Finite Group DatabasebetaFinite Group DatabasebetaOrder 22784.
406 conjugacy classes of subgroups; only the 138 normal subgroups are shown.
This group has 406 conjugacy classes of subgroups — too many to draw. Only its 138 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 | #124 |
| commutator | C22×C4 | C22×C4×C89 |
| Frattini, cosocle | C22×C8 | C23×C89 |
| socle | C2×C89 | #1955 |
| prime radical | #575×C89 | C4 |
| LCS | G ▷ C22×C4 ▷ 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×#2151 | C2 | semidirect |
| 3. C2×#350×C89 | C2 | semidirect |
| 4. C89×#2115 | C2 | semidirect |
| 5. C89×#2110 | C2 | semidirect |
| 107. #13199 | C89 | direct (N × H) |
| 113. #2151 | C2×C89 | semidirect |
| 114. C2×#350 | C2×C89 | semidirect |
| 115. #2115 | C2×C89 | semidirect |
| 116. #2110 | C2×C89 | semidirect |
| 117. C89 | #13199 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C89×#2955 | 2 |
| C89×#2151 | 2 |
| C2×#350×C89 | 2 |
| C89×#2115 | 2 |
| C89×#2110 | 2 |
| #13199 | 89 |