Finite Group DatabasebetaFinite Group DatabasebetaOrder 1920.
142 conjugacy classes of subgroups; only the 16 normal subgroups are shown.
This group has 142 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 | C5⋊(#392⋊C3) | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #10598495 | — | 3 | 1 | no | maximal, Hall |
| 3 | A4⋊#170 | — | 4 | 1 | no | maximal |
| 4 | C5×#95⋊C3 | C4 | 1 | 1 | yes | characteristic, complemented |
| 5 | #68000 | — | 5 | 1 | no | maximal, Hall |
| 6 | #64960 | D3 | 1 | 1 | yes | characteristic |
| 7 | #64917 | — | 3 | 1 | no | — |
| 8 | #64915 | — | 3 | 1 | no | — |
| 9 | A4×Dic5 | — | 4 | 1 | no | — |
| 10 | #392⋊C3 | — | 5 | 1 | no | — |
| 11 | C22×D20 | — | 3 | 1 | no | — |
| 12 | C2×C22⋊AGL(1,𝔽5) | — | 6 | 1 | no | — |
| 13 | C2×#685 | — | 3 | 1 | no | — |
| 14 | C22⋊#170 | — | 6 | 1 | no | — |
| 15 | C5×#95 | Dic3 | 1 | 1 | yes | characteristic |
| 16 | #1343 | — | 15 | 1 | no | Sylow |
| 17 | C2×C5×A4 | — | 4 | 1 | no | — |
| 18 | C3⋊#170 | — | 16 | 1 | no | — |
| 19 | A4⋊C8 | — | 20 | 1 | no | — |
| 20 | #95⋊C3 | AGL(1,𝔽5) | 1 | 1 | yes | characteristic, complemented |
| 21 | C23×D5 | — | 3 | 1 | no | — |
| 22 | C22×AGL(1,𝔽5) | — | 12 | 1 | no | — |
| 23 | C22×C4×C5 | — | 3 | 1 | no | — |
| 24 | C22×Dic5 | S4 | 1 | 1 | yes | characteristic |
| 25 | C2×D20 | — | 12 | 1 | no | — |
| 26 | C2×D20 | — | 12 | 1 | no | — |
| 27 | C2×D20 | — | 12 | 1 | no | — |
| 28 | C22⋊AGL(1,𝔽5) | — | 6 | 2 | no | — |
| 29 | C22⋊AGL(1,𝔽5) | — | 12 | 1 | no | — |
| 30 | C2×#170 | — | 12 | 1 | no | — |
| 31 | #685 | — | 12 | 1 | no | — |
| 32 | #392 | — | 5 | 1 | no | — |
| 33 | #342 | — | 15 | 1 | no | — |
| 34 | #323 | — | 15 | 1 | no | — |
| 35 | C5×A4 | — | 4 | 1 | no | — |
| 36 | C3×Dic5 | — | 16 | 1 | no | — |
| 37 | C4×A4 | — | 20 | 1 | no | — |
| 38 | C23×C5 | SL(2,ℤ/4) | 1 | 1 | yes | characteristic |
| 39 | C22×D5 | — | 3 | 1 | no | — |
| 40 | C22×D5 | — | 3 | 1 | no | — |
| 41 | C22×D5 | — | 6 | 1 | no | — |
| 42 | C22×D5 | — | 6 | 1 | no | — |
| 43 | C22×D5 | — | 12 | 1 | no | — |
| 44 | C22×D5 | — | 12 | 1 | no | — |
| 45 | C2×AGL(1,𝔽5) | — | 12 | 2 | no | — |
| 46 | C2×AGL(1,𝔽5) | — | 24 | 1 | no | — |
| 47 | C2×AGL(1,𝔽5) | — | 24 | 1 | no | — |
| 48 | C2×C4×C5 | — | 6 | 1 | no | — |
| 49 | C2×C4×C5 | — | 12 | 1 | no | — |
| 50 | C2×Dic5 | — | 6 | 1 | no | — |
| 51 | D20 | — | 12 | 1 | no | — |
| 52 | D20 | — | 12 | 1 | no | — |
| 53 | D20 | — | 12 | 2 | no | — |
| 54 | #170 | — | 24 | 1 | no | — |
| 55 | C22×D4 | — | 15 | 1 | no | — |
| 56 | C2×C22⋊C4 | — | 15 | 1 | no | — |
| 57 | C2×C22⋊C4 | — | 30 | 1 | no | — |
| 58 | C22⋊C8 | — | 30 | 1 | no | — |
| 59 | #95 | ΓL(1,𝔽16) | 1 | 1 | yes | characteristic |
| 60 | C2×C3×C5 | — | 16 | 1 | no | — |
| 61 | C2×A4 | — | 4 | 1 | no | — |
| 62 | C3⋊C8 | — | 80 | 1 | no | — |
| 63 | C22×C5 | — | 3 | 1 | no | — |
| 64 | C22×C5 | — | 3 | 1 | no | — |
| 65 | C22×C5 | GU(2,𝔽3) | 1 | 1 | yes | characteristic |
| 66 | C2×D5 | — | 6 | 1 | no | — |
| 67 | C2×D5 | — | 6 | 1 | no | — |
| 68 | C2×D5 | — | 12 | 1 | no | — |
| 69 | C2×D5 | — | 12 | 1 | no | — |
| 70 | C2×D5 | — | 12 | 1 | no | — |
| 71 | C2×D5 | — | 12 | 1 | no | — |
| 72 | C2×D5 | — | 24 | 1 | no | — |
| 73 | AGL(1,𝔽5) | — | 24 | 2 | no | — |
| 74 | C4×C5 | — | 12 | 1 | no | — |
| 75 | Dic5 | — | 4 | 1 | no | — |
| 76 | C24 | — | 15 | 1 | no | — |
| 77 | C2×D4 | — | 60 | 1 | no | — |
| 78 | C2×D4 | — | 60 | 1 | no | — |
| 79 | C2×D4 | — | 60 | 1 | no | — |
| 80 | C22×C4 | — | 3 | 1 | no | — |
| 81 | C22×C4 | — | 5 | 1 | no | — |
| 82 | C22×C4 | — | 60 | 1 | no | — |
| 83 | C2×C8 | — | 60 | 1 | no | — |
| 84 | C22⋊C4 | — | 30 | 2 | no | — |
| 85 | C22⋊C4 | — | 60 | 1 | no | — |
| 86 | C22⋊C4 | — | 60 | 1 | no | — |
| 87 | C3×C5 | — | 16 | 1 | no | Hall |
| 88 | A4 | — | 4 | 1 | no | — |
| 89 | C3×C4 | — | 80 | 1 | no | — |
| 90 | C2×C5 | — | 3 | 1 | no | — |
| 91 | C2×C5 | — | 3 | 1 | no | — |
| 92 | C2×C5 | #4626 | 1 | 1 | yes | characteristic |
| 93 | D5 | — | 12 | 1 | no | — |
| 94 | D5 | — | 12 | 1 | no | — |
| 95 | C23 | — | 15 | 1 | no | — |
| 96 | C23 | — | 15 | 1 | no | — |
| 97 | C23 | — | 30 | 1 | no | — |
| 98 | C23 | — | 30 | 1 | no | — |
| 99 | C23 | — | 60 | 1 | no | — |
| 100 | C23 | — | 60 | 1 | no | — |
| 101 | C23 | A4⋊AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 102 | D4 | — | 60 | 1 | no | — |
| 103 | D4 | — | 60 | 1 | no | — |
| 104 | D4 | — | 60 | 2 | no | — |
| 105 | C2×C4 | — | 6 | 1 | no | — |
| 106 | C2×C4 | — | 12 | 1 | no | — |
| 107 | C2×C4 | — | 30 | 1 | no | — |
| 108 | C2×C4 | — | 60 | 2 | no | — |
| 109 | C2×C4 | — | 120 | 1 | no | — |
| 110 | C2×C4 | — | 120 | 1 | no | — |
| 111 | C8 | — | 120 | 1 | no | — |
| 112 | C2×C3 | — | 16 | 1 | no | — |
| 113 | C5 | #68000 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 114 | C22 | — | 3 | 1 | no | — |
| 115 | C22 | — | 3 | 1 | no | — |
| 116 | C22 | — | 30 | 1 | no | — |
| 117 | C22 | — | 30 | 1 | no | — |
| 118 | C22 | — | 60 | 1 | no | — |
| 119 | C22 | — | 60 | 1 | no | — |
| 120 | C22 | — | 60 | 1 | no | — |
| 121 | C22 | — | 60 | 1 | no | — |
| 122 | C22 | — | 120 | 1 | no | — |
| 123 | C22 | #91885 | 1 | 1 | yes | characteristic |
| 124 | C4 | — | 12 | 1 | no | — |
| 125 | C4 | — | 20 | 1 | no | — |
| 126 | C4 | — | 120 | 2 | no | — |
| 127 | C3 | — | 16 | 1 | no | Sylow |
| 128 | C2 | — | 3 | 1 | no | — |
| 129 | C2 | — | 3 | 1 | no | — |
| 130 | C2 | — | 60 | 1 | no | — |
| 131 | C2 | — | 60 | 1 | no | — |
| 132 | C2 | #11751554 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | #11751554 |
| commutator, hypocenter | C5×#95⋊C3 | C4 |
| Frattini | C23 | A4⋊AGL(1,𝔽5) |
| Fitting | C5×#95 | Dic3 |
| socle | C23×C5 | SL(2,ℤ/4) |
| prime radical, cosocle | C5⋊(#392⋊C3) | C2 |
| LCS | G ▷ C5×#95⋊C3 | |
| UCS | C1 ◁ C2 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 4. C5×#95⋊C3 | C4 | semidirect |
| 20. #95⋊C3 | AGL(1,𝔽5) | semidirect |
| 113. C5 | #68000 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C5⋊(#392⋊C3) | 2 |
| #10598495 | 3 |
| A4⋊#170 | 4 |
| #68000 | 5 |