Finite Group DatabasebetaFinite Group DatabasebetaOrder 1920.
200 conjugacy classes of subgroups; only the 16 normal subgroups are shown.
This group has 200 conjugacy classes of subgroups — too many to draw. Only its 16 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 | Q16×A5 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | D8×A5 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | C16×A5 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | Q8×A5 | — | 2 | 1 | no | — |
| 5 | D4×A5 | — | 2 | 1 | no | — |
| 6 | C8×A5 | C22 | 1 | 1 | yes | characteristic |
| 7 | A4×SD32 | — | 5 | 1 | no | maximal, Hall |
| 8 | D5×SD32 | — | 6 | 1 | no | maximal |
| 9 | C22×A5 | — | 4 | 1 | no | — |
| 10 | C4×A5 | — | 4 | 1 | no | — |
| 11 | C4×A5 | D4 | 1 | 1 | yes | characteristic |
| 12 | A4×Q16 | — | 5 | 1 | no | — |
| 13 | A4×D8 | — | 5 | 1 | no | — |
| 14 | D3×SD32 | — | 10 | 1 | no | maximal |
| 15 | A4×C16 | — | 5 | 1 | no | — |
| 16 | D5×Q16 | — | 6 | 1 | no | — |
| 17 | D5×D8 | — | 6 | 1 | no | — |
| 18 | C5×SD32 | — | 6 | 1 | no | — |
| 19 | #3960 | — | 6 | 1 | no | — |
| 20 | #3959 | — | 6 | 1 | no | — |
| 21 | #3932 | — | 6 | 1 | no | — |
| 22 | D5×C16 | — | 6 | 1 | no | — |
| 23 | C22×SD32 | — | 5 | 1 | no | Sylow |
| 24 | C2×A5 | — | 8 | 1 | no | — |
| 25 | C2×A5 | D8 | 1 | 1 | yes | characteristic |
| 26 | Q8×A4 | — | 10 | 1 | no | — |
| 27 | D4×A4 | — | 10 | 1 | no | — |
| 28 | D3×Q16 | — | 10 | 1 | no | — |
| 29 | D3×D8 | — | 10 | 1 | no | — |
| 30 | C8×A4 | — | 5 | 1 | no | — |
| 31 | C3×SD32 | — | 10 | 1 | no | — |
| 32 | #828 | — | 10 | 1 | no | — |
| 33 | #827 | — | 10 | 1 | no | — |
| 34 | #800 | — | 10 | 1 | no | — |
| 35 | D3×C16 | — | 10 | 1 | no | — |
| 36 | Q8×D5 | — | 12 | 1 | no | — |
| 37 | D4×D5 | — | 12 | 1 | no | — |
| 38 | C5×Q16 | — | 6 | 1 | no | — |
| 39 | C5×D8 | — | 6 | 1 | no | — |
| 40 | #689 | — | 12 | 1 | no | — |
| 41 | #686 | — | 12 | 1 | no | — |
| 42 | Dic20 | — | 6 | 1 | no | — |
| 43 | D40 | — | 6 | 1 | no | — |
| 44 | C8×D5 | — | 6 | 1 | no | — |
| 45 | C5×C16 | — | 6 | 1 | no | — |
| 46 | #672 | — | 6 | 1 | no | — |
| 47 | C22×Q16 | — | 5 | 1 | no | — |
| 48 | C22×D8 | — | 5 | 1 | no | — |
| 49 | C2×SD32 | — | 15 | 1 | no | — |
| 50 | C2×SD32 | — | 15 | 1 | no | — |
| 51 | C2×SD32 | — | 15 | 1 | no | — |
| 52 | C2×SD32 | — | 15 | 1 | no | — |
| 53 | C22×C16 | — | 5 | 1 | no | — |
| 54 | A5 | SD32 | 1 | 1 | yes | characteristic, direct product factor |
| 55 | C22×A4 | — | 20 | 1 | no | — |
| 56 | D3×Q8 | — | 20 | 1 | no | — |
| 57 | D3×D4 | — | 20 | 1 | no | — |
| 58 | C4×A4 | — | 5 | 1 | no | — |
| 59 | C4×A4 | — | 20 | 1 | no | — |
| 60 | C3×Q16 | — | 10 | 1 | no | — |
| 61 | C3×D8 | — | 10 | 1 | no | — |
| 62 | #216 | — | 20 | 1 | no | — |
| 63 | #213 | — | 20 | 1 | no | — |
| 64 | Dic12 | — | 10 | 1 | no | — |
| 65 | D24 | — | 10 | 1 | no | — |
| 66 | D3×C8 | — | 10 | 1 | no | — |
| 67 | C3×C16 | — | 10 | 1 | no | — |
| 68 | C3⋊C16 | — | 10 | 1 | no | — |
| 69 | C22×D5 | — | 24 | 1 | no | — |
| 70 | C5×Q8 | — | 12 | 1 | no | — |
| 71 | C5×D4 | — | 12 | 1 | no | — |
| 72 | C22⋊D5 | — | 24 | 1 | no | — |
| 73 | D20 | — | 12 | 1 | no | — |
| 74 | C4×D5 | — | 6 | 1 | no | — |
| 75 | C4×D5 | — | 24 | 1 | no | — |
| 76 | Dic10 | — | 12 | 1 | no | — |
| 77 | Dic10 | — | 24 | 1 | no | — |
| 78 | C5×C8 | — | 6 | 1 | no | — |
| 79 | #168 | — | 6 | 1 | no | — |
| 80 | C22×Q8 | — | 10 | 1 | no | — |
| 81 | C22×D4 | — | 10 | 1 | no | — |
| 82 | C2×Q16 | — | 15 | 1 | no | — |
| 83 | C2×Q16 | — | 15 | 1 | no | — |
| 84 | C2×Q16 | — | 30 | 1 | no | — |
| 85 | C2×D8 | — | 15 | 1 | no | — |
| 86 | C2×D8 | — | 15 | 1 | no | — |
| 87 | C2×D8 | — | 30 | 1 | no | — |
| 88 | C22×C8 | — | 5 | 1 | no | — |
| 89 | SD32 | — | 15 | 2 | no | — |
| 90 | SD32 | — | 15 | 1 | no | — |
| 91 | SD32 | — | 15 | 1 | no | — |
| 92 | SD32 | — | 15 | 1 | no | — |
| 93 | SD32 | A5 | 1 | 1 | yes | maximal normal, characteristic, direct product factor |
| 94 | C2×C16 | — | 15 | 1 | no | — |
| 95 | C2×C16 | — | 15 | 1 | no | — |
| 96 | C22×D3 | — | 40 | 1 | no | — |
| 97 | C2×A4 | — | 5 | 1 | no | — |
| 98 | C2×A4 | — | 40 | 1 | no | — |
| 99 | C3×Q8 | — | 20 | 1 | no | — |
| 100 | C3×D4 | — | 20 | 1 | no | — |
| 101 | #67 | — | 40 | 1 | no | — |
| 102 | D12 | — | 20 | 1 | no | — |
| 103 | C4×D3 | — | 10 | 1 | no | — |
| 104 | C4×D3 | — | 40 | 1 | no | — |
| 105 | Dic6 | — | 20 | 1 | no | — |
| 106 | Dic6 | — | 40 | 1 | no | — |
| 107 | C3×C8 | — | 10 | 1 | no | — |
| 108 | C3⋊C8 | — | 10 | 1 | no | — |
| 109 | C22×C5 | — | 24 | 1 | no | — |
| 110 | C2×D5 | — | 6 | 1 | no | — |
| 111 | C2×D5 | — | 24 | 1 | no | — |
| 112 | C2×D5 | — | 48 | 1 | no | — |
| 113 | C2×D5 | — | 48 | 1 | no | — |
| 114 | C4×C5 | — | 6 | 1 | no | — |
| 115 | C4×C5 | — | 24 | 1 | no | — |
| 116 | Dic5 | — | 6 | 1 | no | — |
| 117 | Dic5 | — | 24 | 1 | no | — |
| 118 | C24 | — | 20 | 1 | no | — |
| 119 | C2×Q8 | — | 30 | 1 | no | — |
| 120 | C2×Q8 | — | 30 | 1 | no | — |
| 121 | C2×Q8 | — | 60 | 1 | no | — |
| 122 | C2×D4 | — | 30 | 1 | no | — |
| 123 | C2×D4 | — | 30 | 1 | no | — |
| 124 | C2×D4 | — | 60 | 1 | no | — |
| 125 | C22×C4 | — | 5 | 1 | no | — |
| 126 | C22×C4 | — | 20 | 1 | no | — |
| 127 | Q16 | — | 15 | 1 | no | — |
| 128 | Q16 | — | 30 | 1 | no | — |
| 129 | Q16 | — | 30 | 1 | no | — |
| 130 | Q16 | C2×A5 | 1 | 1 | yes | characteristic, complemented |
| 131 | D8 | — | 15 | 1 | no | — |
| 132 | D8 | — | 30 | 1 | no | — |
| 133 | D8 | — | 30 | 1 | no | — |
| 134 | D8 | C2×A5 | 1 | 1 | yes | characteristic |
| 135 | C2×C8 | — | 15 | 1 | no | — |
| 136 | C2×C8 | — | 15 | 1 | no | — |
| 137 | C16 | — | 15 | 1 | no | — |
| 138 | C16 | C2×A5 | 1 | 1 | yes | characteristic, complemented |
| 139 | C22×C3 | — | 40 | 1 | no | — |
| 140 | C2×D3 | — | 10 | 1 | no | — |
| 141 | C2×D3 | — | 40 | 1 | no | — |
| 142 | C2×D3 | — | 80 | 1 | no | — |
| 143 | C2×D3 | — | 80 | 1 | no | — |
| 144 | A4 | — | 5 | 1 | no | — |
| 145 | C3×C4 | — | 10 | 1 | no | — |
| 146 | C3×C4 | — | 40 | 1 | no | — |
| 147 | Dic3 | — | 10 | 1 | no | — |
| 148 | Dic3 | — | 40 | 1 | no | — |
| 149 | C2×C5 | — | 6 | 1 | no | — |
| 150 | C2×C5 | — | 48 | 1 | no | — |
| 151 | D5 | — | 6 | 1 | no | — |
| 152 | D5 | — | 6 | 1 | no | — |
| 153 | D5 | — | 48 | 1 | no | — |
| 154 | C23 | — | 5 | 1 | no | — |
| 155 | C23 | — | 40 | 1 | no | — |
| 156 | C23 | — | 60 | 1 | no | — |
| 157 | C23 | — | 60 | 1 | no | — |
| 158 | C23 | — | 120 | 1 | no | — |
| 159 | Q8 | — | 2 | 1 | no | — |
| 160 | Q8 | — | 30 | 1 | no | — |
| 161 | Q8 | — | 60 | 1 | no | — |
| 162 | Q8 | — | 60 | 1 | no | — |
| 163 | D4 | — | 2 | 1 | no | — |
| 164 | D4 | — | 30 | 1 | no | — |
| 165 | D4 | — | 60 | 1 | no | — |
| 166 | D4 | — | 60 | 1 | no | — |
| 167 | C2×C4 | — | 15 | 1 | no | — |
| 168 | C2×C4 | — | 15 | 1 | no | — |
| 169 | C2×C4 | — | 60 | 1 | no | — |
| 170 | C2×C4 | — | 60 | 1 | no | — |
| 171 | C8 | — | 15 | 1 | no | — |
| 172 | C8 | C22×A5 | 1 | 1 | yes | characteristic |
| 173 | C2×C3 | — | 10 | 1 | no | — |
| 174 | C2×C3 | — | 80 | 1 | no | — |
| 175 | D3 | — | 10 | 1 | no | — |
| 176 | D3 | — | 10 | 1 | no | — |
| 177 | D3 | — | 80 | 1 | no | — |
| 178 | C5 | — | 6 | 1 | no | Sylow |
| 179 | C22 | — | 4 | 1 | no | — |
| 180 | C22 | — | 5 | 1 | no | — |
| 181 | C22 | — | 15 | 1 | no | — |
| 182 | C22 | — | 15 | 1 | no | — |
| 183 | C22 | — | 60 | 1 | no | — |
| 184 | C22 | — | 120 | 1 | no | — |
| 185 | C22 | — | 120 | 1 | no | — |
| 186 | C22 | — | 120 | 1 | no | — |
| 187 | C22 | — | 120 | 1 | no | — |
| 188 | C4 | — | 4 | 1 | no | — |
| 189 | C4 | — | 15 | 1 | no | — |
| 190 | C4 | — | 60 | 1 | no | — |
| 191 | C4 | D4×A5 | 1 | 1 | yes | characteristic |
| 192 | C3 | — | 10 | 1 | no | Sylow |
| 193 | C2 | — | 8 | 1 | no | — |
| 194 | C2 | — | 15 | 1 | no | — |
| 195 | C2 | — | 15 | 1 | no | — |
| 196 | C2 | — | 120 | 1 | no | — |
| 197 | C2 | D8×A5 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center | C2 | D8×A5 |
| commutator | C8×A5 | C22 |
| Frattini, cosocle | C8 | C22×A5 |
| Fitting, hypercenter | SD32 | A5 |
| socle | C2×A5 | D8 |
| prime radical | D8×A5 | C2 |
| hypocenter | A5 | SD32 |
| LCS | G ▷ C8×A5 ▷ C4×A5 ▷ C2×A5 ▷ A5 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 ◁ SD32 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. Q16×A5 | C2 | semidirect |
| 3. C16×A5 | C2 | semidirect |
| 54. A5 | SD32 | direct (N × H) |
| 93. SD32 | A5 | direct (N × H) |
| 130. Q16 | C2×A5 | semidirect |
| 138. C16 | C2×A5 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| Q16×A5 | 2 |
| D8×A5 | 2 |
| C16×A5 | 2 |
| A4×SD32 | 5 |
| D5×SD32 | 6 |
| D3×SD32 | 10 |