Finite Group DatabasebetaFinite Group DatabasebetaOrder 1280.
284 conjugacy classes of subgroups; only the 42 normal subgroups are shown.
This group has 284 conjugacy classes of subgroups — too many to draw. Only its 42 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 | #10604563 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #10604543 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | D5×#354 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | #66262 | C4 | 1 | 1 | yes | characteristic, complemented |
| 5 | D5×#124 | C22 | 1 | 1 | yes | characteristic |
| 6 | D5⋊#101 | C4 | 1 | 1 | yes | characteristic, complemented |
| 7 | #65767 | — | 2 | 1 | no | — |
| 8 | #65683 | — | 2 | 1 | no | — |
| 9 | D5×#99 | — | 2 | 1 | no | — |
| 10 | #64927 | C4 | 1 | 2 | yes | complemented |
| 11 | #64921 | — | 2 | 1 | no | — |
| 12 | #64862 | — | 2 | 1 | no | — |
| 13 | C5×#354 | C4 | 1 | 1 | yes | characteristic, complemented |
| 14 | #64761 | C4 | 1 | 1 | yes | characteristic, complemented |
| 15 | #64693 | — | 2 | 1 | no | — |
| 16 | #7481 | — | 5 | 1 | no | maximal, Sylow |
| 17 | C2×Q8×D5 | C2×C4 | 1 | 1 | yes | characteristic |
| 18 | C2×D4×D5 | D4 | 1 | 1 | yes | characteristic |
| 19 | Q8×AGL(1,𝔽5) | — | 4 | 2 | no | — |
| 20 | #4130 | — | 2 | 1 | no | — |
| 21 | #4130 | — | 4 | 1 | no | — |
| 22 | D5⋊M16 | — | 2 | 1 | no | — |
| 23 | C5×#124 | C2×C4 | 1 | 1 | yes | characteristic |
| 24 | #4093 | C2×C4 | 1 | 1 | yes | characteristic |
| 25 | #4082 | D4 | 1 | 1 | yes | characteristic |
| 26 | #4031 | — | 4 | 1 | no | — |
| 27 | D5×C22⋊C4 | — | 4 | 1 | no | — |
| 28 | D5×C22⋊C4 | — | 4 | 1 | no | — |
| 29 | #4021 | Q8 | 1 | 1 | yes | characteristic |
| 30 | C42×D5 | D4 | 1 | 1 | yes | characteristic |
| 31 | #4007 | — | 4 | 2 | no | — |
| 32 | #4005 | — | 4 | 1 | no | — |
| 33 | #4003 | — | 4 | 1 | no | — |
| 34 | #4002 | — | 2 | 2 | no | — |
| 35 | C4⋊#170 | — | 4 | 1 | no | — |
| 36 | C5×#99 | — | 2 | 1 | no | — |
| 37 | #3966 | — | 2 | 1 | no | — |
| 38 | #3938 | — | 4 | 1 | no | — |
| 39 | C2×#354 | — | 5 | 1 | no | — |
| 40 | #1888 | — | 5 | 1 | no | — |
| 41 | #1533 | — | 5 | 1 | no | — |
| 42 | C23×D5 | — | 2 | 1 | no | — |
| 43 | C2×C5×Q8 | C42 | 1 | 1 | yes | characteristic |
| 44 | C2×C5×D4 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 45 | C2×C22⋊D5 | — | 2 | 1 | no | — |
| 46 | Q8×D5 | — | 4 | 1 | no | — |
| 47 | Q8×D5 | — | 4 | 1 | no | — |
| 48 | D4×D5 | — | 4 | 1 | no | — |
| 49 | D4×D5 | — | 4 | 1 | no | — |
| 50 | C2×D20 | C4⋊C4 | 1 | 1 | yes | characteristic |
| 51 | C2×C4×D5 | — | 2 | 1 | no | — |
| 52 | C2×C4×D5 | — | 2 | 1 | no | — |
| 53 | C2×C4×D5 | — | 8 | 1 | no | — |
| 54 | C2×C4×D5 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 55 | C2×Dic10 | — | 2 | 1 | no | — |
| 56 | C2×Dic10 | C4⋊C4 | 1 | 1 | yes | characteristic |
| 57 | C22⋊AGL(1,𝔽5) | — | 4 | 1 | no | — |
| 58 | #704 | — | 4 | 1 | no | — |
| 59 | C2×#170 | — | 4 | 2 | no | — |
| 60 | C4⋊AGL(1,𝔽5) | — | 2 | 2 | no | — |
| 61 | C4⋊AGL(1,𝔽5) | — | 4 | 2 | no | — |
| 62 | C4×AGL(1,𝔽5) | — | 4 | 1 | no | — |
| 63 | C4×AGL(1,𝔽5) | — | 4 | 2 | no | — |
| 64 | #700 | — | 4 | 1 | no | — |
| 65 | D5⋊C8 | — | 4 | 1 | no | — |
| 66 | C5×C22⋊C4 | — | 4 | 1 | no | — |
| 67 | C5×C22⋊C4 | — | 4 | 1 | no | — |
| 68 | C42×C5 | C4⋊C4 | 1 | 1 | yes | characteristic |
| 69 | C22⋊Dic5 | — | 4 | 1 | no | — |
| 70 | C22⋊Dic5 | — | 4 | 1 | no | — |
| 71 | #685 | — | 4 | 1 | no | — |
| 72 | #685 | — | 4 | 1 | no | — |
| 73 | #685 | — | 8 | 1 | no | — |
| 74 | C4×Dic5 | — | 2 | 1 | no | — |
| 75 | C4×Dic5 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 76 | C2×#124 | — | 5 | 1 | no | — |
| 77 | #519 | — | 5 | 1 | no | — |
| 78 | #432 | — | 10 | 1 | no | — |
| 79 | #418 | — | 10 | 1 | no | — |
| 80 | C2×#101 | — | 5 | 1 | no | — |
| 81 | C2×#99 | — | 10 | 1 | no | — |
| 82 | #354 | — | 5 | 1 | no | — |
| 83 | #354 | — | 10 | 1 | no | — |
| 84 | #354 | AGL(1,𝔽5) | 1 | 1 | yes | characteristic, complemented |
| 85 | #339 | — | 10 | 1 | no | — |
| 86 | #329 | — | 5 | 2 | no | — |
| 87 | #329 | — | 10 | 1 | no | — |
| 88 | C23×C5 | — | 2 | 1 | no | — |
| 89 | C22×D5 | — | 2 | 1 | no | — |
| 90 | C22×D5 | — | 4 | 1 | no | — |
| 91 | C22×D5 | — | 4 | 1 | no | — |
| 92 | C22×D5 | — | 8 | 2 | no | — |
| 93 | C22×D5 | #99 | 1 | 1 | yes | characteristic |
| 94 | C2×AGL(1,𝔽5) | — | 4 | 2 | no | — |
| 95 | C5×Q8 | — | 4 | 1 | no | — |
| 96 | C5×D4 | — | 4 | 1 | no | — |
| 97 | C2×C4×C5 | — | 2 | 1 | no | — |
| 98 | C2×C4×C5 | — | 2 | 1 | no | — |
| 99 | C2×C4×C5 | — | 8 | 1 | no | — |
| 100 | C2×C4×C5 | #95 | 1 | 1 | yes | characteristic |
| 101 | C22⋊D5 | — | 8 | 1 | no | — |
| 102 | C2×Dic5 | — | 2 | 1 | no | — |
| 103 | C2×Dic5 | — | 2 | 1 | no | — |
| 104 | C2×Dic5 | — | 8 | 1 | no | — |
| 105 | C2×Dic5 | #99 | 1 | 1 | yes | characteristic |
| 106 | D20 | — | 4 | 1 | no | — |
| 107 | C4×D5 | — | 2 | 1 | no | — |
| 108 | C4×D5 | — | 2 | 1 | no | — |
| 109 | C4×D5 | — | 4 | 1 | no | — |
| 110 | C4×D5 | — | 4 | 1 | no | — |
| 111 | C4×D5 | — | 8 | 1 | no | — |
| 112 | C4×D5 | — | 16 | 2 | no | — |
| 113 | Dic10 | — | 4 | 2 | no | — |
| 114 | Dic10 | — | 4 | 1 | no | — |
| 115 | #170 | — | 8 | 1 | no | — |
| 116 | #170 | — | 8 | 2 | no | — |
| 117 | C22×Q8 | — | 5 | 1 | no | — |
| 118 | C22×D4 | — | 5 | 1 | no | — |
| 119 | C2×M16 | — | 10 | 1 | no | — |
| 120 | #124 | — | 5 | 1 | no | — |
| 121 | #124 | — | 5 | 1 | no | — |
| 122 | #124 | — | 5 | 1 | no | — |
| 123 | #124 | — | 20 | 1 | no | — |
| 124 | #124 | C2×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 125 | C4×Q8 | — | 20 | 2 | no | — |
| 126 | #117 | — | 10 | 1 | no | — |
| 127 | #117 | — | 20 | 1 | no | — |
| 128 | C2×C22⋊C4 | — | 20 | 1 | no | — |
| 129 | C2×C22⋊C4 | — | 20 | 1 | no | — |
| 130 | C2×C42 | — | 5 | 1 | no | — |
| 131 | C4⋊C8 | — | 20 | 1 | no | — |
| 132 | #102 | — | 20 | 2 | no | — |
| 133 | #101 | — | 20 | 1 | no | — |
| 134 | #99 | — | 2 | 1 | no | — |
| 135 | #99 | — | 10 | 1 | no | — |
| 136 | #99 | — | 20 | 1 | no | — |
| 137 | C22⋊C8 | — | 20 | 1 | no | — |
| 138 | #97 | — | 10 | 2 | no | — |
| 139 | C22×C5 | — | 4 | 1 | no | — |
| 140 | C22×C5 | — | 8 | 1 | no | — |
| 141 | C22×C5 | #342 | 1 | 1 | yes | characteristic |
| 142 | C2×D5 | — | 4 | 1 | no | — |
| 143 | C2×D5 | — | 4 | 1 | no | — |
| 144 | C2×D5 | — | 8 | 2 | no | — |
| 145 | C2×D5 | — | 8 | 1 | no | — |
| 146 | C2×D5 | — | 16 | 1 | no | — |
| 147 | C2×D5 | AΣL(1,𝔽16) | 1 | 1 | yes | characteristic |
| 148 | C2×D5 | #352 | 1 | 1 | yes | characteristic |
| 149 | AGL(1,𝔽5) | — | 8 | 2 | no | — |
| 150 | C4×C5 | — | 2 | 1 | no | — |
| 151 | C4×C5 | — | 4 | 1 | no | — |
| 152 | C4×C5 | — | 4 | 1 | no | — |
| 153 | C4×C5 | — | 16 | 1 | no | — |
| 154 | Dic5 | — | 2 | 1 | no | — |
| 155 | Dic5 | — | 4 | 1 | no | — |
| 156 | Dic5 | — | 4 | 1 | no | — |
| 157 | Dic5 | — | 16 | 1 | no | — |
| 158 | C24 | — | 10 | 1 | no | — |
| 159 | C2×Q8 | — | 5 | 1 | no | — |
| 160 | C2×Q8 | — | 10 | 1 | no | — |
| 161 | C2×Q8 | — | 20 | 1 | no | — |
| 162 | C2×Q8 | — | 20 | 1 | no | — |
| 163 | C2×Q8 | C4×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 164 | C2×D4 | — | 5 | 1 | no | — |
| 165 | C2×D4 | — | 10 | 1 | no | — |
| 166 | C2×D4 | — | 20 | 1 | no | — |
| 167 | C2×D4 | — | 20 | 1 | no | — |
| 168 | C2×D4 | C22⋊AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 169 | C22×C4 | — | 5 | 1 | no | — |
| 170 | C22×C4 | — | 10 | 1 | no | — |
| 171 | C22×C4 | — | 10 | 1 | no | — |
| 172 | C22×C4 | — | 40 | 1 | no | — |
| 173 | M16 | — | 20 | 1 | no | — |
| 174 | M16 | — | 20 | 1 | no | — |
| 175 | C2×C8 | — | 20 | 2 | no | — |
| 176 | C2×C8 | — | 20 | 1 | no | — |
| 177 | C4⋊C4 | — | 10 | 2 | no | — |
| 178 | C4⋊C4 | — | 20 | 2 | no | — |
| 179 | C22⋊C4 | — | 4 | 1 | no | — |
| 180 | C22⋊C4 | — | 4 | 1 | no | — |
| 181 | C22⋊C4 | — | 20 | 1 | no | — |
| 182 | C22⋊C4 | — | 20 | 1 | no | — |
| 183 | C22⋊C4 | — | 20 | 1 | no | — |
| 184 | C22⋊C4 | — | 20 | 1 | no | — |
| 185 | C22⋊C4 | — | 20 | 1 | no | — |
| 186 | C22⋊C4 | — | 40 | 1 | no | — |
| 187 | C42 | — | 5 | 1 | no | — |
| 188 | C42 | — | 10 | 1 | no | — |
| 189 | C42 | — | 20 | 1 | no | — |
| 190 | C42 | — | 20 | 2 | no | — |
| 191 | C42 | C4⋊AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 192 | C2×C5 | — | 2 | 1 | no | — |
| 193 | C2×C5 | — | 8 | 1 | no | — |
| 194 | C2×C5 | #1390 | 1 | 1 | yes | characteristic |
| 195 | D5 | — | 2 | 1 | no | — |
| 196 | D5 | — | 2 | 1 | no | — |
| 197 | D5 | — | 8 | 1 | no | — |
| 198 | C23 | — | 2 | 1 | no | — |
| 199 | C23 | — | 5 | 1 | no | — |
| 200 | C23 | — | 10 | 1 | no | — |
| 201 | C23 | — | 20 | 1 | no | — |
| 202 | C23 | — | 20 | 1 | no | — |
| 203 | C23 | — | 40 | 2 | no | — |
| 204 | Q8 | — | 4 | 1 | no | — |
| 205 | Q8 | — | 20 | 1 | no | — |
| 206 | Q8 | — | 20 | 2 | no | — |
| 207 | D4 | — | 4 | 1 | no | — |
| 208 | D4 | — | 20 | 1 | no | — |
| 209 | D4 | — | 40 | 1 | no | — |
| 210 | C2×C4 | — | 2 | 1 | no | — |
| 211 | C2×C4 | — | 2 | 1 | no | — |
| 212 | C2×C4 | — | 5 | 1 | no | — |
| 213 | C2×C4 | — | 8 | 1 | no | — |
| 214 | C2×C4 | — | 10 | 1 | no | — |
| 215 | C2×C4 | — | 10 | 1 | no | — |
| 216 | C2×C4 | — | 10 | 1 | no | — |
| 217 | C2×C4 | — | 10 | 1 | no | — |
| 218 | C2×C4 | — | 20 | 1 | no | — |
| 219 | C2×C4 | — | 20 | 1 | no | — |
| 220 | C2×C4 | — | 20 | 2 | no | — |
| 221 | C2×C4 | — | 40 | 1 | no | — |
| 222 | C2×C4 | — | 40 | 1 | no | — |
| 223 | C2×C4 | — | 80 | 2 | no | — |
| 224 | C2×C4 | (C2×AGL(1,𝔽5))⋊C4 | 1 | 1 | yes | characteristic |
| 225 | C8 | — | 40 | 1 | no | — |
| 226 | C8 | — | 40 | 2 | no | — |
| 227 | C5 | #7481 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 228 | C22 | — | 4 | 1 | no | — |
| 229 | C22 | — | 5 | 1 | no | — |
| 230 | C22 | — | 5 | 1 | no | — |
| 231 | C22 | — | 8 | 1 | no | — |
| 232 | C22 | — | 20 | 1 | no | — |
| 233 | C22 | — | 20 | 1 | no | — |
| 234 | C22 | — | 40 | 1 | no | — |
| 235 | C22 | — | 40 | 2 | no | — |
| 236 | C22 | — | 80 | 1 | no | — |
| 237 | C22 | #64866 | 1 | 1 | yes | characteristic |
| 238 | C4 | — | 2 | 1 | no | — |
| 239 | C4 | — | 4 | 1 | no | — |
| 240 | C4 | — | 4 | 1 | no | — |
| 241 | C4 | — | 10 | 1 | no | — |
| 242 | C4 | — | 16 | 1 | no | — |
| 243 | C4 | — | 20 | 1 | no | — |
| 244 | C4 | — | 20 | 1 | no | — |
| 245 | C4 | — | 40 | 2 | no | — |
| 246 | C4 | — | 80 | 1 | no | — |
| 247 | C2 | — | 2 | 1 | no | — |
| 248 | C2 | — | 8 | 1 | no | — |
| 249 | C2 | — | 10 | 1 | no | — |
| 250 | C2 | — | 10 | 1 | no | — |
| 251 | C2 | — | 40 | 1 | no | — |
| 252 | C2 | #10598567 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 4. #66262 | C4 | semidirect |
| 6. D5⋊#101 | C4 | semidirect |
| 10. #64927 | C4 | semidirect |
| 13. C5×#354 | C4 | semidirect |
| 14. #64761 | C4 | semidirect |
| 84. #354 | AGL(1,𝔽5) | semidirect |
| 227. C5 | #7481 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #10604563 | 2 |
| #10604543 | 2 |
| D5×#354 | 2 |
| #7481 | 5 |