Finite Group DatabasebetaFinite Group DatabasebetaOrder 16768.
476 conjugacy classes of subgroups; only the 344 normal subgroups are shown.
This group has 476 conjugacy classes of subgroups — too many to draw. Only its 344 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 | C22×D4 |
| commutator | C4 | C23×C4×C131 |
| Frattini, cosocle | C2×C4 | C24×C131 |
| prime radical | C2×#41×C131 | C22 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C131 ◁ C2×#41×C131 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. #428×C131 | C2 | semidirect |
| 4. #427×C131 | C2 | semidirect |
| 5. C2×#106×C131 | C2 | semidirect |
| 7. #131×C131 | C4 | semidirect |
| 24. M16×C131 | C2×C4 | semidirect |
| 25. C2×C8×C131 | C2×C4 | semidirect |
| 56. C131 | #2912 | direct (N × H) |
| 57. #2912 | C131 | direct (N × H) |
| 60. #428 | C2×C131 | semidirect |
| 61. #427 | C2×C131 | semidirect |
| 62. C2×#106 | C2×C131 | semidirect |
| 64. #131 | C4×C131 | semidirect |
| 81. M16 | C2×C4×C131 | semidirect |
| 82. C2×C8 | C2×C4×C131 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#131×C131 | 2 |
| #520×C131 | 2 |
| #428×C131 | 2 |
| #427×C131 | 2 |
| C2×#106×C131 | 2 |
| #2912 | 131 |