Finite Group DatabasebetaFinite Group DatabasebetaOrder 22784.
334 conjugacy classes of subgroups; only the 66 normal subgroups are shown.
This group has 334 conjugacy classes of subgroups — too many to draw. Only its 66 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C89 | #1390 |
| commutator | C2×D4 | C42×C89 |
| Frattini, cosocle | C2×#127 | C22×C89 |
| prime radical | C2×#127×C89 | C22 |
| LCS | G ▷ C2×D4 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C89 ◁ C23×C89 ◁ C22×C4×C89 ◁ C2×#127×C89 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 5. #518×C89 | C4 | semidirect |
| 8. C2×#100×C89 | C4 | semidirect |
| 11. #353×C89 | C4 | semidirect |
| 14. #331×C89 | C4 | semidirect |
| 126. #7493 | C89 | direct (N × H) |
| 138. C89 | #7493 | direct (N × H) |
| 140. #518 | C4×C89 | semidirect |
| 143. C2×#100 | C4×C89 | semidirect |
| 146. #353 | C4×C89 | semidirect |
| 149. #331 | C4×C89 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#353×C89 | 2 |
| C89×#1887 | 2 |
| C89×#1555 | 2 |
| #7493 | 89 |