Finite Group DatabasebetaFinite Group DatabasebetaOrder 480.
96 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C2×S5 | C2 | 1 | 2 | yes | maximal, complemented |
| 2 | C4×A5 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | C2×A5 | C22 | 1 | 1 | yes | characteristic |
| 4 | S5 | — | 2 | 2 | no | — |
| 5 | C4⋊S4 | — | 5 | 1 | no | maximal, Hall |
| 6 | C4⋊AGL(1,𝔽5) | — | 6 | 1 | no | maximal |
| 7 | A5 | D4 | 1 | 1 | yes | characteristic, complemented |
| 8 | C2×S4 | — | 5 | 2 | no | — |
| 9 | D3×D4 | — | 10 | 1 | no | maximal |
| 10 | C4×A4 | — | 5 | 1 | no | — |
| 11 | C2×AGL(1,𝔽5) | — | 6 | 2 | no | — |
| 12 | C4×D5 | — | 6 | 1 | no | — |
| 13 | #121 | — | 15 | 1 | no | Sylow |
| 14 | C22×D3 | — | 10 | 2 | no | — |
| 15 | C2×A4 | — | 5 | 1 | no | — |
| 16 | S4 | — | 10 | 2 | no | — |
| 17 | C3×D4 | — | 10 | 1 | no | — |
| 18 | #67 | — | 10 | 2 | no | — |
| 19 | D12 | — | 10 | 1 | no | — |
| 20 | C4×D3 | — | 10 | 1 | no | — |
| 21 | C2×D5 | — | 6 | 1 | no | — |
| 22 | AGL(1,𝔽5) | — | 12 | 2 | no | — |
| 23 | C4×C5 | — | 6 | 1 | no | — |
| 24 | Dic5 | — | 6 | 1 | no | — |
| 25 | C2×D4 | — | 15 | 1 | no | — |
| 26 | C2×D4 | — | 15 | 2 | no | — |
| 27 | C22×C4 | — | 5 | 1 | no | — |
| 28 | C4⋊C4 | — | 15 | 1 | no | — |
| 29 | C22⋊C4 | — | 15 | 2 | no | — |
| 30 | C22×C3 | — | 10 | 2 | no | — |
| 31 | C2×D3 | — | 10 | 2 | no | — |
| 32 | C2×D3 | — | 10 | 1 | no | — |
| 33 | C2×D3 | — | 20 | 2 | no | — |
| 34 | C2×D3 | — | 20 | 2 | no | — |
| 35 | A4 | — | 5 | 1 | no | — |
| 36 | C3×C4 | — | 10 | 1 | no | — |
| 37 | Dic3 | — | 10 | 1 | no | — |
| 38 | C2×C5 | — | 6 | 1 | no | — |
| 39 | D5 | — | 6 | 1 | no | — |
| 40 | D5 | — | 6 | 1 | no | — |
| 41 | C23 | — | 5 | 1 | no | — |
| 42 | C23 | — | 15 | 2 | no | — |
| 43 | D4 | — | 10 | 1 | no | — |
| 44 | D4 | — | 30 | 1 | no | — |
| 45 | D4 | — | 30 | 2 | no | — |
| 46 | D4 | — | 30 | 2 | no | — |
| 47 | C2×C4 | — | 15 | 1 | no | — |
| 48 | C2×C4 | — | 15 | 1 | no | — |
| 49 | C2×C4 | — | 15 | 2 | no | — |
| 50 | C2×C3 | — | 10 | 1 | no | — |
| 51 | C2×C3 | — | 20 | 2 | no | — |
| 52 | D3 | — | 10 | 1 | no | — |
| 53 | D3 | — | 10 | 1 | no | — |
| 54 | D3 | — | 20 | 2 | no | — |
| 55 | C5 | — | 6 | 1 | no | Sylow |
| 56 | C22 | — | 5 | 1 | no | — |
| 57 | C22 | — | 10 | 2 | no | — |
| 58 | C22 | — | 15 | 1 | no | — |
| 59 | C22 | — | 15 | 1 | no | — |
| 60 | C22 | — | 30 | 2 | no | — |
| 61 | C22 | — | 30 | 2 | no | — |
| 62 | C4 | — | 15 | 1 | no | — |
| 63 | C4 | — | 30 | 2 | no | — |
| 64 | C4 | S5 | 1 | 1 | yes | characteristic, complemented |
| 65 | C3 | — | 10 | 1 | no | Sylow |
| 66 | C2 | — | 15 | 1 | no | — |
| 67 | C2 | — | 15 | 1 | no | — |
| 68 | C2 | — | 20 | 2 | no | — |
| 69 | C2 | C2×S5 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C2×S5 |
| commutator, socle, cosocle | C2×A5 | C22 |
| Fitting, hypercenter | C4 | S5 |
| hypocenter | A5 | D4 |
| LCS | G ▷ C2×A5 ▷ A5 | |
| UCS | C1 ◁ C2 ◁ C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×S5 | C2 | semidirect |
| 2. C4×A5 | C2 | semidirect |
| 7. A5 | D4 | semidirect |
| 64. C4 | S5 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×S5 | 2 |
| C4×A5 | 2 |
| C4⋊S4 | 5 |
| C4⋊AGL(1,𝔽5) | 6 |
| D3×D4 | 10 |