Finite Group DatabasebetaFinite Group DatabasebetaOrder 22784.
734 conjugacy classes of subgroups; only the 186 normal subgroups are shown.
This group has 734 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 | C23×C89 | #127 |
| commutator, Frattini, cosocle | C2×C42 | C23×C89 |
| prime radical | C2×#127×C89 | C22 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C89 ◁ C2×C42×C89 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#498×C89 | C2 | semidirect |
| 3. C89×#1990 | C2 | semidirect |
| 5. C89×#1273 | C2 | semidirect |
| 65. C2×C8×C89 | SD16 | semidirect |
| 66. C2×C8×C89 | SD16 | semidirect |
| 119. #10315 | C89 | direct (N × H) |
| 126. C2×#498 | C2×C89 | semidirect |
| 128. #1990 | C2×C89 | semidirect |
| 130. #1273 | C2×C89 | semidirect |
| 131. C89 | #10315 | direct (N × H) |
| 191. C2×C8 | SD16×C89 | semidirect |
| 192. C2×C8 | SD16×C89 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#498×C89 | 2 |
| C2×#492×C89 | 2 |
| C89×#1990 | 2 |
| C89×#1957 | 2 |
| C89×#1273 | 2 |
| #10315 | 89 |