Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
342 conjugacy classes of subgroups; only the 50 normal subgroups are shown.
This group has 342 conjugacy classes of subgroups — too many to draw. Only its 50 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 | AΣL(1,𝔽16) |
| commutator | C2×Q8 | C2×C8×C53 |
| Frattini, cosocle | #391 | C22×C53 |
| prime radical | C2×2−1+4×C53 | C4 |
| LCS | G ▷ C2×Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C53 ◁ C23×C53 ◁ C22×C4×C53 ◁ C53×#391 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C53×#1319 | C2 | semidirect |
| 3. C53×#1272 | C2 | semidirect |
| 14. 2−1+4×C53 | C8 | semidirect |
| 106. #7345 | C53 | direct (N × H) |
| 124. #1319 | C2×C53 | semidirect |
| 125. #1272 | C2×C53 | semidirect |
| 139. C53 | #7345 | direct (N × H) |
| 143. 2−1+4 | C8×C53 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C53×#1793 | 2 |
| C53×#1319 | 2 |
| C53×#1272 | 2 |
| #7345 | 53 |