Finite Group DatabasebetaFinite Group DatabasebetaOrder 13504.
294 conjugacy classes of subgroups; only the 103 normal subgroups are shown.
This group has 294 conjugacy classes of subgroups — too many to draw. Only its 103 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 | C22⋊Q8×C211 | C2 |
| hypercenter | C22⋊Q8 | D211 |
| hypocenter | C211 | #537 |
| LCS | G ▷ C22×C211 ▷ C211 | |
| UCS | C1 ◁ C22 ◁ C22⋊Q8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#16809963 | C2 | semidirect |
| 2. C2×#16809962 | C2 | semidirect |
| 3. #20269955 | C2 | semidirect |
| 4. #20269953 | C2 | semidirect |
| 5. #20269952 | C2 | semidirect |
| 6. #20269952 | C2 | semidirect |
| 7. #20269951 | C2 | semidirect |
| 8. #20269950 | C2 | semidirect |
| 9. #20269950 | C2 | semidirect |
| 10. C4⋊C4×D211 | C2 | semidirect |
| 11. C22⋊Q8×C211 | C2 | semidirect |
| 54. C4×D211 | D4 | semidirect |
| 89. C211 | #537 | semidirect |
| 95. C22⋊Q8 | D211 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#16809963 | 2 |
| C2×#16809962 | 2 |
| #20269955 | 2 |
| #20269953 | 2 |
| #20269952 | 2 |
| #20269951 | 2 |
| #20269950 | 2 |
| C4⋊C4×D211 | 2 |
| C22⋊Q8×C211 | 2 |
| #537 | 211 |