Finite Group DatabasebetaFinite Group DatabasebetaOrder 16768.
564 conjugacy classes of subgroups; only the 151 normal subgroups are shown.
This group has 564 conjugacy classes of subgroups — too many to draw. Only its 151 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 | #393×C131 | C2 |
| prime radical | C22×C22⋊D131 | C22 |
| hypercenter | #393 | D131 |
| hypocenter | C131 | #2563 |
| LCS | G ▷ C23×C131 ▷ C131 | |
| UCS | C1 ◁ C23 ◁ #393 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#16862244 | C2 | semidirect |
| 3. #22553225 | C2 | semidirect |
| 4. #22553224 | C2 | semidirect |
| 5. #22553223 | C2 | semidirect |
| 7. #22553176 | C2 | semidirect |
| 8. #22553171 | C2 | semidirect |
| 9. #22553170 | C2 | semidirect |
| 10. #22553163 | C2 | semidirect |
| 11. #393×C131 | C2 | semidirect |
| 42. C2×C22⋊D131 | Q8 | semidirect |
| 141. C131 | #2563 | semidirect |
| 148. #393 | D131 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#16862244 | 2 |
| C2×(C22⋊D131)⋊C4 | 2 |
| #22553225 | 2 |
| #22553224 | 2 |
| #22553223 | 2 |
| #22553212 | 2 |
| #22553176 | 2 |
| #22553171 | 2 |
| #22553170 | 2 |
| #22553163 | 2 |
| #393×C131 | 2 |
| #2563 | 131 |