Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
906 conjugacy classes of subgroups; only the 167 normal subgroups are shown.
This group has 906 conjugacy classes of subgroups — too many to draw. Only its 167 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 | #20285730 | C2 | 1 | 3 | yes | maximal, complemented |
| 2 | #20274017 | C2 | 1 | 2 | yes | maximal, complemented |
| 3 | #20274008 | C2 | 1 | 6 | yes | maximal, complemented |
| 4 | #20273346 | C2 | 1 | 3 | yes | maximal, complemented |
| 5 | C53×#3567 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | C2×#13133214 | C22 | 1 | 1 | yes | characteristic |
| 7 | #16811013 | C22 | 1 | 6 | yes | — |
| 8 | #16810989 | C22 | 1 | 6 | yes | — |
| 9 | #16810823 | — | 2 | 3 | no | — |
| 10 | #16810818 | — | 2 | 3 | no | — |
| 11 | #16810773 | — | 2 | 3 | no | — |
| 12 | #16810756 | — | 2 | 3 | no | — |
| 13 | #16810680 | C22 | 1 | 3 | yes | — |
| 14 | #16810365 | C22 | 1 | 6 | yes | complemented |
| 15 | #16810363 | C22 | 1 | 6 | yes | complemented |
| 16 | C53×#564 | C22 | 1 | 1 | yes | characteristic, complemented |
| 17 | C53×#562 | C22 | 1 | 3 | yes | complemented |
| 18 | C53×#553 | — | 2 | 3 | no | — |
| 19 | C53×#552 | — | 2 | 3 | no | — |
| 20 | C53×#540 | — | 2 | 3 | no | — |
| 21 | C53×#539 | — | 2 | 3 | no | — |
| 22 | C53×#529 | C22 | 1 | 3 | yes | — |
| 23 | C22×D212 | — | 2 | 3 | no | — |
| 24 | C2×#11719924 | — | 2 | 6 | no | — |
| 25 | C2×#11719922 | — | 2 | 6 | no | — |
| 26 | #13133214 | — | 2 | 3 | no | — |
| 27 | #13133214 | C23 | 1 | 2 | yes | — |
| 28 | C4⋊#88322 | — | 2 | 3 | no | — |
| 29 | #13133137 | — | 2 | 6 | no | — |
| 30 | #13133137 | — | 2 | 6 | no | — |
| 31 | C22⋊#88322 | — | 2 | 3 | no | — |
| 32 | #13133128 | C23 | 1 | 6 | yes | — |
| 33 | #127×C53 | — | 2 | 1 | no | — |
| 34 | #126×C53 | — | 2 | 1 | no | — |
| 35 | #126×C53 | — | 2 | 3 | no | — |
| 36 | #126×C53 | — | 2 | 3 | no | — |
| 37 | #125×C53 | — | 2 | 3 | no | — |
| 38 | #125×C53 | — | 2 | 3 | no | — |
| 39 | #125×C53 | — | 2 | 3 | no | — |
| 40 | #125×C53 | C23 | 1 | 3 | yes | — |
| 41 | #124×C53 | C23 | 1 | 3 | yes | — |
| 42 | #123×C53 | — | 2 | 3 | no | — |
| 43 | #123×C53 | — | 2 | 3 | no | — |
| 44 | #123×C53 | — | 2 | 3 | no | — |
| 45 | #123×C53 | — | 2 | 3 | no | — |
| 46 | #123×C53 | — | 2 | 3 | no | — |
| 47 | C22⋊Q8×C53 | — | 2 | 3 | no | — |
| 48 | C22⋊Q8×C53 | — | 2 | 3 | no | — |
| 49 | C22⋊Q8×C53 | D4 | 1 | 3 | yes | — |
| 50 | #121×C53 | — | 2 | 3 | no | — |
| 51 | #121×C53 | — | 2 | 3 | no | — |
| 52 | #121×C53 | — | 2 | 3 | no | — |
| 53 | #121×C53 | — | 2 | 3 | no | — |
| 54 | #121×C53 | D4 | 1 | 3 | yes | — |
| 55 | C4×Q8×C53 | D4 | 1 | 3 | yes | — |
| 56 | C4×D4×C53 | — | 2 | 3 | no | — |
| 57 | C4×D4×C53 | — | 2 | 3 | no | — |
| 58 | C4×D4×C53 | — | 2 | 3 | no | — |
| 59 | C4×D4×C53 | D4 | 1 | 3 | yes | — |
| 60 | #117×C53 | — | 2 | 3 | no | — |
| 61 | #117×C53 | — | 2 | 3 | no | — |
| 62 | C2×C4⋊C4×C53 | — | 2 | 3 | no | — |
| 63 | C2×C42×C53 | C23 | 1 | 1 | yes | characteristic |
| 64 | C23×D53 | — | 4 | 1 | no | — |
| 65 | C2×Q8×C53 | C2×D4 | 1 | 3 | yes | — |
| 66 | C2×D4×C53 | — | 2 | 3 | no | — |
| 67 | C2×D4×C53 | — | 2 | 3 | no | — |
| 68 | C2×D4×C53 | — | 2 | 3 | no | — |
| 69 | C2×D4×C53 | C2×D4 | 1 | 3 | yes | — |
| 70 | C22×C4×C53 | — | 2 | 3 | no | — |
| 71 | C22×C4×C53 | — | 2 | 3 | no | — |
| 72 | C22×C4×C53 | — | 2 | 3 | no | — |
| 73 | C22×C4×C53 | C2×D4 | 1 | 3 | yes | — |
| 74 | C2×D212 | — | 2 | 6 | no | — |
| 75 | C2×D212 | — | 4 | 3 | no | — |
| 76 | C2×D212 | — | 8 | 6 | no | — |
| 77 | C4⋊C4×C53 | — | 2 | 3 | no | — |
| 78 | C4⋊C4×C53 | — | 2 | 3 | no | — |
| 79 | C4⋊C4×C53 | — | 2 | 3 | no | — |
| 80 | C4⋊C4×C53 | — | 2 | 3 | no | — |
| 81 | C4⋊C4×C53 | — | 2 | 3 | no | — |
| 82 | C4⋊C4×C53 | — | 2 | 3 | no | — |
| 83 | C4⋊C4×C53 | — | 2 | 3 | no | — |
| 84 | C4⋊C4×C53 | — | 2 | 3 | no | — |
| 85 | C4⋊C4×C53 | C2×D4 | 1 | 3 | yes | — |
| 86 | C4⋊C4×C53 | C2×D4 | 1 | 3 | yes | — |
| 87 | C22⋊C4×C53 | — | 2 | 3 | no | — |
| 88 | C22⋊C4×C53 | — | 2 | 3 | no | — |
| 89 | C22⋊C4×C53 | — | 2 | 3 | no | — |
| 90 | C22⋊C4×C53 | — | 2 | 3 | no | — |
| 91 | C22⋊C4×C53 | — | 2 | 3 | no | — |
| 92 | C22⋊C4×C53 | — | 2 | 3 | no | — |
| 93 | C22⋊C4×C53 | — | 2 | 3 | no | — |
| 94 | C42×C53 | — | 2 | 1 | no | — |
| 95 | C42×C53 | — | 2 | 1 | no | — |
| 96 | C42×C53 | — | 2 | 3 | no | — |
| 97 | C42×C53 | C2×D4 | 1 | 3 | yes | — |
| 98 | C42×C53 | C24 | 1 | 1 | yes | characteristic |
| 99 | #11719924 | — | 8 | 6 | no | — |
| 100 | #11719922 | — | 8 | 6 | no | — |
| 101 | C2×#88322 | — | 2 | 6 | no | — |
| 102 | C23×C53 | — | 2 | 3 | no | — |
| 103 | C23×C53 | #120 | 1 | 1 | yes | characteristic |
| 104 | C22×D53 | — | 4 | 2 | no | — |
| 105 | C22×D53 | — | 8 | 6 | no | — |
| 106 | Q8×C53 | — | 4 | 3 | no | — |
| 107 | D4×C53 | — | 4 | 3 | no | — |
| 108 | D4×C53 | — | 8 | 3 | no | — |
| 109 | D4×C53 | — | 8 | 3 | no | — |
| 110 | D4×C53 | — | 8 | 3 | no | — |
| 111 | C2×C4×C53 | — | 2 | 3 | no | — |
| 112 | C2×C4×C53 | — | 2 | 3 | no | — |
| 113 | C2×C4×C53 | — | 2 | 3 | no | — |
| 114 | C2×C4×C53 | — | 4 | 3 | no | — |
| 115 | C2×C4×C53 | — | 8 | 3 | no | — |
| 116 | C2×C4×C53 | — | 8 | 3 | no | — |
| 117 | C2×C4×C53 | — | 8 | 3 | no | — |
| 118 | C2×C4×C53 | #120 | 1 | 3 | yes | — |
| 119 | C2×C4×C53 | C22×D4 | 1 | 3 | yes | — |
| 120 | D212 | — | 4 | 6 | no | — |
| 121 | D212 | — | 4 | 6 | no | — |
| 122 | D212 | — | 8 | 6 | no | — |
| 123 | #88322 | — | 4 | 6 | no | — |
| 124 | #19096 | — | 53 | 1 | no | maximal, Sylow |
| 125 | C22×C53 | — | 2 | 3 | no | — |
| 126 | C22×C53 | — | 4 | 3 | no | — |
| 127 | C22×C53 | — | 4 | 3 | no | — |
| 128 | C22×C53 | — | 4 | 3 | no | — |
| 129 | C22×C53 | C2×#120 | 1 | 1 | yes | characteristic |
| 130 | C2×D53 | — | 8 | 6 | no | — |
| 131 | C2×D53 | — | 16 | 4 | no | — |
| 132 | C4×C53 | — | 2 | 3 | no | — |
| 133 | C4×C53 | — | 2 | 3 | no | — |
| 134 | C4×C53 | — | 4 | 3 | no | — |
| 135 | C4×C53 | — | 4 | 3 | no | — |
| 136 | C4×C53 | — | 4 | 3 | no | — |
| 137 | #3567 | D53 | 1 | 1 | yes | characteristic, complemented |
| 138 | #3217 | — | 53 | 3 | no | — |
| 139 | #1658 | — | 53 | 2 | no | — |
| 140 | #1656 | — | 53 | 6 | no | — |
| 141 | #1535 | — | 53 | 3 | no | — |
| 142 | C2×C53 | — | 4 | 1 | no | — |
| 143 | C2×C53 | — | 8 | 3 | no | — |
| 144 | C2×C53 | #3021 | 1 | 3 | yes | — |
| 145 | D53 | — | 16 | 2 | no | — |
| 146 | #564 | C2×D53 | 1 | 1 | yes | characteristic, complemented |
| 147 | #562 | C2×D53 | 1 | 3 | yes | complemented |
| 148 | #553 | — | 2 | 3 | no | — |
| 149 | #552 | — | 2 | 3 | no | — |
| 150 | #540 | — | 2 | 3 | no | — |
| 151 | #539 | — | 2 | 3 | no | — |
| 152 | C2×#127 | — | 53 | 1 | no | — |
| 153 | #529 | C2×D53 | 1 | 3 | yes | — |
| 154 | #496 | — | 53 | 6 | no | — |
| 155 | #490 | — | 53 | 6 | no | — |
| 156 | #460 | — | 106 | 3 | no | — |
| 157 | #459 | — | 106 | 3 | no | — |
| 158 | #450 | — | 106 | 3 | no | — |
| 159 | #447 | — | 106 | 3 | no | — |
| 160 | #432 | — | 53 | 3 | no | — |
| 161 | #330 | — | 53 | 6 | no | — |
| 162 | #329 | — | 53 | 6 | no | — |
| 163 | C53 | #19096 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 164 | C22×D4 | — | 106 | 3 | no | — |
| 165 | C2×SD16 | — | 106 | 6 | no | — |
| 166 | C2×D8 | — | 106 | 6 | no | — |
| 167 | #127 | — | 2 | 1 | no | — |
| 168 | #127 | — | 53 | 2 | no | — |
| 169 | #127 | — | 106 | 3 | no | — |
| 170 | #126 | — | 2 | 1 | no | — |
| 171 | #126 | — | 2 | 3 | no | — |
| 172 | #126 | — | 2 | 3 | no | — |
| 173 | #125 | — | 2 | 3 | no | — |
| 174 | #125 | — | 2 | 3 | no | — |
| 175 | #125 | — | 2 | 3 | no | — |
| 176 | #125 | C22×D53 | 1 | 3 | yes | — |
| 177 | #124 | C22×D53 | 1 | 3 | yes | — |
| 178 | #123 | — | 2 | 3 | no | — |
| 179 | #123 | — | 2 | 3 | no | — |
| 180 | #123 | — | 2 | 3 | no | — |
| 181 | #123 | — | 2 | 3 | no | — |
| 182 | #123 | — | 2 | 3 | no | — |
| 183 | C22⋊Q8 | — | 2 | 3 | no | — |
| 184 | C22⋊Q8 | — | 2 | 3 | no | — |
| 185 | C22⋊Q8 | C22⋊D53 | 1 | 3 | yes | — |
| 186 | #121 | — | 2 | 3 | no | — |
| 187 | #121 | — | 2 | 3 | no | — |
| 188 | #121 | — | 2 | 3 | no | — |
| 189 | #121 | — | 2 | 3 | no | — |
| 190 | #121 | C22⋊D53 | 1 | 3 | yes | — |
| 191 | C4×Q8 | C22⋊D53 | 1 | 3 | yes | — |
| 192 | C4×D4 | — | 2 | 3 | no | — |
| 193 | C4×D4 | — | 2 | 3 | no | — |
| 194 | C4×D4 | — | 2 | 3 | no | — |
| 195 | C4×D4 | C22⋊D53 | 1 | 3 | yes | — |
| 196 | #117 | — | 2 | 3 | no | — |
| 197 | #117 | — | 2 | 3 | no | — |
| 198 | C2×C4⋊C4 | — | 2 | 3 | no | — |
| 199 | C2×C42 | C22×D53 | 1 | 1 | yes | characteristic |
| 200 | C4⋊C8 | — | 106 | 3 | no | — |
| 201 | #102 | — | 106 | 6 | no | — |
| 202 | #102 | — | 106 | 6 | no | — |
| 203 | C22⋊C8 | — | 106 | 3 | no | — |
| 204 | #97 | — | 53 | 6 | no | — |
| 205 | C24 | — | 212 | 1 | no | — |
| 206 | C2×Q8 | C2×C22⋊D53 | 1 | 3 | yes | — |
| 207 | C2×D4 | — | 2 | 3 | no | — |
| 208 | C2×D4 | — | 2 | 3 | no | — |
| 209 | C2×D4 | — | 2 | 3 | no | — |
| 210 | C2×D4 | — | 106 | 6 | no | — |
| 211 | C2×D4 | — | 424 | 6 | no | — |
| 212 | C2×D4 | — | 212 | 3 | no | — |
| 213 | C2×D4 | C2×C22⋊D53 | 1 | 3 | yes | — |
| 214 | C22×C4 | — | 2 | 3 | no | — |
| 215 | C22×C4 | — | 2 | 3 | no | — |
| 216 | C22×C4 | — | 2 | 3 | no | — |
| 217 | C22×C4 | C2×C22⋊D53 | 1 | 3 | yes | — |
| 218 | SD16 | — | 424 | 6 | no | — |
| 219 | D8 | — | 424 | 6 | no | — |
| 220 | C2×C8 | — | 106 | 6 | no | — |
| 221 | C4⋊C4 | — | 2 | 3 | no | — |
| 222 | C4⋊C4 | — | 2 | 3 | no | — |
| 223 | C4⋊C4 | — | 2 | 3 | no | — |
| 224 | C4⋊C4 | — | 2 | 3 | no | — |
| 225 | C4⋊C4 | — | 2 | 3 | no | — |
| 226 | C4⋊C4 | — | 2 | 3 | no | — |
| 227 | C4⋊C4 | — | 2 | 3 | no | — |
| 228 | C4⋊C4 | — | 2 | 3 | no | — |
| 229 | C4⋊C4 | C2×C22⋊D53 | 1 | 3 | yes | — |
| 230 | C4⋊C4 | C2×C22⋊D53 | 1 | 3 | yes | — |
| 231 | C22⋊C4 | — | 2 | 3 | no | — |
| 232 | C22⋊C4 | — | 2 | 3 | no | — |
| 233 | C22⋊C4 | — | 2 | 3 | no | — |
| 234 | C22⋊C4 | — | 2 | 3 | no | — |
| 235 | C22⋊C4 | — | 2 | 3 | no | — |
| 236 | C22⋊C4 | — | 2 | 3 | no | — |
| 237 | C22⋊C4 | — | 2 | 3 | no | — |
| 238 | C42 | — | 2 | 1 | no | — |
| 239 | C42 | — | 2 | 1 | no | — |
| 240 | C42 | — | 2 | 3 | no | — |
| 241 | C42 | C2×C22⋊D53 | 1 | 3 | yes | — |
| 242 | C42 | C23×D53 | 1 | 1 | yes | characteristic |
| 243 | C23 | — | 2 | 3 | no | — |
| 244 | C23 | — | 424 | 6 | no | — |
| 245 | C23 | — | 212 | 2 | no | — |
| 246 | C23 | #13133186 | 1 | 1 | yes | characteristic |
| 247 | Q8 | — | 4 | 3 | no | — |
| 248 | D4 | — | 4 | 3 | no | — |
| 249 | D4 | — | 8 | 3 | no | — |
| 250 | D4 | — | 8 | 3 | no | — |
| 251 | D4 | — | 8 | 3 | no | — |
| 252 | D4 | — | 424 | 6 | no | — |
| 253 | D4 | — | 212 | 6 | no | — |
| 254 | D4 | — | 212 | 6 | no | — |
| 255 | C2×C4 | — | 2 | 3 | no | — |
| 256 | C2×C4 | — | 2 | 3 | no | — |
| 257 | C2×C4 | — | 2 | 3 | no | — |
| 258 | C2×C4 | — | 4 | 3 | no | — |
| 259 | C2×C4 | — | 8 | 3 | no | — |
| 260 | C2×C4 | — | 8 | 3 | no | — |
| 261 | C2×C4 | — | 8 | 3 | no | — |
| 262 | C2×C4 | #13133186 | 1 | 3 | yes | — |
| 263 | C2×C4 | C22×C22⋊D53 | 1 | 3 | yes | — |
| 264 | C8 | — | 212 | 6 | no | — |
| 265 | C22 | — | 2 | 3 | no | — |
| 266 | C22 | — | 4 | 3 | no | — |
| 267 | C22 | — | 4 | 3 | no | — |
| 268 | C22 | — | 4 | 3 | no | — |
| 269 | C22 | — | 848 | 4 | no | — |
| 270 | C22 | — | 424 | 6 | no | — |
| 271 | C22 | C2×#13133186 | 1 | 1 | yes | characteristic |
| 272 | C4 | — | 2 | 3 | no | — |
| 273 | C4 | — | 2 | 3 | no | — |
| 274 | C4 | — | 4 | 3 | no | — |
| 275 | C4 | — | 4 | 3 | no | — |
| 276 | C4 | — | 4 | 3 | no | — |
| 277 | C2 | — | 4 | 1 | no | — |
| 278 | C2 | — | 8 | 3 | no | — |
| 279 | C2 | — | 848 | 2 | no | — |
| 280 | C2 | #20284087 | 1 | 3 | yes | — |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #20285730 | C2 | semidirect |
| 2. #20274017 | C2 | semidirect |
| 3. #20274008 | C2 | semidirect |
| 4. #20273346 | C2 | semidirect |
| 5. C53×#3567 | C2 | semidirect |
| 14. #16810365 | C22 | semidirect |
| 15. #16810363 | C22 | semidirect |
| 16. C53×#564 | C22 | semidirect |
| 17. C53×#562 | C22 | semidirect |
| 137. #3567 | D53 | semidirect |
| 146. #564 | C2×D53 | semidirect |
| 147. #562 | C2×D53 | semidirect |
| 163. C53 | #19096 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #20285730 | 2 |
| #20274017 | 2 |
| #20274008 | 2 |
| #20273346 | 2 |
| C53×#3567 | 2 |
| #19096 | 53 |