Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
1630 conjugacy classes of subgroups; only the 171 normal subgroups are shown.
This group has 1630 conjugacy classes of subgroups — too many to draw. Only its 171 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 | C2×#16811258 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #20285089 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #20284815 | C2 | 1 | 2 | yes | maximal, complemented |
| 4 | #20273895 | C2 | 1 | 8 | yes | maximal, complemented |
| 5 | #20273157 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | C53×#1555 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 7 | C22×D4×D53 | — | 2 | 2 | no | — |
| 8 | #16811258 | — | 2 | 4 | no | — |
| 9 | #16811258 | — | 2 | 2 | no | — |
| 10 | #16811258 | C22 | 1 | 4 | yes | — |
| 11 | #16811180 | C22 | 1 | 1 | yes | characteristic, complemented |
| 12 | C2×#13133188 | — | 2 | 2 | no | — |
| 13 | C2×#13133185 | — | 2 | 2 | no | — |
| 14 | C2×C4×D212 | C22 | 1 | 2 | yes | — |
| 15 | #16810987 | — | 2 | 2 | no | — |
| 16 | #16810969 | C22 | 1 | 4 | yes | complemented |
| 17 | #16810959 | — | 2 | 2 | no | — |
| 18 | #16810904 | C22 | 1 | 4 | yes | complemented |
| 19 | #16810829 | C22 | 1 | 4 | yes | complemented |
| 20 | #16810776 | — | 2 | 2 | no | — |
| 21 | #16810725 | — | 2 | 4 | no | — |
| 22 | #16810628 | — | 2 | 2 | no | — |
| 23 | C4×#11719917 | C22 | 1 | 1 | yes | characteristic, complemented |
| 24 | #16810350 | C22 | 1 | 8 | yes | complemented |
| 25 | C2×#127×C53 | C22 | 1 | 1 | yes | characteristic |
| 26 | C53×#432 | C22 | 1 | 2 | yes | complemented |
| 27 | C53×#331 | C22 | 1 | 4 | yes | complemented |
| 28 | C24×D53 | — | 2 | 2 | no | — |
| 29 | C22×D212 | D4 | 1 | 2 | yes | — |
| 30 | C22×C22⋊D53 | — | 2 | 4 | no | — |
| 31 | C2×D4×D53 | — | 2 | 8 | no | — |
| 32 | C2×D4×D53 | — | 2 | 8 | no | — |
| 33 | C2×D4×D53 | — | 4 | 16 | no | — |
| 34 | C22×C4×D53 | — | 2 | 2 | no | — |
| 35 | C2×#11719913 | — | 2 | 4 | no | — |
| 36 | C2×#11719917 | — | 2 | 1 | no | — |
| 37 | #13133217 | C23 | 1 | 2 | yes | complemented |
| 38 | #13133209 | — | 2 | 1 | no | — |
| 39 | #13133196 | — | 2 | 2 | no | — |
| 40 | #13133188 | — | 2 | 8 | no | — |
| 41 | #13133188 | — | 2 | 8 | no | — |
| 42 | #13133185 | — | 2 | 8 | no | — |
| 43 | #13133185 | — | 2 | 8 | no | — |
| 44 | C4×Dic106 | — | 2 | 2 | no | — |
| 45 | C4×D212 | — | 2 | 2 | no | — |
| 46 | C4×D212 | — | 2 | 2 | no | — |
| 47 | C4×D212 | D4 | 1 | 4 | yes | — |
| 48 | C4×D212 | C23 | 1 | 4 | yes | — |
| 49 | #13133176 | — | 2 | 1 | no | — |
| 50 | C2×C4⋊Dic53 | D4 | 1 | 2 | yes | — |
| 51 | C2×C22⋊Dic53 | — | 2 | 2 | no | — |
| 52 | C2×#11719921 | — | 2 | 2 | no | — |
| 53 | #13133151 | — | 2 | 4 | no | — |
| 54 | #13133142 | — | 2 | 4 | no | — |
| 55 | #13133139 | — | 2 | 4 | no | — |
| 56 | #13133138 | — | 2 | 8 | no | — |
| 57 | #13133128 | — | 2 | 1 | no | — |
| 58 | C4×#88322 | C23 | 1 | 2 | yes | complemented |
| 59 | C22×D4×C53 | — | 2 | 2 | no | — |
| 60 | C22×D4×C53 | D4 | 1 | 2 | yes | — |
| 61 | #127×C53 | — | 2 | 2 | no | — |
| 62 | #127×C53 | D4 | 1 | 2 | yes | — |
| 63 | #127×C53 | C23 | 1 | 2 | yes | — |
| 64 | C2×C42×C53 | C23 | 1 | 1 | yes | characteristic |
| 65 | C4⋊C8×C53 | C23 | 1 | 4 | yes | complemented |
| 66 | C22⋊C8×C53 | — | 2 | 2 | no | — |
| 67 | #97×C53 | — | 2 | 2 | no | — |
| 68 | C24×C53 | — | 2 | 2 | no | — |
| 69 | C23×D53 | — | 2 | 2 | no | — |
| 70 | C23×D53 | — | 2 | 8 | no | — |
| 71 | C23×D53 | — | 4 | 16 | no | — |
| 72 | C23×D53 | — | 8 | 4 | no | — |
| 73 | C2×D4×C53 | — | 2 | 4 | no | — |
| 74 | C2×D4×C53 | — | 2 | 2 | no | — |
| 75 | C2×D4×C53 | — | 2 | 4 | no | — |
| 76 | C2×D4×C53 | — | 4 | 8 | no | — |
| 77 | C2×D4×C53 | — | 4 | 4 | no | — |
| 78 | C2×D4×C53 | C2×D4 | 1 | 4 | yes | — |
| 79 | C22×C4×C53 | C2×D4 | 1 | 1 | yes | characteristic |
| 80 | C22×C4×C53 | C2×D4 | 1 | 2 | yes | — |
| 81 | C2×C22⋊D53 | — | 2 | 16 | no | — |
| 82 | C2×C22⋊D53 | — | 4 | 16 | no | — |
| 83 | C22×Dic53 | — | 2 | 2 | no | — |
| 84 | D4×D53 | — | 4 | 32 | no | — |
| 85 | D4×D53 | — | 4 | 16 | no | — |
| 86 | D4×D53 | — | 4 | 16 | no | — |
| 87 | C2×D212 | — | 2 | 2 | no | — |
| 88 | C2×D212 | — | 2 | 8 | no | — |
| 89 | C2×D212 | C2×D4 | 1 | 4 | yes | — |
| 90 | C2×C4×D53 | — | 2 | 4 | no | — |
| 91 | C2×C4×D53 | — | 2 | 4 | no | — |
| 92 | C2×C4×D53 | — | 4 | 8 | no | — |
| 93 | C2×Dic106 | — | 2 | 2 | no | — |
| 94 | C2×C8×C53 | — | 2 | 4 | no | — |
| 95 | C42×C53 | C2×D4 | 1 | 1 | yes | characteristic |
| 96 | C42×C53 | C2×D4 | 1 | 2 | yes | — |
| 97 | C42×C53 | C24 | 1 | 1 | yes | characteristic |
| 98 | C22⋊Dic53 | — | 2 | 8 | no | — |
| 99 | C22⋊Dic53 | — | 4 | 1 | no | — |
| 100 | #11719921 | — | 2 | 4 | no | — |
| 101 | #11719921 | — | 2 | 4 | no | — |
| 102 | C4⋊Dic53 | — | 2 | 2 | no | — |
| 103 | C4⋊Dic53 | — | 2 | 2 | no | — |
| 104 | C4⋊Dic53 | — | 2 | 2 | no | — |
| 105 | C4⋊Dic53 | C2×D4 | 1 | 4 | yes | — |
| 106 | #11719919 | — | 4 | 2 | no | — |
| 107 | C4×Dic53 | — | 4 | 1 | no | — |
| 108 | #11719917 | — | 4 | 2 | no | — |
| 109 | C2×#88322 | — | 2 | 2 | no | — |
| 110 | #11719913 | — | 4 | 8 | no | — |
| 111 | C23×C53 | — | 2 | 4 | no | — |
| 112 | C23×C53 | — | 4 | 8 | no | — |
| 113 | C23×C53 | — | 4 | 4 | no | — |
| 114 | C23×C53 | #120 | 1 | 1 | yes | characteristic |
| 115 | C22×D53 | — | 2 | 8 | no | — |
| 116 | C22×D53 | — | 2 | 4 | no | — |
| 117 | C22×D53 | — | 4 | 32 | no | — |
| 118 | C22×D53 | — | 4 | 4 | no | — |
| 119 | C22×D53 | — | 4 | 4 | no | — |
| 120 | C22×D53 | — | 8 | 8 | no | — |
| 121 | C22×D53 | — | 8 | 32 | no | — |
| 122 | D4×C53 | — | 4 | 4 | no | — |
| 123 | D4×C53 | — | 4 | 8 | no | — |
| 124 | D4×C53 | — | 4 | 8 | no | — |
| 125 | D4×C53 | — | 8 | 2 | no | — |
| 126 | C2×C4×C53 | — | 2 | 2 | no | — |
| 127 | C2×C4×C53 | — | 2 | 4 | no | — |
| 128 | C2×C4×C53 | #120 | 1 | 2 | yes | — |
| 129 | C2×C4×C53 | #120 | 1 | 1 | yes | characteristic |
| 130 | C2×C4×C53 | C22×D4 | 1 | 2 | yes | — |
| 131 | C2×C4×C53 | C22×D4 | 1 | 1 | yes | characteristic |
| 132 | C22⋊D53 | — | 4 | 32 | no | — |
| 133 | C2×Dic53 | — | 2 | 4 | no | — |
| 134 | C2×Dic53 | — | 4 | 4 | no | — |
| 135 | C2×Dic53 | — | 4 | 2 | no | — |
| 136 | D212 | — | 2 | 8 | no | — |
| 137 | D212 | — | 2 | 4 | no | — |
| 138 | D212 | AGL(1,ℤ/8) | 1 | 8 | yes | — |
| 139 | C4×D53 | — | 4 | 8 | no | — |
| 140 | C4×D53 | — | 4 | 8 | no | — |
| 141 | Dic106 | — | 4 | 4 | no | — |
| 142 | C8×C53 | — | 4 | 4 | no | — |
| 143 | #88322 | — | 4 | 2 | no | — |
| 144 | #18529 | — | 53 | 1 | no | maximal, Sylow |
| 145 | C22×C53 | — | 2 | 2 | no | — |
| 146 | C22×C53 | — | 4 | 8 | no | — |
| 147 | C22×C53 | — | 4 | 4 | no | — |
| 148 | C22×C53 | — | 8 | 8 | no | — |
| 149 | C22×C53 | #455 | 1 | 2 | yes | — |
| 150 | C22×C53 | C2×#120 | 1 | 1 | yes | characteristic |
| 151 | C2×D53 | — | 2 | 8 | no | — |
| 152 | C2×D53 | — | 4 | 16 | no | — |
| 153 | C2×D53 | — | 4 | 4 | no | — |
| 154 | C2×D53 | — | 4 | 4 | no | — |
| 155 | C2×D53 | — | 8 | 32 | no | — |
| 156 | C4×C53 | — | 2 | 2 | no | — |
| 157 | C4×C53 | — | 2 | 1 | no | — |
| 158 | C4×C53 | — | 2 | 1 | no | — |
| 159 | C4×C53 | C2×AGL(1,ℤ/8) | 1 | 4 | yes | — |
| 160 | Dic53 | — | 4 | 4 | no | — |
| 161 | Dic53 | — | 8 | 2 | no | — |
| 162 | C2×D42 | — | 53 | 1 | no | — |
| 163 | #3138 | — | 53 | 1 | no | — |
| 164 | #3110 | — | 53 | 2 | no | — |
| 165 | #1639 | — | 53 | 8 | no | — |
| 166 | #1555 | D53 | 1 | 1 | yes | characteristic, complemented |
| 167 | #1490 | — | 53 | 2 | no | — |
| 168 | C2×C53 | — | 2 | 2 | no | — |
| 169 | C2×C53 | — | 8 | 4 | no | — |
| 170 | C2×C53 | #3003 | 1 | 2 | yes | — |
| 171 | C2×C53 | C2×#455 | 1 | 1 | yes | characteristic |
| 172 | D53 | — | 4 | 8 | no | — |
| 173 | C23×D4 | — | 106 | 2 | no | — |
| 174 | D42 | — | 53 | 4 | no | — |
| 175 | D42 | — | 106 | 4 | no | — |
| 176 | D42 | — | 106 | 2 | no | — |
| 177 | #533 | — | 53 | 1 | no | — |
| 178 | C2×#127 | C2×D53 | 1 | 1 | yes | characteristic |
| 179 | C2×#121 | — | 106 | 2 | no | — |
| 180 | C2×#120 | — | 106 | 2 | no | — |
| 181 | C2×C4×D4 | — | 53 | 2 | no | — |
| 182 | #490 | — | 106 | 2 | no | — |
| 183 | #486 | — | 53 | 4 | no | — |
| 184 | #484 | — | 106 | 2 | no | — |
| 185 | #474 | — | 53 | 4 | no | — |
| 186 | #461 | — | 53 | 4 | no | — |
| 187 | #450 | — | 106 | 2 | no | — |
| 188 | #440 | — | 106 | 4 | no | — |
| 189 | #432 | C2×D53 | 1 | 2 | yes | complemented |
| 190 | #418 | — | 106 | 2 | no | — |
| 191 | C4×M16 | — | 53 | 1 | no | — |
| 192 | #331 | C2×D53 | 1 | 4 | yes | complemented |
| 193 | D4⋊C8 | — | 53 | 8 | no | — |
| 194 | C53 | #18529 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 195 | C25 | — | 106 | 2 | no | — |
| 196 | C22×D4 | — | 2 | 2 | no | — |
| 197 | C22×D4 | — | 53 | 2 | no | — |
| 198 | C22×D4 | — | 106 | 8 | no | — |
| 199 | C22×D4 | — | 106 | 8 | no | — |
| 200 | C22×D4 | — | 106 | 4 | no | — |
| 201 | C22×D4 | — | 212 | 16 | no | — |
| 202 | C22×D4 | D212 | 1 | 2 | yes | — |
| 203 | C23×C4 | — | 106 | 2 | no | — |
| 204 | C2×SD16 | — | 106 | 4 | no | — |
| 205 | C2×M16 | — | 106 | 1 | no | — |
| 206 | C4⋊Q8 | — | 53 | 2 | no | — |
| 207 | #127 | — | 2 | 2 | no | — |
| 208 | #127 | D212 | 1 | 2 | yes | — |
| 209 | #127 | C22×D53 | 1 | 2 | yes | — |
| 210 | #125 | — | 106 | 1 | no | — |
| 211 | C22⋊Q8 | — | 106 | 2 | no | — |
| 212 | #121 | — | 106 | 8 | no | — |
| 213 | #121 | — | 106 | 8 | no | — |
| 214 | #120 | — | 106 | 8 | no | — |
| 215 | #120 | — | 106 | 8 | no | — |
| 216 | C4×Q8 | — | 106 | 2 | no | — |
| 217 | C4×D4 | — | 53 | 4 | no | — |
| 218 | C4×D4 | — | 53 | 4 | no | — |
| 219 | C4×D4 | — | 106 | 2 | no | — |
| 220 | C4×D4 | — | 106 | 2 | no | — |
| 221 | #117 | — | 106 | 1 | no | — |
| 222 | C2×C4⋊C4 | — | 53 | 2 | no | — |
| 223 | C2×C22⋊C4 | — | 106 | 2 | no | — |
| 224 | C2×C22⋊C4 | — | 106 | 2 | no | — |
| 225 | C2×C42 | C22×D53 | 1 | 1 | yes | characteristic |
| 226 | #107 | — | 106 | 4 | no | — |
| 227 | C4⋊C8 | C22×D53 | 1 | 4 | yes | complemented |
| 228 | #103 | — | 106 | 4 | no | — |
| 229 | #102 | — | 106 | 4 | no | — |
| 230 | #102 | — | 106 | 8 | no | — |
| 231 | C22⋊C8 | — | 2 | 2 | no | — |
| 232 | #97 | — | 2 | 2 | no | — |
| 233 | #97 | — | 106 | 1 | no | — |
| 234 | C4×C8 | — | 53 | 2 | no | — |
| 235 | C24 | — | 2 | 2 | no | — |
| 236 | C24 | — | 106 | 8 | no | — |
| 237 | C24 | — | 106 | 2 | no | — |
| 238 | C24 | — | 424 | 4 | no | — |
| 239 | C24 | — | 212 | 16 | no | — |
| 240 | C2×Q8 | — | 106 | 2 | no | — |
| 241 | C2×D4 | — | 2 | 4 | no | — |
| 242 | C2×D4 | — | 2 | 2 | no | — |
| 243 | C2×D4 | — | 2 | 4 | no | — |
| 244 | C2×D4 | — | 4 | 8 | no | — |
| 245 | C2×D4 | — | 4 | 4 | no | — |
| 246 | C2×D4 | — | 53 | 4 | no | — |
| 247 | C2×D4 | — | 106 | 8 | no | — |
| 248 | C2×D4 | — | 106 | 16 | no | — |
| 249 | C2×D4 | — | 106 | 2 | no | — |
| 250 | C2×D4 | — | 212 | 16 | no | — |
| 251 | C2×D4 | — | 212 | 32 | no | — |
| 252 | C2×D4 | — | 212 | 16 | no | — |
| 253 | C2×D4 | — | 212 | 16 | no | — |
| 254 | C2×D4 | C2×D212 | 1 | 4 | yes | — |
| 255 | C22×C4 | — | 106 | 2 | no | — |
| 256 | C22×C4 | — | 106 | 4 | no | — |
| 257 | C22×C4 | — | 106 | 4 | no | — |
| 258 | C22×C4 | — | 212 | 8 | no | — |
| 259 | C22×C4 | C2×D212 | 1 | 1 | yes | characteristic |
| 260 | C22×C4 | D4×D53 | 1 | 2 | yes | — |
| 261 | SD16 | — | 212 | 8 | no | — |
| 262 | M16 | — | 212 | 2 | no | — |
| 263 | C2×C8 | — | 2 | 4 | no | — |
| 264 | C2×C8 | — | 106 | 2 | no | — |
| 265 | C4⋊C4 | — | 53 | 4 | no | — |
| 266 | C4⋊C4 | — | 106 | 2 | no | — |
| 267 | C4⋊C4 | — | 106 | 2 | no | — |
| 268 | C4⋊C4 | — | 106 | 2 | no | — |
| 269 | C4⋊C4 | — | 212 | 2 | no | — |
| 270 | C22⋊C4 | — | 106 | 8 | no | — |
| 271 | C22⋊C4 | — | 106 | 4 | no | — |
| 272 | C22⋊C4 | — | 106 | 4 | no | — |
| 273 | C22⋊C4 | — | 212 | 1 | no | — |
| 274 | C42 | — | 212 | 1 | no | — |
| 275 | C42 | C2×D212 | 1 | 1 | yes | characteristic |
| 276 | C42 | D4×D53 | 1 | 2 | yes | — |
| 277 | C42 | C23×D53 | 1 | 1 | yes | characteristic |
| 278 | C23 | — | 2 | 4 | no | — |
| 279 | C23 | — | 4 | 8 | no | — |
| 280 | C23 | — | 4 | 4 | no | — |
| 281 | C23 | — | 106 | 8 | no | — |
| 282 | C23 | — | 106 | 4 | no | — |
| 283 | C23 | — | 424 | 8 | no | — |
| 284 | C23 | — | 424 | 32 | no | — |
| 285 | C23 | — | 212 | 4 | no | — |
| 286 | C23 | — | 212 | 32 | no | — |
| 287 | C23 | — | 212 | 4 | no | — |
| 288 | C23 | #13133187 | 1 | 1 | yes | characteristic |
| 289 | Q8 | — | 212 | 4 | no | — |
| 290 | D4 | — | 4 | 4 | no | — |
| 291 | D4 | — | 4 | 8 | no | — |
| 292 | D4 | — | 4 | 8 | no | — |
| 293 | D4 | — | 8 | 2 | no | — |
| 294 | D4 | — | 53 | 8 | no | — |
| 295 | D4 | — | 106 | 8 | no | — |
| 296 | D4 | — | 106 | 4 | no | — |
| 297 | D4 | — | 212 | 32 | no | — |
| 298 | C2×C4 | — | 2 | 2 | no | — |
| 299 | C2×C4 | — | 2 | 4 | no | — |
| 300 | C2×C4 | — | 106 | 4 | no | — |
| 301 | C2×C4 | — | 212 | 8 | no | — |
| 302 | C2×C4 | — | 212 | 2 | no | — |
| 303 | C2×C4 | — | 212 | 4 | no | — |
| 304 | C2×C4 | — | 212 | 8 | no | — |
| 305 | C2×C4 | #13133187 | 1 | 1 | yes | characteristic |
| 306 | C2×C4 | #13133187 | 1 | 2 | yes | — |
| 307 | C2×C4 | C2×D4×D53 | 1 | 2 | yes | — |
| 308 | C2×C4 | C22×D212 | 1 | 1 | yes | characteristic |
| 309 | C8 | — | 4 | 4 | no | — |
| 310 | C8 | — | 212 | 2 | no | — |
| 311 | C22 | — | 2 | 2 | no | — |
| 312 | C22 | — | 4 | 8 | no | — |
| 313 | C22 | — | 4 | 4 | no | — |
| 314 | C22 | — | 8 | 8 | no | — |
| 315 | C22 | — | 106 | 8 | no | — |
| 316 | C22 | — | 424 | 32 | no | — |
| 317 | C22 | — | 212 | 16 | no | — |
| 318 | C22 | — | 212 | 4 | no | — |
| 319 | C22 | — | 212 | 4 | no | — |
| 320 | C22 | #16810803 | 1 | 2 | yes | — |
| 321 | C22 | C2×#13133187 | 1 | 1 | yes | characteristic |
| 322 | C4 | — | 2 | 2 | no | — |
| 323 | C4 | — | 2 | 1 | no | — |
| 324 | C4 | — | 2 | 1 | no | — |
| 325 | C4 | — | 424 | 2 | no | — |
| 326 | C4 | — | 212 | 4 | no | — |
| 327 | C4 | AGL(1,ℤ/8)×D53 | 1 | 4 | yes | — |
| 328 | C2 | — | 2 | 2 | no | — |
| 329 | C2 | — | 8 | 4 | no | — |
| 330 | C2 | — | 212 | 8 | no | — |
| 331 | C2 | #20283944 | 1 | 2 | yes | — |
| 332 | C2 | C2×#16810803 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #20285089 | C2 | semidirect |
| 3. #20284815 | C2 | semidirect |
| 4. #20273895 | C2 | semidirect |
| 5. #20273157 | C2 | semidirect |
| 6. C53×#1555 | C2 | semidirect |
| 11. #16811180 | C22 | semidirect |
| 16. #16810969 | C22 | semidirect |
| 18. #16810904 | C22 | semidirect |
| 19. #16810829 | C22 | semidirect |
| 23. C4×#11719917 | C22 | semidirect |
| 24. #16810350 | C22 | semidirect |
| 26. C53×#432 | C22 | semidirect |
| 27. C53×#331 | C22 | semidirect |
| 37. #13133217 | C23 | semidirect |
| 58. C4×#88322 | C23 | semidirect |
| 65. C4⋊C8×C53 | C23 | semidirect |
| 166. #1555 | D53 | semidirect |
| 189. #432 | C2×D53 | semidirect |
| 192. #331 | C2×D53 | semidirect |
| 194. C53 | #18529 | semidirect |
| 227. C4⋊C8 | C22×D53 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#16811258 | 2 |
| #20285089 | 2 |
| #20284815 | 2 |
| #20273895 | 2 |
| #20273157 | 2 |
| C53×#1555 | 2 |
| #18529 | 53 |