Finite Group DatabasebetaFinite Group DatabasebetaOrder 1920.
670 conjugacy classes of subgroups; only the 103 normal subgroups are shown.
This group has 670 conjugacy classes of subgroups — too many to draw. Only its 103 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C4 | D5×S4 |
| commutator, hypocenter | C5×A4 | C22×C8 |
| Frattini | C4 | C2×D5×S4 |
| Fitting | C23×C4×C5 | C2×D3 |
| socle | C24×C5 | C4×D3 |
| prime radical | C22×C5×S4 | C4 |
| cosocle | C4×C5×A4 | C23 |
| LCS | G ▷ C5×A4 | |
| UCS | C1 ◁ C2×C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×A4×#168 | C2 | semidirect |
| 3. C2×A4⋊#168 | C2 | semidirect |
| 4. S4×#168 | C2 | direct (N × H) |
| 12. A4×#168 | C22 | semidirect |
| 13. A4⋊#168 | C22 | semidirect |
| 17. C23×#168 | D3 | semidirect |
| 25. C2×C5×S4 | C8 | semidirect |
| 58. C22×#168 | C2×D3 | semidirect |
| 66. C2×C5×A4 | C2×C8 | semidirect |
| 68. C5×S4 | C2×C8 | semidirect |
| 125. C2×#168 | S4 | direct (N × H) |
| 138. C5×A4 | C22×C8 | semidirect |
| 143. C2×S4 | #168 | direct (N × H) |
| 160. C23×C5 | D3×C8 | semidirect |
| 188. #168 | C2×S4 | direct (N × H) |
| 219. C2×A4 | C2×#168 | semidirect |
| 221. S4 | C2×#168 | direct (N × H) |
| 240. C22×C5 | C2×D3×C8 | semidirect |
| 296. A4 | C22×#168 | semidirect |
| 304. C2×C5 | C8×S4 | semidirect |
| 314. C23 | D3×#168 | semidirect |
| 346. C5 | C2×C8×S4 | semidirect |
| 359. C22 | C2×D3×#168 | semidirect |
| 373. C2 | S4×#168 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C4×C5×S4 | 2 |
| C2×A4×#168 | 2 |
| C2×A4⋊#168 | 2 |
| S4×#168 | 2 |
| C2×D4×#168 | 3 |
| C2×D3×#168 | 4 |
| C2×C8×S4 | 5 |