Finite Group DatabasebetaFinite Group DatabasebetaOrder 1440.
143 conjugacy classes of subgroups; only the 7 normal subgroups are shown.
This group has 143 conjugacy classes of subgroups — too many to draw. Only its 7 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 | A4×A5 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | C22⋊S5 | — | 3 | 1 | no | maximal |
| 3 | ΓL(2,𝔽4) | — | 4 | 1 | no | maximal |
| 4 | A4⋊S4 | — | 5 | 1 | no | maximal, Hall |
| 5 | A4⋊AGL(1,𝔽5) | — | 6 | 1 | no | maximal |
| 6 | C22×A5 | D3 | 1 | 1 | yes | characteristic, complemented |
| 7 | C2×S5 | — | 3 | 1 | no | — |
| 8 | A5⋊C4 | — | 3 | 1 | no | — |
| 9 | C3×A5 | — | 4 | 1 | no | — |
| 10 | A42 | — | 5 | 1 | no | — |
| 11 | D3×S4 | — | 10 | 1 | no | maximal |
| 12 | D5×A4 | — | 6 | 1 | no | — |
| 13 | C2×A5 | — | 3 | 1 | no | — |
| 14 | S5 | — | 6 | 1 | no | — |
| 15 | #1020 | — | 15 | 2 | no | — |
| 16 | GL(2,ℤ/4) | — | 15 | 1 | no | — |
| 17 | GL(2,ℤ/4) | — | 15 | 1 | no | — |
| 18 | C22⋊AGL(1,𝔽5) | — | 18 | 1 | no | — |
| 19 | D3×A4 | — | 10 | 1 | no | — |
| 20 | C3⋊S4 | — | 10 | 1 | no | — |
| 21 | C3⋊S4 | — | 20 | 1 | no | — |
| 22 | C3×S4 | — | 10 | 1 | no | — |
| 23 | C5×A4 | — | 6 | 1 | no | — |
| 24 | ΓL(1,𝔽16) | — | 24 | 1 | no | — |
| 25 | A5 | S4 | 1 | 1 | yes | characteristic, complemented |
| 26 | C22⋊A4 | — | 5 | 2 | no | — |
| 27 | C22×A4 | — | 5 | 1 | no | — |
| 28 | C22×A4 | — | 5 | 1 | no | — |
| 29 | C2×S4 | — | 15 | 1 | no | — |
| 30 | C2×S4 | — | 15 | 1 | no | — |
| 31 | D3×D4 | — | 30 | 1 | no | — |
| 32 | SL(2,ℤ/4) | — | 15 | 1 | no | — |
| 33 | SL(2,ℤ/4) | — | 15 | 1 | no | — |
| 34 | C22×D5 | — | 6 | 1 | no | — |
| 35 | C2×AGL(1,𝔽5) | — | 18 | 1 | no | — |
| 36 | C2×AGL(1,𝔽5) | — | 18 | 1 | no | — |
| 37 | C3×A4 | — | 10 | 1 | no | — |
| 38 | C3×A4 | — | 20 | 1 | no | — |
| 39 | D32 | — | 40 | 1 | no | — |
| 40 | #120 | — | 45 | 1 | no | Sylow |
| 41 | C3×D5 | — | 24 | 1 | no | — |
| 42 | C22×D3 | — | 10 | 1 | no | — |
| 43 | C22×D3 | — | 30 | 1 | no | — |
| 44 | C2×A4 | — | 15 | 1 | no | — |
| 45 | C2×A4 | — | 15 | 1 | no | — |
| 46 | S4 | — | 10 | 1 | no | — |
| 47 | S4 | — | 30 | 1 | no | — |
| 48 | S4 | — | 60 | 1 | no | — |
| 49 | S4 | — | 60 | 2 | no | — |
| 50 | S4 | — | 60 | 2 | no | — |
| 51 | C3×D4 | — | 30 | 1 | no | — |
| 52 | #67 | — | 30 | 1 | no | — |
| 53 | #67 | — | 30 | 1 | no | — |
| 54 | #67 | — | 60 | 1 | no | — |
| 55 | D12 | — | 30 | 1 | no | — |
| 56 | C4×D3 | — | 30 | 1 | no | — |
| 57 | C22×C5 | — | 6 | 1 | no | — |
| 58 | C2×D5 | — | 18 | 1 | no | — |
| 59 | C2×D5 | — | 18 | 1 | no | — |
| 60 | AGL(1,𝔽5) | — | 36 | 1 | no | — |
| 61 | AGL(1,𝔽5) | — | 36 | 1 | no | — |
| 62 | C3⋊D3 | — | 40 | 1 | no | — |
| 63 | C3×D3 | — | 40 | 1 | no | — |
| 64 | C3×D3 | — | 40 | 1 | no | — |
| 65 | C24 | — | 5 | 1 | no | — |
| 66 | C2×D4 | — | 45 | 1 | no | — |
| 67 | C2×D4 | — | 45 | 1 | no | — |
| 68 | C2×D4 | — | 45 | 1 | no | — |
| 69 | C22⋊C4 | — | 45 | 1 | no | — |
| 70 | C22⋊C4 | — | 45 | 1 | no | — |
| 71 | C22⋊C4 | — | 45 | 1 | no | — |
| 72 | C3×C5 | — | 24 | 1 | no | — |
| 73 | C22×C3 | — | 10 | 1 | no | — |
| 74 | C22×C3 | — | 20 | 1 | no | — |
| 75 | C22×C3 | — | 30 | 1 | no | — |
| 76 | C2×D3 | — | 30 | 1 | no | — |
| 77 | C2×D3 | — | 30 | 1 | no | — |
| 78 | C2×D3 | — | 30 | 1 | no | — |
| 79 | C2×D3 | — | 60 | 1 | no | — |
| 80 | C2×D3 | — | 60 | 1 | no | — |
| 81 | C2×D3 | — | 60 | 1 | no | — |
| 82 | A4 | — | 5 | 1 | no | — |
| 83 | A4 | — | 20 | 2 | no | — |
| 84 | A4 | — | 20 | 1 | no | — |
| 85 | A4 | — | 60 | 2 | no | — |
| 86 | A4 | S5 | 1 | 1 | yes | characteristic, complemented |
| 87 | C3×C4 | — | 30 | 1 | no | — |
| 88 | Dic3 | — | 30 | 1 | no | — |
| 89 | Dic3 | — | 60 | 1 | no | — |
| 90 | C2×C5 | — | 18 | 1 | no | — |
| 91 | D5 | — | 6 | 1 | no | — |
| 92 | D5 | — | 18 | 1 | no | — |
| 93 | C32 | — | 40 | 1 | no | Sylow |
| 94 | C23 | — | 15 | 1 | no | — |
| 95 | C23 | — | 15 | 1 | no | — |
| 96 | C23 | — | 45 | 1 | no | — |
| 97 | C23 | — | 45 | 1 | no | — |
| 98 | D4 | — | 30 | 1 | no | — |
| 99 | D4 | — | 90 | 1 | no | — |
| 100 | D4 | — | 90 | 1 | no | — |
| 101 | D4 | — | 90 | 2 | no | — |
| 102 | D4 | — | 90 | 1 | no | — |
| 103 | C2×C4 | — | 45 | 1 | no | — |
| 104 | C2×C4 | — | 45 | 1 | no | — |
| 105 | C2×C4 | — | 45 | 1 | no | — |
| 106 | C2×C3 | — | 30 | 1 | no | — |
| 107 | C2×C3 | — | 60 | 1 | no | — |
| 108 | C2×C3 | — | 60 | 1 | no | — |
| 109 | D3 | — | 10 | 1 | no | — |
| 110 | D3 | — | 30 | 1 | no | — |
| 111 | D3 | — | 40 | 1 | no | — |
| 112 | D3 | — | 60 | 1 | no | — |
| 113 | D3 | — | 240 | 1 | no | — |
| 114 | C5 | — | 6 | 1 | no | Sylow |
| 115 | C22 | — | 5 | 1 | no | — |
| 116 | C22 | — | 15 | 2 | no | — |
| 117 | C22 | — | 30 | 1 | no | — |
| 118 | C22 | — | 45 | 1 | no | — |
| 119 | C22 | — | 45 | 1 | no | — |
| 120 | C22 | — | 45 | 1 | no | — |
| 121 | C22 | — | 90 | 1 | no | — |
| 122 | C22 | — | 90 | 1 | no | — |
| 123 | C22 | ΓL(2,𝔽4) | 1 | 1 | yes | characteristic, complemented |
| 124 | C4 | — | 30 | 1 | no | — |
| 125 | C4 | — | 90 | 1 | no | — |
| 126 | C4 | — | 90 | 1 | no | — |
| 127 | C3 | — | 4 | 1 | no | — |
| 128 | C3 | — | 10 | 1 | no | — |
| 129 | C3 | — | 80 | 1 | no | — |
| 130 | C2 | — | 3 | 1 | no | — |
| 131 | C2 | — | 15 | 1 | no | — |
| 132 | C2 | — | 45 | 1 | no | — |
| 133 | C2 | — | 60 | 1 | no | — |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | A4×A5 | C2 |
| Fitting | C22 | ΓL(2,𝔽4) |
| socle | C22×A5 | D3 |
| LCS | G ▷ A4×A5 | |
| UCS | C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. A4×A5 | C2 | semidirect |
| 6. C22×A5 | D3 | semidirect |
| 25. A5 | S4 | semidirect |
| 86. A4 | S5 | semidirect |
| 123. C22 | ΓL(2,𝔽4) | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| A4×A5 | 2 |
| C22⋊S5 | 3 |
| ΓL(2,𝔽4) | 4 |
| A4⋊S4 | 5 |
| A4⋊AGL(1,𝔽5) | 6 |
| D3×S4 | 10 |