Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
1198 conjugacy classes of subgroups; only the 286 normal subgroups are shown.
This group has 1198 conjugacy classes of subgroups — too many to draw. Only its 286 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C22×C8×C53 | C23 |
| commutator | C22 | C22×C16×C53 |
| Frattini, cosocle | C22×C8 | C23×C53 |
| socle | C23×C53 | C22×C8 |
| prime radical | C25×C53 | C8 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C8×C53 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×C53×C22⋊C16 | C2 | semidirect |
| 7. C22×C16×C53 | C22 | semidirect |
| 24. C24×C53 | C16 | semidirect |
| 61. #12679 | C53 | direct (N × H) |
| 67. C22×C53 | C22⋊C16 | semidirect |
| 75. C2×C22⋊C16 | C2×C53 | semidirect |
| 84. C22×C16 | C22×C53 | semidirect |
| 86. C53 | #12679 | direct (N × H) |
| 102. C24 | C16×C53 | semidirect |
| 144. C22 | C53×C22⋊C16 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C24×C8×C53 | 2 |
| C2×C53×C22⋊C16 | 2 |
| #12679 | 53 |