Finite Group DatabasebetaFinite Group DatabasebetaOrder 240.
20 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C3×#168 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #674 | C3 | 1 | 1 | yes | maximal, characteristic, direct product factor, Hall |
| 3 | C3×C4×C5 | C4 | 1 | 1 | yes | characteristic |
| 4 | C3×C16 | — | 5 | 1 | no | maximal, Hall |
| 5 | #168 | C2×C3 | 1 | 1 | yes | characteristic |
| 6 | C2×C3×C5 | C8 | 1 | 1 | yes | characteristic |
| 7 | C3×C8 | — | 5 | 1 | no | — |
| 8 | C4×C5 | C3×C4 | 1 | 1 | yes | characteristic |
| 9 | C16 | — | 5 | 1 | no | Sylow |
| 10 | C3×C5 | C16 | 1 | 1 | yes | characteristic, complemented, Hall |
| 11 | C3×C4 | AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 12 | C2×C5 | C3×C8 | 1 | 1 | yes | characteristic |
| 13 | C8 | — | 5 | 1 | no | — |
| 14 | C2×C3 | #170 | 1 | 1 | yes | characteristic |
| 15 | C5 | C3×C16 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 16 | C4 | C3×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 17 | C3 | #674 | 1 | 1 | yes | characteristic, direct product factor, Sylow |
| 18 | C2 | C3×#170 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3×C4 | AGL(1,𝔽5) |
| commutator, hypocenter | C5 | C3×C16 |
| Frattini | C4 | C3×AGL(1,𝔽5) |
| Fitting | C3×C4×C5 | C4 |
| socle, prime radical | C2×C3×C5 | C8 |
| cosocle | #168 | C2×C3 |
| LCS | G ▷ C5 | |
| UCS | C1 ◁ C3×C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #674 | C3 | direct (N × H) |
| 10. C3×C5 | C16 | semidirect |
| 15. C5 | C3×C16 | semidirect |
| 17. C3 | #674 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×#168 | 2 |
| #674 | 3 |
| C3×C16 | 5 |