Finite Group DatabasebetaFinite Group DatabasebetaOrder 1600.
752 conjugacy classes of subgroups; only the 352 normal subgroups are shown.
This group has 752 conjugacy classes of subgroups — too many to draw. Only its 352 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22×C4 | D52 |
| commutator, hypocenter | C52 | C23×C8 |
| Frattini | C4 | C22×D52 |
| Fitting | C22×C4×C52 | C22 |
| socle | C23×C52 | C2×C4 |
| prime radical | C23×C5×D5 | C4 |
| cosocle | C4×C52 | C24 |
| LCS | G ▷ C52 | |
| UCS | C1 ◁ C22×C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C22×#6029 | C2 | semidirect |
| 3. C22×C5×#168 | C2 | semidirect |
| 4. C2×D5×#168 | C2 | direct (N × H) |
| 10. C2×#6029 | C22 | semidirect |
| 11. C2×C5×#168 | C22 | semidirect |
| 12. D5×#168 | C22 | direct (N × H) |
| 16. C22×C5×D5 | C8 | semidirect |
| 23. #6029 | C23 | semidirect |
| 24. C5×#168 | C23 | semidirect |
| 32. C22×#168 | D5 | direct (N × H) |
| 34. C22×C52 | C2×C8 | semidirect |
| 36. C2×C5×D5 | C2×C8 | semidirect |
| 57. C2×#168 | C2×D5 | direct (N × H) |
| 60. C2×C52 | C22×C8 | semidirect |
| 62. C5×D5 | C22×C8 | semidirect |
| 68. C22×D5 | #168 | direct (N × H) |
| 86. #168 | C22×D5 | direct (N × H) |
| 91. C52 | C23×C8 | semidirect |
| 96. C22×C5 | C8×D5 | semidirect |
| 97. C22×C5 | C2×#168 | semidirect |
| 100. C2×D5 | C2×#168 | direct (N × H) |
| 121. C2×C5 | C2×C8×D5 | semidirect |
| 122. C2×C5 | C22×#168 | semidirect |
| 125. D5 | C22×#168 | direct (N × H) |
| 137. C5 | C22×C8×D5 | semidirect |
| 138. C5 | C23×#168 | semidirect |
| 141. C22 | D5×#168 | direct (N × H) |
| 147. C2 | C2×D5×#168 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C4×C5×D5 | 2 |
| C22×#6029 | 2 |
| C22×C5×#168 | 2 |
| C2×D5×#168 | 2 |
| C23×#168 | 5 |
| C22×C8×D5 | 5 |