Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
302 conjugacy classes of subgroups; only the 142 normal subgroups are shown.
This group has 302 conjugacy classes of subgroups — too many to draw. Only its 142 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 | C23×C53 | D16 |
| commutator | C16 | C22×C4×C53 |
| Frattini, cosocle | C2×C16 | C23×C53 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×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. C53×#1416 | C2 | direct (N × H) |
| 8. C2×Q32×C53 | C4 | semidirect |
| 25. Q32×C53 | C2×C4 | semidirect |
| 67. C2×#1416 | C53 | direct (N × H) |
| 80. #1416 | C2×C53 | direct (N × H) |
| 81. C2×C53 | #1416 | direct (N × H) |
| 88. C2×Q32 | C4×C53 | semidirect |
| 96. C53 | C2×#1416 | direct (N × H) |
| 106. Q32 | C2×C4×C53 | semidirect |
| 157. C2 | C53×#1416 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×Q32×C53 | 2 |
| C22×C32×C53 | 2 |
| C2×C53×#366 | 2 |
| C53×#1416 | 2 |
| C2×#1416 | 53 |