Finite Group DatabasebetaFinite Group DatabasebetaOrder 22784.
414 conjugacy classes of subgroups; only the 90 normal subgroups are shown.
This group has 414 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 | C23×C4 | C23×C89 |
| prime radical | C24×C89 | C2×D4 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C89 ◁ C23×C4×C89 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 26. C2×C4⋊C4×C89 | Q8 | semidirect |
| 121. #10387 | C89 | direct (N × H) |
| 133. C89 | #10387 | direct (N × H) |
| 154. C2×C4⋊C4 | Q8×C89 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C89×#2607 | 2 |
| C89×#1826 | 2 |
| C89×#1820 | 2 |
| C89×#1306 | 2 |
| C89×#1284 | 2 |
| #10387 | 89 |