Finite Group DatabasebetaFinite Group DatabasebetaOrder 15104.
346 conjugacy classes of subgroups; only the 74 normal subgroups are shown.
This group has 346 conjugacy classes of subgroups — too many to draw. Only its 74 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 | B(4,𝔽2) |
| commutator, Frattini, cosocle | C2×C4⋊C4 | C23×C59 |
| prime radical | C23×C59 | #120 |
| LCS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C59 ◁ C23×C59 ◁ C2×C4⋊C4×C59 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 69. C2×Q8×C59 | Q16 | semidirect |
| 138. #12222 | C59 | direct (N × H) |
| 187. C59 | #12222 | direct (N × H) |
| 230. C2×Q8 | Q16×C59 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C59×#2799 | 2 |
| C59×#2028 | 2 |
| C59×#1617 | 2 |
| C59×#1605 | 2 |
| C59×#1354 | 2 |
| C59×#1353 | 2 |
| C59×#1319 | 2 |
| #12222 | 59 |