Finite Group DatabasebetaFinite Group DatabasebetaOrder 22784.
234 conjugacy classes of subgroups; only the 74 normal subgroups are shown.
This group has 234 conjugacy classes of subgroups — too many to draw. Only its 74 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 | #1667 |
| commutator, Frattini, cosocle | C4×C8 | C23×C89 |
| 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 |
|---|---|---|
| 1. C89×#2246 | C2 | semidirect |
| 2. C89×#2213 | C2 | semidirect |
| 3. C89×#2170 | C2 | semidirect |
| 4. C89×#1373 | C2 | semidirect |
| 5. C89×#1357 | C2 | semidirect |
| 80. #12436 | C89 | direct (N × H) |
| 85. #2246 | C2×C89 | semidirect |
| 86. #2213 | C2×C89 | semidirect |
| 87. #2170 | C2×C89 | semidirect |
| 88. #1373 | C2×C89 | semidirect |
| 89. #1357 | C2×C89 | semidirect |
| 90. C89 | #12436 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C89×#2246 | 2 |
| C89×#2213 | 2 |
| C89×#2170 | 2 |
| C89×#1373 | 2 |
| C89×#1357 | 2 |
| #12436 | 89 |