Finite Group DatabasebetaFinite Group DatabasebetaOrder 22784.
600 conjugacy classes of subgroups; only the 226 normal subgroups are shown.
This group has 600 conjugacy classes of subgroups — too many to draw. Only its 226 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C89 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C89 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C89 ◁ C43×C89 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 51. C42×C89 | Q16 | semidirect |
| 76. #32862 | C89 | direct (N × H) |
| 85. C89 | #32862 | direct (N × H) |
| 131. C42 | Q16×C89 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C89×#2863 | 2 |
| C89×#2087 | 2 |
| C89×#2068 | 2 |
| C89×#2056 | 2 |
| C89×#1842 | 2 |
| #32862 | 89 |