Finite Group DatabasebetaFinite Group DatabasebetaOrder 17728.
216 conjugacy classes of subgroups; only the 95 normal subgroups are shown.
This group has 216 conjugacy classes of subgroups — too many to draw. Only its 95 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, prime radical | #121×C277 | C2 |
| hypercenter | #121 | D277 |
| hypocenter | C277 | #542 |
| 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. #22579879 | C2 | semidirect |
| 2. #22579873 | C2 | semidirect |
| 3. #22579867 | C2 | semidirect |
| 4. #22579867 | C2 | semidirect |
| 5. #22579861 | C2 | semidirect |
| 6. #22579859 | C2 | semidirect |
| 7. #22579857 | C2 | semidirect |
| 8. #22579845 | C2 | semidirect |
| 9. D4×Dic277 | C2 | semidirect |
| 10. #22579839 | C2 | semidirect |
| 73. C277 | #542 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #22579879 | 2 |
| #22579873 | 2 |
| #22579867 | 2 |
| #22579861 | 2 |
| #22579859 | 2 |
| #22579857 | 2 |
| #22579845 | 2 |
| D4×Dic277 | 2 |
| #22579839 | 2 |
| #121×C277 | 2 |
| #542 | 277 |