Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
5990 conjugacy classes of subgroups; only the 504 normal subgroups are shown.
This group has 5990 conjugacy classes of subgroups — too many to draw. Only its 504 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, prime radical | C26×C53 | C4 |
| cosocle | C22×Dic53 | C24 |
| hypercenter | C26 | #6157 |
| hypocenter | C53 | #20579 |
| LCS | G ▷ C23×C53 ▷ C53 | |
| UCS | C1 ◁ C23 ◁ C26 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#16811593 | C2 | semidirect |
| 2. #20290537 | C2 | semidirect |
| 4. C22×#11719940 | C22 | semidirect |
| 6. C2×C22⋊#88324 | C22 | semidirect |
| 14. C22×#88324 | C23 | semidirect |
| 39. C24×C53 | M16 | semidirect |
| 92. C22×C53 | #407 | semidirect |
| 114. C53 | #20579 | semidirect |
| 140. C24 | #11719940 | semidirect |
| 193. C22 | #16811593 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#16811593 | 2 |
| #20290537 | 2 |
| C25×Dic53 | 2 |
| #20579 | 53 |