Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
536 conjugacy classes of subgroups; only the 194 normal subgroups are shown.
This group has 536 conjugacy classes of subgroups — too many to draw. Only its 194 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 | H(3,𝔽4) |
| commutator, Frattini, cosocle | C22×C4 | C24×C53 |
| socle | C2×C53 | #2804 |
| prime radical | C53×#532 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C53 ◁ C22×C4×C53 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C53×#2102 | C2 | semidirect |
| 2. C53×#2031 | C2 | semidirect |
| 3. C53×#1995 | C2 | semidirect |
| 6. C53×#1861 | C2 | semidirect |
| 27. C4⋊Q8×C53 | Q8 | semidirect |
| 28. #127×C53 | Q8 | semidirect |
| 30. #124×C53 | Q8 | semidirect |
| 100. #32705 | C53 | direct (N × H) |
| 115. #2102 | C2×C53 | semidirect |
| 116. #2031 | C2×C53 | semidirect |
| 117. #1995 | C2×C53 | semidirect |
| 120. #1861 | C2×C53 | semidirect |
| 139. C53 | #32705 | direct (N × H) |
| 146. C4⋊Q8 | Q8×C53 | semidirect |
| 147. #127 | Q8×C53 | semidirect |
| 149. #124 | Q8×C53 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C53×#2102 | 2 |
| C53×#2031 | 2 |
| C53×#1995 | 2 |
| C53×#1963 | 2 |
| C53×#1884 | 2 |
| C53×#1861 | 2 |
| #32705 | 53 |