Finite Group DatabasebetaFinite Group DatabasebetaOrder 1920.
186 conjugacy classes of subgroups; only the 25 normal subgroups are shown.
This group has 186 conjugacy classes of subgroups — too many to draw. Only its 25 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 | C5×Q8⋊SL(2,𝔽3) | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #10602734 | — | 3 | 1 | no | maximal, Hall |
| 3 | C2×C5⋊#226 | — | 4 | 1 | no | maximal |
| 4 | #91184 | — | 4 | 1 | no | maximal |
| 5 | #91184 | — | 4 | 1 | no | maximal |
| 6 | #85552 | — | 5 | 1 | no | maximal, Hall |
| 7 | C5×Q82 | D3 | 1 | 1 | yes | characteristic |
| 8 | #65370 | — | 3 | 1 | no | — |
| 9 | #65320 | — | 3 | 1 | no | — |
| 10 | #65309 | — | 3 | 1 | no | — |
| 11 | #64766 | — | 3 | 1 | no | — |
| 12 | Q8⋊#168 | — | 3 | 1 | no | — |
| 13 | Q8⋊#168 | — | 3 | 1 | no | — |
| 14 | C2×C5×SL(2,𝔽3) | — | 4 | 1 | no | — |
| 15 | C2×C5×SL(2,𝔽3) | — | 4 | 1 | no | — |
| 16 | C2×C5×SL(2,𝔽3) | — | 4 | 1 | no | — |
| 17 | C2×C5×SL(2,𝔽3) | — | 8 | 1 | no | — |
| 18 | C5⋊#226 | — | 4 | 2 | no | — |
| 19 | Q8⋊SL(2,𝔽3) | D5 | 1 | 1 | yes | characteristic |
| 20 | C5×C4⋊Q8 | — | 3 | 1 | no | — |
| 21 | C5×C4⋊Q8 | — | 6 | 1 | no | — |
| 22 | C4×C5×Q8 | — | 3 | 1 | no | — |
| 23 | C4×C5×Q8 | — | 3 | 1 | no | — |
| 24 | C2×#689 | — | 6 | 1 | no | — |
| 25 | #4015 | — | 3 | 1 | no | — |
| 26 | #3967 | — | 6 | 1 | no | — |
| 27 | #3967 | — | 6 | 1 | no | — |
| 28 | #3967 | — | 6 | 1 | no | — |
| 29 | #3942 | — | 6 | 1 | no | — |
| 30 | #3940 | — | 6 | 1 | no | — |
| 31 | C4⋊#168 | — | 3 | 1 | no | — |
| 32 | C4⋊#168 | — | 3 | 1 | no | — |
| 33 | C4×#168 | — | 3 | 1 | no | — |
| 34 | #1645 | — | 15 | 1 | no | Sylow |
| 35 | C2×Dic15 | — | 16 | 1 | no | — |
| 36 | C5×SL(2,𝔽3) | — | 4 | 1 | no | — |
| 37 | C5×SL(2,𝔽3) | — | 4 | 1 | no | — |
| 38 | C5×SL(2,𝔽3) | — | 4 | 1 | no | — |
| 39 | C5×SL(2,𝔽3) | — | 8 | 1 | no | — |
| 40 | C2×#226 | — | 20 | 1 | no | — |
| 41 | #859 | — | 20 | 1 | no | — |
| 42 | #859 | — | 20 | 1 | no | — |
| 43 | C2×C5×Q8 | — | 2 | 1 | no | — |
| 44 | C2×C5×Q8 | — | 3 | 1 | no | — |
| 45 | C2×C5×Q8 | S4 | 1 | 1 | yes | characteristic |
| 46 | C2×C5×Q8 | S4 | 1 | 1 | yes | characteristic |
| 47 | C2×C5×Q8 | S4 | 1 | 1 | yes | characteristic |
| 48 | C2×Dic10 | — | 6 | 1 | no | — |
| 49 | C5×C4⋊C4 | — | 3 | 1 | no | — |
| 50 | C5×C4⋊C4 | — | 3 | 1 | no | — |
| 51 | C5×C4⋊C4 | — | 6 | 1 | no | — |
| 52 | C5×C4⋊C4 | — | 6 | 1 | no | — |
| 53 | C42×C5 | — | 3 | 1 | no | — |
| 54 | C42×C5 | — | 6 | 1 | no | — |
| 55 | #689 | — | 12 | 1 | no | — |
| 56 | #689 | — | 12 | 1 | no | — |
| 57 | C4⋊Dic5 | — | 6 | 1 | no | — |
| 58 | C4⋊Dic5 | — | 6 | 1 | no | — |
| 59 | C2×#168 | — | 6 | 1 | no | — |
| 60 | C2×#168 | — | 6 | 1 | no | — |
| 61 | C2×#168 | — | 6 | 1 | no | — |
| 62 | Q82 | D15 | 1 | 1 | yes | characteristic |
| 63 | #487 | — | 15 | 1 | no | — |
| 64 | #475 | — | 15 | 1 | no | — |
| 65 | #462 | — | 15 | 1 | no | — |
| 66 | #333 | — | 15 | 1 | no | — |
| 67 | Q8⋊C8 | — | 15 | 1 | no | — |
| 68 | Q8⋊C8 | — | 15 | 1 | no | — |
| 69 | C22×C3×C5 | — | 16 | 1 | no | — |
| 70 | Dic15 | — | 16 | 2 | no | — |
| 71 | C2×SL(2,𝔽3) | — | 4 | 1 | no | — |
| 72 | C2×SL(2,𝔽3) | — | 4 | 1 | no | — |
| 73 | C2×SL(2,𝔽3) | — | 4 | 1 | no | — |
| 74 | C2×SL(2,𝔽3) | — | 8 | 1 | no | — |
| 75 | #226 | — | 20 | 2 | no | — |
| 76 | C5×Q8 | — | 3 | 1 | no | — |
| 77 | C5×Q8 | — | 3 | 1 | no | — |
| 78 | C5×Q8 | — | 4 | 1 | no | — |
| 79 | C5×Q8 | — | 8 | 1 | no | — |
| 80 | C5×Q8 | — | 12 | 1 | no | — |
| 81 | C5×Q8 | #226 | 1 | 1 | yes | characteristic, complemented |
| 82 | C5×Q8 | GL(2,𝔽3) | 1 | 1 | yes | characteristic |
| 83 | C2×C4×C5 | — | 3 | 1 | no | — |
| 84 | C2×C4×C5 | — | 3 | 1 | no | — |
| 85 | C2×C4×C5 | — | 3 | 1 | no | — |
| 86 | C2×C4×C5 | — | 6 | 1 | no | — |
| 87 | C2×Dic5 | — | 12 | 1 | no | — |
| 88 | Dic10 | — | 12 | 1 | no | — |
| 89 | Dic10 | — | 12 | 1 | no | — |
| 90 | #168 | — | 12 | 1 | no | — |
| 91 | #168 | — | 12 | 1 | no | — |
| 92 | #168 | — | 12 | 1 | no | — |
| 93 | C2×Q16 | — | 30 | 1 | no | — |
| 94 | C4⋊Q8 | — | 3 | 1 | no | — |
| 95 | C4⋊Q8 | — | 6 | 1 | no | — |
| 96 | C4⋊Q8 | — | 15 | 1 | no | — |
| 97 | C4×Q8 | — | 3 | 1 | no | — |
| 98 | C4×Q8 | — | 3 | 1 | no | — |
| 99 | #106 | — | 30 | 1 | no | — |
| 100 | C4⋊C8 | — | 15 | 1 | no | — |
| 101 | C4⋊C8 | — | 15 | 1 | no | — |
| 102 | #103 | — | 30 | 1 | no | — |
| 103 | #103 | — | 30 | 1 | no | — |
| 104 | #103 | — | 30 | 1 | no | — |
| 105 | #103 | — | 30 | 1 | no | — |
| 106 | C4×C8 | — | 15 | 1 | no | — |
| 107 | C2×C3×C5 | — | 16 | 1 | no | — |
| 108 | C2×C3×C5 | — | 16 | 1 | no | — |
| 109 | C2×C3×C5 | — | 16 | 1 | no | — |
| 110 | C2×Dic3 | — | 80 | 1 | no | — |
| 111 | SL(2,𝔽3) | — | 4 | 1 | no | — |
| 112 | SL(2,𝔽3) | — | 4 | 1 | no | — |
| 113 | SL(2,𝔽3) | — | 4 | 1 | no | — |
| 114 | SL(2,𝔽3) | — | 8 | 1 | no | — |
| 115 | C22×C5 | #1020 | 1 | 1 | yes | characteristic |
| 116 | C4×C5 | — | 3 | 1 | no | — |
| 117 | C4×C5 | — | 3 | 1 | no | — |
| 118 | C4×C5 | — | 3 | 1 | no | — |
| 119 | C4×C5 | — | 3 | 1 | no | — |
| 120 | C4×C5 | — | 6 | 1 | no | — |
| 121 | C4×C5 | — | 12 | 1 | no | — |
| 122 | Dic5 | — | 24 | 1 | no | — |
| 123 | C2×Q8 | — | 2 | 1 | no | — |
| 124 | C2×Q8 | — | 3 | 1 | no | — |
| 125 | C2×Q8 | — | 30 | 1 | no | — |
| 126 | C2×Q8 | C5⋊S4 | 1 | 1 | yes | characteristic |
| 127 | C2×Q8 | C5⋊S4 | 1 | 1 | yes | characteristic |
| 128 | C2×Q8 | C5⋊S4 | 1 | 1 | yes | characteristic |
| 129 | Q16 | — | 60 | 1 | no | — |
| 130 | Q16 | — | 60 | 1 | no | — |
| 131 | C2×C8 | — | 30 | 1 | no | — |
| 132 | C2×C8 | — | 30 | 1 | no | — |
| 133 | C2×C8 | — | 30 | 1 | no | — |
| 134 | C4⋊C4 | — | 3 | 1 | no | — |
| 135 | C4⋊C4 | — | 3 | 1 | no | — |
| 136 | C4⋊C4 | — | 6 | 1 | no | — |
| 137 | C4⋊C4 | — | 6 | 1 | no | — |
| 138 | C4⋊C4 | — | 30 | 1 | no | — |
| 139 | C4⋊C4 | — | 30 | 1 | no | — |
| 140 | C42 | — | 3 | 1 | no | — |
| 141 | C42 | — | 6 | 1 | no | — |
| 142 | C3×C5 | — | 16 | 1 | no | Hall |
| 143 | C22×C3 | — | 16 | 1 | no | — |
| 144 | Dic3 | — | 80 | 2 | no | — |
| 145 | C2×C5 | #5930 | 1 | 1 | yes | characteristic |
| 146 | C2×C5 | #5931 | 1 | 1 | yes | characteristic |
| 147 | C2×C5 | #5933 | 1 | 1 | yes | characteristic |
| 148 | Q8 | — | 3 | 1 | no | — |
| 149 | Q8 | — | 3 | 1 | no | — |
| 150 | Q8 | — | 4 | 1 | no | — |
| 151 | Q8 | — | 8 | 1 | no | — |
| 152 | Q8 | — | 12 | 1 | no | — |
| 153 | Q8 | — | 60 | 1 | no | — |
| 154 | Q8 | — | 60 | 1 | no | — |
| 155 | Q8 | C5⋊#226 | 1 | 1 | yes | characteristic, complemented |
| 156 | Q8 | C5⋊GL(2,𝔽3) | 1 | 1 | yes | characteristic |
| 157 | C2×C4 | — | 3 | 1 | no | — |
| 158 | C2×C4 | — | 3 | 1 | no | — |
| 159 | C2×C4 | — | 3 | 1 | no | — |
| 160 | C2×C4 | — | 6 | 1 | no | — |
| 161 | C2×C4 | — | 60 | 1 | no | — |
| 162 | C8 | — | 60 | 1 | no | — |
| 163 | C8 | — | 60 | 1 | no | — |
| 164 | C8 | — | 60 | 1 | no | — |
| 165 | C2×C3 | — | 16 | 1 | no | — |
| 166 | C2×C3 | — | 16 | 1 | no | — |
| 167 | C2×C3 | — | 16 | 1 | no | — |
| 168 | C5 | #85552 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 169 | C22 | C5⋊#1020 | 1 | 1 | yes | characteristic |
| 170 | C4 | — | 3 | 1 | no | — |
| 171 | C4 | — | 3 | 1 | no | — |
| 172 | C4 | — | 3 | 1 | no | — |
| 173 | C4 | — | 3 | 1 | no | — |
| 174 | C4 | — | 6 | 1 | no | — |
| 175 | C4 | — | 12 | 1 | no | — |
| 176 | C4 | — | 120 | 1 | no | — |
| 177 | C3 | — | 16 | 1 | no | Sylow |
| 178 | C2 | C5⋊#5930 | 1 | 1 | yes | characteristic |
| 179 | C2 | C5⋊#5931 | 1 | 1 | yes | characteristic |
| 180 | C2 | C5⋊#5933 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C22 | C5⋊#1020 |
| commutator, prime radical, cosocle, hypocenter | C5×Q8⋊SL(2,𝔽3) | C2 |
| Fitting | C5×Q82 | D3 |
| socle | C22×C5 | #1020 |
| LCS | G ▷ C5×Q8⋊SL(2,𝔽3) | |
| UCS | C1 ◁ C22 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 81. C5×Q8 | #226 | semidirect |
| 155. Q8 | C5⋊#226 | semidirect |
| 168. C5 | #85552 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C5×Q8⋊SL(2,𝔽3) | 2 |
| #10602734 | 3 |
| C2×C5⋊#226 | 4 |
| #91184 | 4 |
| #85552 | 5 |