Finite Group DatabasebetaFinite Group DatabasebetaOrder 1280.
442 conjugacy classes of subgroups; only the 72 normal subgroups are shown.
This group has 442 conjugacy classes of subgroups — too many to draw. Only its 72 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 | #10617379 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #10613619 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | #10604638 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #10604627 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | #10604627 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | #10598533 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 7 | #10598528 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 8 | #66059 | C22 | 1 | 1 | yes | characteristic |
| 9 | #66059 | C22 | 1 | 1 | yes | characteristic |
| 10 | #65787 | — | 2 | 2 | no | — |
| 11 | #65704 | — | 2 | 2 | no | — |
| 12 | #65704 | — | 2 | 2 | no | — |
| 13 | #65690 | C22 | 1 | 1 | yes | characteristic |
| 14 | #65684 | C22 | 1 | 1 | yes | characteristic |
| 15 | #65684 | C22 | 1 | 1 | yes | characteristic |
| 16 | #65684 | C22 | 1 | 1 | yes | characteristic |
| 17 | C5×#498 | C4 | 1 | 1 | yes | characteristic, complemented |
| 18 | #65377 | C4 | 1 | 1 | yes | characteristic, complemented |
| 19 | #65377 | C4 | 1 | 1 | yes | characteristic, complemented |
| 20 | #65158 | — | 2 | 2 | no | — |
| 21 | #65154 | — | 2 | 2 | no | — |
| 22 | #64984 | C4 | 1 | 1 | yes | characteristic, complemented |
| 23 | C4×#676 | C22 | 1 | 1 | yes | characteristic |
| 24 | #64927 | — | 2 | 2 | no | — |
| 25 | #64868 | — | 2 | 2 | no | — |
| 26 | #64868 | — | 2 | 2 | no | — |
| 27 | #9417 | — | 5 | 1 | no | maximal, Sylow |
| 28 | C2×#713 | — | 2 | 1 | no | — |
| 29 | C2×#713 | — | 2 | 1 | no | — |
| 30 | C2×#700 | — | 2 | 2 | no | — |
| 31 | C2×D5⋊C8 | — | 2 | 2 | no | — |
| 32 | C5×C4⋊Q8 | C2×C4 | 1 | 1 | yes | characteristic |
| 33 | C5×C4⋊Q8 | C2×C4 | 1 | 1 | yes | characteristic |
| 34 | #4093 | — | 2 | 1 | no | — |
| 35 | #4093 | — | 2 | 1 | no | — |
| 36 | C2×D40 | — | 2 | 2 | no | — |
| 37 | C2×#676 | — | 2 | 1 | no | — |
| 38 | #4041 | — | 2 | 1 | no | — |
| 39 | #4041 | — | 2 | 1 | no | — |
| 40 | #4041 | — | 2 | 1 | no | — |
| 41 | #4041 | D4 | 1 | 2 | yes | — |
| 42 | #4039 | — | 2 | 1 | no | — |
| 43 | #4039 | — | 2 | 1 | no | — |
| 44 | #4039 | — | 2 | 1 | no | — |
| 45 | #4039 | D4 | 1 | 2 | yes | — |
| 46 | #4020 | C2×C4 | 1 | 1 | yes | characteristic |
| 47 | #4020 | C2×C4 | 1 | 1 | yes | characteristic |
| 48 | C42×D5 | C23 | 1 | 1 | yes | characteristic |
| 49 | (C2×AGL(1,𝔽5))⋊C4 | — | 2 | 1 | no | — |
| 50 | (C2×AGL(1,𝔽5))⋊C4 | — | 2 | 1 | no | — |
| 51 | (C2×AGL(1,𝔽5))⋊C4 | — | 2 | 1 | no | — |
| 52 | #4004 | — | 2 | 1 | no | — |
| 53 | #4004 | — | 2 | 1 | no | — |
| 54 | #4004 | — | 2 | 1 | no | — |
| 55 | #4003 | — | 2 | 4 | no | — |
| 56 | #4003 | — | 2 | 1 | no | — |
| 57 | #4003 | — | 2 | 1 | no | — |
| 58 | #4003 | D4 | 1 | 2 | yes | — |
| 59 | #4002 | — | 2 | 1 | no | — |
| 60 | #4002 | — | 2 | 1 | no | — |
| 61 | #4002 | D4 | 1 | 2 | yes | — |
| 62 | C5×#106 | — | 2 | 2 | no | — |
| 63 | C4×C5×C8 | C2×C4 | 1 | 1 | yes | characteristic |
| 64 | #3947 | — | 2 | 1 | no | — |
| 65 | #3941 | — | 2 | 2 | no | — |
| 66 | #3941 | — | 2 | 2 | no | — |
| 67 | C4×#168 | C2×C4 | 1 | 1 | yes | characteristic |
| 68 | #3184 | — | 5 | 1 | no | — |
| 69 | #2980 | — | 5 | 1 | no | — |
| 70 | #1536 | — | 5 | 1 | no | — |
| 71 | #1514 | — | 5 | 1 | no | — |
| 72 | #1514 | — | 5 | 1 | no | — |
| 73 | #1291 | — | 5 | 1 | no | — |
| 74 | #1288 | — | 5 | 1 | no | — |
| 75 | C22×AGL(1,𝔽5) | — | 4 | 1 | no | — |
| 76 | C2×C5×Q8 | — | 2 | 1 | no | — |
| 77 | C2×C5×Q8 | — | 2 | 1 | no | — |
| 78 | #713 | — | 8 | 1 | no | — |
| 79 | #713 | — | 8 | 1 | no | — |
| 80 | #713 | — | 8 | 2 | no | — |
| 81 | C2×D20 | — | 2 | 1 | no | — |
| 82 | C2×D20 | — | 2 | 2 | no | — |
| 83 | C2×D20 | — | 2 | 1 | no | — |
| 84 | C2×D20 | — | 2 | 1 | no | — |
| 85 | C2×D20 | — | 2 | 1 | no | — |
| 86 | C2×D20 | — | 2 | 1 | no | — |
| 87 | C2×D20 | — | 4 | 1 | no | — |
| 88 | C2×D20 | C22⋊C4 | 1 | 2 | yes | — |
| 89 | C2×C4×D5 | — | 2 | 2 | no | — |
| 90 | C2×C4×D5 | — | 4 | 1 | no | — |
| 91 | C2×C4×D5 | C2×D4 | 1 | 1 | yes | characteristic |
| 92 | C2×C4×D5 | C2×D4 | 1 | 1 | yes | characteristic |
| 93 | C2×C4×D5 | #41 | 1 | 1 | yes | characteristic |
| 94 | C2×#170 | — | 2 | 2 | no | — |
| 95 | C2×#170 | — | 2 | 2 | no | — |
| 96 | C2×#170 | — | 4 | 1 | no | — |
| 97 | #700 | — | 8 | 2 | no | — |
| 98 | D5⋊C8 | — | 8 | 2 | no | — |
| 99 | C2×C5×C8 | — | 2 | 1 | no | — |
| 100 | C5×C4⋊C4 | — | 2 | 1 | no | — |
| 101 | C5×C4⋊C4 | — | 2 | 1 | no | — |
| 102 | C5×C4⋊C4 | — | 2 | 1 | no | — |
| 103 | C5×C4⋊C4 | C22⋊C4 | 1 | 2 | yes | — |
| 104 | C42×C5 | C22×C4 | 1 | 1 | yes | characteristic |
| 105 | #685 | — | 2 | 2 | no | — |
| 106 | #685 | — | 2 | 2 | no | — |
| 107 | #685 | — | 4 | 1 | no | — |
| 108 | #685 | — | 4 | 1 | no | — |
| 109 | C4×Dic5 | C2×D4 | 1 | 1 | yes | characteristic |
| 110 | C4×Dic5 | C2×D4 | 1 | 1 | yes | characteristic |
| 111 | C4×Dic5 | #41 | 1 | 1 | yes | characteristic |
| 112 | C2×#168 | — | 2 | 1 | no | — |
| 113 | D40 | — | 4 | 4 | no | — |
| 114 | #676 | — | 4 | 2 | no | — |
| 115 | #532 | — | 5 | 1 | no | — |
| 116 | #532 | — | 5 | 1 | no | — |
| 117 | #498 | AGL(1,𝔽5) | 1 | 1 | yes | characteristic, complemented |
| 118 | #493 | — | 5 | 1 | no | — |
| 119 | #486 | — | 5 | 1 | no | — |
| 120 | #486 | — | 5 | 1 | no | — |
| 121 | #469 | — | 10 | 2 | no | — |
| 122 | #442 | — | 10 | 2 | no | — |
| 123 | #435 | — | 10 | 2 | no | — |
| 124 | #435 | — | 10 | 2 | no | — |
| 125 | #432 | — | 5 | 1 | no | — |
| 126 | #432 | — | 5 | 1 | no | — |
| 127 | #432 | — | 5 | 1 | no | — |
| 128 | #408 | — | 10 | 2 | no | — |
| 129 | C4×M16 | — | 5 | 1 | no | — |
| 130 | #383 | — | 5 | 1 | no | — |
| 131 | #329 | — | 10 | 2 | no | — |
| 132 | #327 | — | 10 | 2 | no | — |
| 133 | #327 | — | 10 | 2 | no | — |
| 134 | C22×D5 | — | 2 | 2 | no | — |
| 135 | C22×D5 | — | 4 | 1 | no | — |
| 136 | C22×D5 | #121 | 1 | 1 | yes | characteristic |
| 137 | C2×AGL(1,𝔽5) | — | 8 | 1 | no | — |
| 138 | C2×AGL(1,𝔽5) | — | 8 | 1 | no | — |
| 139 | C2×AGL(1,𝔽5) | — | 8 | 1 | no | — |
| 140 | C5×Q8 | — | 8 | 1 | no | — |
| 141 | C5×Q8 | — | 8 | 1 | no | — |
| 142 | C2×C4×C5 | — | 2 | 2 | no | — |
| 143 | C2×C4×C5 | — | 4 | 1 | no | — |
| 144 | C2×C4×C5 | C2×C22⋊C4 | 1 | 1 | yes | characteristic |
| 145 | C2×C4×C5 | C4×D4 | 1 | 1 | yes | characteristic |
| 146 | C2×C4×C5 | C4×D4 | 1 | 1 | yes | characteristic |
| 147 | C2×Dic5 | #120 | 1 | 1 | yes | characteristic |
| 148 | C2×Dic5 | #121 | 1 | 1 | yes | characteristic |
| 149 | C2×Dic5 | #123 | 1 | 1 | yes | characteristic |
| 150 | D20 | — | 4 | 2 | no | — |
| 151 | D20 | — | 4 | 2 | no | — |
| 152 | D20 | — | 8 | 1 | no | — |
| 153 | D20 | — | 8 | 1 | no | — |
| 154 | D20 | — | 8 | 1 | no | — |
| 155 | D20 | — | 8 | 1 | no | — |
| 156 | D20 | — | 8 | 2 | no | — |
| 157 | D20 | — | 8 | 2 | no | — |
| 158 | C4×D5 | — | 4 | 1 | no | — |
| 159 | C4×D5 | — | 4 | 1 | no | — |
| 160 | C4×D5 | — | 4 | 1 | no | — |
| 161 | C4×D5 | — | 4 | 2 | no | — |
| 162 | C4×D5 | — | 4 | 2 | no | — |
| 163 | C4×D5 | — | 8 | 2 | no | — |
| 164 | #170 | — | 4 | 2 | no | — |
| 165 | #170 | — | 4 | 2 | no | — |
| 166 | #170 | — | 8 | 1 | no | — |
| 167 | C5×C8 | — | 2 | 2 | no | — |
| 168 | #168 | — | 4 | 1 | no | — |
| 169 | C2×#41 | — | 10 | 1 | no | — |
| 170 | C2×#41 | — | 10 | 1 | no | — |
| 171 | C2×D8 | — | 10 | 2 | no | — |
| 172 | C2×M16 | — | 10 | 1 | no | — |
| 173 | C2×M16 | — | 10 | 2 | no | — |
| 174 | C22×C8 | — | 10 | 2 | no | — |
| 175 | C4⋊Q8 | C2×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 176 | C4⋊Q8 | C2×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 177 | #127 | — | 5 | 1 | no | — |
| 178 | #127 | — | 5 | 1 | no | — |
| 179 | #124 | — | 10 | 1 | no | — |
| 180 | #124 | — | 10 | 1 | no | — |
| 181 | #121 | — | 5 | 2 | no | — |
| 182 | #121 | — | 10 | 1 | no | — |
| 183 | #121 | — | 10 | 1 | no | — |
| 184 | #121 | — | 10 | 1 | no | — |
| 185 | C4×D4 | — | 5 | 2 | no | — |
| 186 | C4×D4 | — | 10 | 1 | no | — |
| 187 | C4×D4 | — | 10 | 1 | no | — |
| 188 | C4×D4 | — | 10 | 1 | no | — |
| 189 | C2×C42 | — | 5 | 1 | no | — |
| 190 | #106 | — | 2 | 2 | no | — |
| 191 | C4⋊C8 | — | 10 | 1 | no | — |
| 192 | C4⋊C8 | — | 10 | 1 | no | — |
| 193 | C4⋊C8 | — | 10 | 1 | no | — |
| 194 | #102 | — | 10 | 2 | no | — |
| 195 | #102 | — | 10 | 2 | no | — |
| 196 | C22⋊C8 | — | 5 | 2 | no | — |
| 197 | C22⋊C8 | — | 10 | 1 | no | — |
| 198 | C22⋊C8 | — | 10 | 4 | no | — |
| 199 | C22⋊C8 | — | 10 | 1 | no | — |
| 200 | #97 | — | 5 | 2 | no | — |
| 201 | #97 | — | 10 | 1 | no | — |
| 202 | #97 | — | 10 | 1 | no | — |
| 203 | #97 | — | 10 | 1 | no | — |
| 204 | C4×C8 | — | 5 | 1 | no | — |
| 205 | C4×C8 | C2×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 206 | #95 | — | 10 | 1 | no | — |
| 207 | #95 | — | 10 | 1 | no | — |
| 208 | #95 | — | 10 | 1 | no | — |
| 209 | C22×C5 | #386 | 1 | 1 | yes | characteristic |
| 210 | C2×D5 | — | 2 | 1 | no | — |
| 211 | C2×D5 | — | 2 | 1 | no | — |
| 212 | C2×D5 | — | 2 | 1 | no | — |
| 213 | C2×D5 | — | 4 | 2 | no | — |
| 214 | C2×D5 | — | 4 | 2 | no | — |
| 215 | C2×D5 | — | 4 | 2 | no | — |
| 216 | C2×D5 | — | 8 | 1 | no | — |
| 217 | C2×D5 | — | 8 | 1 | no | — |
| 218 | C2×D5 | — | 8 | 1 | no | — |
| 219 | AGL(1,𝔽5) | — | 16 | 1 | no | — |
| 220 | C4×C5 | — | 2 | 1 | no | — |
| 221 | C4×C5 | — | 2 | 1 | no | — |
| 222 | C4×C5 | — | 2 | 1 | no | — |
| 223 | C4×C5 | — | 4 | 2 | no | — |
| 224 | C4×C5 | — | 8 | 1 | no | — |
| 225 | Dic5 | — | 2 | 1 | no | — |
| 226 | Dic5 | — | 2 | 1 | no | — |
| 227 | Dic5 | — | 2 | 1 | no | — |
| 228 | #41 | — | 40 | 1 | no | — |
| 229 | #41 | — | 40 | 1 | no | — |
| 230 | #41 | — | 40 | 2 | no | — |
| 231 | C2×Q8 | — | 2 | 1 | no | — |
| 232 | C2×Q8 | — | 2 | 1 | no | — |
| 233 | C2×D4 | — | 5 | 2 | no | — |
| 234 | C2×D4 | — | 10 | 2 | no | — |
| 235 | C2×D4 | — | 10 | 1 | no | — |
| 236 | C2×D4 | — | 10 | 1 | no | — |
| 237 | C2×D4 | — | 10 | 1 | no | — |
| 238 | C2×D4 | — | 10 | 1 | no | — |
| 239 | C2×D4 | — | 10 | 1 | no | — |
| 240 | C2×D4 | — | 20 | 1 | no | — |
| 241 | C22×C4 | — | 5 | 1 | no | — |
| 242 | C22×C4 | — | 5 | 1 | no | — |
| 243 | C22×C4 | — | 5 | 1 | no | — |
| 244 | C22×C4 | — | 10 | 2 | no | — |
| 245 | C22×C4 | — | 20 | 1 | no | — |
| 246 | C22×C4 | — | 20 | 1 | no | — |
| 247 | D8 | — | 20 | 4 | no | — |
| 248 | M16 | — | 20 | 2 | no | — |
| 249 | M16 | — | 40 | 2 | no | — |
| 250 | C2×C8 | — | 2 | 1 | no | — |
| 251 | C2×C8 | — | 10 | 2 | no | — |
| 252 | C2×C8 | — | 10 | 2 | no | — |
| 253 | C2×C8 | — | 10 | 1 | no | — |
| 254 | C2×C8 | — | 20 | 1 | no | — |
| 255 | C2×C8 | — | 40 | 2 | no | — |
| 256 | C4⋊C4 | — | 2 | 1 | no | — |
| 257 | C4⋊C4 | — | 2 | 1 | no | — |
| 258 | C4⋊C4 | — | 2 | 1 | no | — |
| 259 | C4⋊C4 | C22⋊AGL(1,𝔽5) | 1 | 2 | yes | — |
| 260 | C22⋊C4 | — | 10 | 2 | no | — |
| 261 | C22⋊C4 | — | 10 | 2 | no | — |
| 262 | C22⋊C4 | — | 20 | 1 | no | — |
| 263 | C22⋊C4 | — | 20 | 1 | no | — |
| 264 | C42 | — | 5 | 1 | no | — |
| 265 | C42 | — | 5 | 1 | no | — |
| 266 | C42 | — | 5 | 1 | no | — |
| 267 | C42 | C22×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 268 | C2×C5 | #1884 | 1 | 1 | yes | characteristic |
| 269 | C2×C5 | #1903 | 1 | 1 | yes | characteristic |
| 270 | C2×C5 | #1904 | 1 | 1 | yes | characteristic |
| 271 | D5 | — | 4 | 1 | no | — |
| 272 | D5 | — | 8 | 2 | no | — |
| 273 | D5 | — | 16 | 1 | no | — |
| 274 | C23 | — | 5 | 1 | no | — |
| 275 | C23 | — | 10 | 2 | no | — |
| 276 | C23 | — | 20 | 1 | no | — |
| 277 | Q8 | — | 8 | 1 | no | — |
| 278 | Q8 | — | 8 | 1 | no | — |
| 279 | D4 | — | 20 | 2 | no | — |
| 280 | D4 | — | 20 | 2 | no | — |
| 281 | D4 | — | 40 | 1 | no | — |
| 282 | D4 | — | 40 | 1 | no | — |
| 283 | D4 | — | 40 | 1 | no | — |
| 284 | D4 | — | 40 | 1 | no | — |
| 285 | D4 | — | 40 | 2 | no | — |
| 286 | D4 | — | 40 | 2 | no | — |
| 287 | C2×C4 | — | 2 | 2 | no | — |
| 288 | C2×C4 | — | 4 | 1 | no | — |
| 289 | C2×C4 | — | 5 | 1 | no | — |
| 290 | C2×C4 | — | 5 | 1 | no | — |
| 291 | C2×C4 | — | 5 | 1 | no | — |
| 292 | C2×C4 | — | 20 | 1 | no | — |
| 293 | C2×C4 | — | 20 | 1 | no | — |
| 294 | C2×C4 | — | 20 | 1 | no | — |
| 295 | C2×C4 | — | 20 | 2 | no | — |
| 296 | C2×C4 | — | 20 | 2 | no | — |
| 297 | C2×C4 | — | 40 | 2 | no | — |
| 298 | C2×C4 | — | 40 | 1 | no | — |
| 299 | C2×C4 | — | 40 | 1 | no | — |
| 300 | C2×C4 | — | 40 | 1 | no | — |
| 301 | C2×C4 | D4×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 302 | C2×C4 | D4×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 303 | C2×C4 | C2×C22⋊AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 304 | C8 | — | 2 | 2 | no | — |
| 305 | C8 | — | 20 | 1 | no | — |
| 306 | C8 | — | 20 | 2 | no | — |
| 307 | C8 | — | 20 | 2 | no | — |
| 308 | C8 | — | 40 | 1 | no | — |
| 309 | C5 | #9417 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 310 | C22 | — | 10 | 1 | no | — |
| 311 | C22 | — | 10 | 1 | no | — |
| 312 | C22 | — | 10 | 1 | no | — |
| 313 | C22 | — | 20 | 2 | no | — |
| 314 | C22 | — | 20 | 2 | no | — |
| 315 | C22 | — | 20 | 2 | no | — |
| 316 | C22 | — | 40 | 1 | no | — |
| 317 | C22 | — | 40 | 1 | no | — |
| 318 | C22 | — | 40 | 1 | no | — |
| 319 | C22 | #65779 | 1 | 1 | yes | characteristic |
| 320 | C4 | — | 2 | 1 | no | — |
| 321 | C4 | — | 2 | 1 | no | — |
| 322 | C4 | — | 2 | 1 | no | — |
| 323 | C4 | — | 4 | 2 | no | — |
| 324 | C4 | — | 8 | 1 | no | — |
| 325 | C4 | — | 10 | 1 | no | — |
| 326 | C4 | — | 10 | 1 | no | — |
| 327 | C4 | — | 10 | 1 | no | — |
| 328 | C4 | — | 80 | 1 | no | — |
| 329 | C2 | — | 20 | 1 | no | — |
| 330 | C2 | — | 40 | 2 | no | — |
| 331 | C2 | — | 80 | 1 | no | — |
| 332 | C2 | #10604676 | 1 | 1 | yes | characteristic |
| 333 | C2 | #10604819 | 1 | 1 | yes | characteristic |
| 334 | C2 | #10604827 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #10617379 | C2 | semidirect |
| 3. #10604638 | C2 | semidirect |
| 4. #10604627 | C2 | semidirect |
| 5. #10604627 | C2 | semidirect |
| 6. #10598533 | C2 | semidirect |
| 7. #10598528 | C2 | semidirect |
| 17. C5×#498 | C4 | semidirect |
| 18. #65377 | C4 | semidirect |
| 19. #65377 | C4 | semidirect |
| 22. #64984 | C4 | semidirect |
| 117. #498 | AGL(1,𝔽5) | semidirect |
| 309. C5 | #9417 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #10617379 | 2 |
| #10613619 | 2 |
| #10604638 | 2 |
| #10604627 | 2 |
| #10598533 | 2 |
| #10598528 | 2 |
| #9417 | 5 |