Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
1800 conjugacy classes of subgroups; only the 776 normal subgroups are shown.
This group has 1800 conjugacy classes of subgroups — too many to draw. Only its 776 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C24×#6157 |
| commutator, socle | C22×C53 | C24×C4 |
| Fitting | C53×#564 | C4 |
| prime radical, cosocle | C22×D53 | C25 |
| hypercenter | #564 | #6157 |
| hypocenter | C53 | C4×#564 |
| LCS | G ▷ C22×C53 ▷ C53 | |
| UCS | C1 ◁ C22 ◁ #564 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 9. C53⋊#564 | C4 | semidirect |
| 11. C53×#564 | C4 | semidirect |
| 46. #6157 | #564 | direct (N × H) |
| 56. #564 | #6157 | direct (N × H) |
| 63. C53 | C4×#564 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #20291261 | 2 |
| #125×#6157 | 2 |
| #564×D53 | 2 |
| C4×#564 | 53 |