Finite Group DatabasebetaFinite Group DatabasebetaOrder 16430.
16 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C53×ΓL(1,𝔽32) | C2 | 1 | 1 | yes | maximal, characteristic, complemented, Hall |
| 2 | C31×D53 | C5 | 1 | 1 | yes | maximal, characteristic, complemented, Hall |
| 3 | C31×C53 | C2×C5 | 1 | 1 | yes | characteristic, complemented, Hall |
| 4 | C5×D53 | — | 31 | 1 | no | maximal, Hall |
| 5 | C2×ΓL(1,𝔽32) | — | 53 | 1 | no | maximal, Hall |
| 6 | C5×C53 | — | 31 | 1 | no | Hall |
| 7 | ΓL(1,𝔽32) | D53 | 1 | 1 | yes | characteristic, direct product factor, Hall |
| 8 | D53 | ΓL(1,𝔽32) | 1 | 1 | yes | characteristic, direct product factor, Hall |
| 9 | C2×C31 | — | 53 | 1 | no | Hall |
| 10 | C53 | C2×ΓL(1,𝔽32) | 1 | 1 | yes | characteristic, complemented, Sylow |
| 11 | C31 | C5×D53 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 12 | C2×C5 | — | 1643 | 1 | no | Hall |
| 13 | C5 | — | 31 | 1 | no | Sylow |
| 14 | C2 | — | 53 | 1 | no | Sylow |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C31×C53 | C2×C5 |
| LCS | G ▷ C31×C53 | |
| UCS | C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C53×ΓL(1,𝔽32) | C2 | semidirect |
| 2. C31×D53 | C5 | semidirect |
| 3. C31×C53 | C2×C5 | semidirect |
| 7. ΓL(1,𝔽32) | D53 | direct (N × H) |
| 8. D53 | ΓL(1,𝔽32) | direct (N × H) |
| 10. C53 | C2×ΓL(1,𝔽32) | semidirect |
| 11. C31 | C5×D53 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C53×ΓL(1,𝔽32) | 2 |
| C31×D53 | 5 |
| C5×D53 | 31 |
| C2×ΓL(1,𝔽32) | 53 |