3 | no | — |
| 93 | C4×C53 | C22×C16 | 1 | 6 | yes | — |
| 94 | C4×C53 | #520 | 1 | 1 | yes | characteristic |
| 95 | C4×C53 | #568 | 1 | 1 | yes | characteristic |
| 96 | Dic53 | — | 2 | 1 | no | — |
| 97 | Dic53 | — | 2 | 3 | no | — |
| 98 | Dic53 | C22×C16 | 1 | 2 | yes | — |
| 99 | C8×#41 | — | 53 | 1 | no | — |
| 100 | Q8×C16 | — | 53 | 1 | no | — |
| 101 | Q8×C16 | D53 | 1 | 1 | yes | characteristic, complemented |
| 102 | D4×C16 | — | 53 | 3 | no | — |
| 103 | D4×C16 | — | 53 | 3 | no | — |
| 104 | #2162 | — | 53 | 3 | no | — |
| 105 | C2×C4⋊C16 | — | 53 | 3 | no | — |
| 106 | C2×C53 | #2988 | 1 | 1 | yes | characteristic |
| 107 | C2×C53 | C23×C16 | 1 | 1 | yes | characteristic |
| 108 | C2×C53 | #3407 | 1 | 1 | yes | characteristic |
| 109 | D53 | — | 2 | 6 | no | — |
| 110 | C4×#41 | — | 53 | 1 | no | — |
| 111 | C22×C16 | — | 53 | 6 | no | — |
| 112 | C8×Q8 | — | 53 | 1 | no | — |
| 113 | C8×Q8 | C2×D53 | 1 | 1 | yes | characteristic |
| 114 | C8×D4 | — | 53 | 3 | no | — |
| 115 | C8×D4 | — | 53 | 3 | no | — |
| 116 | #431 | — | 53 | 3 | no | — |
| 117 | C2×C4×C8 | — | 53 | 3 | no | — |
| 118 | C4⋊C16 | — | 53 | 3 | no | — |
| 119 | C4⋊C16 | — | 53 | 1 | no | — |
| 120 | C4⋊C16 | — | 53 | 3 | no | — |
| 121 | C4⋊C16 | C2×D53 | 1 | 3 | yes | — |
| 122 | C22⋊C16 | — | 53 | 3 | no | — |
| 123 | C22⋊C16 | — | 53 | 3 | no | — |
| 124 | C4×C16 | — | 53 | 3 | no | — |
| 125 | C4×C16 | C2×D53 | 1 | 3 | yes | — |
| 126 | C53 | #33502 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 127 | C2×#41 | — | 53 | 1 | no | — |
| 128 | C22×C8 | — | 53 | 3 | no | — |
| 129 | C22×C8 | — | 53 | 3 | no | — |
| 130 | C4×Q8 | — | 53 | 1 | no | — |
| 131 | C4×Q8 | C4×D53 | 1 | 1 | yes | characteristic |
| 132 | C4×D4 | — | 53 | 3 | no | — |
| 133 | C4×D4 | — | 53 | 3 | no | — |
| 134 | #117 | — | 53 | 3 | no | — |
| 135 | C2×C42 | — | 53 | 3 | no | — |
| 136 | C2×C16 | — | 53 | 1 | no | — |
| 137 | C2×C16 | — | 53 | 3 | no | — |
| 138 | C2×C16 | — | 212 | 6 | no | — |
| 139 | C2×C16 | C22×D53 | 1 | 1 | yes | characteristic |
| 140 | C2×C16 | C22×D53 | 1 | 3 | yes | — |
| 141 | C4⋊C8 | — | 53 | 3 | no | — |
| 142 | C4⋊C8 | C4×D53 | 1 | 3 | yes | — |
| 143 | C22⋊C8 | — | 53 | 6 | no | — |
| 144 | C4×C8 | — | 53 | 3 | no | — |
| 145 | C4×C8 | — | 53 | 1 | no | — |
| 146 | C4×C8 | — | 53 | 3 | no | — |
| 147 | C4×C8 | C22×D53 | 1 | 3 | yes | — |
| 148 | #41 | — | 53 | 8 | no | — |
| 149 | C2×Q8 | C8×D53 | 1 | 1 | yes | characteristic |
| 150 | C2×D4 | — | 53 | 3 | no | — |
| 151 | C22×C4 | — | 53 | 3 | no | — |
| 152 | C22×C4 | — | 53 | 3 | no | — |
| 153 | C22×C4 | — | 53 | 3 | no | — |
| 154 | C2×C8 | — | 53 | 1 | no | — |
| 155 | C2×C8 | — | 53 | 3 | no | — |
| 156 | C2×C8 | — | 106 | 3 | no | — |
| 157 | C2×C8 | — | 106 | 3 | no | — |
| 158 | C2×C8 | — | 212 | 3 | no | — |
| 159 | C2×C8 | C2×C4×D53 | 1 | 3 | yes | — |
| 160 | C2×C8 | C23×D53 | 1 | 1 | yes | characteristic |
| 161 | C4⋊C4 | — | 53 | 3 | no | — |
| 162 | C4⋊C4 | C8×D53 | 1 | 3 | yes | — |
| 163 | C22⋊C4 | — | 53 | 6 | no | — |
| 164 | C42 | — | 53 | 3 | no | — |
| 165 | C42 | — | 53 | 3 | no | — |
| 166 | C42 | — | 53 | 1 | no | — |
| 167 | C42 | C2×C4×D53 | 1 | 3 | yes | — |
| 168 | C16 | — | 2 | 3 | no | — |
| 169 | C16 | — | 2 | 1 | no | — |
| 170 | C16 | — | 106 | 3 | no | — |
| 171 | C16 | — | 106 | 1 | no | — |
| 172 | C23 | — | 53 | 3 | no | — |
| 173 | Q8 | C16×D53 | 1 | 4 | yes | complemented |
| 174 | D4 | — | 53 | 12 | no | — |
| 175 | C2×C4 | — | 53 | 1 | no | — |
| 176 | C2×C4 | — | 53 | 12 | no | — |
| 177 | C2×C4 | — | 53 | 1 | no | — |
| 178 | C2×C4 | — | 53 | 3 | no | — |
| 179 | C2×C4 | — | 106 | 3 | no | — |
| 180 | C2×C4 | — | 106 | 3 | no | — |
| 181 | C2×C4 | — | 212 | 3 | no | — |
| 182 | C2×C4 | C2×C8×D53 | 1 | 3 | yes | — |
| 183 | C2×C4 | C2×C8×D53 | 1 | 3 | yes | — |
| 184 | C2×C4 | C22×C4×D53 | 1 | 1 | yes | characteristic |
| 185 | C8 | — | 2 | 3 | no | — |
| 186 | C8 | — | 106 | 3 | no | — |
| 187 | C8 | — | 106 | 1 | no | — |
| 188 | C8 | #13133263 | 1 | 1 | yes | characteristic |
| 189 | C8 | #13133265 | 1 | 1 | yes | characteristic |
| 190 | C22 | — | 53 | 6 | no | — |
| 191 | C22 | — | 106 | 3 | no | — |
| 192 | C22 | — | 106 | 3 | no | — |
| 193 | C22 | C22×C8×D53 | 1 | 1 | yes | characteristic |
| 194 | C4 | — | 2 | 3 | no | — |
| 195 | C4 | — | 53 | 2 | no | — |
| 196 | C4 | — | 106 | 1 | no | — |
| 197 | C4 | — | 106 | 3 | no | — |
| 198 | C4 | C2×C16×D53 | 1 | 6 | yes | — |
| 199 | C4 | #16811105 | 1 | 1 | yes | characteristic |
| 200 | C4 | #16811388 | 1 | 1 | yes | characteristic |
| 201 | C2 | — | 106 | 6 | no | — |
| 202 | C2 | #20283825 | 1 | 1 | yes | characteristic |
| 203 | C2 | C22×C16×D53 | 1 | 1 | yes | characteristic |
| 204 | C2 | #20287816 | 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