Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
1566 conjugacy classes of subgroups; only the 89 normal subgroups are shown.
This group has 1566 conjugacy classes of subgroups — too many to draw. Only its 89 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #20278028 | C2 | 1 | 3 | yes | maximal, complemented |
| 2 | #20274629 | C2 | 1 | 3 | yes | maximal, complemented |
| 3 | C53×#1304 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | C23×C22⋊D53 | C22 | 1 | 3 | yes | — |
| 5 | C2×#13133187 | — | 2 | 6 | no | — |
| 6 | C2×#13133131 | — | 2 | 6 | no | — |
| 7 | #16810532 | — | 2 | 6 | no | — |
| 8 | #16810500 | — | 2 | 6 | no | — |
| 9 | #16810479 | C22 | 1 | 1 | yes | characteristic |
| 10 | C53×#379 | C22 | 1 | 3 | yes | — |
| 11 | C24×D53 | D4 | 1 | 3 | yes | complemented |
| 12 | C22×D212 | — | 2 | 6 | no | — |
| 13 | C22×C22⋊D53 | — | 2 | 6 | no | — |
| 14 | C22×C22⋊D53 | — | 2 | 3 | no | — |
| 15 | C22×C22⋊D53 | — | 2 | 6 | no | — |
| 16 | C22×C22⋊D53 | — | 2 | 3 | no | — |
| 17 | C22×C22⋊D53 | — | 4 | 12 | no | — |
| 18 | C23×Dic53 | D4 | 1 | 3 | yes | complemented |
| 19 | #13133187 | — | 4 | 24 | no | — |
| 20 | C2×C22⋊Dic53 | — | 2 | 3 | no | — |
| 21 | C2×#11719921 | — | 2 | 12 | no | — |
| 22 | #13133131 | — | 4 | 12 | no | — |
| 23 | #13133124 | — | 2 | 6 | no | — |
| 24 | C25×C53 | C23 | 1 | 1 | yes | characteristic |
| 25 | C2×C22⋊C4×C53 | — | 2 | 6 | no | — |
| 26 | C2×C22⋊C4×C53 | D4 | 1 | 6 | yes | complemented |
| 27 | C24×C53 | — | 4 | 6 | no | — |
| 28 | C24×C53 | C2×D4 | 1 | 3 | yes | — |
| 29 | C24×C53 | C2×D4 | 1 | 3 | yes | — |
| 30 | C24×C53 | #41 | 1 | 1 | yes | characteristic |
| 31 | C23×D53 | — | 2 | 6 | no | — |
| 32 | C23×D53 | — | 2 | 3 | no | — |
| 33 | C23×D53 | — | 4 | 6 | no | — |
| 34 | C23×D53 | — | 4 | 12 | no | — |
| 35 | C22×C4×C53 | — | 2 | 6 | no | — |
| 36 | C2×C22⋊D53 | — | 2 | 12 | no | — |
| 37 | C2×C22⋊D53 | — | 2 | 24 | no | — |
| 38 | C2×C22⋊D53 | — | 4 | 6 | no | — |
| 39 | C2×C22⋊D53 | — | 4 | 12 | no | — |
| 40 | C2×C22⋊D53 | — | 4 | 12 | no | — |
| 41 | C2×C22⋊D53 | — | 4 | 6 | no | — |
| 42 | C2×C22⋊D53 | — | 4 | 12 | no | — |
| 43 | C2×C22⋊D53 | — | 4 | 6 | no | — |
| 44 | C2×C22⋊D53 | — | 4 | 3 | no | — |
| 45 | C2×C22⋊D53 | — | 4 | 6 | no | — |
| 46 | C2×C22⋊D53 | — | 4 | 3 | no | — |
| 47 | C22×Dic53 | — | 2 | 6 | no | — |
| 48 | C22×Dic53 | — | 2 | 3 | no | — |
| 49 | C22×Dic53 | — | 4 | 6 | no | — |
| 50 | C22×Dic53 | — | 4 | 1 | no | — |
| 51 | C2×D212 | — | 4 | 24 | no | — |
| 52 | C2×D212 | — | 8 | 6 | no | — |
| 53 | C22⋊C4×C53 | — | 4 | 12 | no | — |
| 54 | C22⋊C4×C53 | — | 4 | 6 | no | — |
| 55 | C22⋊Dic53 | — | 4 | 6 | no | — |
| 56 | #11719921 | — | 4 | 24 | no | — |
| 57 | C23×C53 | — | 2 | 12 | no | — |
| 58 | C23×C53 | — | 2 | 6 | no | — |
| 59 | C23×C53 | — | 4 | 3 | no | — |
| 60 | C23×C53 | — | 4 | 6 | no | — |
| 61 | C23×C53 | — | 4 | 3 | no | — |
| 62 | C23×C53 | — | 8 | 8 | no | — |
| 63 | C23×C53 | #120 | 1 | 3 | yes | — |
| 64 | C23×C53 | #121 | 1 | 3 | yes | — |
| 65 | C23×C53 | #127 | 1 | 1 | yes | characteristic |
| 66 | C22×D53 | — | 2 | 12 | no | — |
| 67 | C22×D53 | — | 2 | 6 | no | — |
| 68 | C22×D53 | — | 4 | 12 | no | — |
| 69 | C22×D53 | — | 4 | 6 | no | — |
| 70 | C22×D53 | — | 4 | 3 | no | — |
| 71 | C22×D53 | — | 4 | 3 | no | — |
| 72 | C22×D53 | — | 4 | 12 | no | — |
| 73 | C22×D53 | — | 8 | 24 | no | — |
| 74 | C22×D53 | — | 8 | 6 | no | — |
| 75 | C2×C4×C53 | — | 4 | 12 | no | — |
| 76 | C2×C4×C53 | — | 4 | 6 | no | — |
| 77 | C22⋊D53 | — | 4 | 12 | no | — |
| 78 | C22⋊D53 | — | 4 | 24 | no | — |
| 79 | C22⋊D53 | — | 4 | 12 | no | — |
| 80 | C2×Dic53 | — | 2 | 6 | no | — |
| 81 | C2×Dic53 | — | 2 | 12 | no | — |
| 82 | C2×Dic53 | — | 4 | 6 | no | — |
| 83 | C2×Dic53 | — | 4 | 3 | no | — |
| 84 | C2×Dic53 | — | 4 | 3 | no | — |
| 85 | C2×Dic53 | — | 8 | 3 | no | — |
| 86 | D212 | — | 8 | 24 | no | — |
| 87 | #10336 | — | 53 | 1 | no | maximal, Sylow |
| 88 | C22×C53 | — | 2 | 6 | no | — |
| 89 | C22×C53 | — | 2 | 3 | no | — |
| 90 | C22×C53 | — | 2 | 3 | no | — |
| 91 | C22×C53 | — | 2 | 6 | no | — |
| 92 | C22×C53 | — | 4 | 12 | no | — |
| 93 | C22×C53 | — | 8 | 6 | no | — |
| 94 | C22×C53 | — | 8 | 2 | no | — |
| 95 | C22×C53 | #392 | 1 | 1 | yes | characteristic |
| 96 | C22×C53 | B(4,𝔽2) | 1 | 6 | yes | — |
| 97 | C2×D53 | — | 4 | 24 | no | — |
| 98 | C2×D53 | — | 4 | 6 | no | — |
| 99 | C2×D53 | — | 8 | 12 | no | — |
| 100 | C2×D53 | — | 8 | 6 | no | — |
| 101 | C2×D53 | — | 8 | 6 | no | — |
| 102 | C2×D53 | — | 8 | 6 | no | — |
| 103 | C4×C53 | — | 8 | 6 | no | — |
| 104 | Dic53 | — | 4 | 6 | no | — |
| 105 | Dic53 | — | 16 | 1 | no | — |
| 106 | #2403 | — | 53 | 3 | no | — |
| 107 | #1781 | — | 53 | 3 | no | — |
| 108 | #1304 | D53 | 1 | 1 | yes | characteristic, complemented |
| 109 | C2×C53 | — | 2 | 6 | no | — |
| 110 | C2×C53 | — | 4 | 1 | no | — |
| 111 | C2×C53 | — | 4 | 3 | no | — |
| 112 | C2×C53 | #2021 | 1 | 3 | yes | — |
| 113 | D53 | — | 8 | 6 | no | — |
| 114 | C23×D4 | — | 53 | 3 | no | — |
| 115 | C2×#120 | — | 106 | 6 | no | — |
| 116 | C2×#99 | — | 106 | 6 | no | — |
| 117 | #392 | — | 106 | 6 | no | — |
| 118 | #386 | — | 106 | 6 | no | — |
| 119 | #379 | — | 53 | 1 | no | — |
| 120 | #379 | C2×D53 | 1 | 3 | yes | — |
| 121 | C53 | #10336 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 122 | C25 | — | 53 | 3 | no | — |
| 123 | C25 | C22×D53 | 1 | 1 | yes | characteristic |
| 124 | C22×D4 | — | 106 | 6 | no | — |
| 125 | C22×D4 | — | 106 | 3 | no | — |
| 126 | C22×D4 | — | 106 | 6 | no | — |
| 127 | C22×D4 | — | 106 | 6 | no | — |
| 128 | C22×D4 | — | 106 | 3 | no | — |
| 129 | C22×D4 | — | 212 | 12 | no | — |
| 130 | C23×C4 | — | 53 | 3 | no | — |
| 131 | #120 | — | 212 | 24 | no | — |
| 132 | C2×C22⋊C4 | — | 2 | 6 | no | — |
| 133 | C2×C22⋊C4 | — | 106 | 12 | no | — |
| 134 | C2×C22⋊C4 | — | 106 | 3 | no | — |
| 135 | C2×C22⋊C4 | D212 | 1 | 6 | yes | complemented |
| 136 | #99 | — | 212 | 12 | no | — |
| 137 | #95 | — | 106 | 6 | no | — |
| 138 | C24 | — | 4 | 6 | no | — |
| 139 | C24 | — | 106 | 3 | no | — |
| 140 | C24 | — | 106 | 6 | no | — |
| 141 | C24 | — | 212 | 12 | no | — |
| 142 | C24 | — | 212 | 6 | no | — |
| 143 | C24 | C2×D212 | 1 | 3 | yes | — |
| 144 | C24 | D4×D53 | 1 | 3 | yes | — |
| 145 | C24 | #11719949 | 1 | 1 | yes | characteristic |
| 146 | C2×D4 | — | 106 | 12 | no | — |
| 147 | C2×D4 | — | 106 | 24 | no | — |
| 148 | C2×D4 | — | 424 | 6 | no | — |
| 149 | C2×D4 | — | 212 | 24 | no | — |
| 150 | C2×D4 | — | 212 | 12 | no | — |
| 151 | C2×D4 | — | 212 | 6 | no | — |
| 152 | C2×D4 | — | 212 | 12 | no | — |
| 153 | C2×D4 | — | 212 | 6 | no | — |
| 154 | C2×D4 | — | 212 | 12 | no | — |
| 155 | C2×D4 | — | 212 | 6 | no | — |
| 156 | C2×D4 | — | 212 | 3 | no | — |
| 157 | C2×D4 | — | 212 | 6 | no | — |
| 158 | C2×D4 | — | 212 | 3 | no | — |
| 159 | C22×C4 | — | 2 | 6 | no | — |
| 160 | C22×C4 | — | 106 | 6 | no | — |
| 161 | C22×C4 | — | 106 | 3 | no | — |
| 162 | C22×C4 | — | 212 | 6 | no | — |
| 163 | C22×C4 | — | 212 | 1 | no | — |
| 164 | C22⋊C4 | — | 4 | 12 | no | — |
| 165 | C22⋊C4 | — | 4 | 6 | no | — |
| 166 | C22⋊C4 | — | 212 | 24 | no | — |
| 167 | C22⋊C4 | — | 212 | 6 | no | — |
| 168 | C23 | — | 2 | 12 | no | — |
| 169 | C23 | — | 2 | 6 | no | — |
| 170 | C23 | — | 4 | 3 | no | — |
| 171 | C23 | — | 4 | 6 | no | — |
| 172 | C23 | — | 4 | 3 | no | — |
| 173 | C23 | — | 8 | 8 | no | — |
| 174 | C23 | — | 106 | 6 | no | — |
| 175 | C23 | — | 106 | 12 | no | — |
| 176 | C23 | — | 424 | 6 | no | — |
| 177 | C23 | — | 424 | 24 | no | — |
| 178 | C23 | — | 212 | 12 | no | — |
| 179 | C23 | — | 212 | 6 | no | — |
| 180 | C23 | — | 212 | 3 | no | — |
| 181 | C23 | — | 212 | 3 | no | — |
| 182 | C23 | — | 212 | 12 | no | — |
| 183 | C23 | #13133187 | 1 | 3 | yes | — |
| 184 | C23 | #13133192 | 1 | 3 | yes | — |
| 185 | C23 | #13133214 | 1 | 1 | yes | characteristic |
| 186 | D4 | — | 424 | 24 | no | — |
| 187 | D4 | — | 212 | 12 | no | — |
| 188 | D4 | — | 212 | 24 | no | — |
| 189 | D4 | — | 212 | 12 | no | — |
| 190 | C2×C4 | — | 4 | 12 | no | — |
| 191 | C2×C4 | — | 4 | 6 | no | — |
| 192 | C2×C4 | — | 106 | 12 | no | — |
| 193 | C2×C4 | — | 106 | 6 | no | — |
| 194 | C2×C4 | — | 424 | 3 | no | — |
| 195 | C2×C4 | — | 212 | 6 | no | — |
| 196 | C2×C4 | — | 212 | 3 | no | — |
| 197 | C2×C4 | — | 212 | 3 | no | — |
| 198 | C22 | — | 2 | 6 | no | — |
| 199 | C22 | — | 2 | 3 | no | — |
| 200 | C22 | — | 2 | 3 | no | — |
| 201 | C22 | — | 2 | 6 | no | — |
| 202 | C22 | — | 4 | 12 | no | — |
| 203 | C22 | — | 8 | 6 | no | — |
| 204 | C22 | — | 8 | 2 | no | — |
| 205 | C22 | — | 424 | 12 | no | — |
| 206 | C22 | — | 424 | 6 | no | — |
| 207 | C22 | — | 424 | 6 | no | — |
| 208 | C22 | — | 424 | 6 | no | — |
| 209 | C22 | — | 212 | 24 | no | — |
| 210 | C22 | — | 212 | 6 | no | — |
| 211 | C22 | #16810533 | 1 | 1 | yes | characteristic |
| 212 | C22 | #16810811 | 1 | 6 | yes | — |
| 213 | C4 | — | 8 | 6 | no | — |
| 214 | C4 | — | 848 | 1 | no | — |
| 215 | C4 | — | 212 | 6 | no | — |
| 216 | C2 | — | 2 | 6 | no | — |
| 217 | C2 | — | 4 | 1 | no | — |
| 218 | C2 | — | 4 | 3 | no | — |
| 219 | C2 | — | 424 | 6 | no | — |
| 220 | C2 | #20275842 | 1 | 3 | yes | — |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #20278028 | C2 | semidirect |
| 2. #20274629 | C2 | semidirect |
| 3. C53×#1304 | C2 | semidirect |
| 11. C24×D53 | D4 | semidirect |
| 18. C23×Dic53 | D4 | semidirect |
| 26. C2×C22⋊C4×C53 | D4 | semidirect |
| 108. #1304 | D53 | semidirect |
| 121. C53 | #10336 | semidirect |
| 135. C2×C22⋊C4 | D212 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #20278028 | 2 |
| #20274629 | 2 |
| C53×#1304 | 2 |
| #10336 | 53 |