Finite Group DatabasebetaFinite Group DatabasebetaOrder 13504.
240 conjugacy classes of subgroups; only the 93 normal subgroups are shown.
This group has 240 conjugacy classes of subgroups — too many to draw. Only its 93 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 | #123×C211 | C2 |
| hypercenter | #123 | D211 |
| hypocenter | C211 | #540 |
| LCS | G ▷ C22×C211 ▷ C211 | |
| UCS | C1 ◁ C22 ◁ #123 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #20269973 | C2 | semidirect |
| 2. #20269967 | C2 | semidirect |
| 3. #20269962 | C2 | semidirect |
| 4. #20269961 | C2 | semidirect |
| 5. #20269959 | C2 | semidirect |
| 6. #20269952 | C2 | semidirect |
| 7. #20269951 | C2 | semidirect |
| 8. #20269950 | C2 | semidirect |
| 9. #20269950 | C2 | semidirect |
| 10. #20269950 | C2 | semidirect |
| 11. #20269949 | C2 | semidirect |
| 12. C4×C22⋊D211 | C2 | semidirect |
| 13. (C22⋊D211)⋊C4 | C2 | semidirect |
| 14. #20269935 | C2 | semidirect |
| 15. #123×C211 | C2 | semidirect |
| 115. C211 | #540 | semidirect |
| 122. #123 | D211 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #20269973 | 2 |
| #20269967 | 2 |
| #20269962 | 2 |
| #20269961 | 2 |
| #20269959 | 2 |
| #20269952 | 2 |
| #20269951 | 2 |
| #20269950 | 2 |
| #20269949 | 2 |
| C4×C22⋊D211 | 2 |
| (C22⋊D211)⋊C4 | 2 |
| #20269935 | 2 |
| #123×C211 | 2 |
| #540 | 211 |