Finite Group DatabasebetaFinite Group DatabasebetaOrder 1200.
116 conjugacy classes of subgroups; only the 10 normal subgroups are shown.
This group has 116 conjugacy classes of subgroups — too many to draw. Only its 10 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | D5×A5 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | C5⋊S5 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | C5×S5 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | C5×A5 | C22 | 1 | 1 | yes | characteristic, complemented |
| 5 | D5×S4 | — | 5 | 1 | no | maximal |
| 6 | C2×S5 | — | 5 | 1 | no | maximal |
| 7 | D5×AGL(1,𝔽5) | — | 6 | 1 | no | maximal |
| 8 | C2×D3×D5 | — | 10 | 1 | no | maximal |
| 9 | D5×A4 | — | 5 | 1 | no | — |
| 10 | C5⋊S4 | — | 5 | 1 | no | — |
| 11 | C5×S4 | — | 5 | 1 | no | — |
| 12 | C2×A5 | — | 5 | 1 | no | — |
| 13 | S5 | — | 5 | 1 | no | — |
| 14 | S5 | D5 | 1 | 1 | yes | characteristic, direct product factor |
| 15 | D52 | — | 6 | 1 | no | — |
| 16 | #1042 | — | 6 | 1 | no | — |
| 17 | C5×AGL(1,𝔽5) | — | 6 | 1 | no | — |
| 18 | D4×D5 | — | 15 | 1 | no | — |
| 19 | C2×D15 | — | 10 | 1 | no | — |
| 20 | C2×C5×D3 | — | 10 | 1 | no | — |
| 21 | C2×C3×D5 | — | 10 | 1 | no | — |
| 22 | C5×A4 | — | 5 | 1 | no | — |
| 23 | D3×D5 | — | 10 | 1 | no | — |
| 24 | D3×D5 | — | 10 | 1 | no | — |
| 25 | D3×D5 | — | 10 | 1 | no | — |
| 26 | D3×D5 | — | 10 | 1 | no | — |
| 27 | A5 | C2×D5 | 1 | 1 | yes | characteristic, complemented |
| 28 | C5⋊D5 | — | 6 | 1 | no | — |
| 29 | C5×D5 | — | 6 | 1 | no | — |
| 30 | C5×D5 | — | 6 | 1 | no | — |
| 31 | C2×S4 | — | 25 | 1 | no | Hall |
| 32 | C22×D5 | — | 5 | 1 | no | — |
| 33 | C22×D5 | — | 15 | 1 | no | — |
| 34 | C2×AGL(1,𝔽5) | — | 30 | 1 | no | — |
| 35 | C5×D4 | — | 15 | 1 | no | — |
| 36 | C22⋊D5 | — | 15 | 1 | no | — |
| 37 | C22⋊D5 | — | 15 | 1 | no | — |
| 38 | D20 | — | 15 | 1 | no | — |
| 39 | C4×D5 | — | 15 | 1 | no | — |
| 40 | C2×C3×C5 | — | 10 | 1 | no | — |
| 41 | D15 | — | 10 | 1 | no | — |
| 42 | D15 | — | 10 | 1 | no | — |
| 43 | C3×D5 | — | 10 | 1 | no | — |
| 44 | C3×D5 | — | 10 | 1 | no | — |
| 45 | C5×D3 | — | 10 | 1 | no | — |
| 46 | C5×D3 | — | 10 | 1 | no | — |
| 47 | C52 | — | 6 | 1 | no | Sylow |
| 48 | C22×D3 | — | 50 | 1 | no | — |
| 49 | C2×A4 | — | 25 | 1 | no | — |
| 50 | S4 | — | 5 | 1 | no | — |
| 51 | S4 | — | 25 | 1 | no | — |
| 52 | C22×C5 | — | 5 | 1 | no | — |
| 53 | C22×C5 | — | 15 | 1 | no | — |
| 54 | C2×D5 | — | 10 | 1 | no | — |
| 55 | C2×D5 | — | 15 | 1 | no | — |
| 56 | C2×D5 | — | 15 | 1 | no | — |
| 57 | C2×D5 | — | 15 | 1 | no | — |
| 58 | C2×D5 | — | 30 | 1 | no | — |
| 59 | C2×D5 | — | 30 | 1 | no | — |
| 60 | AGL(1,𝔽5) | — | 6 | 1 | no | — |
| 61 | AGL(1,𝔽5) | — | 30 | 1 | no | — |
| 62 | C4×C5 | — | 15 | 1 | no | — |
| 63 | Dic5 | — | 15 | 1 | no | — |
| 64 | C2×D4 | — | 75 | 1 | no | Sylow |
| 65 | C3×C5 | — | 10 | 1 | no | — |
| 66 | C22×C3 | — | 50 | 1 | no | — |
| 67 | C2×D3 | — | 10 | 1 | no | — |
| 68 | C2×D3 | — | 50 | 1 | no | — |
| 69 | C2×D3 | — | 50 | 1 | no | — |
| 70 | C2×D3 | — | 50 | 1 | no | — |
| 71 | C2×D3 | — | 50 | 1 | no | — |
| 72 | C2×D3 | — | 50 | 1 | no | — |
| 73 | A4 | — | 5 | 1 | no | — |
| 74 | C2×C5 | — | 10 | 1 | no | — |
| 75 | C2×C5 | — | 15 | 1 | no | — |
| 76 | C2×C5 | — | 30 | 1 | no | — |
| 77 | D5 | — | 6 | 1 | no | — |
| 78 | D5 | — | 10 | 1 | no | — |
| 79 | D5 | — | 15 | 1 | no | — |
| 80 | D5 | — | 30 | 1 | no | — |
| 81 | D5 | — | 120 | 1 | no | — |
| 82 | D5 | S5 | 1 | 1 | yes | characteristic, direct product factor |
| 83 | C23 | — | 25 | 1 | no | — |
| 84 | C23 | — | 75 | 1 | no | — |
| 85 | D4 | — | 15 | 1 | no | — |
| 86 | D4 | — | 75 | 1 | no | — |
| 87 | D4 | — | 75 | 1 | no | — |
| 88 | D4 | — | 75 | 1 | no | — |
| 89 | C2×C4 | — | 75 | 1 | no | — |
| 90 | C2×C3 | — | 10 | 1 | no | — |
| 91 | C2×C3 | — | 50 | 1 | no | — |
| 92 | C2×C3 | — | 50 | 1 | no | — |
| 93 | D3 | — | 10 | 1 | no | — |
| 94 | D3 | — | 10 | 1 | no | — |
| 95 | D3 | — | 50 | 1 | no | — |
| 96 | D3 | — | 50 | 1 | no | — |
| 97 | C5 | — | 6 | 1 | no | — |
| 98 | C5 | — | 24 | 1 | no | — |
| 99 | C5 | C2×S5 | 1 | 1 | yes | characteristic, complemented |
| 100 | C22 | — | 5 | 1 | no | — |
| 101 | C22 | — | 15 | 1 | no | — |
| 102 | C22 | — | 50 | 1 | no | — |
| 103 | C22 | — | 75 | 1 | no | — |
| 104 | C22 | — | 75 | 1 | no | — |
| 105 | C22 | — | 75 | 1 | no | — |
| 106 | C22 | — | 150 | 1 | no | — |
| 107 | C4 | — | 15 | 1 | no | — |
| 108 | C4 | — | 75 | 1 | no | — |
| 109 | C3 | — | 10 | 1 | no | Sylow |
| 110 | C2 | — | 5 | 1 | no | — |
| 111 | C2 | — | 10 | 1 | no | — |
| 112 | C2 | — | 15 | 1 | no | — |
| 113 | C2 | — | 50 | 1 | no | — |
| 114 | C2 | — | 75 | 1 | no | — |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, socle, cosocle, hypocenter | C5×A5 | C22 |
| Fitting | C5 | C2×S5 |
| LCS | G ▷ C5×A5 | |
| UCS | C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. D5×A5 | C2 | semidirect |
| 2. C5⋊S5 | C2 | semidirect |
| 3. C5×S5 | C2 | semidirect |
| 4. C5×A5 | C22 | semidirect |
| 14. S5 | D5 | direct (N × H) |
| 27. A5 | C2×D5 | semidirect |
| 82. D5 | S5 | direct (N × H) |
| 99. C5 | C2×S5 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| D5×A5 | 2 |
| C5⋊S5 | 2 |
| C5×S5 | 2 |
| D5×S4 | 5 |
| C2×S5 | 5 |
| D5×AGL(1,𝔽5) | 6 |
| C2×D3×D5 | 10 |