Finite Group DatabasebetaFinite Group DatabasebetaOrder 21376.
318 conjugacy classes of subgroups; only the 100 normal subgroups are shown.
This group has 318 conjugacy classes of subgroups — too many to draw. Only its 100 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C167 | C2×D4 |
| commutator, Frattini, cosocle | C22×C4 | C23×C167 |
| prime radical | C22×D4×C167 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C167 ◁ C2×C42×C167 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×#103×C167 | C2 | semidirect |
| 3. C2×#103×C167 | C2 | semidirect |
| 6. #389×C167 | C2 | semidirect |
| 7. #336×C167 | C2 | semidirect |
| 129. C167 | #2042 | direct (N × H) |
| 130. #2042 | C167 | direct (N × H) |
| 132. C2×#103 | C2×C167 | semidirect |
| 133. C2×#103 | C2×C167 | semidirect |
| 136. #389 | C2×C167 | semidirect |
| 137. #336 | C2×C167 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#124×C167 | 2 |
| C2×#103×C167 | 2 |
| C2×#102×C167 | 2 |
| #389×C167 | 2 |
| #336×C167 | 2 |
| #2042 | 167 |