Finite Group DatabasebetaFinite Group DatabasebetaOrder 13952.
1058 conjugacy classes of subgroups; only the 451 normal subgroups are shown.
This group has 1058 conjugacy classes of subgroups — too many to draw. Only its 451 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 | C24×D109 |
| commutator, socle, cosocle | C22×C109 | C25 |
| Fitting | H(3,𝔽4)×C109 | C2 |
| hypercenter | H(3,𝔽4) | D109 |
| hypocenter | C109 | #3529 |
| LCS | G ▷ C22×C109 ▷ C109 | |
| UCS | C1 ◁ C22 ◁ H(3,𝔽4) | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #20295776 | C2 | semidirect |
| 2. #20295770 | C2 | semidirect |
| 3. #20295765 | C2 | semidirect |
| 4. #20295575 | C2 | semidirect |
| 5. #20295555 | C2 | semidirect |
| 6. #20295532 | C2 | semidirect |
| 7. H(3,𝔽4)×C109 | C2 | semidirect |
| 8. C2×#13182030 | C22 | semidirect |
| 10. C23×Dic109 | C22 | semidirect |
| 11. #16812697 | C22 | semidirect |
| 13. #16812692 | C22 | semidirect |
| 14. #16812691 | C22 | semidirect |
| 15. #16812687 | C22 | semidirect |
| 16. #16812683 | C22 | semidirect |
| 17. #16812681 | C22 | semidirect |
| 19. #16812676 | C22 | semidirect |
| 20. #16812675 | C22 | semidirect |
| 21. #16812672 | C22 | semidirect |
| 24. #16812668 | C22 | semidirect |
| 25. Q8×Dic109 | C22 | semidirect |
| 27. #16812659 | C22 | semidirect |
| 28. #16812655 | C22 | semidirect |
| 29. C42×D109 | C22 | semidirect |
| 98. C109 | #3529 | semidirect |
| 101. H(3,𝔽4) | D109 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #20295776 | 2 |
| #20295770 | 2 |
| #20295765 | 2 |
| #20295575 | 2 |
| #20295555 | 2 |
| #20295532 | 2 |
| H(3,𝔽4)×C109 | 2 |
| #3529 | 109 |