Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
812 conjugacy classes of subgroups; only the 234 normal subgroups are shown.
This group has 812 conjugacy classes of subgroups — too many to draw. Only its 234 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 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C53 |
| prime radical | C24×C53 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C53 ◁ C22×C42×C53 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C53×#2072 | C2 | semidirect |
| 3. C53×#2016 | C2 | semidirect |
| 56. C22×C4×C53 | Q16 | semidirect |
| 101. #32762 | C53 | direct (N × H) |
| 119. #2072 | C2×C53 | semidirect |
| 120. #2016 | C2×C53 | semidirect |
| 143. C53 | #32762 | direct (N × H) |
| 180. C22×C4 | Q16×C53 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C53×#2857 | 2 |
| C53×#2072 | 2 |
| C53×#2016 | 2 |
| C53×#2001 | 2 |
| C53×#1817 | 2 |
| #32762 | 53 |