Finite Group DatabasebetaFinite Group DatabasebetaOrder 13504.
196 conjugacy classes of subgroups; only the 91 normal subgroups are shown.
This group has 196 conjugacy classes of subgroups — too many to draw. Only its 91 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×D211 |
| commutator, socle, cosocle | C22×C211 | C24 |
| Fitting | #126×C211 | C2 |
| prime radical | C2×C22⋊D211 | C22 |
| hypercenter | #126 | D211 |
| hypocenter | C211 | #563 |
| LCS | G ▷ C22×C211 ▷ C211 | |
| UCS | C1 ◁ C22 ◁ #126 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #20269977 | C2 | semidirect |
| 2. #20269976 | C2 | semidirect |
| 3. #20269972 | C2 | semidirect |
| 4. #20269971 | C2 | semidirect |
| 5. #20269970 | C2 | semidirect |
| 6. #20269968 | C2 | semidirect |
| 7. #20269967 | C2 | semidirect |
| 11. #20269957 | C2 | semidirect |
| 12. #20269955 | C2 | semidirect |
| 13. #20269954 | C2 | semidirect |
| 14. #20269954 | C2 | semidirect |
| 15. #126×C211 | C2 | semidirect |
| 97. C211 | #563 | semidirect |
| 104. #126 | D211 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #20269977 | 2 |
| #20269976 | 2 |
| #20269972 | 2 |
| #20269971 | 2 |
| #20269970 | 2 |
| #20269968 | 2 |
| #20269967 | 2 |
| #20269963 | 2 |
| #20269961 | 2 |
| #20269959 | 2 |
| #20269957 | 2 |
| #20269955 | 2 |
| #20269954 | 2 |
| #126×C211 | 2 |
| #563 | 211 |