Finite Group DatabasebetaFinite Group DatabasebetaOrder 3392.
214 conjugacy classes of subgroups; only the 97 normal subgroups are shown.
This group has 214 conjugacy classes of subgroups — too many to draw. Only its 97 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 | #123×C53 | C2 |
| prime radical | C2×D4×C53 | C22 |
| hypercenter | #123 | D53 |
| hypocenter | C53 | #552 |
| LCS | G ▷ C22×C53 ▷ C53 | |
| UCS | C1 ◁ C22 ◁ #123 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #13133210 | C2 | semidirect |
| 2. #13133210 | C2 | semidirect |
| 3. #13133208 | C2 | semidirect |
| 4. #13133207 | C2 | semidirect |
| 5. #13133207 | C2 | semidirect |
| 7. #13133194 | C2 | semidirect |
| 8. #13133194 | C2 | semidirect |
| 9. #13133182 | C2 | semidirect |
| 11. #13133176 | C2 | semidirect |
| 12. #13133173 | C2 | semidirect |
| 13. #13133173 | C2 | semidirect |
| 14. C2×#11719919 | C2 | semidirect |
| 98. C53 | #552 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #13133210 | 2 |
| #13133208 | 2 |
| #13133207 | 2 |
| #13133198 | 2 |
| #13133194 | 2 |
| #13133182 | 2 |
| D4×Dic53 | 2 |
| #13133176 | 2 |
| #13133173 | 2 |
| C2×#11719919 | 2 |
| #123×C53 | 2 |
| #552 | 53 |