Finite Group DatabasebetaFinite Group DatabasebetaOrder 25216.
564 conjugacy classes of subgroups; only the 165 normal subgroups are shown.
This group has 564 conjugacy classes of subgroups — too many to draw. Only its 165 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D197 |
| commutator, socle, cosocle | C23×C197 | C24 |
| Fitting | #385×C197 | C2 |
| prime radical | C24×C197 | C23 |
| hypercenter | #385 | D197 |
| hypocenter | C197 | #2855 |
| LCS | G ▷ C23×C197 ▷ C197 | |
| UCS | C1 ◁ C23 ◁ #385 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. #27160983 | C2 | semidirect |
| 4. #27160974 | C2 | semidirect |
| 5. #27160965 | C2 | semidirect |
| 6. #27160961 | C2 | semidirect |
| 43. C22×Dic197 | Q8 | semidirect |
| 129. C197 | #2855 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#20263303 | 2 |
| C22×C4×Dic197 | 2 |
| #27160983 | 2 |
| #27160974 | 2 |
| #27160965 | 2 |
| #27160961 | 2 |
| #27160952 | 2 |
| #27160940 | 2 |
| #27160937 | 2 |
| #385×C197 | 2 |
| #2855 | 197 |