Finite Group DatabasebetaFinite Group DatabasebetaOrder 60.
12 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C3×D5 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | AGL(1,𝔽5) | C3 | 1 | 1 | yes | maximal, characteristic, direct product factor, Hall |
| 3 | C3×C5 | C4 | 1 | 1 | yes | characteristic, complemented, Hall |
| 4 | C3×C4 | — | 5 | 1 | no | maximal, Hall |
| 5 | D5 | C2×C3 | 1 | 1 | yes | characteristic |
| 6 | C2×C3 | — | 5 | 1 | no | — |
| 7 | C5 | C3×C4 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 8 | C4 | — | 5 | 1 | no | Sylow |
| 9 | C3 | AGL(1,𝔽5) | 1 | 1 | yes | characteristic, direct product factor, Sylow |
| 10 | C2 | — | 5 | 1 | no | — |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3 | AGL(1,𝔽5) |
| commutator, hypocenter | C5 | C3×C4 |
| Frattini | C1 | G |
| Fitting, socle | C3×C5 | C4 |
| prime radical | C3×D5 | C2 |
| cosocle | D5 | C2×C3 |
| LCS | G ▷ C5 | |
| UCS | C1 ◁ C3 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. AGL(1,𝔽5) | C3 | direct (N × H) |
| 3. C3×C5 | C4 | semidirect |
| 7. C5 | C3×C4 | semidirect |
| 9. C3 | AGL(1,𝔽5) | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×D5 | 2 |
| AGL(1,𝔽5) | 3 |
| C3×C4 | 5 |