Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
1422 conjugacy classes of subgroups; only the 360 normal subgroups are shown.
This group has 1422 conjugacy classes of subgroups — too many to draw. Only its 360 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×#6157 |
| commutator, socle | C23×C53 | C23×C4 |
| Fitting | C43×C53 | C4 |
| prime radical, cosocle | C23×D53 | C24 |
| hypercenter | C43 | #6157 |
| hypocenter | C53 | #14689 |
| LCS | G ▷ C23×C53 ▷ C53 | |
| UCS | C1 ◁ C23 ◁ C43 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 12. C42×Dic53 | C4 | semidirect |
| 13. C42×Dic53 | C4 | semidirect |
| 14. C42×Dic53 | C4 | semidirect |
| 15. C43×C53 | C4 | semidirect |
| 130. C43 | #6157 | semidirect |
| 131. C53 | #14689 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#16811567 | 2 |
| #20289893 | 2 |
| #20289891 | 2 |
| C43×D53 | 2 |
| #14689 | 53 |