Finite Group DatabasebetaFinite Group DatabasebetaOrder 1280.
314 conjugacy classes of subgroups; only the 61 normal subgroups are shown.
This group has 314 conjugacy classes of subgroups — too many to draw. Only its 61 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 | #10613660 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #10602145 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #10600079 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #10599333 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | C5×#1363 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | #10598048 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 7 | #10597956 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 8 | #66029 | — | 2 | 1 | no | — |
| 9 | C5×#499 | C22 | 1 | 1 | yes | characteristic |
| 10 | #65355 | — | 2 | 1 | no | — |
| 11 | C2×D80 | — | 2 | 1 | no | — |
| 12 | #65174 | — | 2 | 1 | no | — |
| 13 | #65171 | — | 2 | 1 | no | — |
| 14 | #65154 | — | 2 | 1 | no | — |
| 15 | #65150 | — | 2 | 1 | no | — |
| 16 | #65132 | — | 2 | 1 | no | — |
| 17 | #64984 | C22 | 1 | 1 | yes | characteristic |
| 18 | C4×D40 | C22 | 1 | 1 | yes | characteristic |
| 19 | #64977 | C22 | 1 | 1 | yes | characteristic |
| 20 | C5×C4⋊C16 | C22 | 1 | 1 | yes | characteristic |
| 21 | C5×#334 | C22 | 1 | 1 | yes | characteristic |
| 22 | #64729 | — | 2 | 1 | no | — |
| 23 | #64711 | — | 2 | 1 | no | — |
| 24 | #64711 | — | 2 | 1 | no | — |
| 25 | #64705 | — | 2 | 1 | no | — |
| 26 | #64681 | C22 | 1 | 1 | yes | characteristic |
| 27 | #12348 | — | 5 | 1 | no | maximal, Sylow |
| 28 | C5×#125 | — | 2 | 1 | no | — |
| 29 | C2×D40 | — | 2 | 1 | no | — |
| 30 | C2×D40 | — | 2 | 1 | no | — |
| 31 | C2×D40 | — | 4 | 1 | no | — |
| 32 | C2×D40 | D4 | 1 | 1 | yes | characteristic |
| 33 | C2×#676 | — | 2 | 1 | no | — |
| 34 | #4044 | — | 4 | 1 | no | — |
| 35 | #4041 | — | 2 | 1 | no | — |
| 36 | #4041 | — | 2 | 1 | no | — |
| 37 | #4041 | — | 4 | 1 | no | — |
| 38 | #4040 | — | 4 | 1 | no | — |
| 39 | #4039 | — | 4 | 1 | no | — |
| 40 | #4038 | — | 2 | 1 | no | — |
| 41 | D5×C4⋊C4 | — | 2 | 1 | no | — |
| 42 | #4020 | — | 2 | 1 | no | — |
| 43 | C4×D20 | — | 2 | 1 | no | — |
| 44 | C4×D20 | D4 | 1 | 1 | yes | characteristic |
| 45 | C2×C5×C16 | — | 2 | 1 | no | — |
| 46 | C5×#107 | D4 | 1 | 1 | yes | characteristic |
| 47 | C5×#107 | D4 | 1 | 1 | yes | characteristic |
| 48 | C5×#106 | — | 2 | 1 | no | — |
| 49 | C5×C4⋊C8 | — | 2 | 1 | no | — |
| 50 | C4×C5×C8 | C23 | 1 | 1 | yes | characteristic |
| 51 | #3953 | — | 2 | 1 | no | — |
| 52 | #3953 | — | 4 | 1 | no | — |
| 53 | #3952 | — | 2 | 1 | no | — |
| 54 | #3950 | D4 | 1 | 1 | yes | characteristic |
| 55 | C2×#672 | — | 2 | 1 | no | — |
| 56 | #3941 | — | 2 | 1 | no | — |
| 57 | #3941 | — | 2 | 1 | no | — |
| 58 | #3940 | — | 2 | 1 | no | — |
| 59 | #3939 | — | 2 | 1 | no | — |
| 60 | C4⋊#168 | D4 | 1 | 1 | yes | characteristic |
| 61 | D80 | — | 4 | 2 | no | — |
| 62 | #3355 | — | 5 | 1 | no | — |
| 63 | #2244 | — | 5 | 1 | no | — |
| 64 | #2206 | — | 5 | 1 | no | — |
| 65 | #1687 | — | 5 | 1 | no | — |
| 66 | #1363 | D5 | 1 | 1 | yes | characteristic, complemented |
| 67 | #1333 | — | 5 | 1 | no | — |
| 68 | #1330 | — | 5 | 1 | no | — |
| 69 | C2×D20 | — | 2 | 1 | no | — |
| 70 | C2×D20 | — | 4 | 1 | no | — |
| 71 | C2×D20 | — | 4 | 1 | no | — |
| 72 | C2×D20 | — | 4 | 1 | no | — |
| 73 | C2×D20 | — | 4 | 1 | no | — |
| 74 | C2×D20 | — | 4 | 1 | no | — |
| 75 | C2×D20 | D8 | 1 | 1 | yes | characteristic |
| 76 | C2×C4×D5 | — | 2 | 1 | no | — |
| 77 | C2×C4×D5 | — | 4 | 1 | no | — |
| 78 | C2×C4×D5 | — | 4 | 1 | no | — |
| 79 | C2×C5×C8 | — | 4 | 1 | no | — |
| 80 | C2×C5×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 81 | C2×C5×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 82 | C5×C4⋊C4 | — | 2 | 1 | no | — |
| 83 | C5×C4⋊C4 | — | 2 | 1 | no | — |
| 84 | C5×C4⋊C4 | — | 4 | 1 | no | — |
| 85 | C5×C4⋊C4 | — | 4 | 1 | no | — |
| 86 | C42×C5 | C2×D4 | 1 | 1 | yes | characteristic |
| 87 | #685 | — | 2 | 1 | no | — |
| 88 | #685 | — | 4 | 1 | no | — |
| 89 | #685 | — | 4 | 1 | no | — |
| 90 | #685 | — | 4 | 1 | no | — |
| 91 | #685 | — | 4 | 1 | no | — |
| 92 | C4⋊Dic5 | — | 2 | 1 | no | — |
| 93 | C4⋊Dic5 | D8 | 1 | 1 | yes | characteristic |
| 94 | #683 | — | 4 | 1 | no | — |
| 95 | C4×Dic5 | — | 4 | 1 | no | — |
| 96 | C2×#168 | — | 2 | 1 | no | — |
| 97 | D40 | — | 2 | 1 | no | — |
| 98 | D40 | — | 2 | 1 | no | — |
| 99 | D40 | — | 4 | 2 | no | — |
| 100 | D40 | — | 4 | 1 | no | — |
| 101 | D40 | — | 4 | 1 | no | — |
| 102 | D40 | — | 8 | 2 | no | — |
| 103 | #676 | — | 4 | 1 | no | — |
| 104 | #676 | — | 4 | 1 | no | — |
| 105 | C5×C16 | — | 4 | 1 | no | — |
| 106 | #672 | — | 4 | 1 | no | — |
| 107 | #553 | — | 10 | 1 | no | — |
| 108 | C2×D16 | — | 10 | 1 | no | — |
| 109 | #499 | C2×D5 | 1 | 1 | yes | characteristic |
| 110 | #493 | — | 5 | 1 | no | — |
| 111 | #482 | — | 10 | 1 | no | — |
| 112 | #480 | — | 10 | 1 | no | — |
| 113 | #478 | — | 10 | 1 | no | — |
| 114 | #469 | — | 10 | 1 | no | — |
| 115 | #459 | — | 10 | 1 | no | — |
| 116 | C4×D8 | — | 5 | 1 | no | — |
| 117 | #436 | — | 5 | 1 | no | — |
| 118 | #428 | — | 10 | 1 | no | — |
| 119 | C4⋊C16 | C2×D5 | 1 | 1 | yes | characteristic |
| 120 | #357 | — | 10 | 1 | no | — |
| 121 | #357 | — | 10 | 1 | no | — |
| 122 | #357 | — | 10 | 1 | no | — |
| 123 | #346 | — | 5 | 1 | no | — |
| 124 | #334 | C2×D5 | 1 | 1 | yes | characteristic |
| 125 | #331 | — | 10 | 1 | no | — |
| 126 | C22×D5 | — | 2 | 1 | no | — |
| 127 | C22×D5 | — | 4 | 1 | no | — |
| 128 | C22×D5 | — | 8 | 1 | no | — |
| 129 | C2×C4×C5 | — | 4 | 1 | no | — |
| 130 | C2×C4×C5 | — | 4 | 1 | no | — |
| 131 | C2×C4×C5 | #120 | 1 | 1 | yes | characteristic |
| 132 | C2×C4×C5 | C2×D8 | 1 | 1 | yes | characteristic |
| 133 | C2×C4×C5 | AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 134 | C2×Dic5 | — | 2 | 1 | no | — |
| 135 | C2×Dic5 | — | 4 | 1 | no | — |
| 136 | D20 | — | 4 | 1 | no | — |
| 137 | D20 | — | 4 | 1 | no | — |
| 138 | D20 | — | 4 | 1 | no | — |
| 139 | D20 | — | 8 | 1 | no | — |
| 140 | D20 | — | 8 | 1 | no | — |
| 141 | D20 | — | 8 | 2 | no | — |
| 142 | D20 | — | 8 | 1 | no | — |
| 143 | D20 | — | 8 | 1 | no | — |
| 144 | D20 | — | 16 | 1 | no | — |
| 145 | D20 | — | 16 | 1 | no | — |
| 146 | C4×D5 | — | 4 | 1 | no | — |
| 147 | C4×D5 | — | 4 | 1 | no | — |
| 148 | C4×D5 | — | 16 | 1 | no | — |
| 149 | C4×D5 | — | 16 | 1 | no | — |
| 150 | C5×C8 | — | 2 | 1 | no | — |
| 151 | C5×C8 | — | 8 | 1 | no | — |
| 152 | C5×C8 | #135 | 1 | 1 | yes | characteristic |
| 153 | C5×C8 | AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 154 | #168 | — | 4 | 1 | no | — |
| 155 | C2×D8 | — | 5 | 1 | no | — |
| 156 | C2×D8 | — | 10 | 1 | no | — |
| 157 | C2×D8 | — | 10 | 1 | no | — |
| 158 | C2×D8 | — | 20 | 1 | no | — |
| 159 | C2×M16 | — | 10 | 1 | no | — |
| 160 | #127 | — | 10 | 1 | no | — |
| 161 | #126 | — | 20 | 1 | no | — |
| 162 | #125 | — | 2 | 1 | no | — |
| 163 | #123 | — | 20 | 1 | no | — |
| 164 | #121 | — | 10 | 1 | no | — |
| 165 | #121 | — | 10 | 1 | no | — |
| 166 | #121 | — | 20 | 1 | no | — |
| 167 | C4×D4 | — | 5 | 1 | no | — |
| 168 | C4×D4 | — | 10 | 1 | no | — |
| 169 | C4×D4 | — | 20 | 1 | no | — |
| 170 | #117 | — | 10 | 1 | no | — |
| 171 | C2×C4⋊C4 | — | 10 | 1 | no | — |
| 172 | D16 | — | 20 | 2 | no | — |
| 173 | C2×C16 | — | 2 | 1 | no | — |
| 174 | C2×C16 | — | 10 | 1 | no | — |
| 175 | #107 | — | 5 | 1 | no | — |
| 176 | #107 | — | 10 | 1 | no | — |
| 177 | #107 | D20 | 1 | 1 | yes | characteristic |
| 178 | #107 | D20 | 1 | 1 | yes | characteristic |
| 179 | #106 | — | 2 | 1 | no | — |
| 180 | #106 | — | 10 | 1 | no | — |
| 181 | C4⋊C8 | — | 2 | 1 | no | — |
| 182 | C4⋊C8 | — | 5 | 1 | no | — |
| 183 | #102 | — | 10 | 1 | no | — |
| 184 | #102 | — | 10 | 1 | no | — |
| 185 | #102 | — | 10 | 1 | no | — |
| 186 | #102 | — | 20 | 1 | no | — |
| 187 | C22⋊C8 | — | 10 | 1 | no | — |
| 188 | C4×C8 | C22×D5 | 1 | 1 | yes | characteristic |
| 189 | C22×C5 | #447 | 1 | 1 | yes | characteristic |
| 190 | C2×D5 | — | 4 | 1 | no | — |
| 191 | C2×D5 | — | 4 | 1 | no | — |
| 192 | C2×D5 | — | 4 | 1 | no | — |
| 193 | C2×D5 | — | 8 | 1 | no | — |
| 194 | C2×D5 | — | 8 | 1 | no | — |
| 195 | C2×D5 | — | 8 | 1 | no | — |
| 196 | C2×D5 | — | 16 | 1 | no | — |
| 197 | C2×D5 | — | 16 | 1 | no | — |
| 198 | C2×D5 | — | 16 | 1 | no | — |
| 199 | C4×C5 | — | 2 | 1 | no | — |
| 200 | C4×C5 | — | 2 | 1 | no | — |
| 201 | C4×C5 | — | 8 | 1 | no | — |
| 202 | C4×C5 | — | 8 | 1 | no | — |
| 203 | C4×C5 | #449 | 1 | 1 | yes | characteristic |
| 204 | C4×C5 | #454 | 1 | 1 | yes | characteristic |
| 205 | Dic5 | — | 4 | 1 | no | — |
| 206 | Dic5 | — | 8 | 1 | no | — |
| 207 | C2×D4 | — | 5 | 1 | no | — |
| 208 | C2×D4 | — | 10 | 1 | no | — |
| 209 | C2×D4 | — | 20 | 1 | no | — |
| 210 | C2×D4 | — | 20 | 1 | no | — |
| 211 | C2×D4 | — | 20 | 1 | no | — |
| 212 | C2×D4 | — | 20 | 1 | no | — |
| 213 | C2×D4 | — | 20 | 1 | no | — |
| 214 | C22×C4 | — | 10 | 1 | no | — |
| 215 | C22×C4 | — | 20 | 1 | no | — |
| 216 | C22×C4 | — | 20 | 1 | no | — |
| 217 | D8 | — | 10 | 1 | no | — |
| 218 | D8 | — | 10 | 1 | no | — |
| 219 | D8 | — | 20 | 2 | no | — |
| 220 | D8 | — | 20 | 1 | no | — |
| 221 | D8 | — | 20 | 1 | no | — |
| 222 | D8 | — | 40 | 2 | no | — |
| 223 | M16 | — | 20 | 1 | no | — |
| 224 | M16 | — | 20 | 1 | no | — |
| 225 | C2×C8 | — | 4 | 1 | no | — |
| 226 | C2×C8 | — | 10 | 1 | no | — |
| 227 | C2×C8 | C2×D20 | 1 | 1 | yes | characteristic |
| 228 | C2×C8 | D4×D5 | 1 | 1 | yes | characteristic |
| 229 | C4⋊C4 | — | 2 | 1 | no | — |
| 230 | C4⋊C4 | — | 2 | 1 | no | — |
| 231 | C4⋊C4 | — | 4 | 1 | no | — |
| 232 | C4⋊C4 | — | 4 | 1 | no | — |
| 233 | C4⋊C4 | — | 5 | 1 | no | — |
| 234 | C4⋊C4 | — | 10 | 1 | no | — |
| 235 | C4⋊C4 | — | 20 | 1 | no | — |
| 236 | C22⋊C4 | — | 10 | 1 | no | — |
| 237 | C22⋊C4 | — | 20 | 1 | no | — |
| 238 | C22⋊C4 | — | 20 | 1 | no | — |
| 239 | C22⋊C4 | — | 20 | 1 | no | — |
| 240 | C22⋊C4 | — | 20 | 1 | no | — |
| 241 | C42 | — | 20 | 1 | no | — |
| 242 | C42 | D4×D5 | 1 | 1 | yes | characteristic |
| 243 | C16 | — | 4 | 1 | no | — |
| 244 | C16 | — | 20 | 1 | no | — |
| 245 | C2×C5 | #1625 | 1 | 1 | yes | characteristic |
| 246 | C2×C5 | #2190 | 1 | 1 | yes | characteristic |
| 247 | C2×C5 | #2192 | 1 | 1 | yes | characteristic |
| 248 | D5 | — | 8 | 1 | no | — |
| 249 | D5 | — | 16 | 1 | no | — |
| 250 | D5 | — | 32 | 1 | no | — |
| 251 | C23 | — | 10 | 1 | no | — |
| 252 | C23 | — | 20 | 1 | no | — |
| 253 | C23 | — | 40 | 1 | no | — |
| 254 | D4 | — | 20 | 1 | no | — |
| 255 | D4 | — | 20 | 1 | no | — |
| 256 | D4 | — | 20 | 1 | no | — |
| 257 | D4 | — | 40 | 1 | no | — |
| 258 | D4 | — | 40 | 1 | no | — |
| 259 | D4 | — | 40 | 2 | no | — |
| 260 | D4 | — | 40 | 1 | no | — |
| 261 | D4 | — | 40 | 1 | no | — |
| 262 | D4 | — | 80 | 1 | no | — |
| 263 | D4 | — | 80 | 1 | no | — |
| 264 | C2×C4 | — | 4 | 1 | no | — |
| 265 | C2×C4 | — | 4 | 1 | no | — |
| 266 | C2×C4 | — | 10 | 1 | no | — |
| 267 | C2×C4 | — | 20 | 1 | no | — |
| 268 | C2×C4 | — | 20 | 1 | no | — |
| 269 | C2×C4 | — | 20 | 1 | no | — |
| 270 | C2×C4 | — | 80 | 1 | no | — |
| 271 | C2×C4 | — | 80 | 1 | no | — |
| 272 | C2×C4 | #4028 | 1 | 1 | yes | characteristic |
| 273 | C2×C4 | D5×D8 | 1 | 1 | yes | characteristic |
| 274 | C2×C4 | #4060 | 1 | 1 | yes | characteristic |
| 275 | C8 | — | 2 | 1 | no | — |
| 276 | C8 | — | 8 | 1 | no | — |
| 277 | C8 | — | 20 | 1 | no | — |
| 278 | C8 | #4060 | 1 | 1 | yes | characteristic |
| 279 | C8 | #4065 | 1 | 1 | yes | characteristic |
| 280 | C5 | #12348 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 281 | C22 | — | 20 | 1 | no | — |
| 282 | C22 | — | 20 | 1 | no | — |
| 283 | C22 | — | 20 | 1 | no | — |
| 284 | C22 | — | 40 | 1 | no | — |
| 285 | C22 | — | 40 | 1 | no | — |
| 286 | C22 | — | 40 | 1 | no | — |
| 287 | C22 | — | 80 | 1 | no | — |
| 288 | C22 | — | 80 | 1 | no | — |
| 289 | C22 | — | 80 | 1 | no | — |
| 290 | C22 | #65062 | 1 | 1 | yes | characteristic |
| 291 | C4 | — | 2 | 1 | no | — |
| 292 | C4 | — | 2 | 1 | no | — |
| 293 | C4 | — | 8 | 1 | no | — |
| 294 | C4 | — | 8 | 1 | no | — |
| 295 | C4 | — | 20 | 1 | no | — |
| 296 | C4 | — | 40 | 1 | no | — |
| 297 | C4 | #65042 | 1 | 1 | yes | characteristic |
| 298 | C4 | #65100 | 1 | 1 | yes | characteristic |
| 299 | C2 | — | 40 | 1 | no | — |
| 300 | C2 | — | 80 | 1 | no | — |
| 301 | C2 | — | 160 | 1 | no | — |
| 302 | C2 | #10599225 | 1 | 1 | yes | characteristic |
| 303 | C2 | #10600035 | 1 | 1 | yes | characteristic |
| 304 | C2 | #10600045 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #10613660 | C2 | semidirect |
| 2. #10602145 | C2 | semidirect |
| 3. #10600079 | C2 | semidirect |
| 4. #10599333 | C2 | semidirect |
| 5. C5×#1363 | C2 | semidirect |
| 6. #10598048 | C2 | semidirect |
| 7. #10597956 | C2 | semidirect |
| 66. #1363 | D5 | semidirect |
| 280. C5 | #12348 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #10613660 | 2 |
| #10602145 | 2 |
| #10600079 | 2 |
| #10599333 | 2 |
| C5×#1363 | 2 |
| #10598048 | 2 |
| #10597956 | 2 |
| #12348 | 5 |