Finite Group DatabasebetaFinite Group DatabasebetaOrder 1440.
154 conjugacy classes of subgroups; only the 8 normal subgroups are shown.
This group has 154 conjugacy classes of subgroups — too many to draw. Only its 8 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 | D4×A5 | — | 3 | 1 | no | maximal |
| 3 | D3×A5 | — | 4 | 1 | no | maximal |
| 4 | A4×S4 | — | 5 | 1 | no | maximal, Hall |
| 5 | D5×S4 | — | 6 | 1 | no | maximal |
| 6 | C22×A5 | — | 3 | 1 | no | — |
| 7 | C22×A5 | D3 | 1 | 1 | yes | characteristic, complemented |
| 8 | C4×A5 | — | 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 | C5⋊S4 | — | 6 | 1 | no | — |
| 14 | C5×S4 | — | 6 | 1 | no | — |
| 15 | C2×A5 | — | 3 | 1 | no | — |
| 16 | C2×A5 | — | 6 | 1 | no | — |
| 17 | C22×S4 | — | 5 | 1 | no | — |
| 18 | D4×A4 | — | 15 | 1 | no | — |
| 19 | D4×D5 | — | 18 | 1 | no | — |
| 20 | D3×A4 | — | 10 | 1 | no | — |
| 21 | D3×A4 | — | 20 | 1 | no | — |
| 22 | C3⋊S4 | — | 10 | 1 | no | — |
| 23 | C3×S4 | — | 10 | 1 | no | — |
| 24 | C5×A4 | — | 6 | 1 | no | — |
| 25 | D3×D5 | — | 24 | 1 | no | — |
| 26 | A5 | S4 | 1 | 1 | yes | characteristic, direct product factor |
| 27 | C22⋊A4 | — | 10 | 1 | no | — |
| 28 | C22×A4 | — | 5 | 1 | no | — |
| 29 | C22×A4 | — | 5 | 1 | no | — |
| 30 | C22×A4 | — | 15 | 1 | no | — |
| 31 | C2×S4 | — | 15 | 1 | no | — |
| 32 | C2×S4 | — | 15 | 1 | no | — |
| 33 | D3×D4 | — | 30 | 1 | no | — |
| 34 | C4×A4 | — | 15 | 1 | no | — |
| 35 | C22×D5 | — | 6 | 1 | no | — |
| 36 | C22×D5 | — | 18 | 1 | no | — |
| 37 | C5×D4 | — | 18 | 1 | no | — |
| 38 | C22⋊D5 | — | 18 | 1 | no | — |
| 39 | C22⋊D5 | — | 18 | 1 | no | — |
| 40 | D20 | — | 18 | 1 | no | — |
| 41 | C4×D5 | — | 18 | 1 | no | — |
| 42 | C3×A4 | — | 10 | 1 | no | — |
| 43 | C3×A4 | — | 20 | 1 | no | — |
| 44 | D32 | — | 40 | 1 | no | — |
| 45 | C22×D4 | — | 15 | 1 | no | Sylow |
| 46 | D15 | — | 24 | 1 | no | — |
| 47 | C3×D5 | — | 24 | 1 | no | — |
| 48 | C5×D3 | — | 24 | 1 | no | — |
| 49 | C22×D3 | — | 10 | 1 | no | — |
| 50 | C22×D3 | — | 20 | 1 | no | — |
| 51 | C22×D3 | — | 30 | 1 | no | — |
| 52 | C2×A4 | — | 15 | 1 | no | — |
| 53 | C2×A4 | — | 15 | 1 | no | — |
| 54 | C2×A4 | — | 30 | 1 | no | — |
| 55 | S4 | — | 15 | 1 | no | — |
| 56 | S4 | — | 60 | 1 | no | — |
| 57 | S4 | A5 | 1 | 1 | yes | maximal normal, characteristic, direct product factor |
| 58 | C3×D4 | — | 30 | 1 | no | — |
| 59 | #67 | — | 30 | 1 | no | — |
| 60 | #67 | — | 30 | 1 | no | — |
| 61 | D12 | — | 30 | 1 | no | — |
| 62 | C4×D3 | — | 30 | 1 | no | — |
| 63 | C22×C5 | — | 6 | 1 | no | — |
| 64 | C22×C5 | — | 18 | 1 | no | — |
| 65 | C2×D5 | — | 18 | 1 | no | — |
| 66 | C2×D5 | — | 18 | 1 | no | — |
| 67 | C2×D5 | — | 18 | 1 | no | — |
| 68 | C2×D5 | — | 36 | 1 | no | — |
| 69 | C2×D5 | — | 36 | 1 | no | — |
| 70 | C4×C5 | — | 18 | 1 | no | — |
| 71 | Dic5 | — | 18 | 1 | no | — |
| 72 | C3⋊D3 | — | 40 | 1 | no | — |
| 73 | C3×D3 | — | 40 | 1 | no | — |
| 74 | C3×D3 | — | 40 | 1 | no | — |
| 75 | C24 | — | 5 | 1 | no | — |
| 76 | C24 | — | 15 | 1 | no | — |
| 77 | C2×D4 | — | 45 | 1 | no | — |
| 78 | C2×D4 | — | 45 | 1 | no | — |
| 79 | C2×D4 | — | 45 | 1 | no | — |
| 80 | C2×D4 | — | 45 | 1 | no | — |
| 81 | C22×C4 | — | 15 | 1 | no | — |
| 82 | C3×C5 | — | 24 | 1 | no | — |
| 83 | C22×C3 | — | 10 | 1 | no | — |
| 84 | C22×C3 | — | 20 | 1 | no | — |
| 85 | C22×C3 | — | 30 | 1 | no | — |
| 86 | C2×D3 | — | 30 | 1 | no | — |
| 87 | C2×D3 | — | 30 | 1 | no | — |
| 88 | C2×D3 | — | 30 | 1 | no | — |
| 89 | C2×D3 | — | 60 | 1 | no | — |
| 90 | C2×D3 | — | 60 | 1 | no | — |
| 91 | C2×D3 | — | 60 | 1 | no | — |
| 92 | C2×D3 | — | 60 | 1 | no | — |
| 93 | A4 | — | 5 | 1 | no | — |
| 94 | A4 | — | 20 | 1 | no | — |
| 95 | A4 | — | 40 | 1 | no | — |
| 96 | A4 | — | 120 | 1 | no | — |
| 97 | A4 | C2×A5 | 1 | 1 | yes | characteristic, complemented |
| 98 | C3×C4 | — | 30 | 1 | no | — |
| 99 | Dic3 | — | 30 | 1 | no | — |
| 100 | C2×C5 | — | 18 | 1 | no | — |
| 101 | C2×C5 | — | 36 | 1 | no | — |
| 102 | D5 | — | 6 | 1 | no | — |
| 103 | D5 | — | 18 | 1 | no | — |
| 104 | D5 | — | 36 | 1 | no | — |
| 105 | C32 | — | 40 | 1 | no | Sylow |
| 106 | C23 | — | 15 | 1 | no | — |
| 107 | C23 | — | 15 | 1 | no | — |
| 108 | C23 | — | 30 | 1 | no | — |
| 109 | C23 | — | 45 | 1 | no | — |
| 110 | C23 | — | 45 | 1 | no | — |
| 111 | C23 | — | 45 | 1 | no | — |
| 112 | C23 | — | 90 | 1 | no | — |
| 113 | D4 | — | 3 | 1 | no | — |
| 114 | D4 | — | 45 | 1 | no | — |
| 115 | D4 | — | 45 | 1 | no | — |
| 116 | D4 | — | 45 | 2 | no | — |
| 117 | D4 | — | 45 | 1 | no | — |
| 118 | C2×C4 | — | 45 | 1 | no | — |
| 119 | C2×C4 | — | 45 | 1 | no | — |
| 120 | C2×C3 | — | 30 | 1 | no | — |
| 121 | C2×C3 | — | 60 | 1 | no | — |
| 122 | C2×C3 | — | 60 | 1 | no | — |
| 123 | D3 | — | 4 | 1 | no | — |
| 124 | D3 | — | 10 | 1 | no | — |
| 125 | D3 | — | 30 | 1 | no | — |
| 126 | D3 | — | 60 | 1 | no | — |
| 127 | D3 | — | 60 | 1 | no | — |
| 128 | D3 | — | 240 | 1 | no | — |
| 129 | C5 | — | 6 | 1 | no | Sylow |
| 130 | C22 | — | 3 | 1 | no | — |
| 131 | C22 | — | 5 | 1 | no | — |
| 132 | C22 | — | 30 | 1 | no | — |
| 133 | C22 | — | 45 | 1 | no | — |
| 134 | C22 | — | 45 | 1 | no | — |
| 135 | C22 | — | 45 | 1 | no | — |
| 136 | C22 | — | 45 | 1 | no | — |
| 137 | C22 | — | 90 | 1 | no | — |
| 138 | C22 | — | 90 | 1 | no | — |
| 139 | C22 | — | 90 | 1 | no | — |
| 140 | C22 | — | 90 | 1 | no | — |
| 141 | C22 | D3×A5 | 1 | 1 | yes | characteristic, complemented |
| 142 | C4 | — | 3 | 1 | no | — |
| 143 | C4 | — | 45 | 1 | no | — |
| 144 | C3 | — | 4 | 1 | no | — |
| 145 | C3 | — | 10 | 1 | no | — |
| 146 | C3 | — | 80 | 1 | no | — |
| 147 | C2 | — | 3 | 1 | no | — |
| 148 | C2 | — | 6 | 1 | no | — |
| 149 | C2 | — | 15 | 1 | no | — |
| 150 | C2 | — | 45 | 1 | no | — |
| 151 | C2 | — | 90 | 1 | no | — |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, hypocenter | A4×A5 | C2 |
| Fitting | C22 | D3×A5 |
| socle | C22×A5 | D3 |
| cosocle | A4 | C2×A5 |
| 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 |
| 7. C22×A5 | D3 | semidirect |
| 26. A5 | S4 | direct (N × H) |
| 57. S4 | A5 | direct (N × H) |
| 97. A4 | C2×A5 | semidirect |
| 141. C22 | D3×A5 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| A4×A5 | 2 |
| D4×A5 | 3 |
| D3×A5 | 4 |
| A4×S4 | 5 |
| D5×S4 | 6 |
| D3×S4 | 10 |