Finite Group DatabasebetaFinite Group DatabasebetaOrder 22144.
256 conjugacy classes of subgroups; only the 72 normal subgroups are shown.
This group has 256 conjugacy classes of subgroups — too many to draw. Only its 72 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C173 | #120 |
| commutator, Frattini, cosocle | C42 | C23×C173 |
| prime radical | #550×C173 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C173 ◁ C42×C173 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #492×C173 | C2 | semidirect |
| 3. #477×C173 | C2 | semidirect |
| 4. #476×C173 | C2 | semidirect |
| 5. #331×C173 | C2 | semidirect |
| 6. Q8⋊C8×C173 | C2 | semidirect |
| 7. D4⋊C8×C173 | C2 | semidirect |
| 54. Q8×C173 | SD16 | semidirect |
| 98. C173 | #1631 | direct (N × H) |
| 99. #1631 | C173 | direct (N × H) |
| 101. #492 | C2×C173 | semidirect |
| 102. #477 | C2×C173 | semidirect |
| 103. #476 | C2×C173 | semidirect |
| 104. #331 | C2×C173 | semidirect |
| 105. Q8⋊C8 | C2×C173 | semidirect |
| 106. D4⋊C8 | C2×C173 | semidirect |
| 153. Q8 | SD16×C173 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #550×C173 | 2 |
| #492×C173 | 2 |
| #477×C173 | 2 |
| #476×C173 | 2 |
| #331×C173 | 2 |
| Q8⋊C8×C173 | 2 |
| D4⋊C8×C173 | 2 |
| #1631 | 173 |