Finite Group DatabasebetaFinite Group DatabasebetaOrder 21376.
244 conjugacy classes of subgroups; only the 80 normal subgroups are shown.
This group has 244 conjugacy classes of subgroups — too many to draw. Only its 80 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C4×C167 | #124 |
| commutator, Frattini, cosocle | C22×C4 | C23×C167 |
| socle | C2×C167 | #393 |
| prime radical | C2×#41×C167 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C167 ◁ C22×C4×C167 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #533×C167 | C2 | semidirect |
| 5. #344×C167 | C2 | semidirect |
| 6. #339×C167 | C2 | semidirect |
| 7. H(3,ℤ/4)×C167 | C2 | semidirect |
| 28. C2×Q8×C167 | Q8 | semidirect |
| 30. C2×D4×C167 | Q8 | semidirect |
| 101. C167 | #2031 | direct (N × H) |
| 102. #2031 | C167 | direct (N × H) |
| 103. #533 | C2×C167 | semidirect |
| 107. #344 | C2×C167 | semidirect |
| 108. #339 | C2×C167 | semidirect |
| 109. H(3,ℤ/4) | C2×C167 | semidirect |
| 130. C2×Q8 | Q8×C167 | semidirect |
| 132. C2×D4 | Q8×C167 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #533×C167 | 2 |
| #421×C167 | 2 |
| C2×#104×C167 | 2 |
| #410×C167 | 2 |
| #344×C167 | 2 |
| #339×C167 | 2 |
| H(3,ℤ/4)×C167 | 2 |
| #2031 | 167 |