Finite Group DatabasebetaFinite Group DatabasebetaOrder 22784.
372 conjugacy classes of subgroups; only the 78 normal subgroups are shown.
This group has 372 conjugacy classes of subgroups — too many to draw. Only its 78 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C89 | #1619 |
| commutator, Frattini, cosocle | C4×C8 | C23×C89 |
| prime radical | C89×#3292 | C2 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C89 ◁ C2×C4×C89 ◁ C4×C8×C89 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C89×#2214 | C2 | semidirect |
| 3. C89×#1709 | C2 | semidirect |
| 4. C89×#1357 | C2 | semidirect |
| 5. C89×#1335 | C2 | semidirect |
| 8. #494×C89 | C22 | semidirect |
| 14. #364×C89 | C22 | semidirect |
| 16. C82×C89 | C22 | semidirect |
| 88. #12312 | C89 | direct (N × H) |
| 94. #2214 | C2×C89 | semidirect |
| 95. #1709 | C2×C89 | semidirect |
| 96. #1357 | C2×C89 | semidirect |
| 97. #1335 | C2×C89 | semidirect |
| 98. C89 | #12312 | direct (N × H) |
| 101. #494 | C22×C89 | semidirect |
| 107. #364 | C22×C89 | semidirect |
| 109. C82 | C22×C89 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C89×#3292 | 2 |
| C89×#2214 | 2 |
| C89×#1709 | 2 |
| C89×#1357 | 2 |
| C89×#1335 | 2 |
| #12312 | 89 |