Finite Group DatabasebetaFinite Group DatabasebetaOrder 12416.
360 conjugacy classes of subgroups; only the 209 normal subgroups are shown.
This group has 360 conjugacy classes of subgroups — too many to draw. Only its 209 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C8 | C23×D97 |
| commutator, socle | C2×C97 | C23×C8 |
| Fitting | #504×C97 | C2 |
| prime radical | #41×D97 | C4 |
| cosocle | C8×C97 | C24 |
| hypercenter | #504 | D97 |
| hypocenter | C97 | #3407 |
| LCS | G ▷ C2×C97 ▷ C97 | |
| UCS | C1 ◁ C8 ◁ #504 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #20261153 | C2 | semidirect |
| 3. #20261152 | C2 | semidirect |
| 4. #20261151 | C2 | semidirect |
| 5. C2×#15690749 | C2 | semidirect |
| 6. M32×D97 | C2 | semidirect |
| 7. #504×C97 | C2 | semidirect |
| 14. #15690750 | C22 | semidirect |
| 17. #15690749 | C22 | semidirect |
| 18. #15690749 | C22 | semidirect |
| 19. C2×#13095486 | C22 | semidirect |
| 20. C16×D97 | C22 | semidirect |
| 22. M32×C97 | C22 | semidirect |
| 23. C2×C16×C97 | C22 | semidirect |
| 73. C97 | #3407 | semidirect |
| 78. #504 | D97 | semidirect |
| 93. M32 | C2×D97 | semidirect |
| 96. C2×C16 | C2×D97 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #131×D97 | 2 |
| #20261153 | 2 |
| #20261152 | 2 |
| #20261151 | 2 |
| C2×#15690749 | 2 |
| M32×D97 | 2 |
| #504×C97 | 2 |
| #3407 | 97 |