Finite Group DatabasebetaFinite Group DatabasebetaOrder 3392.
216 conjugacy classes of subgroups; only the 95 normal subgroups are shown.
This group has 216 conjugacy classes of subgroups — too many to draw. Only its 95 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 | #124×C53 | C2 |
| prime radical | #13133188 | C2 |
| hypercenter | #124 | D53 |
| hypocenter | C53 | #542 |
| LCS | G ▷ C22×C53 ▷ C53 | |
| UCS | C1 ◁ C22 ◁ #124 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #13133217 | C2 | semidirect |
| 2. #13133210 | C2 | semidirect |
| 3. #13133203 | C2 | semidirect |
| 4. #13133201 | C2 | semidirect |
| 5. #13133194 | C2 | semidirect |
| 6. #13133193 | C2 | semidirect |
| 8. Q8×Dic53 | C2 | semidirect |
| 9. D4×Dic53 | C2 | semidirect |
| 10. #13133174 | C2 | semidirect |
| 11. #124×C53 | C2 | semidirect |
| 74. C53 | #542 | semidirect |
| 78. #124 | D53 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #13133217 | 2 |
| #13133210 | 2 |
| #13133203 | 2 |
| #13133201 | 2 |
| #13133194 | 2 |
| #13133193 | 2 |
| #13133188 | 2 |
| Q8×Dic53 | 2 |
| D4×Dic53 | 2 |
| #13133174 | 2 |
| #124×C53 | 2 |
| #542 | 53 |