Finite Group DatabasebetaFinite Group DatabasebetaOrder 3392.
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×D53 |
| commutator, socle, cosocle | C22×C53 | C24 |
| Fitting | #126×C53 | C2 |
| hypercenter | #126 | D53 |
| hypocenter | C53 | #560 |
| LCS | G ▷ C22×C53 ▷ C53 | |
| UCS | C1 ◁ C22 ◁ #126 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #13133214 | C2 | semidirect |
| 2. #13133211 | C2 | semidirect |
| 3. #13133202 | C2 | semidirect |
| 4. #13133192 | C2 | semidirect |
| 5. #13133190 | C2 | semidirect |
| 6. #13133187 | C2 | semidirect |
| 7. #126×C53 | C2 | semidirect |
| 12. C2×D212 | C22 | semidirect |
| 13. C2×C4×D53 | C22 | semidirect |
| 14. C4⋊C4×C53 | C22 | semidirect |
| 16. C42×C53 | C22 | semidirect |
| 19. #11719921 | C22 | semidirect |
| 20. #11719919 | C22 | semidirect |
| 51. C53 | #560 | semidirect |
| 54. #126 | D53 | semidirect |
| 66. C4⋊C4 | C2×D53 | semidirect |
| 71. C42 | C2×D53 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #13133214 | 2 |
| #13133211 | 2 |
| #13133202 | 2 |
| #13133192 | 2 |
| #13133190 | 2 |
| #13133187 | 2 |
| #126×C53 | 2 |
| #560 | 53 |