Finite Group DatabasebetaFinite Group DatabasebetaOrder 22784.
304 conjugacy classes of subgroups; only the 90 normal subgroups are shown.
This group has 304 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 | #447 |
| commutator, Frattini, cosocle | C4×C8 | C23×C89 |
| prime radical | #493×C89 | C22 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C89 ◁ C42×C89 ◁ C8×Q8×C89 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. Q8×Q16×C89 | C2 | semidirect |
| 5. C89×#1365 | C2 | semidirect |
| 6. C89×Q8⋊C16 | C2 | semidirect |
| 7. C89×#1332 | C2 | semidirect |
| 61. Q16×C89 | SD16 | semidirect |
| 90. Q8×C89 | SD32 | semidirect |
| 123. #12354 | C89 | direct (N × H) |
| 128. Q8×Q16 | C2×C89 | semidirect |
| 132. #1365 | C2×C89 | semidirect |
| 133. Q8⋊C16 | C2×C89 | semidirect |
| 134. #1332 | C2×C89 | semidirect |
| 135. C89 | #12354 | direct (N × H) |
| 189. Q16 | SD16×C89 | semidirect |
| 218. Q8 | SD32×C89 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| Q8×Q16×C89 | 2 |
| C89×#2240 | 2 |
| C89×#2207 | 2 |
| C89×#1668 | 2 |
| C89×#1365 | 2 |
| C89×Q8⋊C16 | 2 |
| C89×#1332 | 2 |
| #12354 | 89 |