Finite Group DatabasebetaFinite Group DatabasebetaOrder 25216.
528 conjugacy classes of subgroups; only the 155 normal subgroups are shown.
This group has 528 conjugacy classes of subgroups — too many to draw. Only its 155 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 | #399×C197 | C2 |
| prime radical | C24×C197 | C23 |
| hypercenter | #399 | D197 |
| hypocenter | C197 | #2396 |
| LCS | G ▷ C23×C197 ▷ C197 | |
| UCS | C1 ◁ C23 ◁ #399 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#20263291 | C2 | semidirect |
| 3. #27161017 | C2 | semidirect |
| 4. #27161010 | C2 | semidirect |
| 6. #27161000 | C2 | semidirect |
| 8. C4⋊C4×Dic197 | C2 | semidirect |
| 9. #27160930 | C2 | semidirect |
| 41. C22×Dic197 | Q8 | semidirect |
| 128. C197 | #2396 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#20263291 | 2 |
| C2×#20263282 | 2 |
| #27161017 | 2 |
| #27161010 | 2 |
| #27161002 | 2 |
| #27161000 | 2 |
| #27160991 | 2 |
| C4⋊C4×Dic197 | 2 |
| #27160930 | 2 |
| C22⋊C4×Dic197 | 2 |
| #399×C197 | 2 |
| #2396 | 197 |