Finite Group DatabasebetaFinite Group DatabasebetaOrder 720.
75 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C2×C3×A5 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | A5⋊C4 | C3 | 1 | 1 | yes | maximal, characteristic, direct product factor |
| 3 | C3×A5 | C4 | 1 | 1 | yes | characteristic, complemented |
| 4 | C3×SL(2,ℤ/4) | — | 5 | 1 | no | maximal, Hall |
| 5 | C2×C3×AGL(1,𝔽5) | — | 6 | 1 | no | maximal |
| 6 | C2×A5 | C2×C3 | 1 | 1 | yes | characteristic |
| 7 | C2×C3×A4 | — | 5 | 1 | no | — |
| 8 | C3×C4×D3 | — | 10 | 1 | no | maximal |
| 9 | C2×C3×D5 | — | 6 | 1 | no | — |
| 10 | C3×AGL(1,𝔽5) | — | 6 | 2 | no | — |
| 11 | A5 | C3×C4 | 1 | 1 | yes | characteristic, complemented |
| 12 | SL(2,ℤ/4) | — | 5 | 1 | no | — |
| 13 | C3×C22⋊C4 | — | 15 | 1 | no | — |
| 14 | C2×AGL(1,𝔽5) | — | 6 | 1 | no | — |
| 15 | C2×C3×D3 | — | 10 | 1 | no | — |
| 16 | C3×A4 | — | 5 | 1 | no | — |
| 17 | C32×C4 | — | 10 | 1 | no | — |
| 18 | C3×Dic3 | — | 10 | 1 | no | — |
| 19 | C2×C3×C5 | — | 6 | 1 | no | — |
| 20 | C3×D5 | — | 6 | 1 | no | — |
| 21 | C3×D5 | — | 6 | 1 | no | — |
| 22 | C23×C3 | — | 5 | 1 | no | — |
| 23 | C2×A4 | — | 5 | 1 | no | — |
| 24 | C2×A4 | — | 10 | 1 | no | — |
| 25 | C2×C3×C4 | — | 15 | 1 | no | — |
| 26 | C2×C3×C4 | — | 15 | 1 | no | — |
| 27 | C4×D3 | — | 10 | 1 | no | — |
| 28 | C2×D5 | — | 6 | 1 | no | — |
| 29 | AGL(1,𝔽5) | — | 6 | 2 | no | — |
| 30 | C2×C32 | — | 10 | 1 | no | — |
| 31 | C3×D3 | — | 10 | 1 | no | — |
| 32 | C3×D3 | — | 10 | 1 | no | — |
| 33 | C22⋊C4 | — | 15 | 1 | no | Sylow |
| 34 | C3×C5 | — | 6 | 1 | no | — |
| 35 | C22×C3 | — | 5 | 1 | no | — |
| 36 | C22×C3 | — | 15 | 1 | no | — |
| 37 | C22×C3 | — | 15 | 1 | no | — |
| 38 | C2×D3 | — | 10 | 1 | no | — |
| 39 | A4 | — | 5 | 1 | no | — |
| 40 | A4 | — | 10 | 1 | no | — |
| 41 | C3×C4 | — | 10 | 1 | no | — |
| 42 | C3×C4 | — | 10 | 1 | no | — |
| 43 | C3×C4 | — | 20 | 1 | no | — |
| 44 | C3×C4 | — | 30 | 1 | no | — |
| 45 | Dic3 | — | 10 | 1 | no | — |
| 46 | C2×C5 | — | 6 | 1 | no | — |
| 47 | D5 | — | 6 | 1 | no | — |
| 48 | D5 | — | 6 | 1 | no | — |
| 49 | C32 | — | 10 | 1 | no | Sylow |
| 50 | C23 | — | 5 | 1 | no | — |
| 51 | C2×C4 | — | 15 | 1 | no | — |
| 52 | C2×C4 | — | 15 | 1 | no | — |
| 53 | C2×C3 | — | 10 | 1 | no | — |
| 54 | C2×C3 | — | 15 | 1 | no | — |
| 55 | C2×C3 | — | 15 | 1 | no | — |
| 56 | C2×C3 | — | 20 | 1 | no | — |
| 57 | C2×C3 | S5 | 1 | 1 | yes | characteristic |
| 58 | D3 | — | 10 | 1 | no | — |
| 59 | D3 | — | 10 | 1 | no | — |
| 60 | C5 | — | 6 | 1 | no | Sylow |
| 61 | C22 | — | 5 | 1 | no | — |
| 62 | C22 | — | 15 | 1 | no | — |
| 63 | C22 | — | 15 | 1 | no | — |
| 64 | C4 | — | 10 | 1 | no | — |
| 65 | C4 | — | 30 | 1 | no | — |
| 66 | C3 | — | 10 | 1 | no | — |
| 67 | C3 | — | 20 | 1 | no | — |
| 68 | C3 | A5⋊C4 | 1 | 1 | yes | characteristic, direct product factor |
| 69 | C2 | — | 15 | 1 | no | — |
| 70 | C2 | — | 15 | 1 | no | — |
| 71 | C2 | C3×S5 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, Fitting, hypercenter | C2×C3 | S5 |
| commutator, hypocenter | A5 | C3×C4 |
| Frattini | C2 | C3×S5 |
| socle, prime radical | C2×C3×A5 | C2 |
| cosocle | C2×A5 | C2×C3 |
| LCS | G ▷ A5 | |
| UCS | C1 ◁ C2×C3 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. A5⋊C4 | C3 | direct (N × H) |
| 3. C3×A5 | C4 | semidirect |
| 11. A5 | C3×C4 | semidirect |
| 68. C3 | A5⋊C4 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C3×A5 | 2 |
| A5⋊C4 | 3 |
| C3×SL(2,ℤ/4) | 5 |
| C2×C3×AGL(1,𝔽5) | 6 |
| C3×C4×D3 | 10 |