Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
152 conjugacy classes of subgroups; only the 54 normal subgroups are shown.
This group has 152 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 | C22×C16 | C22×C53 |
| prime radical | C23×C53 | D16 |
| LCS | G ▷ C2×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C53 ◁ C22×C4×C53 ◁ C22×C8×C53 ◁ C22×C16×C53 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 4. C2×Q32×C53 | C4 | semidirect |
| 9. C53×#366 | C4 | semidirect |
| 55. #7419 | C53 | direct (N × H) |
| 75. C2×Q32 | C4×C53 | semidirect |
| 80. #366 | C4×C53 | semidirect |
| 82. C53 | #7419 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C53×#2216 | 2 |
| C53×C22⋊C32 | 2 |
| C53×#1380 | 2 |
| #7419 | 53 |