Finite Group DatabasebetaFinite Group DatabasebetaOrder 32235.
16 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C5×C7×C307 | C3 | 1 | 1 | yes | maximal, characteristic, complemented, Hall |
| 2 | #20264516 | C5 | 1 | 1 | yes | maximal, characteristic, direct product factor, Hall |
| 3 | C5×C307⋊C3 | — | 7 | 1 | no | maximal, Hall |
| 4 | C7×C307 | C3×C5 | 1 | 1 | yes | characteristic, complemented, Hall |
| 5 | C5×C307 | ΓL(1,𝔽8) | 1 | 1 | yes | characteristic, complemented, Hall |
| 6 | C307⋊C3 | — | 7 | 1 | no | Hall |
| 7 | C307 | C5×ΓL(1,𝔽8) | 1 | 1 | yes | characteristic, complemented, Sylow |
| 8 | C5×ΓL(1,𝔽8) | — | 307 | 1 | no | maximal, Hall |
| 9 | C5×C7 | C307⋊C3 | 1 | 1 | yes | characteristic, complemented, Hall |
| 10 | ΓL(1,𝔽8) | — | 307 | 1 | no | Hall |
| 11 | C3×C5 | — | 2149 | 1 | no | Hall |
| 12 | C7 | C5×C307⋊C3 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 13 | C5 | #20264516 | 1 | 1 | yes | characteristic, direct product factor, Sylow |
| 14 | C3 | — | 2149 | 1 | no | Sylow |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C5 | #20264516 |
| commutator, cosocle, hypocenter | C7×C307 | C3×C5 |
| Frattini | C1 | G |
| Fitting, socle | C5×C7×C307 | C3 |
| LCS | G ▷ C7×C307 | |
| UCS | C1 ◁ C5 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C5×C7×C307 | C3 | semidirect |
| 2. #20264516 | C5 | direct (N × H) |
| 4. C7×C307 | C3×C5 | semidirect |
| 5. C5×C307 | ΓL(1,𝔽8) | semidirect |
| 7. C307 | C5×ΓL(1,𝔽8) | semidirect |
| 9. C5×C7 | C307⋊C3 | semidirect |
| 12. C7 | C5×C307⋊C3 | semidirect |
| 13. C5 | #20264516 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C5×C7×C307 | 3 |
| #20264516 | 5 |
| C5×C307⋊C3 | 7 |
| C5×ΓL(1,𝔽8) | 307 |