Finite Group DatabasebetaFinite Group DatabasebetaOrder 1920.
462 conjugacy classes of subgroups; only the 32 normal subgroups are shown.
This group has 462 conjugacy classes of subgroups — too many to draw. Only its 32 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 | C2×C5×#997 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C5×#1790 | — | 3 | 1 | no | maximal, Hall |
| 3 | C5×#997 | C4 | 1 | 1 | yes | characteristic, complemented |
| 4 | C2×C5×SL(2,ℤ/4) | — | 4 | 3 | no | maximal |
| 5 | #85564 | C5 | 1 | 1 | yes | maximal, characteristic, direct product factor, Hall |
| 6 | C2×C5×H(5,𝔽2) | D3 | 1 | 1 | yes | characteristic |
| 7 | C2×C5×#99 | — | 3 | 3 | no | — |
| 8 | C5×#386 | — | 3 | 3 | no | — |
| 9 | C22×C5×A4 | — | 4 | 3 | no | — |
| 10 | C2×C5×SL(2,𝔽3) | — | 8 | 1 | no | — |
| 11 | C5×SL(2,ℤ/4) | — | 4 | 6 | no | — |
| 12 | C2×#997 | C2×C5 | 1 | 1 | yes | characteristic |
| 13 | C5×H(5,𝔽2) | — | 3 | 3 | no | — |
| 14 | C5×H(5,𝔽2) | — | 6 | 1 | no | — |
| 15 | C5×H(5,𝔽2) | Dic3 | 1 | 1 | yes | characteristic, complemented |
| 16 | C2×C5×#41 | — | 6 | 1 | no | — |
| 17 | C22×C5×D4 | — | 3 | 3 | no | — |
| 18 | C23×C4×C5 | — | 3 | 1 | no | — |
| 19 | C2×C5×C22⋊C4 | — | 3 | 3 | no | — |
| 20 | C2×C5×C22⋊C4 | — | 6 | 3 | no | — |
| 21 | C5×#99 | — | 6 | 3 | no | — |
| 22 | C5×#99 | — | 6 | 3 | no | — |
| 23 | C5×#95 | — | 6 | 3 | no | — |
| 24 | #1790 | — | 3 | 1 | no | Sylow |
| 25 | C2×C5×A4 | — | 4 | 3 | no | — |
| 26 | C2×C5×A4 | — | 4 | 3 | no | — |
| 27 | C2×C5×A4 | — | 4 | 3 | no | — |
| 28 | C2×C5×Dic3 | — | 16 | 1 | no | — |
| 29 | C5×SL(2,𝔽3) | — | 8 | 1 | no | — |
| 30 | #997 | C4×C5 | 1 | 1 | yes | characteristic, complemented |
| 31 | C2×SL(2,ℤ/4) | — | 4 | 3 | no | — |
| 32 | C24×C5 | — | 3 | 1 | no | — |
| 33 | C24×C5 | S4 | 1 | 3 | yes | — |
| 34 | C5×#41 | — | 6 | 1 | no | — |
| 35 | C5×#41 | — | 6 | 3 | no | — |
| 36 | C5×#41 | — | 6 | 3 | no | — |
| 37 | C5×#41 | — | 6 | 1 | no | — |
| 38 | C2×C5×Q8 | — | 2 | 1 | no | — |
| 39 | C2×C5×D4 | — | 3 | 3 | no | — |
| 40 | C2×C5×D4 | — | 3 | 3 | no | — |
| 41 | C2×C5×D4 | — | 3 | 3 | no | — |
| 42 | C2×C5×D4 | — | 3 | 3 | no | — |
| 43 | C2×C5×D4 | — | 6 | 3 | no | — |
| 44 | C2×C5×D4 | — | 6 | 6 | no | — |
| 45 | C22×C4×C5 | — | 3 | 3 | no | — |
| 46 | C22×C4×C5 | — | 6 | 3 | no | — |
| 47 | C22×C4×C5 | — | 6 | 3 | no | — |
| 48 | C22×C4×C5 | — | 12 | 2 | no | — |
| 49 | C5×C22⋊C4 | — | 6 | 6 | no | — |
| 50 | C5×C22⋊C4 | — | 12 | 6 | no | — |
| 51 | C2×H(5,𝔽2) | C5×D3 | 1 | 1 | yes | characteristic |
| 52 | C2×#99 | — | 3 | 3 | no | — |
| 53 | #386 | — | 3 | 3 | no | — |
| 54 | C22×C3×C5 | — | 16 | 1 | no | — |
| 55 | C5×A4 | — | 4 | 3 | no | — |
| 56 | C5×Dic3 | — | 16 | 2 | no | — |
| 57 | C22×A4 | — | 4 | 3 | no | — |
| 58 | C2×SL(2,𝔽3) | — | 8 | 1 | no | — |
| 59 | SL(2,ℤ/4) | — | 4 | 6 | no | — |
| 60 | C23×C5 | — | 3 | 3 | no | — |
| 61 | C23×C5 | — | 3 | 1 | no | — |
| 62 | C23×C5 | — | 3 | 3 | no | — |
| 63 | C23×C5 | — | 3 | 3 | no | — |
| 64 | C23×C5 | — | 4 | 3 | no | — |
| 65 | C23×C5 | — | 4 | 3 | no | — |
| 66 | C23×C5 | — | 12 | 1 | no | — |
| 67 | C23×C5 | — | 12 | 1 | no | — |
| 68 | C23×C5 | SL(2,ℤ/4) | 1 | 3 | yes | complemented |
| 69 | C5×Q8 | — | 2 | 1 | no | — |
| 70 | C5×Q8 | — | 6 | 1 | no | — |
| 71 | C5×D4 | — | 6 | 3 | no | — |
| 72 | C5×D4 | — | 6 | 3 | no | — |
| 73 | C5×D4 | — | 6 | 6 | no | — |
| 74 | C2×C4×C5 | — | 3 | 3 | no | — |
| 75 | C2×C4×C5 | — | 3 | 3 | no | — |
| 76 | C2×C4×C5 | — | 6 | 3 | no | — |
| 77 | C2×C4×C5 | — | 6 | 1 | no | — |
| 78 | C2×C4×C5 | — | 6 | 6 | no | — |
| 79 | C2×C4×C5 | — | 12 | 3 | no | — |
| 80 | C2×C4×C5 | — | 12 | 1 | no | — |
| 81 | C2×C4×C5 | — | 12 | 3 | no | — |
| 82 | C2×C4×C5 | — | 12 | 3 | no | — |
| 83 | C2×C4×C5 | — | 12 | 3 | no | — |
| 84 | H(5,𝔽2) | — | 3 | 3 | no | — |
| 85 | H(5,𝔽2) | — | 6 | 1 | no | — |
| 86 | H(5,𝔽2) | C5×Dic3 | 1 | 1 | yes | characteristic, complemented |
| 87 | C2×#41 | — | 6 | 1 | no | — |
| 88 | C22×D4 | — | 3 | 3 | no | — |
| 89 | C23×C4 | — | 3 | 1 | no | — |
| 90 | C2×C22⋊C4 | — | 3 | 3 | no | — |
| 91 | C2×C22⋊C4 | — | 6 | 3 | no | — |
| 92 | #99 | — | 6 | 3 | no | — |
| 93 | #99 | — | 6 | 3 | no | — |
| 94 | #95 | — | 6 | 3 | no | — |
| 95 | C2×C3×C5 | — | 16 | 1 | no | — |
| 96 | C2×C3×C5 | — | 16 | 1 | no | — |
| 97 | C2×C3×C5 | — | 16 | 1 | no | — |
| 98 | C2×A4 | — | 4 | 3 | no | — |
| 99 | C2×A4 | — | 4 | 3 | no | — |
| 100 | C2×A4 | — | 4 | 3 | no | — |
| 101 | C2×Dic3 | — | 16 | 1 | no | — |
| 102 | SL(2,𝔽3) | — | 8 | 1 | no | — |
| 103 | C22×C5 | — | 3 | 3 | no | — |
| 104 | C22×C5 | — | 3 | 3 | no | — |
| 105 | C22×C5 | — | 4 | 3 | no | — |
| 106 | C22×C5 | — | 6 | 3 | no | — |
| 107 | C22×C5 | — | 6 | 3 | no | — |
| 108 | C22×C5 | — | 12 | 3 | no | — |
| 109 | C22×C5 | — | 12 | 3 | no | — |
| 110 | C22×C5 | — | 12 | 1 | no | — |
| 111 | C22×C5 | #1020 | 1 | 1 | yes | characteristic |
| 112 | C4×C5 | — | 6 | 1 | no | — |
| 113 | C4×C5 | — | 6 | 1 | no | — |
| 114 | C4×C5 | — | 12 | 2 | no | — |
| 115 | C4×C5 | — | 24 | 3 | no | — |
| 116 | C24 | — | 3 | 1 | no | — |
| 117 | C24 | C5×S4 | 1 | 3 | yes | — |
| 118 | #41 | — | 6 | 1 | no | — |
| 119 | #41 | — | 6 | 3 | no | — |
| 120 | #41 | — | 6 | 3 | no | — |
| 121 | #41 | — | 6 | 1 | no | — |
| 122 | C2×Q8 | — | 2 | 1 | no | — |
| 123 | C2×D4 | — | 3 | 3 | no | — |
| 124 | C2×D4 | — | 3 | 3 | no | — |
| 125 | C2×D4 | — | 3 | 3 | no | — |
| 126 | C2×D4 | — | 3 | 3 | no | — |
| 127 | C2×D4 | — | 6 | 3 | no | — |
| 128 | C2×D4 | — | 6 | 6 | no | — |
| 129 | C22×C4 | — | 3 | 3 | no | — |
| 130 | C22×C4 | — | 6 | 3 | no | — |
| 131 | C22×C4 | — | 6 | 3 | no | — |
| 132 | C22×C4 | — | 12 | 2 | no | — |
| 133 | C22⋊C4 | — | 6 | 6 | no | — |
| 134 | C22⋊C4 | — | 12 | 6 | no | — |
| 135 | C3×C5 | — | 16 | 1 | no | Hall |
| 136 | C22×C3 | — | 16 | 1 | no | — |
| 137 | A4 | — | 4 | 3 | no | — |
| 138 | Dic3 | — | 16 | 2 | no | — |
| 139 | C2×C5 | — | 6 | 3 | no | — |
| 140 | C2×C5 | — | 6 | 3 | no | — |
| 141 | C2×C5 | #5932 | 1 | 1 | yes | characteristic |
| 142 | C2×C5 | #5934 | 1 | 1 | yes | characteristic |
| 143 | C2×C5 | #5936 | 1 | 1 | yes | characteristic |
| 144 | C23 | — | 3 | 3 | no | — |
| 145 | C23 | — | 3 | 1 | no | — |
| 146 | C23 | — | 3 | 3 | no | — |
| 147 | C23 | — | 3 | 3 | no | — |
| 148 | C23 | — | 4 | 3 | no | — |
| 149 | C23 | — | 4 | 3 | no | — |
| 150 | C23 | — | 12 | 1 | no | — |
| 151 | C23 | — | 12 | 1 | no | — |
| 152 | C23 | C5×SL(2,ℤ/4) | 1 | 3 | yes | complemented |
| 153 | Q8 | — | 2 | 1 | no | — |
| 154 | Q8 | — | 6 | 1 | no | — |
| 155 | D4 | — | 6 | 3 | no | — |
| 156 | D4 | — | 6 | 3 | no | — |
| 157 | D4 | — | 6 | 6 | no | — |
| 158 | C2×C4 | — | 3 | 3 | no | — |
| 159 | C2×C4 | — | 3 | 3 | no | — |
| 160 | C2×C4 | — | 6 | 3 | no | — |
| 161 | C2×C4 | — | 6 | 1 | no | — |
| 162 | C2×C4 | — | 6 | 6 | no | — |
| 163 | C2×C4 | — | 12 | 3 | no | — |
| 164 | C2×C4 | — | 12 | 1 | no | — |
| 165 | C2×C4 | — | 12 | 3 | no | — |
| 166 | C2×C4 | — | 12 | 3 | no | — |
| 167 | C2×C4 | — | 12 | 3 | no | — |
| 168 | C2×C3 | — | 16 | 1 | no | — |
| 169 | C2×C3 | — | 16 | 1 | no | — |
| 170 | C2×C3 | — | 16 | 1 | no | — |
| 171 | C5 | #85564 | 1 | 1 | yes | characteristic, direct product factor, Sylow |
| 172 | C22 | — | 3 | 3 | no | — |
| 173 | C22 | — | 3 | 3 | no | — |
| 174 | C22 | — | 4 | 3 | no | — |
| 175 | C22 | — | 6 | 3 | no | — |
| 176 | C22 | — | 6 | 3 | no | — |
| 177 | C22 | — | 12 | 3 | no | — |
| 178 | C22 | — | 12 | 3 | no | — |
| 179 | C22 | — | 12 | 1 | no | — |
| 180 | C22 | C5×#1020 | 1 | 1 | yes | characteristic |
| 181 | C4 | — | 6 | 1 | no | — |
| 182 | C4 | — | 6 | 1 | no | — |
| 183 | C4 | — | 12 | 2 | no | — |
| 184 | C4 | — | 24 | 3 | no | — |
| 185 | C3 | — | 16 | 1 | no | Sylow |
| 186 | C2 | — | 6 | 3 | no | — |
| 187 | C2 | — | 6 | 3 | no | — |
| 188 | C2 | C5×#5932 | 1 | 1 | yes | characteristic |
| 189 | C2 | C5×#5934 | 1 | 1 | yes | characteristic |
| 190 | C2 | C5×#5936 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle, hypercenter | C22×C5 | #1020 |
| commutator, hypocenter | #997 | C4×C5 |
| Frattini | C22 | C5×#1020 |
| Fitting | C2×C5×H(5,𝔽2) | D3 |
| prime radical | C2×C5×#997 | C2 |
| cosocle | C2×#997 | C2×C5 |
| LCS | G ▷ #997 | |
| UCS | C1 ◁ C22×C5 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. C5×#997 | C4 | semidirect |
| 5. #85564 | C5 | direct (N × H) |
| 15. C5×H(5,𝔽2) | Dic3 | semidirect |
| 30. #997 | C4×C5 | semidirect |
| 68. C23×C5 | SL(2,ℤ/4) | semidirect |
| 86. H(5,𝔽2) | C5×Dic3 | semidirect |
| 152. C23 | C5×SL(2,ℤ/4) | semidirect |
| 171. C5 | #85564 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C5×#997 | 2 |
| C5×#1790 | 3 |
| C2×C5×SL(2,ℤ/4) | 4 |
| #85564 | 5 |