Finite Group DatabasebetaFinite Group DatabasebetaOrder 22144.
578 conjugacy classes of subgroups; only the 112 normal subgroups are shown.
This group has 578 conjugacy classes of subgroups — too many to draw. Only its 112 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 | C23 | C22×C4×C173 |
| Frattini, cosocle | C24 | C23×C173 |
| prime radical | C2×#120×C173 | C2 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C173 ◁ C24×C173 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #518×C173 | C2 | semidirect |
| 3. C2×#99×C173 | C2 | semidirect |
| 4. #379×C173 | C2 | semidirect |
| 5. #342×C173 | C2 | semidirect |
| 10. #120×C173 | C4 | semidirect |
| 25. C24×C173 | C2×C4 | semidirect |
| 38. C22×C4×C173 | D4 | semidirect |
| 48. C22⋊C4×C173 | C2×C4 | semidirect |
| 66. C23×C173 | C22⋊C4 | semidirect |
| 84. C2×C4×C173 | C22⋊C4 | semidirect |
| 119. C173 | #1889 | direct (N × H) |
| 120. #1889 | C173 | direct (N × H) |
| 122. #518 | C2×C173 | semidirect |
| 123. C2×#99 | C2×C173 | semidirect |
| 124. #379 | C2×C173 | semidirect |
| 125. #342 | C2×C173 | semidirect |
| 130. #120 | C4×C173 | semidirect |
| 145. C24 | C2×C4×C173 | semidirect |
| 158. C22×C4 | D4×C173 | semidirect |
| 168. C22⋊C4 | C2×C4×C173 | semidirect |
| 186. C23 | C22⋊C4×C173 | semidirect |
| 204. C2×C4 | C22⋊C4×C173 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#120×C173 | 2 |
| #518×C173 | 2 |
| C2×#99×C173 | 2 |
| #379×C173 | 2 |
| #342×C173 | 2 |
| #1889 | 173 |