Finite Group DatabasebetaFinite Group DatabasebetaOrder 7200.
246 conjugacy classes of subgroups; only the 5 normal subgroups are shown.
This group has 246 conjugacy classes of subgroups — too many to draw. Only its 5 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 | A52 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | A4⋊S5 | — | 5 | 2 | no | maximal |
| 3 | A5⋊AGL(1,𝔽5) | — | 6 | 2 | no | maximal |
| 4 | A4×A5 | — | 5 | 2 | no | — |
| 5 | D3×S5 | — | 10 | 2 | no | maximal |
| 6 | D5×A5 | — | 6 | 2 | no | — |
| 7 | C22⋊S5 | — | 15 | 2 | no | — |
| 8 | D3×A5 | — | 10 | 2 | no | — |
| 9 | ΓL(2,𝔽4) | — | 10 | 2 | no | — |
| 10 | C3×S5 | — | 10 | 2 | no | — |
| 11 | C5×A5 | — | 6 | 2 | no | — |
| 12 | A4⋊S4 | — | 25 | 1 | no | Hall |
| 13 | A4⋊AGL(1,𝔽5) | — | 30 | 2 | no | — |
| 14 | C22×A5 | — | 5 | 2 | no | — |
| 15 | C2×S5 | — | 15 | 2 | no | — |
| 16 | A5⋊C4 | — | 15 | 2 | no | — |
| 17 | D5⋊AGL(1,𝔽5) | — | 36 | 1 | no | — |
| 18 | C3×A5 | — | 10 | 2 | no | — |
| 19 | A42 | — | 25 | 1 | no | — |
| 20 | D3×S4 | — | 50 | 2 | no | — |
| 21 | D5×A4 | — | 30 | 2 | no | — |
| 22 | D3×AGL(1,𝔽5) | — | 60 | 2 | no | — |
| 23 | C2×A5 | — | 15 | 2 | no | — |
| 24 | S5 | — | 10 | 2 | no | — |
| 25 | S5 | — | 60 | 2 | no | maximal |
| 26 | D52 | — | 36 | 1 | no | — |
| 27 | #1044 | — | 36 | 1 | no | — |
| 28 | #1043 | — | 36 | 1 | no | — |
| 29 | #1020 | — | 75 | 2 | no | — |
| 30 | GL(2,ℤ/4) | — | 75 | 2 | no | — |
| 31 | C22⋊AGL(1,𝔽5) | — | 90 | 2 | no | — |
| 32 | C2×D32 | — | 100 | 1 | no | — |
| 33 | D3×A4 | — | 50 | 2 | no | — |
| 34 | C3⋊S4 | — | 50 | 2 | no | — |
| 35 | C3×S4 | — | 50 | 2 | no | — |
| 36 | C5×A4 | — | 30 | 2 | no | — |
| 37 | D3×D5 | — | 60 | 2 | no | — |
| 38 | ΓL(1,𝔽16) | — | 60 | 2 | no | — |
| 39 | C3×AGL(1,𝔽5) | — | 60 | 2 | no | — |
| 40 | A5 | — | 60 | 2 | no | — |
| 41 | A5 | S5 | 1 | 2 | yes | complemented |
| 42 | C5⋊D5 | — | 36 | 1 | no | — |
| 43 | C5×D5 | — | 36 | 2 | no | — |
| 44 | C22⋊A4 | — | 25 | 2 | no | — |
| 45 | C22×A4 | — | 25 | 2 | no | — |
| 46 | C2×S4 | — | 75 | 2 | no | — |
| 47 | D3×D4 | — | 150 | 2 | no | — |
| 48 | SL(2,ℤ/4) | — | 75 | 2 | no | — |
| 49 | C22×D5 | — | 30 | 2 | no | — |
| 50 | C2×AGL(1,𝔽5) | — | 90 | 2 | no | — |
| 51 | C2×AGL(1,𝔽5) | — | 90 | 2 | no | — |
| 52 | C2×C3⋊D3 | — | 100 | 1 | no | — |
| 53 | C2×C3×D3 | — | 100 | 2 | no | — |
| 54 | C3×A4 | — | 50 | 2 | no | — |
| 55 | D32 | — | 100 | 1 | no | — |
| 56 | D32 | — | 100 | 2 | no | — |
| 57 | D32 | — | 100 | 1 | no | — |
| 58 | #120 | — | 225 | 1 | no | Sylow |
| 59 | D15 | — | 60 | 2 | no | — |
| 60 | C3×D5 | — | 60 | 2 | no | — |
| 61 | C5×D3 | — | 60 | 2 | no | — |
| 62 | C52 | — | 36 | 1 | no | Sylow |
| 63 | C22×D3 | — | 50 | 2 | no | — |
| 64 | C22×D3 | — | 150 | 2 | no | — |
| 65 | C2×A4 | — | 75 | 2 | no | — |
| 66 | S4 | — | 300 | 2 | no | — |
| 67 | S4 | — | 300 | 2 | no | — |
| 68 | S4 | — | 50 | 2 | no | — |
| 69 | C3×D4 | — | 150 | 2 | no | — |
| 70 | #67 | — | 150 | 2 | no | — |
| 71 | #67 | — | 150 | 2 | no | — |
| 72 | D12 | — | 150 | 2 | no | — |
| 73 | C4×D3 | — | 150 | 2 | no | — |
| 74 | C22×C5 | — | 30 | 2 | no | — |
| 75 | C2×D5 | — | 90 | 2 | no | — |
| 76 | C2×D5 | — | 90 | 2 | no | — |
| 77 | AGL(1,𝔽5) | — | 60 | 2 | no | — |
| 78 | AGL(1,𝔽5) | — | 360 | 2 | no | — |
| 79 | AGL(1,𝔽5) | — | 180 | 2 | no | — |
| 80 | C2×C32 | — | 100 | 1 | no | — |
| 81 | C3⋊D3 | — | 100 | 1 | no | — |
| 82 | C3⋊D3 | — | 100 | 1 | no | — |
| 83 | C3×D3 | — | 100 | 2 | no | — |
| 84 | C3×D3 | — | 100 | 2 | no | — |
| 85 | C24 | — | 25 | 1 | no | — |
| 86 | C2×D4 | — | 225 | 2 | no | — |
| 87 | C2×D4 | — | 225 | 1 | no | — |
| 88 | C22⋊C4 | — | 225 | 2 | no | — |
| 89 | C22⋊C4 | — | 225 | 1 | no | — |
| 90 | C3×C5 | — | 60 | 2 | no | — |
| 91 | C22×C3 | — | 50 | 2 | no | — |
| 92 | C22×C3 | — | 150 | 2 | no | — |
| 93 | C2×D3 | — | 300 | 2 | no | — |
| 94 | C2×D3 | — | 600 | 1 | no | — |
| 95 | C2×D3 | — | 100 | 2 | no | — |
| 96 | C2×D3 | — | 150 | 2 | no | — |
| 97 | C2×D3 | — | 150 | 2 | no | — |
| 98 | C2×D3 | — | 150 | 2 | no | — |
| 99 | A4 | — | 5 | 2 | no | — |
| 100 | A4 | — | 300 | 2 | no | — |
| 101 | A4 | — | 100 | 2 | no | — |
| 102 | C3×C4 | — | 150 | 2 | no | — |
| 103 | Dic3 | — | 150 | 2 | no | — |
| 104 | C2×C5 | — | 90 | 2 | no | — |
| 105 | D5 | — | 6 | 2 | no | — |
| 106 | D5 | — | 90 | 2 | no | — |
| 107 | D5 | — | 360 | 2 | no | — |
| 108 | C32 | — | 100 | 1 | no | Sylow |
| 109 | C23 | — | 75 | 2 | no | — |
| 110 | C23 | — | 225 | 1 | no | — |
| 111 | C23 | — | 225 | 1 | no | — |
| 112 | D4 | — | 150 | 2 | no | — |
| 113 | D4 | — | 450 | 2 | no | — |
| 114 | D4 | — | 450 | 2 | no | — |
| 115 | C2×C4 | — | 225 | 2 | no | — |
| 116 | C2×C4 | — | 225 | 1 | no | — |
| 117 | C2×C3 | — | 100 | 2 | no | — |
| 118 | C2×C3 | — | 150 | 2 | no | — |
| 119 | C2×C3 | — | 200 | 1 | no | — |
| 120 | D3 | — | 10 | 2 | no | — |
| 121 | D3 | — | 600 | 1 | no | — |
| 122 | D3 | — | 600 | 1 | no | — |
| 123 | D3 | — | 100 | 2 | no | — |
| 124 | D3 | — | 150 | 2 | no | — |
| 125 | C5 | — | 6 | 2 | no | — |
| 126 | C5 | — | 72 | 2 | no | — |
| 127 | C22 | — | 5 | 2 | no | — |
| 128 | C22 | — | 75 | 2 | no | — |
| 129 | C22 | — | 150 | 2 | no | — |
| 130 | C22 | — | 450 | 1 | no | — |
| 131 | C22 | — | 225 | 1 | no | — |
| 132 | C22 | — | 225 | 2 | no | — |
| 133 | C4 | — | 150 | 2 | no | — |
| 134 | C4 | — | 450 | 1 | no | — |
| 135 | C3 | — | 10 | 2 | no | — |
| 136 | C3 | — | 200 | 1 | no | — |
| 137 | C2 | — | 15 | 2 | no | — |
| 138 | C2 | — | 100 | 1 | no | — |
| 139 | C2 | — | 225 | 1 | no | — |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, socle, cosocle, hypocenter | A52 | C2 |
| LCS | G ▷ A52 | |
| UCS | C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. A52 | C2 | semidirect |
| 41. A5 | S5 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| A52 | 2 |
| A4⋊S5 | 5 |
| A5⋊AGL(1,𝔽5) | 6 |
| D3×S5 | 10 |
| S5 | 60 |