Finite Group DatabasebetaFinite Group DatabasebetaOrder 640.
504 conjugacy classes of subgroups; only the 81 normal subgroups are shown.
This group has 504 conjugacy classes of subgroups — too many to draw. Only its 81 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×D5×C22⋊C4 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #65244 | C2 | 1 | 2 | yes | maximal |
| 3 | C5×#342 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #64747 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | #64746 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | C24×D5 | C22 | 1 | 1 | yes | characteristic |
| 7 | C22×C4×D5 | C4 | 1 | 2 | yes | complemented |
| 8 | C2×C5×C22⋊C4 | C22 | 1 | 2 | yes | — |
| 9 | C2×C5×C22⋊C4 | C22 | 1 | 1 | yes | characteristic |
| 10 | C2×C22⋊Dic5 | C22 | 1 | 2 | yes | — |
| 11 | C2×C22⋊Dic5 | C22 | 1 | 1 | yes | characteristic |
| 12 | C2×#685 | — | 2 | 4 | no | — |
| 13 | C2×#685 | C4 | 1 | 2 | yes | complemented |
| 14 | D5×C22⋊C4 | — | 2 | 4 | no | — |
| 15 | #1819 | — | 5 | 1 | no | maximal, Sylow |
| 16 | C24×C5 | C23 | 1 | 1 | yes | characteristic |
| 17 | C23×D5 | — | 2 | 2 | no | — |
| 18 | C23×D5 | — | 2 | 1 | no | — |
| 19 | C23×D5 | — | 2 | 1 | no | — |
| 20 | C23×D5 | — | 4 | 4 | no | — |
| 21 | C23×D5 | C2×C4 | 1 | 1 | yes | characteristic |
| 22 | C23×D5 | C2×C4 | 1 | 1 | yes | characteristic |
| 23 | C23×D5 | D4 | 1 | 4 | yes | — |
| 24 | C22×C4×C5 | — | 2 | 2 | no | — |
| 25 | C22×C4×C5 | C2×C4 | 1 | 2 | yes | — |
| 26 | C22×Dic5 | — | 2 | 2 | no | — |
| 27 | C22×Dic5 | C2×C4 | 1 | 2 | yes | — |
| 28 | C2×C4×D5 | — | 2 | 2 | no | — |
| 29 | C2×C4×D5 | — | 2 | 2 | no | — |
| 30 | C2×C4×D5 | — | 4 | 4 | no | — |
| 31 | C5×C22⋊C4 | — | 2 | 2 | no | — |
| 32 | C5×C22⋊C4 | — | 2 | 2 | no | — |
| 33 | C5×C22⋊C4 | — | 2 | 2 | no | — |
| 34 | C22⋊Dic5 | — | 2 | 2 | no | — |
| 35 | C22⋊Dic5 | — | 2 | 2 | no | — |
| 36 | C22⋊Dic5 | — | 2 | 2 | no | — |
| 37 | #685 | — | 2 | 4 | no | — |
| 38 | #685 | — | 4 | 8 | no | — |
| 39 | C22×C22⋊C4 | — | 5 | 1 | no | — |
| 40 | #379 | — | 5 | 2 | no | — |
| 41 | #342 | — | 5 | 1 | no | — |
| 42 | #342 | — | 5 | 2 | no | — |
| 43 | #342 | D5 | 1 | 1 | yes | characteristic, complemented |
| 44 | C23×C5 | — | 4 | 2 | no | — |
| 45 | C23×C5 | C22×C4 | 1 | 1 | yes | characteristic |
| 46 | C23×C5 | C2×D4 | 1 | 2 | yes | — |
| 47 | C23×C5 | #41 | 1 | 2 | yes | — |
| 48 | C23×C5 | #41 | 1 | 1 | yes | characteristic |
| 49 | C23×C5 | #41 | 1 | 1 | yes | characteristic |
| 50 | C22×D5 | — | 2 | 1 | no | — |
| 51 | C22×D5 | — | 2 | 1 | no | — |
| 52 | C22×D5 | — | 2 | 4 | no | — |
| 53 | C22×D5 | — | 2 | 1 | no | — |
| 54 | C22×D5 | — | 2 | 1 | no | — |
| 55 | C22×D5 | — | 2 | 4 | no | — |
| 56 | C22×D5 | — | 2 | 4 | no | — |
| 57 | C22×D5 | — | 2 | 8 | no | — |
| 58 | C22×D5 | — | 4 | 2 | no | — |
| 59 | C22×D5 | — | 4 | 4 | no | — |
| 60 | C22×D5 | — | 4 | 8 | no | — |
| 61 | C22×D5 | — | 4 | 4 | no | — |
| 62 | C22×D5 | — | 4 | 4 | no | — |
| 63 | C22×D5 | C22⋊C4 | 1 | 2 | yes | — |
| 64 | C22×D5 | C22⋊C4 | 1 | 2 | yes | — |
| 65 | C2×C4×C5 | — | 2 | 2 | no | — |
| 66 | C2×C4×C5 | — | 4 | 2 | no | — |
| 67 | C2×C4×C5 | — | 4 | 4 | no | — |
| 68 | C2×C4×C5 | — | 4 | 2 | no | — |
| 69 | C2×Dic5 | — | 2 | 2 | no | — |
| 70 | C2×Dic5 | — | 4 | 2 | no | — |
| 71 | C2×Dic5 | — | 4 | 4 | no | — |
| 72 | C2×Dic5 | — | 4 | 2 | no | — |
| 73 | C4×D5 | — | 4 | 4 | no | — |
| 74 | C4×D5 | — | 4 | 4 | no | — |
| 75 | C25 | — | 5 | 1 | no | — |
| 76 | C23×C4 | — | 5 | 2 | no | — |
| 77 | C2×C22⋊C4 | — | 5 | 1 | no | — |
| 78 | C2×C22⋊C4 | — | 5 | 2 | no | — |
| 79 | C2×C22⋊C4 | — | 5 | 2 | no | — |
| 80 | C2×C22⋊C4 | — | 10 | 4 | no | — |
| 81 | C2×C22⋊C4 | — | 10 | 4 | no | — |
| 82 | C2×C22⋊C4 | C2×D5 | 1 | 2 | yes | — |
| 83 | C2×C22⋊C4 | C2×D5 | 1 | 1 | yes | characteristic |
| 84 | C22×C5 | — | 2 | 2 | no | — |
| 85 | C22×C5 | — | 2 | 4 | no | — |
| 86 | C22×C5 | — | 4 | 2 | no | — |
| 87 | C22×C5 | — | 4 | 1 | no | — |
| 88 | C22×C5 | — | 4 | 1 | no | — |
| 89 | C22×C5 | C2×C22⋊C4 | 1 | 1 | yes | characteristic |
| 90 | C22×C5 | #117 | 1 | 1 | yes | characteristic |
| 91 | C22×C5 | #117 | 1 | 1 | yes | characteristic |
| 92 | C22×C5 | #123 | 1 | 2 | yes | — |
| 93 | C22×C5 | #123 | 1 | 2 | yes | — |
| 94 | C2×D5 | — | 2 | 4 | no | — |
| 95 | C2×D5 | — | 2 | 4 | no | — |
| 96 | C2×D5 | — | 2 | 2 | no | — |
| 97 | C2×D5 | — | 2 | 2 | no | — |
| 98 | C2×D5 | — | 2 | 2 | no | — |
| 99 | C2×D5 | — | 2 | 4 | no | — |
| 100 | C2×D5 | — | 4 | 8 | no | — |
| 101 | C2×D5 | — | 4 | 8 | no | — |
| 102 | C2×D5 | — | 4 | 2 | no | — |
| 103 | C2×D5 | — | 4 | 2 | no | — |
| 104 | C2×D5 | #99 | 1 | 4 | yes | — |
| 105 | C4×C5 | — | 4 | 2 | no | — |
| 106 | C4×C5 | — | 8 | 2 | no | — |
| 107 | Dic5 | — | 4 | 2 | no | — |
| 108 | Dic5 | — | 8 | 2 | no | — |
| 109 | C24 | — | 5 | 1 | no | — |
| 110 | C24 | — | 5 | 4 | no | — |
| 111 | C24 | — | 5 | 1 | no | — |
| 112 | C24 | — | 10 | 2 | no | — |
| 113 | C24 | — | 10 | 1 | no | — |
| 114 | C24 | — | 10 | 1 | no | — |
| 115 | C24 | — | 20 | 4 | no | — |
| 116 | C24 | C22×D5 | 1 | 1 | yes | characteristic |
| 117 | C22×C4 | — | 2 | 2 | no | — |
| 118 | C22×C4 | — | 5 | 2 | no | — |
| 119 | C22×C4 | — | 10 | 2 | no | — |
| 120 | C22×C4 | — | 10 | 2 | no | — |
| 121 | C22×C4 | — | 10 | 2 | no | — |
| 122 | C22×C4 | — | 20 | 4 | no | — |
| 123 | C22×C4 | C4×D5 | 1 | 2 | yes | — |
| 124 | C22⋊C4 | — | 2 | 2 | no | — |
| 125 | C22⋊C4 | — | 2 | 2 | no | — |
| 126 | C22⋊C4 | — | 2 | 2 | no | — |
| 127 | C22⋊C4 | — | 10 | 4 | no | — |
| 128 | C22⋊C4 | — | 10 | 2 | no | — |
| 129 | C22⋊C4 | — | 10 | 2 | no | — |
| 130 | C22⋊C4 | — | 10 | 2 | no | — |
| 131 | C22⋊C4 | — | 20 | 8 | no | — |
| 132 | C2×C5 | — | 2 | 2 | no | — |
| 133 | C2×C5 | — | 2 | 2 | no | — |
| 134 | C2×C5 | — | 2 | 1 | no | — |
| 135 | C2×C5 | — | 2 | 1 | no | — |
| 136 | C2×C5 | #381 | 1 | 1 | yes | characteristic |
| 137 | C2×C5 | C2×#99 | 1 | 2 | yes | — |
| 138 | D5 | — | 2 | 4 | no | — |
| 139 | D5 | — | 4 | 2 | no | — |
| 140 | C23 | — | 4 | 2 | no | — |
| 141 | C23 | — | 5 | 2 | no | — |
| 142 | C23 | — | 5 | 2 | no | — |
| 143 | C23 | — | 10 | 4 | no | — |
| 144 | C23 | — | 10 | 8 | no | — |
| 145 | C23 | — | 10 | 1 | no | — |
| 146 | C23 | — | 10 | 1 | no | — |
| 147 | C23 | — | 10 | 1 | no | — |
| 148 | C23 | — | 10 | 4 | no | — |
| 149 | C23 | — | 10 | 1 | no | — |
| 150 | C23 | — | 10 | 4 | no | — |
| 151 | C23 | — | 20 | 2 | no | — |
| 152 | C23 | — | 20 | 4 | no | — |
| 153 | C23 | — | 20 | 8 | no | — |
| 154 | C23 | — | 20 | 4 | no | — |
| 155 | C23 | — | 20 | 4 | no | — |
| 156 | C23 | C2×C4×D5 | 1 | 1 | yes | characteristic |
| 157 | C23 | #709 | 1 | 2 | yes | — |
| 158 | C23 | D4×D5 | 1 | 2 | yes | — |
| 159 | C23 | #711 | 1 | 1 | yes | characteristic |
| 160 | C23 | #713 | 1 | 1 | yes | characteristic |
| 161 | C2×C4 | — | 2 | 2 | no | — |
| 162 | C2×C4 | — | 4 | 2 | no | — |
| 163 | C2×C4 | — | 4 | 4 | no | — |
| 164 | C2×C4 | — | 4 | 2 | no | — |
| 165 | C2×C4 | — | 10 | 2 | no | — |
| 166 | C2×C4 | — | 20 | 4 | no | — |
| 167 | C2×C4 | — | 20 | 4 | no | — |
| 168 | C2×C4 | — | 20 | 2 | no | — |
| 169 | C2×C4 | — | 20 | 4 | no | — |
| 170 | C2×C4 | — | 20 | 2 | no | — |
| 171 | C5 | #1819 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 172 | C22 | — | 2 | 2 | no | — |
| 173 | C22 | — | 2 | 4 | no | — |
| 174 | C22 | — | 4 | 2 | no | — |
| 175 | C22 | — | 4 | 1 | no | — |
| 176 | C22 | — | 4 | 1 | no | — |
| 177 | C22 | — | 5 | 4 | no | — |
| 178 | C22 | — | 10 | 4 | no | — |
| 179 | C22 | — | 10 | 4 | no | — |
| 180 | C22 | — | 10 | 2 | no | — |
| 181 | C22 | — | 10 | 2 | no | — |
| 182 | C22 | — | 10 | 4 | no | — |
| 183 | C22 | — | 10 | 2 | no | — |
| 184 | C22 | — | 20 | 8 | no | — |
| 185 | C22 | — | 20 | 8 | no | — |
| 186 | C22 | — | 20 | 2 | no | — |
| 187 | C22 | — | 20 | 2 | no | — |
| 188 | C22 | #4018 | 1 | 1 | yes | characteristic |
| 189 | C22 | D5×C22⋊C4 | 1 | 1 | yes | characteristic |
| 190 | C22 | #4029 | 1 | 2 | yes | — |
| 191 | C22 | #4038 | 1 | 1 | yes | characteristic |
| 192 | C22 | #4040 | 1 | 2 | yes | — |
| 193 | C4 | — | 4 | 2 | no | — |
| 194 | C4 | — | 8 | 2 | no | — |
| 195 | C4 | — | 20 | 2 | no | — |
| 196 | C4 | — | 40 | 2 | no | — |
| 197 | C2 | — | 2 | 2 | no | — |
| 198 | C2 | — | 2 | 2 | no | — |
| 199 | C2 | — | 2 | 1 | no | — |
| 200 | C2 | — | 2 | 1 | no | — |
| 201 | C2 | — | 10 | 4 | no | — |
| 202 | C2 | — | 20 | 2 | no | — |
| 203 | C2 | #64953 | 1 | 1 | yes | characteristic |
| 204 | C2 | D5×#99 | 1 | 2 | yes | — |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. C5×#342 | C2 | semidirect |
| 4. #64747 | C2 | semidirect |
| 5. #64746 | C2 | semidirect |
| 7. C22×C4×D5 | C4 | semidirect |
| 13. C2×#685 | C4 | semidirect |
| 43. #342 | D5 | semidirect |
| 171. C5 | #1819 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×D5×C22⋊C4 | 2 |
| #65244 | 2 |
| C5×#342 | 2 |
| #64747 | 2 |
| #64746 | 2 |
| #1819 | 5 |