Finite Group DatabasebetaFinite Group DatabasebetaOrder 16768.
752 conjugacy classes of subgroups; only the 264 normal subgroups are shown.
This group has 752 conjugacy classes of subgroups — too many to draw. Only its 264 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C131 | C24 |
| commutator, Frattini, cosocle | C23 | C24×C131 |
| prime radical | C2×#127×C131 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C131 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×C4⋊Q8×C131 | C2 | semidirect |
| 3. C2×#124×C131 | C2 | semidirect |
| 4. #390×C131 | C2 | semidirect |
| 5. C43×C131 | C2 | semidirect |
| 32. C42×C131 | D4 | semidirect |
| 56. C4×C131 | #124 | semidirect |
| 62. C131 | #2858 | direct (N × H) |
| 63. #2858 | C131 | direct (N × H) |
| 64. C2×C4⋊Q8 | C2×C131 | semidirect |
| 66. C2×#124 | C2×C131 | semidirect |
| 67. #390 | C2×C131 | semidirect |
| 68. C43 | C2×C131 | semidirect |
| 95. C42 | D4×C131 | semidirect |
| 119. C4 | #124×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 |
| C2×#127×C131 | 2 |
| C2×#124×C131 | 2 |
| #390×C131 | 2 |
| C43×C131 | 2 |
| #2858 | 131 |