Finite Group DatabasebetaFinite Group DatabasebetaOrder 960.
150 conjugacy classes of subgroups; only the 11 normal subgroups are shown.
This group has 150 conjugacy classes of subgroups — too many to draw. Only its 11 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 | D4×A5 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | C4⋊S5 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | A5⋊C8 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | C22×A5 | — | 2 | 1 | no | — |
| 5 | C2×S5 | — | 2 | 1 | no | — |
| 6 | C4×A5 | C22 | 1 | 1 | yes | characteristic |
| 7 | #5415 | — | 5 | 1 | no | maximal, Hall |
| 8 | #4007 | — | 6 | 1 | no | maximal |
| 9 | C2×A5 | — | 4 | 1 | no | — |
| 10 | C2×A5 | D4 | 1 | 1 | yes | characteristic |
| 11 | S5 | — | 4 | 1 | no | — |
| 12 | D4×A4 | — | 5 | 1 | no | — |
| 13 | C4⋊S4 | — | 5 | 1 | no | — |
| 14 | D3×D8 | — | 10 | 1 | no | maximal |
| 15 | A4⋊C8 | — | 5 | 1 | no | — |
| 16 | D4×D5 | — | 6 | 1 | no | — |
| 17 | C4⋊AGL(1,𝔽5) | — | 6 | 1 | no | — |
| 18 | D5⋊C8 | — | 6 | 1 | no | — |
| 19 | #447 | — | 15 | 1 | no | Sylow |
| 20 | A5 | D8 | 1 | 1 | yes | characteristic, complemented |
| 21 | C22×A4 | — | 10 | 1 | no | — |
| 22 | C2×S4 | — | 10 | 1 | no | — |
| 23 | D3×D4 | — | 10 | 1 | no | — |
| 24 | D3×D4 | — | 10 | 1 | no | — |
| 25 | C4×A4 | — | 5 | 1 | no | — |
| 26 | C3×D8 | — | 10 | 1 | no | — |
| 27 | #213 | — | 10 | 1 | no | — |
| 28 | #213 | — | 10 | 1 | no | — |
| 29 | D24 | — | 10 | 1 | no | — |
| 30 | D3×C8 | — | 10 | 1 | no | — |
| 31 | C22×D5 | — | 12 | 1 | no | — |
| 32 | C2×AGL(1,𝔽5) | — | 12 | 1 | no | — |
| 33 | C5×D4 | — | 6 | 1 | no | — |
| 34 | C22⋊D5 | — | 12 | 1 | no | — |
| 35 | D20 | — | 6 | 1 | no | — |
| 36 | C4×D5 | — | 6 | 1 | no | — |
| 37 | #170 | — | 12 | 1 | no | — |
| 38 | C22×D4 | — | 5 | 1 | no | — |
| 39 | C2×D8 | — | 15 | 1 | no | — |
| 40 | C2×D8 | — | 15 | 1 | no | — |
| 41 | #121 | — | 15 | 1 | no | — |
| 42 | #102 | — | 15 | 1 | no | — |
| 43 | #102 | — | 15 | 1 | no | — |
| 44 | C22⋊C8 | — | 15 | 1 | no | — |
| 45 | C22×D3 | — | 20 | 1 | no | — |
| 46 | C22×D3 | — | 20 | 1 | no | — |
| 47 | C2×A4 | — | 5 | 1 | no | — |
| 48 | C2×A4 | — | 20 | 1 | no | — |
| 49 | S4 | — | 20 | 1 | no | — |
| 50 | C3×D4 | — | 10 | 1 | no | — |
| 51 | C3×D4 | — | 10 | 1 | no | — |
| 52 | #67 | — | 20 | 1 | no | — |
| 53 | #67 | — | 20 | 1 | no | — |
| 54 | D12 | — | 10 | 1 | no | — |
| 55 | D12 | — | 10 | 1 | no | — |
| 56 | C4×D3 | — | 10 | 1 | no | — |
| 57 | C3×C8 | — | 10 | 1 | no | — |
| 58 | C3⋊C8 | — | 10 | 1 | no | — |
| 59 | C22×C5 | — | 12 | 1 | no | — |
| 60 | C2×D5 | — | 6 | 1 | no | — |
| 61 | C2×D5 | — | 12 | 1 | no | — |
| 62 | C2×D5 | — | 24 | 1 | no | — |
| 63 | C2×D5 | — | 24 | 1 | no | — |
| 64 | AGL(1,𝔽5) | — | 24 | 1 | no | — |
| 65 | C4×C5 | — | 6 | 1 | no | — |
| 66 | Dic5 | — | 6 | 1 | no | — |
| 67 | C24 | — | 10 | 1 | no | — |
| 68 | C2×D4 | — | 15 | 1 | no | — |
| 69 | C2×D4 | — | 15 | 1 | no | — |
| 70 | C2×D4 | — | 15 | 1 | no | — |
| 71 | C2×D4 | — | 30 | 1 | no | — |
| 72 | C2×D4 | — | 30 | 1 | no | — |
| 73 | C22×C4 | — | 5 | 1 | no | — |
| 74 | D8 | — | 10 | 1 | no | — |
| 75 | D8 | — | 30 | 1 | no | — |
| 76 | D8 | — | 30 | 2 | no | — |
| 77 | C2×C8 | — | 15 | 1 | no | — |
| 78 | C2×C8 | — | 15 | 1 | no | — |
| 79 | C4⋊C4 | — | 15 | 1 | no | — |
| 80 | C22⋊C4 | — | 30 | 1 | no | — |
| 81 | C22×C3 | — | 20 | 1 | no | — |
| 82 | C22×C3 | — | 20 | 1 | no | — |
| 83 | C2×D3 | — | 10 | 1 | no | — |
| 84 | C2×D3 | — | 20 | 1 | no | — |
| 85 | C2×D3 | — | 20 | 1 | no | — |
| 86 | C2×D3 | — | 40 | 1 | no | — |
| 87 | C2×D3 | — | 40 | 1 | no | — |
| 88 | C2×D3 | — | 40 | 1 | no | — |
| 89 | C2×D3 | — | 40 | 1 | no | — |
| 90 | A4 | — | 5 | 1 | no | — |
| 91 | C3×C4 | — | 10 | 1 | no | — |
| 92 | Dic3 | — | 10 | 1 | no | — |
| 93 | C2×C5 | — | 6 | 1 | no | — |
| 94 | C2×C5 | — | 24 | 1 | no | — |
| 95 | D5 | — | 6 | 1 | no | — |
| 96 | D5 | — | 6 | 1 | no | — |
| 97 | D5 | — | 24 | 1 | no | — |
| 98 | C23 | — | 5 | 1 | no | — |
| 99 | C23 | — | 20 | 1 | no | — |
| 100 | C23 | — | 30 | 1 | no | — |
| 101 | C23 | — | 30 | 1 | no | — |
| 102 | C23 | — | 30 | 1 | no | — |
| 103 | C23 | — | 60 | 1 | no | — |
| 104 | D4 | — | 10 | 1 | no | — |
| 105 | D4 | — | 15 | 2 | no | — |
| 106 | D4 | — | 15 | 1 | no | — |
| 107 | D4 | — | 30 | 1 | no | — |
| 108 | D4 | — | 30 | 1 | no | — |
| 109 | D4 | — | 60 | 1 | no | — |
| 110 | D4 | — | 60 | 1 | no | — |
| 111 | D4 | S5 | 1 | 1 | yes | characteristic, complemented |
| 112 | C2×C4 | — | 15 | 1 | no | — |
| 113 | C2×C4 | — | 15 | 1 | no | — |
| 114 | C2×C4 | — | 30 | 1 | no | — |
| 115 | C8 | — | 10 | 1 | no | — |
| 116 | C8 | — | 30 | 1 | no | — |
| 117 | C2×C3 | — | 10 | 1 | no | — |
| 118 | C2×C3 | — | 40 | 1 | no | — |
| 119 | C2×C3 | — | 40 | 1 | no | — |
| 120 | D3 | — | 10 | 1 | no | — |
| 121 | D3 | — | 10 | 1 | no | — |
| 122 | D3 | — | 40 | 1 | no | — |
| 123 | D3 | — | 40 | 1 | no | — |
| 124 | C5 | — | 6 | 1 | no | Sylow |
| 125 | C22 | — | 2 | 1 | no | — |
| 126 | C22 | — | 5 | 1 | no | — |
| 127 | C22 | — | 15 | 1 | no | — |
| 128 | C22 | — | 15 | 1 | no | — |
| 129 | C22 | — | 20 | 1 | no | — |
| 130 | C22 | — | 30 | 1 | no | — |
| 131 | C22 | — | 60 | 1 | no | — |
| 132 | C22 | — | 60 | 1 | no | — |
| 133 | C22 | — | 60 | 1 | no | — |
| 134 | C22 | — | 60 | 1 | no | — |
| 135 | C22 | — | 60 | 1 | no | — |
| 136 | C22 | — | 60 | 1 | no | — |
| 137 | C4 | — | 15 | 1 | no | — |
| 138 | C4 | — | 60 | 1 | no | — |
| 139 | C4 | C2×S5 | 1 | 1 | yes | characteristic |
| 140 | C3 | — | 10 | 1 | no | Sylow |
| 141 | C2 | — | 4 | 1 | no | — |
| 142 | C2 | — | 15 | 1 | no | — |
| 143 | C2 | — | 15 | 1 | no | — |
| 144 | C2 | — | 40 | 1 | no | — |
| 145 | C2 | — | 60 | 1 | no | — |
| 146 | C2 | C22⋊S5 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C22⋊S5 |
| commutator, cosocle | C4×A5 | C22 |
| Frattini | C4 | C2×S5 |
| Fitting, hypercenter | D4 | S5 |
| socle | C2×A5 | D4 |
| hypocenter | A5 | D8 |
| LCS | G ▷ C4×A5 ▷ C2×A5 ▷ A5 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ D4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. D4×A5 | C2 | semidirect |
| 2. C4⋊S5 | C2 | semidirect |
| 3. A5⋊C8 | C2 | semidirect |
| 20. A5 | D8 | semidirect |
| 111. D4 | S5 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| D4×A5 | 2 |
| C4⋊S5 | 2 |
| A5⋊C8 | 2 |
| #5415 | 5 |
| #4007 | 6 |
| D3×D8 | 10 |