Finite Group DatabasebetaFinite Group DatabasebetaOrder 16768.
564 conjugacy classes of subgroups; only the 165 normal subgroups are shown.
This group has 564 conjugacy classes of subgroups — too many to draw. Only its 165 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D131 |
| commutator, socle, cosocle | C23×C131 | C24 |
| Fitting | #390×C131 | C2 |
| prime radical | C22×D4×C131 | C22 |
| hypercenter | #390 | D131 |
| hypocenter | C131 | #2563 |
| LCS | G ▷ C23×C131 ▷ C131 | |
| UCS | C1 ◁ C23 ◁ #390 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#16862245 | C2 | semidirect |
| 3. #22553224 | C2 | semidirect |
| 5. #22553178 | C2 | semidirect |
| 6. #22553177 | C2 | semidirect |
| 7. #22553170 | C2 | semidirect |
| 8. C4×Dic131⋊C4 | C2 | semidirect |
| 49. C2×D4×C131 | Q8 | semidirect |
| 108. C131 | #2563 | semidirect |
| 143. C2×D4 | Dic262 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#16862245 | 2 |
| C2×D4×Dic131 | 2 |
| #22553224 | 2 |
| #22553215 | 2 |
| #22553178 | 2 |
| #22553177 | 2 |
| #22553170 | 2 |
| C4×Dic131⋊C4 | 2 |
| #390×C131 | 2 |
| #2563 | 131 |