Finite Group DatabasebetaFinite Group DatabasebetaOrder 22784.
1358 conjugacy classes of subgroups; only the 270 normal subgroups are shown.
This group has 1358 conjugacy classes of subgroups — too many to draw. Only its 270 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×C89 | #120 |
| commutator | C24 | C22×C4×C89 |
| Frattini, cosocle | C25 | C23×C89 |
| prime radical | C25×C89 | C23 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C89 ◁ C25×C89 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 6. #543×C89 | C4 | semidirect |
| 19. C23×C4×C89 | Q8 | semidirect |
| 30. C2×C22⋊C4×C89 | Q8 | semidirect |
| 129. #11390 | C89 | direct (N × H) |
| 142. C89 | #11390 | direct (N × H) |
| 145. #543 | C4×C89 | semidirect |
| 158. C23×C4 | Q8×C89 | semidirect |
| 169. C2×C22⋊C4 | Q8×C89 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#543×C89 | 2 |
| C2×#342×C89 | 2 |
| C89×C22⋊#95 | 2 |
| #11390 | 89 |