Finite Group DatabasebetaFinite Group DatabasebetaOrder 14415.
18 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C31×ΓL(1,𝔽32) | C3 | 1 | 1 | yes | maximal, characteristic, complemented, Hall |
| 2 | #15668333 | C5 | 1 | 1 | yes | maximal, characteristic, complemented, Hall |
| 3 | C312 | C3×C5 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 4 | C5×C31⋊C3 | — | 31 | 1 | no | maximal |
| 5 | (C31⋊C3)⋊C5 | — | 31 | 1 | no | maximal |
| 6 | C5×C31 | — | 31 | 1 | no | — |
| 7 | ΓL(1,𝔽32) | C31⋊C3 | 1 | 1 | yes | characteristic, complemented |
| 8 | C31⋊C3 | — | 31 | 1 | no | — |
| 9 | C31⋊C3 | — | 155 | 1 | no | — |
| 10 | C31 | — | 15 | 2 | no | — |
| 11 | C31 | (C31⋊C3)⋊C5 | 1 | 1 | yes | characteristic, complemented |
| 12 | C31 | C5×C31⋊C3 | 1 | 1 | yes | characteristic, complemented |
| 13 | C3×C5 | — | 961 | 1 | no | Hall |
| 14 | C5 | — | 31 | 1 | no | Sylow |
| 15 | C3 | — | 4805 | 1 | no | Sylow |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C312 | C3×C5 |
| LCS | G ▷ C312 | |
| UCS | C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C31×ΓL(1,𝔽32) | C3 | semidirect |
| 2. #15668333 | C5 | semidirect |
| 3. C312 | C3×C5 | semidirect |
| 7. ΓL(1,𝔽32) | C31⋊C3 | semidirect |
| 11. C31 | (C31⋊C3)⋊C5 | semidirect |
| 12. C31 | C5×C31⋊C3 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C31×ΓL(1,𝔽32) | 3 |
| #15668333 | 5 |
| C5×C31⋊C3 | 31 |
| (C31⋊C3)⋊C5 | 31 |