Finite Group DatabasebetaFinite Group DatabasebetaOrder 23940.
112 conjugacy classes of subgroups; only the 35 normal subgroups are shown.
This group has 112 conjugacy classes of subgroups — too many to draw. Only its 35 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, hypocenter | C3×C5×C7×C19 | C3×C4 |
| prime radical | C3×D5×#87705 | C2 |
| cosocle | C3×C7×D5×C19 | C2×C3 |
| LCS | G ▷ C3×C5×C7×C19 | |
| UCS | C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C7×C19×ΓL(1,𝔽16) | C3 | semidirect |
| 4. C3×C5×#87705 | C4 | semidirect |
| 12. C3×C5×C7×C19 | C3×C4 | semidirect |
| 14. C5×#87705 | Dic3 | semidirect |
| 20. C3×#87705 | AGL(1,𝔽5) | semidirect |
| 21. C19×ΓL(1,𝔽16) | ΓL(1,𝔽8) | semidirect |
| 28. C5×C7×C19 | C3×Dic3 | semidirect |
| 34. C7×ΓL(1,𝔽16) | C19⋊C3 | semidirect |
| 36. C3×C7×C19 | C3×AGL(1,𝔽5) | semidirect |
| 38. #87705 | ΓL(1,𝔽16) | direct (N × H) |
| 42. C3×C5×C19 | C4×ΓL(1,𝔽8) | semidirect |
| 56. C7×C19 | C3×ΓL(1,𝔽16) | semidirect |
| 61. C3×C5×C7 | C4×C19⋊C3 | semidirect |
| 64. C5×C19 | Dic3×ΓL(1,𝔽8) | semidirect |
| 71. ΓL(1,𝔽16) | #87705 | direct (N × H) |
| 73. C3×C19 | AGL(1,𝔽5)×ΓL(1,𝔽8) | semidirect |
| 82. C5×C7 | Dic3×C19⋊C3 | semidirect |
| 87. C3×C7 | AGL(1,𝔽5)×C19⋊C3 | semidirect |
| 91. C19 | ΓL(1,𝔽8)×ΓL(1,𝔽16) | semidirect |
| 95. C3×C5 | C4×#87705 | semidirect |
| 101. C7 | C19⋊C3×ΓL(1,𝔽16) | semidirect |
| 105. C5 | Dic3×#87705 | semidirect |
| 109. C3 | AGL(1,𝔽5)×#87705 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×D5×#87705 | 2 |
| C7×C19×ΓL(1,𝔽16) | 3 |
| AGL(1,𝔽5)×#87705 | 3 |
| Dic3×#87705 | 5 |
| C19⋊C3×ΓL(1,𝔽16) | 7 |
| ΓL(1,𝔽8)×ΓL(1,𝔽16) | 19 |