Finite Group DatabasebetaFinite Group DatabasebetaOrder 22784.
374 conjugacy classes of subgroups; only the 78 normal subgroups are shown.
This group has 374 conjugacy classes of subgroups — too many to draw. Only its 78 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 | #342 |
| commutator | C22×C4 | C42×C89 |
| Frattini, cosocle | C22×C4⋊C4 | C22×C89 |
| prime radical | C24×C89 | C22⋊C4 |
| LCS | G ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C89 ◁ C24×C89 ◁ C22×C4⋊C4×C89 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 9. #324×C89 | C4 | semidirect |
| 107. #7326 | C89 | direct (N × H) |
| 118. C89 | #7326 | direct (N × H) |
| 125. #324 | C4×C89 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C89×#1463 | 2 |
| C89×#1445 | 2 |
| #7326 | 89 |