Finite Group DatabasebetaFinite Group DatabasebetaOrder 640.
932 conjugacy classes of subgroups; only the 153 normal subgroups are shown.
This group has 932 conjugacy classes of subgroups — too many to draw. Only its 153 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 | C22×#713 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | C2×#4030 | C2 | 1 | 3 | yes | maximal, complemented |
| 3 | C5×#393 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #65520 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | #65249 | C2 | 1 | 3 | yes | maximal, complemented |
| 6 | #64960 | C2 | 1 | 3 | yes | maximal, complemented |
| 7 | #64956 | C2 | 1 | 3 | yes | maximal, complemented |
| 8 | C22×C5×Q8 | C22 | 1 | 1 | yes | characteristic |
| 9 | C22×C22⋊D5 | C22 | 1 | 1 | yes | characteristic |
| 10 | C22×C22⋊D5 | C22 | 1 | 3 | yes | — |
| 11 | C2×#713 | — | 2 | 12 | no | — |
| 12 | C22×D20 | C22 | 1 | 3 | yes | — |
| 13 | C22×D20 | C22 | 1 | 3 | yes | — |
| 14 | C22×C4×D5 | C22 | 1 | 3 | yes | — |
| 15 | C2×C5×C22⋊C4 | C22 | 1 | 3 | yes | — |
| 16 | C2×#685 | C22 | 1 | 3 | yes | — |
| 17 | C2×#685 | C22 | 1 | 6 | yes | — |
| 18 | C2×#683 | C22 | 1 | 3 | yes | — |
| 19 | #4030 | — | 2 | 12 | no | — |
| 20 | C5×#95 | C22 | 1 | 3 | yes | — |
| 21 | #3963 | C22 | 1 | 3 | yes | — |
| 22 | #2615 | — | 5 | 1 | no | maximal, Sylow |
| 23 | C24×C5 | C23 | 1 | 1 | yes | characteristic |
| 24 | C23×D5 | C23 | 1 | 1 | yes | characteristic |
| 25 | C23×D5 | C23 | 1 | 3 | yes | — |
| 26 | C2×C5×Q8 | — | 2 | 6 | no | — |
| 27 | C22×C4×C5 | C23 | 1 | 3 | yes | — |
| 28 | C22×C4×C5 | C23 | 1 | 3 | yes | — |
| 29 | C2×C22⋊D5 | — | 2 | 6 | no | — |
| 30 | C2×C22⋊D5 | — | 2 | 6 | no | — |
| 31 | C2×C22⋊D5 | — | 4 | 6 | no | — |
| 32 | C22×Dic5 | C23 | 1 | 3 | yes | — |
| 33 | C22×Dic5 | C23 | 1 | 1 | yes | characteristic |
| 34 | #713 | — | 2 | 32 | no | — |
| 35 | C2×D20 | — | 2 | 12 | no | — |
| 36 | C2×D20 | — | 2 | 6 | no | — |
| 37 | C2×D20 | — | 2 | 6 | no | — |
| 38 | C2×D20 | — | 4 | 6 | no | — |
| 39 | C2×C4×D5 | — | 2 | 12 | no | — |
| 40 | C2×C4×D5 | D4 | 1 | 12 | yes | complemented |
| 41 | C5×C22⋊C4 | — | 2 | 6 | no | — |
| 42 | #685 | — | 2 | 6 | no | — |
| 43 | #685 | — | 4 | 6 | no | — |
| 44 | #683 | — | 2 | 6 | no | — |
| 45 | C22×#41 | — | 5 | 1 | no | — |
| 46 | C2×#121 | — | 5 | 3 | no | — |
| 47 | #393 | — | 5 | 1 | no | — |
| 48 | #393 | D5 | 1 | 1 | yes | characteristic, complemented |
| 49 | #392 | — | 5 | 3 | no | — |
| 50 | #392 | — | 5 | 3 | no | — |
| 51 | #380 | — | 5 | 3 | no | — |
| 52 | C23×C5 | — | 2 | 3 | no | — |
| 53 | C23×C5 | — | 2 | 3 | no | — |
| 54 | C23×C5 | — | 2 | 1 | no | — |
| 55 | C23×C5 | C24 | 1 | 1 | yes | characteristic |
| 56 | C22×D5 | — | 2 | 6 | no | — |
| 57 | C22×D5 | — | 2 | 3 | no | — |
| 58 | C22×D5 | — | 2 | 6 | no | — |
| 59 | C22×D5 | — | 2 | 3 | no | — |
| 60 | C22×D5 | — | 2 | 3 | no | — |
| 61 | C22×D5 | — | 2 | 3 | no | — |
| 62 | C22×D5 | — | 2 | 1 | no | — |
| 63 | C22×D5 | C2×D4 | 1 | 6 | yes | — |
| 64 | C5×Q8 | — | 2 | 8 | no | — |
| 65 | C2×C4×C5 | — | 2 | 6 | no | — |
| 66 | C2×C4×C5 | — | 2 | 3 | no | — |
| 67 | C2×C4×C5 | — | 2 | 6 | no | — |
| 68 | C2×C4×C5 | C2×D4 | 1 | 6 | yes | — |
| 69 | C22⋊D5 | — | 4 | 12 | no | — |
| 70 | C22⋊D5 | — | 4 | 4 | no | — |
| 71 | C2×Dic5 | — | 2 | 3 | no | — |
| 72 | C2×Dic5 | — | 2 | 6 | no | — |
| 73 | C2×Dic5 | C2×D4 | 1 | 6 | yes | — |
| 74 | D20 | — | 2 | 24 | no | — |
| 75 | D20 | — | 4 | 12 | no | — |
| 76 | C4×D5 | — | 2 | 24 | no | — |
| 77 | C2×#41 | — | 10 | 12 | no | — |
| 78 | C22×Q8 | C2×D5 | 1 | 1 | yes | characteristic |
| 79 | C22×D4 | — | 5 | 1 | no | — |
| 80 | C22×D4 | — | 5 | 3 | no | — |
| 81 | C22×D4 | — | 5 | 3 | no | — |
| 82 | C22×D4 | — | 5 | 3 | no | — |
| 83 | C23×C4 | — | 5 | 3 | no | — |
| 84 | #121 | — | 10 | 12 | no | — |
| 85 | C2×C4⋊C4 | — | 5 | 3 | no | — |
| 86 | C2×C22⋊C4 | — | 5 | 3 | no | — |
| 87 | C2×C22⋊C4 | — | 5 | 6 | no | — |
| 88 | C2×C22⋊C4 | C2×D5 | 1 | 3 | yes | — |
| 89 | #95 | — | 5 | 3 | no | — |
| 90 | #95 | C2×D5 | 1 | 3 | yes | — |
| 91 | C22×C5 | — | 4 | 1 | no | — |
| 92 | C22×C5 | — | 4 | 3 | no | — |
| 93 | C22×C5 | — | 4 | 3 | no | — |
| 94 | C22×C5 | C22×D4 | 1 | 3 | yes | — |
| 95 | C22×C5 | H(5,𝔽2) | 1 | 3 | yes | — |
| 96 | C22×C5 | 2−1+4 | 1 | 1 | yes | characteristic |
| 97 | C2×D5 | — | 2 | 6 | no | — |
| 98 | C2×D5 | — | 2 | 6 | no | — |
| 99 | C2×D5 | — | 2 | 6 | no | — |
| 100 | C2×D5 | — | 4 | 6 | no | — |
| 101 | C2×D5 | — | 4 | 6 | no | — |
| 102 | C2×D5 | — | 4 | 1 | no | — |
| 103 | C2×D5 | — | 4 | 3 | no | — |
| 104 | C2×D5 | — | 4 | 3 | no | — |
| 105 | C4×C5 | — | 2 | 6 | no | — |
| 106 | C4×C5 | — | 4 | 3 | no | — |
| 107 | Dic5 | — | 4 | 3 | no | — |
| 108 | Dic5 | #120 | 1 | 4 | yes | — |
| 109 | C24 | — | 5 | 3 | no | — |
| 110 | C24 | — | 5 | 1 | no | — |
| 111 | C24 | C22×D5 | 1 | 1 | yes | characteristic |
| 112 | #41 | — | 10 | 32 | no | — |
| 113 | C2×Q8 | — | 2 | 6 | no | — |
| 114 | C2×D4 | — | 10 | 12 | no | — |
| 115 | C2×D4 | — | 10 | 6 | no | — |
| 116 | C2×D4 | — | 10 | 6 | no | — |
| 117 | C2×D4 | — | 10 | 6 | no | — |
| 118 | C2×D4 | — | 10 | 6 | no | — |
| 119 | C2×D4 | — | 20 | 6 | no | — |
| 120 | C2×D4 | — | 20 | 6 | no | — |
| 121 | C22×C4 | — | 5 | 1 | no | — |
| 122 | C22×C4 | — | 5 | 12 | no | — |
| 123 | C22×C4 | — | 5 | 3 | no | — |
| 124 | C22×C4 | — | 10 | 12 | no | — |
| 125 | C22×C4 | C22×D5 | 1 | 3 | yes | — |
| 126 | C22×C4 | C22×D5 | 1 | 3 | yes | — |
| 127 | C4⋊C4 | — | 10 | 6 | no | — |
| 128 | C22⋊C4 | — | 2 | 6 | no | — |
| 129 | C22⋊C4 | — | 10 | 6 | no | — |
| 130 | C22⋊C4 | — | 20 | 6 | no | — |
| 131 | C2×C5 | — | 8 | 1 | no | — |
| 132 | C2×C5 | C2×#120 | 1 | 1 | yes | characteristic |
| 133 | C2×C5 | #535 | 1 | 3 | yes | — |
| 134 | C2×C5 | #537 | 1 | 3 | yes | — |
| 135 | D5 | — | 4 | 6 | no | — |
| 136 | D5 | — | 8 | 1 | no | — |
| 137 | C23 | — | 2 | 3 | no | — |
| 138 | C23 | — | 2 | 1 | no | — |
| 139 | C23 | — | 2 | 3 | no | — |
| 140 | C23 | — | 5 | 6 | no | — |
| 141 | C23 | — | 10 | 6 | no | — |
| 142 | C23 | — | 10 | 3 | no | — |
| 143 | C23 | — | 10 | 3 | no | — |
| 144 | C23 | — | 10 | 6 | no | — |
| 145 | C23 | — | 10 | 3 | no | — |
| 146 | C23 | — | 10 | 1 | no | — |
| 147 | C23 | — | 10 | 3 | no | — |
| 148 | C23 | C23×D5 | 1 | 1 | yes | characteristic |
| 149 | Q8 | — | 2 | 8 | no | — |
| 150 | D4 | — | 10 | 24 | no | — |
| 151 | D4 | — | 20 | 12 | no | — |
| 152 | D4 | — | 20 | 4 | no | — |
| 153 | D4 | — | 20 | 12 | no | — |
| 154 | C2×C4 | — | 2 | 6 | no | — |
| 155 | C2×C4 | — | 2 | 3 | no | — |
| 156 | C2×C4 | — | 2 | 6 | no | — |
| 157 | C2×C4 | — | 5 | 6 | no | — |
| 158 | C2×C4 | — | 10 | 24 | no | — |
| 159 | C2×C4 | — | 10 | 3 | no | — |
| 160 | C2×C4 | — | 10 | 6 | no | — |
| 161 | C2×C4 | D4×D5 | 1 | 6 | yes | — |
| 162 | C5 | #2615 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 163 | C22 | — | 4 | 1 | no | — |
| 164 | C22 | — | 4 | 3 | no | — |
| 165 | C22 | — | 4 | 3 | no | — |
| 166 | C22 | — | 10 | 6 | no | — |
| 167 | C22 | — | 10 | 6 | no | — |
| 168 | C22 | — | 10 | 6 | no | — |
| 169 | C22 | — | 20 | 6 | no | — |
| 170 | C22 | — | 20 | 6 | no | — |
| 171 | C22 | — | 20 | 1 | no | — |
| 172 | C22 | — | 20 | 3 | no | — |
| 173 | C22 | — | 20 | 3 | no | — |
| 174 | C22 | C2×D4×D5 | 1 | 3 | yes | — |
| 175 | C22 | #4147 | 1 | 1 | yes | characteristic |
| 176 | C22 | #4149 | 1 | 3 | yes | — |
| 177 | C4 | — | 2 | 6 | no | — |
| 178 | C4 | — | 4 | 3 | no | — |
| 179 | C4 | — | 5 | 4 | no | — |
| 180 | C4 | — | 20 | 3 | no | — |
| 181 | C2 | — | 8 | 1 | no | — |
| 182 | C2 | — | 20 | 6 | no | — |
| 183 | C2 | — | 40 | 1 | no | — |
| 184 | C2 | D5×#120 | 1 | 1 | yes | characteristic |
| 185 | C2 | #65963 | 1 | 3 | yes | — |
| 186 | C2 | #66008 | 1 | 3 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D5 |
| commutator, socle, cosocle | C23×C5 | C24 |
| Fitting | C5×#393 | C2 |
| hypercenter | #393 | D5 |
| hypocenter | C5 | #2615 |
| LCS | G ▷ C23×C5 ▷ C5 | |
| UCS | C1 ◁ C23 ◁ #393 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×#713 | C2 | semidirect |
| 2. C2×#4030 | C2 | semidirect |
| 3. C5×#393 | C2 | semidirect |
| 4. #65520 | C2 | semidirect |
| 5. #65249 | C2 | semidirect |
| 6. #64960 | C2 | semidirect |
| 7. #64956 | C2 | semidirect |
| 40. C2×C4×D5 | D4 | semidirect |
| 48. #393 | D5 | semidirect |
| 162. C5 | #2615 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×#713 | 2 |
| C2×#4030 | 2 |
| C5×#393 | 2 |
| #65520 | 2 |
| #65249 | 2 |
| #64960 | 2 |
| #64956 | 2 |
| #2615 | 5 |