Finite Group DatabasebetaFinite Group DatabasebetaOrder 16768.
948 conjugacy classes of subgroups; only the 171 normal subgroups are shown.
This group has 948 conjugacy classes of subgroups — too many to draw. Only its 171 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 | #392×C131 | C2 |
| hypercenter | #392 | D131 |
| hypocenter | C131 | #2541 |
| LCS | G ▷ C23×C131 ▷ C131 | |
| UCS | C1 ◁ C23 ◁ #392 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#16862242 | C2 | semidirect |
| 2. C2×#16862242 | C2 | semidirect |
| 3. C2×#16862219 | C2 | semidirect |
| 4. C2×#16862218 | C2 | semidirect |
| 5. C2×(C22⋊D131)⋊C4 | C2 | semidirect |
| 6. #22553222 | C2 | semidirect |
| 7. #22553221 | C2 | semidirect |
| 8. #22553220 | C2 | semidirect |
| 9. #22553212 | C2 | semidirect |
| 10. #22553163 | C2 | semidirect |
| 11. #392×C131 | C2 | semidirect |
| 61. C2×C22⋊D131 | D4 | semidirect |
| 71. Dic131⋊C4 | D4 | semidirect |
| 77. C4×Dic131 | D4 | semidirect |
| 177. Dic131 | #121 | semidirect |
| 178. Dic131 | #127 | semidirect |
| 188. C131 | #2541 | semidirect |
| 198. #392 | D131 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#16862242 | 2 |
| C2×#16862219 | 2 |
| C2×#16862218 | 2 |
| C2×(C22⋊D131)⋊C4 | 2 |
| #22553222 | 2 |
| #22553221 | 2 |
| #22553220 | 2 |
| #22553212 | 2 |
| #22553163 | 2 |
| #392×C131 | 2 |
| #2541 | 131 |