Finite Group DatabasebetaFinite Group DatabasebetaOrder 15104.
514 conjugacy classes of subgroups; only the 94 normal subgroups are shown.
This group has 514 conjugacy classes of subgroups — too many to draw. Only its 94 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×D4 | C22×C4×C59 |
| Frattini, cosocle | #124 | C23×C59 |
| prime radical | C2×H(5,𝔽2)×C59 | C4 |
| LCS | G ▷ C2×D4 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C59 ◁ C23×C59 ◁ C22×C4×C59 ◁ C2×#124×C59 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×C59×#355 | C2 | semidirect |
| 3. C2×C59×#329 | C2 | semidirect |
| 4. C59×#1403 | C2 | direct (N × H) |
| 17. C59×#355 | C22 | semidirect |
| 19. C59×#329 | C22 | semidirect |
| 150. C2×#1403 | C59 | direct (N × H) |
| 173. C2×#355 | C2×C59 | semidirect |
| 174. C2×#329 | C2×C59 | semidirect |
| 175. #1403 | C2×C59 | direct (N × H) |
| 181. C2×C59 | #1403 | direct (N × H) |
| 195. #355 | C22×C59 | semidirect |
| 197. #329 | C22×C59 | semidirect |
| 198. C59 | C2×#1403 | direct (N × H) |
| 355. C2 | C59×#1403 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C59×#455 | 2 |
| C2×C59×#355 | 2 |
| C2×C59×#329 | 2 |
| C59×#1403 | 2 |
| C2×#1403 | 59 |