Finite Group DatabasebetaFinite Group DatabasebetaOrder 1920.
286 conjugacy classes of subgroups; only the 42 normal subgroups are shown.
This group has 286 conjugacy classes of subgroups — too many to draw. Only its 42 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⋊(#131⋊C3) | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #11756725 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | D5×GU(2,𝔽3) | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | #10604006 | — | 3 | 1 | no | maximal, Hall |
| 5 | D5×#41⋊C3 | C22 | 1 | 1 | yes | characteristic |
| 6 | #4133⋊C3 | C4 | 1 | 2 | yes | complemented |
| 7 | #91886 | C4 | 1 | 2 | yes | complemented |
| 8 | #91185 | C4 | 1 | 1 | yes | characteristic, complemented |
| 9 | C5×GU(2,𝔽3) | C4 | 1 | 1 | yes | characteristic, complemented |
| 10 | #91150 | — | 4 | 1 | no | maximal |
| 11 | #73123 | — | 5 | 1 | no | maximal, Hall |
| 12 | D5⋊#131 | D3 | 1 | 1 | yes | characteristic |
| 13 | #65794 | — | 3 | 1 | no | — |
| 14 | #65728 | — | 3 | 1 | no | — |
| 15 | #65676 | — | 3 | 1 | no | — |
| 16 | D5×#104 | — | 3 | 1 | no | — |
| 17 | #64901 | — | 3 | 1 | no | — |
| 18 | #64862 | — | 3 | 1 | no | — |
| 19 | C5×#41⋊C3 | C2×C4 | 1 | 1 | yes | characteristic |
| 20 | C4×ΓL(1,𝔽16) | — | 4 | 1 | no | — |
| 21 | C3×D5⋊C8 | — | 4 | 1 | no | — |
| 22 | D5×SL(2,𝔽3) | C2×C4 | 1 | 1 | yes | characteristic |
| 23 | C5⋊(#41⋊C3) | C2×C4 | 1 | 1 | yes | characteristic |
| 24 | D5×C3⋊C8 | — | 4 | 1 | no | — |
| 25 | C2×#131⋊C3 | — | 5 | 1 | no | — |
| 26 | #5424 | — | 5 | 1 | no | — |
| 27 | C2×GU(2,𝔽3) | — | 5 | 1 | no | — |
| 28 | D5×#41 | C2×D3 | 1 | 1 | yes | characteristic |
| 29 | #4133 | Dic3 | 1 | 2 | yes | complemented |
| 30 | #4131 | — | 3 | 2 | no | — |
| 31 | #4130 | — | 3 | 1 | no | — |
| 32 | D5⋊M16 | — | 3 | 1 | no | — |
| 33 | C2×D5⋊C8 | — | 3 | 1 | no | — |
| 34 | C2×D5⋊C8 | — | 3 | 1 | no | — |
| 35 | D5×M16 | — | 3 | 1 | no | — |
| 36 | C42×D5 | — | 3 | 1 | no | — |
| 37 | #4009 | — | 3 | 2 | no | — |
| 38 | #4007 | — | 3 | 2 | no | — |
| 39 | #4002 | — | 6 | 2 | no | — |
| 40 | #3992 | — | 6 | 1 | no | — |
| 41 | C8×AGL(1,𝔽5) | — | 6 | 1 | no | — |
| 42 | C5×#104 | — | 3 | 1 | no | — |
| 43 | #3969 | — | 3 | 1 | no | — |
| 44 | #3957 | — | 3 | 1 | no | — |
| 45 | #3937 | — | 3 | 1 | no | — |
| 46 | #1765 | — | 15 | 1 | no | Sylow |
| 47 | C2×ΓL(1,𝔽16) | — | 4 | 2 | no | — |
| 48 | C3×C4×D5 | — | 4 | 1 | no | — |
| 49 | C5×SL(2,𝔽3) | C42 | 1 | 1 | yes | characteristic |
| 50 | C3×#170 | — | 4 | 2 | no | — |
| 51 | C3⋊#168 | — | 4 | 1 | no | — |
| 52 | C5×C3⋊C8 | — | 4 | 1 | no | — |
| 53 | C2×#41⋊C3 | — | 5 | 1 | no | — |
| 54 | #131⋊C3 | — | 5 | 2 | no | — |
| 55 | GU(2,𝔽3) | — | 5 | 1 | no | — |
| 56 | GU(2,𝔽3) | AGL(1,𝔽5) | 1 | 1 | yes | characteristic, complemented |
| 57 | #859 | — | 5 | 2 | no | — |
| 58 | #815 | — | 20 | 1 | no | — |
| 59 | C5×#41 | C4×D3 | 1 | 1 | yes | characteristic |
| 60 | #713 | C2×Dic3 | 1 | 1 | yes | characteristic |
| 61 | Q8×D5 | C4×D3 | 1 | 1 | yes | characteristic |
| 62 | #711 | — | 3 | 1 | no | — |
| 63 | D4×D5 | — | 3 | 1 | no | — |
| 64 | #709 | — | 3 | 1 | no | — |
| 65 | C2×C4×D5 | — | 3 | 1 | no | — |
| 66 | C2×C4×D5 | — | 6 | 1 | no | — |
| 67 | C22⋊AGL(1,𝔽5) | — | 6 | 1 | no | — |
| 68 | #704 | — | 6 | 1 | no | — |
| 69 | C2×#170 | — | 3 | 2 | no | — |
| 70 | C2×#170 | — | 6 | 1 | no | — |
| 71 | C4⋊AGL(1,𝔽5) | — | 3 | 2 | no | — |
| 72 | C4×AGL(1,𝔽5) | — | 6 | 1 | no | — |
| 73 | #700 | — | 3 | 2 | no | — |
| 74 | D5⋊C8 | — | 3 | 1 | no | — |
| 75 | D5⋊C8 | — | 3 | 2 | no | — |
| 76 | D5⋊C8 | — | 3 | 2 | no | — |
| 77 | D5⋊C8 | — | 6 | 1 | no | — |
| 78 | D5⋊C8 | S4 | 1 | 1 | yes | characteristic |
| 79 | C5×M16 | — | 3 | 1 | no | — |
| 80 | C42×C5 | — | 3 | 1 | no | — |
| 81 | C4×Dic5 | — | 3 | 1 | no | — |
| 82 | C4×Dic5 | — | 6 | 1 | no | — |
| 83 | #681 | — | 3 | 1 | no | — |
| 84 | #676 | — | 6 | 1 | no | — |
| 85 | C8×D5 | — | 6 | 1 | no | — |
| 86 | C2×#131 | — | 5 | 1 | no | — |
| 87 | C2×#104 | — | 15 | 1 | no | — |
| 88 | #416 | — | 15 | 1 | no | — |
| 89 | #405 | — | 15 | 1 | no | — |
| 90 | C2×#97 | — | 15 | 1 | no | — |
| 91 | #341 | — | 15 | 1 | no | — |
| 92 | #339 | — | 15 | 1 | no | — |
| 93 | C2×C3×D5 | — | 4 | 1 | no | — |
| 94 | ΓL(1,𝔽16) | — | 8 | 2 | no | — |
| 95 | C3×C4×C5 | — | 4 | 1 | no | — |
| 96 | C3×Dic5 | — | 4 | 1 | no | — |
| 97 | #41⋊C3 | — | 5 | 1 | no | — |
| 98 | #41⋊C3 | C2×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 99 | C2×SL(2,𝔽3) | — | 5 | 1 | no | — |
| 100 | C2×C3×C8 | — | 20 | 1 | no | — |
| 101 | C4×Dic3 | — | 20 | 1 | no | — |
| 102 | C2×C3⋊C8 | — | 20 | 1 | no | — |
| 103 | C22×D5 | — | 3 | 1 | no | — |
| 104 | C2×AGL(1,𝔽5) | — | 6 | 2 | no | — |
| 105 | C5×Q8 | C4×Dic3 | 1 | 1 | yes | characteristic |
| 106 | C5×D4 | — | 3 | 1 | no | — |
| 107 | C2×C4×C5 | — | 3 | 1 | no | — |
| 108 | C2×C4×C5 | — | 6 | 1 | no | — |
| 109 | C22⋊D5 | — | 6 | 1 | no | — |
| 110 | C2×Dic5 | — | 3 | 1 | no | — |
| 111 | C2×Dic5 | — | 6 | 1 | no | — |
| 112 | D20 | — | 3 | 1 | no | — |
| 113 | C4×D5 | — | 3 | 1 | no | — |
| 114 | C4×D5 | — | 3 | 1 | no | — |
| 115 | C4×D5 | — | 3 | 1 | no | — |
| 116 | C4×D5 | — | 12 | 2 | no | — |
| 117 | C4×D5 | C2×S4 | 1 | 1 | yes | characteristic |
| 118 | Dic10 | — | 3 | 1 | no | — |
| 119 | #170 | — | 3 | 2 | no | — |
| 120 | #170 | — | 6 | 2 | no | — |
| 121 | #170 | SL(2,ℤ/4) | 1 | 2 | yes | — |
| 122 | C5×C8 | — | 6 | 1 | no | — |
| 123 | #168 | — | 6 | 1 | no | — |
| 124 | C2×#41 | — | 5 | 1 | no | — |
| 125 | #131 | — | 5 | 2 | no | — |
| 126 | #131 | — | 15 | 2 | no | — |
| 127 | C2×M16 | — | 15 | 1 | no | — |
| 128 | C2×M16 | — | 15 | 1 | no | — |
| 129 | C22×C8 | — | 15 | 1 | no | — |
| 130 | C22×C8 | — | 15 | 1 | no | — |
| 131 | #117 | — | 15 | 1 | no | — |
| 132 | C2×C42 | — | 15 | 1 | no | — |
| 133 | #104 | — | 3 | 1 | no | — |
| 134 | #104 | — | 15 | 1 | no | — |
| 135 | #104 | — | 15 | 1 | no | — |
| 136 | #104 | — | 15 | 1 | no | — |
| 137 | #103 | — | 15 | 2 | no | — |
| 138 | #102 | — | 15 | 2 | no | — |
| 139 | #97 | — | 30 | 1 | no | — |
| 140 | #97 | — | 30 | 2 | no | — |
| 141 | C4×C8 | — | 30 | 1 | no | — |
| 142 | C2×C3×C5 | — | 4 | 1 | no | — |
| 143 | C3×D5 | — | 4 | 2 | no | — |
| 144 | C2×C3×C4 | — | 20 | 1 | no | — |
| 145 | C2×Dic3 | — | 20 | 2 | no | — |
| 146 | SL(2,𝔽3) | C4×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 147 | C3×C8 | — | 20 | 2 | no | — |
| 148 | C3⋊C8 | — | 4 | 1 | no | — |
| 149 | C3⋊C8 | — | 20 | 1 | no | — |
| 150 | C22×C5 | — | 3 | 1 | no | — |
| 151 | C2×D5 | — | 3 | 1 | no | — |
| 152 | C2×D5 | — | 6 | 2 | no | — |
| 153 | C2×D5 | C4×S4 | 1 | 1 | yes | characteristic |
| 154 | AGL(1,𝔽5) | — | 12 | 2 | no | — |
| 155 | C4×C5 | — | 3 | 1 | no | — |
| 156 | C4×C5 | — | 12 | 1 | no | — |
| 157 | C4×C5 | C4×S4 | 1 | 1 | yes | characteristic |
| 158 | Dic5 | — | 3 | 1 | no | — |
| 159 | Dic5 | — | 12 | 1 | no | — |
| 160 | Dic5 | C2×SL(2,ℤ/4) | 1 | 1 | yes | characteristic |
| 161 | #41 | — | 5 | 1 | no | — |
| 162 | #41 | — | 15 | 1 | no | — |
| 163 | #41 | — | 15 | 1 | no | — |
| 164 | #41 | D3×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 165 | C2×Q8 | — | 5 | 1 | no | — |
| 166 | C2×D4 | — | 15 | 1 | no | — |
| 167 | C22×C4 | — | 15 | 1 | no | — |
| 168 | C22×C4 | — | 30 | 1 | no | — |
| 169 | M16 | — | 3 | 1 | no | — |
| 170 | M16 | — | 15 | 2 | no | — |
| 171 | M16 | — | 15 | 1 | no | — |
| 172 | M16 | — | 30 | 1 | no | — |
| 173 | M16 | — | 30 | 1 | no | — |
| 174 | C2×C8 | — | 5 | 1 | no | — |
| 175 | C2×C8 | — | 15 | 2 | no | — |
| 176 | C2×C8 | — | 15 | 2 | no | — |
| 177 | C2×C8 | — | 15 | 1 | no | — |
| 178 | C2×C8 | — | 15 | 2 | no | — |
| 179 | C2×C8 | — | 30 | 1 | no | — |
| 180 | C2×C8 | — | 30 | 1 | no | — |
| 181 | C2×C8 | — | 30 | 1 | no | — |
| 182 | C4⋊C4 | — | 15 | 2 | no | — |
| 183 | C22⋊C4 | — | 30 | 1 | no | — |
| 184 | C42 | — | 3 | 1 | no | — |
| 185 | C42 | — | 15 | 1 | no | — |
| 186 | C42 | — | 30 | 1 | no | — |
| 187 | C42 | — | 30 | 1 | no | — |
| 188 | C3×C5 | — | 4 | 1 | no | Hall |
| 189 | C22×C3 | — | 20 | 1 | no | — |
| 190 | C3×C4 | — | 4 | 1 | no | — |
| 191 | C3×C4 | — | 20 | 1 | no | — |
| 192 | Dic3 | — | 40 | 2 | no | — |
| 193 | C2×C5 | — | 6 | 1 | no | — |
| 194 | C2×C5 | C4×SL(2,ℤ/4) | 1 | 1 | yes | characteristic |
| 195 | D5 | — | 6 | 1 | no | — |
| 196 | D5 | #5405 | 1 | 2 | yes | — |
| 197 | C23 | — | 15 | 1 | no | — |
| 198 | Q8 | — | 15 | 1 | no | — |
| 199 | Q8 | Dic3×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 200 | D4 | — | 3 | 1 | no | — |
| 201 | D4 | — | 15 | 1 | no | — |
| 202 | D4 | — | 30 | 1 | no | — |
| 203 | C2×C4 | — | 3 | 1 | no | — |
| 204 | C2×C4 | — | 5 | 1 | no | — |
| 205 | C2×C4 | — | 6 | 1 | no | — |
| 206 | C2×C4 | — | 15 | 1 | no | — |
| 207 | C2×C4 | — | 15 | 1 | no | — |
| 208 | C2×C4 | — | 15 | 1 | no | — |
| 209 | C2×C4 | — | 15 | 1 | no | — |
| 210 | C2×C4 | — | 30 | 1 | no | — |
| 211 | C2×C4 | — | 30 | 2 | no | — |
| 212 | C2×C4 | — | 60 | 2 | no | — |
| 213 | C8 | — | 5 | 2 | no | — |
| 214 | C8 | — | 6 | 1 | no | — |
| 215 | C8 | — | 15 | 2 | no | — |
| 216 | C8 | — | 30 | 2 | no | — |
| 217 | C8 | — | 30 | 1 | no | — |
| 218 | C2×C3 | — | 4 | 1 | no | — |
| 219 | C2×C3 | — | 20 | 2 | no | — |
| 220 | C5 | #73123 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 221 | C22 | — | 3 | 1 | no | — |
| 222 | C22 | — | 5 | 1 | no | — |
| 223 | C22 | — | 15 | 1 | no | — |
| 224 | C22 | — | 30 | 2 | no | — |
| 225 | C4 | — | 3 | 1 | no | — |
| 226 | C4 | — | 5 | 1 | no | — |
| 227 | C4 | — | 12 | 1 | no | — |
| 228 | C4 | — | 15 | 1 | no | — |
| 229 | C4 | — | 60 | 2 | no | — |
| 230 | C4 | — | 60 | 1 | no | — |
| 231 | C4 | AGL(1,𝔽5)×S4 | 1 | 1 | yes | characteristic |
| 232 | C3 | — | 4 | 1 | no | Sylow |
| 233 | C2 | — | 5 | 2 | no | — |
| 234 | C2 | — | 6 | 1 | no | — |
| 235 | C2 | — | 30 | 1 | no | — |
| 236 | C2 | AGL(1,𝔽5)×SL(2,ℤ/4) | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C4 | AGL(1,𝔽5)×S4 |
| commutator, hypocenter | C5×SL(2,𝔽3) | C42 |
| Fitting | C5×#41 | C4×D3 |
| socle | C2×C5 | C4×SL(2,ℤ/4) |
| prime radical, cosocle | D5×#41⋊C3 | C22 |
| LCS | G ▷ C5×SL(2,𝔽3) | |
| UCS | C1 ◁ C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 6. #4133⋊C3 | C4 | semidirect |
| 7. #91886 | C4 | semidirect |
| 8. #91185 | C4 | semidirect |
| 9. C5×GU(2,𝔽3) | C4 | semidirect |
| 29. #4133 | Dic3 | semidirect |
| 56. GU(2,𝔽3) | AGL(1,𝔽5) | semidirect |
| 220. C5 | #73123 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| D5⋊(#131⋊C3) | 2 |
| #11756725 | 2 |
| D5×GU(2,𝔽3) | 2 |
| #10604006 | 3 |
| #91150 | 4 |
| #73123 | 5 |