Finite Group DatabasebetaFinite Group DatabasebetaOrder 16768.
732 conjugacy classes of subgroups; only the 169 normal subgroups are shown.
This group has 732 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 | C2×C4×D4×C131 | C2 |
| hypercenter | C2×C4×D4 | D131 |
| hypocenter | C131 | #2535 |
| LCS | G ▷ C23×C131 ▷ C131 | |
| UCS | C1 ◁ C23 ◁ C2×C4×D4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#16862217 | C2 | semidirect |
| 2. C2×C4×D524 | C2 | semidirect |
| 3. #22553237 | C2 | semidirect |
| 4. #22553221 | C2 | semidirect |
| 5. #22553191 | C2 | semidirect |
| 6. #22553178 | C2 | semidirect |
| 7. #22553169 | C2 | semidirect |
| 8. C2×C4×D4×C131 | C2 | semidirect |
| 47. C2×D4×C131 | D4 | semidirect |
| 56. C42×C131 | D4 | semidirect |
| 120. C131 | #2535 | semidirect |
| 124. C2×C4×D4 | D131 | semidirect |
| 158. C2×D4 | D524 | semidirect |
| 177. C42 | C22⋊D131 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#16862217 | 2 |
| C2×C4×D524 | 2 |
| #22553237 | 2 |
| #22553221 | 2 |
| #22553191 | 2 |
| #22553178 | 2 |
| #22553169 | 2 |
| C2×C4×D4×C131 | 2 |
| #2535 | 131 |