Finite Group DatabasebetaFinite Group DatabasebetaOrder 3392.
270 conjugacy classes of subgroups; only the 103 normal subgroups are shown.
This group has 270 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×D53 |
| commutator, socle, cosocle | C22×C53 | C24 |
| Fitting | C22⋊Q8×C53 | C2 |
| prime radical | C2×#11719945 | C2 |
| hypercenter | C22⋊Q8 | D53 |
| hypocenter | C53 | #536 |
| 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 |
|---|---|---|
| 2. C2×Q8×D53 | C2 | semidirect |
| 3. #13133216 | C2 | semidirect |
| 4. #13133203 | C2 | semidirect |
| 5. #13133202 | C2 | semidirect |
| 6. #13133199 | C2 | semidirect |
| 7. #13133197 | C2 | semidirect |
| 8. #13133196 | C2 | semidirect |
| 9. #13133193 | C2 | semidirect |
| 10. #13133175 | C2 | semidirect |
| 11. C22⋊Q8×C53 | C2 | semidirect |
| 84. C53 | #536 | semidirect |
| 94. C22⋊Q8 | D53 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#11719945 | 2 |
| C2×Q8×D53 | 2 |
| #13133216 | 2 |
| #13133203 | 2 |
| #13133202 | 2 |
| #13133199 | 2 |
| #13133197 | 2 |
| #13133196 | 2 |
| #13133193 | 2 |
| #13133175 | 2 |
| C22⋊Q8×C53 | 2 |
| #536 | 53 |