Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
424 conjugacy classes of subgroups; only the 72 normal subgroups are shown.
This group has 424 conjugacy classes of subgroups — too many to draw. Only its 72 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 | #20291260 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #20290721 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | #20290671 | C2 | 1 | 2 | yes | maximal |
| 4 | #20289811 | C2 | 1 | 2 | yes | maximal |
| 5 | #356×D53 | C2 | 1 | 1 | yes | maximal, characteristic |
| 6 | #16811691 | C22 | 1 | 1 | yes | characteristic |
| 7 | #16811634 | — | 2 | 2 | no | — |
| 8 | #16811616 | — | 2 | 1 | no | — |
| 9 | D53⋊#101 | C22 | 1 | 1 | yes | characteristic |
| 10 | (C2×#6157)⋊Q8 | — | 2 | 1 | no | — |
| 11 | (C2×#6157)⋊Q8 | C22 | 1 | 2 | yes | — |
| 12 | #16811572 | — | 2 | 2 | no | — |
| 13 | #16811572 | — | 2 | 2 | no | — |
| 14 | #16811568 | — | 2 | 1 | no | — |
| 15 | #16811495 | — | 2 | 2 | no | — |
| 16 | #16811494 | — | 2 | 1 | no | — |
| 17 | C4⋊Q8×D53 | C22 | 1 | 1 | yes | characteristic |
| 18 | #101×D53 | C22 | 1 | 2 | yes | — |
| 19 | #16810422 | C4 | 1 | 1 | yes | characteristic, complemented |
| 20 | #16810421 | C4 | 1 | 2 | yes | complemented |
| 21 | C53×#356 | C4 | 1 | 1 | yes | characteristic, complemented |
| 22 | D53⋊M16 | — | 2 | 1 | no | — |
| 23 | Q8×#6157 | — | 4 | 2 | no | — |
| 24 | #13133296 | — | 2 | 1 | no | — |
| 25 | #13133296 | — | 2 | 2 | no | — |
| 26 | C2×C4⋊#6157 | — | 2 | 1 | no | — |
| 27 | C2×C4⋊#6157 | D4 | 1 | 2 | yes | — |
| 28 | C2×C4×#6157 | — | 2 | 1 | no | — |
| 29 | C2×C4×#6157 | D4 | 1 | 2 | yes | — |
| 30 | #13133286 | — | 4 | 2 | no | — |
| 31 | #13133285 | — | 4 | 2 | no | — |
| 32 | #13133280 | — | 4 | 2 | no | — |
| 33 | #13133278 | — | 2 | 2 | no | — |
| 34 | (C2×#6157)⋊C4 | — | 2 | 1 | no | — |
| 35 | (C2×#6157)⋊C4 | — | 2 | 2 | no | — |
| 36 | (C2×#6157)⋊C4 | — | 2 | 1 | no | — |
| 37 | (C2×#6157)⋊C4 | — | 2 | 2 | no | — |
| 38 | (C2×#6157)⋊C4 | — | 2 | 2 | no | — |
| 39 | (C2×#6157)⋊C4 | — | 2 | 2 | no | — |
| 40 | C2×Q8×D53 | D4 | 1 | 1 | yes | characteristic |
| 41 | C2×Q8×D53 | C23 | 1 | 1 | yes | characteristic |
| 42 | M16×D53 | — | 2 | 2 | no | — |
| 43 | #13133217 | Q8 | 1 | 1 | yes | characteristic |
| 44 | #13133216 | — | 2 | 2 | no | — |
| 45 | #13133215 | C2×C4 | 1 | 1 | yes | characteristic |
| 46 | #13133215 | Q8 | 1 | 1 | yes | characteristic |
| 47 | C4⋊C4×D53 | — | 2 | 2 | no | — |
| 48 | C42×D53 | D4 | 1 | 1 | yes | characteristic |
| 49 | #13133136 | C2×C4 | 1 | 2 | yes | — |
| 50 | #13133135 | — | 2 | 2 | no | — |
| 51 | C4⋊Q8×C53 | C2×C4 | 1 | 1 | yes | characteristic |
| 52 | #101×C53 | C2×C4 | 1 | 2 | yes | — |
| 53 | C22×#6157 | — | 2 | 2 | no | — |
| 54 | C22×#6157 | — | 2 | 2 | no | — |
| 55 | C2×Q8×C53 | C4⋊C4 | 1 | 1 | yes | characteristic |
| 56 | C2×Q8×C53 | C22×C4 | 1 | 1 | yes | characteristic |
| 57 | Q8×D53 | — | 4 | 1 | no | — |
| 58 | Q8×D53 | — | 4 | 1 | no | — |
| 59 | Q8×D53 | — | 4 | 1 | no | — |
| 60 | Q8×D53 | — | 4 | 1 | no | — |
| 61 | C2×C4×D53 | — | 2 | 1 | no | — |
| 62 | C2×C4×D53 | — | 2 | 1 | no | — |
| 63 | C2×C4×D53 | C2×D4 | 1 | 2 | yes | — |
| 64 | C2×C4×D53 | C2×D4 | 1 | 1 | yes | characteristic |
| 65 | C2×Dic106 | — | 2 | 1 | no | — |
| 66 | C2×Dic106 | C4⋊C4 | 1 | 1 | yes | characteristic |
| 67 | C2×Dic106 | C2×Q8 | 1 | 1 | yes | characteristic |
| 68 | C2×Dic106 | #41 | 1 | 2 | yes | — |
| 69 | C22⋊#6157 | — | 4 | 1 | no | — |
| 70 | #11719940 | — | 4 | 1 | no | — |
| 71 | C4⋊#6157 | — | 2 | 2 | no | — |
| 72 | C4⋊#6157 | — | 2 | 2 | no | — |
| 73 | C4⋊#6157 | — | 2 | 2 | no | — |
| 74 | C4⋊#6157 | — | 4 | 2 | no | — |
| 75 | C4⋊#6157 | — | 8 | 1 | no | — |
| 76 | C4×#6157 | — | 4 | 2 | no | — |
| 77 | C4×#6157 | — | 4 | 1 | no | — |
| 78 | C4×#6157 | — | 4 | 2 | no | — |
| 79 | C4×#6157 | — | 8 | 1 | no | — |
| 80 | #11719936 | — | 4 | 1 | no | — |
| 81 | D53⋊C8 | — | 4 | 1 | no | — |
| 82 | M16×C53 | — | 2 | 2 | no | — |
| 83 | C4⋊C4×C53 | — | 2 | 2 | no | — |
| 84 | C42×C53 | C4⋊C4 | 1 | 1 | yes | characteristic |
| 85 | C4⋊Dic53 | — | 2 | 2 | no | — |
| 86 | #11719919 | — | 2 | 2 | no | — |
| 87 | #11719919 | — | 2 | 2 | no | — |
| 88 | C4×Dic53 | — | 2 | 1 | no | — |
| 89 | C4×Dic53 | C4⋊C4 | 1 | 1 | yes | characteristic |
| 90 | #11719917 | — | 2 | 2 | no | — |
| 91 | #11719912 | — | 4 | 2 | no | — |
| 92 | C8×D53 | — | 4 | 2 | no | — |
| 93 | C22×D53 | #120 | 1 | 1 | yes | characteristic |
| 94 | C2×#6157 | — | 4 | 4 | no | — |
| 95 | C2×#6157 | — | 4 | 4 | no | — |
| 96 | C2×#6157 | — | 4 | 2 | no | — |
| 97 | C2×#6157 | — | 4 | 2 | no | — |
| 98 | C2×#6157 | — | 4 | 2 | no | — |
| 99 | Q8×C53 | — | 4 | 1 | no | — |
| 100 | Q8×C53 | — | 4 | 1 | no | — |
| 101 | C2×C4×C53 | — | 2 | 1 | no | — |
| 102 | C2×C4×C53 | — | 2 | 1 | no | — |
| 103 | C2×C4×C53 | C2×C4⋊C4 | 1 | 1 | yes | characteristic |
| 104 | C2×C4×C53 | C4×D4 | 1 | 2 | yes | — |
| 105 | C2×Dic53 | — | 2 | 1 | no | — |
| 106 | C2×Dic53 | — | 2 | 1 | no | — |
| 107 | C2×Dic53 | C22⋊Q8 | 1 | 2 | yes | — |
| 108 | C2×Dic53 | #123 | 1 | 1 | yes | characteristic |
| 109 | C4×D53 | — | 2 | 2 | no | — |
| 110 | C4×D53 | — | 2 | 1 | no | — |
| 111 | C4×D53 | — | 2 | 1 | no | — |
| 112 | C4×D53 | — | 2 | 2 | no | — |
| 113 | C4×D53 | — | 4 | 2 | no | — |
| 114 | C4×D53 | — | 4 | 1 | no | — |
| 115 | C4×D53 | — | 4 | 1 | no | — |
| 116 | Dic106 | — | 4 | 1 | no | — |
| 117 | Dic106 | — | 4 | 2 | no | — |
| 118 | Dic106 | — | 4 | 1 | no | — |
| 119 | Dic106 | — | 8 | 1 | no | — |
| 120 | #88324 | — | 8 | 1 | no | — |
| 121 | C8×C53 | — | 4 | 2 | no | — |
| 122 | #88322 | — | 4 | 2 | no | — |
| 123 | #12971 | — | 53 | 1 | no | maximal, Sylow |
| 124 | C22×C53 | #385 | 1 | 1 | yes | characteristic |
| 125 | C2×D53 | — | 2 | 2 | no | — |
| 126 | C2×D53 | B(4,𝔽2) | 1 | 1 | yes | characteristic |
| 127 | C2×D53 | #458 | 1 | 1 | yes | characteristic |
| 128 | #6157 | — | 8 | 2 | no | — |
| 129 | #6157 | — | 8 | 2 | no | — |
| 130 | #6157 | — | 8 | 2 | no | — |
| 131 | C4×C53 | — | 2 | 2 | no | — |
| 132 | C4×C53 | — | 2 | 1 | no | — |
| 133 | C4×C53 | — | 4 | 1 | no | — |
| 134 | C4×C53 | — | 4 | 1 | no | — |
| 135 | Dic53 | — | 2 | 1 | no | — |
| 136 | Dic53 | — | 2 | 2 | no | — |
| 137 | Dic53 | — | 4 | 1 | no | — |
| 138 | Dic53 | — | 4 | 1 | no | — |
| 139 | #2381 | — | 53 | 1 | no | — |
| 140 | C2×#356 | — | 53 | 1 | no | — |
| 141 | #1886 | — | 53 | 1 | no | — |
| 142 | #1857 | — | 53 | 2 | no | — |
| 143 | #1394 | — | 53 | 2 | no | — |
| 144 | C2×C53 | — | 2 | 1 | no | — |
| 145 | C2×C53 | #1867 | 1 | 1 | yes | characteristic |
| 146 | D53 | — | 2 | 1 | no | — |
| 147 | D53 | #2203 | 1 | 2 | yes | — |
| 148 | C2×C4⋊Q8 | — | 53 | 1 | no | — |
| 149 | #519 | — | 53 | 1 | no | — |
| 150 | #428 | — | 106 | 2 | no | — |
| 151 | #419 | — | 106 | 1 | no | — |
| 152 | C2×#101 | — | 53 | 2 | no | — |
| 153 | C2×#101 | — | 53 | 1 | no | — |
| 154 | #391 | — | 53 | 2 | no | — |
| 155 | #391 | — | 106 | 1 | no | — |
| 156 | #387 | — | 106 | 2 | no | — |
| 157 | #387 | — | 106 | 2 | no | — |
| 158 | #383 | — | 106 | 1 | no | — |
| 159 | #356 | — | 53 | 1 | no | — |
| 160 | #356 | — | 53 | 2 | no | — |
| 161 | #356 | #6157 | 1 | 1 | yes | characteristic, complemented |
| 162 | #339 | — | 106 | 2 | no | — |
| 163 | #339 | — | 106 | 1 | no | — |
| 164 | C53 | #12971 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 165 | C22×Q8 | — | 53 | 1 | no | — |
| 166 | C22×Q8 | — | 53 | 1 | no | — |
| 167 | C2×M16 | — | 106 | 2 | no | — |
| 168 | C2×M16 | — | 106 | 1 | no | — |
| 169 | C4⋊Q8 | — | 53 | 1 | no | — |
| 170 | C4⋊Q8 | — | 53 | 1 | no | — |
| 171 | C4⋊Q8 | — | 53 | 1 | no | — |
| 172 | C4⋊Q8 | — | 106 | 2 | no | — |
| 173 | C4⋊Q8 | C2×#6157 | 1 | 1 | yes | characteristic |
| 174 | C4×Q8 | — | 212 | 2 | no | — |
| 175 | #117 | — | 106 | 1 | no | — |
| 176 | #117 | — | 106 | 2 | no | — |
| 177 | C2×C4⋊C4 | — | 53 | 2 | no | — |
| 178 | C2×C4⋊C4 | — | 106 | 2 | no | — |
| 179 | C2×C4⋊C4 | — | 106 | 1 | no | — |
| 180 | C2×C42 | — | 53 | 1 | no | — |
| 181 | C2×C42 | — | 53 | 2 | no | — |
| 182 | C2×C42 | — | 106 | 1 | no | — |
| 183 | #107 | — | 212 | 2 | no | — |
| 184 | #106 | — | 212 | 2 | no | — |
| 185 | #103 | — | 212 | 2 | no | — |
| 186 | #101 | — | 53 | 2 | no | — |
| 187 | #101 | — | 106 | 2 | no | — |
| 188 | #101 | — | 106 | 2 | no | — |
| 189 | #101 | C2×#6157 | 1 | 2 | yes | — |
| 190 | #95 | — | 106 | 2 | no | — |
| 191 | #95 | — | 106 | 2 | no | — |
| 192 | #95 | — | 106 | 1 | no | — |
| 193 | #95 | — | 106 | 2 | no | — |
| 194 | #95 | — | 106 | 1 | no | — |
| 195 | #95 | — | 106 | 2 | no | — |
| 196 | C2×Q8 | — | 53 | 1 | no | — |
| 197 | C2×Q8 | — | 53 | 2 | no | — |
| 198 | C2×Q8 | — | 53 | 1 | no | — |
| 199 | C2×Q8 | — | 106 | 1 | no | — |
| 200 | C2×Q8 | — | 212 | 1 | no | — |
| 201 | C2×Q8 | — | 212 | 1 | no | — |
| 202 | C2×Q8 | — | 212 | 1 | no | — |
| 203 | C2×Q8 | — | 212 | 1 | no | — |
| 204 | C2×Q8 | C4⋊#6157 | 1 | 1 | yes | characteristic |
| 205 | C2×Q8 | C22×#6157 | 1 | 1 | yes | characteristic |
| 206 | C22×C4 | — | 53 | 1 | no | — |
| 207 | C22×C4 | — | 53 | 2 | no | — |
| 208 | C22×C4 | — | 106 | 1 | no | — |
| 209 | C22×C4 | — | 106 | 1 | no | — |
| 210 | C22×C4 | — | 106 | 2 | no | — |
| 211 | C22×C4 | — | 106 | 2 | no | — |
| 212 | M16 | — | 2 | 2 | no | — |
| 213 | M16 | — | 106 | 2 | no | — |
| 214 | M16 | — | 212 | 2 | no | — |
| 215 | M16 | — | 212 | 1 | no | — |
| 216 | M16 | — | 212 | 1 | no | — |
| 217 | C2×C8 | — | 212 | 2 | no | — |
| 218 | C2×C8 | — | 212 | 1 | no | — |
| 219 | C4⋊C4 | — | 2 | 2 | no | — |
| 220 | C4⋊C4 | — | 106 | 2 | no | — |
| 221 | C4⋊C4 | — | 106 | 2 | no | — |
| 222 | C4⋊C4 | — | 106 | 2 | no | — |
| 223 | C4⋊C4 | — | 106 | 2 | no | — |
| 224 | C4⋊C4 | — | 106 | 2 | no | — |
| 225 | C4⋊C4 | — | 106 | 2 | no | — |
| 226 | C4⋊C4 | — | 424 | 1 | no | — |
| 227 | C4⋊C4 | — | 212 | 2 | no | — |
| 228 | C22⋊C4 | — | 212 | 1 | no | — |
| 229 | C42 | — | 53 | 1 | no | — |
| 230 | C42 | — | 106 | 1 | no | — |
| 231 | C42 | — | 424 | 1 | no | — |
| 232 | C42 | — | 212 | 2 | no | — |
| 233 | C42 | — | 212 | 2 | no | — |
| 234 | C42 | — | 212 | 1 | no | — |
| 235 | C42 | C4⋊#6157 | 1 | 1 | yes | characteristic |
| 236 | C23 | — | 53 | 1 | no | — |
| 237 | Q8 | — | 4 | 1 | no | — |
| 238 | Q8 | — | 4 | 1 | no | — |
| 239 | Q8 | — | 424 | 1 | no | — |
| 240 | Q8 | — | 212 | 2 | no | — |
| 241 | Q8 | — | 212 | 1 | no | — |
| 242 | Q8 | — | 212 | 1 | no | — |
| 243 | C2×C4 | — | 2 | 1 | no | — |
| 244 | C2×C4 | — | 2 | 1 | no | — |
| 245 | C2×C4 | — | 53 | 2 | no | — |
| 246 | C2×C4 | — | 53 | 1 | no | — |
| 247 | C2×C4 | — | 106 | 2 | no | — |
| 248 | C2×C4 | — | 106 | 1 | no | — |
| 249 | C2×C4 | — | 106 | 2 | no | — |
| 250 | C2×C4 | — | 106 | 1 | no | — |
| 251 | C2×C4 | — | 106 | 1 | no | — |
| 252 | C2×C4 | — | 106 | 1 | no | — |
| 253 | C2×C4 | — | 212 | 1 | no | — |
| 254 | C2×C4 | — | 212 | 1 | no | — |
| 255 | C2×C4 | — | 212 | 2 | no | — |
| 256 | C2×C4 | — | 212 | 2 | no | — |
| 257 | C2×C4 | — | 212 | 2 | no | — |
| 258 | C2×C4 | — | 212 | 2 | no | — |
| 259 | C2×C4 | — | 212 | 4 | no | — |
| 260 | C2×C4 | — | 212 | 4 | no | — |
| 261 | C2×C4 | C2×C4⋊#6157 | 1 | 1 | yes | characteristic |
| 262 | C2×C4 | D4×#6157 | 1 | 2 | yes | — |
| 263 | C8 | — | 4 | 2 | no | — |
| 264 | C8 | — | 424 | 1 | no | — |
| 265 | C8 | — | 212 | 2 | no | — |
| 266 | C22 | — | 53 | 1 | no | — |
| 267 | C22 | — | 53 | 1 | no | — |
| 268 | C22 | — | 106 | 2 | no | — |
| 269 | C22 | #16811570 | 1 | 1 | yes | characteristic |
| 270 | C4 | — | 2 | 2 | no | — |
| 271 | C4 | — | 2 | 1 | no | — |
| 272 | C4 | — | 4 | 1 | no | — |
| 273 | C4 | — | 4 | 1 | no | — |
| 274 | C4 | — | 106 | 1 | no | — |
| 275 | C4 | — | 106 | 2 | no | — |
| 276 | C4 | — | 424 | 2 | no | — |
| 277 | C4 | — | 424 | 2 | no | — |
| 278 | C4 | — | 424 | 2 | no | — |
| 279 | C4 | — | 212 | 1 | no | — |
| 280 | C4 | — | 212 | 1 | no | — |
| 281 | C2 | — | 2 | 1 | no | — |
| 282 | C2 | — | 53 | 2 | no | — |
| 283 | C2 | — | 106 | 1 | no | — |
| 284 | C2 | #20290686 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 19. #16810422 | C4 | semidirect |
| 20. #16810421 | C4 | semidirect |
| 21. C53×#356 | C4 | semidirect |
| 161. #356 | #6157 | semidirect |
| 164. C53 | #12971 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #20291260 | 2 |
| #20290721 | 2 |
| #20290671 | 2 |
| #20289811 | 2 |
| #356×D53 | 2 |
| #12971 | 53 |