Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
1634 conjugacy classes of subgroups; only the 272 normal subgroups are shown.
This group has 1634 conjugacy classes of subgroups — too many to draw. Only its 272 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 | C53×#395 | C4 |
| prime radical, cosocle | C23×D53 | C24 |
| hypercenter | #395 | #6157 |
| hypocenter | C53 | #14302 |
| LCS | G ▷ C23×C53 ▷ C53 | |
| UCS | C1 ◁ C23 ◁ #395 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 21. #16810544 | C4 | semidirect |
| 22. #16810543 | C4 | semidirect |
| 23. #16810542 | C4 | semidirect |
| 25. C53×#395 | C4 | semidirect |
| 197. #395 | #6157 | semidirect |
| 206. C53 | #14302 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×Q8×#6157 | 2 |
| C2×#16811572 | 2 |
| #20289905 | 2 |
| #20289901 | 2 |
| #20289900 | 2 |
| #20289890 | 2 |
| #395×D53 | 2 |
| #14302 | 53 |