Finite Group DatabasebetaFinite Group DatabasebetaOrder 480.
98 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C2×S5 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C4×A5 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | A5⋊C4 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | C2×A5 | C22 | 1 | 1 | yes | characteristic |
| 5 | S5 | C4 | 1 | 2 | yes | direct product factor |
| 6 | C4×S4 | — | 5 | 1 | no | maximal, Hall |
| 7 | C4×AGL(1,𝔽5) | — | 6 | 1 | no | maximal |
| 8 | A5 | C2×C4 | 1 | 1 | yes | characteristic, complemented |
| 9 | C2×S4 | — | 5 | 1 | no | — |
| 10 | C2×C4×D3 | — | 10 | 1 | no | maximal |
| 11 | C4×A4 | — | 5 | 1 | no | — |
| 12 | SL(2,ℤ/4) | — | 5 | 1 | no | — |
| 13 | C2×AGL(1,𝔽5) | — | 6 | 1 | no | — |
| 14 | C2×AGL(1,𝔽5) | — | 6 | 1 | no | — |
| 15 | C4×D5 | — | 6 | 1 | no | — |
| 16 | C4×D4 | — | 15 | 1 | no | Sylow |
| 17 | C22×D3 | — | 10 | 1 | no | — |
| 18 | C2×A4 | — | 5 | 1 | no | — |
| 19 | S4 | — | 5 | 2 | no | — |
| 20 | C2×C3×C4 | — | 10 | 1 | no | — |
| 21 | C2×Dic3 | — | 10 | 1 | no | — |
| 22 | C4×D3 | — | 10 | 1 | no | — |
| 23 | C4×D3 | — | 10 | 1 | no | — |
| 24 | C4×D3 | — | 10 | 1 | no | — |
| 25 | C4×D3 | — | 10 | 1 | no | — |
| 26 | C2×D5 | — | 6 | 1 | no | — |
| 27 | AGL(1,𝔽5) | — | 6 | 2 | no | — |
| 28 | AGL(1,𝔽5) | — | 6 | 2 | no | — |
| 29 | C4×C5 | — | 6 | 1 | no | — |
| 30 | Dic5 | — | 6 | 1 | no | — |
| 31 | C2×D4 | — | 15 | 1 | no | — |
| 32 | C22×C4 | — | 5 | 1 | no | — |
| 33 | C22×C4 | — | 15 | 1 | no | — |
| 34 | C4⋊C4 | — | 15 | 1 | no | — |
| 35 | C22⋊C4 | — | 15 | 1 | no | — |
| 36 | C22⋊C4 | — | 15 | 1 | no | — |
| 37 | C42 | — | 15 | 1 | no | — |
| 38 | C22×C3 | — | 10 | 1 | no | — |
| 39 | C2×D3 | — | 10 | 1 | no | — |
| 40 | C2×D3 | — | 10 | 1 | no | — |
| 41 | C2×D3 | — | 10 | 2 | no | — |
| 42 | C2×D3 | — | 10 | 2 | no | — |
| 43 | A4 | — | 5 | 1 | no | — |
| 44 | C3×C4 | — | 10 | 1 | no | — |
| 45 | C3×C4 | — | 10 | 1 | no | — |
| 46 | Dic3 | — | 10 | 1 | no | — |
| 47 | Dic3 | — | 10 | 1 | no | — |
| 48 | C2×C5 | — | 6 | 1 | no | — |
| 49 | D5 | — | 6 | 1 | no | — |
| 50 | D5 | — | 6 | 1 | no | — |
| 51 | C23 | — | 5 | 1 | no | — |
| 52 | C23 | — | 15 | 1 | no | — |
| 53 | D4 | — | 15 | 2 | no | — |
| 54 | D4 | — | 15 | 2 | no | — |
| 55 | C2×C4 | — | 10 | 1 | no | — |
| 56 | C2×C4 | — | 15 | 1 | no | — |
| 57 | C2×C4 | — | 15 | 1 | no | — |
| 58 | C2×C4 | — | 15 | 1 | no | — |
| 59 | C2×C4 | — | 15 | 1 | no | — |
| 60 | C2×C4 | — | 15 | 1 | no | — |
| 61 | C2×C4 | — | 30 | 1 | no | — |
| 62 | C2×C3 | — | 10 | 2 | no | — |
| 63 | C2×C3 | — | 10 | 1 | no | — |
| 64 | D3 | — | 10 | 2 | no | — |
| 65 | D3 | — | 10 | 1 | no | — |
| 66 | D3 | — | 10 | 1 | no | — |
| 67 | C5 | — | 6 | 1 | no | Sylow |
| 68 | C22 | — | 5 | 1 | no | — |
| 69 | C22 | — | 10 | 1 | no | — |
| 70 | C22 | — | 15 | 2 | no | — |
| 71 | C22 | — | 15 | 1 | no | — |
| 72 | C22 | — | 15 | 1 | no | — |
| 73 | C22 | — | 30 | 1 | no | — |
| 74 | C4 | — | 10 | 1 | no | — |
| 75 | C4 | — | 15 | 2 | no | — |
| 76 | C4 | — | 15 | 1 | no | — |
| 77 | C4 | — | 30 | 1 | no | — |
| 78 | C4 | S5 | 1 | 1 | yes | characteristic, direct product factor |
| 79 | C3 | — | 10 | 1 | no | Sylow |
| 80 | C2 | — | 10 | 2 | no | — |
| 81 | C2 | — | 15 | 1 | no | — |
| 82 | C2 | — | 15 | 1 | no | — |
| 83 | C2 | C2×S5 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, Fitting, hypercenter | C4 | S5 |
| commutator, hypocenter | A5 | C2×C4 |
| Frattini | C2 | C2×S5 |
| socle, cosocle | C2×A5 | C22 |
| prime radical | C2×S5 | C2 |
| LCS | G ▷ A5 | |
| UCS | C1 ◁ C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C4×A5 | C2 | semidirect |
| 3. A5⋊C4 | C2 | semidirect |
| 5. S5 | C4 | direct (N × H) |
| 8. A5 | C2×C4 | semidirect |
| 78. C4 | S5 | direct (N × H) |
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 |
| A5⋊C4 | 2 |
| C4×S4 | 5 |
| C4×AGL(1,𝔽5) | 6 |
| C2×C4×D3 | 10 |