Finite Group DatabasebetaFinite Group DatabasebetaOrder 3392.
228 conjugacy classes of subgroups; only the 107 normal subgroups are shown.
This group has 228 conjugacy classes of subgroups — too many to draw. Only its 107 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×D53 |
| commutator, socle, cosocle | C22×C53 | C24 |
| Fitting | C22⋊Q8×C53 | C2 |
| prime radical | C2×C22⋊D53 | C22 |
| hypercenter | C22⋊Q8 | D53 |
| hypocenter | C53 | #554 |
| LCS | G ▷ C22×C53 ▷ C53 | |
| UCS | C1 ◁ C22 ◁ C22⋊Q8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #13133216 | C2 | semidirect |
| 2. #13133207 | C2 | semidirect |
| 3. #13133197 | C2 | semidirect |
| 4. #13133194 | C2 | semidirect |
| 5. #13133193 | C2 | semidirect |
| 6. Q8×Dic53 | C2 | semidirect |
| 9. C2×C4⋊Dic53 | C2 | semidirect |
| 10. C4⋊C4×D53 | C2 | semidirect |
| 11. C22⋊Q8×C53 | C2 | semidirect |
| 43. C22⋊D53 | Q8 | semidirect |
| 75. C53 | #554 | semidirect |
| 81. C22⋊Q8 | D53 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #13133216 | 2 |
| #13133207 | 2 |
| #13133197 | 2 |
| #13133194 | 2 |
| #13133193 | 2 |
| Q8×Dic53 | 2 |
| C4×C22⋊D53 | 2 |
| #13133177 | 2 |
| C2×C4⋊Dic53 | 2 |
| C4⋊C4×D53 | 2 |
| C22⋊Q8×C53 | 2 |
| #554 | 53 |