Finite Group DatabasebetaFinite Group DatabasebetaOrder 3392.
196 conjugacy classes of subgroups; only the 91 normal subgroups are shown.
This group has 196 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 |
| prime radical | C2×C22⋊D53 | C22 |
| hypercenter | #126 | D53 |
| hypocenter | C53 | #563 |
| 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. #13133217 | C2 | semidirect |
| 2. #13133216 | C2 | semidirect |
| 3. #13133212 | C2 | semidirect |
| 4. #13133211 | C2 | semidirect |
| 5. #13133210 | C2 | semidirect |
| 6. #13133208 | C2 | semidirect |
| 7. #13133207 | C2 | semidirect |
| 11. #13133197 | C2 | semidirect |
| 12. #13133195 | C2 | semidirect |
| 13. #13133194 | C2 | semidirect |
| 14. #13133194 | C2 | semidirect |
| 15. #126×C53 | C2 | semidirect |
| 98. C53 | #563 | semidirect |
| 104. #126 | D53 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #13133217 | 2 |
| #13133216 | 2 |
| #13133212 | 2 |
| #13133211 | 2 |
| #13133210 | 2 |
| #13133208 | 2 |
| #13133207 | 2 |
| #13133203 | 2 |
| #13133201 | 2 |
| #13133199 | 2 |
| #13133197 | 2 |
| #13133195 | 2 |
| #13133194 | 2 |
| #126×C53 | 2 |
| #563 | 53 |