Finite Group DatabasebetaFinite Group DatabasebetaOrder 22784.
384 conjugacy classes of subgroups; only the 86 normal subgroups are shown.
This group has 384 conjugacy classes of subgroups — too many to draw. Only its 86 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 | C89×#3287 | C2 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C89 ◁ C42×C89 ◁ C8×D4×C89 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C89×#2248 | C2 | semidirect |
| 3. C89×#2209 | C2 | semidirect |
| 4. C89×#1669 | C2 | semidirect |
| 5. C89×#1361 | C2 | semidirect |
| 6. C89×#1332 | C2 | semidirect |
| 7. C89×D4⋊C16 | C2 | semidirect |
| 140. #12321 | C89 | direct (N × H) |
| 147. #2248 | C2×C89 | semidirect |
| 148. #2209 | C2×C89 | semidirect |
| 149. #1669 | C2×C89 | semidirect |
| 150. #1361 | C2×C89 | semidirect |
| 151. #1332 | C2×C89 | semidirect |
| 152. D4⋊C16 | C2×C89 | semidirect |
| 153. C89 | #12321 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C89×#3287 | 2 |
| C89×#2248 | 2 |
| C89×#2209 | 2 |
| C89×#1669 | 2 |
| C89×#1361 | 2 |
| C89×#1332 | 2 |
| C89×D4⋊C16 | 2 |
| #12321 | 89 |