Finite Group DatabasebetaFinite Group DatabasebetaOrder 16768.
1564 conjugacy classes of subgroups; only the 197 normal subgroups are shown.
This group has 1564 conjugacy classes of subgroups — too many to draw. Only its 197 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 | #379×C131 | C2 |
| hypercenter | #379 | D131 |
| hypocenter | C131 | #2403 |
| LCS | G ▷ C23×C131 ▷ C131 | |
| UCS | C1 ◁ C23 ◁ #379 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C23×C22⋊D131 | C2 | semidirect |
| 2. C2×#16862216 | C2 | semidirect |
| 3. #22553221 | C2 | semidirect |
| 4. #22553189 | C2 | semidirect |
| 5. #379×C131 | C2 | semidirect |
| 6. C24×D131 | C22 | semidirect |
| 7. C22×D524 | C22 | semidirect |
| 11. C22×C22⋊D131 | C22 | semidirect |
| 12. C23×Dic131 | C22 | semidirect |
| 14. C2×(C2×D131)⋊C4 | C22 | semidirect |
| 15. C131⋊#95 | C22 | semidirect |
| 17. C2×C22⋊C4×C131 | C22 | semidirect |
| 18. C2×C22⋊C4×C131 | C22 | semidirect |
| 32. C2×C22⋊D131 | D4 | semidirect |
| 39. C24×C131 | D4 | semidirect |
| 53. C22×D131 | C2×D4 | semidirect |
| 60. C2×Dic131 | C2×D4 | semidirect |
| 81. C22×C131 | #120 | semidirect |
| 99. C131 | #2403 | semidirect |
| 105. #379 | D131 | semidirect |
| 116. C2×C22⋊C4 | C2×D131 | semidirect |
| 117. C2×C22⋊C4 | C2×D131 | semidirect |
| 126. C24 | D524 | semidirect |
| 187. C22 | #16862216 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×C22⋊D131 | 2 |
| C2×#16862216 | 2 |
| #22553221 | 2 |
| #22553189 | 2 |
| #379×C131 | 2 |
| #2403 | 131 |