Finite Group DatabasebetaFinite Group DatabasebetaOrder 15104.
700 conjugacy classes of subgroups; only the 118 normal subgroups are shown.
This group has 700 conjugacy classes of subgroups — too many to draw. Only its 118 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 | C2×Q8 | C22×C4×C59 |
| Frattini, cosocle | C22×Q8 | C23×C59 |
| prime radical | C2×C59×#454 | C2 |
| LCS | G ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C59 ◁ C23×C59 ◁ C22×Q8×C59 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C59×#2372 | C2 | semidirect |
| 3. C2×C59×#355 | C2 | semidirect |
| 4. C59×#1793 | C2 | semidirect |
| 5. C59×#1394 | C2 | semidirect |
| 10. C59×#454 | C4 | semidirect |
| 25. 2−1+4×C59 | C2×C4 | semidirect |
| 56. #101×C59 | C2×C4 | semidirect |
| 162. #13057 | C59 | direct (N × H) |
| 183. #2372 | C2×C59 | semidirect |
| 184. C2×#355 | C2×C59 | semidirect |
| 185. #1793 | C2×C59 | semidirect |
| 186. #1394 | C2×C59 | semidirect |
| 197. #454 | C4×C59 | semidirect |
| 209. C59 | #13057 | direct (N × H) |
| 213. 2−1+4 | C2×C4×C59 | semidirect |
| 244. #101 | C2×C4×C59 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C59×#454 | 2 |
| C59×#2372 | 2 |
| C2×C59×#355 | 2 |
| C59×#1793 | 2 |
| C59×#1394 | 2 |
| #13057 | 59 |