Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
786 conjugacy classes of subgroups; only the 76 normal subgroups are shown.
This group has 786 conjugacy classes of subgroups — too many to draw. Only its 76 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 | #20290541 | C2 | 1 | 2 | yes | maximal |
| 2 | #20289614 | C2 | 1 | 4 | yes | maximal, complemented |
| 3 | #20277296 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | C2×#13133275 | — | 2 | 4 | no | — |
| 5 | #16811564 | C22 | 1 | 4 | yes | — |
| 6 | C23×C22⋊D53 | C22 | 1 | 1 | yes | characteristic |
| 7 | C2×#13133173 | C4 | 1 | 2 | yes | complemented |
| 8 | #16810523 | — | 2 | 4 | no | — |
| 9 | #16810479 | C4 | 1 | 1 | yes | characteristic, complemented |
| 10 | #16810478 | C22 | 1 | 2 | yes | — |
| 11 | C53×#379 | C4 | 1 | 1 | yes | characteristic, complemented |
| 12 | C2×C22⋊#6157 | — | 2 | 4 | no | — |
| 13 | C2×C22⋊#6157 | — | 2 | 8 | no | — |
| 14 | #13133275 | — | 4 | 8 | no | — |
| 15 | C24×D53 | C23 | 1 | 1 | yes | characteristic |
| 16 | C22×C22⋊D53 | — | 2 | 4 | no | — |
| 17 | C22×C22⋊D53 | — | 4 | 4 | no | — |
| 18 | C22×C22⋊D53 | D4 | 1 | 4 | yes | — |
| 19 | C23×Dic53 | C2×C4 | 1 | 1 | yes | characteristic |
| 20 | #13133173 | — | 4 | 4 | no | — |
| 21 | C2×#11719919 | — | 2 | 2 | no | — |
| 22 | C2×C22⋊Dic53 | — | 2 | 2 | no | — |
| 23 | C2×C22⋊Dic53 | C2×C4 | 1 | 2 | yes | — |
| 24 | C2×#11719921 | — | 2 | 4 | no | — |
| 25 | C2×C4×Dic53 | — | 2 | 2 | no | — |
| 26 | C25×C53 | C2×C4 | 1 | 1 | yes | characteristic |
| 27 | C2×C22⋊C4×C53 | — | 2 | 2 | no | — |
| 28 | C2×C22⋊C4×C53 | C2×C4 | 1 | 2 | yes | — |
| 29 | C24×C53 | — | 2 | 2 | no | — |
| 30 | C24×C53 | — | 4 | 2 | no | — |
| 31 | C24×C53 | — | 8 | 2 | no | — |
| 32 | C24×C53 | C22⋊C4 | 1 | 2 | yes | — |
| 33 | C24×C53 | C22×C4 | 1 | 1 | yes | characteristic |
| 34 | C23×D53 | — | 4 | 2 | no | — |
| 35 | C23×D53 | — | 8 | 2 | no | — |
| 36 | C23×D53 | C2×D4 | 1 | 2 | yes | — |
| 37 | C23×D53 | #41 | 1 | 4 | yes | — |
| 38 | C22×#6157 | — | 4 | 4 | no | — |
| 39 | C22×C4×C53 | — | 2 | 2 | no | — |
| 40 | C2×C22⋊D53 | — | 2 | 8 | no | — |
| 41 | C2×C22⋊D53 | — | 2 | 8 | no | — |
| 42 | C2×C22⋊D53 | — | 4 | 4 | no | — |
| 43 | C2×C22⋊D53 | — | 4 | 16 | no | — |
| 44 | C22×Dic53 | — | 2 | 2 | no | — |
| 45 | C22×Dic53 | — | 2 | 4 | no | — |
| 46 | C22×Dic53 | — | 2 | 2 | no | — |
| 47 | C22×Dic53 | C22⋊C4 | 1 | 2 | yes | — |
| 48 | C22⋊#6157 | — | 4 | 8 | no | — |
| 49 | C22⋊#6157 | — | 8 | 8 | no | — |
| 50 | C22⋊C4×C53 | — | 4 | 2 | no | — |
| 51 | C22⋊C4×C53 | — | 8 | 2 | no | — |
| 52 | C22⋊Dic53 | — | 4 | 2 | no | — |
| 53 | C22⋊Dic53 | — | 8 | 2 | no | — |
| 54 | #11719921 | — | 8 | 4 | no | — |
| 55 | #11719919 | — | 4 | 4 | no | — |
| 56 | C4×Dic53 | — | 4 | 4 | no | — |
| 57 | C23×C53 | — | 2 | 2 | no | — |
| 58 | C23×C53 | — | 2 | 2 | no | — |
| 59 | C23×C53 | — | 2 | 4 | no | — |
| 60 | C23×C53 | — | 4 | 2 | no | — |
| 61 | C23×C53 | — | 4 | 4 | no | — |
| 62 | C23×C53 | — | 4 | 2 | no | — |
| 63 | C23×C53 | — | 4 | 2 | no | — |
| 64 | C23×C53 | — | 8 | 4 | no | — |
| 65 | C23×C53 | — | 8 | 8 | no | — |
| 66 | C23×C53 | C2×C22⋊C4 | 1 | 1 | yes | characteristic |
| 67 | C23×C53 | #117 | 1 | 2 | yes | — |
| 68 | C22×D53 | — | 2 | 4 | no | — |
| 69 | C22×D53 | — | 4 | 8 | no | — |
| 70 | C22×D53 | — | 4 | 4 | no | — |
| 71 | C22×D53 | — | 4 | 4 | no | — |
| 72 | C22×D53 | — | 8 | 8 | no | — |
| 73 | C22×D53 | #123 | 1 | 4 | yes | — |
| 74 | C2×#6157 | — | 8 | 4 | no | — |
| 75 | C2×#6157 | — | 8 | 8 | no | — |
| 76 | C2×C4×C53 | — | 4 | 4 | no | — |
| 77 | C2×C4×C53 | — | 4 | 2 | no | — |
| 78 | C22⋊D53 | — | 4 | 16 | no | — |
| 79 | C2×Dic53 | — | 2 | 8 | no | — |
| 80 | C2×Dic53 | — | 2 | 2 | no | — |
| 81 | C2×Dic53 | — | 2 | 2 | no | — |
| 82 | C2×Dic53 | — | 4 | 4 | no | — |
| 83 | C2×Dic53 | — | 4 | 2 | no | — |
| 84 | C2×Dic53 | #99 | 1 | 4 | yes | — |
| 85 | #8749 | — | 53 | 1 | no | maximal, Sylow |
| 86 | C22×C53 | — | 2 | 2 | no | — |
| 87 | C22×C53 | — | 2 | 2 | no | — |
| 88 | C22×C53 | — | 2 | 4 | no | — |
| 89 | C22×C53 | — | 4 | 4 | no | — |
| 90 | C22×C53 | — | 4 | 2 | no | — |
| 91 | C22×C53 | — | 4 | 2 | no | — |
| 92 | C22×C53 | — | 4 | 2 | no | — |
| 93 | C22×C53 | — | 8 | 4 | no | — |
| 94 | C22×C53 | — | 8 | 8 | no | — |
| 95 | C22×C53 | #381 | 1 | 1 | yes | characteristic |
| 96 | C22×C53 | C2×#99 | 1 | 2 | yes | — |
| 97 | C2×D53 | — | 2 | 4 | no | — |
| 98 | C2×D53 | — | 2 | 8 | no | — |
| 99 | C2×D53 | — | 8 | 4 | no | — |
| 100 | C2×D53 | — | 8 | 8 | no | — |
| 101 | #6157 | — | 16 | 4 | no | — |
| 102 | C4×C53 | — | 8 | 2 | no | — |
| 103 | Dic53 | — | 2 | 4 | no | — |
| 104 | Dic53 | — | 8 | 2 | no | — |
| 105 | #2309 | — | 53 | 1 | no | — |
| 106 | #1781 | — | 53 | 2 | no | — |
| 107 | #1304 | — | 53 | 4 | no | — |
| 108 | C2×C53 | — | 2 | 2 | no | — |
| 109 | C2×C53 | — | 4 | 2 | no | — |
| 110 | C2×C53 | — | 8 | 2 | no | — |
| 111 | C2×C53 | #1819 | 1 | 1 | yes | characteristic |
| 112 | C2×C53 | #1828 | 1 | 2 | yes | — |
| 113 | D53 | — | 4 | 4 | no | — |
| 114 | C23×D4 | — | 53 | 1 | no | — |
| 115 | C2×#117 | — | 53 | 2 | no | — |
| 116 | C2×#99 | — | 106 | 4 | no | — |
| 117 | #390 | — | 106 | 4 | no | — |
| 118 | #379 | — | 53 | 1 | no | — |
| 119 | #379 | — | 53 | 2 | no | — |
| 120 | #379 | — | 53 | 4 | no | — |
| 121 | #379 | #6157 | 1 | 1 | yes | characteristic, complemented |
| 122 | C53 | #8749 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 123 | C25 | — | 53 | 1 | no | — |
| 124 | C25 | C2×#6157 | 1 | 1 | yes | characteristic |
| 125 | C22×D4 | — | 53 | 4 | no | — |
| 126 | C22×D4 | — | 106 | 4 | no | — |
| 127 | C22×D4 | — | 212 | 4 | no | — |
| 128 | C23×C4 | — | 53 | 1 | no | — |
| 129 | #117 | — | 212 | 4 | no | — |
| 130 | C2×C4⋊C4 | — | 106 | 2 | no | — |
| 131 | C2×C22⋊C4 | — | 2 | 2 | no | — |
| 132 | C2×C22⋊C4 | — | 53 | 2 | no | — |
| 133 | C2×C22⋊C4 | — | 106 | 4 | no | — |
| 134 | C2×C22⋊C4 | — | 106 | 2 | no | — |
| 135 | C2×C22⋊C4 | — | 106 | 8 | no | — |
| 136 | C2×C22⋊C4 | — | 106 | 4 | no | — |
| 137 | C2×C22⋊C4 | C2×#6157 | 1 | 2 | yes | — |
| 138 | C2×C42 | — | 106 | 2 | no | — |
| 139 | #99 | — | 212 | 8 | no | — |
| 140 | C24 | — | 2 | 2 | no | — |
| 141 | C24 | — | 4 | 2 | no | — |
| 142 | C24 | — | 8 | 2 | no | — |
| 143 | C24 | — | 53 | 2 | no | — |
| 144 | C24 | — | 53 | 4 | no | — |
| 145 | C24 | — | 424 | 2 | no | — |
| 146 | C24 | — | 212 | 2 | no | — |
| 147 | C24 | C22⋊#6157 | 1 | 2 | yes | — |
| 148 | C24 | C22×#6157 | 1 | 1 | yes | characteristic |
| 149 | C2×D4 | — | 106 | 8 | no | — |
| 150 | C2×D4 | — | 106 | 8 | no | — |
| 151 | C2×D4 | — | 212 | 4 | no | — |
| 152 | C2×D4 | — | 212 | 16 | no | — |
| 153 | C22×C4 | — | 2 | 2 | no | — |
| 154 | C22×C4 | — | 53 | 2 | no | — |
| 155 | C22×C4 | — | 106 | 4 | no | — |
| 156 | C22×C4 | — | 106 | 2 | no | — |
| 157 | C22×C4 | — | 106 | 2 | no | — |
| 158 | C22×C4 | — | 212 | 4 | no | — |
| 159 | C4⋊C4 | — | 212 | 4 | no | — |
| 160 | C22⋊C4 | — | 4 | 2 | no | — |
| 161 | C22⋊C4 | — | 8 | 2 | no | — |
| 162 | C22⋊C4 | — | 424 | 4 | no | — |
| 163 | C22⋊C4 | — | 424 | 8 | no | — |
| 164 | C22⋊C4 | — | 424 | 2 | no | — |
| 165 | C22⋊C4 | — | 212 | 8 | no | — |
| 166 | C22⋊C4 | — | 212 | 2 | no | — |
| 167 | C42 | — | 212 | 4 | no | — |
| 168 | C23 | — | 2 | 2 | no | — |
| 169 | C23 | — | 2 | 2 | no | — |
| 170 | C23 | — | 2 | 4 | no | — |
| 171 | C23 | — | 4 | 2 | no | — |
| 172 | C23 | — | 4 | 4 | no | — |
| 173 | C23 | — | 4 | 2 | no | — |
| 174 | C23 | — | 4 | 2 | no | — |
| 175 | C23 | — | 8 | 4 | no | — |
| 176 | C23 | — | 8 | 8 | no | — |
| 177 | C23 | — | 53 | 4 | no | — |
| 178 | C23 | — | 106 | 4 | no | — |
| 179 | C23 | — | 424 | 8 | no | — |
| 180 | C23 | — | 212 | 4 | no | — |
| 181 | C23 | — | 212 | 8 | no | — |
| 182 | C23 | — | 212 | 4 | no | — |
| 183 | C23 | C2×C22⋊#6157 | 1 | 1 | yes | characteristic |
| 184 | C23 | #13133296 | 1 | 2 | yes | — |
| 185 | D4 | — | 212 | 16 | no | — |
| 186 | C2×C4 | — | 4 | 4 | no | — |
| 187 | C2×C4 | — | 4 | 2 | no | — |
| 188 | C2×C4 | — | 53 | 4 | no | — |
| 189 | C2×C4 | — | 106 | 8 | no | — |
| 190 | C2×C4 | — | 106 | 2 | no | — |
| 191 | C2×C4 | — | 106 | 2 | no | — |
| 192 | C2×C4 | — | 424 | 8 | no | — |
| 193 | C2×C4 | — | 424 | 4 | no | — |
| 194 | C2×C4 | — | 212 | 4 | no | — |
| 195 | C2×C4 | — | 212 | 2 | no | — |
| 196 | C22 | — | 2 | 2 | no | — |
| 197 | C22 | — | 2 | 2 | no | — |
| 198 | C22 | — | 2 | 4 | no | — |
| 199 | C22 | — | 4 | 4 | no | — |
| 200 | C22 | — | 4 | 2 | no | — |
| 201 | C22 | — | 4 | 2 | no | — |
| 202 | C22 | — | 4 | 2 | no | — |
| 203 | C22 | — | 8 | 4 | no | — |
| 204 | C22 | — | 8 | 8 | no | — |
| 205 | C22 | — | 106 | 4 | no | — |
| 206 | C22 | — | 106 | 8 | no | — |
| 207 | C22 | — | 424 | 4 | no | — |
| 208 | C22 | — | 424 | 8 | no | — |
| 209 | C22 | #16811566 | 1 | 1 | yes | characteristic |
| 210 | C22 | D53⋊#99 | 1 | 2 | yes | — |
| 211 | C4 | — | 8 | 2 | no | — |
| 212 | C4 | — | 848 | 4 | no | — |
| 213 | C4 | — | 106 | 4 | no | — |
| 214 | C4 | — | 424 | 2 | no | — |
| 215 | C2 | — | 2 | 2 | no | — |
| 216 | C2 | — | 4 | 2 | no | — |
| 217 | C2 | — | 8 | 2 | no | — |
| 218 | C2 | — | 212 | 4 | no | — |
| 219 | C2 | #20290611 | 1 | 1 | yes | characteristic |
| 220 | C2 | #20290626 | 1 | 2 | yes | — |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #20289614 | C2 | semidirect |
| 7. C2×#13133173 | C4 | semidirect |
| 9. #16810479 | C4 | semidirect |
| 11. C53×#379 | C4 | semidirect |
| 121. #379 | #6157 | semidirect |
| 122. C53 | #8749 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #20290541 | 2 |
| #20289614 | 2 |
| #20277296 | 2 |
| #8749 | 53 |