Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
4390 conjugacy classes of subgroups; only the 467 normal subgroups are shown.
This group has 4390 conjugacy classes of subgroups — too many to draw. Only its 467 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 | C23×#11719949 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C2×#16810482 | C2 | 1 | 6 | yes | maximal, complemented |
| 3 | #20272986 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #20272955 | C2 | 1 | 6 | yes | maximal, complemented |
| 5 | C53×#1446 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | C22×#11719949 | — | 2 | 24 | no | — |
| 7 | C22×#11719949 | C4 | 1 | 8 | yes | complemented |
| 8 | C23×D212 | C22 | 1 | 3 | yes | — |
| 9 | C23×C4×D53 | C22 | 1 | 3 | yes | — |
| 10 | C22×#11719919 | C22 | 1 | 4 | yes | — |
| 11 | C22×#11719921 | C22 | 1 | 12 | yes | — |
| 12 | #16810482 | — | 2 | 24 | no | — |
| 13 | C2×#13133124 | C22 | 1 | 6 | yes | — |
| 14 | C23×Q8×C53 | C22 | 1 | 1 | yes | characteristic |
| 15 | C2×#95×C53 | C22 | 1 | 6 | yes | — |
| 16 | C24×D53 | C23 | 1 | 3 | yes | — |
| 17 | C2×#11719949 | — | 2 | 128 | no | — |
| 18 | C2×#11719949 | — | 2 | 96 | no | — |
| 19 | C22×D212 | — | 2 | 24 | no | — |
| 20 | C22×D212 | — | 2 | 12 | no | — |
| 21 | C22×D212 | C2×C4 | 1 | 12 | yes | — |
| 22 | C23×Dic53 | C23 | 1 | 4 | yes | — |
| 23 | C23×Dic53 | C23 | 1 | 1 | yes | characteristic |
| 24 | C22×C4×D53 | — | 2 | 24 | no | — |
| 25 | C22×C4×D53 | C2×C4 | 1 | 12 | yes | — |
| 26 | C22×C4×D53 | D4 | 1 | 24 | yes | — |
| 27 | C2×#11719919 | — | 2 | 24 | no | — |
| 28 | C2×#11719921 | — | 2 | 24 | no | — |
| 29 | C2×#11719921 | — | 4 | 24 | no | — |
| 30 | #13133124 | — | 2 | 12 | no | — |
| 31 | C22×Q8×C53 | — | 2 | 12 | no | — |
| 32 | C22×Q8×C53 | C2×C4 | 1 | 4 | yes | — |
| 33 | C23×C4×C53 | C23 | 1 | 4 | yes | — |
| 34 | C23×C4×C53 | C23 | 1 | 3 | yes | — |
| 35 | #95×C53 | — | 2 | 12 | no | — |
| 36 | C24×C53 | C24 | 1 | 1 | yes | characteristic |
| 37 | C23×D53 | — | 2 | 12 | no | — |
| 38 | C23×D53 | — | 2 | 12 | no | — |
| 39 | C23×D53 | — | 2 | 12 | no | — |
| 40 | C23×D53 | C22×C4 | 1 | 6 | yes | — |
| 41 | C23×D53 | C2×D4 | 1 | 12 | yes | — |
| 42 | C2×Q8×C53 | — | 2 | 32 | no | — |
| 43 | C2×Q8×C53 | — | 2 | 24 | no | — |
| 44 | C22×C4×C53 | — | 2 | 12 | no | — |
| 45 | C22×C4×C53 | — | 2 | 12 | no | — |
| 46 | C22×C4×C53 | — | 2 | 12 | no | — |
| 47 | C22×C4×C53 | — | 2 | 4 | no | — |
| 48 | C22×C4×C53 | C22×C4 | 1 | 6 | yes | — |
| 49 | C22×C4×C53 | C2×D4 | 1 | 12 | yes | — |
| 50 | C22×Dic53 | — | 2 | 12 | no | — |
| 51 | C22×Dic53 | — | 2 | 12 | no | — |
| 52 | C22×Dic53 | — | 2 | 4 | no | — |
| 53 | C22×Dic53 | C22×C4 | 1 | 2 | yes | — |
| 54 | C22×Dic53 | C2×D4 | 1 | 12 | yes | — |
| 55 | #11719949 | — | 2 | 256 | no | — |
| 56 | C2×D212 | — | 2 | 96 | no | — |
| 57 | C2×D212 | — | 2 | 48 | no | — |
| 58 | C2×D212 | — | 2 | 24 | no | — |
| 59 | C2×C4×D53 | — | 2 | 48 | no | — |
| 60 | C2×C4×D53 | — | 2 | 96 | no | — |
| 61 | C2×C4×D53 | C22⋊C4 | 1 | 48 | yes | complemented |
| 62 | #11719921 | — | 4 | 48 | no | — |
| 63 | #11719919 | — | 4 | 16 | no | — |
| 64 | C23×C53 | C23×C4 | 1 | 1 | yes | characteristic |
| 65 | C23×C53 | C22×D4 | 1 | 6 | yes | — |
| 66 | C23×C53 | H(5,𝔽2) | 1 | 4 | yes | — |
| 67 | C23×C53 | 2−1+4 | 1 | 4 | yes | — |
| 68 | C22×D53 | — | 2 | 12 | no | — |
| 69 | C22×D53 | — | 2 | 24 | no | — |
| 70 | C22×D53 | — | 2 | 24 | no | — |
| 71 | C22×D53 | — | 2 | 12 | no | — |
| 72 | C22×D53 | — | 2 | 12 | no | — |
| 73 | C22×D53 | — | 2 | 24 | no | — |
| 74 | C22×D53 | — | 4 | 12 | no | — |
| 75 | C22×D53 | — | 4 | 12 | no | — |
| 76 | C22×D53 | — | 4 | 24 | no | — |
| 77 | C22×D53 | C2×C22⋊C4 | 1 | 12 | yes | — |
| 78 | Q8×C53 | — | 2 | 32 | no | — |
| 79 | C2×C4×C53 | — | 2 | 12 | no | — |
| 80 | C2×C4×C53 | — | 2 | 24 | no | — |
| 81 | C2×C4×C53 | — | 4 | 4 | no | — |
| 82 | C2×C4×C53 | — | 4 | 12 | no | — |
| 83 | C2×C4×C53 | — | 4 | 12 | no | — |
| 84 | C2×C4×C53 | C2×C22⋊C4 | 1 | 12 | yes | — |
| 85 | C2×Dic53 | — | 4 | 4 | no | — |
| 86 | C2×Dic53 | — | 4 | 12 | no | — |
| 87 | C2×Dic53 | — | 4 | 12 | no | — |
| 88 | C2×Dic53 | C2×C22⋊C4 | 1 | 12 | yes | — |
| 89 | C2×Dic53 | #120 | 1 | 16 | yes | — |
| 90 | D212 | — | 2 | 96 | no | — |
| 91 | C4×D53 | — | 2 | 96 | no | — |
| 92 | #14337 | — | 53 | 1 | no | maximal, Sylow |
| 93 | C22×C53 | C22×C22⋊C4 | 1 | 3 | yes | — |
| 94 | C22×C53 | #520 | 1 | 4 | yes | — |
| 95 | C22×C53 | C2×#120 | 1 | 4 | yes | — |
| 96 | C22×C53 | #535 | 1 | 6 | yes | — |
| 97 | C22×C53 | #536 | 1 | 6 | yes | — |
| 98 | C22×C53 | #537 | 1 | 12 | yes | — |
| 99 | C2×D53 | — | 2 | 24 | no | — |
| 100 | C2×D53 | — | 2 | 12 | no | — |
| 101 | C2×D53 | — | 4 | 24 | no | — |
| 102 | C2×D53 | — | 4 | 24 | no | — |
| 103 | C2×D53 | — | 4 | 24 | no | — |
| 104 | C4×C53 | — | 2 | 12 | no | — |
| 105 | C4×C53 | — | 8 | 4 | no | — |
| 106 | Dic53 | — | 8 | 4 | no | — |
| 107 | Dic53 | #379 | 1 | 8 | yes | — |
| 108 | C23×#41 | — | 53 | 1 | no | — |
| 109 | C2×#380 | — | 53 | 6 | no | — |
| 110 | #1446 | — | 53 | 1 | no | — |
| 111 | #1446 | D53 | 1 | 1 | yes | characteristic, complemented |
| 112 | #1438 | — | 53 | 6 | no | — |
| 113 | C2×C53 | C2×#379 | 1 | 1 | yes | characteristic |
| 114 | C2×C53 | #2311 | 1 | 6 | yes | — |
| 115 | C2×C53 | #2615 | 1 | 4 | yes | — |
| 116 | C2×C53 | #2616 | 1 | 4 | yes | — |
| 117 | D53 | — | 4 | 12 | no | — |
| 118 | C22×#41 | — | 53 | 8 | no | — |
| 119 | C22×#41 | — | 106 | 24 | no | — |
| 120 | C23×Q8 | C2×D53 | 1 | 1 | yes | characteristic |
| 121 | C23×D4 | — | 53 | 3 | no | — |
| 122 | C24×C4 | — | 53 | 3 | no | — |
| 123 | C22×C4⋊C4 | — | 53 | 4 | no | — |
| 124 | C22×C22⋊C4 | — | 53 | 12 | no | — |
| 125 | #380 | — | 106 | 24 | no | — |
| 126 | C2×#95 | — | 53 | 6 | no | — |
| 127 | C2×#95 | C2×D53 | 1 | 6 | yes | — |
| 128 | C53 | #14337 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 129 | C25 | — | 53 | 3 | no | — |
| 130 | C2×#41 | — | 106 | 96 | no | — |
| 131 | C2×#41 | — | 106 | 128 | no | — |
| 132 | C22×Q8 | — | 2 | 12 | no | — |
| 133 | C22×Q8 | C4×D53 | 1 | 4 | yes | — |
| 134 | C22×D4 | — | 53 | 12 | no | — |
| 135 | C22×D4 | — | 106 | 24 | no | — |
| 136 | C22×D4 | — | 106 | 12 | no | — |
| 137 | C23×C4 | — | 53 | 1 | no | — |
| 138 | C23×C4 | — | 53 | 12 | no | — |
| 139 | C23×C4 | — | 53 | 24 | no | — |
| 140 | C23×C4 | — | 53 | 4 | no | — |
| 141 | C23×C4 | — | 106 | 24 | no | — |
| 142 | C23×C4 | C22×D53 | 1 | 4 | yes | — |
| 143 | C23×C4 | C22×D53 | 1 | 3 | yes | — |
| 144 | C2×C4⋊C4 | — | 106 | 24 | no | — |
| 145 | C2×C22⋊C4 | — | 106 | 24 | no | — |
| 146 | C2×C22⋊C4 | — | 212 | 24 | no | — |
| 147 | #95 | — | 2 | 12 | no | — |
| 148 | #95 | — | 106 | 12 | no | — |
| 149 | C24 | — | 53 | 6 | no | — |
| 150 | C24 | — | 53 | 12 | no | — |
| 151 | C24 | — | 106 | 12 | no | — |
| 152 | C24 | — | 106 | 12 | no | — |
| 153 | C24 | — | 106 | 12 | no | — |
| 154 | C24 | C23×D53 | 1 | 1 | yes | characteristic |
| 155 | #41 | — | 106 | 256 | no | — |
| 156 | C2×Q8 | — | 2 | 32 | no | — |
| 157 | C2×Q8 | — | 2 | 24 | no | — |
| 158 | C2×D4 | — | 106 | 48 | no | — |
| 159 | C2×D4 | — | 106 | 96 | no | — |
| 160 | C2×D4 | — | 106 | 24 | no | — |
| 161 | C22×C4 | — | 2 | 12 | no | — |
| 162 | C22×C4 | — | 2 | 12 | no | — |
| 163 | C22×C4 | — | 2 | 12 | no | — |
| 164 | C22×C4 | — | 2 | 4 | no | — |
| 165 | C22×C4 | — | 53 | 2 | no | — |
| 166 | C22×C4 | — | 53 | 12 | no | — |
| 167 | C22×C4 | — | 53 | 48 | no | — |
| 168 | C22×C4 | — | 106 | 48 | no | — |
| 169 | C22×C4 | — | 106 | 96 | no | — |
| 170 | C22×C4 | — | 106 | 12 | no | — |
| 171 | C22×C4 | — | 106 | 12 | no | — |
| 172 | C22×C4 | — | 106 | 4 | no | — |
| 173 | C22×C4 | C2×C4×D53 | 1 | 6 | yes | — |
| 174 | C22×C4 | D4×D53 | 1 | 12 | yes | — |
| 175 | C4⋊C4 | — | 212 | 16 | no | — |
| 176 | C22⋊C4 | — | 212 | 48 | no | — |
| 177 | C23 | — | 53 | 12 | no | — |
| 178 | C23 | — | 106 | 12 | no | — |
| 179 | C23 | — | 106 | 24 | no | — |
| 180 | C23 | — | 106 | 24 | no | — |
| 181 | C23 | — | 106 | 12 | no | — |
| 182 | C23 | — | 106 | 12 | no | — |
| 183 | C23 | — | 106 | 24 | no | — |
| 184 | C23 | — | 212 | 12 | no | — |
| 185 | C23 | — | 212 | 12 | no | — |
| 186 | C23 | — | 212 | 24 | no | — |
| 187 | C23 | C22×C4×D53 | 1 | 1 | yes | characteristic |
| 188 | C23 | C2×D4×D53 | 1 | 6 | yes | — |
| 189 | C23 | #13133263 | 1 | 4 | yes | — |
| 190 | C23 | #13133265 | 1 | 4 | yes | — |
| 191 | Q8 | — | 2 | 32 | no | — |
| 192 | D4 | — | 106 | 96 | no | — |
| 193 | C2×C4 | — | 2 | 12 | no | — |
| 194 | C2×C4 | — | 2 | 24 | no | — |
| 195 | C2×C4 | — | 4 | 4 | no | — |
| 196 | C2×C4 | — | 4 | 12 | no | — |
| 197 | C2×C4 | — | 4 | 12 | no | — |
| 198 | C2×C4 | — | 53 | 12 | no | — |
| 199 | C2×C4 | — | 53 | 16 | no | — |
| 200 | C2×C4 | — | 106 | 96 | no | — |
| 201 | C2×C4 | — | 212 | 4 | no | — |
| 202 | C2×C4 | — | 212 | 12 | no | — |
| 203 | C2×C4 | — | 212 | 12 | no | — |
| 204 | C2×C4 | C22⋊C4×D53 | 1 | 12 | yes | — |
| 205 | C22 | — | 106 | 24 | no | — |
| 206 | C22 | — | 106 | 12 | no | — |
| 207 | C22 | — | 212 | 24 | no | — |
| 208 | C22 | — | 212 | 24 | no | — |
| 209 | C22 | — | 212 | 24 | no | — |
| 210 | C22 | C2×C22⋊C4×D53 | 1 | 3 | yes | — |
| 211 | C22 | #16811105 | 1 | 4 | yes | — |
| 212 | C22 | #120×D53 | 1 | 4 | yes | — |
| 213 | C22 | #16811191 | 1 | 6 | yes | — |
| 214 | C22 | #16811198 | 1 | 6 | yes | — |
| 215 | C22 | #16811203 | 1 | 12 | yes | — |
| 216 | C4 | — | 2 | 12 | no | — |
| 217 | C4 | — | 8 | 4 | no | — |
| 218 | C4 | — | 53 | 8 | no | — |
| 219 | C4 | — | 424 | 4 | no | — |
| 220 | C2 | — | 212 | 12 | no | — |
| 221 | C2 | #379×D53 | 1 | 1 | yes | characteristic |
| 222 | C2 | #20277306 | 1 | 6 | yes | — |
| 223 | C2 | #20280442 | 1 | 4 | yes | — |
| 224 | C2 | #20280447 | 1 | 4 | yes | — |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×#16810482 | C2 | semidirect |
| 3. #20272986 | C2 | semidirect |
| 4. #20272955 | C2 | semidirect |
| 5. C53×#1446 | C2 | semidirect |
| 7. C22×#11719949 | C4 | semidirect |
| 61. C2×C4×D53 | C22⋊C4 | semidirect |
| 111. #1446 | D53 | semidirect |
| 128. C53 | #14337 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×#11719949 | 2 |
| C2×#16810482 | 2 |
| #20272986 | 2 |
| #20272955 | 2 |
| C53×#1446 | 2 |
| #14337 | 53 |