Finite Group DatabasebetaFinite Group DatabasebetaOrder 800.
196 conjugacy classes of subgroups; only the 63 normal subgroups are shown.
This group has 196 conjugacy classes of subgroups — too many to draw. Only its 63 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 | C22×#1042 |
| commutator, socle | C2×C52 | C22×C4 |
| Fitting | C52×D4 | C4 |
| prime radical | C5×D4×D5 | C2 |
| cosocle | C2×C5×D5 | C23 |
| hypercenter | D4 | #1042 |
| hypocenter | C52 | C4×D4 |
| LCS | G ▷ C2×C52 ▷ C52 | |
| UCS | C1 ◁ C2 ◁ D4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×#1042 | C2 | semidirect |
| 3. C22⋊#1042 | C2 | semidirect |
| 4. C4⋊#1042 | C2 | semidirect |
| 5. C4×#1042 | C2 | semidirect |
| 11. C52×D4 | C4 | semidirect |
| 12. C5×C22⋊D5 | C4 | semidirect |
| 13. C5×D20 | C4 | semidirect |
| 17. C22×C52 | C2×C4 | semidirect |
| 20. C2×C5×D5 | C2×C4 | semidirect |
| 24. #1042 | D4 | direct (N × H) |
| 25. C4×C52 | C2×C4 | semidirect |
| 26. C5×Dic5 | C2×C4 | semidirect |
| 51. C5×D4 | Dic5 | semidirect |
| 52. C5×D4 | AGL(1,𝔽5) | semidirect |
| 54. C22⋊D5 | Dic5 | semidirect |
| 59. D20 | Dic5 | semidirect |
| 62. C52 | C4×D4 | semidirect |
| 68. C22×C5 | C2×Dic5 | semidirect |
| 69. C22×C5 | C2×AGL(1,𝔽5) | semidirect |
| 72. C2×D5 | C2×Dic5 | semidirect |
| 79. C4×C5 | C2×Dic5 | semidirect |
| 80. C4×C5 | C2×AGL(1,𝔽5) | semidirect |
| 84. Dic5 | C2×Dic5 | semidirect |
| 105. D4 | #1042 | direct (N × H) |
| 112. C5 | D4×Dic5 | semidirect |
| 113. C5 | D4×AGL(1,𝔽5) | semidirect |
| 118. C22 | C2×#1042 | semidirect |
| 123. C4 | C2×#1042 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×#1042 | 2 |
| C5×D4×D5 | 2 |
| C22⋊#1042 | 2 |
| C4⋊#1042 | 2 |
| C4×#1042 | 2 |
| D4×AGL(1,𝔽5) | 5 |
| D4×Dic5 | 5 |