Finite Group DatabasebetaFinite Group DatabasebetaOrder 22784.
274 conjugacy classes of subgroups; only the 70 normal subgroups are shown.
This group has 274 conjugacy classes of subgroups — too many to draw. Only its 70 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 | #1616 |
| commutator, Frattini, cosocle | C2×M16 | C23×C89 |
| prime radical | C23×C89 | AGL(1,ℤ/8) |
| LCS | G ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C89 ◁ C2×C4×C89 ◁ C2×M16×C89 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 97. #12308 | C89 | direct (N × H) |
| 106. C89 | #12308 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C89×#3255 | 2 |
| C89×#2239 | 2 |
| C89×#2102 | 2 |
| C89×#1340 | 2 |
| C89×#1327 | 2 |
| #12308 | 89 |