Finite Group DatabasebetaFinite Group DatabasebetaOrder 16768.
240 conjugacy classes of subgroups; only the 112 normal subgroups are shown.
This group has 240 conjugacy classes of subgroups — too many to draw. Only its 112 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C131 | C2×C22⋊C4 |
| commutator | C2×C4 | C22×C4×C131 |
| Frattini, cosocle | C42 | C23×C131 |
| prime radical | C23×C131 | C22⋊C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C131 ◁ C2×C42×C131 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 4. #332×C131 | C2 | semidirect |
| 72. C131 | #1546 | direct (N × H) |
| 73. #1546 | C131 | direct (N × H) |
| 77. #332 | C2×C131 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C4⋊Q8×C131 | 2 |
| #431×C131 | 2 |
| #423×C131 | 2 |
| #332×C131 | 2 |
| #1546 | 131 |