Finite Group DatabasebetaFinite Group DatabasebetaOrder 17728.
312 conjugacy classes of subgroups; only the 105 normal subgroups are shown.
This group has 312 conjugacy classes of subgroups — too many to draw. Only its 105 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 | #121×C277 | C2 |
| hypercenter | #121 | D277 |
| hypocenter | C277 | #550 |
| LCS | G ▷ C22×C277 ▷ C277 | |
| UCS | C1 ◁ C22 ◁ #121 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#17984563 | C2 | semidirect |
| 2. C2×#17984562 | C2 | semidirect |
| 3. #22579876 | C2 | semidirect |
| 4. #22579876 | C2 | semidirect |
| 5. #22579855 | C2 | semidirect |
| 6. #22579853 | C2 | semidirect |
| 7. #22579852 | C2 | semidirect |
| 8. #22579845 | C2 | semidirect |
| 9. C4×C22⋊D277 | C2 | semidirect |
| 10. #22579840 | C2 | semidirect |
| 11. #121×C277 | C2 | semidirect |
| 39. Dic554 | D4 | semidirect |
| 81. Dic277 | #41 | semidirect |
| 89. C277 | #550 | semidirect |
| 98. #121 | D277 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#17984563 | 2 |
| C2×#17984562 | 2 |
| #22579876 | 2 |
| #22579855 | 2 |
| #22579853 | 2 |
| #22579852 | 2 |
| #22579845 | 2 |
| C4×C22⋊D277 | 2 |
| #22579840 | 2 |
| #121×C277 | 2 |
| #550 | 277 |