Finite Group DatabasebetaFinite Group DatabasebetaOrder 22784.
854 conjugacy classes of subgroups; only the 186 normal subgroups are shown.
This group has 854 conjugacy classes of subgroups — too many to draw. Only its 186 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C89 | #2840 |
| commutator, Frattini, cosocle | C24 | C24×C89 |
| prime radical | C23×D4×C89 | C22 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C89 ◁ C24×C89 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C89×#2032 | C2 | semidirect |
| 2. C89×#2029 | C2 | semidirect |
| 5. #543×C89 | C22 | semidirect |
| 8. #342×C89 | C22 | semidirect |
| 9. C25×C89 | Q8 | semidirect |
| 12. C22×D4×C89 | Q8 | semidirect |
| 14. C23×C4×C89 | Q8 | semidirect |
| 74. #32873 | C89 | direct (N × H) |
| 81. #2032 | C2×C89 | semidirect |
| 82. #2029 | C2×C89 | semidirect |
| 84. C89 | #32873 | direct (N × H) |
| 86. #543 | C22×C89 | semidirect |
| 89. #342 | C22×C89 | semidirect |
| 90. C25 | Q8×C89 | semidirect |
| 93. C22×D4 | Q8×C89 | semidirect |
| 95. C23×C4 | Q8×C89 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C89×#2032 | 2 |
| C89×#2029 | 2 |
| C89×#1781 | 2 |
| #32873 | 89 |