Finite Group DatabasebetaFinite Group DatabasebetaOrder 22784.
710 conjugacy classes of subgroups; only the 214 normal subgroups are shown.
This group has 710 conjugacy classes of subgroups — too many to draw. Only its 214 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 | C42 | C22×C4×C89 |
| Frattini, cosocle | C2×C42 | C23×C89 |
| prime radical | C2×#493×C89 | C2 |
| LCS | G ▷ 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 |
|---|---|---|
| 2. C89×#1957 | C2 | semidirect |
| 3. C89×#1844 | C2 | semidirect |
| 4. C2×#322×C89 | C2 | semidirect |
| 8. #493×C89 | C4 | semidirect |
| 77. C8×C89 | #102 | semidirect |
| 92. #9837 | C89 | direct (N × H) |
| 99. #1957 | C2×C89 | semidirect |
| 100. #1844 | C2×C89 | semidirect |
| 101. C2×#322 | C2×C89 | semidirect |
| 102. C89 | #9837 | direct (N × H) |
| 106. #493 | C4×C89 | semidirect |
| 175. C8 | #102×C89 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#493×C89 | 2 |
| C89×#1957 | 2 |
| C89×#1844 | 2 |
| C2×#322×C89 | 2 |
| #9837 | 89 |