Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
306 conjugacy classes of subgroups; only the 222 normal subgroups are shown.
This group has 306 conjugacy classes of subgroups — too many to draw. Only its 222 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C8×C53 | C22 |
| commutator | C2 | C8×C16×C53 |
| Frattini, cosocle | C2×C4×C8 | C22×C53 |
| socle, prime radical | C23×C53 | C4×C8 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C8×C53 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 14. C2×C16×C53 | C8 | semidirect |
| 24. C2×C8×C53 | C16 | semidirect |
| 45. #7136 | C53 | direct (N × H) |
| 68. C53 | #7136 | direct (N × H) |
| 74. C2×C16 | C8×C53 | semidirect |
| 84. C2×C8 | C16×C53 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C4×C16×C53 | 2 |
| C2×C82×C53 | 2 |
| #7136 | 53 |