Finite Group DatabasebetaFinite Group DatabasebetaOrder 1920.
174 conjugacy classes of subgroups; only the 28 normal subgroups are shown.
This group has 174 conjugacy classes of subgroups — too many to draw. Only its 28 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 | #11751616 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #65791⋊C3 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #11751558 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #10603999 | — | 3 | 1 | no | maximal, Hall |
| 5 | #91955 | C4 | 1 | 1 | yes | characteristic |
| 6 | #4094⋊C3 | C22 | 1 | 1 | yes | characteristic |
| 7 | C5×GU(2,𝔽3) | C4 | 1 | 1 | yes | characteristic |
| 8 | #91164 | — | 4 | 1 | no | maximal |
| 9 | #73038 | — | 5 | 1 | no | maximal, Hall |
| 10 | #65791 | D3 | 1 | 1 | yes | characteristic, complemented |
| 11 | #65725 | — | 3 | 1 | no | — |
| 12 | #65113 | — | 3 | 1 | no | — |
| 13 | #64932 | — | 3 | 1 | no | — |
| 14 | #64898 | — | 3 | 1 | no | — |
| 15 | #64861 | — | 3 | 1 | no | — |
| 16 | C4×#674 | — | 3 | 1 | no | — |
| 17 | C5×#41⋊C3 | C2×C4 | 1 | 1 | yes | characteristic |
| 18 | C5⋊GL(2,𝔽3) | C8 | 1 | 1 | yes | characteristic |
| 19 | C5⋊#226 | C8 | 1 | 1 | yes | characteristic |
| 20 | #6648 | — | 4 | 1 | no | — |
| 21 | C3⋊#674 | — | 4 | 1 | no | — |
| 22 | C3×#674 | — | 4 | 1 | no | — |
| 23 | #5404 | — | 5 | 1 | no | — |
| 24 | #504⋊C3 | — | 5 | 1 | no | — |
| 25 | #4628 | — | 5 | 1 | no | — |
| 26 | #4096 | — | 3 | 1 | no | — |
| 27 | #4094 | C2×D3 | 1 | 1 | yes | characteristic |
| 28 | #4053 | — | 3 | 1 | no | — |
| 29 | #3998 | — | 3 | 1 | no | — |
| 30 | #3998 | — | 3 | 1 | no | — |
| 31 | C2×#674 | — | 3 | 1 | no | — |
| 32 | C2×#674 | — | 3 | 1 | no | — |
| 33 | #3990 | — | 6 | 1 | no | — |
| 34 | D5⋊C16 | — | 6 | 1 | no | — |
| 35 | C5×#104 | — | 3 | 1 | no | — |
| 36 | #3954 | — | 3 | 1 | no | — |
| 37 | #3937 | — | 3 | 1 | no | — |
| 38 | C4×#168 | — | 3 | 1 | no | — |
| 39 | #2170 | — | 15 | 1 | no | Sylow |
| 40 | C4×D15 | — | 4 | 1 | no | — |
| 41 | C5×SL(2,𝔽3) | C2×C8 | 1 | 1 | yes | characteristic |
| 42 | C3×#168 | — | 4 | 1 | no | — |
| 43 | C5×C3⋊C8 | — | 4 | 1 | no | — |
| 44 | #985 | — | 5 | 1 | no | — |
| 45 | #131⋊C3 | — | 5 | 1 | no | — |
| 46 | GU(2,𝔽3) | AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 47 | D3×C16 | — | 20 | 1 | no | — |
| 48 | C5×#41 | C4×D3 | 1 | 1 | yes | characteristic |
| 49 | #709 | — | 3 | 1 | no | — |
| 50 | C5×M16 | — | 3 | 1 | no | — |
| 51 | C42×C5 | — | 3 | 1 | no | — |
| 52 | #689 | — | 3 | 1 | no | — |
| 53 | #688 | — | 3 | 1 | no | — |
| 54 | #687 | — | 3 | 1 | no | — |
| 55 | #686 | — | 3 | 1 | no | — |
| 56 | #681 | — | 3 | 1 | no | — |
| 57 | C2×#168 | — | 3 | 1 | no | — |
| 58 | C2×#168 | — | 3 | 1 | no | — |
| 59 | #676 | — | 6 | 1 | no | — |
| 60 | C8×D5 | — | 6 | 1 | no | — |
| 61 | #674 | — | 3 | 1 | no | — |
| 62 | #674 | — | 6 | 1 | no | — |
| 63 | #674 | — | 6 | 1 | no | — |
| 64 | #674 | S4 | 1 | 1 | yes | characteristic |
| 65 | #504 | — | 5 | 1 | no | — |
| 66 | #504 | — | 15 | 1 | no | — |
| 67 | #443 | — | 15 | 1 | no | — |
| 68 | #364 | — | 15 | 1 | no | — |
| 69 | #350 | — | 15 | 1 | no | — |
| 70 | #350 | — | 15 | 1 | no | — |
| 71 | C4×C16 | — | 15 | 1 | no | — |
| 72 | C2×D15 | — | 4 | 1 | no | — |
| 73 | C3×C4×C5 | — | 4 | 1 | no | — |
| 74 | Dic15 | — | 4 | 1 | no | — |
| 75 | #41⋊C3 | C2×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 76 | GL(2,𝔽3) | — | 5 | 1 | no | — |
| 77 | #226 | — | 5 | 1 | no | — |
| 78 | D3×C8 | — | 20 | 1 | no | — |
| 79 | C3×C16 | — | 20 | 1 | no | — |
| 80 | C3⋊C16 | — | 20 | 1 | no | — |
| 81 | C5×Q8 | D3×C8 | 1 | 1 | yes | characteristic |
| 82 | C5×D4 | — | 3 | 1 | no | — |
| 83 | C2×C4×C5 | — | 3 | 1 | no | — |
| 84 | C2×C4×C5 | — | 6 | 1 | no | — |
| 85 | C22⋊D5 | — | 6 | 1 | no | — |
| 86 | D20 | — | 3 | 1 | no | — |
| 87 | C4×D5 | — | 6 | 1 | no | — |
| 88 | Dic10 | — | 3 | 1 | no | — |
| 89 | C5×C8 | — | 6 | 1 | no | — |
| 90 | #168 | — | 3 | 1 | no | — |
| 91 | #168 | — | 3 | 1 | no | — |
| 92 | #168 | — | 3 | 1 | no | — |
| 93 | #168 | C2×S4 | 1 | 1 | yes | characteristic |
| 94 | #135 | — | 15 | 1 | no | — |
| 95 | #131 | — | 5 | 1 | no | — |
| 96 | #131 | — | 15 | 1 | no | — |
| 97 | M32 | — | 15 | 1 | no | — |
| 98 | M32 | — | 15 | 1 | no | — |
| 99 | M32 | — | 30 | 1 | no | — |
| 100 | C2×C16 | — | 15 | 1 | no | — |
| 101 | C2×C16 | — | 15 | 1 | no | — |
| 102 | C2×C16 | — | 30 | 1 | no | — |
| 103 | #108 | — | 15 | 1 | no | — |
| 104 | #104 | — | 3 | 1 | no | — |
| 105 | #104 | — | 15 | 1 | no | — |
| 106 | C4×C8 | — | 15 | 1 | no | — |
| 107 | C2×C3×C5 | — | 4 | 1 | no | — |
| 108 | D15 | — | 4 | 2 | no | — |
| 109 | C4×D3 | — | 20 | 1 | no | — |
| 110 | SL(2,𝔽3) | D5⋊C8 | 1 | 1 | yes | characteristic |
| 111 | C3×C8 | — | 20 | 1 | no | — |
| 112 | C3⋊C8 | — | 4 | 1 | no | — |
| 113 | C22×C5 | — | 3 | 1 | no | — |
| 114 | C2×D5 | — | 6 | 1 | no | — |
| 115 | C4×C5 | — | 3 | 1 | no | — |
| 116 | C4×C5 | — | 6 | 2 | no | — |
| 117 | C4×C5 | C4×S4 | 1 | 1 | yes | characteristic |
| 118 | Dic5 | — | 6 | 1 | no | — |
| 119 | #41 | — | 15 | 1 | no | — |
| 120 | #41 | D3×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 121 | Q16 | — | 15 | 1 | no | — |
| 122 | SD16 | — | 15 | 1 | no | — |
| 123 | SD16 | — | 15 | 1 | no | — |
| 124 | D8 | — | 15 | 1 | no | — |
| 125 | M16 | — | 3 | 1 | no | — |
| 126 | M16 | — | 15 | 1 | no | — |
| 127 | M16 | — | 30 | 1 | no | — |
| 128 | C2×C8 | — | 15 | 1 | no | — |
| 129 | C2×C8 | — | 15 | 1 | no | — |
| 130 | C2×C8 | — | 30 | 1 | no | — |
| 131 | C42 | — | 3 | 1 | no | — |
| 132 | C16 | — | 5 | 1 | no | — |
| 133 | C16 | — | 15 | 1 | no | — |
| 134 | C16 | — | 30 | 1 | no | — |
| 135 | C16 | — | 30 | 1 | no | — |
| 136 | C3×C5 | — | 4 | 1 | no | Hall |
| 137 | C2×D3 | — | 20 | 1 | no | — |
| 138 | C3×C4 | — | 4 | 1 | no | — |
| 139 | Dic3 | — | 20 | 1 | no | — |
| 140 | C2×C5 | — | 6 | 1 | no | — |
| 141 | C2×C5 | C8×S4 | 1 | 1 | yes | characteristic |
| 142 | D5 | — | 12 | 1 | no | — |
| 143 | Q8 | — | 15 | 1 | no | — |
| 144 | Q8 | #6738 | 1 | 1 | yes | characteristic |
| 145 | D4 | — | 3 | 1 | no | — |
| 146 | D4 | — | 15 | 1 | no | — |
| 147 | D4 | — | 30 | 1 | no | — |
| 148 | C2×C4 | — | 3 | 1 | no | — |
| 149 | C2×C4 | — | 6 | 1 | no | — |
| 150 | C2×C4 | — | 30 | 1 | no | — |
| 151 | C8 | — | 5 | 1 | no | — |
| 152 | C8 | — | 6 | 1 | no | — |
| 153 | C8 | — | 15 | 1 | no | — |
| 154 | C8 | — | 15 | 1 | no | — |
| 155 | C8 | — | 15 | 1 | no | — |
| 156 | C2×C3 | — | 4 | 1 | no | — |
| 157 | D3 | — | 20 | 2 | no | — |
| 158 | C5 | #73038 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 159 | C22 | — | 3 | 1 | no | — |
| 160 | C22 | — | 30 | 1 | no | — |
| 161 | C4 | — | 3 | 1 | no | — |
| 162 | C4 | — | 6 | 2 | no | — |
| 163 | C4 | — | 30 | 1 | no | — |
| 164 | C4 | AGL(1,𝔽5)×S4 | 1 | 1 | yes | characteristic |
| 165 | C3 | — | 4 | 1 | no | Sylow |
| 166 | C2 | — | 6 | 1 | no | — |
| 167 | C2 | — | 60 | 1 | no | — |
| 168 | C2 | #11756708 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C4 | AGL(1,𝔽5)×S4 |
| commutator, hypocenter | C5×SL(2,𝔽3) | C2×C8 |
| Fitting | C5×#41 | C4×D3 |
| socle | C2×C5 | C8×S4 |
| prime radical | #91955 | C4 |
| cosocle | #4094⋊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 |
|---|---|---|
| 2. #65791⋊C3 | C2 | semidirect |
| 3. #11751558 | C2 | semidirect |
| 10. #65791 | D3 | semidirect |
| 158. C5 | #73038 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #11751616 | 2 |
| #65791⋊C3 | 2 |
| #11751558 | 2 |
| #10603999 | 3 |
| #91164 | 4 |
| #73038 | 5 |