Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
1154 conjugacy classes of subgroups; only the 390 normal subgroups are shown.
This group has 1154 conjugacy classes of subgroups — too many to draw. Only its 390 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C53 | C24 |
| commutator | C23 | C23×C4×C53 |
| Frattini, cosocle | C22×C4 | C24×C53 |
| socle | C23×C53 | C23×C4 |
| prime radical | C53×#2397 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C53 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×C53×#436 | C2 | semidirect |
| 3. C53×#1955 | C2 | semidirect |
| 4. C53×#1843 | C2 | semidirect |
| 5. C53×#1726 | C2 | semidirect |
| 30. C4×C8×C53 | D4 | semidirect |
| 64. #21684 | C53 | direct (N × H) |
| 74. C2×#436 | C2×C53 | semidirect |
| 76. #1955 | C2×C53 | semidirect |
| 77. #1843 | C2×C53 | semidirect |
| 78. #1726 | C2×C53 | semidirect |
| 92. C53 | #21684 | direct (N × H) |
| 107. C4×C8 | D4×C53 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C53×#436 | 2 |
| C53×#2397 | 2 |
| C53×#1955 | 2 |
| C53×#1843 | 2 |
| C53×#1726 | 2 |
| #21684 | 53 |