Finite Group DatabasebetaFinite Group DatabasebetaOrder 15104.
1550 conjugacy classes of subgroups; only the 298 normal subgroups are shown.
This group has 1550 conjugacy classes of subgroups — too many to draw. Only its 298 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 | C2×#120 |
| commutator, Frattini, cosocle | C24 | C24×C59 |
| prime radical | C2×H(5,𝔽2)×C59 | C22 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C59 ◁ C23×C4×C59 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C59×#2409 | C2 | semidirect |
| 3. C59×#1867 | C2 | semidirect |
| 4. C59×#1799 | C2 | semidirect |
| 7. C23×Q8×C59 | C22 | semidirect |
| 20. C59×#385 | C22 | semidirect |
| 21. C59×#380 | C22 | semidirect |
| 22. C59×#342 | C22 | semidirect |
| 24. H(5,𝔽2)×C59 | Q8 | semidirect |
| 119. C23×C59 | C22⋊Q8 | semidirect |
| 150. C2×C4×C59 | C22⋊Q8 | semidirect |
| 151. #32975 | C59 | direct (N × H) |
| 173. #2409 | C2×C59 | semidirect |
| 174. #1867 | C2×C59 | semidirect |
| 175. #1799 | C2×C59 | semidirect |
| 184. C23×Q8 | C22×C59 | semidirect |
| 197. #385 | C22×C59 | semidirect |
| 198. #380 | C22×C59 | semidirect |
| 199. #342 | C22×C59 | semidirect |
| 200. C59 | #32975 | direct (N × H) |
| 202. H(5,𝔽2) | Q8×C59 | semidirect |
| 297. C23 | C22⋊Q8×C59 | semidirect |
| 328. C2×C4 | C22⋊Q8×C59 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C4×H(5,𝔽2)×C59 | 2 |
| C59×#2409 | 2 |
| C59×#1867 | 2 |
| C59×#1799 | 2 |
| C59×#1790 | 2 |
| #32975 | 59 |