Finite Group DatabasebetaFinite Group DatabasebetaOrder 22144.
944 conjugacy classes of subgroups; only the 163 normal subgroups are shown.
This group has 944 conjugacy classes of subgroups — too many to draw. Only its 163 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 | #393×C173 | C2 |
| hypercenter | #393 | D173 |
| hypocenter | C173 | #2388 |
| LCS | G ▷ C23×C173 ▷ C173 | |
| UCS | C1 ◁ C23 ◁ #393 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×#14583827 | C2 | semidirect |
| 2. C2×#19149639 | C2 | semidirect |
| 3. #26011410 | C2 | semidirect |
| 4. #26011409 | C2 | semidirect |
| 5. #26011400 | C2 | semidirect |
| 6. #26011338 | C2 | semidirect |
| 7. #393×C173 | C2 | semidirect |
| 34. C2×C4×D173 | D4 | semidirect |
| 56. C2×Dic173 | #41 | semidirect |
| 60. C23×C173 | #41 | semidirect |
| 88. C173 | #2388 | semidirect |
| 94. #393 | D173 | semidirect |
| 140. C23 | #14583827 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×#14583827 | 2 |
| C2×#19149639 | 2 |
| #26011410 | 2 |
| #26011409 | 2 |
| #26011400 | 2 |
| #26011338 | 2 |
| #393×C173 | 2 |
| #2388 | 173 |