Finite Group DatabasebetaFinite Group DatabasebetaOrder 3392.
216 conjugacy classes of subgroups; only the 93 normal subgroups are shown.
This group has 216 conjugacy classes of subgroups — too many to draw. Only its 93 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 | #13133189 | C2 |
| hypercenter | #123 | D53 |
| hypocenter | C53 | #542 |
| 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. #13133216 | C2 | semidirect |
| 2. #13133212 | C2 | semidirect |
| 3. #13133210 | C2 | semidirect |
| 4. #13133204 | C2 | semidirect |
| 5. #13133203 | C2 | semidirect |
| 6. #13133203 | C2 | semidirect |
| 7. #13133199 | C2 | semidirect |
| 8. #13133196 | C2 | semidirect |
| 9. #13133195 | C2 | semidirect |
| 10. #13133194 | C2 | semidirect |
| 12. #13133182 | C2 | semidirect |
| 13. C4×C22⋊D53 | C2 | semidirect |
| 14. #13133173 | C2 | semidirect |
| 15. #123×C53 | C2 | semidirect |
| 104. C53 | #542 | semidirect |
| 112. #123 | D53 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #13133216 | 2 |
| #13133212 | 2 |
| #13133210 | 2 |
| #13133204 | 2 |
| #13133203 | 2 |
| #13133199 | 2 |
| #13133196 | 2 |
| #13133195 | 2 |
| #13133194 | 2 |
| #13133189 | 2 |
| #13133182 | 2 |
| C4×C22⋊D53 | 2 |
| #13133173 | 2 |
| #123×C53 | 2 |
| #542 | 53 |