Finite Group DatabasebetaFinite Group DatabasebetaOrder 1280.
882 conjugacy classes of subgroups; only the 171 normal subgroups are shown.
This group has 882 conjugacy classes of subgroups — too many to draw. Only its 171 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 | C2×C5⋊#564 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | C2×#65356 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | C2×#65002 | C2 | 1 | 2 | yes | maximal |
| 4 | C2×C5×#330 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | C2×#64685 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | #10599168 | C2 | 1 | 8 | yes | maximal, direct product factor |
| 7 | C2×C5×#125 | C22 | 1 | 1 | yes | characteristic |
| 8 | C5⋊#564 | — | 2 | 4 | no | — |
| 9 | C5⋊#564 | — | 2 | 2 | no | — |
| 10 | C5⋊#564 | C22 | 1 | 4 | yes | complemented |
| 11 | C2×#4036 | — | 2 | 2 | no | — |
| 12 | C2×#4035 | — | 2 | 2 | no | — |
| 13 | C2×#4035 | — | 2 | 2 | no | — |
| 14 | C2×#4020 | C22 | 1 | 1 | yes | characteristic |
| 15 | C2×#4016 | C22 | 1 | 2 | yes | — |
| 16 | C2×C5×#97 | C22 | 1 | 2 | yes | — |
| 17 | C2×#3953 | — | 2 | 2 | no | — |
| 18 | C2×#3953 | — | 2 | 2 | no | — |
| 19 | #65356 | — | 2 | 2 | no | — |
| 20 | #65356 | C22 | 1 | 4 | yes | — |
| 21 | C2×#3941 | — | 2 | 2 | no | — |
| 22 | C2×#3935 | C22 | 1 | 1 | yes | characteristic |
| 23 | #65002 | — | 2 | 4 | no | — |
| 24 | #65002 | C22 | 1 | 8 | yes | — |
| 25 | C5×#330 | C22 | 1 | 4 | yes | complemented |
| 26 | #64685 | C22 | 1 | 8 | yes | complemented |
| 27 | C2×#1658 | — | 5 | 1 | no | maximal, Sylow |
| 28 | C22×D20 | — | 2 | 2 | no | — |
| 29 | C22×D20 | — | 2 | 1 | no | — |
| 30 | C22×C5×C8 | — | 2 | 2 | no | — |
| 31 | C5×#125 | — | 2 | 2 | no | — |
| 32 | C5×#125 | D4 | 1 | 2 | yes | — |
| 33 | C5×#125 | C23 | 1 | 2 | yes | — |
| 34 | C2×C5×C4⋊C4 | — | 2 | 2 | no | — |
| 35 | C2×C5×C4⋊C4 | D4 | 1 | 2 | yes | — |
| 36 | C2×C42×C5 | C23 | 1 | 1 | yes | characteristic |
| 37 | C2×C4⋊Dic5 | — | 2 | 2 | no | — |
| 38 | C2×C4⋊Dic5 | D4 | 1 | 2 | yes | — |
| 39 | C2×C4⋊Dic5 | D4 | 1 | 2 | yes | — |
| 40 | C2×#683 | — | 2 | 2 | no | — |
| 41 | C2×#683 | — | 2 | 2 | no | — |
| 42 | C2×#683 | — | 2 | 2 | no | — |
| 43 | C2×#683 | — | 2 | 2 | no | — |
| 44 | C2×C4×Dic5 | — | 2 | 2 | no | — |
| 45 | C22×#168 | — | 2 | 1 | no | — |
| 46 | #4036 | — | 2 | 8 | no | — |
| 47 | #4036 | — | 2 | 8 | no | — |
| 48 | #4035 | — | 2 | 8 | no | — |
| 49 | #4035 | — | 2 | 8 | no | — |
| 50 | #4035 | — | 2 | 8 | no | — |
| 51 | #4035 | — | 2 | 8 | no | — |
| 52 | #4020 | — | 2 | 1 | no | — |
| 53 | #4020 | — | 2 | 2 | no | — |
| 54 | #4020 | C23 | 1 | 2 | yes | — |
| 55 | #4016 | — | 2 | 2 | no | — |
| 56 | #4016 | — | 2 | 2 | no | — |
| 57 | #4016 | D4 | 1 | 4 | yes | — |
| 58 | #4016 | C23 | 1 | 4 | yes | — |
| 59 | C5×#97 | — | 2 | 2 | no | — |
| 60 | C5×#97 | C23 | 1 | 4 | yes | — |
| 61 | #3953 | — | 2 | 8 | no | — |
| 62 | #3953 | — | 2 | 8 | no | — |
| 63 | #3941 | — | 2 | 8 | no | — |
| 64 | #3935 | — | 2 | 1 | no | — |
| 65 | #3935 | C23 | 1 | 2 | yes | — |
| 66 | C2×#564 | — | 5 | 1 | no | — |
| 67 | C2×#490 | — | 5 | 2 | no | — |
| 68 | C2×#490 | — | 5 | 1 | no | — |
| 69 | #1658 | — | 5 | 8 | no | — |
| 70 | C2×#330 | — | 5 | 2 | no | — |
| 71 | C2×#330 | D5 | 1 | 1 | yes | characteristic, complemented |
| 72 | C23×D5 | — | 4 | 1 | no | — |
| 73 | C22×C4×C5 | — | 2 | 2 | no | — |
| 74 | C22×C4×C5 | C2×D4 | 1 | 1 | yes | characteristic |
| 75 | C22×C4×C5 | C2×D4 | 1 | 2 | yes | — |
| 76 | C22×Dic5 | — | 2 | 2 | no | — |
| 77 | C22×Dic5 | — | 2 | 2 | no | — |
| 78 | C2×D20 | — | 2 | 2 | no | — |
| 79 | C2×D20 | — | 2 | 4 | no | — |
| 80 | C2×D20 | — | 4 | 4 | no | — |
| 81 | C2×D20 | — | 4 | 8 | no | — |
| 82 | C2×D20 | — | 4 | 1 | no | — |
| 83 | C2×D20 | — | 4 | 2 | no | — |
| 84 | C2×C5×C8 | — | 2 | 4 | no | — |
| 85 | C2×C5×C8 | — | 4 | 4 | no | — |
| 86 | C5×C4⋊C4 | — | 2 | 4 | no | — |
| 87 | C5×C4⋊C4 | — | 2 | 2 | no | — |
| 88 | C5×C4⋊C4 | — | 2 | 4 | no | — |
| 89 | C5×C4⋊C4 | C2×D4 | 1 | 4 | yes | — |
| 90 | C42×C5 | C2×D4 | 1 | 1 | yes | characteristic |
| 91 | C42×C5 | C2×D4 | 1 | 2 | yes | — |
| 92 | C42×C5 | C24 | 1 | 1 | yes | characteristic |
| 93 | C4⋊Dic5 | — | 2 | 8 | no | — |
| 94 | C4⋊Dic5 | — | 2 | 2 | no | — |
| 95 | C4⋊Dic5 | — | 2 | 2 | no | — |
| 96 | C4⋊Dic5 | C2×D4 | 1 | 4 | yes | — |
| 97 | C4⋊Dic5 | C2×D4 | 1 | 4 | yes | — |
| 98 | #683 | — | 2 | 8 | no | — |
| 99 | #683 | — | 2 | 8 | no | — |
| 100 | #683 | — | 2 | 4 | no | — |
| 101 | #683 | — | 2 | 4 | no | — |
| 102 | #683 | — | 2 | 4 | no | — |
| 103 | #683 | — | 2 | 4 | no | — |
| 104 | C4×Dic5 | — | 2 | 8 | no | — |
| 105 | C2×#168 | — | 2 | 2 | no | — |
| 106 | C2×#168 | — | 4 | 2 | no | — |
| 107 | #564 | — | 5 | 4 | no | — |
| 108 | #564 | — | 10 | 4 | no | — |
| 109 | #564 | — | 10 | 2 | no | — |
| 110 | C2×#127 | — | 5 | 1 | no | — |
| 111 | C2×#125 | — | 5 | 2 | no | — |
| 112 | C2×#125 | — | 10 | 2 | no | — |
| 113 | C2×#125 | — | 10 | 2 | no | — |
| 114 | C2×#125 | — | 10 | 2 | no | — |
| 115 | C2×#125 | C2×D5 | 1 | 1 | yes | characteristic |
| 116 | #490 | — | 5 | 4 | no | — |
| 117 | #490 | — | 5 | 8 | no | — |
| 118 | #490 | — | 10 | 4 | no | — |
| 119 | #490 | — | 10 | 2 | no | — |
| 120 | C2×#102 | — | 10 | 2 | no | — |
| 121 | C2×#102 | — | 10 | 2 | no | — |
| 122 | C2×#102 | — | 10 | 2 | no | — |
| 123 | C2×#97 | — | 5 | 1 | no | — |
| 124 | C2×#97 | C2×D5 | 1 | 2 | yes | — |
| 125 | #330 | — | 5 | 8 | no | — |
| 126 | #330 | C2×D5 | 1 | 4 | yes | complemented |
| 127 | C23×C5 | #120 | 1 | 1 | yes | characteristic |
| 128 | C22×D5 | — | 4 | 2 | no | — |
| 129 | C22×D5 | — | 8 | 2 | no | — |
| 130 | C22×D5 | — | 8 | 4 | no | — |
| 131 | C2×C4×C5 | — | 2 | 2 | no | — |
| 132 | C2×C4×C5 | — | 2 | 4 | no | — |
| 133 | C2×C4×C5 | — | 2 | 4 | no | — |
| 134 | C2×C4×C5 | — | 4 | 4 | no | — |
| 135 | C2×C4×C5 | #120 | 1 | 1 | yes | characteristic |
| 136 | C2×C4×C5 | #120 | 1 | 2 | yes | — |
| 137 | C2×C4×C5 | C22×D4 | 1 | 1 | yes | characteristic |
| 138 | C2×C4×C5 | C22×D4 | 1 | 2 | yes | — |
| 139 | C2×Dic5 | — | 2 | 4 | no | — |
| 140 | C2×Dic5 | — | 2 | 4 | no | — |
| 141 | C2×Dic5 | — | 4 | 4 | no | — |
| 142 | C2×Dic5 | — | 4 | 4 | no | — |
| 143 | D20 | — | 4 | 4 | no | — |
| 144 | D20 | — | 4 | 8 | no | — |
| 145 | D20 | — | 8 | 2 | no | — |
| 146 | D20 | — | 8 | 4 | no | — |
| 147 | C5×C8 | — | 4 | 4 | no | — |
| 148 | #168 | — | 4 | 2 | no | — |
| 149 | C22×D4 | — | 10 | 1 | no | — |
| 150 | C22×D4 | — | 10 | 2 | no | — |
| 151 | C22×C8 | — | 2 | 2 | no | — |
| 152 | C22×C8 | — | 10 | 1 | no | — |
| 153 | #127 | — | 5 | 2 | no | — |
| 154 | #127 | — | 10 | 1 | no | — |
| 155 | #127 | — | 10 | 2 | no | — |
| 156 | #125 | — | 2 | 2 | no | — |
| 157 | #125 | — | 5 | 4 | no | — |
| 158 | #125 | — | 5 | 4 | no | — |
| 159 | #125 | — | 10 | 8 | no | — |
| 160 | #125 | — | 10 | 8 | no | — |
| 161 | #125 | — | 10 | 8 | no | — |
| 162 | #125 | — | 10 | 8 | no | — |
| 163 | #125 | — | 10 | 8 | no | — |
| 164 | #125 | — | 10 | 8 | no | — |
| 165 | #125 | — | 10 | 2 | no | — |
| 166 | #125 | — | 10 | 2 | no | — |
| 167 | #125 | D20 | 1 | 2 | yes | — |
| 168 | #125 | C22×D5 | 1 | 2 | yes | — |
| 169 | C2×C4⋊C4 | — | 2 | 2 | no | — |
| 170 | C2×C4⋊C4 | — | 5 | 2 | no | — |
| 171 | C2×C4⋊C4 | — | 5 | 2 | no | — |
| 172 | C2×C4⋊C4 | — | 10 | 2 | no | — |
| 173 | C2×C4⋊C4 | — | 10 | 2 | no | — |
| 174 | C2×C4⋊C4 | — | 10 | 2 | no | — |
| 175 | C2×C4⋊C4 | — | 10 | 2 | no | — |
| 176 | C2×C4⋊C4 | — | 10 | 2 | no | — |
| 177 | C2×C4⋊C4 | D20 | 1 | 2 | yes | — |
| 178 | C2×C42 | — | 10 | 2 | no | — |
| 179 | C2×C42 | C22×D5 | 1 | 1 | yes | characteristic |
| 180 | #102 | — | 10 | 8 | no | — |
| 181 | #102 | — | 10 | 8 | no | — |
| 182 | #102 | — | 10 | 8 | no | — |
| 183 | #97 | — | 2 | 2 | no | — |
| 184 | #97 | — | 5 | 2 | no | — |
| 185 | #97 | — | 10 | 1 | no | — |
| 186 | #97 | C22×D5 | 1 | 4 | yes | — |
| 187 | C22×C5 | #454 | 1 | 2 | yes | — |
| 188 | C22×C5 | #454 | 1 | 4 | yes | — |
| 189 | C22×C5 | C2×#120 | 1 | 1 | yes | characteristic |
| 190 | C2×D5 | — | 8 | 2 | no | — |
| 191 | C2×D5 | — | 8 | 4 | no | — |
| 192 | C2×D5 | — | 16 | 4 | no | — |
| 193 | C4×C5 | — | 2 | 1 | no | — |
| 194 | C4×C5 | — | 2 | 2 | no | — |
| 195 | C4×C5 | — | 2 | 1 | no | — |
| 196 | C4×C5 | — | 2 | 2 | no | — |
| 197 | C4×C5 | — | 4 | 4 | no | — |
| 198 | Dic5 | — | 4 | 4 | no | — |
| 199 | Dic5 | — | 4 | 4 | no | — |
| 200 | C24 | — | 20 | 1 | no | — |
| 201 | C2×D4 | — | 10 | 2 | no | — |
| 202 | C2×D4 | — | 10 | 4 | no | — |
| 203 | C2×D4 | — | 20 | 4 | no | — |
| 204 | C2×D4 | — | 20 | 8 | no | — |
| 205 | C2×D4 | — | 20 | 2 | no | — |
| 206 | C2×D4 | — | 20 | 1 | no | — |
| 207 | C22×C4 | — | 2 | 2 | no | — |
| 208 | C22×C4 | — | 10 | 2 | no | — |
| 209 | C22×C4 | — | 10 | 2 | no | — |
| 210 | C22×C4 | C2×D20 | 1 | 1 | yes | characteristic |
| 211 | C22×C4 | D4×D5 | 1 | 2 | yes | — |
| 212 | C2×C8 | — | 2 | 4 | no | — |
| 213 | C2×C8 | — | 4 | 4 | no | — |
| 214 | C2×C8 | — | 10 | 2 | no | — |
| 215 | C2×C8 | — | 20 | 2 | no | — |
| 216 | C4⋊C4 | — | 2 | 4 | no | — |
| 217 | C4⋊C4 | — | 2 | 4 | no | — |
| 218 | C4⋊C4 | — | 2 | 2 | no | — |
| 219 | C4⋊C4 | — | 5 | 4 | no | — |
| 220 | C4⋊C4 | — | 5 | 4 | no | — |
| 221 | C4⋊C4 | — | 10 | 8 | no | — |
| 222 | C4⋊C4 | — | 10 | 8 | no | — |
| 223 | C4⋊C4 | — | 10 | 8 | no | — |
| 224 | C4⋊C4 | — | 10 | 4 | no | — |
| 225 | C4⋊C4 | — | 10 | 4 | no | — |
| 226 | C4⋊C4 | — | 10 | 2 | no | — |
| 227 | C4⋊C4 | — | 10 | 2 | no | — |
| 228 | C4⋊C4 | — | 10 | 4 | no | — |
| 229 | C4⋊C4 | — | 10 | 4 | no | — |
| 230 | C4⋊C4 | C2×D20 | 1 | 4 | yes | — |
| 231 | C42 | — | 10 | 8 | no | — |
| 232 | C42 | C2×D20 | 1 | 1 | yes | characteristic |
| 233 | C42 | D4×D5 | 1 | 2 | yes | — |
| 234 | C42 | C23×D5 | 1 | 1 | yes | characteristic |
| 235 | C2×C5 | #1658 | 1 | 4 | yes | complemented |
| 236 | C2×C5 | C2×#454 | 1 | 1 | yes | characteristic |
| 237 | C2×C5 | C2×#454 | 1 | 2 | yes | — |
| 238 | D5 | — | 16 | 2 | no | — |
| 239 | C23 | — | 20 | 2 | no | — |
| 240 | C23 | — | 40 | 2 | no | — |
| 241 | C23 | — | 40 | 4 | no | — |
| 242 | C23 | #4028 | 1 | 1 | yes | characteristic |
| 243 | D4 | — | 20 | 4 | no | — |
| 244 | D4 | — | 20 | 8 | no | — |
| 245 | D4 | — | 40 | 2 | no | — |
| 246 | D4 | — | 40 | 4 | no | — |
| 247 | C2×C4 | — | 2 | 2 | no | — |
| 248 | C2×C4 | — | 2 | 4 | no | — |
| 249 | C2×C4 | — | 2 | 4 | no | — |
| 250 | C2×C4 | — | 4 | 4 | no | — |
| 251 | C2×C4 | — | 10 | 4 | no | — |
| 252 | C2×C4 | — | 10 | 4 | no | — |
| 253 | C2×C4 | — | 20 | 4 | no | — |
| 254 | C2×C4 | — | 20 | 4 | no | — |
| 255 | C2×C4 | #4028 | 1 | 2 | yes | — |
| 256 | C2×C4 | #4028 | 1 | 1 | yes | characteristic |
| 257 | C2×C4 | C22×D20 | 1 | 1 | yes | characteristic |
| 258 | C2×C4 | C2×D4×D5 | 1 | 2 | yes | — |
| 259 | C8 | — | 4 | 4 | no | — |
| 260 | C8 | — | 20 | 2 | no | — |
| 261 | C5 | C2×#1658 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 262 | C22 | — | 40 | 2 | no | — |
| 263 | C22 | — | 40 | 4 | no | — |
| 264 | C22 | — | 80 | 4 | no | — |
| 265 | C22 | #65042 | 1 | 2 | yes | — |
| 266 | C22 | #65116 | 1 | 4 | yes | — |
| 267 | C22 | C2×#4028 | 1 | 1 | yes | characteristic |
| 268 | C4 | — | 2 | 1 | no | — |
| 269 | C4 | — | 2 | 1 | no | — |
| 270 | C4 | — | 2 | 2 | no | — |
| 271 | C4 | — | 2 | 2 | no | — |
| 272 | C4 | — | 4 | 4 | no | — |
| 273 | C4 | — | 20 | 4 | no | — |
| 274 | C4 | — | 20 | 4 | no | — |
| 275 | C2 | — | 80 | 2 | no | — |
| 276 | C2 | #10599168 | 1 | 4 | yes | direct product factor |
| 277 | C2 | C2×#65042 | 1 | 1 | yes | characteristic |
| 278 | C2 | C2×#65116 | 1 | 2 | yes | — |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×C5⋊#564 | C2 | semidirect |
| 4. C2×C5×#330 | C2 | semidirect |
| 5. C2×#64685 | C2 | semidirect |
| 6. #10599168 | C2 | direct (N × H) |
| 10. C5⋊#564 | C22 | semidirect |
| 25. C5×#330 | C22 | semidirect |
| 26. #64685 | C22 | semidirect |
| 71. C2×#330 | D5 | semidirect |
| 126. #330 | C2×D5 | semidirect |
| 235. C2×C5 | #1658 | semidirect |
| 261. C5 | C2×#1658 | semidirect |
| 276. C2 | #10599168 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C5⋊#564 | 2 |
| C2×#65356 | 2 |
| C2×#65002 | 2 |
| C2×C5×#330 | 2 |
| C2×#64685 | 2 |
| #10599168 | 2 |
| C2×#1658 | 5 |