Finite Group DatabasebetaFinite Group DatabasebetaOrder 22784.
126 conjugacy classes of subgroups; only the 62 normal subgroups are shown.
This group has 126 conjugacy classes of subgroups — too many to draw. Only its 62 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 | D16 |
| commutator | C16 | C2×C8×C89 |
| Frattini, cosocle | C4×C16 | C22×C89 |
| socle, prime radical | C22×C89 | #357 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C89 ◁ C42×C89 ◁ C4×C8×C89 ◁ C4×C16×C89 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 14. Q32×C89 | C8 | semidirect |
| 54. #7384 | C89 | direct (N × H) |
| 61. C89 | #7384 | direct (N × H) |
| 72. Q32 | C8×C89 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C4×Q32×C89 | 2 |
| C4×C32×C89 | 2 |
| C89×#1371 | 2 |
| #7384 | 89 |