Finite Group DatabasebetaFinite Group DatabasebetaOrder 16768.
1224 conjugacy classes of subgroups; only the 237 normal subgroups are shown.
This group has 1224 conjugacy classes of subgroups — too many to draw. Only its 237 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 | C24×C4×C131 | C2 |
| hypercenter | C24×C4 | D131 |
| hypocenter | C131 | #2847 |
| LCS | G ▷ C23×C131 ▷ C131 | |
| UCS | C1 ◁ C23 ◁ C24×C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#16862220 | C2 | semidirect |
| 2. C2×#16862215 | C2 | semidirect |
| 3. #22553191 | C2 | semidirect |
| 4. #22553172 | C2 | semidirect |
| 5. C24×C4×C131 | C2 | semidirect |
| 6. C22×D524 | C22 | semidirect |
| 10. C2×C4⋊Dic131 | C22 | semidirect |
| 12. C2×(C2×D131)⋊C4 | C22 | semidirect |
| 13. C131⋊#95 | C22 | semidirect |
| 29. C24×C131 | D4 | semidirect |
| 35. C22×C4×C131 | D4 | semidirect |
| 65. C22×C131 | #121 | semidirect |
| 73. C4×C131 | #120 | semidirect |
| 80. C131 | #2847 | semidirect |
| 82. C24×C4 | D131 | semidirect |
| 102. C24 | D524 | semidirect |
| 113. C22×C4 | C22⋊D131 | semidirect |
| 149. C22 | #16862220 | semidirect |
| 155. C4 | #16862215 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#16862220 | 2 |
| C2×#16862215 | 2 |
| #22553191 | 2 |
| #22553172 | 2 |
| C24×C4×C131 | 2 |
| #2847 | 131 |