Finite Group DatabasebetaFinite Group DatabasebetaOrder 240.
30 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | SL(2,𝔽5) | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #41⋊C3 | — | 5 | 1 | no | maximal, Hall |
| 3 | C4×D5 | — | 6 | 1 | no | maximal |
| 4 | C4×D3 | — | 10 | 1 | no | maximal |
| 5 | SL(2,𝔽3) | — | 5 | 1 | no | — |
| 6 | C2×D5 | — | 6 | 1 | no | — |
| 7 | C4×C5 | — | 6 | 1 | no | — |
| 8 | Dic5 | — | 6 | 1 | no | — |
| 9 | #41 | — | 5 | 1 | no | Sylow |
| 10 | C2×D3 | — | 10 | 1 | no | — |
| 11 | C3×C4 | — | 10 | 1 | no | — |
| 12 | Dic3 | — | 10 | 1 | no | — |
| 13 | C2×C5 | — | 6 | 1 | no | — |
| 14 | D5 | — | 6 | 2 | no | — |
| 15 | Q8 | — | 5 | 1 | no | — |
| 16 | D4 | — | 15 | 1 | no | — |
| 17 | C2×C4 | — | 15 | 1 | no | — |
| 18 | C2×C3 | — | 10 | 1 | no | — |
| 19 | D3 | — | 10 | 2 | no | — |
| 20 | C5 | — | 6 | 1 | no | Sylow |
| 21 | C22 | — | 15 | 1 | no | — |
| 22 | C4 | — | 15 | 1 | no | — |
| 23 | C4 | A5 | 1 | 1 | yes | maximal normal, characteristic |
| 24 | C3 | — | 10 | 1 | no | Sylow |
| 25 | C2 | — | 30 | 1 | no | — |
| 26 | C2 | C2×A5 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, Fitting, hypercenter | C4 | A5 |
| commutator, hypocenter | SL(2,𝔽5) | C2 |
| Frattini, socle, cosocle | C2 | C2×A5 |
| LCS | G ▷ SL(2,𝔽5) | |
| UCS | C1 ◁ C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. SL(2,𝔽5) | C2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| SL(2,𝔽5) | 2 |
| #41⋊C3 | 5 |
| C4×D5 | 6 |
| C4×D3 | 10 |