Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
10894 conjugacy classes of subgroups; only the 3815 normal subgroups are shown.
This group has 10894 conjugacy classes of subgroups — too many to draw. Only its 3815 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2 | C24×D212 |
| commutator, cosocle | C4×C53 | C26 |
| Frattini | C4 | C25×D53 |
| Fitting | C53×#3573 | C2 |
| socle | C2×C53 | C24×D4 |
| hypercenter | #3573 | D53 |
| hypocenter | C53 | #63091 |
| LCS | G ▷ C4×C53 ▷ C2×C53 ▷ C53 | |
| UCS | C1 ◁ C2 ◁ #585 ◁ #3573 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #20289526 | C2 | semidirect |
| 2. #20289496 | C2 | semidirect |
| 3. #20289489 | C2 | semidirect |
| 4. C2×#16811428 | C2 | semidirect |
| 5. C53×#3573 | C2 | semidirect |
| 7. #16811459 | C22 | semidirect |
| 8. C2×#13133263 | C22 | semidirect |
| 10. #16811440 | C22 | semidirect |
| 11. #16811435 | C22 | semidirect |
| 12. #16811428 | C22 | semidirect |
| 13. #16811428 | C22 | semidirect |
| 14. #16811423 | C22 | semidirect |
| 15. C2×#13133245 | C22 | semidirect |
| 16. C2×#13133239 | C22 | semidirect |
| 17. C22×D424 | C22 | semidirect |
| 19. C53×#568 | C22 | semidirect |
| 20. C2×#131×C53 | C22 | semidirect |
| 22. #13133264 | C23 | semidirect |
| 24. #13133263 | C23 | semidirect |
| 27. C2×#11719945 | C23 | semidirect |
| 29. C22×D212 | C23 | semidirect |
| 31. #13133250 | C23 | semidirect |
| 32. #13133245 | C23 | semidirect |
| 33. #13133239 | C23 | semidirect |
| 34. #13133239 | C23 | semidirect |
| 35. C2×Dic212 | C23 | semidirect |
| 36. C2×#11719913 | C23 | semidirect |
| 37. C2×D424 | C23 | semidirect |
| 38. C2×D424 | C23 | semidirect |
| 42. #131×C53 | C23 | semidirect |
| 43. #131×C53 | C23 | semidirect |
| 44. C2×M16×C53 | C23 | semidirect |
| 45. C22×C8×C53 | C23 | semidirect |
| 99. #3573 | D53 | semidirect |
| 116. #568 | C2×D53 | semidirect |
| 117. C2×#131 | C2×D53 | semidirect |
| 118. C53 | #63091 | semidirect |
| 140. #131 | C22×D53 | semidirect |
| 141. #131 | C22×D53 | semidirect |
| 142. C2×M16 | C22×D53 | semidirect |
| 143. C22×C8 | C22×D53 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #20289526 | 2 |
| #20289496 | 2 |
| #20289489 | 2 |
| C2×#16811428 | 2 |
| C53×#3573 | 2 |
| #63091 | 53 |