Finite Group DatabasebetaFinite Group DatabasebetaOrder 13504.
234 conjugacy classes of subgroups; only the 99 normal subgroups are shown.
This group has 234 conjugacy classes of subgroups — too many to draw. Only its 99 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×D211 |
| commutator, socle, cosocle | C22×C211 | C24 |
| Fitting | #124×C211 | C2 |
| prime radical | #20269948 | C2 |
| hypercenter | #124 | D211 |
| hypocenter | C211 | #529 |
| LCS | G ▷ C22×C211 ▷ C211 | |
| UCS | C1 ◁ C22 ◁ #124 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #20269970 | C2 | semidirect |
| 2. #20269969 | C2 | semidirect |
| 3. #20269959 | C2 | semidirect |
| 4. #20269953 | C2 | semidirect |
| 6. Q8×Dic211 | C2 | semidirect |
| 7. D4×Dic211 | C2 | semidirect |
| 8. (C22⋊D211)⋊C4 | C2 | semidirect |
| 9. #20269933 | C2 | semidirect |
| 10. C42×D211 | C2 | semidirect |
| 11. #124×C211 | C2 | semidirect |
| 74. C211 | #529 | semidirect |
| 78. #124 | D211 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #20269970 | 2 |
| #20269969 | 2 |
| #20269959 | 2 |
| #20269953 | 2 |
| #20269948 | 2 |
| Q8×Dic211 | 2 |
| D4×Dic211 | 2 |
| (C22⋊D211)⋊C4 | 2 |
| #20269933 | 2 |
| C42×D211 | 2 |
| #124×C211 | 2 |
| #529 | 211 |