Finite Group DatabasebetaFinite Group DatabasebetaOrder 13504.
252 conjugacy classes of subgroups; only the 91 normal subgroups are shown.
This group has 252 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 |
| hypercenter | #126 | D211 |
| hypocenter | C211 | #560 |
| 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. #20269974 | C2 | semidirect |
| 2. #20269971 | C2 | semidirect |
| 3. #20269962 | C2 | semidirect |
| 4. #20269952 | C2 | semidirect |
| 5. #20269950 | C2 | semidirect |
| 6. #20269947 | C2 | semidirect |
| 7. #126×C211 | C2 | semidirect |
| 11. C2×D844 | C22 | semidirect |
| 13. C2×C4×D211 | C22 | semidirect |
| 14. Dic211⋊C4 | C22 | semidirect |
| 17. (C2×D211)⋊C4 | C22 | semidirect |
| 18. C4⋊C4×C211 | C22 | semidirect |
| 20. C42×C211 | C22 | semidirect |
| 50. C211 | #560 | semidirect |
| 54. #126 | D211 | semidirect |
| 66. C4⋊C4 | C2×D211 | semidirect |
| 71. C42 | C2×D211 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #20269974 | 2 |
| #20269971 | 2 |
| #20269962 | 2 |
| #20269952 | 2 |
| #20269950 | 2 |
| #20269947 | 2 |
| #126×C211 | 2 |
| #560 | 211 |