Finite Group DatabasebetaFinite Group DatabasebetaOrder 27392.
342 conjugacy classes of subgroups; only the 110 normal subgroups are shown.
This group has 342 conjugacy classes of subgroups — too many to draw. Only its 110 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C107 | #99 |
| commutator | C2×C4 | C4×C8×C107 |
| Frattini, cosocle | C22×M16 | C22×C107 |
| socle | C22×C107 | #324 |
| prime radical | C24×C107 | M16 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C107 ◁ C22×C4×C107 ◁ C22×M16×C107 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 8. #349×C107 | C4 | semidirect |
| 108. #7193 | C107 | direct (N × H) |
| 117. C107 | #7193 | direct (N × H) |
| 123. #349 | C4×C107 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#349×C107 | 2 |
| C107×#1728 | 2 |
| #7193 | 107 |