Finite Group DatabasebetaFinite Group DatabasebetaOrder 16768.
5872 conjugacy classes of subgroups; only the 3221 normal subgroups are shown.
This group has 5872 conjugacy classes of subgroups — too many to draw. Only its 3221 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C23 | C23×D131 |
| commutator, cosocle | C2×C131 | C26 |
| Frattini | C2 | C25×D131 |
| Fitting | C22×#41×C131 | C2 |
| socle | C23×C131 | C24 |
| hypercenter | C22×#41 | D131 |
| hypocenter | C131 | C22×2−1+4 |
| LCS | G ▷ C2×C131 ▷ C131 | |
| UCS | C1 ◁ C23 ◁ C22×#41 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#16862293 | C2 | direct (N × H) |
| 2. C22×#14523513 | C2 | semidirect |
| 3. C22×#14523512 | C2 | semidirect |
| 4. C23×Dic262 | C2 | semidirect |
| 5. C22×Q8×D131 | C2 | semidirect |
| 6. C22×#41×C131 | C2 | semidirect |
| 7. #16862293 | C22 | direct (N × H) |
| 8. C2×#14523513 | C22 | semidirect |
| 9. C2×#14523512 | C22 | semidirect |
| 12. C22×Dic262 | C22 | semidirect |
| 13. C2×Q8×D131 | C22 | semidirect |
| 18. C2×#41×C131 | C22 | semidirect |
| 23. #14523513 | C23 | semidirect |
| 24. #14523512 | C23 | semidirect |
| 27. C2×Dic262 | C23 | semidirect |
| 28. Q8×D131 | C23 | semidirect |
| 35. #41×C131 | C23 | semidirect |
| 59. C22×C131 | 2−1+4 | semidirect |
| 68. C2×C131 | C2×2−1+4 | semidirect |
| 71. C131 | C22×2−1+4 | semidirect |
| 76. C22×#41 | D131 | semidirect |
| 82. C2×#41 | C2×D131 | semidirect |
| 98. #41 | C22×D131 | semidirect |
| 133. C22 | #16862293 | direct (N × H) |
| 141. C2 | C2×#16862293 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#16862293 | 2 |
| C22×#14523513 | 2 |
| C22×#14523512 | 2 |
| C23×Dic262 | 2 |
| C22×Q8×D131 | 2 |
| C22×#41×C131 | 2 |
| C22×2−1+4 | 131 |