Finite Group DatabasebetaFinite Group DatabasebetaOrder 1280.
508 conjugacy classes of subgroups; only the 74 normal subgroups are shown.
This group has 508 conjugacy classes of subgroups — too many to draw. Only its 74 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 | #10617390 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #10613531 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | #10604662 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #10604627 | C2 | 1 | 2 | yes | maximal |
| 5 | #10598528 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | C2×#4064 | — | 2 | 2 | no | — |
| 7 | #66059 | C22 | 1 | 1 | yes | characteristic |
| 8 | D5×C4⋊Q8 | C22 | 1 | 1 | yes | characteristic |
| 9 | (C2×AGL(1,𝔽5))⋊Q8 | — | 2 | 2 | no | — |
| 10 | #65711 | — | 2 | 4 | no | — |
| 11 | #65690 | C22 | 1 | 2 | yes | — |
| 12 | #65684 | C22 | 1 | 2 | yes | — |
| 13 | C5×#494 | C4 | 1 | 1 | yes | characteristic, complemented |
| 14 | #65479 | — | 2 | 2 | no | — |
| 15 | #65381 | C4 | 1 | 1 | yes | characteristic, complemented |
| 16 | #65374 | C4 | 1 | 1 | yes | characteristic, complemented |
| 17 | #64982 | C4 | 1 | 1 | yes | characteristic, complemented |
| 18 | C4×#676 | C22 | 1 | 1 | yes | characteristic |
| 19 | #64931 | — | 2 | 2 | no | — |
| 20 | #64868 | — | 2 | 4 | no | — |
| 21 | #9429 | — | 5 | 1 | no | maximal, Sylow |
| 22 | C2×#713 | — | 2 | 1 | no | — |
| 23 | C2×Q8×D5 | D4 | 1 | 2 | yes | — |
| 24 | C2×C4⋊AGL(1,𝔽5) | — | 2 | 2 | no | — |
| 25 | C2×C4×AGL(1,𝔽5) | — | 2 | 2 | no | — |
| 26 | C2×C5×Q16 | — | 2 | 2 | no | — |
| 27 | C5×C4⋊Q8 | C2×C4 | 1 | 1 | yes | characteristic |
| 28 | C5×C4⋊Q8 | C2×C4 | 1 | 1 | yes | characteristic |
| 29 | #4093 | — | 2 | 1 | no | — |
| 30 | #4090 | D4 | 1 | 2 | yes | — |
| 31 | C2×#689 | — | 2 | 2 | no | — |
| 32 | C2×#688 | — | 2 | 2 | no | — |
| 33 | #4064 | — | 4 | 4 | no | — |
| 34 | #4064 | — | 4 | 4 | no | — |
| 35 | C2×#677 | — | 2 | 2 | no | — |
| 36 | C2×#676 | — | 2 | 1 | no | — |
| 37 | #4041 | — | 2 | 2 | no | — |
| 38 | #4039 | — | 2 | 2 | no | — |
| 39 | D5×C4⋊C4 | — | 2 | 2 | no | — |
| 40 | #4034 | — | 2 | 2 | no | — |
| 41 | #4020 | C2×C4 | 1 | 1 | yes | characteristic |
| 42 | C42×D5 | C23 | 1 | 1 | yes | characteristic |
| 43 | #4015 | C2×C4 | 1 | 1 | yes | characteristic |
| 44 | (C2×AGL(1,𝔽5))⋊C4 | — | 2 | 4 | no | — |
| 45 | (C2×AGL(1,𝔽5))⋊C4 | — | 2 | 2 | no | — |
| 46 | (C2×AGL(1,𝔽5))⋊C4 | — | 2 | 2 | no | — |
| 47 | (C2×AGL(1,𝔽5))⋊C4 | D4 | 1 | 4 | yes | — |
| 48 | #4004 | — | 2 | 2 | no | — |
| 49 | #4003 | — | 2 | 2 | no | — |
| 50 | #4002 | — | 2 | 2 | no | — |
| 51 | C4×C5×C8 | C2×C4 | 1 | 1 | yes | characteristic |
| 52 | #3947 | — | 2 | 1 | no | — |
| 53 | C4×#168 | C2×C4 | 1 | 1 | yes | characteristic |
| 54 | #3152 | — | 5 | 1 | no | — |
| 55 | #2381 | — | 5 | 1 | no | — |
| 56 | #1558 | — | 5 | 1 | no | — |
| 57 | #1514 | — | 5 | 2 | no | — |
| 58 | #1288 | — | 5 | 2 | no | — |
| 59 | C22×AGL(1,𝔽5) | — | 2 | 4 | no | — |
| 60 | C2×C5×Q8 | — | 2 | 1 | no | — |
| 61 | C2×C5×Q8 | C22⋊C4 | 1 | 2 | yes | — |
| 62 | #713 | — | 4 | 2 | no | — |
| 63 | #713 | — | 4 | 2 | no | — |
| 64 | Q8×D5 | — | 4 | 2 | no | — |
| 65 | Q8×D5 | — | 4 | 2 | no | — |
| 66 | C2×D20 | — | 2 | 1 | no | — |
| 67 | C2×D20 | — | 2 | 2 | no | — |
| 68 | C2×D20 | — | 4 | 1 | no | — |
| 69 | C2×C4×D5 | — | 2 | 2 | no | — |
| 70 | C2×C4×D5 | — | 4 | 1 | no | — |
| 71 | C2×C4×D5 | C2×D4 | 1 | 1 | yes | characteristic |
| 72 | C2×C4×D5 | C2×D4 | 1 | 2 | yes | — |
| 73 | C2×Dic10 | — | 2 | 2 | no | — |
| 74 | C2×Dic10 | C22⋊C4 | 1 | 2 | yes | — |
| 75 | C2×#170 | — | 4 | 2 | no | — |
| 76 | C4⋊AGL(1,𝔽5) | — | 8 | 2 | no | — |
| 77 | C4×AGL(1,𝔽5) | — | 8 | 2 | no | — |
| 78 | C5×Q16 | — | 4 | 4 | no | — |
| 79 | C2×C5×C8 | — | 2 | 1 | no | — |
| 80 | C5×C4⋊C4 | — | 2 | 2 | no | — |
| 81 | C5×C4⋊C4 | — | 2 | 2 | no | — |
| 82 | C42×C5 | C22×C4 | 1 | 1 | yes | characteristic |
| 83 | #689 | — | 4 | 4 | no | — |
| 84 | #688 | — | 4 | 4 | no | — |
| 85 | #685 | — | 4 | 2 | no | — |
| 86 | C4⋊Dic5 | — | 2 | 2 | no | — |
| 87 | #683 | — | 2 | 4 | no | — |
| 88 | C4×Dic5 | C2×D4 | 1 | 1 | yes | characteristic |
| 89 | C4×Dic5 | #41 | 1 | 2 | yes | — |
| 90 | C2×#168 | — | 2 | 1 | no | — |
| 91 | #677 | — | 4 | 4 | no | — |
| 92 | #676 | — | 2 | 4 | no | — |
| 93 | C2×#137 | — | 10 | 2 | no | — |
| 94 | #532 | — | 5 | 1 | no | — |
| 95 | C2×C4⋊Q8 | — | 5 | 1 | no | — |
| 96 | #497 | — | 10 | 2 | no | — |
| 97 | #494 | — | 5 | 1 | no | — |
| 98 | #494 | AGL(1,𝔽5) | 1 | 1 | yes | characteristic, complemented |
| 99 | #492 | — | 5 | 1 | no | — |
| 100 | #492 | — | 5 | 1 | no | — |
| 101 | #432 | — | 5 | 2 | no | — |
| 102 | C4×M16 | — | 5 | 1 | no | — |
| 103 | #391 | — | 10 | 2 | no | — |
| 104 | #387 | — | 10 | 4 | no | — |
| 105 | #383 | — | 5 | 2 | no | — |
| 106 | #333 | — | 10 | 2 | no | — |
| 107 | #327 | — | 10 | 4 | no | — |
| 108 | C22×D5 | — | 4 | 1 | no | — |
| 109 | C22×D5 | #120 | 1 | 1 | yes | characteristic |
| 110 | C2×AGL(1,𝔽5) | — | 4 | 8 | no | — |
| 111 | C2×AGL(1,𝔽5) | — | 4 | 4 | no | — |
| 112 | C5×Q8 | — | 4 | 2 | no | — |
| 113 | C5×Q8 | — | 4 | 2 | no | — |
| 114 | C2×C4×C5 | — | 2 | 2 | no | — |
| 115 | C2×C4×C5 | — | 4 | 1 | no | — |
| 116 | C2×C4×C5 | C2×C22⋊C4 | 1 | 1 | yes | characteristic |
| 117 | C2×C4×C5 | C4×D4 | 1 | 2 | yes | — |
| 118 | C2×Dic5 | — | 2 | 2 | no | — |
| 119 | C2×Dic5 | #121 | 1 | 2 | yes | — |
| 120 | C2×Dic5 | #123 | 1 | 1 | yes | characteristic |
| 121 | D20 | — | 4 | 2 | no | — |
| 122 | D20 | — | 8 | 2 | no | — |
| 123 | D20 | — | 8 | 2 | no | — |
| 124 | C4×D5 | — | 2 | 2 | no | — |
| 125 | C4×D5 | — | 4 | 2 | no | — |
| 126 | C4×D5 | — | 4 | 2 | no | — |
| 127 | C4×D5 | — | 4 | 2 | no | — |
| 128 | C4×D5 | — | 8 | 1 | no | — |
| 129 | C4×D5 | — | 8 | 1 | no | — |
| 130 | Dic10 | — | 4 | 2 | no | — |
| 131 | Dic10 | — | 8 | 2 | no | — |
| 132 | #170 | — | 8 | 2 | no | — |
| 133 | C5×C8 | — | 2 | 2 | no | — |
| 134 | #168 | — | 2 | 2 | no | — |
| 135 | C2×#41 | — | 10 | 1 | no | — |
| 136 | C22×Q8 | — | 5 | 2 | no | — |
| 137 | #137 | — | 20 | 4 | no | — |
| 138 | #137 | — | 20 | 4 | no | — |
| 139 | C2×Q16 | — | 2 | 2 | no | — |
| 140 | C2×Q16 | — | 10 | 2 | no | — |
| 141 | C2×SD16 | — | 10 | 2 | no | — |
| 142 | C2×SD16 | — | 10 | 2 | no | — |
| 143 | C2×M16 | — | 10 | 1 | no | — |
| 144 | C4⋊Q8 | — | 5 | 2 | no | — |
| 145 | C4⋊Q8 | — | 5 | 1 | no | — |
| 146 | C4⋊Q8 | — | 10 | 2 | no | — |
| 147 | C4⋊Q8 | C2×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 148 | C4⋊Q8 | C2×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 149 | #127 | — | 5 | 1 | no | — |
| 150 | #124 | — | 10 | 1 | no | — |
| 151 | #121 | — | 10 | 2 | no | — |
| 152 | C4×D4 | — | 10 | 2 | no | — |
| 153 | C2×C4⋊C4 | — | 10 | 2 | no | — |
| 154 | C2×C4⋊C4 | — | 10 | 2 | no | — |
| 155 | C2×C42 | — | 5 | 1 | no | — |
| 156 | C2×C42 | — | 10 | 2 | no | — |
| 157 | C4⋊C8 | — | 10 | 2 | no | — |
| 158 | C22⋊C8 | — | 10 | 2 | no | — |
| 159 | #97 | — | 10 | 1 | no | — |
| 160 | #97 | — | 10 | 2 | no | — |
| 161 | C4×C8 | — | 5 | 1 | no | — |
| 162 | C4×C8 | C2×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 163 | #95 | — | 5 | 4 | no | — |
| 164 | #95 | — | 10 | 2 | no | — |
| 165 | #95 | — | 10 | 2 | no | — |
| 166 | #95 | — | 10 | 4 | no | — |
| 167 | C22×C5 | #386 | 1 | 1 | yes | characteristic |
| 168 | C2×D5 | — | 2 | 2 | no | — |
| 169 | C2×D5 | — | 8 | 1 | no | — |
| 170 | C2×D5 | — | 8 | 2 | no | — |
| 171 | C2×D5 | #453 | 1 | 2 | yes | — |
| 172 | AGL(1,𝔽5) | — | 8 | 4 | no | — |
| 173 | C4×C5 | — | 2 | 1 | no | — |
| 174 | C4×C5 | — | 2 | 2 | no | — |
| 175 | C4×C5 | — | 4 | 2 | no | — |
| 176 | C4×C5 | — | 8 | 1 | no | — |
| 177 | Dic5 | — | 2 | 1 | no | — |
| 178 | Dic5 | — | 2 | 2 | no | — |
| 179 | Dic5 | — | 4 | 2 | no | — |
| 180 | #41 | — | 20 | 2 | no | — |
| 181 | #41 | — | 20 | 2 | no | — |
| 182 | C2×Q8 | — | 2 | 1 | no | — |
| 183 | C2×Q8 | — | 5 | 2 | no | — |
| 184 | C2×Q8 | — | 10 | 2 | no | — |
| 185 | C2×Q8 | — | 20 | 2 | no | — |
| 186 | C2×Q8 | — | 20 | 2 | no | — |
| 187 | C2×Q8 | C22⋊AGL(1,𝔽5) | 1 | 2 | yes | — |
| 188 | C2×D4 | — | 10 | 2 | no | — |
| 189 | C2×D4 | — | 10 | 1 | no | — |
| 190 | C2×D4 | — | 20 | 1 | no | — |
| 191 | C22×C4 | — | 5 | 1 | no | — |
| 192 | C22×C4 | — | 5 | 2 | no | — |
| 193 | C22×C4 | — | 10 | 2 | no | — |
| 194 | C22×C4 | — | 10 | 4 | no | — |
| 195 | C22×C4 | — | 20 | 1 | no | — |
| 196 | Q16 | — | 4 | 4 | no | — |
| 197 | Q16 | — | 20 | 4 | no | — |
| 198 | SD16 | — | 20 | 4 | no | — |
| 199 | SD16 | — | 20 | 4 | no | — |
| 200 | M16 | — | 10 | 4 | no | — |
| 201 | C2×C8 | — | 2 | 1 | no | — |
| 202 | C2×C8 | — | 10 | 1 | no | — |
| 203 | C2×C8 | — | 20 | 2 | no | — |
| 204 | C4⋊C4 | — | 2 | 2 | no | — |
| 205 | C4⋊C4 | — | 2 | 2 | no | — |
| 206 | C4⋊C4 | — | 10 | 4 | no | — |
| 207 | C4⋊C4 | — | 10 | 2 | no | — |
| 208 | C4⋊C4 | — | 40 | 2 | no | — |
| 209 | C22⋊C4 | — | 20 | 2 | no | — |
| 210 | C42 | — | 5 | 2 | no | — |
| 211 | C42 | — | 5 | 1 | no | — |
| 212 | C42 | — | 40 | 2 | no | — |
| 213 | C42 | C22×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 214 | C2×C5 | #1887 | 1 | 1 | yes | characteristic |
| 215 | C2×C5 | #1905 | 1 | 2 | yes | — |
| 216 | D5 | — | 2 | 2 | no | — |
| 217 | D5 | — | 16 | 1 | no | — |
| 218 | C23 | — | 5 | 1 | no | — |
| 219 | C23 | — | 20 | 1 | no | — |
| 220 | Q8 | — | 4 | 2 | no | — |
| 221 | Q8 | — | 4 | 2 | no | — |
| 222 | Q8 | — | 20 | 2 | no | — |
| 223 | Q8 | — | 40 | 2 | no | — |
| 224 | D4 | — | 20 | 2 | no | — |
| 225 | D4 | — | 40 | 2 | no | — |
| 226 | D4 | — | 40 | 2 | no | — |
| 227 | C2×C4 | — | 2 | 2 | no | — |
| 228 | C2×C4 | — | 4 | 1 | no | — |
| 229 | C2×C4 | — | 5 | 2 | no | — |
| 230 | C2×C4 | — | 5 | 1 | no | — |
| 231 | C2×C4 | — | 10 | 2 | no | — |
| 232 | C2×C4 | — | 10 | 2 | no | — |
| 233 | C2×C4 | — | 20 | 2 | no | — |
| 234 | C2×C4 | — | 20 | 2 | no | — |
| 235 | C2×C4 | — | 20 | 2 | no | — |
| 236 | C2×C4 | — | 20 | 4 | no | — |
| 237 | C2×C4 | — | 20 | 8 | no | — |
| 238 | C2×C4 | — | 40 | 1 | no | — |
| 239 | C2×C4 | — | 40 | 1 | no | — |
| 240 | C2×C4 | D4×AGL(1,𝔽5) | 1 | 2 | yes | — |
| 241 | C2×C4 | C2×C22⋊AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 242 | C8 | — | 2 | 2 | no | — |
| 243 | C8 | — | 10 | 2 | no | — |
| 244 | C8 | — | 40 | 2 | no | — |
| 245 | C5 | #9429 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 246 | C22 | — | 5 | 2 | no | — |
| 247 | C22 | — | 10 | 2 | no | — |
| 248 | C22 | — | 40 | 2 | no | — |
| 249 | C22 | — | 40 | 1 | no | — |
| 250 | C22 | #65779 | 1 | 1 | yes | characteristic |
| 251 | C4 | — | 2 | 2 | no | — |
| 252 | C4 | — | 2 | 1 | no | — |
| 253 | C4 | — | 4 | 2 | no | — |
| 254 | C4 | — | 8 | 1 | no | — |
| 255 | C4 | — | 10 | 2 | no | — |
| 256 | C4 | — | 10 | 1 | no | — |
| 257 | C4 | — | 20 | 2 | no | — |
| 258 | C4 | — | 40 | 4 | no | — |
| 259 | C2 | — | 10 | 2 | no | — |
| 260 | C2 | — | 80 | 1 | no | — |
| 261 | C2 | #10604614 | 1 | 1 | yes | characteristic |
| 262 | C2 | #10604846 | 1 | 2 | yes | — |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #10617390 | C2 | semidirect |
| 3. #10604662 | C2 | semidirect |
| 5. #10598528 | C2 | semidirect |
| 13. C5×#494 | C4 | semidirect |
| 15. #65381 | C4 | semidirect |
| 16. #65374 | C4 | semidirect |
| 17. #64982 | C4 | semidirect |
| 98. #494 | AGL(1,𝔽5) | semidirect |
| 245. C5 | #9429 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #10617390 | 2 |
| #10613531 | 2 |
| #10604662 | 2 |
| #10604627 | 2 |
| #10598528 | 2 |
| #9429 | 5 |