Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
1178 conjugacy classes of subgroups; only the 209 normal subgroups are shown.
This group has 1178 conjugacy classes of subgroups — too many to draw. Only its 209 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 | #20288117 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #20277300 | C2 | 1 | 3 | yes | maximal |
| 3 | #20275883 | C2 | 1 | 3 | yes | maximal |
| 4 | #20275882 | C2 | 1 | 3 | yes | maximal |
| 5 | #20275881 | C2 | 1 | 1 | yes | maximal, characteristic |
| 6 | #20274446 | C2 | 1 | 3 | yes | maximal |
| 7 | C53×#2030 | C2 | 1 | 1 | yes | maximal, characteristic |
| 8 | C23×Dic106 | C22 | 1 | 1 | yes | characteristic |
| 9 | #16811252 | — | 2 | 6 | no | — |
| 10 | #16811249 | C22 | 1 | 3 | yes | — |
| 11 | #16811175 | — | 2 | 6 | no | — |
| 12 | C2×#13133173 | C22 | 1 | 3 | yes | — |
| 13 | C2×#13133173 | C22 | 1 | 3 | yes | — |
| 14 | #16810527 | — | 2 | 6 | no | — |
| 15 | #16810527 | — | 2 | 6 | no | — |
| 16 | #16810486 | C22 | 1 | 3 | yes | — |
| 17 | #16810484 | C22 | 1 | 3 | yes | — |
| 18 | #16810484 | C22 | 1 | 3 | yes | — |
| 19 | #16810390 | C22 | 1 | 3 | yes | — |
| 20 | #16810389 | C22 | 1 | 6 | yes | — |
| 21 | C53×#543 | C22 | 1 | 1 | yes | characteristic |
| 22 | C53×#381 | C22 | 1 | 3 | yes | — |
| 23 | C53×#342 | C22 | 1 | 3 | yes | — |
| 24 | C22×Dic106 | — | 2 | 3 | no | — |
| 25 | C22×Dic106 | — | 2 | 3 | no | — |
| 26 | C22×Dic106 | — | 2 | 6 | no | — |
| 27 | C22×Dic106 | Q8 | 1 | 4 | yes | complemented |
| 28 | C23×Dic53 | C23 | 1 | 1 | yes | characteristic |
| 29 | C23×Dic53 | C23 | 1 | 1 | yes | characteristic |
| 30 | #13133216 | — | 2 | 6 | no | — |
| 31 | #13133215 | — | 2 | 6 | no | — |
| 32 | #13133207 | — | 2 | 12 | no | — |
| 33 | #13133196 | — | 2 | 6 | no | — |
| 34 | #13133194 | — | 2 | 12 | no | — |
| 35 | Q8×Dic53 | — | 2 | 2 | no | — |
| 36 | #13133182 | — | 2 | 6 | no | — |
| 37 | #13133173 | — | 2 | 6 | no | — |
| 38 | #13133173 | — | 2 | 6 | no | — |
| 39 | #13133173 | — | 2 | 12 | no | — |
| 40 | C2×#11719919 | — | 2 | 3 | no | — |
| 41 | C2×#11719919 | D4 | 1 | 6 | yes | — |
| 42 | C2×C22⋊Dic53 | C23 | 1 | 3 | yes | — |
| 43 | C2×C22⋊Dic53 | C23 | 1 | 3 | yes | — |
| 44 | C2×C4×Dic53 | — | 2 | 3 | no | — |
| 45 | C2×C4×Dic53 | D4 | 1 | 6 | yes | — |
| 46 | #13133124 | — | 2 | 6 | no | — |
| 47 | #13133124 | — | 2 | 6 | no | — |
| 48 | #13133124 | — | 2 | 6 | no | — |
| 49 | C23×C4×C53 | C23 | 1 | 1 | yes | characteristic |
| 50 | C22⋊Q8×C53 | — | 2 | 6 | no | — |
| 51 | C2×C22⋊C4×C53 | C23 | 1 | 3 | yes | — |
| 52 | C2×C22⋊C4×C53 | C23 | 1 | 3 | yes | — |
| 53 | #95×C53 | — | 2 | 6 | no | — |
| 54 | C24×C53 | C24 | 1 | 1 | yes | characteristic |
| 55 | C2×Q8×C53 | — | 2 | 2 | no | — |
| 56 | C22×C4×C53 | — | 2 | 3 | no | — |
| 57 | C22×C4×C53 | — | 2 | 3 | no | — |
| 58 | C22×C4×C53 | — | 2 | 3 | no | — |
| 59 | C22×C4×C53 | C2×D4 | 1 | 6 | yes | — |
| 60 | C22×C4×C53 | C2×Q8 | 1 | 2 | yes | — |
| 61 | C22×Dic53 | — | 2 | 3 | no | — |
| 62 | C22×Dic53 | — | 2 | 3 | no | — |
| 63 | C22×Dic53 | — | 2 | 3 | no | — |
| 64 | C22×Dic53 | C2×D4 | 1 | 6 | yes | — |
| 65 | C22×Dic53 | C2×D4 | 1 | 3 | yes | — |
| 66 | C22×Dic53 | C2×D4 | 1 | 3 | yes | — |
| 67 | C22×Dic53 | C2×Q8 | 1 | 2 | yes | — |
| 68 | C22×Dic53 | C2×Q8 | 1 | 2 | yes | — |
| 69 | C22×Dic53 | #41 | 1 | 6 | yes | — |
| 70 | C2×Dic106 | — | 2 | 12 | no | — |
| 71 | C2×Dic106 | — | 2 | 12 | no | — |
| 72 | C2×Dic106 | — | 2 | 6 | no | — |
| 73 | C2×Dic106 | — | 4 | 1 | no | — |
| 74 | C2×Dic106 | — | 4 | 3 | no | — |
| 75 | C2×Dic106 | — | 4 | 6 | no | — |
| 76 | C2×Dic106 | — | 4 | 6 | no | — |
| 77 | C4⋊C4×C53 | — | 2 | 6 | no | — |
| 78 | C22⋊C4×C53 | — | 2 | 6 | no | — |
| 79 | C22⋊C4×C53 | — | 2 | 6 | no | — |
| 80 | C22⋊Dic53 | — | 2 | 6 | no | — |
| 81 | C22⋊Dic53 | — | 2 | 6 | no | — |
| 82 | C4⋊Dic53 | — | 2 | 6 | no | — |
| 83 | #11719919 | — | 2 | 12 | no | — |
| 84 | #11719919 | — | 2 | 12 | no | — |
| 85 | #11719919 | — | 4 | 6 | no | — |
| 86 | C4×Dic53 | — | 2 | 6 | no | — |
| 87 | C4×Dic53 | — | 2 | 6 | no | — |
| 88 | C4×Dic53 | — | 4 | 6 | no | — |
| 89 | C23×C53 | — | 4 | 2 | no | — |
| 90 | C23×C53 | C22×D4 | 1 | 3 | yes | — |
| 91 | C23×C53 | C22×Q8 | 1 | 1 | yes | characteristic |
| 92 | C23×C53 | C2×#41 | 1 | 3 | yes | — |
| 93 | Q8×C53 | — | 8 | 2 | no | — |
| 94 | C2×C4×C53 | — | 2 | 6 | no | — |
| 95 | C2×C4×C53 | — | 2 | 6 | no | — |
| 96 | C2×C4×C53 | — | 4 | 1 | no | — |
| 97 | C2×C4×C53 | — | 4 | 3 | no | — |
| 98 | C2×C4×C53 | — | 4 | 6 | no | — |
| 99 | C2×C4×C53 | — | 4 | 6 | no | — |
| 100 | C2×C4×C53 | — | 4 | 3 | no | — |
| 101 | C2×Dic53 | — | 2 | 6 | no | — |
| 102 | C2×Dic53 | — | 2 | 6 | no | — |
| 103 | C2×Dic53 | — | 4 | 1 | no | — |
| 104 | C2×Dic53 | — | 4 | 3 | no | — |
| 105 | C2×Dic53 | — | 4 | 6 | no | — |
| 106 | C2×Dic53 | — | 4 | 6 | no | — |
| 107 | C2×Dic53 | — | 4 | 3 | no | — |
| 108 | C2×Dic53 | #120 | 1 | 1 | yes | characteristic |
| 109 | C2×Dic53 | #120 | 1 | 3 | yes | — |
| 110 | C2×Dic53 | C22⋊Q8 | 1 | 6 | yes | — |
| 111 | C2×Dic53 | C22⋊Q8 | 1 | 6 | yes | — |
| 112 | C2×Dic53 | #124 | 1 | 6 | yes | — |
| 113 | C2×Dic53 | #124 | 1 | 6 | yes | — |
| 114 | Dic106 | — | 4 | 4 | no | — |
| 115 | Dic106 | — | 4 | 12 | no | — |
| 116 | Dic106 | — | 8 | 6 | no | — |
| 117 | #23470 | — | 53 | 1 | no | maximal, Sylow |
| 118 | C22×C53 | — | 2 | 6 | no | — |
| 119 | C22×C53 | — | 4 | 1 | no | — |
| 120 | C22×C53 | — | 4 | 3 | no | — |
| 121 | C22×C53 | C2×#120 | 1 | 1 | yes | characteristic |
| 122 | C22×C53 | C2×C22⋊Q8 | 1 | 3 | yes | — |
| 123 | C22×C53 | C2×#124 | 1 | 3 | yes | — |
| 124 | C4×C53 | — | 4 | 2 | no | — |
| 125 | C4×C53 | — | 4 | 6 | no | — |
| 126 | C4×C53 | — | 8 | 3 | no | — |
| 127 | Dic53 | — | 4 | 2 | no | — |
| 128 | Dic53 | — | 4 | 6 | no | — |
| 129 | Dic53 | — | 8 | 3 | no | — |
| 130 | Dic53 | #393 | 1 | 2 | yes | — |
| 131 | Dic53 | #393 | 1 | 6 | yes | — |
| 132 | #3458 | — | 53 | 1 | no | — |
| 133 | #2310 | — | 53 | 3 | no | — |
| 134 | #2030 | — | 53 | 3 | no | — |
| 135 | #2030 | — | 53 | 3 | no | — |
| 136 | #2030 | — | 53 | 1 | no | — |
| 137 | #2030 | D53 | 1 | 1 | yes | characteristic |
| 138 | #1740 | — | 53 | 3 | no | — |
| 139 | C2×C53 | — | 2 | 3 | no | — |
| 140 | C2×C53 | — | 2 | 3 | no | — |
| 141 | C2×C53 | C2×#393 | 1 | 1 | yes | characteristic |
| 142 | C2×C53 | #3028 | 1 | 2 | yes | — |
| 143 | C23×Q8 | — | 53 | 1 | no | — |
| 144 | #544 | — | 106 | 6 | no | — |
| 145 | #543 | — | 53 | 3 | no | — |
| 146 | #543 | C2×D53 | 1 | 1 | yes | characteristic |
| 147 | #533 | — | 106 | 6 | no | — |
| 148 | C2×#117 | — | 53 | 3 | no | — |
| 149 | C2×#117 | — | 53 | 3 | no | — |
| 150 | #391 | — | 106 | 6 | no | — |
| 151 | #391 | — | 106 | 6 | no | — |
| 152 | #381 | — | 53 | 3 | no | — |
| 153 | #381 | — | 53 | 3 | no | — |
| 154 | #381 | — | 53 | 3 | no | — |
| 155 | #381 | C2×D53 | 1 | 3 | yes | — |
| 156 | #342 | — | 53 | 6 | no | — |
| 157 | #342 | — | 53 | 3 | no | — |
| 158 | #342 | C2×D53 | 1 | 3 | yes | — |
| 159 | C53 | #23470 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 160 | C22×Q8 | — | 53 | 4 | no | — |
| 161 | C22×Q8 | — | 106 | 3 | no | — |
| 162 | C22×Q8 | — | 106 | 3 | no | — |
| 163 | C22×Q8 | — | 106 | 6 | no | — |
| 164 | C23×C4 | — | 53 | 1 | no | — |
| 165 | C23×C4 | — | 53 | 1 | no | — |
| 166 | C23×C4 | C22×D53 | 1 | 1 | yes | characteristic |
| 167 | C4⋊Q8 | — | 106 | 6 | no | — |
| 168 | C4⋊Q8 | — | 106 | 6 | no | — |
| 169 | #125 | — | 106 | 12 | no | — |
| 170 | C22⋊Q8 | — | 2 | 6 | no | — |
| 171 | C22⋊Q8 | — | 106 | 12 | no | — |
| 172 | C22⋊Q8 | — | 106 | 6 | no | — |
| 173 | C4×Q8 | — | 106 | 6 | no | — |
| 174 | C4×Q8 | — | 106 | 2 | no | — |
| 175 | #117 | — | 106 | 6 | no | — |
| 176 | #117 | — | 106 | 6 | no | — |
| 177 | #117 | — | 106 | 12 | no | — |
| 178 | C2×C4⋊C4 | — | 53 | 6 | no | — |
| 179 | C2×C4⋊C4 | — | 106 | 3 | no | — |
| 180 | C2×C22⋊C4 | — | 53 | 3 | no | — |
| 181 | C2×C22⋊C4 | — | 53 | 3 | no | — |
| 182 | C2×C22⋊C4 | C22×D53 | 1 | 3 | yes | — |
| 183 | C2×C22⋊C4 | C22×D53 | 1 | 3 | yes | — |
| 184 | C2×C42 | — | 53 | 6 | no | — |
| 185 | C2×C42 | — | 106 | 3 | no | — |
| 186 | #95 | — | 2 | 6 | no | — |
| 187 | #95 | — | 106 | 6 | no | — |
| 188 | #95 | — | 106 | 6 | no | — |
| 189 | #95 | — | 106 | 6 | no | — |
| 190 | C24 | C23×D53 | 1 | 1 | yes | characteristic |
| 191 | C2×Q8 | — | 2 | 2 | no | — |
| 192 | C2×Q8 | — | 106 | 12 | no | — |
| 193 | C2×Q8 | — | 106 | 12 | no | — |
| 194 | C2×Q8 | — | 106 | 6 | no | — |
| 195 | C2×Q8 | — | 212 | 1 | no | — |
| 196 | C2×Q8 | — | 212 | 3 | no | — |
| 197 | C2×Q8 | — | 212 | 6 | no | — |
| 198 | C2×Q8 | — | 212 | 6 | no | — |
| 199 | C22×C4 | — | 2 | 3 | no | — |
| 200 | C22×C4 | — | 2 | 3 | no | — |
| 201 | C22×C4 | — | 2 | 3 | no | — |
| 202 | C22×C4 | — | 53 | 3 | no | — |
| 203 | C22×C4 | — | 53 | 3 | no | — |
| 204 | C22×C4 | — | 53 | 2 | no | — |
| 205 | C22×C4 | — | 53 | 6 | no | — |
| 206 | C22×C4 | — | 53 | 2 | no | — |
| 207 | C22×C4 | — | 53 | 6 | no | — |
| 208 | C22×C4 | — | 106 | 3 | no | — |
| 209 | C22×C4 | — | 106 | 3 | no | — |
| 210 | C22×C4 | — | 106 | 3 | no | — |
| 211 | C22×C4 | D4×D53 | 1 | 6 | yes | — |
| 212 | C22×C4 | Q8×D53 | 1 | 2 | yes | — |
| 213 | C4⋊C4 | — | 2 | 6 | no | — |
| 214 | C4⋊C4 | — | 106 | 12 | no | — |
| 215 | C4⋊C4 | — | 106 | 12 | no | — |
| 216 | C4⋊C4 | — | 106 | 6 | no | — |
| 217 | C4⋊C4 | — | 212 | 6 | no | — |
| 218 | C22⋊C4 | — | 2 | 6 | no | — |
| 219 | C22⋊C4 | — | 2 | 6 | no | — |
| 220 | C22⋊C4 | — | 106 | 6 | no | — |
| 221 | C22⋊C4 | — | 106 | 6 | no | — |
| 222 | C42 | — | 106 | 6 | no | — |
| 223 | C42 | — | 106 | 6 | no | — |
| 224 | C42 | — | 212 | 6 | no | — |
| 225 | C23 | — | 4 | 2 | no | — |
| 226 | C23 | C2×D4×D53 | 1 | 3 | yes | — |
| 227 | C23 | C2×Q8×D53 | 1 | 1 | yes | characteristic |
| 228 | C23 | #41×D53 | 1 | 3 | yes | — |
| 229 | Q8 | — | 8 | 2 | no | — |
| 230 | Q8 | — | 424 | 6 | no | — |
| 231 | Q8 | — | 212 | 4 | no | — |
| 232 | Q8 | — | 212 | 12 | no | — |
| 233 | C2×C4 | — | 2 | 6 | no | — |
| 234 | C2×C4 | — | 2 | 6 | no | — |
| 235 | C2×C4 | — | 4 | 1 | no | — |
| 236 | C2×C4 | — | 4 | 3 | no | — |
| 237 | C2×C4 | — | 4 | 6 | no | — |
| 238 | C2×C4 | — | 4 | 6 | no | — |
| 239 | C2×C4 | — | 4 | 3 | no | — |
| 240 | C2×C4 | — | 53 | 1 | no | — |
| 241 | C2×C4 | — | 53 | 3 | no | — |
| 242 | C2×C4 | — | 53 | 6 | no | — |
| 243 | C2×C4 | — | 53 | 6 | no | — |
| 244 | C2×C4 | — | 53 | 6 | no | — |
| 245 | C2×C4 | — | 53 | 6 | no | — |
| 246 | C2×C4 | — | 106 | 6 | no | — |
| 247 | C2×C4 | — | 106 | 6 | no | — |
| 248 | C2×C4 | — | 212 | 6 | no | — |
| 249 | C2×C4 | — | 212 | 3 | no | — |
| 250 | C2×C4 | — | 212 | 1 | no | — |
| 251 | C2×C4 | — | 212 | 3 | no | — |
| 252 | C2×C4 | — | 212 | 6 | no | — |
| 253 | C22 | — | 2 | 6 | no | — |
| 254 | C22 | — | 4 | 1 | no | — |
| 255 | C22 | — | 4 | 3 | no | — |
| 256 | C22 | #120×D53 | 1 | 1 | yes | characteristic |
| 257 | C22 | C22⋊Q8×D53 | 1 | 3 | yes | — |
| 258 | C22 | #124×D53 | 1 | 3 | yes | — |
| 259 | C4 | — | 4 | 2 | no | — |
| 260 | C4 | — | 4 | 6 | no | — |
| 261 | C4 | — | 8 | 3 | no | — |
| 262 | C4 | — | 53 | 2 | no | — |
| 263 | C4 | — | 53 | 6 | no | — |
| 264 | C4 | — | 424 | 3 | no | — |
| 265 | C4 | — | 212 | 2 | no | — |
| 266 | C4 | — | 212 | 6 | no | — |
| 267 | C2 | — | 2 | 3 | no | — |
| 268 | C2 | — | 2 | 3 | no | — |
| 269 | C2 | #393×D53 | 1 | 1 | yes | characteristic |
| 270 | C2 | #20284119 | 1 | 2 | yes | — |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 27. C22×Dic106 | Q8 | semidirect |
| 159. C53 | #23470 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #20288117 | 2 |
| #20277300 | 2 |
| #20275883 | 2 |
| #20275882 | 2 |
| #20275881 | 2 |
| #20274446 | 2 |
| C53×#2030 | 2 |
| #23470 | 53 |