Finite Group DatabasebetaFinite Group DatabasebetaOrder 22144.
328 conjugacy classes of subgroups; only the 136 normal subgroups are shown.
This group has 328 conjugacy classes of subgroups — too many to draw. Only its 136 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C173 | C22⋊C4 |
| commutator | C22 | C22×C8×C173 |
| Frattini, cosocle | C22×C4 | C23×C173 |
| prime radical | C24×C173 | C2×C4 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C173 ◁ C23×C4×C173 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. #324×C173 | C2 | direct (N × H) |
| 22. C22×C4×C173 | C8 | semidirect |
| 53. C2×C4×C173 | C2×C8 | semidirect |
| 76. C2×C173 | #324 | direct (N × H) |
| 80. C173 | C2×#324 | direct (N × H) |
| 81. C2×#324 | C173 | direct (N × H) |
| 84. #324 | C2×C173 | direct (N × H) |
| 103. C22×C4 | C8×C173 | semidirect |
| 134. C2×C4 | C2×C8×C173 | semidirect |
| 157. C2 | #324×C173 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C4⋊C4×C173 | 2 |
| C2×C22⋊C8×C173 | 2 |
| #324×C173 | 2 |
| C2×#324 | 173 |