Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
832 conjugacy classes of subgroups; only the 86 normal subgroups are shown.
This group has 832 conjugacy classes of subgroups — too many to draw. Only its 86 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 | #20290989 | C2 | 1 | 3 | yes | maximal, complemented |
| 2 | #20290717 | C2 | 1 | 3 | yes | maximal, complemented |
| 3 | #20288342 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | #16811703 | C22 | 1 | 1 | yes | characteristic |
| 5 | #16811622 | — | 2 | 3 | no | — |
| 6 | #16811604 | C22 | 1 | 3 | yes | — |
| 7 | #16811529 | — | 2 | 3 | no | — |
| 8 | #16811528 | — | 2 | 3 | no | — |
| 9 | #16811494 | — | 2 | 3 | no | — |
| 10 | #16811339 | — | 2 | 3 | no | — |
| 11 | #16811339 | C4 | 1 | 3 | yes | complemented |
| 12 | #16811295 | — | 2 | 3 | no | — |
| 13 | #16811174 | — | 2 | 3 | no | — |
| 14 | #16811174 | C22 | 1 | 3 | yes | — |
| 15 | C4×#11719949 | — | 2 | 3 | no | — |
| 16 | Q82×C53 | C4 | 1 | 1 | yes | characteristic, complemented |
| 17 | #13133305 | — | 4 | 2 | no | — |
| 18 | D53⋊M16 | — | 2 | 3 | no | — |
| 19 | #13133296 | — | 2 | 3 | no | — |
| 20 | #13133282 | — | 4 | 6 | no | — |
| 21 | #13133276 | — | 2 | 6 | no | — |
| 22 | C2×#11719949 | — | 2 | 3 | no | — |
| 23 | C2×#11719949 | — | 2 | 1 | no | — |
| 24 | C2×#11719949 | D4 | 1 | 3 | yes | — |
| 25 | C2×#11719949 | C23 | 1 | 1 | yes | characteristic |
| 26 | #13133214 | — | 2 | 3 | no | — |
| 27 | #13133214 | D4 | 1 | 3 | yes | — |
| 28 | #13133206 | — | 2 | 3 | no | — |
| 29 | #13133206 | — | 2 | 3 | no | — |
| 30 | #13133206 | C2×C4 | 1 | 3 | yes | — |
| 31 | #13133206 | D4 | 1 | 3 | yes | — |
| 32 | #13133192 | — | 2 | 6 | no | — |
| 33 | #13133192 | — | 2 | 6 | no | — |
| 34 | #13133192 | — | 2 | 6 | no | — |
| 35 | Q8×Dic53 | — | 2 | 1 | no | — |
| 36 | C4×D212 | — | 2 | 6 | no | — |
| 37 | C4×D212 | — | 2 | 3 | no | — |
| 38 | #13133179 | — | 2 | 3 | no | — |
| 39 | #13133179 | — | 2 | 6 | no | — |
| 40 | #13133175 | — | 2 | 3 | no | — |
| 41 | #13133175 | — | 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 | #11719949 | — | 4 | 1 | no | — |
| 52 | #11719949 | — | 4 | 1 | no | — |
| 53 | #11719949 | — | 4 | 3 | no | — |
| 54 | #11719949 | — | 4 | 3 | no | — |
| 55 | #11719949 | — | 4 | 12 | no | — |
| 56 | #11719949 | — | 8 | 1 | no | — |
| 57 | #11719949 | — | 8 | 1 | no | — |
| 58 | C2×D212 | — | 2 | 3 | no | — |
| 59 | C2×D212 | — | 2 | 6 | no | — |
| 60 | C2×D212 | — | 2 | 3 | no | — |
| 61 | C2×D212 | — | 2 | 3 | no | — |
| 62 | C2×D212 | C22⋊C4 | 1 | 3 | yes | — |
| 63 | C2×D212 | C2×D4 | 1 | 3 | yes | — |
| 64 | C2×C4×D53 | — | 2 | 6 | no | — |
| 65 | C2×C4×D53 | — | 2 | 3 | no | — |
| 66 | C2×C4×D53 | — | 2 | 3 | no | — |
| 67 | C2×C4×D53 | C2×D4 | 1 | 3 | yes | — |
| 68 | C22⋊#6157 | — | 4 | 3 | no | — |
| 69 | #11719940 | — | 4 | 1 | no | — |
| 70 | C4⋊#6157 | — | 4 | 3 | no | — |
| 71 | C4×#6157 | — | 4 | 3 | no | — |
| 72 | #11719936 | — | 2 | 6 | no | — |
| 73 | D53⋊C8 | — | 4 | 3 | no | — |
| 74 | C4⋊C4×C53 | — | 2 | 3 | no | — |
| 75 | C4⋊C4×C53 | — | 2 | 6 | no | — |
| 76 | C42×C53 | — | 2 | 3 | no | — |
| 77 | C42×C53 | C22⋊C4 | 1 | 3 | yes | — |
| 78 | #11719921 | — | 2 | 6 | no | — |
| 79 | #11719921 | — | 2 | 6 | no | — |
| 80 | #11719921 | — | 2 | 6 | no | — |
| 81 | C4⋊Dic53 | — | 2 | 3 | no | — |
| 82 | C4×Dic53 | — | 2 | 3 | no | — |
| 83 | C4×Dic53 | — | 2 | 3 | no | — |
| 84 | C4×Dic53 | C22⋊C4 | 1 | 3 | yes | — |
| 85 | C22×D53 | — | 2 | 3 | no | — |
| 86 | C22×D53 | #120 | 1 | 3 | yes | — |
| 87 | C2×#6157 | — | 8 | 3 | no | — |
| 88 | Q8×C53 | — | 2 | 4 | no | — |
| 89 | Q8×C53 | — | 4 | 1 | no | — |
| 90 | Q8×C53 | — | 4 | 3 | no | — |
| 91 | Q8×C53 | — | 8 | 1 | no | — |
| 92 | C2×C4×C53 | — | 2 | 3 | no | — |
| 93 | C2×C4×C53 | — | 2 | 3 | no | — |
| 94 | C2×C4×C53 | C2×C22⋊C4 | 1 | 3 | yes | — |
| 95 | C2×Dic53 | — | 2 | 3 | no | — |
| 96 | C2×Dic53 | #120 | 1 | 1 | yes | characteristic |
| 97 | D212 | — | 4 | 3 | no | — |
| 98 | D212 | — | 4 | 3 | no | — |
| 99 | D212 | — | 4 | 12 | no | — |
| 100 | D212 | — | 4 | 6 | no | — |
| 101 | D212 | — | 8 | 3 | no | — |
| 102 | D212 | — | 8 | 3 | no | — |
| 103 | C4×D53 | — | 2 | 3 | no | — |
| 104 | C4×D53 | — | 2 | 3 | no | — |
| 105 | C4×D53 | — | 4 | 6 | no | — |
| 106 | C4×D53 | — | 4 | 12 | no | — |
| 107 | C4×D53 | — | 4 | 3 | no | — |
| 108 | C4×D53 | — | 4 | 3 | no | — |
| 109 | #88324 | — | 4 | 2 | no | — |
| 110 | #12773 | — | 53 | 1 | no | maximal, Sylow |
| 111 | C22×C53 | #379 | 1 | 1 | yes | characteristic |
| 112 | C2×D53 | — | 2 | 3 | no | — |
| 113 | C2×D53 | — | 4 | 3 | no | — |
| 114 | C2×D53 | — | 4 | 6 | no | — |
| 115 | C2×D53 | — | 4 | 3 | no | — |
| 116 | #6157 | — | 16 | 3 | no | — |
| 117 | C4×C53 | — | 2 | 6 | no | — |
| 118 | C4×C53 | — | 2 | 3 | no | — |
| 119 | C4×C53 | — | 4 | 3 | no | — |
| 120 | Dic53 | — | 4 | 3 | no | — |
| 121 | Dic53 | B(4,𝔽2) | 1 | 1 | yes | characteristic |
| 122 | Dic53 | #458 | 1 | 1 | yes | characteristic |
| 123 | #3483 | — | 53 | 1 | no | — |
| 124 | #2126 | — | 53 | 3 | no | — |
| 125 | #1884 | — | 53 | 3 | no | — |
| 126 | C2×C53 | — | 2 | 1 | no | — |
| 127 | C2×C53 | #1790 | 1 | 1 | yes | characteristic |
| 128 | D53 | — | 4 | 3 | no | — |
| 129 | D53 | — | 8 | 3 | no | — |
| 130 | #568 | — | 53 | 1 | no | — |
| 131 | Q82 | #6157 | 1 | 1 | yes | characteristic, complemented |
| 132 | #555 | — | 53 | 3 | no | — |
| 133 | #555 | — | 106 | 3 | no | — |
| 134 | #550 | — | 106 | 3 | no | — |
| 135 | #532 | — | 53 | 3 | no | — |
| 136 | #532 | — | 106 | 3 | no | — |
| 137 | C4×#41 | — | 106 | 3 | no | — |
| 138 | #421 | — | 106 | 3 | no | — |
| 139 | #410 | — | 53 | 3 | no | — |
| 140 | #354 | — | 106 | 3 | no | — |
| 141 | #353 | — | 106 | 3 | no | — |
| 142 | #339 | — | 106 | 3 | no | — |
| 143 | C53 | #12773 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 144 | C2×#41 | — | 53 | 1 | no | — |
| 145 | C2×#41 | — | 53 | 3 | no | — |
| 146 | C2×#41 | — | 106 | 1 | no | — |
| 147 | C2×#41 | — | 106 | 3 | no | — |
| 148 | #131 | — | 212 | 2 | no | — |
| 149 | C2×M16 | — | 106 | 3 | no | — |
| 150 | C4⋊Q8 | — | 2 | 3 | no | — |
| 151 | C4⋊Q8 | C2×#6157 | 1 | 3 | yes | — |
| 152 | #127 | — | 53 | 3 | no | — |
| 153 | #127 | — | 106 | 3 | no | — |
| 154 | #124 | — | 53 | 3 | no | — |
| 155 | #124 | — | 53 | 3 | no | — |
| 156 | #124 | — | 106 | 3 | no | — |
| 157 | #124 | — | 106 | 3 | no | — |
| 158 | #121 | — | 106 | 6 | no | — |
| 159 | #121 | — | 106 | 6 | no | — |
| 160 | #121 | — | 106 | 6 | no | — |
| 161 | C4×Q8 | — | 2 | 3 | no | — |
| 162 | C4×Q8 | — | 106 | 1 | no | — |
| 163 | C4×D4 | — | 106 | 6 | no | — |
| 164 | C4×D4 | — | 106 | 6 | no | — |
| 165 | C4×D4 | — | 106 | 3 | no | — |
| 166 | C4×D4 | — | 106 | 3 | no | — |
| 167 | #117 | — | 106 | 6 | no | — |
| 168 | #117 | — | 106 | 3 | no | — |
| 169 | #117 | — | 106 | 3 | no | — |
| 170 | C2×C42 | — | 53 | 3 | no | — |
| 171 | C2×C42 | — | 106 | 3 | no | — |
| 172 | #104 | — | 212 | 6 | no | — |
| 173 | #99 | — | 106 | 6 | no | — |
| 174 | #41 | — | 424 | 1 | no | — |
| 175 | #41 | — | 424 | 1 | no | — |
| 176 | #41 | — | 212 | 1 | no | — |
| 177 | #41 | — | 212 | 1 | no | — |
| 178 | #41 | — | 212 | 3 | no | — |
| 179 | #41 | — | 212 | 3 | no | — |
| 180 | #41 | — | 212 | 12 | no | — |
| 181 | C2×Q8 | — | 2 | 1 | no | — |
| 182 | C2×Q8 | — | 2 | 1 | no | — |
| 183 | C2×Q8 | C22⋊#6157 | 1 | 3 | yes | — |
| 184 | C2×Q8 | C22×#6157 | 1 | 1 | yes | characteristic |
| 185 | C2×D4 | — | 53 | 3 | no | — |
| 186 | C2×D4 | — | 53 | 3 | no | — |
| 187 | C2×D4 | — | 106 | 3 | no | — |
| 188 | C2×D4 | — | 106 | 3 | no | — |
| 189 | C2×D4 | — | 106 | 6 | no | — |
| 190 | C2×D4 | — | 106 | 3 | no | — |
| 191 | C22×C4 | — | 53 | 3 | no | — |
| 192 | C22×C4 | — | 106 | 3 | no | — |
| 193 | C22×C4 | — | 106 | 3 | no | — |
| 194 | C22×C4 | — | 106 | 6 | no | — |
| 195 | M16 | — | 106 | 6 | no | — |
| 196 | M16 | — | 212 | 1 | no | — |
| 197 | C2×C8 | — | 212 | 3 | no | — |
| 198 | C4⋊C4 | — | 2 | 3 | no | — |
| 199 | C4⋊C4 | — | 2 | 6 | no | — |
| 200 | C4⋊C4 | — | 106 | 3 | no | — |
| 201 | C4⋊C4 | — | 212 | 3 | no | — |
| 202 | C22⋊C4 | — | 106 | 6 | no | — |
| 203 | C22⋊C4 | — | 106 | 6 | no | — |
| 204 | C22⋊C4 | — | 106 | 6 | no | — |
| 205 | C22⋊C4 | — | 212 | 3 | no | — |
| 206 | C42 | — | 2 | 3 | no | — |
| 207 | C42 | — | 53 | 3 | no | — |
| 208 | C42 | — | 106 | 3 | no | — |
| 209 | C42 | — | 106 | 3 | no | — |
| 210 | C42 | — | 212 | 3 | no | — |
| 211 | C42 | C22⋊#6157 | 1 | 3 | yes | — |
| 212 | C23 | — | 53 | 3 | no | — |
| 213 | C23 | — | 106 | 3 | no | — |
| 214 | Q8 | — | 2 | 4 | no | — |
| 215 | Q8 | — | 4 | 1 | no | — |
| 216 | Q8 | — | 4 | 3 | no | — |
| 217 | Q8 | — | 8 | 1 | no | — |
| 218 | D4 | — | 424 | 3 | no | — |
| 219 | D4 | — | 424 | 3 | no | — |
| 220 | D4 | — | 212 | 3 | no | — |
| 221 | D4 | — | 212 | 3 | no | — |
| 222 | D4 | — | 212 | 6 | no | — |
| 223 | D4 | — | 212 | 12 | no | — |
| 224 | C2×C4 | — | 2 | 3 | no | — |
| 225 | C2×C4 | — | 2 | 3 | no | — |
| 226 | C2×C4 | — | 53 | 1 | no | — |
| 227 | C2×C4 | — | 106 | 3 | no | — |
| 228 | C2×C4 | — | 106 | 3 | no | — |
| 229 | C2×C4 | — | 106 | 3 | no | — |
| 230 | C2×C4 | — | 424 | 3 | no | — |
| 231 | C2×C4 | — | 212 | 12 | no | — |
| 232 | C2×C4 | — | 212 | 6 | no | — |
| 233 | C2×C4 | — | 212 | 3 | no | — |
| 234 | C2×C4 | — | 212 | 3 | no | — |
| 235 | C2×C4 | C2×C22⋊#6157 | 1 | 3 | yes | — |
| 236 | C8 | — | 212 | 2 | no | — |
| 237 | C22 | — | 106 | 3 | no | — |
| 238 | C22 | — | 212 | 6 | no | — |
| 239 | C22 | — | 212 | 3 | no | — |
| 240 | C22 | — | 212 | 3 | no | — |
| 241 | C22 | #16811564 | 1 | 1 | yes | characteristic |
| 242 | C4 | — | 2 | 6 | no | — |
| 243 | C4 | — | 2 | 3 | no | — |
| 244 | C4 | — | 4 | 3 | no | — |
| 245 | C4 | — | 53 | 1 | no | — |
| 246 | C4 | — | 53 | 1 | no | — |
| 247 | C4 | — | 848 | 3 | no | — |
| 248 | C4 | — | 212 | 3 | no | — |
| 249 | C2 | — | 2 | 1 | no | — |
| 250 | C2 | — | 424 | 3 | no | — |
| 251 | C2 | — | 212 | 3 | no | — |
| 252 | C2 | #20290559 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #20290989 | C2 | semidirect |
| 2. #20290717 | C2 | semidirect |
| 11. #16811339 | C4 | semidirect |
| 16. Q82×C53 | C4 | semidirect |
| 131. Q82 | #6157 | semidirect |
| 143. C53 | #12773 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #20290989 | 2 |
| #20290717 | 2 |
| #20288342 | 2 |
| #12773 | 53 |