Finite Group DatabasebetaFinite Group DatabasebetaOrder 17728.
270 conjugacy classes of subgroups; only the 103 normal subgroups are shown.
This group has 270 conjugacy classes of subgroups — too many to draw. Only its 103 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×D277 |
| commutator, socle, cosocle | C22×C277 | C24 |
| Fitting | #123×C277 | C2 |
| prime radical | C2×#17984562 | C2 |
| hypercenter | #123 | D277 |
| hypocenter | C277 | #536 |
| LCS | G ▷ C22×C277 ▷ C277 | |
| UCS | C1 ◁ C22 ◁ #123 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C22×Dic554 | C2 | semidirect |
| 3. #22579879 | C2 | semidirect |
| 4. #22579866 | C2 | semidirect |
| 5. #22579864 | C2 | semidirect |
| 6. #22579863 | C2 | semidirect |
| 7. #22579861 | C2 | semidirect |
| 8. #22579858 | C2 | semidirect |
| 9. #22579857 | C2 | semidirect |
| 10. #22579836 | C2 | semidirect |
| 11. #123×C277 | C2 | semidirect |
| 79. C277 | #536 | semidirect |
| 88. #123 | D277 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#17984562 | 2 |
| C22×Dic554 | 2 |
| #22579879 | 2 |
| #22579866 | 2 |
| #22579864 | 2 |
| #22579863 | 2 |
| #22579861 | 2 |
| #22579858 | 2 |
| #22579857 | 2 |
| #22579836 | 2 |
| #123×C277 | 2 |
| #536 | 277 |