Finite Group DatabasebetaFinite Group DatabasebetaOrder 22784.
234 conjugacy classes of subgroups; only the 98 normal subgroups are shown.
This group has 234 conjugacy classes of subgroups — too many to draw. Only its 98 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C89 | #102 |
| commutator | C2×C8 | C42×C89 |
| Frattini, cosocle | C2×C4×C8 | C22×C89 |
| socle | C22×C89 | #340 |
| prime radical | C23×C89 | #102 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C89 ◁ C2×C42×C89 ◁ C2×C4×C8×C89 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 8. C4⋊C16×C89 | C4 | semidirect |
| 12. #335×C89 | C4 | semidirect |
| 79. #7299 | C89 | direct (N × H) |
| 87. C89 | #7299 | direct (N × H) |
| 92. C4⋊C16 | C4×C89 | semidirect |
| 96. #335 | C4×C89 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C89×#2162 | 2 |
| C4×#106×C89 | 2 |
| C89×#1568 | 2 |
| #7299 | 89 |