Finite Group DatabasebetaFinite Group DatabasebetaOrder 60.
12 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C2×C3×C5 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C4×C5 | — | 3 | 1 | no | maximal, Hall |
| 3 | C3×C5 | C4 | 1 | 1 | yes | characteristic, complemented, Hall |
| 4 | Dic3 | C5 | 1 | 1 | yes | maximal, characteristic, direct product factor, Hall |
| 5 | C2×C5 | D3 | 1 | 1 | yes | characteristic |
| 6 | C2×C3 | C2×C5 | 1 | 1 | yes | characteristic |
| 7 | C5 | Dic3 | 1 | 1 | yes | characteristic, direct product factor, Sylow |
| 8 | C4 | — | 3 | 1 | no | Sylow |
| 9 | C3 | C4×C5 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 10 | C2 | C5×D3 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C5 | D3 |
| commutator, hypocenter | C3 | C4×C5 |
| Frattini | C2 | C5×D3 |
| Fitting, socle, prime radical | C2×C3×C5 | C2 |
| cosocle | C2×C3 | C2×C5 |
| LCS | G ▷ C3 | |
| UCS | C1 ◁ C2×C5 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. C3×C5 | C4 | semidirect |
| 4. Dic3 | C5 | direct (N × H) |
| 7. C5 | Dic3 | direct (N × H) |
| 9. C3 | C4×C5 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C3×C5 | 2 |
| C4×C5 | 3 |
| Dic3 | 5 |