Finite Group DatabasebetaFinite Group DatabasebetaOrder 1280.
1078 conjugacy classes of subgroups; only the 207 normal subgroups are shown.
This group has 1078 conjugacy classes of subgroups — too many to draw. Only its 207 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 | D5×#430 | C2 | 1 | 3 | yes | maximal, complemented |
| 2 | C2×D5×#100 | C2 | 1 | 3 | yes | maximal |
| 3 | C5×#2069 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #10601523 | C2 | 1 | 3 | yes | maximal, complemented |
| 5 | #10600870 | C2 | 1 | 3 | yes | maximal, complemented |
| 6 | #10599515 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 7 | D5×#338 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 8 | C22×D4×D5 | C22 | 1 | 1 | yes | characteristic |
| 9 | C2×D5×M16 | C22 | 1 | 3 | yes | — |
| 10 | C2×D5×M16 | C22 | 1 | 3 | yes | — |
| 11 | C5×#430 | C22 | 1 | 3 | yes | complemented |
| 12 | C2×C5×#100 | C22 | 1 | 3 | yes | — |
| 13 | C2×#3965 | C22 | 1 | 3 | yes | — |
| 14 | C2×#3955 | C22 | 1 | 3 | yes | — |
| 15 | C2×#3955 | C22 | 1 | 3 | yes | — |
| 16 | #65414 | C22 | 1 | 3 | yes | complemented |
| 17 | #65413 | C22 | 1 | 3 | yes | complemented |
| 18 | #65413 | C22 | 1 | 3 | yes | complemented |
| 19 | #65182 | — | 2 | 6 | no | — |
| 20 | D5×#108 | — | 2 | 6 | no | — |
| 21 | D5×#100 | — | 2 | 6 | no | — |
| 22 | D5×#100 | — | 2 | 6 | no | — |
| 23 | C5×#338 | C22 | 1 | 1 | yes | characteristic, complemented |
| 24 | #64780 | C22 | 1 | 3 | yes | complemented |
| 25 | C2×#2069 | — | 5 | 1 | no | maximal, Sylow |
| 26 | C24×D5 | Q8 | 1 | 2 | yes | — |
| 27 | C22×C5×D4 | C23 | 1 | 1 | yes | characteristic |
| 28 | C22×C22⋊D5 | Q8 | 1 | 2 | yes | — |
| 29 | C2×D4×D5 | — | 2 | 3 | no | — |
| 30 | C2×D4×D5 | — | 2 | 3 | no | — |
| 31 | C2×D4×D5 | — | 2 | 3 | no | — |
| 32 | C2×D4×D5 | — | 2 | 3 | no | — |
| 33 | C22×D20 | C23 | 1 | 1 | yes | characteristic |
| 34 | C22×C4×D5 | C23 | 1 | 1 | yes | characteristic |
| 35 | C2×C5×M16 | C23 | 1 | 3 | yes | — |
| 36 | C2×C5×M16 | C23 | 1 | 3 | yes | — |
| 37 | C2×#681 | C23 | 1 | 3 | yes | — |
| 38 | C2×#681 | C23 | 1 | 3 | yes | — |
| 39 | D5×M16 | — | 2 | 3 | no | — |
| 40 | D5×M16 | — | 2 | 3 | no | — |
| 41 | D5×M16 | — | 2 | 3 | no | — |
| 42 | D5×M16 | — | 2 | 3 | no | — |
| 43 | D5×M16 | — | 2 | 3 | no | — |
| 44 | D5×M16 | — | 2 | 3 | no | — |
| 45 | D5×M16 | D4 | 1 | 6 | yes | — |
| 46 | D5×M16 | D4 | 1 | 6 | yes | — |
| 47 | C2×#676 | — | 2 | 3 | no | — |
| 48 | C2×#676 | — | 2 | 3 | no | — |
| 49 | C2×C8×D5 | — | 2 | 3 | no | — |
| 50 | C2×C8×D5 | — | 2 | 3 | no | — |
| 51 | C5×#108 | — | 2 | 6 | no | — |
| 52 | C5×#100 | — | 2 | 6 | no | — |
| 53 | #3965 | — | 2 | 6 | no | — |
| 54 | #3955 | — | 2 | 6 | no | — |
| 55 | #3955 | — | 2 | 6 | no | — |
| 56 | #3954 | — | 2 | 6 | no | — |
| 57 | #3954 | — | 2 | 6 | no | — |
| 58 | #3951 | — | 2 | 6 | no | — |
| 59 | C2×#430 | — | 5 | 3 | no | — |
| 60 | C22×#100 | — | 5 | 3 | no | — |
| 61 | #2069 | — | 5 | 3 | no | — |
| 62 | #2069 | — | 5 | 3 | no | — |
| 63 | #2069 | — | 5 | 1 | no | — |
| 64 | #2069 | D5 | 1 | 1 | yes | characteristic, direct product factor |
| 65 | C2×#338 | — | 5 | 1 | no | — |
| 66 | C24×C5 | C2×Q8 | 1 | 2 | yes | — |
| 67 | C23×D5 | — | 2 | 6 | no | — |
| 68 | C23×D5 | — | 2 | 6 | no | — |
| 69 | C23×D5 | — | 8 | 4 | no | — |
| 70 | C23×D5 | C2×Q8 | 1 | 2 | yes | — |
| 71 | C23×D5 | C2×Q8 | 1 | 1 | yes | characteristic |
| 72 | C2×C5×D4 | — | 2 | 3 | no | — |
| 73 | C2×C5×D4 | — | 2 | 3 | no | — |
| 74 | C22×C4×C5 | C24 | 1 | 1 | yes | characteristic |
| 75 | C2×C22⋊D5 | — | 2 | 6 | no | — |
| 76 | C2×C22⋊D5 | — | 2 | 6 | no | — |
| 77 | C22×Dic5 | C2×Q8 | 1 | 1 | yes | characteristic |
| 78 | D4×D5 | — | 4 | 3 | no | — |
| 79 | D4×D5 | — | 4 | 3 | no | — |
| 80 | D4×D5 | — | 4 | 3 | no | — |
| 81 | D4×D5 | — | 4 | 3 | no | — |
| 82 | D4×D5 | — | 4 | 3 | no | — |
| 83 | D4×D5 | — | 4 | 1 | no | — |
| 84 | C2×D20 | — | 2 | 3 | no | — |
| 85 | C2×D20 | — | 2 | 3 | no | — |
| 86 | C2×C4×D5 | C2×D4 | 1 | 3 | yes | — |
| 87 | C2×C4×D5 | C2×D4 | 1 | 3 | yes | — |
| 88 | C2×C4×D5 | #41 | 1 | 3 | yes | — |
| 89 | C2×C4×D5 | #41 | 1 | 3 | yes | — |
| 90 | C5×M16 | — | 2 | 3 | no | — |
| 91 | C5×M16 | — | 2 | 3 | no | — |
| 92 | C5×M16 | — | 2 | 3 | no | — |
| 93 | C5×M16 | C2×D4 | 1 | 6 | yes | — |
| 94 | C2×C5×C8 | — | 2 | 3 | no | — |
| 95 | C2×C5×C8 | — | 2 | 3 | no | — |
| 96 | #681 | — | 2 | 3 | no | — |
| 97 | #681 | — | 2 | 3 | no | — |
| 98 | #681 | — | 2 | 3 | no | — |
| 99 | #681 | C2×D4 | 1 | 6 | yes | — |
| 100 | C2×#168 | — | 2 | 3 | no | — |
| 101 | C2×#168 | — | 2 | 3 | no | — |
| 102 | #676 | — | 2 | 6 | no | — |
| 103 | #676 | — | 2 | 6 | no | — |
| 104 | #676 | — | 4 | 3 | no | — |
| 105 | #676 | — | 4 | 3 | no | — |
| 106 | C8×D5 | — | 2 | 6 | no | — |
| 107 | C8×D5 | — | 2 | 6 | no | — |
| 108 | C8×D5 | — | 4 | 3 | no | — |
| 109 | C8×D5 | — | 4 | 3 | no | — |
| 110 | C23×D4 | — | 5 | 1 | no | — |
| 111 | C22×M16 | — | 5 | 3 | no | — |
| 112 | C22×M16 | — | 5 | 3 | no | — |
| 113 | #430 | — | 5 | 3 | no | — |
| 114 | #430 | — | 5 | 3 | no | — |
| 115 | #430 | — | 5 | 3 | no | — |
| 116 | #430 | — | 10 | 6 | no | — |
| 117 | #430 | C2×D5 | 1 | 3 | yes | complemented |
| 118 | C2×#108 | — | 10 | 6 | no | — |
| 119 | C2×#100 | — | 5 | 3 | no | — |
| 120 | C2×#100 | — | 5 | 3 | no | — |
| 121 | C2×#100 | — | 5 | 3 | no | — |
| 122 | C2×#100 | — | 10 | 6 | no | — |
| 123 | C2×#100 | — | 10 | 6 | no | — |
| 124 | C2×#100 | C2×D5 | 1 | 3 | yes | — |
| 125 | #338 | — | 5 | 3 | no | — |
| 126 | #338 | C2×D5 | 1 | 1 | yes | characteristic, complemented |
| 127 | C23×C5 | — | 2 | 6 | no | — |
| 128 | C23×C5 | — | 8 | 2 | no | — |
| 129 | C23×C5 | C22×Q8 | 1 | 1 | yes | characteristic |
| 130 | C22×D5 | — | 2 | 6 | no | — |
| 131 | C22×D5 | — | 4 | 2 | no | — |
| 132 | C22×D5 | — | 4 | 6 | no | — |
| 133 | C22×D5 | — | 4 | 2 | no | — |
| 134 | C22×D5 | — | 8 | 12 | no | — |
| 135 | C22×D5 | — | 8 | 12 | no | — |
| 136 | C22×D5 | — | 8 | 2 | no | — |
| 137 | C22×D5 | C22⋊Q8 | 1 | 3 | yes | — |
| 138 | C22×D5 | C22⋊Q8 | 1 | 3 | yes | — |
| 139 | C5×D4 | — | 4 | 3 | no | — |
| 140 | C5×D4 | — | 4 | 1 | no | — |
| 141 | C2×C4×C5 | C22×D4 | 1 | 3 | yes | — |
| 142 | C2×C4×C5 | C2×#41 | 1 | 3 | yes | — |
| 143 | C22⋊D5 | — | 4 | 6 | no | — |
| 144 | C22⋊D5 | — | 4 | 2 | no | — |
| 145 | C2×Dic5 | C22⋊Q8 | 1 | 3 | yes | — |
| 146 | C2×Dic5 | C22⋊Q8 | 1 | 3 | yes | — |
| 147 | D20 | — | 4 | 3 | no | — |
| 148 | D20 | — | 4 | 1 | no | — |
| 149 | C4×D5 | #121 | 1 | 3 | yes | — |
| 150 | C4×D5 | #121 | 1 | 3 | yes | — |
| 151 | C4×D5 | #121 | 1 | 3 | yes | — |
| 152 | C4×D5 | #121 | 1 | 3 | yes | — |
| 153 | C4×D5 | #126 | 1 | 3 | yes | — |
| 154 | C4×D5 | #126 | 1 | 1 | yes | characteristic |
| 155 | C5×C8 | — | 2 | 6 | no | — |
| 156 | C5×C8 | — | 4 | 3 | no | — |
| 157 | #168 | — | 2 | 6 | no | — |
| 158 | #168 | — | 4 | 3 | no | — |
| 159 | C25 | — | 5 | 2 | no | — |
| 160 | C22×D4 | — | 5 | 2 | no | — |
| 161 | C22×D4 | — | 5 | 1 | no | — |
| 162 | C22×D4 | — | 10 | 3 | no | — |
| 163 | C22×D4 | — | 10 | 3 | no | — |
| 164 | C22×D4 | — | 10 | 3 | no | — |
| 165 | C22×D4 | — | 10 | 3 | no | — |
| 166 | C22×D4 | C22×D5 | 1 | 1 | yes | characteristic |
| 167 | C23×C4 | — | 5 | 1 | no | — |
| 168 | C2×M16 | — | 5 | 3 | no | — |
| 169 | C2×M16 | — | 5 | 6 | no | — |
| 170 | C2×M16 | — | 5 | 6 | no | — |
| 171 | C2×M16 | — | 5 | 3 | no | — |
| 172 | C2×M16 | — | 10 | 3 | no | — |
| 173 | C2×M16 | — | 10 | 3 | no | — |
| 174 | C2×M16 | — | 10 | 3 | no | — |
| 175 | C2×M16 | — | 10 | 3 | no | — |
| 176 | C2×M16 | — | 10 | 3 | no | — |
| 177 | C2×M16 | — | 10 | 3 | no | — |
| 178 | C2×M16 | — | 10 | 3 | no | — |
| 179 | C2×M16 | — | 10 | 3 | no | — |
| 180 | C2×M16 | C22×D5 | 1 | 3 | yes | — |
| 181 | C2×M16 | C22×D5 | 1 | 3 | yes | — |
| 182 | C22×C8 | — | 10 | 3 | no | — |
| 183 | C22×C8 | — | 10 | 3 | no | — |
| 184 | #108 | — | 2 | 6 | no | — |
| 185 | #108 | — | 10 | 6 | no | — |
| 186 | #108 | — | 10 | 6 | no | — |
| 187 | #108 | — | 10 | 6 | no | — |
| 188 | #100 | — | 2 | 6 | no | — |
| 189 | #100 | — | 10 | 6 | no | — |
| 190 | #100 | — | 10 | 6 | no | — |
| 191 | #100 | — | 10 | 6 | no | — |
| 192 | C22×C5 | — | 4 | 1 | no | — |
| 193 | C22×C5 | — | 4 | 2 | no | — |
| 194 | C22×C5 | — | 8 | 6 | no | — |
| 195 | C22×C5 | C2×C22⋊Q8 | 1 | 3 | yes | — |
| 196 | C2×D5 | — | 2 | 6 | no | — |
| 197 | C2×D5 | — | 4 | 3 | no | — |
| 198 | C2×D5 | — | 4 | 2 | no | — |
| 199 | C2×D5 | — | 8 | 12 | no | — |
| 200 | C2×D5 | — | 8 | 4 | no | — |
| 201 | C2×D5 | — | 8 | 6 | no | — |
| 202 | C2×D5 | #396 | 1 | 3 | yes | — |
| 203 | C2×D5 | #396 | 1 | 1 | yes | characteristic |
| 204 | C4×C5 | C2×#121 | 1 | 3 | yes | — |
| 205 | C4×C5 | C2×#126 | 1 | 1 | yes | characteristic |
| 206 | Dic5 | #396 | 1 | 3 | yes | — |
| 207 | Dic5 | #396 | 1 | 1 | yes | characteristic |
| 208 | C24 | — | 5 | 1 | no | — |
| 209 | C24 | — | 5 | 2 | no | — |
| 210 | C24 | — | 10 | 6 | no | — |
| 211 | C24 | — | 10 | 6 | no | — |
| 212 | C24 | — | 40 | 4 | no | — |
| 213 | C24 | Q8×D5 | 1 | 2 | yes | — |
| 214 | C2×D4 | — | 2 | 3 | no | — |
| 215 | C2×D4 | — | 2 | 3 | no | — |
| 216 | C2×D4 | — | 10 | 6 | no | — |
| 217 | C2×D4 | — | 10 | 6 | no | — |
| 218 | C2×D4 | — | 10 | 3 | no | — |
| 219 | C2×D4 | — | 10 | 3 | no | — |
| 220 | C2×D4 | — | 20 | 3 | no | — |
| 221 | C2×D4 | — | 20 | 3 | no | — |
| 222 | C2×D4 | — | 20 | 3 | no | — |
| 223 | C2×D4 | — | 20 | 3 | no | — |
| 224 | C2×D4 | — | 20 | 1 | no | — |
| 225 | C2×D4 | — | 20 | 3 | no | — |
| 226 | C22×C4 | — | 5 | 1 | no | — |
| 227 | C22×C4 | — | 5 | 3 | no | — |
| 228 | C22×C4 | — | 5 | 3 | no | — |
| 229 | C22×C4 | — | 5 | 3 | no | — |
| 230 | C22×C4 | — | 5 | 3 | no | — |
| 231 | C22×C4 | C23×D5 | 1 | 1 | yes | characteristic |
| 232 | M16 | — | 2 | 3 | no | — |
| 233 | M16 | — | 2 | 3 | no | — |
| 234 | M16 | — | 2 | 3 | no | — |
| 235 | M16 | — | 5 | 6 | no | — |
| 236 | M16 | — | 10 | 6 | no | — |
| 237 | M16 | — | 10 | 6 | no | — |
| 238 | M16 | — | 10 | 3 | no | — |
| 239 | M16 | — | 10 | 3 | no | — |
| 240 | M16 | — | 10 | 3 | no | — |
| 241 | M16 | — | 20 | 3 | no | — |
| 242 | M16 | — | 20 | 3 | no | — |
| 243 | M16 | D4×D5 | 1 | 6 | yes | — |
| 244 | C2×C8 | — | 2 | 3 | no | — |
| 245 | C2×C8 | — | 2 | 3 | no | — |
| 246 | C2×C8 | — | 10 | 6 | no | — |
| 247 | C2×C8 | — | 10 | 6 | no | — |
| 248 | C2×C8 | — | 10 | 3 | no | — |
| 249 | C2×C8 | — | 10 | 3 | no | — |
| 250 | C2×C8 | — | 20 | 3 | no | — |
| 251 | C2×C8 | — | 20 | 3 | no | — |
| 252 | C2×C5 | — | 2 | 3 | no | — |
| 253 | C2×C5 | — | 8 | 2 | no | — |
| 254 | C2×C5 | C2×#396 | 1 | 1 | yes | characteristic |
| 255 | D5 | — | 2 | 3 | no | — |
| 256 | D5 | — | 8 | 2 | no | — |
| 257 | D5 | #2069 | 1 | 2 | yes | direct product factor |
| 258 | C23 | — | 2 | 6 | no | — |
| 259 | C23 | — | 5 | 3 | no | — |
| 260 | C23 | — | 5 | 3 | no | — |
| 261 | C23 | — | 8 | 2 | no | — |
| 262 | C23 | — | 10 | 6 | no | — |
| 263 | C23 | — | 20 | 2 | no | — |
| 264 | C23 | — | 20 | 6 | no | — |
| 265 | C23 | — | 20 | 2 | no | — |
| 266 | C23 | — | 40 | 2 | no | — |
| 267 | C23 | — | 40 | 12 | no | — |
| 268 | C23 | — | 40 | 12 | no | — |
| 269 | C23 | C2×Q8×D5 | 1 | 1 | yes | characteristic |
| 270 | D4 | — | 4 | 3 | no | — |
| 271 | D4 | — | 4 | 1 | no | — |
| 272 | D4 | — | 20 | 6 | no | — |
| 273 | D4 | — | 20 | 2 | no | — |
| 274 | D4 | — | 20 | 3 | no | — |
| 275 | D4 | — | 20 | 1 | no | — |
| 276 | C2×C4 | — | 5 | 3 | no | — |
| 277 | C2×C4 | — | 5 | 3 | no | — |
| 278 | C2×C4 | — | 5 | 3 | no | — |
| 279 | C2×C4 | — | 5 | 3 | no | — |
| 280 | C2×C4 | — | 5 | 3 | no | — |
| 281 | C2×C4 | — | 5 | 3 | no | — |
| 282 | C2×C4 | — | 5 | 1 | no | — |
| 283 | C2×C4 | — | 5 | 3 | no | — |
| 284 | C2×C4 | C2×D4×D5 | 1 | 3 | yes | — |
| 285 | C2×C4 | D5×#41 | 1 | 3 | yes | — |
| 286 | C8 | — | 2 | 6 | no | — |
| 287 | C8 | — | 4 | 3 | no | — |
| 288 | C8 | — | 10 | 6 | no | — |
| 289 | C8 | — | 20 | 3 | no | — |
| 290 | C5 | C2×#2069 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 291 | C22 | — | 4 | 1 | no | — |
| 292 | C22 | — | 4 | 2 | no | — |
| 293 | C22 | — | 5 | 1 | no | — |
| 294 | C22 | — | 5 | 3 | no | — |
| 295 | C22 | — | 8 | 6 | no | — |
| 296 | C22 | — | 10 | 6 | no | — |
| 297 | C22 | — | 20 | 3 | no | — |
| 298 | C22 | — | 20 | 2 | no | — |
| 299 | C22 | — | 40 | 6 | no | — |
| 300 | C22 | — | 40 | 4 | no | — |
| 301 | C22 | — | 40 | 12 | no | — |
| 302 | C22 | D5×C22⋊Q8 | 1 | 3 | yes | — |
| 303 | C4 | — | 5 | 3 | no | — |
| 304 | C4 | — | 5 | 1 | no | — |
| 305 | C4 | D5×#121 | 1 | 3 | yes | — |
| 306 | C4 | D5×#126 | 1 | 1 | yes | characteristic |
| 307 | C2 | — | 2 | 3 | no | — |
| 308 | C2 | — | 5 | 2 | no | — |
| 309 | C2 | — | 8 | 2 | no | — |
| 310 | C2 | — | 10 | 3 | no | — |
| 311 | C2 | — | 40 | 2 | no | — |
| 312 | C2 | D5×#396 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. D5×#430 | C2 | semidirect |
| 3. C5×#2069 | C2 | semidirect |
| 4. #10601523 | C2 | semidirect |
| 5. #10600870 | C2 | semidirect |
| 6. #10599515 | C2 | semidirect |
| 7. D5×#338 | C2 | semidirect |
| 11. C5×#430 | C22 | semidirect |
| 16. #65414 | C22 | semidirect |
| 17. #65413 | C22 | semidirect |
| 18. #65413 | C22 | semidirect |
| 23. C5×#338 | C22 | semidirect |
| 24. #64780 | C22 | semidirect |
| 64. #2069 | D5 | direct (N × H) |
| 117. #430 | C2×D5 | semidirect |
| 126. #338 | C2×D5 | semidirect |
| 257. D5 | #2069 | direct (N × H) |
| 290. C5 | C2×#2069 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| D5×#430 | 2 |
| C2×D5×#100 | 2 |
| C5×#2069 | 2 |
| #10601523 | 2 |
| #10600870 | 2 |
| #10599515 | 2 |
| D5×#338 | 2 |
| C2×#2069 | 5 |