Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
334 conjugacy classes of subgroups; only the 102 normal subgroups are shown.
This group has 334 conjugacy classes of subgroups — too many to draw. Only its 102 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 | #410 |
| commutator | C22×C4 | C22×C4×C53 |
| Frattini, cosocle | C2×M16 | C23×C53 |
| socle | C2×C53 | #1466 |
| prime radical | C53×#525 | C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C53 ◁ C2×C8×C53 ◁ C53×#405 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C53×#2166 | C2 | semidirect |
| 3. C53×#1321 | C2 | semidirect |
| 4. C53×#1320 | C2 | semidirect |
| 102. #11538 | C53 | direct (N × H) |
| 119. #2166 | C2×C53 | semidirect |
| 120. #1321 | C2×C53 | semidirect |
| 121. #1320 | C2×C53 | semidirect |
| 139. C53 | #11538 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C53×#2930 | 2 |
| C53×#2166 | 2 |
| C53×#1321 | 2 |
| C53×#1320 | 2 |
| #11538 | 53 |