Finite Group DatabasebetaFinite Group DatabasebetaOrder 16768.
4544 conjugacy classes of subgroups; only the 805 normal subgroups are shown.
This group has 4544 conjugacy classes of subgroups — too many to draw. Only its 805 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C24 | C22×D131 |
| commutator, cosocle | C22×C131 | C25 |
| Frattini | C22 | C24×D131 |
| Fitting | C26×C131 | C2 |
| socle | C24×C131 | C23 |
| hypercenter | C26 | D131 |
| hypocenter | C131 | C22×#120 |
| LCS | G ▷ C22×C131 ▷ C131 | |
| UCS | C1 ◁ C24 ◁ C26 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C23×C22⋊D131 | C2 | semidirect |
| 2. C2×#16862215 | C2 | direct (N × H) |
| 3. C22×C22⋊Dic131 | C2 | semidirect |
| 4. C26×C131 | C2 | semidirect |
| 5. C24×D131 | C22 | semidirect |
| 7. C22×C22⋊D131 | C22 | semidirect |
| 8. C23×Dic131 | C22 | semidirect |
| 9. #16862215 | C22 | direct (N × H) |
| 10. C2×C22⋊Dic131 | C22 | semidirect |
| 13. C25×C131 | C22 | semidirect |
| 15. C23×D131 | C23 | semidirect |
| 17. C2×C22⋊D131 | C23 | semidirect |
| 19. C22×Dic131 | C23 | semidirect |
| 20. C22⋊Dic131 | C23 | semidirect |
| 24. C24×C131 | D4 | semidirect |
| 27. C24×C131 | C23 | semidirect |
| 30. C22×D131 | C24 | semidirect |
| 33. C2×Dic131 | C24 | semidirect |
| 39. C23×C131 | C2×D4 | semidirect |
| 45. C22×C131 | #120 | semidirect |
| 47. C22×C131 | C22×D4 | semidirect |
| 53. C2×C131 | C2×#120 | semidirect |
| 56. C131 | C22×#120 | semidirect |
| 58. C26 | D131 | semidirect |
| 65. C25 | C2×D131 | semidirect |
| 76. C24 | C22⋊D131 | semidirect |
| 79. C24 | C22×D131 | semidirect |
| 93. C23 | C2×C22⋊D131 | semidirect |
| 104. C22 | #16862215 | direct (N × H) |
| 106. C22 | C22×C22⋊D131 | semidirect |
| 111. C2 | C2×#16862215 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×C22⋊D131 | 2 |
| C2×#16862215 | 2 |
| C22×C22⋊Dic131 | 2 |
| C26×C131 | 2 |
| C22×#120 | 131 |