Finite Group DatabasebetaFinite Group DatabasebetaOrder 15104.
678 conjugacy classes of subgroups; only the 270 normal subgroups are shown.
This group has 678 conjugacy classes of subgroups — too many to draw. Only its 270 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C59 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C59 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C59 ◁ C43×C59 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 51. C42×C59 | Q16 | semidirect |
| 68. #33267 | C59 | direct (N × H) |
| 97. C59 | #33267 | direct (N × H) |
| 132. C42 | Q16×C59 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C59×#2863 | 2 |
| C59×#1969 | 2 |
| C59×#1942 | 2 |
| C59×#1933 | 2 |
| C42×C8×C59 | 2 |
| #33267 | 59 |