Finite Group DatabasebetaFinite Group DatabasebetaOrder 1472.
196 conjugacy classes of subgroups; only the 91 normal subgroups are shown.
This group has 196 conjugacy classes of subgroups — too many to draw. Only its 91 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×D23 |
| commutator, socle, cosocle | C22×C23 | C24 |
| Fitting | C23×#126 | C2 |
| prime radical | C2×C22⋊D23 | C22 |
| hypercenter | #126 | D23 |
| hypocenter | C23 | #563 |
| LCS | G ▷ C22×C23 ▷ C23 | |
| UCS | C1 ◁ C22 ◁ #126 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C23×#126 | C2 | semidirect |
| 2. #10625069 | C2 | semidirect |
| 3. #10625068 | C2 | semidirect |
| 4. #10625067 | C2 | semidirect |
| 5. #10625061 | C2 | semidirect |
| 6. #10625060 | C2 | semidirect |
| 7. #10625059 | C2 | semidirect |
| 11. #10625050 | C2 | semidirect |
| 12. #10625049 | C2 | semidirect |
| 13. #10625049 | C2 | semidirect |
| 14. #10625047 | C2 | semidirect |
| 15. #10625040 | C2 | semidirect |
| 103. #126 | D23 | semidirect |
| 113. C23 | #563 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×#126 | 2 |
| #10625069 | 2 |
| #10625068 | 2 |
| #10625067 | 2 |
| #10625061 | 2 |
| #10625060 | 2 |
| #10625059 | 2 |
| #10625057 | 2 |
| #10625056 | 2 |
| #10625054 | 2 |
| #10625050 | 2 |
| #10625049 | 2 |
| #10625047 | 2 |
| #10625040 | 2 |
| #563 | 23 |