Finite Group DatabasebetaFinite Group DatabasebetaOrder 32487.
80 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C72×C13×C17 | C3 | 1 | 1 | yes | maximal, characteristic, complemented, Hall |
| 2 | C13×C17×ΓL(1,𝔽8) | — | 7 | 8 | no | maximal |
| 3 | C17×#3861 | C13 | 1 | 1 | yes | maximal, characteristic, direct product factor, Hall |
| 4 | C13×#3861 | C17 | 1 | 1 | yes | maximal, characteristic, direct product factor, Hall |
| 5 | C7×C13×C17 | ΓL(1,𝔽8) | 1 | 8 | yes | complemented |
| 6 | C72×C17 | C3×C13 | 1 | 1 | yes | characteristic, complemented, Hall |
| 7 | C3×C13×C17 | — | 49 | 1 | no | Hall |
| 8 | C72×C13 | C3×C17 | 1 | 1 | yes | characteristic, complemented, Hall |
| 9 | C17×ΓL(1,𝔽8) | — | 7 | 8 | no | — |
| 10 | C13×ΓL(1,𝔽8) | — | 7 | 8 | no | — |
| 11 | C13×C17 | #3861 | 1 | 1 | yes | characteristic, direct product factor, Hall |
| 12 | #3861 | C13×C17 | 1 | 1 | yes | characteristic, direct product factor, Hall |
| 13 | C7×C17 | C13×ΓL(1,𝔽8) | 1 | 8 | yes | complemented |
| 14 | C7×C13 | C17×ΓL(1,𝔽8) | 1 | 8 | yes | complemented |
| 15 | C3×C17 | — | 49 | 1 | no | Hall |
| 16 | C72 | C3×C13×C17 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 17 | C3×C13 | — | 49 | 1 | no | Hall |
| 18 | ΓL(1,𝔽8) | — | 7 | 8 | no | — |
| 19 | C17 | C13×#3861 | 1 | 1 | yes | characteristic, direct product factor, Sylow |
| 20 | C13 | C17×#3861 | 1 | 1 | yes | characteristic, direct product factor, Sylow |
| 21 | C7 | C13×C17×ΓL(1,𝔽8) | 1 | 8 | yes | complemented |
| 22 | C3 | — | 49 | 1 | no | Sylow |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C13×C17 | #3861 |
| commutator, cosocle, hypocenter | C72 | C3×C13×C17 |
| Frattini | C1 | G |
| Fitting, socle | C72×C13×C17 | C3 |
| LCS | G ▷ C72 | |
| UCS | C1 ◁ C13×C17 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C72×C13×C17 | C3 | semidirect |
| 3. C17×#3861 | C13 | direct (N × H) |
| 4. C13×#3861 | C17 | direct (N × H) |
| 5. C7×C13×C17 | ΓL(1,𝔽8) | semidirect |
| 6. C72×C17 | C3×C13 | semidirect |
| 8. C72×C13 | C3×C17 | semidirect |
| 11. C13×C17 | #3861 | direct (N × H) |
| 12. #3861 | C13×C17 | direct (N × H) |
| 13. C7×C17 | C13×ΓL(1,𝔽8) | semidirect |
| 14. C7×C13 | C17×ΓL(1,𝔽8) | semidirect |
| 16. C72 | C3×C13×C17 | semidirect |
| 19. C17 | C13×#3861 | direct (N × H) |
| 20. C13 | C17×#3861 | direct (N × H) |
| 21. C7 | C13×C17×ΓL(1,𝔽8) | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C72×C13×C17 | 3 |
| C13×C17×ΓL(1,𝔽8) | 7 |
| C17×#3861 | 13 |
| C13×#3861 | 17 |