Finite Group DatabasebetaFinite Group DatabasebetaOrder 4352.
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×C17 | AΣL(1,𝔽16) |
| commutator | C2×Q8 | C42×C17 |
| Frattini, cosocle | C2×#124 | C22×C17 |
| prime radical | C22×D4×C17 | D4 |
| LCS | G ▷ C2×Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C17 ◁ C23×C17 ◁ C22×C4×C17 ◁ C2×C17×#124 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 5. C17×#519 | C4 | semidirect |
| 7. C2×C17×#101 | C4 | semidirect |
| 11. C17×#354 | C4 | semidirect |
| 14. C17×#329 | C4 | semidirect |
| 70. #7489 | C17 | direct (N × H) |
| 121. #519 | C4×C17 | semidirect |
| 123. C2×#101 | C4×C17 | semidirect |
| 127. #354 | C4×C17 | semidirect |
| 130. #329 | C4×C17 | semidirect |
| 158. C17 | #7489 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C17×#354 | 2 |
| C17×#1888 | 2 |
| C2×C17×#329 | 2 |
| #7489 | 17 |