Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
136 conjugacy classes of subgroups; only the 54 normal subgroups are shown.
This group has 136 conjugacy classes of subgroups — too many to draw. Only its 54 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 | #357 |
| commutator | C2×C16 | C2×C4×C53 |
| Frattini, cosocle | C4×C16 | C22×C53 |
| prime radical | C2×D16×C53 | C4 |
| LCS | G ▷ C2×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C53 ◁ C42×C53 ◁ C4×C8×C53 ◁ C4×C16×C53 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C53×C4⋊C32 | C2 | semidirect |
| 3. C53×#1372 | C2 | semidirect |
| 53. #7446 | C53 | direct (N × H) |
| 64. C4⋊C32 | C2×C53 | semidirect |
| 65. #1372 | C2×C53 | semidirect |
| 77. C53 | #7446 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C53×#2249 | 2 |
| C53×C4⋊C32 | 2 |
| C53×#1372 | 2 |
| #7446 | 53 |