Finite Group DatabasebetaFinite Group DatabasebetaOrder 448.
170 conjugacy classes of subgroups; only the 106 normal subgroups are shown.
This group has 170 conjugacy classes of subgroups — too many to draw. Only its 106 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C7 | C23 |
| commutator | C22 | C22×C4×C7 |
| Frattini, cosocle | C23 | C23×C7 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C7 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 13. C7×C4⋊C4 | C4 | semidirect |
| 15. #389 | C7 | direct (N × H) |
| 48. C4×C7 | C4⋊C4 | semidirect |
| 56. C4⋊C4 | C4×C7 | semidirect |
| 81. C7 | #389 | direct (N × H) |
| 93. C4 | C7×C4⋊C4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C7×C4⋊C4 | 2 |
| C2×C42×C7 | 2 |
| C7×#95 | 2 |
| #389 | 7 |