Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
3550 conjugacy classes of subgroups; only the 579 normal subgroups are shown.
This group has 3550 conjugacy classes of subgroups — too many to draw. Only its 579 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 |
| prime radical | C25×D53 | C22 |
| hypercenter | #2840 | D53 |
| hypocenter | C53 | C2×#2840 |
| 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. #20282968 | C2 | semidirect |
| 2. #20282967 | C2 | semidirect |
| 3. #393×D53 | C2 | semidirect |
| 5. C53×#2840 | C2 | semidirect |
| 7. C22×Q8×D53 | C22 | semidirect |
| 9. #16810536 | C22 | semidirect |
| 10. #16810535 | C22 | semidirect |
| 11. #16810534 | C22 | semidirect |
| 14. #95×D53 | C22 | semidirect |
| 15. C53×#393 | C22 | semidirect |
| 18. C24×D53 | Q8 | semidirect |
| 21. C22×Dic106 | C23 | semidirect |
| 27. #13133124 | C23 | semidirect |
| 29. C22×Q8×C53 | C23 | semidirect |
| 31. #95×C53 | C23 | semidirect |
| 33. C24×C53 | C2×Q8 | semidirect |
| 40. C23×D53 | C2×Q8 | semidirect |
| 87. #2840 | D53 | direct (N × H) |
| 94. D53 | #2840 | direct (N × H) |
| 101. #393 | C2×D53 | semidirect |
| 106. C53 | C2×#2840 | semidirect |
| 114. C22×Q8 | C22×D53 | semidirect |
| 121. #95 | C22×D53 | semidirect |
| 131. C24 | Q8×D53 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #20282968 | 2 |
| #20282967 | 2 |
| #393×D53 | 2 |
| #379×D53 | 2 |
| C53×#2840 | 2 |
| C2×#2840 | 53 |