Finite Group DatabasebetaFinite Group DatabasebetaOrder 1280.
434 conjugacy classes of subgroups; only the 73 normal subgroups are shown.
This group has 434 conjugacy classes of subgroups — too many to draw. Only its 73 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×#3199 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #10602655 | C2 | 1 | 2 | yes | maximal, complemented |
| 3 | #10602648 | C2 | 1 | 2 | yes | maximal, complemented |
| 4 | #10600794 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | #10598184 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | C5×#535 | C22 | 1 | 2 | yes | — |
| 7 | #66153 | C22 | 1 | 1 | yes | characteristic |
| 8 | C5×#482 | — | 2 | 2 | no | — |
| 9 | C5×#469 | — | 2 | 1 | no | — |
| 10 | C5×#466 | — | 2 | 1 | no | — |
| 11 | C5×#447 | — | 2 | 2 | no | — |
| 12 | C5×#408 | C22 | 1 | 1 | yes | characteristic |
| 13 | #65526 | C22 | 1 | 2 | yes | — |
| 14 | #65526 | C22 | 1 | 1 | yes | characteristic |
| 15 | C2×#3965 | — | 2 | 2 | no | — |
| 16 | #65413 | — | 2 | 1 | no | — |
| 17 | #64772 | — | 2 | 1 | no | — |
| 18 | #64762 | — | 2 | 4 | no | — |
| 19 | #64753 | — | 2 | 2 | no | — |
| 20 | #13429 | — | 5 | 1 | no | maximal, Sylow |
| 21 | C2×C5×#41 | C23 | 1 | 1 | yes | characteristic |
| 22 | C22×C5×D4 | D4 | 1 | 2 | yes | — |
| 23 | C2×C5×D8 | — | 2 | 1 | no | — |
| 24 | C2×C5×D8 | — | 2 | 1 | no | — |
| 25 | C2×C5×M16 | D4 | 1 | 1 | yes | characteristic |
| 26 | C22×C5×C8 | D4 | 1 | 1 | yes | characteristic |
| 27 | C5×#127 | — | 2 | 2 | no | — |
| 28 | C5×#124 | — | 2 | 2 | no | — |
| 29 | C5×#121 | — | 2 | 2 | no | — |
| 30 | C5×#121 | — | 2 | 2 | no | — |
| 31 | C5×#120 | — | 2 | 2 | no | — |
| 32 | C5×#120 | — | 2 | 2 | no | — |
| 33 | C5×#117 | D4 | 1 | 2 | yes | — |
| 34 | #4094 | — | 4 | 2 | no | — |
| 35 | C2×#681 | — | 2 | 1 | no | — |
| 36 | C2×#681 | — | 2 | 1 | no | — |
| 37 | C2×#681 | — | 2 | 2 | no | — |
| 38 | C2×#681 | — | 2 | 2 | no | — |
| 39 | C5×#107 | — | 2 | 1 | no | — |
| 40 | C5×#106 | — | 2 | 1 | no | — |
| 41 | C5×#102 | — | 2 | 2 | no | — |
| 42 | C5×#102 | — | 2 | 2 | no | — |
| 43 | C5×C22⋊C8 | — | 2 | 2 | no | — |
| 44 | #3968 | — | 2 | 2 | no | — |
| 45 | #3968 | D4 | 1 | 2 | yes | — |
| 46 | #3965 | — | 2 | 1 | no | — |
| 47 | #3965 | — | 2 | 2 | no | — |
| 48 | #3965 | — | 4 | 4 | no | — |
| 49 | #3954 | — | 4 | 2 | no | — |
| 50 | #3199 | D5 | 1 | 1 | yes | characteristic |
| 51 | #2134 | — | 5 | 2 | no | — |
| 52 | #1914 | — | 5 | 2 | no | — |
| 53 | #1858 | — | 5 | 1 | no | — |
| 54 | #1392 | — | 5 | 1 | no | — |
| 55 | C24×C5 | — | 2 | 2 | no | — |
| 56 | C5×#41 | — | 4 | 1 | no | — |
| 57 | C5×#41 | — | 4 | 1 | no | — |
| 58 | C2×C5×Q8 | C2×D4 | 1 | 1 | yes | characteristic |
| 59 | C2×C5×D4 | — | 2 | 2 | no | — |
| 60 | C2×C5×D4 | — | 2 | 2 | no | — |
| 61 | C2×C5×D4 | — | 4 | 2 | no | — |
| 62 | C2×C5×D4 | — | 4 | 2 | no | — |
| 63 | C2×C5×D4 | — | 4 | 2 | no | — |
| 64 | C2×C5×D4 | — | 4 | 2 | no | — |
| 65 | C2×C5×D4 | #41 | 1 | 1 | yes | characteristic |
| 66 | C2×C5×D4 | #41 | 1 | 2 | yes | — |
| 67 | C22×C4×C5 | C2×D4 | 1 | 2 | yes | — |
| 68 | C22×C4×C5 | C2×D4 | 1 | 1 | yes | characteristic |
| 69 | C5×D8 | — | 8 | 1 | no | — |
| 70 | C5×D8 | — | 8 | 1 | no | — |
| 71 | C5×M16 | — | 4 | 1 | no | — |
| 72 | C2×C5×C8 | — | 2 | 1 | no | — |
| 73 | C2×C5×C8 | — | 2 | 1 | no | — |
| 74 | C2×C5×C8 | — | 4 | 1 | no | — |
| 75 | C5×C4⋊C4 | — | 2 | 2 | no | — |
| 76 | C5×C22⋊C4 | — | 2 | 2 | no | — |
| 77 | C5×C22⋊C4 | — | 4 | 2 | no | — |
| 78 | C5×C22⋊C4 | — | 4 | 2 | no | — |
| 79 | C42×C5 | — | 2 | 2 | no | — |
| 80 | #681 | — | 2 | 2 | no | — |
| 81 | #681 | — | 2 | 4 | no | — |
| 82 | #681 | — | 2 | 2 | no | — |
| 83 | #681 | — | 4 | 1 | no | — |
| 84 | #681 | — | 4 | 1 | no | — |
| 85 | #681 | — | 4 | 2 | no | — |
| 86 | #681 | — | 4 | 2 | no | — |
| 87 | C2×#168 | — | 4 | 1 | no | — |
| 88 | C2×#168 | — | 4 | 1 | no | — |
| 89 | C2×#168 | — | 4 | 2 | no | — |
| 90 | C2×#168 | — | 4 | 2 | no | — |
| 91 | #568 | — | 5 | 1 | no | — |
| 92 | #535 | C2×D5 | 1 | 2 | yes | — |
| 93 | #482 | — | 2 | 2 | no | — |
| 94 | #469 | — | 2 | 1 | no | — |
| 95 | #466 | — | 2 | 1 | no | — |
| 96 | #447 | — | 2 | 2 | no | — |
| 97 | #430 | — | 10 | 1 | no | — |
| 98 | #413 | — | 5 | 1 | no | — |
| 99 | #413 | — | 5 | 2 | no | — |
| 100 | C2×#100 | — | 10 | 2 | no | — |
| 101 | #408 | C2×D5 | 1 | 1 | yes | characteristic |
| 102 | #355 | — | 10 | 4 | no | — |
| 103 | #344 | — | 10 | 2 | no | — |
| 104 | #341 | — | 10 | 1 | no | — |
| 105 | C23×C5 | — | 4 | 4 | no | — |
| 106 | C23×C5 | — | 4 | 2 | no | — |
| 107 | C23×C5 | — | 16 | 2 | no | — |
| 108 | C23×C5 | #123 | 1 | 2 | yes | — |
| 109 | C23×C5 | #124 | 1 | 1 | yes | characteristic |
| 110 | C5×Q8 | — | 4 | 1 | no | — |
| 111 | C5×D4 | — | 4 | 1 | no | — |
| 112 | C5×D4 | — | 4 | 2 | no | — |
| 113 | C5×D4 | — | 4 | 2 | no | — |
| 114 | C5×D4 | — | 4 | 2 | no | — |
| 115 | C5×D4 | — | 8 | 2 | no | — |
| 116 | C5×D4 | — | 16 | 2 | no | — |
| 117 | C5×D4 | — | 16 | 2 | no | — |
| 118 | C2×C4×C5 | — | 2 | 1 | no | — |
| 119 | C2×C4×C5 | — | 2 | 2 | no | — |
| 120 | C2×C4×C5 | — | 2 | 2 | no | — |
| 121 | C2×C4×C5 | — | 2 | 1 | no | — |
| 122 | C2×C4×C5 | — | 4 | 2 | no | — |
| 123 | C2×C4×C5 | #120 | 1 | 1 | yes | characteristic |
| 124 | C2×C4×C5 | #121 | 1 | 1 | yes | characteristic |
| 125 | C2×C4×C5 | #121 | 1 | 2 | yes | — |
| 126 | C5×C8 | — | 4 | 1 | no | — |
| 127 | C5×C8 | — | 4 | 1 | no | — |
| 128 | #168 | — | 4 | 2 | no | — |
| 129 | #168 | — | 4 | 2 | no | — |
| 130 | #168 | — | 8 | 2 | no | — |
| 131 | C2×#41 | C22×D5 | 1 | 1 | yes | characteristic |
| 132 | C22×D4 | C22⋊D5 | 1 | 2 | yes | — |
| 133 | C2×D8 | — | 2 | 1 | no | — |
| 134 | C2×D8 | — | 2 | 1 | no | — |
| 135 | #131 | — | 20 | 2 | no | — |
| 136 | C2×M16 | — | 10 | 1 | no | — |
| 137 | C2×M16 | — | 10 | 2 | no | — |
| 138 | C2×M16 | — | 10 | 1 | no | — |
| 139 | C2×M16 | — | 10 | 2 | no | — |
| 140 | C2×M16 | C22⋊D5 | 1 | 1 | yes | characteristic |
| 141 | C22×C8 | C22⋊D5 | 1 | 1 | yes | characteristic |
| 142 | #127 | — | 2 | 2 | no | — |
| 143 | #124 | — | 2 | 2 | no | — |
| 144 | #121 | — | 2 | 2 | no | — |
| 145 | #121 | — | 2 | 2 | no | — |
| 146 | #120 | — | 2 | 2 | no | — |
| 147 | #120 | — | 2 | 2 | no | — |
| 148 | #117 | C22⋊D5 | 1 | 2 | yes | — |
| 149 | #108 | — | 20 | 2 | no | — |
| 150 | #107 | — | 2 | 1 | no | — |
| 151 | #106 | — | 2 | 1 | no | — |
| 152 | #102 | — | 2 | 2 | no | — |
| 153 | #102 | — | 2 | 2 | no | — |
| 154 | #101 | — | 5 | 2 | no | — |
| 155 | #101 | — | 10 | 2 | no | — |
| 156 | #100 | — | 10 | 1 | no | — |
| 157 | #100 | — | 10 | 2 | no | — |
| 158 | #100 | — | 20 | 4 | no | — |
| 159 | C22⋊C8 | — | 2 | 2 | no | — |
| 160 | C22×C5 | — | 2 | 2 | no | — |
| 161 | C22×C5 | — | 2 | 1 | no | — |
| 162 | C22×C5 | — | 4 | 1 | no | — |
| 163 | C22×C5 | — | 4 | 2 | no | — |
| 164 | C22×C5 | — | 8 | 2 | no | — |
| 165 | C22×C5 | — | 8 | 2 | no | — |
| 166 | C22×C5 | — | 8 | 2 | no | — |
| 167 | C22×C5 | — | 16 | 4 | no | — |
| 168 | C22×C5 | #394 | 1 | 1 | yes | characteristic |
| 169 | C4×C5 | — | 2 | 1 | no | — |
| 170 | C4×C5 | — | 2 | 2 | no | — |
| 171 | C4×C5 | — | 8 | 2 | no | — |
| 172 | C4×C5 | B(4,𝔽2) | 1 | 1 | yes | characteristic |
| 173 | C4×C5 | #458 | 1 | 1 | yes | characteristic |
| 174 | C24 | — | 2 | 2 | no | — |
| 175 | #41 | — | 4 | 1 | no | — |
| 176 | #41 | — | 4 | 1 | no | — |
| 177 | C2×Q8 | D4×D5 | 1 | 1 | yes | characteristic |
| 178 | C2×D4 | — | 2 | 2 | no | — |
| 179 | C2×D4 | — | 2 | 2 | no | — |
| 180 | C2×D4 | — | 4 | 2 | no | — |
| 181 | C2×D4 | — | 4 | 2 | no | — |
| 182 | C2×D4 | — | 4 | 2 | no | — |
| 183 | C2×D4 | — | 4 | 2 | no | — |
| 184 | C2×D4 | #711 | 1 | 1 | yes | characteristic |
| 185 | C2×D4 | #711 | 1 | 2 | yes | — |
| 186 | C22×C4 | C2×C22⋊D5 | 1 | 1 | yes | characteristic |
| 187 | C22×C4 | C2×C22⋊D5 | 1 | 2 | yes | — |
| 188 | D8 | — | 8 | 1 | no | — |
| 189 | D8 | — | 8 | 1 | no | — |
| 190 | M16 | — | 4 | 1 | no | — |
| 191 | M16 | — | 10 | 4 | no | — |
| 192 | M16 | — | 10 | 2 | no | — |
| 193 | M16 | — | 10 | 2 | no | — |
| 194 | M16 | — | 20 | 1 | no | — |
| 195 | M16 | — | 20 | 1 | no | — |
| 196 | M16 | — | 20 | 2 | no | — |
| 197 | M16 | — | 20 | 2 | no | — |
| 198 | C2×C8 | — | 2 | 1 | no | — |
| 199 | C2×C8 | — | 2 | 1 | no | — |
| 200 | C2×C8 | — | 4 | 1 | no | — |
| 201 | C2×C8 | — | 20 | 1 | no | — |
| 202 | C2×C8 | — | 20 | 1 | no | — |
| 203 | C2×C8 | — | 20 | 2 | no | — |
| 204 | C2×C8 | — | 20 | 2 | no | — |
| 205 | C4⋊C4 | — | 2 | 2 | no | — |
| 206 | C22⋊C4 | — | 2 | 2 | no | — |
| 207 | C22⋊C4 | — | 4 | 2 | no | — |
| 208 | C22⋊C4 | — | 4 | 2 | no | — |
| 209 | C42 | — | 2 | 2 | no | — |
| 210 | C2×C5 | — | 2 | 1 | no | — |
| 211 | C2×C5 | — | 4 | 1 | no | — |
| 212 | C2×C5 | — | 4 | 2 | no | — |
| 213 | C2×C5 | — | 16 | 2 | no | — |
| 214 | C2×C5 | #2044 | 1 | 1 | yes | characteristic |
| 215 | C23 | — | 4 | 4 | no | — |
| 216 | C23 | — | 4 | 2 | no | — |
| 217 | C23 | — | 16 | 2 | no | — |
| 218 | C23 | #4081 | 1 | 2 | yes | — |
| 219 | C23 | #4082 | 1 | 1 | yes | characteristic |
| 220 | Q8 | — | 4 | 1 | no | — |
| 221 | D4 | — | 4 | 1 | no | — |
| 222 | D4 | — | 4 | 2 | no | — |
| 223 | D4 | — | 4 | 2 | no | — |
| 224 | D4 | — | 4 | 2 | no | — |
| 225 | D4 | — | 8 | 2 | no | — |
| 226 | D4 | — | 16 | 2 | no | — |
| 227 | D4 | — | 16 | 2 | no | — |
| 228 | C2×C4 | — | 2 | 1 | no | — |
| 229 | C2×C4 | — | 2 | 2 | no | — |
| 230 | C2×C4 | — | 2 | 1 | no | — |
| 231 | C2×C4 | — | 2 | 2 | no | — |
| 232 | C2×C4 | — | 4 | 2 | no | — |
| 233 | C2×C4 | #4084 | 1 | 1 | yes | characteristic |
| 234 | C2×C4 | #4085 | 1 | 2 | yes | — |
| 235 | C2×C4 | #4099 | 1 | 1 | yes | characteristic |
| 236 | C8 | — | 4 | 1 | no | — |
| 237 | C8 | — | 4 | 1 | no | — |
| 238 | C8 | — | 20 | 2 | no | — |
| 239 | C8 | — | 20 | 2 | no | — |
| 240 | C8 | — | 40 | 2 | no | — |
| 241 | C5 | #13429 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 242 | C22 | — | 2 | 2 | no | — |
| 243 | C22 | — | 2 | 1 | no | — |
| 244 | C22 | — | 4 | 1 | no | — |
| 245 | C22 | — | 4 | 2 | no | — |
| 246 | C22 | — | 8 | 2 | no | — |
| 247 | C22 | — | 8 | 2 | no | — |
| 248 | C22 | — | 8 | 2 | no | — |
| 249 | C22 | — | 16 | 4 | no | — |
| 250 | C22 | #65511 | 1 | 1 | yes | characteristic |
| 251 | C4 | — | 2 | 1 | no | — |
| 252 | C4 | — | 2 | 2 | no | — |
| 253 | C4 | — | 8 | 2 | no | — |
| 254 | C4 | #65532 | 1 | 1 | yes | characteristic |
| 255 | C4 | #65533 | 1 | 1 | yes | characteristic |
| 256 | C2 | — | 2 | 1 | no | — |
| 257 | C2 | — | 4 | 2 | no | — |
| 258 | C2 | — | 4 | 1 | no | — |
| 259 | C2 | — | 16 | 2 | no | — |
| 260 | C2 | #10602643 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #10602655 | C2 | semidirect |
| 3. #10602648 | C2 | semidirect |
| 4. #10600794 | C2 | semidirect |
| 5. #10598184 | C2 | semidirect |
| 241. C5 | #13429 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C5×#3199 | 2 |
| #10602655 | 2 |
| #10602648 | 2 |
| #10600794 | 2 |
| #10598184 | 2 |
| #13429 | 5 |