Finite Group DatabasebetaFinite Group DatabasebetaOrder 448.
258 conjugacy classes of subgroups; only the 87 normal subgroups are shown.
This group has 258 conjugacy classes of subgroups — too many to draw. Only its 87 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C22×D7 |
| commutator | C22×C7 | C22×C4 |
| Fitting | C2×C42×C7 | C2 |
| socle, cosocle | C23×C7 | C23 |
| prime radical | C22×D28 | C2 |
| hypercenter | C2×C42 | D7 |
| hypocenter | C7 | #390 |
| LCS | G ▷ C22×C7 ▷ C7 | |
| UCS | C1 ◁ C23 ◁ C2×C42 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×C42×C7 | C2 | semidirect |
| 3. C2×(C2×D7)⋊C4 | C2 | semidirect |
| 4. C2×C4⋊Dic7 | C2 | semidirect |
| 11. C2×D28 | C4 | semidirect |
| 27. C2×C4×C7 | D4 | semidirect |
| 34. C2×C42 | D7 | semidirect |
| 46. C4×C7 | C22⋊C4 | semidirect |
| 77. C2×C4 | D28 | semidirect |
| 79. C7 | #390 | semidirect |
| 92. C4 | (C2×D7)⋊C4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×D28 | 2 |
| C2×C42×C7 | 2 |
| C2×(C2×D7)⋊C4 | 2 |
| C2×C4⋊Dic7 | 2 |
| #390 | 7 |