Finite Group DatabasebetaFinite Group DatabasebetaOrder 22784.
126 conjugacy classes of subgroups; only the 54 normal subgroups are shown.
This group has 126 conjugacy classes of subgroups — too many to draw. Only its 54 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C8×C89 | #99 |
| commutator | C2×C4 | C2×C16×C89 |
| Frattini, cosocle | C2×M32 | C22×C89 |
| socle | C2×C89 | #1322 |
| prime radical | C23×C89 | M32 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8×C89 ◁ C2×C8×C89 ◁ C2×M32×C89 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 51. #7364 | C89 | direct (N × H) |
| 57. C89 | #7364 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C89×#2150 | 2 |
| C89×#1400 | 2 |
| #7364 | 89 |