Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
2070 conjugacy classes of subgroups; only the 501 normal subgroups are shown.
This group has 2070 conjugacy classes of subgroups — too many to draw. Only its 501 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 | C24×D53 |
| commutator, socle, cosocle | C23×C53 | C25 |
| Fitting | C53×#2840 | C2 |
| hypercenter | #2840 | D53 |
| hypocenter | C53 | #36675 |
| LCS | G ▷ C23×C53 ▷ C53 | |
| UCS | C1 ◁ C23 ◁ #2840 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #20282906 | C2 | semidirect |
| 2. #20282799 | C2 | semidirect |
| 3. #20282712 | C2 | semidirect |
| 4. #20277997 | C2 | semidirect |
| 5. #20277932 | C2 | semidirect |
| 6. #20277214 | C2 | semidirect |
| 7. C53×#2840 | C2 | semidirect |
| 8. C22×#11719949 | C22 | semidirect |
| 11. C2×Q8×Dic53 | C22 | semidirect |
| 13. #16810565 | C22 | semidirect |
| 14. #16810550 | C22 | semidirect |
| 16. #16810542 | C22 | semidirect |
| 17. #16810541 | C22 | semidirect |
| 19. #16810534 | C22 | semidirect |
| 24. #16810470 | C22 | semidirect |
| 25. #16810468 | C22 | semidirect |
| 57. C24×C53 | #41 | semidirect |
| 126. #2840 | D53 | semidirect |
| 159. C53 | #36675 | semidirect |
| 189. C24 | #11719949 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #20282906 | 2 |
| #20282799 | 2 |
| #20282712 | 2 |
| #20277997 | 2 |
| #20277932 | 2 |
| #20277214 | 2 |
| C53×#2840 | 2 |
| #36675 | 53 |