Finite Group DatabasebetaFinite Group DatabasebetaOrder 320.
294 conjugacy classes of subgroups; only the 138 normal subgroups are shown.
This group has 294 conjugacy classes of subgroups — too many to draw. Only its 138 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C23×AGL(1,𝔽5) |
| commutator, socle | C2×C5 | C23×C4 |
| Fitting | C5×#41 | C4 |
| prime radical | D5×#41 | C2 |
| cosocle | C2×D5 | C24 |
| hypercenter | #41 | AGL(1,𝔽5) |
| hypocenter | C5 | #520 |
| LCS | G ▷ C2×C5 ▷ C5 | |
| UCS | C1 ◁ C2 ◁ #41 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. Q8×AGL(1,𝔽5) | C2 | semidirect |
| 3. D4×AGL(1,𝔽5) | C2 | semidirect |
| 4. #4130 | C2 | semidirect |
| 5. C2×C4⋊AGL(1,𝔽5) | C2 | semidirect |
| 7. C5×#41 | C4 | semidirect |
| 8. #713 | C4 | semidirect |
| 10. #711 | C4 | semidirect |
| 12. #709 | C4 | semidirect |
| 24. C5×Q8 | C2×C4 | semidirect |
| 25. C5×D4 | C2×C4 | semidirect |
| 26. C2×C4×C5 | C2×C4 | semidirect |
| 27. C22⋊D5 | C2×C4 | semidirect |
| 28. C2×Dic5 | C2×C4 | semidirect |
| 29. D20 | C2×C4 | semidirect |
| 30. C4×D5 | C2×C4 | semidirect |
| 31. C4×D5 | C2×C4 | semidirect |
| 34. Dic10 | C2×C4 | semidirect |
| 54. #41 | AGL(1,𝔽5) | semidirect |
| 71. Q8 | C2×AGL(1,𝔽5) | semidirect |
| 74. D4 | C2×AGL(1,𝔽5) | semidirect |
| 83. C2×C4 | C2×AGL(1,𝔽5) | semidirect |
| 84. C5 | #520 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| D5×#41 | 2 |
| Q8×AGL(1,𝔽5) | 2 |
| D4×AGL(1,𝔽5) | 2 |
| #4130 | 2 |
| C2×C4⋊AGL(1,𝔽5) | 2 |
| #520 | 5 |