Finite Group DatabasebetaFinite Group DatabasebetaOrder 12416.
716 conjugacy classes of subgroups; only the 212 normal subgroups are shown.
This group has 716 conjugacy classes of subgroups — too many to draw. Only its 212 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C97 | C2×#120 |
| commutator, Frattini, cosocle | C23 | C24×C97 |
| prime radical | #585×C97 | C2 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C97 ◁ C22×C4×C97 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #536×C97 | C2 | semidirect |
| 3. #458×C97 | C2 | semidirect |
| 4. #410×C97 | C2 | semidirect |
| 10. C22×Q8×C97 | C22 | semidirect |
| 13. #123×C97 | C22 | semidirect |
| 15. #117×C97 | C22 | semidirect |
| 16. #99×C97 | C22 | semidirect |
| 70. #3028 | C97 | direct (N × H) |
| 71. C97 | #3028 | direct (N × H) |
| 73. #536 | C2×C97 | semidirect |
| 74. #458 | C2×C97 | semidirect |
| 75. #410 | C2×C97 | semidirect |
| 81. C22×Q8 | C22×C97 | semidirect |
| 84. #123 | C22×C97 | semidirect |
| 86. #117 | C22×C97 | semidirect |
| 87. #99 | C22×C97 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #585×C97 | 2 |
| #536×C97 | 2 |
| #458×C97 | 2 |
| #410×C97 | 2 |
| #3028 | 97 |