Finite Group DatabasebetaFinite Group DatabasebetaOrder 16768.
1056 conjugacy classes of subgroups; only the 185 normal subgroups are shown.
This group has 1056 conjugacy classes of subgroups — too many to draw. Only its 185 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 | C22×#41×C131 | C2 |
| hypercenter | C22×#41 | D131 |
| hypocenter | C131 | #2404 |
| LCS | G ▷ C23×C131 ▷ C131 | |
| UCS | C1 ◁ C23 ◁ C22×#41 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#16862214 | C2 | semidirect |
| 2. C22×(C2×D131)⋊C4 | C2 | semidirect |
| 3. #22553223 | C2 | semidirect |
| 4. #22553220 | C2 | semidirect |
| 5. #22553192 | C2 | semidirect |
| 6. #22553191 | C2 | semidirect |
| 7. #22553175 | C2 | semidirect |
| 8. C22×#41×C131 | C2 | semidirect |
| 14. C2×C22⋊Dic131 | C22 | semidirect |
| 19. C131⋊#95 | C22 | semidirect |
| 20. C131⋊#95 | C22 | semidirect |
| 21. C131⋊#95 | C22 | semidirect |
| 24. C22×Q8×C131 | C22 | semidirect |
| 26. C22×D4×C131 | C22 | semidirect |
| 28. C23×C4×C131 | C22 | semidirect |
| 55. C2×D4×C131 | D4 | semidirect |
| 60. C22×C4×C131 | D4 | semidirect |
| 72. C22×D131 | #41 | semidirect |
| 133. C131 | #2404 | semidirect |
| 135. C22×#41 | D131 | semidirect |
| 146. C22×Q8 | C2×D131 | semidirect |
| 149. C22×D4 | C2×D131 | semidirect |
| 152. C23×C4 | C2×D131 | semidirect |
| 180. C2×D4 | C22⋊D131 | semidirect |
| 189. C22×C4 | C22⋊D131 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#16862214 | 2 |
| C22×(C2×D131)⋊C4 | 2 |
| #22553223 | 2 |
| #22553220 | 2 |
| #22553192 | 2 |
| #22553191 | 2 |
| #22553175 | 2 |
| C22×#41×C131 | 2 |
| #2404 | 131 |