Finite Group DatabasebetaFinite Group DatabasebetaOrder 16768.
388 conjugacy classes of subgroups; only the 216 normal subgroups are shown.
This group has 388 conjugacy classes of subgroups — too many to draw. Only its 216 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C131 | C22⋊C4 |
| commutator | C22 | C2×C42×C131 |
| Frattini, cosocle | C22×C4 | C23×C131 |
| socle | C22×C131 | C2×C22⋊C4 |
| prime radical | C24×C131 | C2×C4 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C131 ◁ C23×C4×C131 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. #343×C131 | C2 | semidirect |
| 7. C2×M16×C131 | C4 | semidirect |
| 71. C131 | #1742 | direct (N × H) |
| 72. #1742 | C131 | direct (N × H) |
| 75. #343 | C2×C131 | semidirect |
| 79. C2×M16 | C4×C131 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×M16×C131 | 2 |
| C2×#117×C131 | 2 |
| #343×C131 | 2 |
| #1742 | 131 |