Finite Group DatabasebetaFinite Group DatabasebetaOrder 1280.
260 conjugacy classes of subgroups; only the 44 normal subgroups are shown.
This group has 260 conjugacy classes of subgroups — too many to draw. Only its 44 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 | #10604673 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | D5⋊#356 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | D5×#356 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | D5×C4⋊Q8 | C22 | 1 | 1 | yes | characteristic |
| 5 | D5⋊#101 | — | 2 | 1 | no | — |
| 6 | C2×#4009 | — | 2 | 1 | no | — |
| 7 | C2×#4009 | — | 2 | 1 | no | — |
| 8 | #65689 | C4 | 1 | 1 | yes | characteristic |
| 9 | #65676 | C4 | 1 | 1 | yes | characteristic |
| 10 | D5×#101 | — | 2 | 1 | no | — |
| 11 | #64929 | — | 2 | 1 | no | — |
| 12 | #64855 | C4 | 1 | 2 | yes | complemented |
| 13 | C5×#356 | C4 | 1 | 1 | yes | characteristic, complemented |
| 14 | #64768 | C4 | 1 | 1 | yes | characteristic, complemented |
| 15 | #64698 | — | 2 | 1 | no | — |
| 16 | #7505 | — | 5 | 1 | no | maximal, Sylow |
| 17 | C2×Q8×D5 | D4 | 1 | 1 | yes | characteristic |
| 18 | C2×Q8×D5 | D4 | 1 | 1 | yes | characteristic |
| 19 | C2×C4⋊AGL(1,𝔽5) | — | 2 | 1 | no | — |
| 20 | D5⋊M16 | — | 4 | 1 | no | — |
| 21 | C2×D5⋊C8 | — | 2 | 1 | no | — |
| 22 | C5×C4⋊Q8 | C2×C4 | 1 | 1 | yes | characteristic |
| 23 | #4090 | D4 | 1 | 1 | yes | characteristic |
| 24 | #4090 | Q8 | 1 | 1 | yes | characteristic |
| 25 | D5×M16 | — | 4 | 1 | no | — |
| 26 | D5×C4⋊C4 | — | 4 | 1 | no | — |
| 27 | #4034 | — | 4 | 1 | no | — |
| 28 | C42×D5 | C2×C4 | 1 | 1 | yes | characteristic |
| 29 | #4015 | C2×C4 | 1 | 1 | yes | characteristic |
| 30 | #4009 | — | 4 | 2 | no | — |
| 31 | #4009 | — | 4 | 2 | no | — |
| 32 | (C2×AGL(1,𝔽5))⋊C4 | — | 4 | 1 | no | — |
| 33 | #4005 | — | 2 | 2 | no | — |
| 34 | #4005 | — | 4 | 1 | no | — |
| 35 | #4002 | — | 4 | 1 | no | — |
| 36 | C5×#101 | — | 2 | 1 | no | — |
| 37 | #3968 | — | 2 | 1 | no | — |
| 38 | #3956 | — | 4 | 1 | no | — |
| 39 | C2×#356 | — | 5 | 1 | no | — |
| 40 | C2×#356 | — | 5 | 1 | no | — |
| 41 | #1960 | — | 5 | 1 | no | — |
| 42 | C22×AGL(1,𝔽5) | — | 4 | 1 | no | — |
| 43 | C2×C5×Q8 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 44 | C2×C5×Q8 | C4⋊C4 | 1 | 1 | yes | characteristic |
| 45 | Q8×D5 | — | 4 | 1 | no | — |
| 46 | Q8×D5 | — | 4 | 1 | no | — |
| 47 | Q8×D5 | — | 4 | 1 | no | — |
| 48 | Q8×D5 | — | 4 | 1 | no | — |
| 49 | C2×C4×D5 | — | 2 | 1 | no | — |
| 50 | C2×C4×D5 | — | 2 | 1 | no | — |
| 51 | C2×C4×D5 | — | 2 | 1 | no | — |
| 52 | C2×C4×D5 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 53 | C2×Dic10 | — | 2 | 1 | no | — |
| 54 | C2×Dic10 | — | 2 | 1 | no | — |
| 55 | C2×Dic10 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 56 | C2×Dic10 | C4⋊C4 | 1 | 1 | yes | characteristic |
| 57 | #704 | — | 4 | 2 | no | — |
| 58 | C2×#170 | — | 4 | 1 | no | — |
| 59 | C4⋊AGL(1,𝔽5) | — | 4 | 1 | no | — |
| 60 | C4⋊AGL(1,𝔽5) | — | 4 | 1 | no | — |
| 61 | #700 | — | 8 | 1 | no | — |
| 62 | D5⋊C8 | — | 2 | 2 | no | — |
| 63 | D5⋊C8 | — | 4 | 1 | no | — |
| 64 | D5⋊C8 | — | 8 | 1 | no | — |
| 65 | C5×M16 | — | 4 | 1 | no | — |
| 66 | C5×C4⋊C4 | — | 4 | 1 | no | — |
| 67 | C42×C5 | C42 | 1 | 1 | yes | characteristic |
| 68 | C4⋊Dic5 | — | 4 | 1 | no | — |
| 69 | #683 | — | 4 | 1 | no | — |
| 70 | #683 | — | 4 | 1 | no | — |
| 71 | C4×Dic5 | — | 2 | 1 | no | — |
| 72 | C4×Dic5 | C4⋊C4 | 1 | 1 | yes | characteristic |
| 73 | #681 | — | 4 | 1 | no | — |
| 74 | #676 | — | 8 | 1 | no | — |
| 75 | C8×D5 | — | 8 | 1 | no | — |
| 76 | C2×C4⋊Q8 | — | 5 | 1 | no | — |
| 77 | C2×#103 | — | 10 | 1 | no | — |
| 78 | C2×#103 | — | 10 | 1 | no | — |
| 79 | C2×#101 | — | 10 | 1 | no | — |
| 80 | C2×#101 | — | 10 | 1 | no | — |
| 81 | C2×#97 | — | 5 | 1 | no | — |
| 82 | #382 | — | 5 | 1 | no | — |
| 83 | #356 | — | 5 | 2 | no | — |
| 84 | #356 | — | 5 | 1 | no | — |
| 85 | #356 | — | 10 | 1 | no | — |
| 86 | #356 | — | 10 | 1 | no | — |
| 87 | #356 | AGL(1,𝔽5) | 1 | 1 | yes | characteristic, complemented |
| 88 | C22×D5 | #99 | 1 | 1 | yes | characteristic |
| 89 | C2×AGL(1,𝔽5) | — | 8 | 2 | no | — |
| 90 | C2×AGL(1,𝔽5) | — | 8 | 1 | no | — |
| 91 | C5×Q8 | — | 4 | 1 | no | — |
| 92 | C5×Q8 | — | 4 | 1 | no | — |
| 93 | C2×C4×C5 | — | 2 | 1 | no | — |
| 94 | C2×C4×C5 | — | 2 | 1 | no | — |
| 95 | C2×C4×C5 | — | 2 | 1 | no | — |
| 96 | C2×C4×C5 | #95 | 1 | 1 | yes | characteristic |
| 97 | C2×Dic5 | — | 2 | 1 | no | — |
| 98 | C2×Dic5 | — | 2 | 1 | no | — |
| 99 | C2×Dic5 | — | 2 | 1 | no | — |
| 100 | C2×Dic5 | #99 | 1 | 1 | yes | characteristic |
| 101 | C4×D5 | — | 2 | 1 | no | — |
| 102 | C4×D5 | — | 2 | 1 | no | — |
| 103 | C4×D5 | — | 4 | 2 | no | — |
| 104 | C4×D5 | — | 4 | 1 | no | — |
| 105 | C4×D5 | — | 4 | 1 | no | — |
| 106 | C4×D5 | — | 4 | 1 | no | — |
| 107 | C4×D5 | — | 4 | 1 | no | — |
| 108 | Dic10 | — | 4 | 1 | no | — |
| 109 | Dic10 | — | 4 | 1 | no | — |
| 110 | Dic10 | — | 8 | 1 | no | — |
| 111 | Dic10 | — | 8 | 1 | no | — |
| 112 | #170 | — | 4 | 2 | no | — |
| 113 | #170 | — | 8 | 2 | no | — |
| 114 | C5×C8 | — | 8 | 1 | no | — |
| 115 | #168 | — | 8 | 1 | no | — |
| 116 | C22×Q8 | — | 5 | 1 | no | — |
| 117 | C22×Q8 | — | 5 | 1 | no | — |
| 118 | C2×M16 | — | 20 | 1 | no | — |
| 119 | C2×M16 | — | 20 | 1 | no | — |
| 120 | C22×C8 | — | 10 | 1 | no | — |
| 121 | C4⋊Q8 | — | 5 | 1 | no | — |
| 122 | C4⋊Q8 | — | 5 | 1 | no | — |
| 123 | C4⋊Q8 | — | 5 | 1 | no | — |
| 124 | C4⋊Q8 | — | 20 | 1 | no | — |
| 125 | C4⋊Q8 | C2×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 126 | C2×C4⋊C4 | — | 10 | 1 | no | — |
| 127 | C2×C4⋊C4 | — | 20 | 1 | no | — |
| 128 | C2×C42 | — | 5 | 1 | no | — |
| 129 | #103 | — | 20 | 2 | no | — |
| 130 | #103 | — | 20 | 2 | no | — |
| 131 | #101 | — | 2 | 1 | no | — |
| 132 | #101 | — | 10 | 2 | no | — |
| 133 | #101 | — | 10 | 1 | no | — |
| 134 | #101 | — | 20 | 1 | no | — |
| 135 | #101 | — | 20 | 1 | no | — |
| 136 | #97 | — | 20 | 1 | no | — |
| 137 | #95 | — | 20 | 1 | no | — |
| 138 | C22×C5 | #342 | 1 | 1 | yes | characteristic |
| 139 | C2×D5 | — | 2 | 2 | no | — |
| 140 | C2×D5 | #353 | 1 | 1 | yes | characteristic |
| 141 | C2×D5 | #354 | 1 | 1 | yes | characteristic |
| 142 | AGL(1,𝔽5) | — | 16 | 1 | no | — |
| 143 | C4×C5 | — | 2 | 1 | no | — |
| 144 | C4×C5 | — | 4 | 1 | no | — |
| 145 | C4×C5 | — | 4 | 1 | no | — |
| 146 | C4×C5 | — | 4 | 1 | no | — |
| 147 | Dic5 | — | 2 | 1 | no | — |
| 148 | Dic5 | — | 4 | 1 | no | — |
| 149 | Dic5 | — | 4 | 1 | no | — |
| 150 | Dic5 | — | 4 | 1 | no | — |
| 151 | C2×Q8 | — | 5 | 1 | no | — |
| 152 | C2×Q8 | — | 5 | 1 | no | — |
| 153 | C2×Q8 | — | 10 | 1 | no | — |
| 154 | C2×Q8 | — | 10 | 1 | no | — |
| 155 | C2×Q8 | — | 20 | 1 | no | — |
| 156 | C2×Q8 | — | 20 | 1 | no | — |
| 157 | C2×Q8 | — | 20 | 1 | no | — |
| 158 | C2×Q8 | — | 20 | 1 | no | — |
| 159 | C2×Q8 | C4⋊AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 160 | C2×Q8 | C22⋊AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 161 | C22×C4 | — | 5 | 1 | no | — |
| 162 | C22×C4 | — | 10 | 1 | no | — |
| 163 | C22×C4 | — | 10 | 1 | no | — |
| 164 | C22×C4 | — | 10 | 1 | no | — |
| 165 | C22×C4 | — | 20 | 1 | no | — |
| 166 | M16 | — | 4 | 1 | no | — |
| 167 | M16 | — | 20 | 1 | no | — |
| 168 | M16 | — | 20 | 2 | no | — |
| 169 | M16 | — | 40 | 1 | no | — |
| 170 | M16 | — | 40 | 1 | no | — |
| 171 | C2×C8 | — | 10 | 2 | no | — |
| 172 | C2×C8 | — | 20 | 1 | no | — |
| 173 | C2×C8 | — | 20 | 1 | no | — |
| 174 | C2×C8 | — | 40 | 1 | no | — |
| 175 | C2×C8 | — | 40 | 1 | no | — |
| 176 | C4⋊C4 | — | 4 | 1 | no | — |
| 177 | C4⋊C4 | — | 20 | 1 | no | — |
| 178 | C4⋊C4 | — | 20 | 1 | no | — |
| 179 | C4⋊C4 | — | 20 | 1 | no | — |
| 180 | C4⋊C4 | — | 20 | 1 | no | — |
| 181 | C4⋊C4 | — | 20 | 1 | no | — |
| 182 | C42 | — | 5 | 1 | no | — |
| 183 | C42 | — | 10 | 1 | no | — |
| 184 | C42 | C4×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 185 | C2×C5 | — | 2 | 1 | no | — |
| 186 | C2×C5 | #1393 | 1 | 1 | yes | characteristic |
| 187 | D5 | — | 2 | 1 | no | — |
| 188 | D5 | #1413 | 1 | 2 | yes | — |
| 189 | C23 | — | 5 | 1 | no | — |
| 190 | Q8 | — | 4 | 1 | no | — |
| 191 | Q8 | — | 4 | 1 | no | — |
| 192 | Q8 | — | 20 | 1 | no | — |
| 193 | Q8 | — | 20 | 1 | no | — |
| 194 | Q8 | — | 40 | 1 | no | — |
| 195 | Q8 | — | 40 | 1 | no | — |
| 196 | C2×C4 | — | 2 | 1 | no | — |
| 197 | C2×C4 | — | 2 | 1 | no | — |
| 198 | C2×C4 | — | 2 | 1 | no | — |
| 199 | C2×C4 | — | 5 | 1 | no | — |
| 200 | C2×C4 | — | 10 | 1 | no | — |
| 201 | C2×C4 | — | 10 | 1 | no | — |
| 202 | C2×C4 | — | 10 | 1 | no | — |
| 203 | C2×C4 | — | 10 | 1 | no | — |
| 204 | C2×C4 | — | 10 | 1 | no | — |
| 205 | C2×C4 | — | 20 | 1 | no | — |
| 206 | C2×C4 | — | 20 | 1 | no | — |
| 207 | C2×C4 | — | 20 | 1 | no | — |
| 208 | C2×C4 | — | 20 | 1 | no | — |
| 209 | C2×C4 | — | 20 | 2 | no | — |
| 210 | C2×C4 | — | 40 | 1 | no | — |
| 211 | C2×C4 | — | 40 | 2 | no | — |
| 212 | C2×C4 | (C2×AGL(1,𝔽5))⋊C4 | 1 | 1 | yes | characteristic |
| 213 | C8 | — | 8 | 1 | no | — |
| 214 | C8 | — | 20 | 2 | no | — |
| 215 | C8 | — | 40 | 1 | no | — |
| 216 | C8 | — | 40 | 2 | no | — |
| 217 | C5 | #7505 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 218 | C22 | — | 5 | 1 | no | — |
| 219 | C22 | — | 5 | 1 | no | — |
| 220 | C22 | — | 10 | 2 | no | — |
| 221 | C22 | #64866 | 1 | 1 | yes | characteristic |
| 222 | C4 | — | 2 | 1 | no | — |
| 223 | C4 | — | 4 | 1 | no | — |
| 224 | C4 | — | 4 | 1 | no | — |
| 225 | C4 | — | 4 | 1 | no | — |
| 226 | C4 | — | 10 | 1 | no | — |
| 227 | C4 | — | 20 | 1 | no | — |
| 228 | C4 | — | 20 | 1 | no | — |
| 229 | C4 | — | 20 | 1 | no | — |
| 230 | C4 | — | 80 | 1 | no | — |
| 231 | C2 | — | 2 | 1 | no | — |
| 232 | C2 | — | 5 | 2 | no | — |
| 233 | C2 | — | 10 | 1 | no | — |
| 234 | C2 | #10598568 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 12. #64855 | C4 | semidirect |
| 13. C5×#356 | C4 | semidirect |
| 14. #64768 | C4 | semidirect |
| 87. #356 | AGL(1,𝔽5) | semidirect |
| 217. C5 | #7505 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #10604673 | 2 |
| D5⋊#356 | 2 |
| D5×#356 | 2 |
| #7505 | 5 |