Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
240 conjugacy classes of subgroups; only the 42 normal subgroups are shown.
This group has 240 conjugacy classes of subgroups — too many to draw. Only its 42 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C4×C53 | #353 |
| commutator | C2×M16 | C2×C4×C53 |
| Frattini, cosocle | #408 | C22×C53 |
| socle | C2×C53 | #1343 |
| prime radical | C53×#2017 | C2 |
| LCS | G ▷ C2×M16 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C53 ◁ C2×C4×C53 ◁ C22×C4×C53 ◁ C53×#408 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C53×#1392 | C2 | semidirect |
| 3. C53×#1315 | C2 | semidirect |
| 89. #7407 | C53 | direct (N × H) |
| 103. #1392 | C2×C53 | semidirect |
| 104. #1315 | C2×C53 | semidirect |
| 117. C53 | #7407 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C53×#2017 | 2 |
| C53×#1392 | 2 |
| C53×#1315 | 2 |
| #7407 | 53 |