Finite Group DatabasebetaFinite Group DatabasebetaOrder 1216.
220 conjugacy classes of subgroups; only the 91 normal subgroups are shown.
This group has 220 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×D19 |
| commutator, socle, cosocle | C22×C19 | C24 |
| Fitting | C19×#123 | C2 |
| hypercenter | #123 | D19 |
| hypocenter | C19 | #562 |
| LCS | G ▷ C22×C19 ▷ C19 | |
| UCS | C1 ◁ C22 ◁ #123 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C19×#123 | C2 | semidirect |
| 2. #10597385 | C2 | semidirect |
| 3. #10597384 | C2 | semidirect |
| 4. #10597376 | C2 | semidirect |
| 5. #10597370 | C2 | semidirect |
| 6. #10597343 | C2 | semidirect |
| 7. #10597342 | C2 | semidirect |
| 8. #10597340 | C2 | semidirect |
| 9. #10597336 | C2 | semidirect |
| 10. #10597332 | C2 | semidirect |
| 11. #10597331 | C2 | semidirect |
| 12. #10597331 | C2 | semidirect |
| 13. #10597330 | C2 | semidirect |
| 14. #10597329 | C2 | semidirect |
| 15. #10597324 | C2 | semidirect |
| 116. #123 | D19 | semidirect |
| 125. C19 | #562 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C19×#123 | 2 |
| #10597385 | 2 |
| #10597384 | 2 |
| #10597376 | 2 |
| #10597370 | 2 |
| #10597343 | 2 |
| #10597342 | 2 |
| #10597340 | 2 |
| #10597336 | 2 |
| #10597332 | 2 |
| #10597331 | 2 |
| #10597330 | 2 |
| #10597329 | 2 |
| #10597324 | 2 |
| #562 | 19 |