Finite Group DatabasebetaFinite Group DatabasebetaOrder 16768.
556 conjugacy classes of subgroups; only the 169 normal subgroups are shown.
This group has 556 conjugacy classes of subgroups — too many to draw. Only its 169 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 | #382×C131 | C2 |
| prime radical | C23×D131 | C23 |
| hypercenter | #382 | D131 |
| hypocenter | C131 | #2851 |
| LCS | G ▷ C23×C131 ▷ C131 | |
| UCS | C1 ◁ C23 ◁ #382 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#16862238 | C2 | semidirect |
| 3. #22553211 | C2 | semidirect |
| 4. #22553197 | C2 | semidirect |
| 7. #382×C131 | C2 | semidirect |
| 79. C2×D131 | #125 | semidirect |
| 92. C131 | #2851 | semidirect |
| 99. #382 | D131 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#16862238 | 2 |
| C2×C42×D131 | 2 |
| #22553211 | 2 |
| #22553197 | 2 |
| #22553186 | 2 |
| #22553169 | 2 |
| #382×C131 | 2 |
| #2851 | 131 |