Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
1482 conjugacy classes of subgroups; only the 350 normal subgroups are shown.
This group has 1482 conjugacy classes of subgroups — too many to draw. Only its 350 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C53 | #120 |
| commutator, Frattini, cosocle | C24 | C24×C53 |
| prime radical | C25×C53 | C23 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C53 ◁ C25×C53 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. C53×#2030 | C2 | direct (N × H) |
| 22. C23×C4×C53 | Q8 | semidirect |
| 58. C22×C4×C53 | C2×Q8 | semidirect |
| 91. C2×#2030 | C53 | direct (N × H) |
| 111. #2030 | C2×C53 | direct (N × H) |
| 116. C2×C53 | #2030 | direct (N × H) |
| 129. C53 | C2×#2030 | direct (N × H) |
| 137. C23×C4 | Q8×C53 | semidirect |
| 173. C22×C4 | C2×Q8×C53 | semidirect |
| 226. C2 | C53×#2030 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C53×#543 | 2 |
| C2×C53×#381 | 2 |
| C53×#2030 | 2 |
| C2×C53×#342 | 2 |
| C2×#2030 | 53 |