Finite Group DatabasebetaFinite Group DatabasebetaOrder 12416.
1412 conjugacy classes of subgroups; only the 545 normal subgroups are shown.
This group has 1412 conjugacy classes of subgroups — too many to draw. Only its 545 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2 | C23×C22⋊D97 |
| commutator, cosocle | C4×C97 | C25 |
| Frattini | C4 | C24×D97 |
| Fitting | #585×C97 | C2 |
| socle | C2×C97 | C23×D4 |
| hypercenter | #585 | D97 |
| hypocenter | C97 | #3586 |
| LCS | G ▷ C4×C97 ▷ C2×C97 ▷ C97 | |
| UCS | C1 ◁ C2 ◁ C2×#41 ◁ #585 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#15690857 | C2 | semidirect |
| 2. #20261550 | C2 | semidirect |
| 3. #20261546 | C2 | semidirect |
| 4. #20261533 | C2 | semidirect |
| 5. C2×#15690840 | C2 | semidirect |
| 6. #20261502 | C2 | semidirect |
| 7. #585×C97 | C2 | semidirect |
| 9. #15690857 | C22 | semidirect |
| 10. C2×#13095511 | C22 | semidirect |
| 12. C22×Dic194 | C22 | semidirect |
| 14. #15690841 | C22 | semidirect |
| 15. #15690840 | C22 | semidirect |
| 16. #15690840 | C22 | semidirect |
| 17. #15690836 | C22 | semidirect |
| 18. #15690835 | C22 | semidirect |
| 19. #15690830 | C22 | semidirect |
| 20. C2×#13095501 | C22 | semidirect |
| 21. C2×#13095499 | C22 | semidirect |
| 22. #15690817 | C22 | semidirect |
| 23. C2×#13095495 | C22 | semidirect |
| 25. 2−1+4×C97 | C22 | semidirect |
| 27. H(5,𝔽2)×C97 | C22 | semidirect |
| 113. C97 | #3586 | semidirect |
| 114. #585 | D97 | semidirect |
| 124. 2−1+4 | C2×D97 | semidirect |
| 126. H(5,𝔽2) | C2×D97 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#15690857 | 2 |
| #20261550 | 2 |
| #20261546 | 2 |
| #20261533 | 2 |
| C2×#15690840 | 2 |
| #20261502 | 2 |
| #585×C97 | 2 |
| #3586 | 97 |