Finite Group DatabasebetaFinite Group DatabasebetaOrder 15104.
202 conjugacy classes of subgroups; only the 50 normal subgroups are shown.
This group has 202 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×C59 | AΣL(1,𝔽16) |
| commutator | C2×Q8 | C2×C8×C59 |
| Frattini, cosocle | #391 | C22×C59 |
| prime radical | C23×C59 | #99 |
| LCS | G ▷ C2×Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C59 ◁ C23×C59 ◁ C22×C4×C59 ◁ C59×#391 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 23. #101×C59 | C8 | semidirect |
| 74. #7376 | C59 | direct (N × H) |
| 104. C59 | #7376 | direct (N × H) |
| 117. #101 | C8×C59 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C59×#1857 | 2 |
| C59×#1319 | 2 |
| C59×#1272 | 2 |
| #7376 | 59 |