Finite Group DatabasebetaFinite Group DatabasebetaOrder 22784.
846 conjugacy classes of subgroups; only the 166 normal subgroups are shown.
This group has 846 conjugacy classes of subgroups — too many to draw. Only its 166 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 | C2×C22⋊C4 |
| commutator | C2×C4 | C22×C8×C89 |
| Frattini, cosocle | C2×C42 | C23×C89 |
| prime radical | C23×D4×C89 | C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C89 ◁ C22×C42×C89 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C89×#1817 | C2 | semidirect |
| 3. C89×#1270 | C2 | semidirect |
| 13. #336×C89 | C22 | semidirect |
| 19. C22×D4×C89 | C8 | semidirect |
| 137. #7592 | C89 | direct (N × H) |
| 146. #1817 | C2×C89 | semidirect |
| 147. #1270 | C2×C89 | semidirect |
| 148. C89 | #7592 | direct (N × H) |
| 158. #336 | C22×C89 | semidirect |
| 164. C22×D4 | C8×C89 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#390×C89 | 2 |
| C89×#1817 | 2 |
| C89×#1270 | 2 |
| #7592 | 89 |