Finite Group DatabasebetaFinite Group DatabasebetaOrder 22784.
270 conjugacy classes of subgroups; only the 66 normal subgroups are shown.
This group has 270 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 | C4×C89 | AΣL(1,𝔽16) |
| commutator | C2×D4 | C42×C89 |
| Frattini, cosocle | #532 | C22×C89 |
| socle | C2×C89 | #1390 |
| prime radical | #532×C89 | C22 |
| LCS | G ▷ C2×D4 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C89 ◁ C2×C4×C89 ◁ C22×C4×C89 ◁ #532×C89 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 11. #355×C89 | C4 | semidirect |
| 14. #331×C89 | C4 | semidirect |
| 111. #7488 | C89 | direct (N × H) |
| 120. C89 | #7488 | direct (N × H) |
| 128. #355 | C4×C89 | semidirect |
| 131. #331 | C4×C89 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C89×#2132 | 2 |
| C89×#1884 | 2 |
| C89×#1543 | 2 |
| #7488 | 89 |