Finite Group DatabasebetaFinite Group DatabasebetaOrder 1600.
672 conjugacy classes of subgroups; only the 216 normal subgroups are shown.
This group has 672 conjugacy classes of subgroups — too many to draw. Only its 216 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C52 | #120 |
| commutator, Frattini, cosocle | C2×C4 | C23×C52 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C52 ◁ C2×C4×C52 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C52×H(5,𝔽2) | C2 | semidirect |
| 2. C52×AGL(1,ℤ/8) | C2 | semidirect |
| 3. C52×#127 | C2 | semidirect |
| 4. C52×#104 | C2 | semidirect |
| 5. C52×#100 | C2 | semidirect |
| 12. C2×C52×D4 | C22 | semidirect |
| 15. C52×M16 | C22 | semidirect |
| 16. C42×C52 | C22 | semidirect |
| 17. C5×#453 | C5 | direct (N × H) |
| 21. C52×Q8 | D4 | semidirect |
| 27. C52×D4 | D4 | semidirect |
| 33. C5×H(5,𝔽2) | C2×C5 | semidirect |
| 34. C5×AGL(1,ℤ/8) | C2×C5 | semidirect |
| 35. C5×#127 | C2×C5 | semidirect |
| 36. C5×#104 | C2×C5 | semidirect |
| 37. C5×#100 | C2×C5 | semidirect |
| 54. C2×C5×D4 | C22×C5 | semidirect |
| 57. C5×M16 | C22×C5 | semidirect |
| 58. C42×C5 | C22×C5 | semidirect |
| 59. #453 | C52 | direct (N × H) |
| 68. C5×Q8 | C5×D4 | semidirect |
| 74. C5×D4 | C5×D4 | semidirect |
| 80. H(5,𝔽2) | C2×C52 | semidirect |
| 81. AGL(1,ℤ/8) | C2×C52 | semidirect |
| 82. #127 | C2×C52 | semidirect |
| 83. #104 | C2×C52 | semidirect |
| 84. #100 | C2×C52 | semidirect |
| 85. C52 | #453 | direct (N × H) |
| 102. C2×D4 | C22×C52 | semidirect |
| 105. M16 | C22×C52 | semidirect |
| 106. C42 | C22×C52 | semidirect |
| 115. Q8 | C52×D4 | semidirect |
| 121. D4 | C52×D4 | semidirect |
| 127. C5 | C5×#453 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C52×H(5,𝔽2) | 2 |
| C52×AGL(1,ℤ/8) | 2 |
| C52×#127 | 2 |
| C52×#104 | 2 |
| C52×#100 | 2 |
| C5×#453 | 5 |