Finite Group DatabasebetaFinite Group DatabasebetaOrder 22784.
210 conjugacy classes of subgroups; only the 30 normal subgroups are shown.
This group has 210 conjugacy classes of subgroups — too many to draw. Only its 30 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 | #1402 |
| commutator | #127 | C2×C4×C89 |
| Frattini, cosocle | #493 | C22×C89 |
| prime radical | C89×#1619 | C2 |
| LCS | G ▷ #127 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C89 ◁ C22×C89 ◁ C2×C4×C89 ◁ C42×C89 ◁ #493×C89 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C89×#1408 | C2 | semidirect |
| 3. C89×#1357 | C2 | semidirect |
| 4. D42×C89 | C4 | semidirect |
| 9. #331×C89 | C4 | semidirect |
| 83. #7515 | C89 | direct (N × H) |
| 90. #1408 | C2×C89 | semidirect |
| 91. #1357 | C2×C89 | semidirect |
| 92. C89 | #7515 | direct (N × H) |
| 93. D42 | C4×C89 | semidirect |
| 98. #331 | C4×C89 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C89×#1619 | 2 |
| C89×#1408 | 2 |
| C89×#1357 | 2 |
| #7515 | 89 |