Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
302 conjugacy classes of subgroups; only the 70 normal subgroups are shown.
This group has 302 conjugacy classes of subgroups — too many to draw. Only its 70 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 | C42×C53 |
| Frattini, cosocle | C2×#124 | C22×C53 |
| prime radical | C22×D4×C53 | D4 |
| LCS | G ▷ C2×Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C53 ◁ C23×C53 ◁ C22×C4×C53 ◁ C2×#124×C53 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 5. C53×#519 | C4 | semidirect |
| 7. C2×#101×C53 | C4 | semidirect |
| 11. C53×#354 | C4 | semidirect |
| 14. C53×#329 | C4 | semidirect |
| 98. #7489 | C53 | direct (N × H) |
| 127. #519 | C4×C53 | semidirect |
| 129. C2×#101 | C4×C53 | semidirect |
| 133. #354 | C4×C53 | semidirect |
| 136. #329 | C4×C53 | semidirect |
| 137. C53 | #7489 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C53×#354 | 2 |
| C53×#1888 | 2 |
| C2×C53×#329 | 2 |
| #7489 | 53 |