Finite Group DatabasebetaFinite Group DatabasebetaOrder 1280.
468 conjugacy classes of subgroups; only the 76 normal subgroups are shown.
This group has 468 conjugacy classes of subgroups — too many to draw. Only its 76 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 | #10610914 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #10604727 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #10604726 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #10604382 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | #10604354 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | C22×D40 | C4 | 1 | 1 | yes | characteristic, complemented |
| 7 | C2×#4047 | C22 | 1 | 1 | yes | characteristic |
| 8 | #65859 | C22 | 1 | 2 | yes | — |
| 9 | #65761 | C22 | 1 | 1 | yes | characteristic |
| 10 | #65745 | C22 | 1 | 2 | yes | — |
| 11 | #65745 | C22 | 1 | 1 | yes | characteristic |
| 12 | #65721 | — | 2 | 1 | no | — |
| 13 | #65718 | — | 2 | 1 | no | — |
| 14 | C5×#427 | C4 | 1 | 1 | yes | characteristic, complemented |
| 15 | #65290 | C4 | 1 | 2 | yes | complemented |
| 16 | #65172 | — | 2 | 1 | no | — |
| 17 | #65172 | — | 2 | 1 | no | — |
| 18 | #65154 | — | 2 | 1 | no | — |
| 19 | #65154 | — | 2 | 1 | no | — |
| 20 | #64895 | — | 2 | 1 | no | — |
| 21 | #64894 | — | 2 | 1 | no | — |
| 22 | #64864 | — | 2 | 4 | no | — |
| 23 | #64853 | — | 2 | 4 | no | — |
| 24 | #13231 | — | 5 | 1 | no | maximal, Sylow |
| 25 | C2×#709 | C23 | 1 | 1 | yes | characteristic |
| 26 | C22×D20 | C2×C4 | 1 | 2 | yes | — |
| 27 | #4130 | — | 2 | 1 | no | — |
| 28 | #4130 | — | 2 | 2 | no | — |
| 29 | D5⋊M16 | — | 2 | 1 | no | — |
| 30 | C22×C5×C8 | C2×C4 | 1 | 1 | yes | characteristic |
| 31 | C5×#117 | C2×C4 | 1 | 2 | yes | — |
| 32 | C2×#681 | C2×C4 | 1 | 1 | yes | characteristic |
| 33 | C2×D40 | — | 2 | 1 | no | — |
| 34 | C2×D40 | — | 2 | 1 | no | — |
| 35 | C2×D40 | — | 4 | 1 | no | — |
| 36 | C2×D40 | — | 4 | 1 | no | — |
| 37 | #4047 | — | 4 | 1 | no | — |
| 38 | #4047 | — | 4 | 1 | no | — |
| 39 | C2×#676 | D4 | 1 | 1 | yes | characteristic |
| 40 | C2×#676 | D4 | 1 | 1 | yes | characteristic |
| 41 | C2×C8×D5 | D4 | 1 | 1 | yes | characteristic |
| 42 | C2×C8×D5 | D4 | 1 | 1 | yes | characteristic |
| 43 | #4041 | — | 2 | 2 | no | — |
| 44 | #4041 | — | 2 | 2 | no | — |
| 45 | #4028 | — | 2 | 2 | no | — |
| 46 | #4028 | — | 2 | 2 | no | — |
| 47 | #4021 | — | 2 | 2 | no | — |
| 48 | #4020 | — | 2 | 2 | no | — |
| 49 | #4013 | D4 | 1 | 2 | yes | — |
| 50 | #4011 | — | 2 | 1 | no | — |
| 51 | #4011 | — | 2 | 2 | no | — |
| 52 | #4005 | — | 2 | 1 | no | — |
| 53 | #3999 | — | 2 | 2 | no | — |
| 54 | #3999 | D4 | 1 | 2 | yes | — |
| 55 | #3994 | — | 4 | 1 | no | — |
| 56 | #3993 | — | 4 | 1 | no | — |
| 57 | #3992 | — | 4 | 1 | no | — |
| 58 | C8×AGL(1,𝔽5) | — | 4 | 1 | no | — |
| 59 | C5×#107 | — | 2 | 1 | no | — |
| 60 | C5×#106 | — | 2 | 1 | no | — |
| 61 | #3941 | — | 2 | 2 | no | — |
| 62 | #3941 | — | 2 | 2 | no | — |
| 63 | #3058 | — | 5 | 1 | no | — |
| 64 | #2128 | — | 5 | 2 | no | — |
| 65 | #2122 | — | 5 | 2 | no | — |
| 66 | #1812 | — | 5 | 1 | no | — |
| 67 | #1757 | — | 5 | 1 | no | — |
| 68 | C23×D5 | — | 2 | 2 | no | — |
| 69 | C22×C4×C5 | C22×C4 | 1 | 1 | yes | characteristic |
| 70 | C2×C22⋊D5 | C2×D4 | 1 | 1 | yes | characteristic |
| 71 | C2×C22⋊D5 | #41 | 1 | 1 | yes | characteristic |
| 72 | #709 | — | 4 | 1 | no | — |
| 73 | #709 | — | 4 | 1 | no | — |
| 74 | C2×D20 | — | 2 | 2 | no | — |
| 75 | C2×D20 | — | 2 | 2 | no | — |
| 76 | C2×D20 | — | 4 | 2 | no | — |
| 77 | C2×D20 | — | 4 | 2 | no | — |
| 78 | C2×D20 | — | 4 | 2 | no | — |
| 79 | C2×D20 | — | 4 | 2 | no | — |
| 80 | C2×D20 | C2×D4 | 1 | 1 | yes | characteristic |
| 81 | C2×C4×D5 | C2×D4 | 1 | 1 | yes | characteristic |
| 82 | C2×C4×D5 | C2×D4 | 1 | 1 | yes | characteristic |
| 83 | C2×Dic10 | #41 | 1 | 1 | yes | characteristic |
| 84 | C22⋊AGL(1,𝔽5) | — | 2 | 2 | no | — |
| 85 | C22⋊AGL(1,𝔽5) | — | 4 | 2 | no | — |
| 86 | #704 | — | 2 | 2 | no | — |
| 87 | C4⋊AGL(1,𝔽5) | — | 4 | 1 | no | — |
| 88 | C4⋊AGL(1,𝔽5) | — | 4 | 2 | no | — |
| 89 | C4×AGL(1,𝔽5) | — | 4 | 1 | no | — |
| 90 | C4×AGL(1,𝔽5) | — | 4 | 2 | no | — |
| 91 | #700 | — | 4 | 1 | no | — |
| 92 | D5⋊C8 | — | 4 | 1 | no | — |
| 93 | C2×C5×C8 | — | 2 | 1 | no | — |
| 94 | C2×C5×C8 | — | 2 | 1 | no | — |
| 95 | C2×C5×C8 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 96 | C2×C5×C8 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 97 | C5×C4⋊C4 | — | 2 | 2 | no | — |
| 98 | C5×C22⋊C4 | — | 2 | 2 | no | — |
| 99 | C42×C5 | — | 2 | 2 | no | — |
| 100 | #685 | — | 4 | 2 | no | — |
| 101 | #685 | — | 4 | 2 | no | — |
| 102 | #681 | — | 4 | 1 | no | — |
| 103 | C2×#168 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 104 | C2×#168 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 105 | D40 | — | 4 | 2 | no | — |
| 106 | D40 | — | 4 | 1 | no | — |
| 107 | D40 | — | 4 | 1 | no | — |
| 108 | #676 | — | 4 | 1 | no | — |
| 109 | #676 | — | 4 | 1 | no | — |
| 110 | C8×D5 | — | 2 | 1 | no | — |
| 111 | C8×D5 | — | 2 | 1 | no | — |
| 112 | C8×D5 | — | 2 | 1 | no | — |
| 113 | C8×D5 | — | 2 | 1 | no | — |
| 114 | C22×D8 | — | 5 | 1 | no | — |
| 115 | C2×#131 | — | 5 | 1 | no | — |
| 116 | #535 | — | 5 | 2 | no | — |
| 117 | #469 | — | 10 | 1 | no | — |
| 118 | #469 | — | 10 | 1 | no | — |
| 119 | #466 | — | 10 | 1 | no | — |
| 120 | #466 | — | 10 | 1 | no | — |
| 121 | #427 | — | 10 | 1 | no | — |
| 122 | #427 | AGL(1,𝔽5) | 1 | 1 | yes | characteristic, complemented |
| 123 | #418 | — | 5 | 2 | no | — |
| 124 | #413 | — | 5 | 1 | no | — |
| 125 | #410 | — | 5 | 1 | no | — |
| 126 | #410 | — | 5 | 2 | no | — |
| 127 | #405 | — | 10 | 1 | no | — |
| 128 | #353 | — | 10 | 4 | no | — |
| 129 | AΣL(1,𝔽16) | — | 10 | 4 | no | — |
| 130 | #344 | — | 10 | 1 | no | — |
| 131 | H(3,ℤ/4) | — | 10 | 1 | no | — |
| 132 | C23×C5 | C4×D4 | 1 | 1 | yes | characteristic |
| 133 | C22×D5 | — | 4 | 2 | no | — |
| 134 | C22×D5 | — | 8 | 2 | no | — |
| 135 | C22×D5 | — | 16 | 2 | no | — |
| 136 | C22×D5 | #121 | 1 | 1 | yes | characteristic |
| 137 | C22×D5 | #127 | 1 | 1 | yes | characteristic |
| 138 | C2×AGL(1,𝔽5) | — | 4 | 2 | no | — |
| 139 | C2×AGL(1,𝔽5) | — | 8 | 2 | no | — |
| 140 | C2×C4×C5 | — | 2 | 1 | no | — |
| 141 | C2×C4×C5 | — | 2 | 1 | no | — |
| 142 | C2×C4×C5 | — | 4 | 2 | no | — |
| 143 | C2×C4×C5 | C2×C22⋊C4 | 1 | 1 | yes | characteristic |
| 144 | C2×C4×C5 | C4×D4 | 1 | 1 | yes | characteristic |
| 145 | C22⋊D5 | — | 4 | 1 | no | — |
| 146 | C22⋊D5 | — | 4 | 1 | no | — |
| 147 | C2×Dic5 | #121 | 1 | 1 | yes | characteristic |
| 148 | C2×Dic5 | #124 | 1 | 1 | yes | characteristic |
| 149 | D20 | — | 4 | 1 | no | — |
| 150 | D20 | — | 4 | 2 | no | — |
| 151 | D20 | — | 4 | 2 | no | — |
| 152 | D20 | — | 8 | 2 | no | — |
| 153 | D20 | — | 16 | 2 | no | — |
| 154 | D20 | — | 16 | 2 | no | — |
| 155 | C4×D5 | — | 2 | 1 | no | — |
| 156 | C4×D5 | — | 2 | 1 | no | — |
| 157 | C4×D5 | — | 2 | 1 | no | — |
| 158 | C4×D5 | — | 2 | 1 | no | — |
| 159 | Dic10 | — | 4 | 1 | no | — |
| 160 | #170 | — | 4 | 2 | no | — |
| 161 | C5×C8 | — | 2 | 1 | no | — |
| 162 | C5×C8 | #99 | 1 | 1 | yes | characteristic |
| 163 | C5×C8 | #99 | 1 | 1 | yes | characteristic |
| 164 | #168 | — | 2 | 1 | no | — |
| 165 | #168 | — | 2 | 1 | no | — |
| 166 | C2×#41 | — | 5 | 1 | no | — |
| 167 | C22×D4 | — | 5 | 2 | no | — |
| 168 | C2×D8 | — | 10 | 1 | no | — |
| 169 | C2×D8 | — | 10 | 1 | no | — |
| 170 | C2×D8 | — | 20 | 1 | no | — |
| 171 | C2×D8 | — | 20 | 1 | no | — |
| 172 | #131 | — | 20 | 1 | no | — |
| 173 | #131 | — | 20 | 1 | no | — |
| 174 | C2×M16 | — | 5 | 1 | no | — |
| 175 | C2×M16 | — | 5 | 1 | no | — |
| 176 | C2×M16 | — | 5 | 1 | no | — |
| 177 | C2×M16 | — | 10 | 1 | no | — |
| 178 | C22×C8 | — | 5 | 1 | no | — |
| 179 | C22×C8 | — | 5 | 1 | no | — |
| 180 | C22×C8 | C2×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 181 | #127 | — | 10 | 2 | no | — |
| 182 | #124 | — | 10 | 2 | no | — |
| 183 | #121 | — | 10 | 2 | no | — |
| 184 | #121 | — | 10 | 2 | no | — |
| 185 | #120 | — | 10 | 2 | no | — |
| 186 | #120 | — | 10 | 2 | no | — |
| 187 | #117 | — | 10 | 1 | no | — |
| 188 | #117 | — | 10 | 2 | no | — |
| 189 | #117 | C2×AGL(1,𝔽5) | 1 | 2 | yes | — |
| 190 | #107 | — | 2 | 1 | no | — |
| 191 | #107 | — | 20 | 1 | no | — |
| 192 | #106 | — | 2 | 1 | no | — |
| 193 | #106 | — | 20 | 1 | no | — |
| 194 | #102 | — | 10 | 2 | no | — |
| 195 | #102 | — | 10 | 2 | no | — |
| 196 | #101 | — | 10 | 1 | no | — |
| 197 | #100 | — | 5 | 2 | no | — |
| 198 | #99 | — | 5 | 2 | no | — |
| 199 | #99 | — | 10 | 1 | no | — |
| 200 | #99 | — | 10 | 2 | no | — |
| 201 | #99 | — | 10 | 2 | no | — |
| 202 | #97 | — | 20 | 1 | no | — |
| 203 | C4×C8 | — | 20 | 1 | no | — |
| 204 | C22×C5 | — | 2 | 1 | no | — |
| 205 | C22×C5 | — | 4 | 1 | no | — |
| 206 | C22×C5 | #390 | 1 | 1 | yes | characteristic |
| 207 | C2×D5 | — | 2 | 1 | no | — |
| 208 | C2×D5 | — | 2 | 1 | no | — |
| 209 | C2×D5 | — | 4 | 1 | no | — |
| 210 | C2×D5 | — | 4 | 1 | no | — |
| 211 | C2×D5 | — | 8 | 2 | no | — |
| 212 | C2×D5 | — | 8 | 2 | no | — |
| 213 | C2×D5 | — | 8 | 2 | no | — |
| 214 | C2×D5 | — | 16 | 4 | no | — |
| 215 | AGL(1,𝔽5) | — | 8 | 2 | no | — |
| 216 | AGL(1,𝔽5) | — | 16 | 2 | no | — |
| 217 | C4×C5 | — | 2 | 1 | no | — |
| 218 | C4×C5 | — | 8 | 2 | no | — |
| 219 | C4×C5 | C2×#99 | 1 | 1 | yes | characteristic |
| 220 | C4×C5 | #410 | 1 | 1 | yes | characteristic |
| 221 | Dic5 | — | 2 | 1 | no | — |
| 222 | Dic5 | — | 2 | 1 | no | — |
| 223 | C24 | — | 10 | 2 | no | — |
| 224 | #41 | — | 20 | 1 | no | — |
| 225 | #41 | — | 20 | 1 | no | — |
| 226 | C2×Q8 | — | 5 | 1 | no | — |
| 227 | C2×D4 | — | 5 | 1 | no | — |
| 228 | C2×D4 | — | 5 | 1 | no | — |
| 229 | C2×D4 | — | 5 | 1 | no | — |
| 230 | C2×D4 | — | 10 | 2 | no | — |
| 231 | C2×D4 | — | 10 | 2 | no | — |
| 232 | C2×D4 | — | 20 | 2 | no | — |
| 233 | C2×D4 | — | 20 | 2 | no | — |
| 234 | C2×D4 | — | 20 | 2 | no | — |
| 235 | C2×D4 | — | 20 | 2 | no | — |
| 236 | C22×C4 | — | 5 | 1 | no | — |
| 237 | C22×C4 | — | 5 | 1 | no | — |
| 238 | C22×C4 | C22×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 239 | D8 | — | 20 | 2 | no | — |
| 240 | D8 | — | 20 | 1 | no | — |
| 241 | D8 | — | 20 | 1 | no | — |
| 242 | M16 | — | 10 | 2 | no | — |
| 243 | M16 | — | 20 | 1 | no | — |
| 244 | M16 | — | 20 | 1 | no | — |
| 245 | M16 | — | 20 | 1 | no | — |
| 246 | M16 | — | 20 | 1 | no | — |
| 247 | C2×C8 | — | 2 | 1 | no | — |
| 248 | C2×C8 | — | 2 | 1 | no | — |
| 249 | C2×C8 | — | 5 | 1 | no | — |
| 250 | C2×C8 | — | 5 | 1 | no | — |
| 251 | C2×C8 | — | 10 | 1 | no | — |
| 252 | C2×C8 | — | 10 | 1 | no | — |
| 253 | C2×C8 | — | 10 | 1 | no | — |
| 254 | C2×C8 | — | 10 | 1 | no | — |
| 255 | C2×C8 | — | 20 | 1 | no | — |
| 256 | C2×C8 | C22⋊AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 257 | C2×C8 | C22⋊AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 258 | C4⋊C4 | — | 2 | 2 | no | — |
| 259 | C4⋊C4 | — | 20 | 1 | no | — |
| 260 | C4⋊C4 | — | 20 | 2 | no | — |
| 261 | C22⋊C4 | — | 2 | 2 | no | — |
| 262 | C22⋊C4 | — | 10 | 2 | no | — |
| 263 | C22⋊C4 | — | 20 | 2 | no | — |
| 264 | C22⋊C4 | — | 20 | 2 | no | — |
| 265 | C22⋊C4 | — | 20 | 2 | no | — |
| 266 | C42 | — | 2 | 2 | no | — |
| 267 | C42 | — | 20 | 1 | no | — |
| 268 | C42 | — | 20 | 2 | no | — |
| 269 | C2×C5 | — | 2 | 1 | no | — |
| 270 | C2×C5 | — | 4 | 1 | no | — |
| 271 | C2×C5 | #1964 | 1 | 1 | yes | characteristic |
| 272 | D5 | — | 4 | 1 | no | — |
| 273 | D5 | — | 4 | 1 | no | — |
| 274 | D5 | — | 16 | 2 | no | — |
| 275 | C23 | — | 5 | 1 | no | — |
| 276 | C23 | — | 5 | 1 | no | — |
| 277 | C23 | — | 20 | 2 | no | — |
| 278 | C23 | — | 40 | 2 | no | — |
| 279 | C23 | — | 80 | 2 | no | — |
| 280 | C23 | D4×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 281 | Q8 | — | 20 | 1 | no | — |
| 282 | D4 | — | 20 | 1 | no | — |
| 283 | D4 | — | 20 | 1 | no | — |
| 284 | D4 | — | 20 | 1 | no | — |
| 285 | D4 | — | 20 | 2 | no | — |
| 286 | D4 | — | 20 | 2 | no | — |
| 287 | D4 | — | 40 | 2 | no | — |
| 288 | D4 | — | 80 | 2 | no | — |
| 289 | D4 | — | 80 | 2 | no | — |
| 290 | C2×C4 | — | 2 | 1 | no | — |
| 291 | C2×C4 | — | 2 | 1 | no | — |
| 292 | C2×C4 | — | 4 | 2 | no | — |
| 293 | C2×C4 | — | 5 | 1 | no | — |
| 294 | C2×C4 | — | 5 | 1 | no | — |
| 295 | C2×C4 | — | 10 | 1 | no | — |
| 296 | C2×C4 | — | 10 | 1 | no | — |
| 297 | C2×C4 | — | 10 | 1 | no | — |
| 298 | C2×C4 | — | 10 | 1 | no | — |
| 299 | C2×C4 | — | 20 | 2 | no | — |
| 300 | C2×C4 | — | 40 | 2 | no | — |
| 301 | C2×C4 | D4×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 302 | C2×C4 | C2×C22⋊AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 303 | C8 | — | 2 | 1 | no | — |
| 304 | C8 | — | 10 | 1 | no | — |
| 305 | C8 | — | 10 | 1 | no | — |
| 306 | C8 | — | 20 | 2 | no | — |
| 307 | C8 | #3999 | 1 | 1 | yes | characteristic |
| 308 | C8 | #3999 | 1 | 1 | yes | characteristic |
| 309 | C5 | #13231 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 310 | C22 | — | 2 | 1 | no | — |
| 311 | C22 | — | 4 | 1 | no | — |
| 312 | C22 | — | 10 | 1 | no | — |
| 313 | C22 | — | 10 | 1 | no | — |
| 314 | C22 | — | 20 | 1 | no | — |
| 315 | C22 | — | 20 | 1 | no | — |
| 316 | C22 | — | 40 | 2 | no | — |
| 317 | C22 | — | 40 | 2 | no | — |
| 318 | C22 | — | 40 | 2 | no | — |
| 319 | C22 | — | 80 | 4 | no | — |
| 320 | C22 | #65780 | 1 | 1 | yes | characteristic |
| 321 | C4 | — | 2 | 1 | no | — |
| 322 | C4 | — | 8 | 2 | no | — |
| 323 | C4 | — | 10 | 1 | no | — |
| 324 | C4 | — | 10 | 1 | no | — |
| 325 | C4 | — | 40 | 2 | no | — |
| 326 | C4 | — | 80 | 2 | no | — |
| 327 | C4 | C2×#3999 | 1 | 1 | yes | characteristic |
| 328 | C4 | #65785 | 1 | 1 | yes | characteristic |
| 329 | C2 | — | 2 | 1 | no | — |
| 330 | C2 | — | 4 | 1 | no | — |
| 331 | C2 | — | 20 | 1 | no | — |
| 332 | C2 | — | 20 | 1 | no | — |
| 333 | C2 | — | 80 | 2 | no | — |
| 334 | C2 | #10604309 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #10604727 | C2 | semidirect |
| 3. #10604726 | C2 | semidirect |
| 4. #10604382 | C2 | semidirect |
| 5. #10604354 | C2 | semidirect |
| 6. C22×D40 | C4 | semidirect |
| 14. C5×#427 | C4 | semidirect |
| 15. #65290 | C4 | semidirect |
| 122. #427 | AGL(1,𝔽5) | semidirect |
| 309. C5 | #13231 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #10610914 | 2 |
| #10604727 | 2 |
| #10604726 | 2 |
| #10604382 | 2 |
| #10604354 | 2 |
| #13231 | 5 |