Finite Group DatabasebetaFinite Group DatabasebetaOrder 16768.
924 conjugacy classes of subgroups; only the 189 normal subgroups are shown.
This group has 924 conjugacy classes of subgroups — too many to draw. Only its 189 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 | #390×C131 | C2 |
| hypercenter | #390 | D131 |
| hypocenter | C131 | #2856 |
| LCS | G ▷ C23×C131 ▷ C131 | |
| UCS | C1 ◁ C23 ◁ #390 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#16862242 | C2 | semidirect |
| 2. C2×#16862221 | C2 | semidirect |
| 3. C2×#16862219 | C2 | semidirect |
| 4. C2×#16862217 | C2 | semidirect |
| 5. C2×C42×D131 | C2 | semidirect |
| 6. #22553214 | C2 | semidirect |
| 7. #22553212 | C2 | semidirect |
| 8. #22553181 | C2 | semidirect |
| 9. #390×C131 | C2 | semidirect |
| 51. C2×C4×D131 | D4 | semidirect |
| 52. C2×C4×D131 | D4 | semidirect |
| 59. C4×Dic131 | D4 | semidirect |
| 128. C2×D131 | #127 | semidirect |
| 131. C4×C131 | #121 | semidirect |
| 134. Dic131 | #121 | semidirect |
| 143. C131 | #2856 | semidirect |
| 152. #390 | D131 | semidirect |
| 278. C4 | #16862219 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#16862242 | 2 |
| C2×#16862221 | 2 |
| C2×#16862219 | 2 |
| C2×#16862217 | 2 |
| C2×C42×D131 | 2 |
| #22553214 | 2 |
| #22553212 | 2 |
| #22553181 | 2 |
| #390×C131 | 2 |
| #2856 | 131 |