Finite Group DatabasebetaFinite Group DatabasebetaOrder 3360.
224 conjugacy classes of subgroups; only the 28 normal subgroups are shown.
This group has 224 conjugacy classes of subgroups — too many to draw. Only its 28 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, hypocenter | C5×C7×A4 | C2×C4 |
| Fitting, socle | C22×C5×C7 | C4×D3 |
| prime radical | D5×C7⋊S4 | C2 |
| cosocle | C7×D5×A4 | C22 |
| LCS | G ▷ C5×C7×A4 | |
| UCS | C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. A4×C7⋊AGL(1,𝔽5) | C2 | semidirect |
| 3. C7×A4⋊AGL(1,𝔽5) | C2 | semidirect |
| 6. C5⋊(C7⋊S4) | C4 | semidirect |
| 7. C5×C7⋊S4 | C4 | semidirect |
| 10. C22×C7⋊AGL(1,𝔽5) | D3 | semidirect |
| 18. C5×C7×A4 | C2×C4 | semidirect |
| 40. A4⋊AGL(1,𝔽5) | D7 | semidirect |
| 47. C7⋊S4 | AGL(1,𝔽5) | semidirect |
| 50. C22×C5×C7 | C4×D3 | semidirect |
| 58. C7⋊AGL(1,𝔽5) | S4 | semidirect |
| 77. C7×A4 | C2×AGL(1,𝔽5) | semidirect |
| 92. C5×A4 | C4×D7 | semidirect |
| 128. C5×C7 | C4×S4 | semidirect |
| 135. C22×C7 | D3×AGL(1,𝔽5) | semidirect |
| 152. C22×C5 | D21⋊C4 | semidirect |
| 178. A4 | D7×AGL(1,𝔽5) | semidirect |
| 199. C7 | AGL(1,𝔽5)×S4 | semidirect |
| 203. C5 | (C7⋊S4)⋊C4 | semidirect |
| 210. C22 | D21⋊AGL(1,𝔽5) | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| D5×C7⋊S4 | 2 |
| A4×C7⋊AGL(1,𝔽5) | 2 |
| C7×A4⋊AGL(1,𝔽5) | 2 |
| (C22⋊D7)⋊AGL(1,𝔽5) | 3 |
| D21⋊AGL(1,𝔽5) | 4 |
| (C7⋊S4)⋊C4 | 5 |
| AGL(1,𝔽5)×S4 | 7 |