Finite Group DatabasebetaFinite Group DatabasebetaOrder 17728.
312 conjugacy classes of subgroups; only the 107 normal subgroups are shown.
This group has 312 conjugacy classes of subgroups — too many to draw. Only its 107 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 | C4×D4×C277 | C2 |
| hypercenter | C4×D4 | D277 |
| hypocenter | C277 | #550 |
| LCS | G ▷ C22×C277 ▷ C277 | |
| UCS | C1 ◁ C22 ◁ C4×D4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#17984563 | C2 | semidirect |
| 2. #22579877 | C2 | semidirect |
| 3. #22579876 | C2 | semidirect |
| 4. #22579854 | C2 | semidirect |
| 5. #22579853 | C2 | semidirect |
| 6. C4×Dic554 | C2 | semidirect |
| 7. #22579842 | C2 | semidirect |
| 8. C4×D4×C277 | C2 | semidirect |
| 27. Dic554 | D4 | semidirect |
| 53. C4×C277 | #41 | semidirect |
| 61. C277 | #550 | semidirect |
| 70. C4×D4 | D277 | semidirect |
| 117. C4 | #17984563 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#17984563 | 2 |
| #22579877 | 2 |
| #22579876 | 2 |
| #22579854 | 2 |
| #22579853 | 2 |
| C4×Dic554 | 2 |
| #22579842 | 2 |
| C4×D4×C277 | 2 |
| #550 | 277 |