Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
3694 conjugacy classes of subgroups; only the 827 normal subgroups are shown.
This group has 3694 conjugacy classes of subgroups — too many to draw. Only its 827 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 | C23×D53 |
| commutator, socle, cosocle | C23×C53 | C25 |
| Frattini | C23 | C24×D53 |
| Fitting | C2×C53×#376 | C2 |
| hypercenter | C2×#376 | D53 |
| hypocenter | C53 | #34641 |
| LCS | G ▷ C23×C53 ▷ C53 | |
| UCS | C1 ◁ C22×C4 ◁ C2×#376 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#16811173 | C2 | semidirect |
| 2. #20280825 | C2 | semidirect |
| 3. #20280821 | C2 | semidirect |
| 4. #20280817 | C2 | semidirect |
| 5. #20277996 | C2 | semidirect |
| 6. C2×C53×#376 | C2 | semidirect |
| 35. C2×C42×C53 | D4 | semidirect |
| 92. C2×#376 | D53 | semidirect |
| 110. C53 | #34641 | semidirect |
| 126. C2×C42 | D212 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#16811173 | 2 |
| #20280825 | 2 |
| #20280821 | 2 |
| #20280817 | 2 |
| #20277996 | 2 |
| C2×C53×#376 | 2 |
| #34641 | 53 |