Finite Group DatabasebetaFinite Group DatabasebetaOrder 240.
26 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C5×SL(2,𝔽3) | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #689 | — | 3 | 1 | no | maximal, Hall |
| 3 | Dic15 | — | 4 | 1 | no | maximal |
| 4 | #226 | — | 5 | 1 | no | maximal, Hall |
| 5 | C5×Q8 | D3 | 1 | 1 | yes | characteristic |
| 6 | Dic10 | — | 3 | 1 | no | — |
| 7 | #168 | — | 3 | 1 | no | — |
| 8 | C2×C3×C5 | — | 4 | 1 | no | — |
| 9 | SL(2,𝔽3) | D5 | 1 | 1 | yes | characteristic |
| 10 | C4×C5 | — | 3 | 1 | no | — |
| 11 | Dic5 | — | 6 | 1 | no | — |
| 12 | Q16 | — | 15 | 1 | no | Sylow |
| 13 | C3×C5 | — | 4 | 1 | no | Hall |
| 14 | Dic3 | — | 20 | 1 | no | — |
| 15 | C2×C5 | S4 | 1 | 1 | yes | characteristic |
| 16 | Q8 | — | 15 | 1 | no | — |
| 17 | Q8 | D15 | 1 | 1 | yes | characteristic |
| 18 | C8 | — | 15 | 1 | no | — |
| 19 | C2×C3 | — | 4 | 1 | no | — |
| 20 | C5 | #226 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 21 | C4 | — | 3 | 1 | no | — |
| 22 | C4 | — | 30 | 1 | no | — |
| 23 | C3 | — | 4 | 1 | no | Sylow |
| 24 | C2 | C5⋊S4 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | C5⋊S4 |
| commutator, prime radical, cosocle, hypocenter | C5×SL(2,𝔽3) | C2 |
| Fitting | C5×Q8 | D3 |
| socle | C2×C5 | S4 |
| LCS | G ▷ C5×SL(2,𝔽3) | |
| UCS | C1 ◁ C2 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 20. C5 | #226 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C5×SL(2,𝔽3) | 2 |
| #689 | 3 |
| Dic15 | 4 |
| #226 | 5 |