Finite Group DatabasebetaFinite Group DatabasebetaOrder 480.
106 conjugacy classes of subgroups; only the 9 normal subgroups are shown.
This group has 106 conjugacy classes of subgroups — too many to draw. Only its 9 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 | C22×A5 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | C2×S5 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | A5⋊C4 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | C2×A5 | — | 2 | 1 | no | — |
| 5 | C2×A5 | C22 | 1 | 1 | yes | characteristic |
| 6 | S5 | — | 2 | 1 | no | — |
| 7 | GL(2,ℤ/4) | — | 5 | 1 | no | maximal, Hall |
| 8 | C22⋊AGL(1,𝔽5) | — | 6 | 1 | no | maximal |
| 9 | A5 | D4 | 1 | 1 | yes | characteristic, complemented |
| 10 | C22×A4 | — | 5 | 1 | no | — |
| 11 | C2×S4 | — | 5 | 1 | no | — |
| 12 | D3×D4 | — | 10 | 1 | no | maximal |
| 13 | SL(2,ℤ/4) | — | 5 | 1 | no | — |
| 14 | C22×D5 | — | 6 | 1 | no | — |
| 15 | C2×AGL(1,𝔽5) | — | 6 | 1 | no | — |
| 16 | C2×AGL(1,𝔽5) | — | 6 | 1 | no | — |
| 17 | #120 | — | 15 | 1 | no | Sylow |
| 18 | C22×D3 | — | 10 | 1 | no | — |
| 19 | C22×D3 | — | 10 | 1 | no | — |
| 20 | C2×A4 | — | 5 | 1 | no | — |
| 21 | C2×A4 | — | 10 | 1 | no | — |
| 22 | S4 | — | 10 | 1 | no | — |
| 23 | C3×D4 | — | 10 | 1 | no | — |
| 24 | #67 | — | 10 | 1 | no | — |
| 25 | #67 | — | 10 | 1 | no | — |
| 26 | D12 | — | 10 | 1 | no | — |
| 27 | C4×D3 | — | 10 | 1 | no | — |
| 28 | C22×C5 | — | 6 | 1 | no | — |
| 29 | C2×D5 | — | 6 | 1 | no | — |
| 30 | C2×D5 | — | 6 | 1 | no | — |
| 31 | C2×D5 | — | 12 | 1 | no | — |
| 32 | C2×D5 | — | 12 | 1 | no | — |
| 33 | AGL(1,𝔽5) | — | 12 | 1 | no | — |
| 34 | AGL(1,𝔽5) | — | 12 | 1 | no | — |
| 35 | C24 | — | 5 | 1 | no | — |
| 36 | C2×D4 | — | 15 | 1 | no | — |
| 37 | C2×D4 | — | 15 | 1 | no | — |
| 38 | C2×D4 | — | 15 | 1 | no | — |
| 39 | C22⋊C4 | — | 15 | 1 | no | — |
| 40 | C22⋊C4 | — | 15 | 1 | no | — |
| 41 | C22⋊C4 | — | 15 | 1 | no | — |
| 42 | C22×C3 | — | 10 | 1 | no | — |
| 43 | C22×C3 | — | 10 | 1 | no | — |
| 44 | C2×D3 | — | 10 | 1 | no | — |
| 45 | C2×D3 | — | 10 | 1 | no | — |
| 46 | C2×D3 | — | 10 | 1 | no | — |
| 47 | C2×D3 | — | 20 | 1 | no | — |
| 48 | C2×D3 | — | 20 | 1 | no | — |
| 49 | C2×D3 | — | 20 | 1 | no | — |
| 50 | C2×D3 | — | 20 | 1 | no | — |
| 51 | A4 | — | 5 | 1 | no | — |
| 52 | C3×C4 | — | 10 | 1 | no | — |
| 53 | Dic3 | — | 10 | 1 | no | — |
| 54 | C2×C5 | — | 6 | 1 | no | — |
| 55 | C2×C5 | — | 12 | 1 | no | — |
| 56 | D5 | — | 6 | 1 | no | — |
| 57 | D5 | — | 6 | 1 | no | — |
| 58 | D5 | — | 12 | 1 | no | — |
| 59 | C23 | — | 5 | 1 | no | — |
| 60 | C23 | — | 10 | 1 | no | — |
| 61 | C23 | — | 15 | 1 | no | — |
| 62 | C23 | — | 15 | 1 | no | — |
| 63 | C23 | — | 15 | 1 | no | — |
| 64 | C23 | — | 30 | 1 | no | — |
| 65 | D4 | — | 10 | 1 | no | — |
| 66 | D4 | — | 30 | 1 | no | — |
| 67 | D4 | — | 30 | 1 | no | — |
| 68 | D4 | — | 30 | 1 | no | — |
| 69 | D4 | — | 30 | 2 | no | — |
| 70 | C2×C4 | — | 15 | 1 | no | — |
| 71 | C2×C4 | — | 15 | 1 | no | — |
| 72 | C2×C4 | — | 15 | 1 | no | — |
| 73 | C2×C3 | — | 10 | 1 | no | — |
| 74 | C2×C3 | — | 20 | 1 | no | — |
| 75 | C2×C3 | — | 20 | 1 | no | — |
| 76 | D3 | — | 10 | 1 | no | — |
| 77 | D3 | — | 10 | 1 | no | — |
| 78 | D3 | — | 20 | 1 | no | — |
| 79 | D3 | — | 20 | 1 | no | — |
| 80 | C5 | — | 6 | 1 | no | Sylow |
| 81 | C22 | — | 5 | 1 | no | — |
| 82 | C22 | — | 10 | 1 | no | — |
| 83 | C22 | — | 15 | 1 | no | — |
| 84 | C22 | — | 15 | 1 | no | — |
| 85 | C22 | — | 15 | 2 | no | — |
| 86 | C22 | — | 15 | 1 | no | — |
| 87 | C22 | — | 30 | 1 | no | — |
| 88 | C22 | — | 30 | 1 | no | — |
| 89 | C22 | — | 30 | 1 | no | — |
| 90 | C22 | — | 30 | 1 | no | — |
| 91 | C22 | — | 30 | 1 | no | — |
| 92 | C22 | S5 | 1 | 1 | yes | characteristic, complemented |
| 93 | C4 | — | 10 | 1 | no | — |
| 94 | C4 | — | 30 | 1 | no | — |
| 95 | C4 | — | 30 | 1 | no | — |
| 96 | C3 | — | 10 | 1 | no | Sylow |
| 97 | C2 | — | 2 | 1 | no | — |
| 98 | C2 | — | 15 | 1 | no | — |
| 99 | C2 | — | 15 | 1 | no | — |
| 100 | C2 | — | 20 | 1 | no | — |
| 101 | C2 | — | 30 | 1 | no | — |
| 102 | C2 | C2×S5 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C2×S5 |
| commutator, socle, cosocle | C2×A5 | C22 |
| Fitting, hypercenter | C22 | S5 |
| hypocenter | A5 | D4 |
| LCS | G ▷ C2×A5 ▷ A5 | |
| UCS | C1 ◁ C2 ◁ C22 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×A5 | C2 | semidirect |
| 2. C2×S5 | C2 | semidirect |
| 3. A5⋊C4 | C2 | semidirect |
| 9. A5 | D4 | semidirect |
| 92. C22 | S5 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×A5 | 2 |
| C2×S5 | 2 |
| A5⋊C4 | 2 |
| GL(2,ℤ/4) | 5 |
| C22⋊AGL(1,𝔽5) | 6 |
| D3×D4 | 10 |