Finite Group DatabasebetaFinite Group DatabasebetaOrder 16768.
950 conjugacy classes of subgroups; only the 449 normal subgroups are shown.
This group has 950 conjugacy classes of subgroups — too many to draw. Only its 449 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 | C24×D131 |
| commutator, socle, cosocle | C22×C131 | C25 |
| Fitting | #564×C131 | C2 |
| hypercenter | #564 | D131 |
| hypocenter | C131 | #3567 |
| LCS | G ▷ C22×C131 ▷ C131 | |
| UCS | C1 ◁ C22 ◁ #564 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #22554060 | C2 | semidirect |
| 2. #22554015 | C2 | semidirect |
| 3. #22553994 | C2 | semidirect |
| 4. #22553918 | C2 | semidirect |
| 5. #22553910 | C2 | semidirect |
| 6. #22553847 | C2 | semidirect |
| 7. #564×C131 | C2 | semidirect |
| 99. C131 | #3567 | semidirect |
| 101. #564 | D131 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #22554060 | 2 |
| #22554015 | 2 |
| #22553994 | 2 |
| #22553918 | 2 |
| #22553910 | 2 |
| #22553847 | 2 |
| #564×C131 | 2 |
| #3567 | 131 |