← #27295892
Subgroups — #27295892
Order 13568.
206 conjugacy classes of subgroups.
The subgroup lattice has 196 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | #20276624 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | Q8×#11719908 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | #20272785 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | Q8⋊#11719908 | C2 | 1 | 1 | yes | maximal, characteristic |
| 5 | Q8⋊#11719908 | C2 | 1 | 1 | yes | maximal, characteristic |
| 6 | #20272632 | C2 | 1 | 1 | yes | maximal, characteristic |
| 7 | C8×Q16×C53 | C2 | 1 | 1 | yes | maximal, characteristic |
| 8 | C4⋊#11719908 | — | 2 | 1 | no | — |
| 9 | C4⋊#11719908 | — | 2 | 1 | no | — |
| 10 | C4⋊#11719908 | C22 | 1 | 1 | yes | characteristic |
| 11 | C4⋊#11719908 | C22 | 1 | 1 | yes | characteristic |
| 12 | #16810399 | — | 2 | 1 | no | — |
| 13 | C4×#11719908 | — | 2 | 1 | no | — |
| 14 | C4×#11719908 | C22 | 1 | 1 | yes | characteristic |
| 15 | C4×#11719908 | C22 | 1 | 1 | yes | characteristic |
| 16 | C8×Q8×C53 | C22 | 1 | 1 | yes | characteristic |
| 17 | C8×Q8×C53 | C22 | 1 | 1 | yes | characteristic |
| 18 | C4×Q16×C53 | C4 | 1 | 1 | yes | characteristic |
| 19 | C53×#335 | C4 | 1 | 1 | yes | characteristic |
| 20 | C53×Q8⋊C8 | C4 | 1 | 1 | yes | characteristic |
| 21 | C53×Q8⋊C8 | C4 | 1 | 1 | yes | characteristic |
| 22 | C82×C53 | C22 | 1 | 1 | yes | characteristic |
| 23 | C2×#11719908 | — | 2 | 1 | no | — |
| 24 | C2×#11719908 | — | 2 | 1 | no | — |
| 25 | C2×#11719908 | — | 2 | 1 | no | — |
| 26 | C2×#11719908 | D4 | 1 | 1 | yes | characteristic |
| 27 | C2×#11719908 | D4 | 1 | 1 | yes | characteristic |
| 28 | C2×Q16×C53 | C8 | 1 | 1 | yes | characteristic |
| 29 | C4×Q8×C53 | C2×C4 | 1 | 1 | yes | characteristic |
| 30 | C4×Q8×C53 | C2×C4 | 1 | 1 | yes | characteristic |
| 31 | #107×C53 | C8 | 1 | 1 | yes | characteristic |
| 32 | C4⋊C8×C53 | — | 2 | 1 | no | — |
| 33 | C4⋊C8×C53 | — | 2 | 1 | no | — |
| 34 | C4⋊C8×C53 | C2×C4 | 1 | 1 | yes | characteristic |
| 35 | C4⋊C8×C53 | C2×C4 | 1 | 1 | yes | characteristic |
| 36 | #103×C53 | C8 | 1 | 1 | yes | characteristic |
| 37 | #103×C53 | C8 | 1 | 1 | yes | characteristic |
| 38 | C4×C8×C53 | — | 2 | 1 | no | — |
| 39 | C4×C8×C53 | — | 2 | 1 | no | — |
| 40 | C4×C8×C53 | C2×C4 | 1 | 1 | yes | characteristic |
| 41 | C4×C8×C53 | C2×C4 | 1 | 1 | yes | characteristic |
| 42 | C4×C8×C53 | C23 | 1 | 1 | yes | characteristic |
| 43 | C2×Q8×C53 | C2×C8 | 1 | 1 | yes | characteristic |
| 44 | C2×Q8×C53 | C2×C8 | 1 | 1 | yes | characteristic |
| 45 | Q16×C53 | — | 2 | 2 | no | — |
| 46 | C2×C8×C53 | — | 2 | 1 | no | — |
| 47 | C2×C8×C53 | — | 2 | 1 | no | — |
| 48 | C2×C8×C53 | — | 2 | 1 | no | — |
| 49 | C2×C8×C53 | C2×C8 | 1 | 1 | yes | characteristic |
| 50 | C2×C8×C53 | C2×C8 | 1 | 1 | yes | characteristic |
| 51 | C2×C8×C53 | C2×D4 | 1 | 1 | yes | characteristic |
| 52 | C2×C8×C53 | #41 | 1 | 1 | yes | characteristic |
| 53 | C4⋊C4×C53 | — | 2 | 1 | no | — |
| 54 | C4⋊C4×C53 | — | 2 | 1 | no | — |
| 55 | C4⋊C4×C53 | C2×C8 | 1 | 1 | yes | characteristic |
| 56 | C4⋊C4×C53 | C2×C8 | 1 | 1 | yes | characteristic |
| 57 | C42×C53 | — | 2 | 1 | no | — |
| 58 | C42×C53 | — | 2 | 1 | no | — |
| 59 | C42×C53 | C22×C4 | 1 | 1 | yes | characteristic |
| 60 | #11719908 | — | 2 | 2 | no | — |
| 61 | #11719908 | — | 2 | 1 | no | — |
| 62 | #11719908 | — | 4 | 1 | no | — |
| 63 | #11719908 | — | 4 | 1 | no | — |
| 64 | #11719908 | Q16 | 1 | 2 | yes | complemented |
| 65 | Q8×C53 | — | 2 | 1 | no | — |
| 66 | Q8×C53 | — | 2 | 1 | no | — |
| 67 | Q8×C53 | — | 2 | 1 | no | — |
| 68 | Q8×C53 | M32 | 1 | 2 | yes | — |
| 69 | C2×C4×C53 | — | 2 | 1 | no | — |
| 70 | C2×C4×C53 | — | 2 | 1 | no | — |
| 71 | C2×C4×C53 | — | 2 | 1 | no | — |
| 72 | C2×C4×C53 | — | 2 | 1 | no | — |
| 73 | C2×C4×C53 | C4×D4 | 1 | 1 | yes | characteristic |
| 74 | C2×C4×C53 | C22×C8 | 1 | 1 | yes | characteristic |
| 75 | C2×C4×C53 | #131 | 1 | 1 | yes | characteristic |
| 76 | C8×C53 | — | 2 | 1 | no | — |
| 77 | C8×C53 | — | 2 | 1 | no | — |
| 78 | C8×C53 | — | 2 | 1 | no | — |
| 79 | C8×C53 | — | 4 | 1 | no | — |
| 80 | C8×C53 | — | 4 | 1 | no | — |
| 81 | C8×C53 | — | 4 | 1 | no | — |
| 82 | C8×C53 | C2×Q16 | 1 | 1 | yes | characteristic |
| 83 | C8×C53 | #135 | 1 | 1 | yes | characteristic |
| 84 | #12053 | — | 53 | 1 | no | maximal, Sylow |
| 85 | C22×C53 | C8×D4 | 1 | 1 | yes | characteristic |
| 86 | C4×C53 | — | 2 | 2 | no | — |
| 87 | C4×C53 | — | 2 | 1 | no | — |
| 88 | C4×C53 | — | 4 | 1 | no | — |
| 89 | C4×C53 | — | 4 | 1 | no | — |
| 90 | C4×C53 | — | 4 | 1 | no | — |
| 91 | C4×C53 | C4×Q16 | 1 | 1 | yes | characteristic |
| 92 | C4×C53 | #443 | 1 | 1 | yes | characteristic |
| 93 | C4×C53 | C2×M32 | 1 | 1 | yes | characteristic |
| 94 | C4×C53 | #504 | 1 | 1 | yes | characteristic |
| 95 | #2183 | — | 53 | 1 | no | — |
| 96 | Q8×C16 | — | 53 | 1 | no | — |
| 97 | C8×Q16 | D53 | 1 | 1 | yes | characteristic |
| 98 | #1370 | — | 53 | 1 | no | — |
| 99 | Q8⋊C16 | — | 53 | 1 | no | — |
| 100 | Q8⋊C16 | — | 53 | 1 | no | — |
| 101 | #1311 | — | 53 | 1 | no | — |
| 102 | C2×C53 | C8×Q16 | 1 | 1 | yes | characteristic |
| 103 | C2×C53 | #2168 | 1 | 1 | yes | characteristic |
| 104 | C2×C53 | #2171 | 1 | 1 | yes | characteristic |
| 105 | C8×Q8 | C2×D53 | 1 | 1 | yes | characteristic |
| 106 | C8×Q8 | C2×D53 | 1 | 1 | yes | characteristic |
| 107 | C4×Q16 | Dic53 | 1 | 1 | yes | characteristic |
| 108 | C4⋊C16 | — | 53 | 1 | no | — |
| 109 | C4⋊C16 | — | 53 | 1 | no | — |
| 110 | C4⋊C16 | — | 106 | 1 | no | — |
| 111 | C4⋊C16 | — | 106 | 1 | no | — |
| 112 | #346 | — | 106 | 1 | no | — |
| 113 | C4×C16 | — | 53 | 1 | no | — |
| 114 | C4×C16 | — | 53 | 1 | no | — |
| 115 | C4×C16 | — | 106 | 1 | no | — |
| 116 | #335 | Dic53 | 1 | 1 | yes | characteristic |
| 117 | Q8⋊C8 | Dic53 | 1 | 1 | yes | characteristic |
| 118 | Q8⋊C8 | Dic53 | 1 | 1 | yes | characteristic |
| 119 | C82 | C2×D53 | 1 | 1 | yes | characteristic |
| 120 | C53 | #12053 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 121 | C2×Q16 | #88322 | 1 | 1 | yes | characteristic |
| 122 | C4×Q8 | C2×Dic53 | 1 | 1 | yes | characteristic |
| 123 | C4×Q8 | C2×Dic53 | 1 | 1 | yes | characteristic |
| 124 | C2×C16 | — | 53 | 1 | no | — |
| 125 | C2×C16 | — | 53 | 1 | no | — |
| 126 | C2×C16 | — | 106 | 1 | no | — |
| 127 | C2×C16 | — | 106 | 1 | no | — |
| 128 | C2×C16 | — | 106 | 1 | no | — |
| 129 | #107 | #88322 | 1 | 1 | yes | characteristic |
| 130 | C4⋊C8 | — | 2 | 1 | no | — |
| 131 | C4⋊C8 | — | 2 | 1 | no | — |
| 132 | C4⋊C8 | C2×Dic53 | 1 | 1 | yes | characteristic |
| 133 | C4⋊C8 | C2×Dic53 | 1 | 1 | yes | characteristic |
| 134 | #103 | #88322 | 1 | 1 | yes | characteristic |
| 135 | #103 | #88322 | 1 | 1 | yes | characteristic |
| 136 | C4×C8 | — | 2 | 1 | no | — |
| 137 | C4×C8 | — | 2 | 1 | no | — |
| 138 | C4×C8 | C2×Dic53 | 1 | 1 | yes | characteristic |
| 139 | C4×C8 | C2×Dic53 | 1 | 1 | yes | characteristic |
| 140 | C4×C8 | C22×D53 | 1 | 1 | yes | characteristic |
| 141 | C2×Q8 | C2×#88322 | 1 | 1 | yes | characteristic |
| 142 | C2×Q8 | C2×#88322 | 1 | 1 | yes | characteristic |
| 143 | Q16 | — | 2 | 2 | no | — |
| 144 | C2×C8 | — | 2 | 1 | no | — |
| 145 | C2×C8 | — | 2 | 1 | no | — |
| 146 | C2×C8 | — | 2 | 1 | no | — |
| 147 | C2×C8 | C2×#88322 | 1 | 1 | yes | characteristic |
| 148 | C2×C8 | C2×#88322 | 1 | 1 | yes | characteristic |
| 149 | C2×C8 | D4×D53 | 1 | 1 | yes | characteristic |
| 150 | C2×C8 | #11719947 | 1 | 1 | yes | characteristic |
| 151 | C4⋊C4 | — | 2 | 1 | no | — |
| 152 | C4⋊C4 | — | 2 | 1 | no | — |
| 153 | C4⋊C4 | C2×#88322 | 1 | 1 | yes | characteristic |
| 154 | C4⋊C4 | C2×#88322 | 1 | 1 | yes | characteristic |
| 155 | C42 | — | 2 | 1 | no | — |
| 156 | C42 | — | 2 | 1 | no | — |
| 157 | C42 | C22×Dic53 | 1 | 1 | yes | characteristic |
| 158 | C16 | — | 53 | 2 | no | — |
| 159 | C16 | — | 106 | 2 | no | — |
| 160 | C16 | — | 106 | 1 | no | — |
| 161 | C16 | — | 212 | 1 | no | — |
| 162 | C16 | — | 212 | 1 | no | — |
| 163 | Q8 | — | 2 | 1 | no | — |
| 164 | Q8 | — | 2 | 1 | no | — |
| 165 | Q8 | — | 2 | 1 | no | — |
| 166 | Q8 | #13133157 | 1 | 2 | yes | — |
| 167 | C2×C4 | — | 2 | 1 | no | — |
| 168 | C2×C4 | — | 2 | 1 | no | — |
| 169 | C2×C4 | — | 2 | 1 | no | — |
| 170 | C2×C4 | — | 2 | 1 | no | — |
| 171 | C2×C4 | D4×Dic53 | 1 | 1 | yes | characteristic |
| 172 | C2×C4 | C22×#88322 | 1 | 1 | yes | characteristic |
| 173 | C2×C4 | #13133224 | 1 | 1 | yes | characteristic |
| 174 | C8 | — | 2 | 1 | no | — |
| 175 | C8 | — | 2 | 1 | no | — |
| 176 | C8 | — | 2 | 1 | no | — |
| 177 | C8 | — | 4 | 1 | no | — |
| 178 | C8 | — | 4 | 1 | no | — |
| 179 | C8 | — | 4 | 1 | no | — |
| 180 | C8 | Q16×D53 | 1 | 1 | yes | characteristic |
| 181 | C8 | #13133240 | 1 | 1 | yes | characteristic |
| 182 | C22 | D4×#88322 | 1 | 1 | yes | characteristic |
| 183 | C4 | — | 2 | 2 | no | — |
| 184 | C4 | — | 2 | 1 | no | — |
| 185 | C4 | — | 4 | 1 | no | — |
| 186 | C4 | — | 4 | 1 | no | — |
| 187 | C4 | — | 4 | 1 | no | — |
| 188 | C4 | Q16×Dic53 | 1 | 1 | yes | characteristic |
| 189 | C4 | #16810740 | 1 | 1 | yes | characteristic |
| 190 | C4 | C2×#13133157 | 1 | 1 | yes | characteristic |
| 191 | C4 | #16811040 | 1 | 1 | yes | characteristic |
| 192 | C2 | Q16×#88322 | 1 | 1 | yes | characteristic |
| 193 | C2 | #20276541 | 1 | 1 | yes | characteristic |
| 194 | C2 | #20276557 | 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