Finite Group DatabasebetaFinite Group DatabasebetaOrder 16768.
540 conjugacy classes of subgroups; only the 189 normal subgroups are shown.
This group has 540 conjugacy classes of subgroups — too many to draw. Only its 189 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, prime radical, cosocle | C23×C131 | C24 |
| Fitting | C2×C4⋊Q8×C131 | C2 |
| hypercenter | C2×C4⋊Q8 | D131 |
| hypocenter | C131 | #2546 |
| LCS | G ▷ C23×C131 ▷ C131 | |
| UCS | C1 ◁ C23 ◁ C2×C4⋊Q8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 31. C4⋊Dic131 | Q8 | semidirect |
| 75. C4×C131 | C4⋊Q8 | semidirect |
| 84. C131 | #2546 | semidirect |
| 163. C4 | #16862244 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#16862244 | 2 |
| C2×C4×Dic262 | 2 |
| #22553243 | 2 |
| #22553219 | 2 |
| #22553208 | 2 |
| C4×C4⋊Dic131 | 2 |
| C2×C4⋊Q8×C131 | 2 |
| #2546 | 131 |