Finite Group DatabasebetaFinite Group DatabasebetaOrder 240.
22 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | SL(2,𝔽5) | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #226 | — | 5 | 1 | no | maximal, Hall |
| 3 | #170 | — | 6 | 1 | no | maximal |
| 4 | Dic6 | — | 10 | 1 | no | maximal |
| 5 | SL(2,𝔽3) | — | 5 | 1 | no | — |
| 6 | Dic5 | — | 6 | 1 | no | — |
| 7 | Q16 | — | 15 | 1 | no | Sylow |
| 8 | C3×C4 | — | 10 | 1 | no | — |
| 9 | Dic3 | — | 10 | 1 | no | — |
| 10 | Dic3 | — | 10 | 1 | no | — |
| 11 | C2×C5 | — | 6 | 1 | no | — |
| 12 | Q8 | — | 5 | 1 | no | — |
| 13 | Q8 | — | 15 | 1 | no | — |
| 14 | C8 | — | 15 | 1 | no | — |
| 15 | C2×C3 | — | 10 | 1 | no | — |
| 16 | C5 | — | 6 | 1 | no | Sylow |
| 17 | C4 | — | 10 | 1 | no | — |
| 18 | C4 | — | 15 | 1 | no | — |
| 19 | C3 | — | 10 | 1 | no | Sylow |
| 20 | C2 | S5 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, Fitting, socle, hypercenter | C2 | S5 |
| commutator, prime radical, cosocle, hypocenter | SL(2,𝔽5) | C2 |
| LCS | G ▷ SL(2,𝔽5) | |
| UCS | C1 ◁ C2 | |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| SL(2,𝔽5) | 2 |
| #226 | 5 |
| #170 | 6 |
| Dic6 | 10 |