Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
824 conjugacy classes of subgroups; only the 230 normal subgroups are shown.
This group has 824 conjugacy classes of subgroups — too many to draw. Only its 230 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C53 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C24×C53 |
| prime radical | C23×C53 | #120 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C53 ◁ C2×C42×C53 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 4. C53×#1663 | C2 | semidirect |
| 120. #19084 | C53 | direct (N × H) |
| 135. #1663 | C2×C53 | semidirect |
| 161. C53 | #19084 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C53×#3556 | 2 |
| C53×#3165 | 2 |
| C53×#3139 | 2 |
| C53×#1663 | 2 |
| C53×#1526 | 2 |
| #19084 | 53 |