Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
344 conjugacy classes of subgroups; only the 53 normal subgroups are shown.
This group has 344 conjugacy classes of subgroups — too many to draw. Only its 53 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 | #20286181 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #20276665 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #20276663 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | #20276022 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | #20272699 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | #20272667 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 7 | C53×#1340 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 8 | #16811423 | C22 | 1 | 1 | yes | characteristic |
| 9 | #16811246 | — | 2 | 1 | no | — |
| 10 | #16811064 | — | 2 | 1 | no | — |
| 11 | #16811061 | — | 2 | 1 | no | — |
| 12 | #16810912 | — | 2 | 1 | no | — |
| 13 | #16810906 | — | 2 | 1 | no | — |
| 14 | #16810876 | C22 | 1 | 1 | yes | characteristic |
| 15 | #16810854 | — | 2 | 1 | no | — |
| 16 | #16810847 | — | 2 | 1 | no | — |
| 17 | #16810841 | — | 2 | 1 | no | — |
| 18 | #16810651 | C22 | 1 | 1 | yes | characteristic |
| 19 | #16810617 | — | 2 | 1 | no | — |
| 20 | #16810596 | — | 2 | 1 | no | — |
| 21 | #16810434 | — | 2 | 1 | no | — |
| 22 | #16810432 | — | 2 | 1 | no | — |
| 23 | #16810405 | C22 | 1 | 1 | yes | characteristic |
| 24 | C53×#470 | C22 | 1 | 1 | yes | characteristic |
| 25 | C53×#349 | C22 | 1 | 1 | yes | characteristic |
| 26 | C53×H(3,ℤ/4) | C22 | 1 | 1 | yes | characteristic |
| 27 | C2×#11719945 | — | 2 | 1 | no | — |
| 28 | #13133250 | — | 4 | 1 | no | — |
| 29 | #13133245 | — | 4 | 1 | no | — |
| 30 | #13133239 | — | 2 | 1 | no | — |
| 31 | #13133239 | — | 2 | 1 | no | — |
| 32 | C2×#11719924 | — | 2 | 1 | no | — |
| 33 | C2×D424 | D4 | 1 | 1 | yes | characteristic |
| 34 | C22×#88322 | D4 | 1 | 1 | yes | characteristic |
| 35 | #13133216 | — | 2 | 1 | no | — |
| 36 | #13133212 | — | 4 | 1 | no | — |
| 37 | #13133206 | — | 2 | 1 | no | — |
| 38 | #13133203 | — | 4 | 1 | no | — |
| 39 | #13133199 | — | 4 | 1 | no | — |
| 40 | #13133197 | — | 4 | 1 | no | — |
| 41 | #13133191 | — | 2 | 1 | no | — |
| 42 | Q8×Dic53 | — | 2 | 1 | no | — |
| 43 | #13133179 | — | 2 | 1 | no | — |
| 44 | #13133176 | D4 | 1 | 1 | yes | characteristic |
| 45 | #13133163 | — | 4 | 1 | no | — |
| 46 | #13133162 | — | 4 | 1 | no | — |
| 47 | #13133160 | — | 4 | 1 | no | — |
| 48 | D848 | — | 4 | 1 | no | — |
| 49 | #13133157 | — | 2 | 1 | no | — |
| 50 | #13133149 | D4 | 1 | 1 | yes | characteristic |
| 51 | #13133148 | — | 2 | 1 | no | — |
| 52 | #13133146 | — | 2 | 1 | no | — |
| 53 | #13133146 | — | 2 | 1 | no | — |
| 54 | #13133141 | — | 2 | 1 | no | — |
| 55 | #13133140 | — | 4 | 1 | no | — |
| 56 | #13133139 | — | 2 | 1 | no | — |
| 57 | #13133137 | — | 2 | 1 | no | — |
| 58 | #13133137 | — | 4 | 1 | no | — |
| 59 | #13133131 | — | 4 | 1 | no | — |
| 60 | #13133131 | — | 4 | 1 | no | — |
| 61 | C2×Q16×C53 | D4 | 1 | 1 | yes | characteristic |
| 62 | C2×M16×C53 | C23 | 1 | 1 | yes | characteristic |
| 63 | C22⋊Q8×C53 | — | 2 | 1 | no | — |
| 64 | #117×C53 | — | 2 | 1 | no | — |
| 65 | M32×C53 | — | 2 | 1 | no | — |
| 66 | #106×C53 | D4 | 1 | 1 | yes | characteristic |
| 67 | #103×C53 | — | 2 | 1 | no | — |
| 68 | C2×Q8×C53 | — | 2 | 1 | no | — |
| 69 | C22×C4×C53 | C2×D4 | 1 | 1 | yes | characteristic |
| 70 | C2×C22⋊D53 | — | 4 | 1 | no | — |
| 71 | C2×C22⋊D53 | — | 4 | 1 | no | — |
| 72 | #11719945 | — | 4 | 1 | no | — |
| 73 | #11719945 | — | 4 | 1 | no | — |
| 74 | #11719945 | — | 4 | 1 | no | — |
| 75 | #11719945 | — | 4 | 1 | no | — |
| 76 | C2×D212 | — | 2 | 1 | no | — |
| 77 | C2×D212 | — | 2 | 1 | no | — |
| 78 | C2×C4×D53 | — | 4 | 1 | no | — |
| 79 | C2×C4×D53 | — | 4 | 1 | no | — |
| 80 | C2×Dic106 | — | 2 | 1 | no | — |
| 81 | C2×Dic106 | — | 4 | 1 | no | — |
| 82 | Q16×C53 | — | 4 | 1 | no | — |
| 83 | M16×C53 | — | 4 | 1 | no | — |
| 84 | C2×C8×C53 | C2×D4 | 1 | 1 | yes | characteristic |
| 85 | C2×C8×C53 | C2×D4 | 1 | 1 | yes | characteristic |
| 86 | C4⋊C4×C53 | — | 2 | 1 | no | — |
| 87 | C4⋊C4×C53 | — | 4 | 1 | no | — |
| 88 | C4⋊C4×C53 | — | 4 | 1 | no | — |
| 89 | C22⋊C4×C53 | — | 4 | 1 | no | — |
| 90 | C22⋊C4×C53 | — | 4 | 1 | no | — |
| 91 | C42×C53 | — | 4 | 1 | no | — |
| 92 | C22⋊Dic53 | D8 | 1 | 1 | yes | characteristic |
| 93 | C22⋊Dic53 | D8 | 1 | 1 | yes | characteristic |
| 94 | #11719924 | — | 8 | 1 | no | — |
| 95 | #11719921 | — | 4 | 1 | no | — |
| 96 | #11719921 | — | 4 | 1 | no | — |
| 97 | #11719921 | — | 4 | 1 | no | — |
| 98 | #11719921 | — | 4 | 1 | no | — |
| 99 | C4⋊Dic53 | — | 2 | 1 | no | — |
| 100 | C4⋊Dic53 | — | 2 | 1 | no | — |
| 101 | C4⋊Dic53 | — | 4 | 1 | no | — |
| 102 | #11719919 | — | 4 | 1 | no | — |
| 103 | C4×Dic53 | — | 2 | 1 | no | — |
| 104 | C4×Dic53 | — | 4 | 1 | no | — |
| 105 | C2×#88322 | — | 2 | 1 | no | — |
| 106 | C2×#88322 | — | 4 | 1 | no | — |
| 107 | Dic212 | — | 2 | 1 | no | — |
| 108 | Dic212 | — | 2 | 1 | no | — |
| 109 | D424 | — | 2 | 1 | no | — |
| 110 | D424 | — | 2 | 1 | no | — |
| 111 | D424 | — | 4 | 1 | no | — |
| 112 | #11719913 | — | 4 | 1 | no | — |
| 113 | #11719913 | — | 4 | 1 | no | — |
| 114 | C16×C53 | — | 4 | 1 | no | — |
| 115 | #11719908 | — | 4 | 1 | no | — |
| 116 | C23×C53 | C2×D8 | 1 | 1 | yes | characteristic |
| 117 | C22×D53 | — | 4 | 1 | no | — |
| 118 | C22×D53 | — | 4 | 1 | no | — |
| 119 | Q8×C53 | — | 4 | 1 | no | — |
| 120 | Q8×C53 | — | 4 | 1 | no | — |
| 121 | C2×C4×C53 | — | 2 | 1 | no | — |
| 122 | C2×C4×C53 | — | 2 | 1 | no | — |
| 123 | C2×C4×C53 | — | 4 | 1 | no | — |
| 124 | C2×C4×C53 | — | 4 | 1 | no | — |
| 125 | C2×C4×C53 | — | 8 | 1 | no | — |
| 126 | C2×C4×C53 | #120 | 1 | 1 | yes | characteristic |
| 127 | C2×C4×C53 | AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 128 | C22⋊D53 | — | 8 | 1 | no | — |
| 129 | C22⋊D53 | — | 8 | 1 | no | — |
| 130 | C22⋊D53 | — | 16 | 1 | no | — |
| 131 | C2×Dic53 | — | 2 | 1 | no | — |
| 132 | C2×Dic53 | — | 2 | 1 | no | — |
| 133 | C2×Dic53 | — | 4 | 1 | no | — |
| 134 | C2×Dic53 | — | 4 | 1 | no | — |
| 135 | D212 | — | 4 | 1 | no | — |
| 136 | D212 | — | 4 | 1 | no | — |
| 137 | D212 | — | 4 | 1 | no | — |
| 138 | D212 | — | 4 | 1 | no | — |
| 139 | C4×D53 | — | 8 | 1 | no | — |
| 140 | C4×D53 | — | 8 | 1 | no | — |
| 141 | C4×D53 | — | 16 | 1 | no | — |
| 142 | Dic106 | — | 4 | 1 | no | — |
| 143 | Dic106 | — | 4 | 1 | no | — |
| 144 | Dic106 | — | 16 | 1 | no | — |
| 145 | C8×C53 | — | 2 | 1 | no | — |
| 146 | C8×C53 | — | 2 | 1 | no | — |
| 147 | #88322 | — | 4 | 1 | no | — |
| 148 | #12172 | — | 53 | 1 | no | maximal, Sylow |
| 149 | C22×C53 | — | 2 | 1 | no | — |
| 150 | C22×C53 | — | 4 | 1 | no | — |
| 151 | C22×C53 | #447 | 1 | 1 | yes | characteristic |
| 152 | C2×D53 | — | 8 | 1 | no | — |
| 153 | C2×D53 | — | 8 | 1 | no | — |
| 154 | C2×D53 | — | 8 | 1 | no | — |
| 155 | C2×D53 | — | 8 | 1 | no | — |
| 156 | C2×D53 | — | 16 | 1 | no | — |
| 157 | C4×C53 | — | 2 | 1 | no | — |
| 158 | C4×C53 | — | 8 | 1 | no | — |
| 159 | C4×C53 | — | 8 | 1 | no | — |
| 160 | C4×C53 | — | 16 | 1 | no | — |
| 161 | C4×C53 | #454 | 1 | 1 | yes | characteristic |
| 162 | C4×C53 | B(4,𝔽2) | 1 | 1 | yes | characteristic |
| 163 | Dic53 | — | 4 | 1 | no | — |
| 164 | Dic53 | — | 4 | 1 | no | — |
| 165 | Dic53 | — | 8 | 1 | no | — |
| 166 | Dic53 | — | 8 | 1 | no | — |
| 167 | #3263 | — | 53 | 1 | no | — |
| 168 | #2191 | — | 53 | 1 | no | — |
| 169 | #2190 | — | 53 | 1 | no | — |
| 170 | #2053 | — | 53 | 1 | no | — |
| 171 | #1340 | D53 | 1 | 1 | yes | characteristic, complemented |
| 172 | #1339 | — | 53 | 1 | no | — |
| 173 | #1328 | — | 53 | 1 | no | — |
| 174 | C2×C53 | — | 2 | 1 | no | — |
| 175 | C2×C53 | — | 4 | 1 | no | — |
| 176 | C2×C53 | #1598 | 1 | 1 | yes | characteristic |
| 177 | D53 | — | 16 | 1 | no | — |
| 178 | D53 | — | 16 | 1 | no | — |
| 179 | #575 | — | 53 | 1 | no | — |
| 180 | #542 | — | 106 | 1 | no | — |
| 181 | #510 | — | 106 | 1 | no | — |
| 182 | #509 | — | 106 | 1 | no | — |
| 183 | #476 | — | 106 | 1 | no | — |
| 184 | #475 | — | 106 | 1 | no | — |
| 185 | #470 | C2×D53 | 1 | 1 | yes | characteristic |
| 186 | #469 | — | 53 | 1 | no | — |
| 187 | #465 | — | 106 | 1 | no | — |
| 188 | #464 | — | 106 | 1 | no | — |
| 189 | #463 | — | 106 | 1 | no | — |
| 190 | #424 | — | 53 | 1 | no | — |
| 191 | #416 | — | 106 | 1 | no | — |
| 192 | #410 | — | 106 | 1 | no | — |
| 193 | #360 | — | 106 | 1 | no | — |
| 194 | #360 | — | 106 | 1 | no | — |
| 195 | #349 | — | 53 | 1 | no | — |
| 196 | #349 | C2×D53 | 1 | 1 | yes | characteristic |
| 197 | H(3,ℤ/4) | C2×D53 | 1 | 1 | yes | characteristic |
| 198 | C53 | #12172 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 199 | C2×#41 | — | 106 | 1 | no | — |
| 200 | #137 | — | 212 | 1 | no | — |
| 201 | AGL(1,ℤ/8) | — | 212 | 1 | no | — |
| 202 | #135 | — | 106 | 1 | no | — |
| 203 | #135 | — | 106 | 1 | no | — |
| 204 | C2×Q16 | D212 | 1 | 1 | yes | characteristic |
| 205 | C2×SD16 | — | 106 | 1 | no | — |
| 206 | C2×D8 | — | 53 | 1 | no | — |
| 207 | C2×M16 | C22×D53 | 1 | 1 | yes | characteristic |
| 208 | C22×C8 | — | 53 | 1 | no | — |
| 209 | C4⋊Q8 | — | 106 | 1 | no | — |
| 210 | #126 | — | 212 | 1 | no | — |
| 211 | #124 | — | 106 | 1 | no | — |
| 212 | #124 | — | 212 | 1 | no | — |
| 213 | #123 | — | 212 | 1 | no | — |
| 214 | C22⋊Q8 | — | 2 | 1 | no | — |
| 215 | C22⋊Q8 | — | 212 | 1 | no | — |
| 216 | #121 | — | 106 | 1 | no | — |
| 217 | C4×Q8 | — | 106 | 1 | no | — |
| 218 | C4×D4 | — | 106 | 1 | no | — |
| 219 | #117 | — | 2 | 1 | no | — |
| 220 | #117 | — | 53 | 1 | no | — |
| 221 | Q32 | — | 212 | 1 | no | — |
| 222 | SD32 | — | 212 | 1 | no | — |
| 223 | SD32 | — | 212 | 1 | no | — |
| 224 | D16 | — | 212 | 1 | no | — |
| 225 | M32 | — | 2 | 1 | no | — |
| 226 | M32 | — | 106 | 1 | no | — |
| 227 | #106 | — | 53 | 1 | no | — |
| 228 | #106 | — | 106 | 1 | no | — |
| 229 | #106 | D212 | 1 | 1 | yes | characteristic |
| 230 | C4⋊C8 | — | 106 | 1 | no | — |
| 231 | C4⋊C8 | — | 106 | 1 | no | — |
| 232 | #103 | — | 2 | 1 | no | — |
| 233 | #103 | — | 106 | 1 | no | — |
| 234 | #103 | — | 212 | 1 | no | — |
| 235 | #102 | — | 106 | 1 | no | — |
| 236 | #102 | — | 106 | 1 | no | — |
| 237 | #102 | — | 212 | 1 | no | — |
| 238 | #99 | — | 212 | 1 | no | — |
| 239 | #99 | — | 212 | 1 | no | — |
| 240 | #41 | — | 212 | 1 | no | — |
| 241 | #41 | — | 212 | 1 | no | — |
| 242 | #41 | — | 212 | 1 | no | — |
| 243 | #41 | — | 212 | 1 | no | — |
| 244 | C2×Q8 | — | 2 | 1 | no | — |
| 245 | C2×Q8 | — | 106 | 1 | no | — |
| 246 | C2×Q8 | — | 212 | 1 | no | — |
| 247 | C2×D4 | — | 106 | 1 | no | — |
| 248 | C2×D4 | — | 106 | 1 | no | — |
| 249 | C2×D4 | — | 212 | 1 | no | — |
| 250 | C2×D4 | — | 212 | 1 | no | — |
| 251 | C22×C4 | — | 212 | 1 | no | — |
| 252 | C22×C4 | — | 212 | 1 | no | — |
| 253 | C22×C4 | D4×D53 | 1 | 1 | yes | characteristic |
| 254 | Q16 | — | 4 | 1 | no | — |
| 255 | Q16 | — | 106 | 1 | no | — |
| 256 | Q16 | — | 106 | 1 | no | — |
| 257 | SD16 | — | 424 | 1 | no | — |
| 258 | SD16 | — | 212 | 1 | no | — |
| 259 | SD16 | — | 212 | 1 | no | — |
| 260 | D8 | — | 106 | 1 | no | — |
| 261 | D8 | — | 106 | 1 | no | — |
| 262 | D8 | — | 212 | 1 | no | — |
| 263 | M16 | — | 4 | 1 | no | — |
| 264 | C2×C8 | — | 106 | 1 | no | — |
| 265 | C2×C8 | — | 212 | 1 | no | — |
| 266 | C2×C8 | C2×D212 | 1 | 1 | yes | characteristic |
| 267 | C2×C8 | D4×D53 | 1 | 1 | yes | characteristic |
| 268 | C4⋊C4 | — | 2 | 1 | no | — |
| 269 | C4⋊C4 | — | 4 | 1 | no | — |
| 270 | C4⋊C4 | — | 4 | 1 | no | — |
| 271 | C4⋊C4 | — | 106 | 1 | no | — |
| 272 | C4⋊C4 | — | 106 | 1 | no | — |
| 273 | C4⋊C4 | — | 212 | 1 | no | — |
| 274 | C4⋊C4 | — | 212 | 1 | no | — |
| 275 | C22⋊C4 | — | 4 | 1 | no | — |
| 276 | C22⋊C4 | — | 4 | 1 | no | — |
| 277 | C22⋊C4 | — | 53 | 1 | no | — |
| 278 | C22⋊C4 | — | 53 | 1 | no | — |
| 279 | C22⋊C4 | — | 212 | 1 | no | — |
| 280 | C22⋊C4 | — | 212 | 1 | no | — |
| 281 | C22⋊C4 | — | 212 | 1 | no | — |
| 282 | C22⋊C4 | — | 212 | 1 | no | — |
| 283 | C42 | — | 4 | 1 | no | — |
| 284 | C42 | — | 106 | 1 | no | — |
| 285 | C42 | — | 212 | 1 | no | — |
| 286 | C16 | — | 4 | 1 | no | — |
| 287 | C16 | — | 212 | 1 | no | — |
| 288 | C23 | — | 212 | 1 | no | — |
| 289 | C23 | — | 212 | 1 | no | — |
| 290 | C23 | D8×D53 | 1 | 1 | yes | characteristic |
| 291 | Q8 | — | 4 | 1 | no | — |
| 292 | Q8 | — | 4 | 1 | no | — |
| 293 | Q8 | — | 848 | 1 | no | — |
| 294 | Q8 | — | 212 | 1 | no | — |
| 295 | Q8 | — | 212 | 1 | no | — |
| 296 | D4 | — | 848 | 1 | no | — |
| 297 | D4 | — | 424 | 1 | no | — |
| 298 | D4 | — | 424 | 1 | no | — |
| 299 | D4 | — | 212 | 1 | no | — |
| 300 | D4 | — | 212 | 1 | no | — |
| 301 | D4 | — | 212 | 1 | no | — |
| 302 | D4 | — | 212 | 1 | no | — |
| 303 | C2×C4 | — | 2 | 1 | no | — |
| 304 | C2×C4 | — | 2 | 1 | no | — |
| 305 | C2×C4 | — | 4 | 1 | no | — |
| 306 | C2×C4 | — | 4 | 1 | no | — |
| 307 | C2×C4 | — | 8 | 1 | no | — |
| 308 | C2×C4 | — | 848 | 1 | no | — |
| 309 | C2×C4 | — | 106 | 1 | no | — |
| 310 | C2×C4 | — | 106 | 1 | no | — |
| 311 | C2×C4 | — | 424 | 1 | no | — |
| 312 | C2×C4 | — | 424 | 1 | no | — |
| 313 | C2×C4 | — | 212 | 1 | no | — |
| 314 | C2×C4 | — | 212 | 1 | no | — |
| 315 | C2×C4 | #13133187 | 1 | 1 | yes | characteristic |
| 316 | C2×C4 | #13133244 | 1 | 1 | yes | characteristic |
| 317 | C8 | — | 2 | 1 | no | — |
| 318 | C8 | — | 2 | 1 | no | — |
| 319 | C8 | — | 212 | 1 | no | — |
| 320 | C22 | — | 2 | 1 | no | — |
| 321 | C22 | — | 4 | 1 | no | — |
| 322 | C22 | — | 848 | 1 | no | — |
| 323 | C22 | — | 424 | 1 | no | — |
| 324 | C22 | — | 424 | 1 | no | — |
| 325 | C22 | — | 424 | 1 | no | — |
| 326 | C22 | — | 424 | 1 | no | — |
| 327 | C22 | #16810758 | 1 | 1 | yes | characteristic |
| 328 | C4 | — | 2 | 1 | no | — |
| 329 | C4 | — | 8 | 1 | no | — |
| 330 | C4 | — | 8 | 1 | no | — |
| 331 | C4 | — | 16 | 1 | no | — |
| 332 | C4 | — | 424 | 1 | no | — |
| 333 | C4 | — | 424 | 1 | no | — |
| 334 | C4 | — | 212 | 1 | no | — |
| 335 | C4 | — | 212 | 1 | no | — |
| 336 | C4 | #16810797 | 1 | 1 | yes | characteristic |
| 337 | C4 | #16810811 | 1 | 1 | yes | characteristic |
| 338 | C2 | — | 2 | 1 | no | — |
| 339 | C2 | — | 4 | 1 | no | — |
| 340 | C2 | — | 848 | 1 | no | — |
| 341 | C2 | — | 848 | 1 | no | — |
| 342 | C2 | #20273651 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #20286181 | C2 | semidirect |
| 2. #20276665 | C2 | semidirect |
| 4. #20276022 | C2 | semidirect |
| 5. #20272699 | C2 | semidirect |
| 6. #20272667 | C2 | semidirect |
| 7. C53×#1340 | C2 | semidirect |
| 171. #1340 | D53 | semidirect |
| 198. C53 | #12172 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #20286181 | 2 |
| #20276665 | 2 |
| #20276663 | 2 |
| #20276022 | 2 |
| #20272699 | 2 |
| #20272667 | 2 |
| C53×#1340 | 2 |
| #12172 | 53 |