Finite Group DatabasebetaFinite Group DatabasebetaOrder 22144.
840 conjugacy classes of subgroups; only the 167 normal subgroups are shown.
This group has 840 conjugacy classes of subgroups — too many to draw. Only its 167 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D173 |
| commutator, socle, cosocle | C23×C173 | C24 |
| Fitting | #392×C173 | C2 |
| hypercenter | #392 | D173 |
| hypocenter | C173 | #2397 |
| LCS | G ▷ C23×C173 ▷ C173 | |
| UCS | C1 ◁ C23 ◁ #392 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×D4×Dic173 | C2 | semidirect |
| 2. C2×#19149639 | C2 | semidirect |
| 3. #26011428 | C2 | semidirect |
| 4. #26011427 | C2 | semidirect |
| 5. #26011399 | C2 | semidirect |
| 6. #26011336 | C2 | semidirect |
| 7. #392×C173 | C2 | semidirect |
| 36. C4×Dic173 | D4 | semidirect |
| 86. C173 | #2397 | semidirect |
| 93. #392 | D173 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×D4×Dic173 | 2 |
| C2×#19149639 | 2 |
| #26011428 | 2 |
| #26011427 | 2 |
| #26011399 | 2 |
| #26011336 | 2 |
| #392×C173 | 2 |
| #2397 | 173 |