Finite Group DatabasebetaFinite Group DatabasebetaOrder 1280.
2734 conjugacy classes of subgroups; only the 371 normal subgroups are shown.
This group has 2734 conjugacy classes of subgroups — too many to draw. Only its 371 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×#379 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C2×#65536 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | C2×#65536 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | #10602340 | C2 | 1 | 12 | yes | maximal, complemented |
| 5 | C26×C5 | C22 | 1 | 1 | yes | characteristic |
| 6 | C22×C5×C22⋊C4 | C22 | 1 | 6 | yes | — |
| 7 | C22×C22⋊Dic5 | C22 | 1 | 6 | yes | — |
| 8 | C22×C22⋊Dic5 | C22 | 1 | 6 | yes | — |
| 9 | C22×#683 | C22 | 1 | 4 | yes | complemented |
| 10 | C5×#379 | — | 2 | 4 | no | — |
| 11 | #65536 | — | 2 | 4 | no | — |
| 12 | #65536 | C4 | 1 | 8 | yes | complemented |
| 13 | C2×#3963 | C22 | 1 | 12 | yes | complemented |
| 14 | #14673 | — | 5 | 1 | no | maximal, Sylow |
| 15 | C25×C5 | — | 4 | 6 | no | — |
| 16 | C25×C5 | — | 4 | 6 | no | — |
| 17 | C25×C5 | C2×C4 | 1 | 4 | yes | — |
| 18 | C25×C5 | D4 | 1 | 4 | yes | — |
| 19 | C25×C5 | Q8 | 1 | 4 | yes | — |
| 20 | C25×C5 | C23 | 1 | 3 | yes | — |
| 21 | C23×C4×C5 | C23 | 1 | 4 | yes | — |
| 22 | C23×Dic5 | C23 | 1 | 4 | yes | — |
| 23 | C23×Dic5 | C23 | 1 | 4 | yes | — |
| 24 | C2×C5×C22⋊C4 | — | 2 | 12 | no | — |
| 25 | C2×C5×C22⋊C4 | — | 4 | 6 | no | — |
| 26 | C2×C5×C22⋊C4 | — | 4 | 6 | no | — |
| 27 | C2×C22⋊Dic5 | — | 2 | 12 | no | — |
| 28 | C2×C22⋊Dic5 | — | 4 | 6 | no | — |
| 29 | C2×C22⋊Dic5 | — | 4 | 6 | no | — |
| 30 | C2×C22⋊Dic5 | — | 4 | 6 | no | — |
| 31 | C2×C22⋊Dic5 | — | 4 | 6 | no | — |
| 32 | C2×C22⋊Dic5 | C2×C4 | 1 | 24 | yes | — |
| 33 | C2×#683 | — | 4 | 12 | no | — |
| 34 | #3963 | — | 4 | 12 | no | — |
| 35 | C2×#379 | — | 5 | 1 | no | — |
| 36 | C2×#379 | — | 5 | 1 | no | — |
| 37 | C2×#379 | D5 | 1 | 1 | yes | characteristic |
| 38 | #1439 | — | 5 | 12 | no | — |
| 39 | C24×C5 | — | 2 | 6 | no | — |
| 40 | C24×C5 | — | 2 | 6 | no | — |
| 41 | C24×C5 | — | 2 | 6 | no | — |
| 42 | C24×C5 | — | 2 | 6 | no | — |
| 43 | C24×C5 | — | 2 | 6 | no | — |
| 44 | C24×C5 | — | 2 | 6 | no | — |
| 45 | C24×C5 | — | 4 | 24 | no | — |
| 46 | C24×C5 | — | 4 | 24 | no | — |
| 47 | C24×C5 | — | 4 | 12 | no | — |
| 48 | C24×C5 | — | 4 | 12 | no | — |
| 49 | C24×C5 | — | 4 | 48 | no | — |
| 50 | C24×C5 | — | 4 | 16 | no | — |
| 51 | C24×C5 | C4⋊C4 | 1 | 16 | yes | complemented |
| 52 | C24×C5 | C22×C4 | 1 | 6 | yes | — |
| 53 | C24×C5 | C2×D4 | 1 | 6 | yes | — |
| 54 | C24×C5 | C2×Q8 | 1 | 6 | yes | — |
| 55 | C24×C5 | C24 | 1 | 1 | yes | characteristic |
| 56 | C22×C4×C5 | — | 2 | 4 | no | — |
| 57 | C22×C4×C5 | — | 2 | 12 | no | — |
| 58 | C22×C4×C5 | — | 2 | 12 | no | — |
| 59 | C22×Dic5 | — | 2 | 12 | no | — |
| 60 | C22×Dic5 | — | 2 | 12 | no | — |
| 61 | C22×Dic5 | — | 2 | 4 | no | — |
| 62 | C22×Dic5 | — | 2 | 12 | no | — |
| 63 | C22×Dic5 | — | 2 | 12 | no | — |
| 64 | C22×Dic5 | C22×C4 | 1 | 8 | yes | — |
| 65 | C5×C22⋊C4 | — | 4 | 24 | no | — |
| 66 | C22⋊Dic5 | — | 4 | 24 | no | — |
| 67 | C22⋊Dic5 | — | 4 | 24 | no | — |
| 68 | #683 | — | 4 | 16 | no | — |
| 69 | C26 | C2×D5 | 1 | 1 | yes | characteristic |
| 70 | C22×C4⋊C4 | — | 5 | 4 | no | — |
| 71 | C22×C22⋊C4 | — | 5 | 6 | no | — |
| 72 | C22×C22⋊C4 | — | 5 | 6 | no | — |
| 73 | C22×C22⋊C4 | C2×D5 | 1 | 6 | yes | — |
| 74 | #379 | — | 2 | 4 | no | — |
| 75 | #379 | — | 5 | 8 | no | — |
| 76 | #379 | — | 10 | 4 | no | — |
| 77 | C2×#95 | — | 5 | 12 | no | — |
| 78 | C23×C5 | — | 2 | 12 | no | — |
| 79 | C23×C5 | — | 2 | 12 | no | — |
| 80 | C23×C5 | — | 2 | 12 | no | — |
| 81 | C23×C5 | — | 2 | 12 | no | — |
| 82 | C23×C5 | — | 2 | 12 | no | — |
| 83 | C23×C5 | — | 2 | 12 | no | — |
| 84 | C23×C5 | — | 2 | 6 | no | — |
| 85 | C23×C5 | — | 2 | 6 | no | — |
| 86 | C23×C5 | — | 2 | 6 | no | — |
| 87 | C23×C5 | — | 2 | 6 | no | — |
| 88 | C23×C5 | — | 2 | 6 | no | — |
| 89 | C23×C5 | — | 2 | 6 | no | — |
| 90 | C23×C5 | — | 4 | 48 | no | — |
| 91 | C23×C5 | — | 4 | 16 | no | — |
| 92 | C23×C5 | — | 4 | 48 | no | — |
| 93 | C23×C5 | — | 4 | 64 | no | — |
| 94 | C23×C5 | — | 4 | 64 | no | — |
| 95 | C23×C5 | — | 4 | 12 | no | — |
| 96 | C23×C5 | — | 4 | 12 | no | — |
| 97 | C23×C5 | — | 4 | 12 | no | — |
| 98 | C23×C5 | — | 4 | 12 | no | — |
| 99 | C23×C5 | C2×C4⋊C4 | 1 | 12 | yes | — |
| 100 | C23×C5 | C23×C4 | 1 | 1 | yes | characteristic |
| 101 | C23×C5 | C22×D4 | 1 | 1 | yes | characteristic |
| 102 | C23×C5 | C22×Q8 | 1 | 1 | yes | characteristic |
| 103 | C23×C5 | H(5,𝔽2) | 1 | 6 | yes | — |
| 104 | C23×C5 | H(5,𝔽2) | 1 | 6 | yes | — |
| 105 | C2×C4×C5 | — | 4 | 12 | no | — |
| 106 | C2×C4×C5 | — | 4 | 12 | no | — |
| 107 | C2×C4×C5 | — | 4 | 4 | no | — |
| 108 | C2×Dic5 | — | 2 | 24 | no | — |
| 109 | C2×Dic5 | — | 4 | 12 | no | — |
| 110 | C2×Dic5 | — | 4 | 4 | no | — |
| 111 | C2×Dic5 | — | 4 | 12 | no | — |
| 112 | C2×Dic5 | — | 4 | 12 | no | — |
| 113 | C2×Dic5 | — | 4 | 4 | no | — |
| 114 | C25 | — | 4 | 6 | no | — |
| 115 | C25 | — | 4 | 6 | no | — |
| 116 | C25 | Dic10 | 1 | 4 | yes | — |
| 117 | C25 | C4×D5 | 1 | 4 | yes | — |
| 118 | C25 | C22⋊D5 | 1 | 4 | yes | — |
| 119 | C25 | C22×D5 | 1 | 3 | yes | — |
| 120 | C23×C4 | — | 5 | 4 | no | — |
| 121 | C23×C4 | — | 5 | 4 | no | — |
| 122 | C23×C4 | C22×D5 | 1 | 4 | yes | — |
| 123 | C2×C4⋊C4 | — | 20 | 12 | no | — |
| 124 | C2×C22⋊C4 | — | 2 | 12 | no | — |
| 125 | C2×C22⋊C4 | — | 4 | 6 | no | — |
| 126 | C2×C22⋊C4 | — | 4 | 6 | no | — |
| 127 | C2×C22⋊C4 | — | 5 | 24 | no | — |
| 128 | C2×C22⋊C4 | — | 10 | 12 | no | — |
| 129 | C2×C22⋊C4 | — | 20 | 6 | no | — |
| 130 | C2×C22⋊C4 | — | 20 | 6 | no | — |
| 131 | C2×C22⋊C4 | — | 20 | 6 | no | — |
| 132 | C2×C22⋊C4 | — | 20 | 6 | no | — |
| 133 | #95 | — | 20 | 12 | no | — |
| 134 | C22×C5 | — | 2 | 12 | no | — |
| 135 | C22×C5 | — | 2 | 12 | no | — |
| 136 | C22×C5 | — | 2 | 12 | no | — |
| 137 | C22×C5 | — | 4 | 64 | no | — |
| 138 | C22×C5 | — | 4 | 6 | no | — |
| 139 | C22×C5 | — | 4 | 6 | no | — |
| 140 | C22×C5 | — | 4 | 6 | no | — |
| 141 | C22×C5 | — | 4 | 6 | no | — |
| 142 | C22×C5 | — | 4 | 24 | no | — |
| 143 | C22×C5 | — | 4 | 24 | no | — |
| 144 | C22×C5 | C22×C4⋊C4 | 1 | 1 | yes | characteristic |
| 145 | C22×C5 | #518 | 1 | 6 | yes | — |
| 146 | C22×C5 | #534 | 1 | 6 | yes | — |
| 147 | C22×C5 | #543 | 1 | 3 | yes | — |
| 148 | C22×C5 | #543 | 1 | 3 | yes | — |
| 149 | C22×C5 | #560 | 1 | 12 | yes | — |
| 150 | C22×C5 | H(3,𝔽4) | 1 | 4 | yes | — |
| 151 | C4×C5 | — | 8 | 4 | no | — |
| 152 | Dic5 | — | 4 | 8 | no | — |
| 153 | Dic5 | — | 8 | 4 | no | — |
| 154 | C24 | — | 2 | 6 | no | — |
| 155 | C24 | — | 2 | 6 | no | — |
| 156 | C24 | — | 2 | 6 | no | — |
| 157 | C24 | — | 2 | 6 | no | — |
| 158 | C24 | — | 2 | 6 | no | — |
| 159 | C24 | — | 2 | 6 | no | — |
| 160 | C24 | — | 4 | 12 | no | — |
| 161 | C24 | — | 4 | 12 | no | — |
| 162 | C24 | — | 4 | 24 | no | — |
| 163 | C24 | — | 4 | 24 | no | — |
| 164 | C24 | — | 4 | 16 | no | — |
| 165 | C24 | — | 4 | 48 | no | — |
| 166 | C24 | #683 | 1 | 16 | yes | complemented |
| 167 | C24 | C2×Dic10 | 1 | 6 | yes | — |
| 168 | C24 | C2×C4×D5 | 1 | 6 | yes | — |
| 169 | C24 | C2×C22⋊D5 | 1 | 6 | yes | — |
| 170 | C24 | C23×D5 | 1 | 1 | yes | characteristic |
| 171 | C22×C4 | — | 2 | 12 | no | — |
| 172 | C22×C4 | — | 2 | 12 | no | — |
| 173 | C22×C4 | — | 2 | 4 | no | — |
| 174 | C22×C4 | — | 5 | 8 | no | — |
| 175 | C22×C4 | — | 10 | 12 | no | — |
| 176 | C22×C4 | — | 10 | 12 | no | — |
| 177 | C22×C4 | — | 10 | 12 | no | — |
| 178 | C22×C4 | — | 10 | 12 | no | — |
| 179 | C22×C4 | — | 10 | 4 | no | — |
| 180 | C4⋊C4 | — | 20 | 16 | no | — |
| 181 | C22⋊C4 | — | 4 | 24 | no | — |
| 182 | C22⋊C4 | — | 20 | 24 | no | — |
| 183 | C22⋊C4 | — | 20 | 24 | no | — |
| 184 | C2×C5 | — | 4 | 12 | no | — |
| 185 | C2×C5 | #2315 | 1 | 3 | yes | — |
| 186 | C2×C5 | #2375 | 1 | 4 | yes | — |
| 187 | C2×C5 | #2812 | 1 | 4 | yes | — |
| 188 | C2×C5 | #2840 | 1 | 1 | yes | characteristic |
| 189 | C2×C5 | #2842 | 1 | 3 | yes | — |
| 190 | C23 | — | 2 | 6 | no | — |
| 191 | C23 | — | 2 | 6 | no | — |
| 192 | C23 | — | 2 | 6 | no | — |
| 193 | C23 | — | 2 | 6 | no | — |
| 194 | C23 | — | 2 | 6 | no | — |
| 195 | C23 | — | 2 | 6 | no | — |
| 196 | C23 | — | 2 | 12 | no | — |
| 197 | C23 | — | 2 | 12 | no | — |
| 198 | C23 | — | 2 | 12 | no | — |
| 199 | C23 | — | 2 | 12 | no | — |
| 200 | C23 | — | 2 | 12 | no | — |
| 201 | C23 | — | 2 | 12 | no | — |
| 202 | C23 | — | 4 | 48 | no | — |
| 203 | C23 | — | 4 | 16 | no | — |
| 204 | C23 | — | 4 | 48 | no | — |
| 205 | C23 | — | 4 | 64 | no | — |
| 206 | C23 | — | 4 | 64 | no | — |
| 207 | C23 | — | 4 | 12 | no | — |
| 208 | C23 | — | 4 | 12 | no | — |
| 209 | C23 | — | 4 | 12 | no | — |
| 210 | C23 | — | 4 | 12 | no | — |
| 211 | C23 | C2×#683 | 1 | 12 | yes | — |
| 212 | C23 | C22×Dic10 | 1 | 1 | yes | characteristic |
| 213 | C23 | C22×C4×D5 | 1 | 1 | yes | characteristic |
| 214 | C23 | #4144 | 1 | 6 | yes | — |
| 215 | C23 | #4144 | 1 | 6 | yes | — |
| 216 | C23 | C22×C22⋊D5 | 1 | 1 | yes | characteristic |
| 217 | C2×C4 | — | 4 | 12 | no | — |
| 218 | C2×C4 | — | 4 | 4 | no | — |
| 219 | C2×C4 | — | 4 | 12 | no | — |
| 220 | C2×C4 | — | 10 | 24 | no | — |
| 221 | C2×C4 | — | 20 | 4 | no | — |
| 222 | C2×C4 | — | 20 | 12 | no | — |
| 223 | C2×C4 | — | 20 | 12 | no | — |
| 224 | C2×C4 | — | 20 | 4 | no | — |
| 225 | C2×C4 | — | 20 | 12 | no | — |
| 226 | C5 | #14673 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 227 | C22 | — | 2 | 12 | no | — |
| 228 | C22 | — | 2 | 12 | no | — |
| 229 | C22 | — | 2 | 12 | no | — |
| 230 | C22 | — | 4 | 64 | no | — |
| 231 | C22 | — | 4 | 6 | no | — |
| 232 | C22 | — | 4 | 6 | no | — |
| 233 | C22 | — | 4 | 6 | no | — |
| 234 | C22 | — | 4 | 6 | no | — |
| 235 | C22 | — | 4 | 24 | no | — |
| 236 | C22 | — | 4 | 24 | no | — |
| 237 | C22 | #65817 | 1 | 3 | yes | — |
| 238 | C22 | #65817 | 1 | 3 | yes | — |
| 239 | C22 | #65820 | 1 | 6 | yes | — |
| 240 | C22 | #65928 | 1 | 4 | yes | — |
| 241 | C22 | #65929 | 1 | 12 | yes | — |
| 242 | C22 | C22×#683 | 1 | 1 | yes | characteristic |
| 243 | C22 | #66138 | 1 | 6 | yes | — |
| 244 | C4 | — | 8 | 4 | no | — |
| 245 | C4 | — | 20 | 8 | no | — |
| 246 | C4 | — | 40 | 4 | no | — |
| 247 | C2 | — | 4 | 12 | no | — |
| 248 | C2 | #10605575 | 1 | 1 | yes | characteristic |
| 249 | C2 | #10605576 | 1 | 3 | yes | — |
| 250 | C2 | #10605582 | 1 | 4 | yes | — |
| 251 | C2 | #10613826 | 1 | 3 | yes | — |
| 252 | C2 | #10614342 | 1 | 4 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C24 | C23×D5 |
| commutator | C23×C5 | C23×C4 |
| Fitting | C2×C5×#379 | C2 |
| socle, cosocle | C24×C5 | C24 |
| prime radical | C26×C5 | C22 |
| hypercenter | C2×#379 | D5 |
| hypocenter | C5 | #14673 |
| LCS | G ▷ C23×C5 ▷ C5 | |
| UCS | C1 ◁ C24 ◁ C2×#379 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 4. #10602340 | C2 | semidirect |
| 9. C22×#683 | C22 | semidirect |
| 12. #65536 | C4 | semidirect |
| 13. C2×#3963 | C22 | semidirect |
| 51. C24×C5 | C4⋊C4 | semidirect |
| 166. C24 | #683 | semidirect |
| 226. C5 | #14673 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C5×#379 | 2 |
| C2×#65536 | 2 |
| #10602340 | 2 |
| #14673 | 5 |