Finite Group DatabasebetaFinite Group DatabasebetaOrder 16768.
584 conjugacy classes of subgroups; only the 136 normal subgroups are shown.
This group has 584 conjugacy classes of subgroups — too many to draw. Only its 136 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 | C23 | C22×C4×C131 |
| Frattini, cosocle | C22×C4 | C23×C131 |
| prime radical | C22×#41×C131 | C2 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C131 ◁ C23×C4×C131 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×#101×C131 | C2 | semidirect |
| 3. C2×#100×C131 | C2 | semidirect |
| 4. #407×C131 | C2 | semidirect |
| 86. C131 | #1785 | direct (N × H) |
| 87. #1785 | C131 | direct (N × H) |
| 89. C2×#101 | C2×C131 | semidirect |
| 90. C2×#100 | C2×C131 | semidirect |
| 91. #407 | C2×C131 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×#41×C131 | 2 |
| C2×#101×C131 | 2 |
| C2×#100×C131 | 2 |
| #407×C131 | 2 |
| #1785 | 131 |