Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
684 conjugacy classes of subgroups; only the 124 normal subgroups are shown.
This group has 684 conjugacy classes of subgroups — too many to draw. Only its 124 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 | #127×#6157 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #20290502 | C2 | 1 | 2 | yes | maximal, complemented |
| 3 | D53⋊#331 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | #20289594 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | #331×D53 | C2 | 1 | 1 | yes | maximal, characteristic |
| 6 | C2×D4×#6157 | — | 2 | 2 | no | — |
| 7 | C2×D4×#6157 | C4 | 1 | 2 | yes | — |
| 8 | D53⋊(C4⋊C8) | C22 | 1 | 2 | yes | — |
| 9 | C2×#13133271 | — | 2 | 2 | no | — |
| 10 | #16811575 | — | 2 | 2 | no | — |
| 11 | #16811575 | C4 | 1 | 2 | yes | — |
| 12 | #16811569 | C22 | 1 | 1 | yes | characteristic |
| 13 | C42×#6157 | C22 | 1 | 1 | yes | characteristic |
| 14 | (C2×#6157)⋊C8 | — | 2 | 2 | no | — |
| 15 | #16811481 | C4 | 1 | 4 | yes | complemented |
| 16 | #127×D53 | C22 | 1 | 1 | yes | characteristic |
| 17 | C4⋊C8×D53 | C22 | 1 | 2 | yes | — |
| 18 | #16810367 | C4 | 1 | 1 | yes | characteristic, complemented |
| 19 | #16810366 | C4 | 1 | 2 | yes | complemented |
| 20 | C53×#331 | C4 | 1 | 1 | yes | characteristic, complemented |
| 21 | C23×#6157 | — | 2 | 2 | no | — |
| 22 | C2×D53⋊C8 | — | 2 | 2 | no | — |
| 23 | D4×#6157 | — | 4 | 4 | no | — |
| 24 | D4×#6157 | — | 4 | 4 | no | — |
| 25 | D4×#6157 | — | 4 | 4 | no | — |
| 26 | C2×C4⋊#6157 | — | 2 | 2 | no | — |
| 27 | C2×C4⋊#6157 | C2×C4 | 1 | 2 | yes | — |
| 28 | C2×C22⋊#6157 | — | 2 | 2 | no | — |
| 29 | C2×C22⋊#6157 | — | 2 | 4 | no | — |
| 30 | C2×C4×#6157 | C2×C4 | 1 | 1 | yes | characteristic |
| 31 | C2×C4×#6157 | C2×C4 | 1 | 1 | yes | characteristic |
| 32 | C2×C4×#6157 | D4 | 1 | 2 | yes | — |
| 33 | C2×C4×#6157 | D4 | 1 | 2 | yes | — |
| 34 | #13133284 | C2×C4 | 1 | 4 | yes | complemented |
| 35 | C4⋊#88324 | — | 2 | 2 | no | — |
| 36 | #13133271 | — | 4 | 4 | no | — |
| 37 | C2×D4×D53 | — | 2 | 2 | no | — |
| 38 | C2×D4×D53 | C2×C4 | 1 | 2 | yes | — |
| 39 | C2×C8×D53 | — | 2 | 2 | no | — |
| 40 | #13133214 | C2×C4 | 1 | 1 | yes | characteristic |
| 41 | #13133213 | — | 2 | 2 | no | — |
| 42 | #13133213 | C2×C4 | 1 | 2 | yes | — |
| 43 | C42×D53 | C23 | 1 | 1 | yes | characteristic |
| 44 | C4⋊#88322 | C2×C4 | 1 | 2 | yes | complemented |
| 45 | #13133146 | — | 2 | 2 | no | — |
| 46 | #127×C53 | C2×C4 | 1 | 1 | yes | characteristic |
| 47 | C4⋊C8×C53 | C2×C4 | 1 | 2 | yes | complemented |
| 48 | C23×D53 | — | 2 | 2 | no | — |
| 49 | C22×#6157 | — | 2 | 2 | no | — |
| 50 | C22×#6157 | — | 4 | 8 | no | — |
| 51 | C22×#6157 | — | 4 | 4 | no | — |
| 52 | C22×#6157 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 53 | C22×#6157 | C22⋊C4 | 1 | 2 | yes | — |
| 54 | C22×#6157 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 55 | C2×D4×C53 | — | 2 | 2 | no | — |
| 56 | C2×D4×C53 | C42 | 1 | 2 | yes | — |
| 57 | C2×C22⋊D53 | — | 2 | 2 | no | — |
| 58 | C2×C22⋊D53 | — | 2 | 4 | no | — |
| 59 | D4×D53 | — | 4 | 2 | no | — |
| 60 | D4×D53 | — | 4 | 2 | no | — |
| 61 | D4×D53 | — | 4 | 2 | no | — |
| 62 | D4×D53 | — | 4 | 2 | no | — |
| 63 | D4×D53 | — | 4 | 2 | no | — |
| 64 | D4×D53 | — | 4 | 2 | no | — |
| 65 | C2×D212 | — | 2 | 2 | no | — |
| 66 | C2×D212 | C42 | 1 | 2 | yes | — |
| 67 | C2×C4×D53 | C22×C4 | 1 | 1 | yes | characteristic |
| 68 | C2×C4×D53 | C2×D4 | 1 | 2 | yes | — |
| 69 | C22⋊#6157 | — | 4 | 4 | no | — |
| 70 | C22⋊#6157 | — | 4 | 8 | no | — |
| 71 | C2×#88324 | — | 2 | 4 | no | — |
| 72 | C4⋊#6157 | — | 4 | 2 | no | — |
| 73 | C4⋊#6157 | — | 4 | 2 | no | — |
| 74 | C4⋊#6157 | — | 4 | 2 | no | — |
| 75 | C4×#6157 | — | 2 | 2 | no | — |
| 76 | C4×#6157 | — | 2 | 2 | no | — |
| 77 | C4×#6157 | — | 2 | 2 | no | — |
| 78 | C4×#6157 | — | 2 | 2 | no | — |
| 79 | C4×#6157 | — | 2 | 2 | no | — |
| 80 | C4×#6157 | — | 4 | 1 | no | — |
| 81 | D53⋊C8 | — | 4 | 2 | no | — |
| 82 | D53⋊C8 | — | 4 | 2 | no | — |
| 83 | C2×C8×C53 | — | 2 | 2 | no | — |
| 84 | C42×C53 | C22×C4 | 1 | 1 | yes | characteristic |
| 85 | C4×Dic53 | C22×C4 | 1 | 1 | yes | characteristic |
| 86 | C4×Dic53 | #41 | 1 | 2 | yes | — |
| 87 | C2×#88322 | — | 2 | 2 | no | — |
| 88 | C8×D53 | — | 4 | 2 | no | — |
| 89 | C8×D53 | — | 4 | 2 | no | — |
| 90 | C23×C53 | — | 2 | 2 | no | — |
| 91 | C22×D53 | — | 2 | 2 | no | — |
| 92 | C22×D53 | — | 4 | 2 | no | — |
| 93 | C22×D53 | — | 4 | 2 | no | — |
| 94 | C22×D53 | — | 4 | 4 | no | — |
| 95 | C22×D53 | — | 4 | 4 | no | — |
| 96 | C22×D53 | C2×C22⋊C4 | 1 | 1 | yes | characteristic |
| 97 | C2×#6157 | — | 2 | 2 | no | — |
| 98 | C2×#6157 | — | 2 | 1 | no | — |
| 99 | C2×#6157 | — | 2 | 2 | no | — |
| 100 | C2×#6157 | — | 2 | 2 | no | — |
| 101 | C2×#6157 | — | 2 | 2 | no | — |
| 102 | C2×#6157 | — | 2 | 2 | no | — |
| 103 | C2×#6157 | — | 4 | 2 | no | — |
| 104 | C2×#6157 | — | 4 | 4 | no | — |
| 105 | C2×#6157 | — | 8 | 8 | no | — |
| 106 | C2×#6157 | #100 | 1 | 2 | yes | — |
| 107 | D4×C53 | — | 4 | 2 | no | — |
| 108 | D4×C53 | — | 4 | 2 | no | — |
| 109 | D4×C53 | — | 4 | 2 | no | — |
| 110 | C2×C4×C53 | C2×C42 | 1 | 1 | yes | characteristic |
| 111 | C2×C4×C53 | C4×D4 | 1 | 2 | yes | — |
| 112 | C22⋊D53 | — | 4 | 4 | no | — |
| 113 | C22⋊D53 | — | 4 | 4 | no | — |
| 114 | C22⋊D53 | — | 4 | 4 | no | — |
| 115 | C2×Dic53 | #117 | 1 | 1 | yes | characteristic |
| 116 | C2×Dic53 | C4×D4 | 1 | 2 | yes | — |
| 117 | D212 | — | 4 | 2 | no | — |
| 118 | D212 | — | 4 | 2 | no | — |
| 119 | D212 | — | 4 | 2 | no | — |
| 120 | C4×D53 | — | 2 | 1 | no | — |
| 121 | C4×D53 | — | 2 | 1 | no | — |
| 122 | C4×D53 | AGL(1,ℤ/8) | 1 | 2 | yes | — |
| 123 | C4×D53 | AGL(1,ℤ/8) | 1 | 2 | yes | — |
| 124 | C4×D53 | AGL(1,ℤ/8) | 1 | 2 | yes | — |
| 125 | C4×D53 | #137 | 1 | 2 | yes | — |
| 126 | #88324 | — | 4 | 4 | no | — |
| 127 | C8×C53 | — | 4 | 2 | no | — |
| 128 | #88322 | — | 4 | 2 | no | — |
| 129 | #8231 | — | 53 | 1 | no | maximal, Sylow |
| 130 | C22×C53 | — | 4 | 2 | no | — |
| 131 | C22×C53 | — | 4 | 4 | no | — |
| 132 | C22×C53 | C4×C22⋊C4 | 1 | 1 | yes | characteristic |
| 133 | C2×D53 | — | 4 | 2 | no | — |
| 134 | C2×D53 | — | 4 | 4 | no | — |
| 135 | C2×D53 | — | 8 | 2 | no | — |
| 136 | C2×D53 | — | 8 | 4 | no | — |
| 137 | C2×D53 | — | 8 | 2 | no | — |
| 138 | C2×D53 | C2×#100 | 1 | 1 | yes | characteristic |
| 139 | C2×D53 | #413 | 1 | 1 | yes | characteristic |
| 140 | C2×D53 | #417 | 1 | 2 | yes | — |
| 141 | C2×D53 | #418 | 1 | 2 | yes | — |
| 142 | #6157 | — | 2 | 2 | no | — |
| 143 | #6157 | — | 4 | 2 | no | — |
| 144 | #6157 | — | 4 | 1 | no | — |
| 145 | #6157 | — | 8 | 2 | no | — |
| 146 | C4×C53 | — | 2 | 1 | no | — |
| 147 | C4×C53 | #440 | 1 | 2 | yes | — |
| 148 | C4×C53 | #442 | 1 | 2 | yes | — |
| 149 | Dic53 | — | 2 | 1 | no | — |
| 150 | Dic53 | #440 | 1 | 2 | yes | — |
| 151 | Dic53 | #442 | 1 | 2 | yes | — |
| 152 | C4×#127 | — | 53 | 1 | no | — |
| 153 | #1767 | — | 53 | 2 | no | — |
| 154 | C2×#331 | — | 53 | 1 | no | — |
| 155 | C2×#331 | — | 53 | 1 | no | — |
| 156 | #1296 | — | 53 | 2 | no | — |
| 157 | C2×C53 | — | 8 | 2 | no | — |
| 158 | C2×C53 | C4×#100 | 1 | 1 | yes | characteristic |
| 159 | C2×C53 | #1762 | 1 | 2 | yes | — |
| 160 | D53 | — | 8 | 2 | no | — |
| 161 | D53 | #1555 | 1 | 1 | yes | characteristic |
| 162 | D53 | #1557 | 1 | 2 | yes | — |
| 163 | D53 | #1560 | 1 | 1 | yes | characteristic |
| 164 | C2×#127 | — | 53 | 1 | no | — |
| 165 | C2×C4×D4 | — | 53 | 2 | no | — |
| 166 | C2×C4×D4 | — | 106 | 2 | no | — |
| 167 | C2×C4⋊C8 | — | 53 | 2 | no | — |
| 168 | C2×C4⋊C8 | — | 53 | 2 | no | — |
| 169 | C2×#97 | — | 106 | 2 | no | — |
| 170 | #390 | — | 53 | 2 | no | — |
| 171 | #390 | — | 106 | 2 | no | — |
| 172 | #384 | — | 53 | 1 | no | — |
| 173 | C43 | — | 53 | 1 | no | — |
| 174 | #336 | — | 106 | 2 | no | — |
| 175 | #331 | — | 53 | 4 | no | — |
| 176 | #331 | — | 53 | 1 | no | — |
| 177 | #331 | — | 53 | 2 | no | — |
| 178 | #331 | #6157 | 1 | 1 | yes | characteristic, complemented |
| 179 | C53 | #8231 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 180 | C22×D4 | — | 53 | 2 | no | — |
| 181 | C22×D4 | — | 106 | 2 | no | — |
| 182 | C23×C4 | — | 106 | 2 | no | — |
| 183 | C22×C8 | — | 106 | 2 | no | — |
| 184 | C22×C8 | — | 106 | 2 | no | — |
| 185 | #127 | — | 53 | 2 | no | — |
| 186 | #127 | — | 53 | 1 | no | — |
| 187 | #127 | — | 106 | 2 | no | — |
| 188 | #127 | C2×#6157 | 1 | 1 | yes | characteristic |
| 189 | C4×D4 | — | 212 | 4 | no | — |
| 190 | C4×D4 | — | 212 | 4 | no | — |
| 191 | C4×D4 | — | 212 | 4 | no | — |
| 192 | C2×C4⋊C4 | — | 53 | 2 | no | — |
| 193 | C2×C4⋊C4 | — | 106 | 2 | no | — |
| 194 | C2×C22⋊C4 | — | 106 | 2 | no | — |
| 195 | C2×C22⋊C4 | — | 106 | 4 | no | — |
| 196 | C2×C42 | — | 53 | 1 | no | — |
| 197 | C2×C42 | — | 53 | 1 | no | — |
| 198 | C2×C42 | — | 53 | 2 | no | — |
| 199 | C2×C42 | — | 53 | 2 | no | — |
| 200 | C2×C42 | — | 53 | 1 | no | — |
| 201 | C4⋊C8 | — | 53 | 4 | no | — |
| 202 | C4⋊C8 | — | 53 | 2 | no | — |
| 203 | C4⋊C8 | — | 106 | 2 | no | — |
| 204 | C4⋊C8 | — | 106 | 2 | no | — |
| 205 | C4⋊C8 | C2×#6157 | 1 | 2 | yes | complemented |
| 206 | #97 | — | 212 | 4 | no | — |
| 207 | C24 | — | 106 | 2 | no | — |
| 208 | C2×D4 | — | 2 | 2 | no | — |
| 209 | C2×D4 | — | 53 | 2 | no | — |
| 210 | C2×D4 | — | 106 | 4 | no | — |
| 211 | C2×D4 | — | 106 | 2 | no | — |
| 212 | C2×D4 | — | 106 | 2 | no | — |
| 213 | C2×D4 | — | 212 | 2 | no | — |
| 214 | C2×D4 | — | 212 | 2 | no | — |
| 215 | C2×D4 | — | 212 | 2 | no | — |
| 216 | C2×D4 | — | 212 | 2 | no | — |
| 217 | C2×D4 | — | 212 | 2 | no | — |
| 218 | C2×D4 | — | 212 | 2 | no | — |
| 219 | C2×D4 | C4×#6157 | 1 | 2 | yes | — |
| 220 | C22×C4 | — | 53 | 2 | no | — |
| 221 | C22×C4 | — | 53 | 1 | no | — |
| 222 | C22×C4 | — | 53 | 1 | no | — |
| 223 | C22×C4 | — | 53 | 1 | no | — |
| 224 | C22×C4 | — | 53 | 2 | no | — |
| 225 | C22×C4 | — | 106 | 2 | no | — |
| 226 | C22×C4 | — | 212 | 4 | no | — |
| 227 | C22×C4 | — | 212 | 8 | no | — |
| 228 | C2×C8 | — | 2 | 2 | no | — |
| 229 | C2×C8 | — | 106 | 4 | no | — |
| 230 | C2×C8 | — | 106 | 2 | no | — |
| 231 | C2×C8 | — | 212 | 2 | no | — |
| 232 | C2×C8 | — | 212 | 2 | no | — |
| 233 | C2×C8 | — | 212 | 2 | no | — |
| 234 | C2×C8 | — | 212 | 2 | no | — |
| 235 | C4⋊C4 | — | 212 | 2 | no | — |
| 236 | C4⋊C4 | — | 212 | 2 | no | — |
| 237 | C4⋊C4 | — | 212 | 2 | no | — |
| 238 | C22⋊C4 | — | 212 | 8 | no | — |
| 239 | C22⋊C4 | — | 212 | 4 | no | — |
| 240 | C42 | — | 53 | 1 | no | — |
| 241 | C42 | — | 53 | 2 | no | — |
| 242 | C42 | — | 106 | 2 | no | — |
| 243 | C42 | — | 106 | 2 | no | — |
| 244 | C42 | — | 106 | 2 | no | — |
| 245 | C42 | — | 106 | 2 | no | — |
| 246 | C42 | — | 106 | 2 | no | — |
| 247 | C42 | — | 212 | 1 | no | — |
| 248 | C42 | C22×#6157 | 1 | 1 | yes | characteristic |
| 249 | C23 | — | 2 | 2 | no | — |
| 250 | C23 | — | 53 | 1 | no | — |
| 251 | C23 | — | 106 | 2 | no | — |
| 252 | C23 | — | 212 | 4 | no | — |
| 253 | C23 | — | 212 | 2 | no | — |
| 254 | C23 | — | 212 | 4 | no | — |
| 255 | C23 | — | 212 | 2 | no | — |
| 256 | D4 | — | 4 | 2 | no | — |
| 257 | D4 | — | 4 | 2 | no | — |
| 258 | D4 | — | 4 | 2 | no | — |
| 259 | D4 | — | 212 | 2 | no | — |
| 260 | D4 | — | 212 | 4 | no | — |
| 261 | D4 | — | 212 | 4 | no | — |
| 262 | D4 | — | 212 | 4 | no | — |
| 263 | D4 | — | 212 | 2 | no | — |
| 264 | D4 | — | 212 | 2 | no | — |
| 265 | C2×C4 | — | 53 | 2 | no | — |
| 266 | C2×C4 | — | 53 | 2 | no | — |
| 267 | C2×C4 | — | 53 | 2 | no | — |
| 268 | C2×C4 | — | 53 | 2 | no | — |
| 269 | C2×C4 | — | 53 | 2 | no | — |
| 270 | C2×C4 | — | 53 | 2 | no | — |
| 271 | C2×C4 | — | 53 | 1 | no | — |
| 272 | C2×C4 | — | 106 | 1 | no | — |
| 273 | C2×C4 | — | 106 | 1 | no | — |
| 274 | C2×C4 | — | 106 | 2 | no | — |
| 275 | C2×C4 | — | 106 | 1 | no | — |
| 276 | C2×C4 | — | 106 | 2 | no | — |
| 277 | C2×C4 | — | 106 | 2 | no | — |
| 278 | C2×C4 | — | 106 | 2 | no | — |
| 279 | C2×C4 | — | 106 | 2 | no | — |
| 280 | C2×C4 | — | 424 | 8 | no | — |
| 281 | C2×C4 | — | 212 | 4 | no | — |
| 282 | C2×C4 | — | 212 | 2 | no | — |
| 283 | C2×C4 | C2×C4×#6157 | 1 | 1 | yes | characteristic |
| 284 | C2×C4 | D4×#6157 | 1 | 2 | yes | — |
| 285 | C8 | — | 4 | 2 | no | — |
| 286 | C8 | — | 212 | 4 | no | — |
| 287 | C8 | — | 212 | 2 | no | — |
| 288 | C22 | — | 4 | 2 | no | — |
| 289 | C22 | — | 4 | 4 | no | — |
| 290 | C22 | — | 53 | 1 | no | — |
| 291 | C22 | — | 53 | 1 | no | — |
| 292 | C22 | — | 53 | 2 | no | — |
| 293 | C22 | — | 53 | 2 | no | — |
| 294 | C22 | — | 424 | 2 | no | — |
| 295 | C22 | — | 424 | 4 | no | — |
| 296 | C22 | — | 424 | 2 | no | — |
| 297 | C22 | — | 212 | 2 | no | — |
| 298 | C22 | — | 212 | 4 | no | — |
| 299 | C22 | C22⋊C4×#6157 | 1 | 1 | yes | characteristic |
| 300 | C4 | — | 2 | 1 | no | — |
| 301 | C4 | — | 53 | 2 | no | — |
| 302 | C4 | — | 53 | 2 | no | — |
| 303 | C4 | — | 106 | 2 | no | — |
| 304 | C4 | — | 106 | 1 | no | — |
| 305 | C4 | — | 424 | 2 | no | — |
| 306 | C4 | — | 212 | 2 | no | — |
| 307 | C4 | — | 212 | 1 | no | — |
| 308 | C4 | #16811662 | 1 | 2 | yes | — |
| 309 | C4 | #16811664 | 1 | 2 | yes | — |
| 310 | C2 | — | 8 | 2 | no | — |
| 311 | C2 | — | 53 | 1 | no | — |
| 312 | C2 | — | 53 | 2 | no | — |
| 313 | C2 | — | 53 | 1 | no | — |
| 314 | C2 | — | 424 | 2 | no | — |
| 315 | C2 | #100×#6157 | 1 | 1 | yes | characteristic |
| 316 | C2 | #20290486 | 1 | 2 | yes | — |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #20290502 | C2 | semidirect |
| 4. #20289594 | C2 | semidirect |
| 15. #16811481 | C4 | semidirect |
| 18. #16810367 | C4 | semidirect |
| 19. #16810366 | C4 | semidirect |
| 20. C53×#331 | C4 | semidirect |
| 34. #13133284 | C2×C4 | semidirect |
| 44. C4⋊#88322 | C2×C4 | semidirect |
| 47. C4⋊C8×C53 | C2×C4 | semidirect |
| 178. #331 | #6157 | semidirect |
| 179. C53 | #8231 | semidirect |
| 205. C4⋊C8 | C2×#6157 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #127×#6157 | 2 |
| #20290502 | 2 |
| D53⋊#331 | 2 |
| #20289594 | 2 |
| #331×D53 | 2 |
| #8231 | 53 |