Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
344 conjugacy classes of subgroups; only the 54 normal subgroups are shown.
This group has 344 conjugacy classes of subgroups — too many to draw. Only its 54 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 | #20291015 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #20290991 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #20289861 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #20289860 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | #20289843 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | #20289843 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 7 | #20285766 | C2 | 1 | 1 | yes | maximal, characteristic |
| 8 | #16811610 | — | 2 | 1 | no | — |
| 9 | D53⋊#99 | — | 2 | 1 | no | — |
| 10 | #16811532 | C22 | 1 | 1 | yes | characteristic |
| 11 | #16811532 | C22 | 1 | 1 | yes | characteristic |
| 12 | #16811531 | — | 2 | 1 | no | — |
| 13 | #16811530 | C22 | 1 | 1 | yes | characteristic |
| 14 | #16811530 | C22 | 1 | 1 | yes | characteristic |
| 15 | #16811527 | — | 2 | 1 | no | — |
| 16 | #16811229 | C4 | 1 | 1 | yes | characteristic, complemented |
| 17 | #16811186 | C22 | 1 | 1 | yes | characteristic |
| 18 | #16811012 | C22 | 1 | 1 | yes | characteristic |
| 19 | #16810999 | C4 | 1 | 1 | yes | characteristic, complemented |
| 20 | #16810980 | C22 | 1 | 1 | yes | characteristic |
| 21 | #16810837 | — | 2 | 1 | no | — |
| 22 | #16810817 | — | 2 | 1 | no | — |
| 23 | #16810769 | — | 2 | 1 | no | — |
| 24 | #16810758 | — | 2 | 1 | no | — |
| 25 | #16810685 | C4 | 1 | 1 | yes | characteristic, complemented |
| 26 | C53×#490 | C4 | 1 | 1 | yes | characteristic, complemented |
| 27 | C2×#11719940 | — | 4 | 1 | no | — |
| 28 | C2×C22⋊#6157 | — | 4 | 1 | no | — |
| 29 | #13133278 | — | 2 | 1 | no | — |
| 30 | #13133278 | — | 2 | 1 | no | — |
| 31 | #13133277 | — | 4 | 1 | no | — |
| 32 | #13133276 | — | 2 | 1 | no | — |
| 33 | #13133276 | — | 2 | 1 | no | — |
| 34 | #13133275 | — | 4 | 1 | no | — |
| 35 | C2×#11719947 | D4 | 1 | 1 | yes | characteristic |
| 36 | C2×D4×D53 | D4 | 1 | 1 | yes | characteristic |
| 37 | C2×#11719913 | — | 2 | 1 | no | — |
| 38 | C2×D424 | — | 2 | 1 | no | — |
| 39 | C2×#11719922 | — | 2 | 1 | no | — |
| 40 | C2×#11719922 | — | 2 | 1 | no | — |
| 41 | #13133214 | C2×C4 | 1 | 1 | yes | characteristic, complemented |
| 42 | #13133213 | D4 | 1 | 1 | yes | characteristic |
| 43 | #13133205 | C23 | 1 | 1 | yes | characteristic |
| 44 | #13133204 | D4 | 1 | 1 | yes | characteristic |
| 45 | #13133202 | — | 4 | 1 | no | — |
| 46 | #13133192 | — | 2 | 1 | no | — |
| 47 | #13133192 | — | 4 | 1 | no | — |
| 48 | #13133189 | — | 4 | 1 | no | — |
| 49 | #13133185 | — | 4 | 1 | no | — |
| 50 | #13133179 | — | 2 | 1 | no | — |
| 51 | #13133174 | C2×C4 | 1 | 1 | yes | characteristic |
| 52 | #13133146 | — | 2 | 1 | no | — |
| 53 | #13133140 | — | 2 | 1 | no | — |
| 54 | #13133137 | — | 2 | 1 | no | — |
| 55 | #13133129 | — | 2 | 1 | no | — |
| 56 | C4×#88322 | C2×C4 | 1 | 1 | yes | characteristic, complemented |
| 57 | #127×C53 | C2×C4 | 1 | 1 | yes | characteristic |
| 58 | #125×C53 | C2×C4 | 1 | 1 | yes | characteristic, complemented |
| 59 | #102×C53 | — | 2 | 1 | no | — |
| 60 | #102×C53 | — | 2 | 1 | no | — |
| 61 | #97×C53 | C2×C4 | 1 | 1 | yes | characteristic, complemented |
| 62 | C23×D53 | — | 2 | 1 | no | — |
| 63 | C22×#6157 | — | 8 | 1 | no | — |
| 64 | C2×D4×C53 | — | 4 | 1 | no | — |
| 65 | C2×D4×C53 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 66 | C2×D4×C53 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 67 | C2×C22⋊D53 | — | 2 | 1 | no | — |
| 68 | C2×C22⋊D53 | — | 2 | 1 | no | — |
| 69 | C2×C22⋊D53 | — | 4 | 1 | no | — |
| 70 | C22×Dic53 | — | 2 | 1 | no | — |
| 71 | #11719947 | — | 8 | 1 | no | — |
| 72 | D4×D53 | — | 8 | 1 | no | — |
| 73 | C2×D212 | — | 2 | 1 | no | — |
| 74 | C2×D212 | — | 4 | 1 | no | — |
| 75 | C2×D212 | — | 4 | 1 | no | — |
| 76 | C2×D212 | C2×D4 | 1 | 1 | yes | characteristic |
| 77 | C2×C4×D53 | — | 4 | 1 | no | — |
| 78 | C2×C4×D53 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 79 | C2×Dic106 | C2×D4 | 1 | 1 | yes | characteristic |
| 80 | C22⋊#6157 | — | 4 | 1 | no | — |
| 81 | C22⋊#6157 | — | 4 | 1 | no | — |
| 82 | C22⋊#6157 | — | 8 | 1 | no | — |
| 83 | #11719940 | — | 4 | 1 | no | — |
| 84 | #11719940 | — | 4 | 1 | no | — |
| 85 | #11719940 | — | 8 | 1 | no | — |
| 86 | C2×#88324 | — | 8 | 1 | no | — |
| 87 | C2×C8×C53 | — | 2 | 1 | no | — |
| 88 | C4⋊C4×C53 | — | 2 | 1 | no | — |
| 89 | C4⋊C4×C53 | — | 4 | 1 | no | — |
| 90 | C42×C53 | C22×C4 | 1 | 1 | yes | characteristic |
| 91 | C22⋊Dic53 | — | 4 | 1 | no | — |
| 92 | #11719922 | — | 8 | 1 | no | — |
| 93 | #11719922 | — | 8 | 1 | no | — |
| 94 | #11719921 | — | 2 | 1 | no | — |
| 95 | #11719921 | — | 4 | 1 | no | — |
| 96 | #11719921 | — | 4 | 1 | no | — |
| 97 | #11719921 | — | 4 | 1 | no | — |
| 98 | #11719919 | — | 2 | 1 | no | — |
| 99 | C4×Dic53 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 100 | C2×#88322 | — | 2 | 1 | no | — |
| 101 | D424 | — | 8 | 1 | no | — |
| 102 | #11719913 | — | 8 | 1 | no | — |
| 103 | C23×C53 | — | 2 | 1 | no | — |
| 104 | C23×C53 | — | 2 | 1 | no | — |
| 105 | C22×D53 | — | 2 | 1 | no | — |
| 106 | C22×D53 | — | 4 | 1 | no | — |
| 107 | C22×D53 | — | 4 | 1 | no | — |
| 108 | C22×D53 | — | 4 | 1 | no | — |
| 109 | C22×D53 | — | 16 | 1 | no | — |
| 110 | C22×D53 | #99 | 1 | 1 | yes | characteristic |
| 111 | C2×#6157 | — | 8 | 1 | no | — |
| 112 | C2×#6157 | — | 8 | 1 | no | — |
| 113 | C2×#6157 | — | 16 | 2 | no | — |
| 114 | D4×C53 | — | 4 | 1 | no | — |
| 115 | D4×C53 | — | 4 | 1 | no | — |
| 116 | D4×C53 | — | 16 | 1 | no | — |
| 117 | C2×C4×C53 | — | 2 | 1 | no | — |
| 118 | C2×C4×C53 | — | 4 | 1 | no | — |
| 119 | C2×C4×C53 | C2×C22⋊C4 | 1 | 1 | yes | characteristic |
| 120 | C22⋊D53 | — | 8 | 1 | no | — |
| 121 | C22⋊D53 | — | 8 | 1 | no | — |
| 122 | C22⋊D53 | — | 16 | 1 | no | — |
| 123 | C2×Dic53 | — | 2 | 1 | no | — |
| 124 | C2×Dic53 | — | 2 | 1 | no | — |
| 125 | C2×Dic53 | — | 4 | 1 | no | — |
| 126 | C2×Dic53 | — | 4 | 1 | no | — |
| 127 | C2×Dic53 | #99 | 1 | 1 | yes | characteristic |
| 128 | D212 | — | 4 | 1 | no | — |
| 129 | D212 | — | 4 | 1 | no | — |
| 130 | D212 | — | 4 | 1 | no | — |
| 131 | D212 | — | 16 | 1 | no | — |
| 132 | D212 | — | 16 | 1 | no | — |
| 133 | C4×D53 | — | 4 | 1 | no | — |
| 134 | C4×D53 | — | 16 | 1 | no | — |
| 135 | Dic106 | — | 4 | 1 | no | — |
| 136 | #88324 | — | 8 | 1 | no | — |
| 137 | #88324 | — | 8 | 1 | no | — |
| 138 | C8×C53 | — | 4 | 1 | no | — |
| 139 | #88322 | — | 4 | 1 | no | — |
| 140 | #13605 | — | 53 | 1 | no | maximal, Sylow |
| 141 | C22×C53 | — | 4 | 1 | no | — |
| 142 | C22×C53 | — | 4 | 1 | no | — |
| 143 | C22×C53 | — | 8 | 1 | no | — |
| 144 | C22×C53 | — | 8 | 1 | no | — |
| 145 | C22×C53 | C2×#99 | 1 | 1 | yes | characteristic |
| 146 | C2×D53 | — | 2 | 1 | no | — |
| 147 | C2×D53 | — | 4 | 1 | no | — |
| 148 | C2×D53 | — | 4 | 1 | no | — |
| 149 | C2×D53 | — | 8 | 2 | no | — |
| 150 | C2×D53 | — | 8 | 1 | no | — |
| 151 | C2×D53 | — | 8 | 1 | no | — |
| 152 | C2×D53 | — | 8 | 1 | no | — |
| 153 | C2×D53 | — | 8 | 1 | no | — |
| 154 | C2×D53 | — | 16 | 1 | no | — |
| 155 | #6157 | — | 16 | 1 | no | — |
| 156 | #6157 | — | 16 | 1 | no | — |
| 157 | C4×C53 | — | 2 | 1 | no | — |
| 158 | C4×C53 | — | 4 | 1 | no | — |
| 159 | C4×C53 | — | 8 | 1 | no | — |
| 160 | Dic53 | — | 2 | 1 | no | — |
| 161 | Dic53 | — | 4 | 1 | no | — |
| 162 | Dic53 | — | 4 | 1 | no | — |
| 163 | #3220 | — | 53 | 1 | no | — |
| 164 | #2134 | — | 53 | 1 | no | — |
| 165 | #2127 | — | 53 | 1 | no | — |
| 166 | #1414 | — | 53 | 1 | no | — |
| 167 | #1414 | — | 53 | 1 | no | — |
| 168 | #1406 | — | 53 | 1 | no | — |
| 169 | #1406 | — | 53 | 1 | no | — |
| 170 | C2×C53 | — | 2 | 1 | no | — |
| 171 | C2×C53 | — | 8 | 1 | no | — |
| 172 | C2×C53 | — | 8 | 1 | no | — |
| 173 | C2×C53 | #2127 | 1 | 1 | yes | characteristic |
| 174 | D53 | — | 4 | 1 | no | — |
| 175 | D53 | — | 8 | 1 | no | — |
| 176 | D53 | — | 16 | 1 | no | — |
| 177 | #540 | — | 53 | 1 | no | — |
| 178 | #535 | — | 53 | 1 | no | — |
| 179 | #496 | — | 53 | 1 | no | — |
| 180 | #493 | — | 53 | 1 | no | — |
| 181 | #490 | #6157 | 1 | 1 | yes | characteristic, complemented |
| 182 | #488 | — | 53 | 1 | no | — |
| 183 | #463 | — | 106 | 1 | no | — |
| 184 | #459 | — | 106 | 1 | no | — |
| 185 | #449 | — | 106 | 1 | no | — |
| 186 | #447 | — | 106 | 1 | no | — |
| 187 | #433 | — | 53 | 1 | no | — |
| 188 | #413 | — | 106 | 1 | no | — |
| 189 | C2×#99 | — | 106 | 1 | no | — |
| 190 | #355 | — | 53 | 1 | no | — |
| 191 | #355 | — | 53 | 1 | no | — |
| 192 | #355 | — | 106 | 1 | no | — |
| 193 | #354 | — | 53 | 1 | no | — |
| 194 | #354 | — | 53 | 1 | no | — |
| 195 | #353 | — | 106 | 1 | no | — |
| 196 | C53 | #13605 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 197 | C2×#41 | — | 53 | 1 | no | — |
| 198 | C22×D4 | — | 53 | 1 | no | — |
| 199 | C2×SD16 | — | 106 | 1 | no | — |
| 200 | C2×D8 | — | 106 | 1 | no | — |
| 201 | C2×D8 | — | 106 | 1 | no | — |
| 202 | C2×D8 | — | 106 | 1 | no | — |
| 203 | C2×M16 | — | 212 | 1 | no | — |
| 204 | #127 | — | 53 | 1 | no | — |
| 205 | #127 | — | 53 | 1 | no | — |
| 206 | #127 | C2×#6157 | 1 | 1 | yes | characteristic |
| 207 | #125 | C2×#6157 | 1 | 1 | yes | characteristic, complemented |
| 208 | #124 | — | 53 | 1 | no | — |
| 209 | #124 | — | 53 | 1 | no | — |
| 210 | #123 | — | 212 | 1 | no | — |
| 211 | #121 | — | 106 | 1 | no | — |
| 212 | #121 | — | 212 | 1 | no | — |
| 213 | #121 | — | 212 | 1 | no | — |
| 214 | #120 | — | 212 | 1 | no | — |
| 215 | C4×D4 | — | 106 | 1 | no | — |
| 216 | #117 | — | 53 | 1 | no | — |
| 217 | C2×C22⋊C4 | — | 212 | 1 | no | — |
| 218 | C4⋊C8 | — | 106 | 1 | no | — |
| 219 | #103 | — | 106 | 1 | no | — |
| 220 | #102 | — | 2 | 1 | no | — |
| 221 | #102 | — | 2 | 1 | no | — |
| 222 | #102 | — | 106 | 1 | no | — |
| 223 | #101 | — | 106 | 1 | no | — |
| 224 | #101 | — | 106 | 1 | no | — |
| 225 | #100 | — | 212 | 1 | no | — |
| 226 | #99 | — | 106 | 1 | no | — |
| 227 | #99 | — | 106 | 1 | no | — |
| 228 | #99 | — | 212 | 1 | no | — |
| 229 | C22⋊C8 | — | 106 | 1 | no | — |
| 230 | #97 | C2×#6157 | 1 | 1 | yes | characteristic, complemented |
| 231 | C4×C8 | — | 53 | 1 | no | — |
| 232 | C24 | — | 106 | 1 | no | — |
| 233 | #41 | — | 424 | 1 | no | — |
| 234 | C2×Q8 | — | 53 | 1 | no | — |
| 235 | C2×D4 | — | 4 | 1 | no | — |
| 236 | C2×D4 | — | 53 | 1 | no | — |
| 237 | C2×D4 | — | 106 | 1 | no | — |
| 238 | C2×D4 | — | 106 | 1 | no | — |
| 239 | C2×D4 | — | 106 | 1 | no | — |
| 240 | C2×D4 | — | 424 | 1 | no | — |
| 241 | C2×D4 | — | 212 | 1 | no | — |
| 242 | C2×D4 | — | 212 | 1 | no | — |
| 243 | C2×D4 | — | 212 | 1 | no | — |
| 244 | C2×D4 | C22⋊#6157 | 1 | 1 | yes | characteristic |
| 245 | C2×D4 | C22⋊#6157 | 1 | 1 | yes | characteristic |
| 246 | C22×C4 | — | 53 | 1 | no | — |
| 247 | C22×C4 | — | 106 | 1 | no | — |
| 248 | C22×C4 | — | 424 | 1 | no | — |
| 249 | C22×C4 | — | 212 | 1 | no | — |
| 250 | SD16 | — | 424 | 1 | no | — |
| 251 | D8 | — | 424 | 1 | no | — |
| 252 | D8 | — | 424 | 1 | no | — |
| 253 | D8 | — | 424 | 1 | no | — |
| 254 | M16 | — | 424 | 1 | no | — |
| 255 | M16 | — | 212 | 1 | no | — |
| 256 | M16 | — | 212 | 1 | no | — |
| 257 | C2×C8 | — | 2 | 1 | no | — |
| 258 | C2×C8 | — | 106 | 1 | no | — |
| 259 | C2×C8 | — | 424 | 1 | no | — |
| 260 | C4⋊C4 | — | 2 | 1 | no | — |
| 261 | C4⋊C4 | — | 4 | 1 | no | — |
| 262 | C4⋊C4 | — | 106 | 1 | no | — |
| 263 | C22⋊C4 | — | 106 | 1 | no | — |
| 264 | C22⋊C4 | — | 424 | 1 | no | — |
| 265 | C22⋊C4 | — | 212 | 1 | no | — |
| 266 | C22⋊C4 | — | 212 | 1 | no | — |
| 267 | C22⋊C4 | — | 212 | 1 | no | — |
| 268 | C22⋊C4 | — | 212 | 1 | no | — |
| 269 | C22⋊C4 | — | 212 | 1 | no | — |
| 270 | C22⋊C4 | — | 212 | 1 | no | — |
| 271 | C42 | — | 53 | 1 | no | — |
| 272 | C42 | C22×#6157 | 1 | 1 | yes | characteristic |
| 273 | C23 | — | 2 | 1 | no | — |
| 274 | C23 | — | 2 | 1 | no | — |
| 275 | C23 | — | 53 | 1 | no | — |
| 276 | C23 | — | 848 | 1 | no | — |
| 277 | C23 | — | 106 | 1 | no | — |
| 278 | C23 | — | 212 | 1 | no | — |
| 279 | C23 | — | 212 | 1 | no | — |
| 280 | C23 | — | 212 | 1 | no | — |
| 281 | Q8 | — | 212 | 1 | no | — |
| 282 | D4 | — | 4 | 1 | no | — |
| 283 | D4 | — | 4 | 1 | no | — |
| 284 | D4 | — | 16 | 1 | no | — |
| 285 | D4 | — | 848 | 1 | no | — |
| 286 | D4 | — | 848 | 1 | no | — |
| 287 | D4 | — | 848 | 1 | no | — |
| 288 | D4 | — | 424 | 1 | no | — |
| 289 | D4 | — | 424 | 1 | no | — |
| 290 | D4 | — | 212 | 1 | no | — |
| 291 | D4 | — | 212 | 1 | no | — |
| 292 | D4 | — | 212 | 1 | no | — |
| 293 | C2×C4 | — | 2 | 1 | no | — |
| 294 | C2×C4 | — | 4 | 1 | no | — |
| 295 | C2×C4 | — | 53 | 1 | no | — |
| 296 | C2×C4 | — | 848 | 1 | no | — |
| 297 | C2×C4 | — | 848 | 2 | no | — |
| 298 | C2×C4 | — | 106 | 1 | no | — |
| 299 | C2×C4 | — | 106 | 1 | no | — |
| 300 | C2×C4 | — | 424 | 1 | no | — |
| 301 | C2×C4 | — | 424 | 1 | no | — |
| 302 | C2×C4 | — | 212 | 1 | no | — |
| 303 | C2×C4 | — | 212 | 1 | no | — |
| 304 | C2×C4 | — | 212 | 1 | no | — |
| 305 | C2×C4 | C2×C22⋊#6157 | 1 | 1 | yes | characteristic |
| 306 | C8 | — | 4 | 1 | no | — |
| 307 | C8 | — | 424 | 1 | no | — |
| 308 | C8 | — | 424 | 1 | no | — |
| 309 | C8 | — | 212 | 1 | no | — |
| 310 | C22 | — | 4 | 1 | no | — |
| 311 | C22 | — | 4 | 1 | no | — |
| 312 | C22 | — | 8 | 1 | no | — |
| 313 | C22 | — | 8 | 1 | no | — |
| 314 | C22 | — | 848 | 1 | no | — |
| 315 | C22 | — | 106 | 1 | no | — |
| 316 | C22 | — | 424 | 1 | no | — |
| 317 | C22 | — | 424 | 1 | no | — |
| 318 | C22 | — | 424 | 1 | no | — |
| 319 | C22 | — | 424 | 2 | no | — |
| 320 | C22 | — | 424 | 1 | no | — |
| 321 | C22 | — | 212 | 1 | no | — |
| 322 | C22 | — | 212 | 1 | no | — |
| 323 | C22 | D53⋊#99 | 1 | 1 | yes | characteristic |
| 324 | C4 | — | 2 | 1 | no | — |
| 325 | C4 | — | 4 | 1 | no | — |
| 326 | C4 | — | 8 | 1 | no | — |
| 327 | C4 | — | 848 | 1 | no | — |
| 328 | C4 | — | 848 | 1 | no | — |
| 329 | C4 | — | 106 | 1 | no | — |
| 330 | C4 | — | 212 | 1 | no | — |
| 331 | C4 | — | 212 | 1 | no | — |
| 332 | C2 | — | 2 | 1 | no | — |
| 333 | C2 | — | 8 | 1 | no | — |
| 334 | C2 | — | 8 | 1 | no | — |
| 335 | C2 | — | 848 | 1 | no | — |
| 336 | C2 | — | 424 | 1 | no | — |
| 337 | C2 | — | 212 | 1 | no | — |
| 338 | C2 | #20290993 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #20291015 | C2 | semidirect |
| 2. #20290991 | C2 | semidirect |
| 3. #20289861 | C2 | semidirect |
| 4. #20289860 | C2 | semidirect |
| 5. #20289843 | C2 | semidirect |
| 6. #20289843 | C2 | semidirect |
| 16. #16811229 | C4 | semidirect |
| 19. #16810999 | C4 | semidirect |
| 25. #16810685 | C4 | semidirect |
| 26. C53×#490 | C4 | semidirect |
| 41. #13133214 | C2×C4 | semidirect |
| 56. C4×#88322 | C2×C4 | semidirect |
| 58. #125×C53 | C2×C4 | semidirect |
| 61. #97×C53 | C2×C4 | semidirect |
| 181. #490 | #6157 | semidirect |
| 196. C53 | #13605 | semidirect |
| 207. #125 | C2×#6157 | semidirect |
| 230. #97 | C2×#6157 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #20291015 | 2 |
| #20290991 | 2 |
| #20289861 | 2 |
| #20289860 | 2 |
| #20289843 | 2 |
| #20285766 | 2 |
| #13605 | 53 |