Finite Group DatabasebetaFinite Group DatabasebetaOrder 1280.
728 conjugacy classes of subgroups; only the 163 normal subgroups are shown.
This group has 728 conjugacy classes of subgroups — too many to draw. Only its 163 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 | C5×#3139 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #10615078 | C2 | 1 | 2 | yes | maximal, complemented |
| 3 | #10614930 | C2 | 1 | 2 | yes | maximal, complemented |
| 4 | #10610266 | C2 | 1 | 1 | yes | maximal, characteristic |
| 5 | #10602017 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | #10602012 | C2 | 1 | 2 | yes | maximal, complemented |
| 7 | #10602011 | C2 | 1 | 2 | yes | maximal, complemented |
| 8 | #10602008 | C2 | 1 | 2 | yes | maximal, complemented |
| 9 | #10600642 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 10 | C5×#533 | C22 | 1 | 1 | yes | characteristic |
| 11 | C2×C5×C4⋊Q8 | C22 | 1 | 1 | yes | characteristic |
| 12 | #65811 | C22 | 1 | 2 | yes | — |
| 13 | C5×#491 | — | 2 | 2 | no | — |
| 14 | C5×#487 | C22 | 1 | 2 | yes | — |
| 15 | C5×#487 | C22 | 1 | 2 | yes | — |
| 16 | C5×#419 | — | 2 | 2 | no | — |
| 17 | C4×C5×M16 | C22 | 1 | 1 | yes | characteristic |
| 18 | #65424 | — | 2 | 2 | no | — |
| 19 | #65382 | C22 | 1 | 2 | yes | — |
| 20 | #65382 | C22 | 1 | 2 | yes | — |
| 21 | #65378 | — | 2 | 4 | no | — |
| 22 | #65377 | C22 | 1 | 2 | yes | — |
| 23 | #65377 | C22 | 1 | 2 | yes | — |
| 24 | #65377 | C22 | 1 | 2 | yes | — |
| 25 | #65374 | C22 | 1 | 2 | yes | — |
| 26 | #65370 | C22 | 1 | 2 | yes | — |
| 27 | #65367 | C22 | 1 | 2 | yes | — |
| 28 | #65338 | — | 2 | 4 | no | — |
| 29 | #65315 | — | 2 | 4 | no | — |
| 30 | #65255 | — | 2 | 4 | no | — |
| 31 | C4×#681 | C22 | 1 | 2 | yes | — |
| 32 | #65000 | — | 2 | 2 | no | — |
| 33 | #64984 | C22 | 1 | 1 | yes | characteristic |
| 34 | #64983 | C22 | 1 | 2 | yes | — |
| 35 | #64969 | C22 | 1 | 1 | yes | characteristic |
| 36 | #64675 | C22 | 1 | 1 | yes | characteristic |
| 37 | #64674 | C22 | 1 | 2 | yes | — |
| 38 | C8×#168 | C22 | 1 | 1 | yes | characteristic |
| 39 | #19754 | — | 5 | 1 | no | maximal, Sylow |
| 40 | C22×C5×Q8 | — | 2 | 1 | no | — |
| 41 | C2×#709 | — | 2 | 2 | no | — |
| 42 | C2×C5×M16 | D4 | 1 | 2 | yes | — |
| 43 | C5×C4⋊Q8 | — | 2 | 2 | no | — |
| 44 | C5×C4⋊Q8 | — | 2 | 1 | no | — |
| 45 | C5×C4⋊Q8 | C23 | 1 | 2 | yes | — |
| 46 | C5×C4⋊Q8 | C23 | 1 | 2 | yes | — |
| 47 | C5×#125 | — | 2 | 1 | no | — |
| 48 | C5×C22⋊Q8 | — | 2 | 2 | no | — |
| 49 | C4×C5×Q8 | — | 2 | 2 | no | — |
| 50 | C5×#117 | — | 2 | 1 | no | — |
| 51 | C2×C5×C4⋊C4 | — | 2 | 2 | no | — |
| 52 | C2×C42×C5 | C23 | 1 | 1 | yes | characteristic |
| 53 | C2×#688 | — | 2 | 4 | no | — |
| 54 | #4076 | — | 2 | 2 | no | — |
| 55 | #4076 | — | 2 | 2 | no | — |
| 56 | C2×#681 | D4 | 1 | 4 | yes | — |
| 57 | C2×D40 | — | 2 | 2 | no | — |
| 58 | #4021 | — | 2 | 2 | no | — |
| 59 | #4020 | C23 | 1 | 2 | yes | — |
| 60 | C4×D20 | — | 2 | 2 | no | — |
| 61 | C4×D20 | — | 2 | 2 | no | — |
| 62 | #4015 | C23 | 1 | 2 | yes | — |
| 63 | C5×#103 | — | 2 | 4 | no | — |
| 64 | C5×#103 | — | 2 | 4 | no | — |
| 65 | C5×#97 | D4 | 1 | 2 | yes | — |
| 66 | C4×C5×C8 | C23 | 1 | 1 | yes | characteristic |
| 67 | C4×C5×C8 | C23 | 1 | 1 | yes | characteristic |
| 68 | #3967 | — | 2 | 4 | no | — |
| 69 | #3953 | — | 2 | 4 | no | — |
| 70 | #3949 | — | 2 | 2 | no | — |
| 71 | #3942 | — | 2 | 4 | no | — |
| 72 | #3942 | — | 2 | 4 | no | — |
| 73 | #3941 | — | 2 | 4 | no | — |
| 74 | #3941 | — | 2 | 4 | no | — |
| 75 | #3941 | — | 2 | 4 | no | — |
| 76 | #3939 | — | 2 | 4 | no | — |
| 77 | #3935 | D4 | 1 | 4 | yes | — |
| 78 | C4×#168 | C23 | 1 | 2 | yes | — |
| 79 | C4×#168 | C23 | 1 | 2 | yes | — |
| 80 | #3184 | — | 5 | 1 | no | — |
| 81 | #3182 | — | 5 | 2 | no | — |
| 82 | #3141 | — | 5 | 2 | no | — |
| 83 | #3139 | D5 | 1 | 1 | yes | characteristic, complemented |
| 84 | #1719 | — | 5 | 2 | no | — |
| 85 | #1718 | — | 5 | 2 | no | — |
| 86 | #1717 | — | 5 | 2 | no | — |
| 87 | #1711 | — | 5 | 2 | no | — |
| 88 | #1455 | — | 5 | 1 | no | — |
| 89 | C2×C5×Q8 | — | 2 | 2 | no | — |
| 90 | C2×C5×Q8 | — | 2 | 2 | no | — |
| 91 | C2×C5×Q8 | — | 4 | 4 | no | — |
| 92 | C2×C5×Q8 | — | 4 | 1 | no | — |
| 93 | C22×C4×C5 | — | 4 | 1 | no | — |
| 94 | C22×C4×C5 | C2×D4 | 1 | 2 | yes | — |
| 95 | C22×C4×C5 | C2×D4 | 1 | 1 | yes | characteristic |
| 96 | C2×C22⋊D5 | — | 4 | 2 | no | — |
| 97 | #709 | — | 4 | 8 | no | — |
| 98 | C2×D20 | — | 2 | 2 | no | — |
| 99 | C2×D20 | — | 2 | 2 | no | — |
| 100 | C2×D20 | — | 2 | 2 | no | — |
| 101 | C2×C4×D5 | — | 4 | 2 | no | — |
| 102 | C2×Dic10 | — | 2 | 2 | no | — |
| 103 | C5×M16 | — | 4 | 2 | no | — |
| 104 | C2×C5×C8 | C2×D4 | 1 | 2 | yes | — |
| 105 | C2×C5×C8 | C2×D4 | 1 | 2 | yes | — |
| 106 | C5×C4⋊C4 | — | 2 | 2 | no | — |
| 107 | C5×C4⋊C4 | — | 2 | 2 | no | — |
| 108 | C5×C4⋊C4 | — | 2 | 2 | no | — |
| 109 | C5×C4⋊C4 | — | 2 | 2 | no | — |
| 110 | C5×C4⋊C4 | — | 4 | 2 | no | — |
| 111 | C5×C4⋊C4 | — | 4 | 2 | no | — |
| 112 | C5×C22⋊C4 | — | 4 | 1 | no | — |
| 113 | C42×C5 | — | 4 | 1 | no | — |
| 114 | C42×C5 | C2×D4 | 1 | 2 | yes | — |
| 115 | C42×C5 | C2×D4 | 1 | 1 | yes | characteristic |
| 116 | C42×C5 | C24 | 1 | 1 | yes | characteristic |
| 117 | #688 | — | 4 | 8 | no | — |
| 118 | #685 | — | 4 | 2 | no | — |
| 119 | #685 | — | 4 | 2 | no | — |
| 120 | C4⋊Dic5 | — | 2 | 2 | no | — |
| 121 | C4⋊Dic5 | — | 2 | 2 | no | — |
| 122 | #681 | — | 4 | 4 | no | — |
| 123 | C2×#168 | C2×D4 | 1 | 4 | yes | — |
| 124 | C2×#168 | C2×D4 | 1 | 4 | yes | — |
| 125 | D40 | — | 4 | 4 | no | — |
| 126 | #533 | C2×D5 | 1 | 1 | yes | characteristic |
| 127 | #532 | — | 5 | 2 | no | — |
| 128 | C2×C4⋊Q8 | C2×D5 | 1 | 1 | yes | characteristic |
| 129 | #500 | — | 5 | 2 | no | — |
| 130 | #498 | — | 5 | 1 | no | — |
| 131 | #493 | — | 5 | 1 | no | — |
| 132 | #492 | — | 5 | 2 | no | — |
| 133 | #491 | — | 2 | 2 | no | — |
| 134 | #489 | — | 10 | 4 | no | — |
| 135 | #487 | — | 5 | 2 | no | — |
| 136 | #487 | — | 5 | 2 | no | — |
| 137 | #487 | — | 5 | 2 | no | — |
| 138 | #487 | C2×D5 | 1 | 2 | yes | — |
| 139 | #487 | C2×D5 | 1 | 2 | yes | — |
| 140 | #486 | — | 5 | 2 | no | — |
| 141 | #486 | — | 5 | 2 | no | — |
| 142 | #486 | — | 5 | 2 | no | — |
| 143 | #486 | — | 5 | 2 | no | — |
| 144 | #469 | — | 10 | 2 | no | — |
| 145 | #468 | — | 10 | 4 | no | — |
| 146 | #442 | — | 10 | 2 | no | — |
| 147 | #440 | — | 10 | 4 | no | — |
| 148 | #419 | — | 2 | 2 | no | — |
| 149 | #417 | — | 10 | 4 | no | — |
| 150 | C4×M16 | — | 5 | 2 | no | — |
| 151 | C4×M16 | C2×D5 | 1 | 1 | yes | characteristic |
| 152 | #322 | — | 5 | 2 | no | — |
| 153 | #322 | — | 5 | 1 | no | — |
| 154 | C82 | — | 5 | 1 | no | — |
| 155 | C23×C5 | #127 | 1 | 1 | yes | characteristic |
| 156 | C22×D5 | — | 4 | 2 | no | — |
| 157 | C5×Q8 | — | 4 | 4 | no | — |
| 158 | C5×Q8 | — | 4 | 4 | no | — |
| 159 | C5×Q8 | — | 8 | 2 | no | — |
| 160 | C2×C4×C5 | — | 2 | 4 | no | — |
| 161 | C2×C4×C5 | — | 2 | 2 | no | — |
| 162 | C2×C4×C5 | — | 4 | 2 | no | — |
| 163 | C2×C4×C5 | — | 4 | 2 | no | — |
| 164 | C2×C4×C5 | — | 8 | 2 | no | — |
| 165 | C2×C4×C5 | #127 | 1 | 1 | yes | characteristic |
| 166 | C2×C4×C5 | #127 | 1 | 2 | yes | — |
| 167 | C2×C4×C5 | C22×D4 | 1 | 2 | yes | — |
| 168 | C2×C4×C5 | C22×D4 | 1 | 1 | yes | characteristic |
| 169 | C22⋊D5 | — | 8 | 4 | no | — |
| 170 | C2×Dic5 | — | 4 | 2 | no | — |
| 171 | D20 | — | 4 | 4 | no | — |
| 172 | D20 | — | 4 | 4 | no | — |
| 173 | D20 | — | 4 | 4 | no | — |
| 174 | C4×D5 | — | 8 | 4 | no | — |
| 175 | Dic10 | — | 4 | 4 | no | — |
| 176 | C5×C8 | — | 2 | 2 | no | — |
| 177 | C5×C8 | — | 2 | 2 | no | — |
| 178 | #168 | — | 2 | 4 | no | — |
| 179 | #168 | — | 2 | 4 | no | — |
| 180 | C2×#41 | — | 10 | 2 | no | — |
| 181 | C22×Q8 | — | 2 | 1 | no | — |
| 182 | C2×SD16 | — | 10 | 4 | no | — |
| 183 | C2×D8 | — | 10 | 2 | no | — |
| 184 | C2×M16 | — | 5 | 4 | no | — |
| 185 | C2×M16 | C22⋊D5 | 1 | 2 | yes | — |
| 186 | C4⋊Q8 | — | 2 | 2 | no | — |
| 187 | C4⋊Q8 | — | 2 | 1 | no | — |
| 188 | C4⋊Q8 | — | 5 | 2 | no | — |
| 189 | C4⋊Q8 | C22×D5 | 1 | 2 | yes | — |
| 190 | C4⋊Q8 | C22×D5 | 1 | 2 | yes | — |
| 191 | #127 | — | 5 | 2 | no | — |
| 192 | #125 | — | 2 | 1 | no | — |
| 193 | #124 | — | 10 | 2 | no | — |
| 194 | C22⋊Q8 | — | 2 | 2 | no | — |
| 195 | #121 | — | 10 | 2 | no | — |
| 196 | #121 | — | 10 | 2 | no | — |
| 197 | C4×Q8 | — | 2 | 2 | no | — |
| 198 | C4×D4 | — | 10 | 2 | no | — |
| 199 | C4×D4 | — | 10 | 2 | no | — |
| 200 | #117 | — | 2 | 1 | no | — |
| 201 | C2×C4⋊C4 | — | 2 | 2 | no | — |
| 202 | C2×C42 | C22×D5 | 1 | 1 | yes | characteristic |
| 203 | #107 | — | 10 | 4 | no | — |
| 204 | #106 | — | 10 | 2 | no | — |
| 205 | #103 | — | 2 | 4 | no | — |
| 206 | #103 | — | 2 | 4 | no | — |
| 207 | #103 | — | 10 | 4 | no | — |
| 208 | #103 | — | 10 | 4 | no | — |
| 209 | #103 | — | 10 | 4 | no | — |
| 210 | #102 | — | 10 | 4 | no | — |
| 211 | #102 | — | 10 | 4 | no | — |
| 212 | #102 | — | 10 | 4 | no | — |
| 213 | #102 | — | 10 | 4 | no | — |
| 214 | #97 | — | 5 | 4 | no | — |
| 215 | #97 | C22⋊D5 | 1 | 2 | yes | — |
| 216 | C4×C8 | — | 5 | 2 | no | — |
| 217 | C4×C8 | — | 5 | 2 | no | — |
| 218 | C4×C8 | C22×D5 | 1 | 1 | yes | characteristic |
| 219 | C4×C8 | C22×D5 | 1 | 1 | yes | characteristic |
| 220 | C22×C5 | — | 2 | 2 | no | — |
| 221 | C22×C5 | — | 2 | 1 | no | — |
| 222 | C22×C5 | C2×#127 | 1 | 1 | yes | characteristic |
| 223 | C2×D5 | — | 8 | 2 | no | — |
| 224 | C2×D5 | — | 8 | 2 | no | — |
| 225 | C2×D5 | — | 8 | 2 | no | — |
| 226 | C4×C5 | — | 2 | 2 | no | — |
| 227 | C4×C5 | — | 2 | 1 | no | — |
| 228 | C4×C5 | — | 8 | 2 | no | — |
| 229 | C4×C5 | — | 8 | 2 | no | — |
| 230 | C4×C5 | #575 | 1 | 4 | yes | — |
| 231 | C4×C5 | #575 | 1 | 2 | yes | — |
| 232 | Dic5 | — | 8 | 2 | no | — |
| 233 | #41 | — | 20 | 8 | no | — |
| 234 | C2×Q8 | — | 2 | 2 | no | — |
| 235 | C2×Q8 | — | 2 | 2 | no | — |
| 236 | C2×Q8 | — | 4 | 4 | no | — |
| 237 | C2×Q8 | — | 4 | 1 | no | — |
| 238 | C2×Q8 | — | 10 | 2 | no | — |
| 239 | C2×D4 | — | 10 | 2 | no | — |
| 240 | C2×D4 | — | 10 | 2 | no | — |
| 241 | C2×D4 | — | 10 | 2 | no | — |
| 242 | C2×D4 | — | 20 | 2 | no | — |
| 243 | C22×C4 | — | 4 | 1 | no | — |
| 244 | C22×C4 | — | 20 | 2 | no | — |
| 245 | C22×C4 | D4×D5 | 1 | 2 | yes | — |
| 246 | C22×C4 | C2×C22⋊D5 | 1 | 1 | yes | characteristic |
| 247 | SD16 | — | 20 | 8 | no | — |
| 248 | D8 | — | 20 | 4 | no | — |
| 249 | M16 | — | 4 | 2 | no | — |
| 250 | M16 | — | 20 | 4 | no | — |
| 251 | C2×C8 | — | 5 | 4 | no | — |
| 252 | C2×C8 | — | 5 | 4 | no | — |
| 253 | C2×C8 | C2×C22⋊D5 | 1 | 2 | yes | — |
| 254 | C2×C8 | C2×C22⋊D5 | 1 | 2 | yes | — |
| 255 | C4⋊C4 | — | 2 | 2 | no | — |
| 256 | C4⋊C4 | — | 2 | 2 | no | — |
| 257 | C4⋊C4 | — | 2 | 2 | no | — |
| 258 | C4⋊C4 | — | 2 | 2 | no | — |
| 259 | C4⋊C4 | — | 4 | 2 | no | — |
| 260 | C4⋊C4 | — | 4 | 2 | no | — |
| 261 | C4⋊C4 | — | 10 | 2 | no | — |
| 262 | C4⋊C4 | — | 10 | 2 | no | — |
| 263 | C22⋊C4 | — | 4 | 1 | no | — |
| 264 | C22⋊C4 | — | 20 | 2 | no | — |
| 265 | C22⋊C4 | — | 20 | 2 | no | — |
| 266 | C42 | — | 4 | 1 | no | — |
| 267 | C42 | D4×D5 | 1 | 2 | yes | — |
| 268 | C42 | C2×C22⋊D5 | 1 | 1 | yes | characteristic |
| 269 | C42 | C23×D5 | 1 | 1 | yes | characteristic |
| 270 | C2×C5 | — | 4 | 1 | no | — |
| 271 | C2×C5 | #3153 | 1 | 1 | yes | characteristic |
| 272 | C2×C5 | #3153 | 1 | 2 | yes | — |
| 273 | D5 | — | 16 | 2 | no | — |
| 274 | C23 | — | 20 | 2 | no | — |
| 275 | C23 | #4086 | 1 | 1 | yes | characteristic |
| 276 | Q8 | — | 4 | 4 | no | — |
| 277 | Q8 | — | 4 | 4 | no | — |
| 278 | Q8 | — | 8 | 2 | no | — |
| 279 | Q8 | — | 20 | 4 | no | — |
| 280 | D4 | — | 20 | 4 | no | — |
| 281 | D4 | — | 20 | 4 | no | — |
| 282 | D4 | — | 20 | 4 | no | — |
| 283 | D4 | — | 40 | 4 | no | — |
| 284 | C2×C4 | — | 2 | 4 | no | — |
| 285 | C2×C4 | — | 2 | 2 | no | — |
| 286 | C2×C4 | — | 4 | 2 | no | — |
| 287 | C2×C4 | — | 4 | 2 | no | — |
| 288 | C2×C4 | — | 8 | 2 | no | — |
| 289 | C2×C4 | — | 20 | 2 | no | — |
| 290 | C2×C4 | — | 40 | 4 | no | — |
| 291 | C2×C4 | #4086 | 1 | 1 | yes | characteristic |
| 292 | C2×C4 | #4086 | 1 | 2 | yes | — |
| 293 | C2×C4 | C2×D4×D5 | 1 | 2 | yes | — |
| 294 | C2×C4 | C22×C22⋊D5 | 1 | 1 | yes | characteristic |
| 295 | C8 | — | 2 | 2 | no | — |
| 296 | C8 | — | 2 | 2 | no | — |
| 297 | C8 | — | 10 | 4 | no | — |
| 298 | C8 | — | 10 | 4 | no | — |
| 299 | C5 | #19754 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 300 | C22 | — | 2 | 1 | no | — |
| 301 | C22 | — | 2 | 2 | no | — |
| 302 | C22 | — | 40 | 2 | no | — |
| 303 | C22 | — | 40 | 2 | no | — |
| 304 | C22 | — | 40 | 2 | no | — |
| 305 | C22 | C2×#4086 | 1 | 1 | yes | characteristic |
| 306 | C4 | — | 2 | 1 | no | — |
| 307 | C4 | — | 2 | 2 | no | — |
| 308 | C4 | — | 8 | 2 | no | — |
| 309 | C4 | — | 8 | 2 | no | — |
| 310 | C4 | — | 40 | 2 | no | — |
| 311 | C4 | #66111 | 1 | 4 | yes | — |
| 312 | C4 | #66156 | 1 | 2 | yes | — |
| 313 | C2 | — | 4 | 1 | no | — |
| 314 | C2 | — | 80 | 2 | no | — |
| 315 | C2 | #10614674 | 1 | 1 | yes | characteristic |
| 316 | C2 | #10615704 | 1 | 2 | yes | — |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C5×#3139 | C2 | semidirect |
| 2. #10615078 | C2 | semidirect |
| 3. #10614930 | C2 | semidirect |
| 5. #10602017 | C2 | semidirect |
| 6. #10602012 | C2 | semidirect |
| 7. #10602011 | C2 | semidirect |
| 8. #10602008 | C2 | semidirect |
| 9. #10600642 | C2 | semidirect |
| 83. #3139 | D5 | semidirect |
| 299. C5 | #19754 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C5×#3139 | 2 |
| #10615078 | 2 |
| #10614930 | 2 |
| #10610266 | 2 |
| #10602017 | 2 |
| #10602012 | 2 |
| #10602011 | 2 |
| #10602008 | 2 |
| #10600642 | 2 |
| #19754 | 5 |