Finite Group DatabasebetaFinite Group DatabasebetaOrder 16768.
696 conjugacy classes of subgroups; only the 145 normal subgroups are shown.
This group has 696 conjugacy classes of subgroups — too many to draw. Only its 145 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 | #394×C131 | C2 |
| hypercenter | #394 | D131 |
| hypocenter | C131 | #2555 |
| LCS | G ▷ C23×C131 ▷ C131 | |
| UCS | C1 ◁ C23 ◁ #394 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#16862232 | C2 | semidirect |
| 2. C2×#16862218 | C2 | semidirect |
| 3. #22553232 | C2 | semidirect |
| 4. #22553221 | C2 | semidirect |
| 5. #22553213 | C2 | semidirect |
| 6. #22553201 | C2 | semidirect |
| 7. #22553189 | C2 | semidirect |
| 8. C22⋊C4×Dic131 | C2 | semidirect |
| 9. #394×C131 | C2 | semidirect |
| 52. C22⋊Dic131 | D4 | semidirect |
| 142. C131 | #2555 | semidirect |
| 147. #394 | D131 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#16862232 | 2 |
| C2×#16862218 | 2 |
| #22553232 | 2 |
| #22553221 | 2 |
| #22553213 | 2 |
| #22553201 | 2 |
| #22553189 | 2 |
| C22⋊C4×Dic131 | 2 |
| #394×C131 | 2 |
| #2555 | 131 |