Finite Group DatabasebetaFinite Group DatabasebetaOrder 28365.
16 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #22628162 | C3 | 1 | 1 | yes | maximal, characteristic, complemented, Hall |
| 2 | C61×C31⋊C3 | C5 | 1 | 1 | yes | maximal, characteristic, complemented, Hall |
| 3 | C31×C61 | C3×C5 | 1 | 1 | yes | characteristic, complemented, Hall |
| 4 | C3×C61⋊C5 | — | 31 | 1 | no | maximal, Hall |
| 5 | (C31⋊C3)⋊C5 | — | 61 | 1 | no | maximal, Hall |
| 6 | C61⋊C5 | — | 31 | 1 | no | Hall |
| 7 | C3×C61 | — | 155 | 1 | no | Hall |
| 8 | ΓL(1,𝔽32) | — | 61 | 1 | no | Hall |
| 9 | C31⋊C3 | C61⋊C5 | 1 | 1 | yes | characteristic, complemented, Hall |
| 10 | C61 | (C31⋊C3)⋊C5 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 11 | C31 | C3×C61⋊C5 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 12 | C3×C5 | — | 1891 | 1 | no | Hall |
| 13 | C5 | — | 1891 | 1 | no | Sylow |
| 14 | C3 | — | 155 | 1 | no | Sylow |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C31×C61 | C3×C5 |
| LCS | G ▷ C31×C61 | |
| UCS | C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #22628162 | C3 | semidirect |
| 2. C61×C31⋊C3 | C5 | semidirect |
| 3. C31×C61 | C3×C5 | semidirect |
| 9. C31⋊C3 | C61⋊C5 | semidirect |
| 10. C61 | (C31⋊C3)⋊C5 | semidirect |
| 11. C31 | C3×C61⋊C5 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #22628162 | 3 |
| C61×C31⋊C3 | 5 |
| C3×C61⋊C5 | 31 |
| (C31⋊C3)⋊C5 | 61 |