Finite Group DatabasebetaFinite Group DatabasebetaOrder 22784.
296 conjugacy classes of subgroups; only the 42 normal subgroups are shown.
This group has 296 conjugacy classes of subgroups — too many to draw. Only its 42 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C4×C89 | #353 |
| commutator | C2×M16 | C2×C4×C89 |
| Frattini, cosocle | #407 | C22×C89 |
| socle | C2×C89 | #1343 |
| prime radical | C89×#2010 | C2 |
| LCS | G ▷ C2×M16 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C89 ◁ C2×C4×C89 ◁ C22×C4×C89 ◁ #407×C89 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C89×#1320 | C2 | semidirect |
| 3. C89×#1297 | C2 | semidirect |
| 4. #575×C89 | C4 | semidirect |
| 10. H(3,ℤ/4)×C89 | C4 | semidirect |
| 111. #7396 | C89 | direct (N × H) |
| 118. #1320 | C2×C89 | semidirect |
| 119. #1297 | C2×C89 | semidirect |
| 120. C89 | #7396 | direct (N × H) |
| 121. #575 | C4×C89 | semidirect |
| 127. H(3,ℤ/4) | C4×C89 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C89×#2010 | 2 |
| C89×#1320 | 2 |
| C89×#1297 | 2 |
| #7396 | 89 |