Finite Group DatabasebetaFinite Group DatabasebetaOrder 17728.
292 conjugacy classes of subgroups; only the 105 normal subgroups are shown.
This group has 292 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 | C22⋊Q8×C277 | C2 |
| hypercenter | C22⋊Q8 | D277 |
| hypocenter | C277 | #547 |
| LCS | G ▷ C22×C277 ▷ C277 | |
| UCS | C1 ◁ C22 ◁ C22⋊Q8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#17984564 | C2 | semidirect |
| 2. C2×#17984563 | C2 | semidirect |
| 3. #22579878 | C2 | semidirect |
| 4. #22579865 | C2 | semidirect |
| 5. #22579862 | C2 | semidirect |
| 6. #22579860 | C2 | semidirect |
| 7. #22579853 | C2 | semidirect |
| 8. #22579842 | C2 | semidirect |
| 9. C4×C22⋊D277 | C2 | semidirect |
| 10. C4⋊C4×D277 | C2 | semidirect |
| 11. C22⋊Q8×C277 | C2 | semidirect |
| 41. D1108 | D4 | semidirect |
| 76. Dic277 | #41 | semidirect |
| 83. C277 | #547 | semidirect |
| 91. C22⋊Q8 | D277 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#17984564 | 2 |
| C2×#17984563 | 2 |
| #22579878 | 2 |
| #22579865 | 2 |
| #22579862 | 2 |
| #22579860 | 2 |
| #22579853 | 2 |
| #22579842 | 2 |
| C4×C22⋊D277 | 2 |
| C4⋊C4×D277 | 2 |
| C22⋊Q8×C277 | 2 |
| #547 | 277 |