Finite Group DatabasebetaFinite Group DatabasebetaOrder 3392.
236 conjugacy classes of subgroups; only the 97 normal subgroups are shown.
This group has 236 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 | C4×D4×C53 | C2 |
| prime radical | #13133213 | C2 |
| hypercenter | C4×D4 | D53 |
| hypocenter | C53 | #559 |
| LCS | G ▷ C22×C53 ▷ C53 | |
| UCS | C1 ◁ C22 ◁ C4×D4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #13133205 | C2 | semidirect |
| 3. #13133204 | C2 | semidirect |
| 4. #13133203 | C2 | semidirect |
| 5. #13133200 | C2 | semidirect |
| 6. #13133199 | C2 | semidirect |
| 7. C4×Dic106 | C2 | semidirect |
| 8. #13133182 | C2 | semidirect |
| 9. #13133179 | C2 | semidirect |
| 10. C4×C22⋊D53 | C2 | semidirect |
| 11. C4×D4×C53 | C2 | semidirect |
| 76. C53 | #559 | semidirect |
| 87. C4×D4 | D53 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #13133213 | 2 |
| #13133205 | 2 |
| #13133204 | 2 |
| #13133203 | 2 |
| #13133200 | 2 |
| #13133199 | 2 |
| C4×Dic106 | 2 |
| #13133182 | 2 |
| #13133179 | 2 |
| C4×C22⋊D53 | 2 |
| C4×D4×C53 | 2 |
| #559 | 53 |