Finite Group DatabasebetaFinite Group DatabasebetaOrder 12416.
1488 conjugacy classes of subgroups; only the 813 normal subgroups are shown.
This group has 1488 conjugacy classes of subgroups — too many to draw. Only its 813 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 | C24×D97 |
| commutator | C2×C97 | C24×C4 |
| Fitting | #519×C97 | C2 |
| socle, cosocle | C22×C97 | C25 |
| prime radical | C23×D97 | C23 |
| hypercenter | #519 | D97 |
| hypocenter | C97 | C2×#519 |
| LCS | G ▷ C2×C97 ▷ C97 | |
| UCS | C1 ◁ C22 ◁ #519 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #20261213 | C2 | semidirect |
| 3. #20261212 | C2 | semidirect |
| 4. #20261211 | C2 | semidirect |
| 5. C4×Q8×D97 | C2 | semidirect |
| 7. #519×C97 | C2 | semidirect |
| 9. C2×Q8×D97 | C4 | semidirect |
| 13. Q8×Dic97 | C22 | semidirect |
| 14. C4×Dic194 | C22 | semidirect |
| 15. #15690775 | C22 | semidirect |
| 24. C4×Q8×C97 | C22 | semidirect |
| 27. C2×Dic194 | C2×C4 | semidirect |
| 30. Q8×D97 | C2×C4 | semidirect |
| 42. C2×Q8×C97 | C2×C4 | semidirect |
| 80. D97 | #519 | direct (N × H) |
| 82. C97 | C2×#519 | semidirect |
| 87. #519 | D97 | direct (N × H) |
| 99. C4×Q8 | C2×D97 | semidirect |
| 113. C2×Q8 | C4×D97 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×Q8×D97 | 2 |
| #20261213 | 2 |
| #20261212 | 2 |
| #20261211 | 2 |
| C4×Q8×D97 | 2 |
| #117×D97 | 2 |
| #519×C97 | 2 |
| C2×#519 | 97 |