Finite Group DatabasebetaFinite Group DatabasebetaOrder 22784.
762 conjugacy classes of subgroups; only the 90 normal subgroups are shown.
This group has 762 conjugacy classes of subgroups — too many to draw. Only its 90 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×C89 | #393 |
| commutator, Frattini, cosocle | C25 | C23×C89 |
| prime radical | C23×D4×C89 | C22 |
| LCS | G ▷ C25 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C89 ◁ C25×C89 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C89×#2840 | C2 | semidirect |
| 3. C89×#1304 | C2 | semidirect |
| 9. C25×C89 | Q8 | semidirect |
| 15. C23×C4×C89 | Q8 | semidirect |
| 88. #10390 | C89 | direct (N × H) |
| 95. #2840 | C2×C89 | semidirect |
| 97. #1304 | C2×C89 | semidirect |
| 98. C89 | #10390 | direct (N × H) |
| 104. C25 | Q8×C89 | semidirect |
| 110. C23×C4 | Q8×C89 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C89×#2840 | 2 |
| C89×#1781 | 2 |
| C89×#1304 | 2 |
| #10390 | 89 |