Finite Group DatabasebetaFinite Group DatabasebetaOrder 16768.
296 conjugacy classes of subgroups; only the 100 normal subgroups are shown.
This group has 296 conjugacy classes of subgroups — too many to draw. Only its 100 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C131 | #381 |
| commutator | C23 | C22×C4×C131 |
| Frattini, cosocle | C24 | C23×C131 |
| prime radical | C22×D4×C131 | C22 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C131 ◁ C24×C131 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. #342×C131 | C2 | semidirect |
| 6. #117×C131 | C4 | semidirect |
| 7. C2×C22⋊C4×C131 | C4 | semidirect |
| 58. C131 | #1828 | direct (N × H) |
| 59. #1828 | C131 | direct (N × H) |
| 62. #342 | C2×C131 | semidirect |
| 65. #117 | C4×C131 | semidirect |
| 66. C2×C22⋊C4 | C4×C131 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #518×C131 | 2 |
| C2×#99×C131 | 2 |
| #342×C131 | 2 |
| #1828 | 131 |