Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
3774 conjugacy classes of subgroups; only the 1886 normal subgroups are shown.
This group has 3774 conjugacy classes of subgroups — too many to draw. Only its 1886 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24×C53 | C24 |
| commutator, Frattini, cosocle | C23 | C25×C53 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24×C53 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C53×#2868 | C2 | direct (N × H) |
| 22. C2×C42×C53 | Q8 | semidirect |
| 37. C42×C53 | C2×Q8 | semidirect |
| 45. C2×C4×C53 | C4⋊Q8 | semidirect |
| 49. C2×#2868 | C53 | direct (N × H) |
| 56. C4×C53 | C2×C4⋊Q8 | semidirect |
| 58. #2868 | C2×C53 | direct (N × H) |
| 61. C2×C53 | #2868 | direct (N × H) |
| 72. C53 | C2×#2868 | direct (N × H) |
| 82. C2×C42 | Q8×C53 | semidirect |
| 97. C42 | C2×Q8×C53 | semidirect |
| 105. C2×C4 | C4⋊Q8×C53 | semidirect |
| 115. C4 | C2×C4⋊Q8×C53 | semidirect |
| 116. C2 | C53×#2868 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C4⋊Q8×C53 | 2 |
| C53×#2868 | 2 |
| C2×C53×#384 | 2 |
| C2×C43×C53 | 2 |
| C2×#2868 | 53 |