Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
784 conjugacy classes of subgroups; only the 88 normal subgroups are shown.
This group has 784 conjugacy classes of subgroups — too many to draw. Only its 88 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 | D53⋊#356 | C2 | 1 | 3 | yes | maximal |
| 2 | #20290721 | C2 | 1 | 3 | yes | maximal |
| 3 | Q82×D53 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | #16811691 | C22 | 1 | 1 | yes | characteristic |
| 5 | #16811616 | — | 2 | 3 | no | — |
| 6 | D53⋊#101 | C22 | 1 | 3 | yes | — |
| 7 | #16811534 | — | 2 | 3 | no | — |
| 8 | #16811533 | — | 2 | 3 | no | — |
| 9 | #16811494 | — | 2 | 3 | no | — |
| 10 | Q8×Dic106 | — | 2 | 3 | no | — |
| 11 | #16811353 | — | 2 | 3 | no | — |
| 12 | #16811353 | C4 | 1 | 3 | yes | complemented |
| 13 | C4⋊Q8×D53 | — | 2 | 3 | no | — |
| 14 | C4⋊Q8×D53 | C22 | 1 | 3 | yes | — |
| 15 | C4×Q8×D53 | — | 2 | 3 | no | — |
| 16 | Q82×C53 | C4 | 1 | 1 | yes | characteristic, complemented |
| 17 | D53⋊M16 | — | 2 | 3 | no | — |
| 18 | Q8×#6157 | — | 4 | 2 | no | — |
| 19 | #13133296 | — | 2 | 3 | no | — |
| 20 | #13133280 | — | 4 | 6 | no | — |
| 21 | #13133278 | — | 2 | 6 | no | — |
| 22 | C2×Q8×D53 | — | 2 | 1 | no | — |
| 23 | C2×Q8×D53 | — | 2 | 1 | no | — |
| 24 | C2×Q8×D53 | D4 | 1 | 3 | yes | — |
| 25 | C2×Q8×D53 | C23 | 1 | 1 | yes | characteristic |
| 26 | #13133217 | — | 2 | 3 | no | — |
| 27 | #13133217 | D4 | 1 | 3 | yes | — |
| 28 | #13133216 | — | 2 | 6 | no | — |
| 29 | #13133216 | — | 2 | 6 | no | — |
| 30 | #13133216 | — | 2 | 6 | no | — |
| 31 | #13133215 | — | 2 | 3 | no | — |
| 32 | #13133215 | — | 2 | 3 | no | — |
| 33 | #13133215 | C2×C4 | 1 | 3 | yes | — |
| 34 | #13133215 | D4 | 1 | 3 | yes | — |
| 35 | Q8×Dic53 | — | 2 | 3 | no | — |
| 36 | C4×Dic106 | — | 2 | 6 | no | — |
| 37 | C4×Dic106 | — | 2 | 3 | no | — |
| 38 | #13133182 | — | 2 | 3 | no | — |
| 39 | #13133182 | — | 2 | 6 | no | — |
| 40 | C4⋊C4×D53 | — | 2 | 3 | no | — |
| 41 | C4⋊C4×D53 | — | 2 | 6 | no | — |
| 42 | C42×D53 | — | 2 | 3 | no | — |
| 43 | C42×D53 | D4 | 1 | 3 | yes | — |
| 44 | C4⋊Q8×C53 | — | 2 | 3 | no | — |
| 45 | C4⋊Q8×C53 | C2×C4 | 1 | 3 | yes | — |
| 46 | C4×Q8×C53 | — | 2 | 3 | no | — |
| 47 | C2×Q8×C53 | — | 2 | 1 | no | — |
| 48 | C2×Q8×C53 | — | 2 | 1 | no | — |
| 49 | C2×Q8×C53 | C22⋊C4 | 1 | 3 | yes | — |
| 50 | C2×Q8×C53 | C22×C4 | 1 | 1 | yes | characteristic |
| 51 | Q8×D53 | — | 2 | 8 | no | — |
| 52 | Q8×D53 | — | 4 | 1 | no | — |
| 53 | Q8×D53 | — | 4 | 1 | no | — |
| 54 | Q8×D53 | — | 4 | 3 | no | — |
| 55 | Q8×D53 | — | 4 | 3 | no | — |
| 56 | Q8×D53 | — | 8 | 1 | no | — |
| 57 | Q8×D53 | — | 8 | 1 | no | — |
| 58 | C2×C4×D53 | — | 2 | 3 | no | — |
| 59 | C2×C4×D53 | — | 2 | 3 | no | — |
| 60 | C2×C4×D53 | C2×D4 | 1 | 3 | yes | — |
| 61 | C2×Dic106 | — | 2 | 3 | no | — |
| 62 | C2×Dic106 | — | 2 | 3 | no | — |
| 63 | C2×Dic106 | — | 2 | 3 | no | — |
| 64 | C2×Dic106 | C22⋊C4 | 1 | 3 | yes | — |
| 65 | C2×Dic106 | C2×D4 | 1 | 3 | yes | — |
| 66 | C22⋊#6157 | — | 4 | 1 | no | — |
| 67 | #11719940 | — | 4 | 3 | no | — |
| 68 | C4⋊#6157 | — | 2 | 6 | no | — |
| 69 | C4×#6157 | — | 4 | 3 | no | — |
| 70 | #11719936 | — | 4 | 3 | no | — |
| 71 | D53⋊C8 | — | 4 | 3 | no | — |
| 72 | C4⋊C4×C53 | — | 2 | 3 | no | — |
| 73 | C4⋊C4×C53 | — | 2 | 6 | no | — |
| 74 | C42×C53 | — | 2 | 3 | no | — |
| 75 | C42×C53 | C22⋊C4 | 1 | 3 | yes | — |
| 76 | C4⋊Dic53 | — | 2 | 3 | no | — |
| 77 | C4⋊Dic53 | — | 2 | 6 | no | — |
| 78 | #11719919 | — | 2 | 6 | no | — |
| 79 | #11719919 | — | 2 | 6 | no | — |
| 80 | #11719919 | — | 2 | 6 | no | — |
| 81 | C4×Dic53 | — | 2 | 3 | no | — |
| 82 | C4×Dic53 | — | 2 | 6 | no | — |
| 83 | C4×Dic53 | — | 2 | 3 | no | — |
| 84 | C4×Dic53 | C22⋊C4 | 1 | 3 | yes | — |
| 85 | C22×D53 | #120 | 1 | 1 | yes | characteristic |
| 86 | C2×#6157 | — | 4 | 2 | no | — |
| 87 | Q8×C53 | — | 2 | 4 | no | — |
| 88 | Q8×C53 | — | 4 | 1 | no | — |
| 89 | Q8×C53 | — | 4 | 3 | no | — |
| 90 | Q8×C53 | — | 8 | 1 | no | — |
| 91 | C2×C4×C53 | — | 2 | 3 | no | — |
| 92 | C2×C4×C53 | — | 2 | 3 | no | — |
| 93 | C2×C4×C53 | C2×C22⋊C4 | 1 | 3 | yes | — |
| 94 | C2×Dic53 | — | 2 | 3 | no | — |
| 95 | C2×Dic53 | — | 2 | 3 | no | — |
| 96 | C2×Dic53 | #120 | 1 | 3 | yes | — |
| 97 | C4×D53 | — | 2 | 12 | no | — |
| 98 | C4×D53 | — | 2 | 3 | no | — |
| 99 | C4×D53 | — | 2 | 3 | no | — |
| 100 | C4×D53 | — | 4 | 3 | no | — |
| 101 | C4×D53 | — | 4 | 3 | no | — |
| 102 | Dic106 | — | 2 | 12 | no | — |
| 103 | Dic106 | — | 4 | 3 | no | — |
| 104 | Dic106 | — | 4 | 3 | no | — |
| 105 | Dic106 | — | 8 | 3 | no | — |
| 106 | Dic106 | — | 8 | 3 | no | — |
| 107 | #88324 | — | 8 | 3 | no | — |
| 108 | #12777 | — | 53 | 1 | no | maximal, Sylow |
| 109 | C22×C53 | #379 | 1 | 1 | yes | characteristic |
| 110 | C2×D53 | — | 2 | 2 | no | — |
| 111 | C2×D53 | B(4,𝔽2) | 1 | 1 | yes | characteristic |
| 112 | C2×D53 | #458 | 1 | 1 | yes | characteristic |
| 113 | #6157 | — | 8 | 2 | no | — |
| 114 | C4×C53 | — | 2 | 6 | no | — |
| 115 | C4×C53 | — | 2 | 3 | no | — |
| 116 | C4×C53 | — | 4 | 3 | no | — |
| 117 | Dic53 | — | 2 | 6 | no | — |
| 118 | Dic53 | — | 2 | 3 | no | — |
| 119 | Dic53 | — | 4 | 3 | no | — |
| 120 | C2×Q82 | — | 53 | 1 | no | — |
| 121 | C2×#356 | — | 53 | 3 | no | — |
| 122 | #1886 | — | 53 | 3 | no | — |
| 123 | C2×C53 | — | 2 | 1 | no | — |
| 124 | C2×C53 | #1790 | 1 | 1 | yes | characteristic |
| 125 | D53 | — | 2 | 1 | no | — |
| 126 | D53 | #2205 | 1 | 2 | yes | — |
| 127 | Q82 | — | 53 | 3 | no | — |
| 128 | Q82 | — | 106 | 3 | no | — |
| 129 | Q82 | — | 106 | 3 | no | — |
| 130 | Q82 | #6157 | 1 | 1 | yes | characteristic, complemented |
| 131 | C2×C4⋊Q8 | — | 53 | 3 | no | — |
| 132 | C2×C4⋊Q8 | — | 106 | 3 | no | — |
| 133 | #519 | — | 53 | 1 | no | — |
| 134 | C2×C4×Q8 | — | 106 | 3 | no | — |
| 135 | #419 | — | 106 | 3 | no | — |
| 136 | C2×#101 | — | 53 | 3 | no | — |
| 137 | #356 | — | 106 | 3 | no | — |
| 138 | #356 | — | 106 | 3 | no | — |
| 139 | #339 | — | 106 | 3 | no | — |
| 140 | C53 | #12777 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 141 | C22×Q8 | — | 53 | 1 | no | — |
| 142 | C22×Q8 | — | 53 | 3 | no | — |
| 143 | C22×Q8 | — | 106 | 1 | no | — |
| 144 | C22×Q8 | — | 106 | 1 | no | — |
| 145 | C2×M16 | — | 106 | 3 | no | — |
| 146 | C4⋊Q8 | — | 2 | 3 | no | — |
| 147 | C4⋊Q8 | — | 53 | 3 | no | — |
| 148 | C4⋊Q8 | — | 53 | 3 | no | — |
| 149 | C4⋊Q8 | — | 53 | 3 | no | — |
| 150 | C4⋊Q8 | — | 106 | 3 | no | — |
| 151 | C4⋊Q8 | — | 106 | 6 | no | — |
| 152 | C4⋊Q8 | — | 106 | 6 | no | — |
| 153 | C4⋊Q8 | — | 106 | 6 | no | — |
| 154 | C4⋊Q8 | — | 106 | 3 | no | — |
| 155 | C4⋊Q8 | — | 106 | 3 | no | — |
| 156 | C4⋊Q8 | C2×#6157 | 1 | 3 | yes | — |
| 157 | C4×Q8 | — | 2 | 3 | no | — |
| 158 | C4×Q8 | — | 106 | 6 | no | — |
| 159 | C4×Q8 | — | 106 | 6 | no | — |
| 160 | C4×Q8 | — | 106 | 3 | no | — |
| 161 | C4×Q8 | — | 106 | 3 | no | — |
| 162 | C4×Q8 | — | 106 | 3 | no | — |
| 163 | C4×Q8 | — | 212 | 2 | no | — |
| 164 | #117 | — | 106 | 3 | no | — |
| 165 | C2×C4⋊C4 | — | 106 | 6 | no | — |
| 166 | C2×C4⋊C4 | — | 106 | 3 | no | — |
| 167 | C2×C42 | — | 53 | 3 | no | — |
| 168 | C2×C42 | — | 106 | 3 | no | — |
| 169 | #103 | — | 212 | 6 | no | — |
| 170 | #101 | — | 106 | 6 | no | — |
| 171 | C2×Q8 | — | 2 | 1 | no | — |
| 172 | C2×Q8 | — | 2 | 1 | no | — |
| 173 | C2×Q8 | — | 53 | 3 | no | — |
| 174 | C2×Q8 | — | 53 | 3 | no | — |
| 175 | C2×Q8 | — | 106 | 3 | no | — |
| 176 | C2×Q8 | — | 106 | 3 | no | — |
| 177 | C2×Q8 | — | 106 | 3 | no | — |
| 178 | C2×Q8 | — | 106 | 8 | no | — |
| 179 | C2×Q8 | — | 424 | 1 | no | — |
| 180 | C2×Q8 | — | 424 | 1 | no | — |
| 181 | C2×Q8 | — | 212 | 1 | no | — |
| 182 | C2×Q8 | — | 212 | 1 | no | — |
| 183 | C2×Q8 | — | 212 | 3 | no | — |
| 184 | C2×Q8 | — | 212 | 3 | no | — |
| 185 | C2×Q8 | C22⋊#6157 | 1 | 3 | yes | — |
| 186 | C2×Q8 | C22×#6157 | 1 | 1 | yes | characteristic |
| 187 | C22×C4 | — | 53 | 3 | no | — |
| 188 | C22×C4 | — | 106 | 3 | no | — |
| 189 | C22×C4 | — | 106 | 3 | no | — |
| 190 | M16 | — | 212 | 3 | no | — |
| 191 | M16 | — | 212 | 3 | no | — |
| 192 | C2×C8 | — | 212 | 3 | no | — |
| 193 | C4⋊C4 | — | 2 | 3 | no | — |
| 194 | C4⋊C4 | — | 2 | 6 | no | — |
| 195 | C4⋊C4 | — | 106 | 6 | no | — |
| 196 | C4⋊C4 | — | 106 | 6 | no | — |
| 197 | C4⋊C4 | — | 106 | 6 | no | — |
| 198 | C4⋊C4 | — | 106 | 3 | no | — |
| 199 | C4⋊C4 | — | 106 | 6 | no | — |
| 200 | C4⋊C4 | — | 106 | 6 | no | — |
| 201 | C22⋊C4 | — | 212 | 1 | no | — |
| 202 | C42 | — | 2 | 3 | no | — |
| 203 | C42 | — | 53 | 3 | no | — |
| 204 | C42 | — | 106 | 3 | no | — |
| 205 | C42 | — | 106 | 6 | no | — |
| 206 | C42 | — | 106 | 3 | no | — |
| 207 | C42 | — | 212 | 3 | no | — |
| 208 | C42 | C22⋊#6157 | 1 | 3 | yes | — |
| 209 | C23 | — | 53 | 1 | no | — |
| 210 | Q8 | — | 2 | 4 | no | — |
| 211 | Q8 | — | 4 | 1 | no | — |
| 212 | Q8 | — | 4 | 3 | no | — |
| 213 | Q8 | — | 8 | 1 | no | — |
| 214 | Q8 | — | 106 | 12 | no | — |
| 215 | Q8 | — | 424 | 3 | no | — |
| 216 | Q8 | — | 424 | 3 | no | — |
| 217 | Q8 | — | 212 | 3 | no | — |
| 218 | Q8 | — | 212 | 3 | no | — |
| 219 | C2×C4 | — | 2 | 3 | no | — |
| 220 | C2×C4 | — | 2 | 3 | no | — |
| 221 | C2×C4 | — | 53 | 3 | no | — |
| 222 | C2×C4 | — | 106 | 3 | no | — |
| 223 | C2×C4 | — | 106 | 3 | no | — |
| 224 | C2×C4 | — | 106 | 12 | no | — |
| 225 | C2×C4 | — | 106 | 3 | no | — |
| 226 | C2×C4 | — | 106 | 3 | no | — |
| 227 | C2×C4 | — | 212 | 3 | no | — |
| 228 | C2×C4 | — | 212 | 3 | no | — |
| 229 | C2×C4 | — | 212 | 2 | no | — |
| 230 | C2×C4 | C2×C22⋊#6157 | 1 | 3 | yes | — |
| 231 | C8 | — | 424 | 3 | no | — |
| 232 | C22 | — | 53 | 1 | no | — |
| 233 | C22 | — | 53 | 1 | no | — |
| 234 | C22 | — | 106 | 2 | no | — |
| 235 | C22 | #16811564 | 1 | 1 | yes | characteristic |
| 236 | C4 | — | 2 | 6 | no | — |
| 237 | C4 | — | 2 | 3 | no | — |
| 238 | C4 | — | 4 | 3 | no | — |
| 239 | C4 | — | 106 | 6 | no | — |
| 240 | C4 | — | 106 | 3 | no | — |
| 241 | C4 | — | 424 | 2 | no | — |
| 242 | C4 | — | 212 | 3 | no | — |
| 243 | C2 | — | 2 | 1 | no | — |
| 244 | C2 | — | 53 | 2 | no | — |
| 245 | C2 | — | 106 | 1 | no | — |
| 246 | C2 | #20290559 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 12. #16811353 | C4 | semidirect |
| 16. Q82×C53 | C4 | semidirect |
| 130. Q82 | #6157 | semidirect |
| 140. C53 | #12777 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| D53⋊#356 | 2 |
| #20290721 | 2 |
| Q82×D53 | 2 |
| #12777 | 53 |