Finite Group DatabasebetaFinite Group DatabasebetaOrder 120.
12 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | SL(2,𝔽3) | — | 5 | 1 | no | maximal, Hall |
| 2 | Dic5 | — | 6 | 1 | no | maximal |
| 3 | Dic3 | — | 10 | 1 | no | maximal |
| 4 | C2×C5 | — | 6 | 1 | no | — |
| 5 | Q8 | — | 5 | 1 | no | Sylow |
| 6 | C2×C3 | — | 10 | 1 | no | — |
| 7 | C5 | — | 6 | 1 | no | Sylow |
| 8 | C4 | — | 15 | 1 | no | — |
| 9 | C3 | — | 10 | 1 | no | Sylow |
| 10 | C2 | A5 | 1 | 1 | yes | maximal normal, characteristic |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, Fitting, socle, cosocle, hypercenter | C2 | A5 |
| commutator, hypocenter | G | C1 |
| LCS | G | |
| UCS | C1 ◁ C2 | |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| SL(2,𝔽3) | 5 |
| Dic5 | 6 |
| Dic3 | 10 |