Finite Group DatabasebetaFinite Group DatabasebetaOrder 22784.
894 conjugacy classes of subgroups; only the 210 normal subgroups are shown.
This group has 894 conjugacy classes of subgroups — too many to draw. Only its 210 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C8×C89 | #120 |
| commutator | C23 | C22×C8×C89 |
| Frattini, cosocle | C22×C8 | C23×C89 |
| socle | C2×C89 | #1779 |
| prime radical | #585×C89 | C4 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8×C89 ◁ C22×C8×C89 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C89×#2115 | C2 | semidirect |
| 3. C89×#2113 | C2 | semidirect |
| 9. C2×M32×C89 | C22 | semidirect |
| 10. C22×C16×C89 | C22 | semidirect |
| 83. #12684 | C89 | direct (N × H) |
| 88. #2115 | C2×C89 | semidirect |
| 89. #2113 | C2×C89 | semidirect |
| 90. C89 | #12684 | direct (N × H) |
| 96. C2×M32 | C22×C89 | semidirect |
| 97. C22×C16 | C22×C89 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C89×#3573 | 2 |
| C89×#2115 | 2 |
| C89×#2113 | 2 |
| #12684 | 89 |