Finite Group Databasebeta

← C2×C3×C72×D51 #44473015

Subgroups — C2×C3×C72×D51 #44473015

Order 29988.

440 conjugacy classes of subgroups; only the 220 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 220 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C72×D51C212yesmaximal, direct product factor
2C2×C32×C72×C17C211yesmaximal, characteristic, complemented
3C2×C3×C72×D17—31nomaximal
4C2×C72×D51C311yesmaximal, characteristic, direct product factor
5C32×C72×C17C2211yescharacteristic, complemented, Hall
6C72×D51C2×C312yesdirect product factor
7C3×C72×D17—32no—
8C2×C3×C72×C17—21no—
9C2×C3×C72×C17D311yescharacteristic, complemented
10C2×C3×C72×C17C2×C311yescharacteristic, complemented
11C2×C3×C7×D51C718yesmaximal, direct product factor
12C2×C72×D17—31noHall
13C3×C72×C17—21no—
14C3×C72×C17C2×D311yescharacteristic, complemented
15C3×C72×C17C22×C311yescharacteristic, complemented
16C3×C7×D51C2×C7116yesdirect product factor
17C2×C32×C7×C17C2×C718yescomplemented
18C2×C3×D3×C72—171nomaximal, Hall
19C2×C72×C17C3×D311yescharacteristic, complemented
20C72×D17—32no—
21C2×C7×D51C3×C718yesdirect product factor
22C2×C3×C7×D17—38no—
23C32×C7×C17C22×C718yescomplemented
24C2×C32×C72D1711yescharacteristic, complemented
25C3×D3×C72—172no—
26C72×C17C2×C3×D311yescharacteristic, complemented, Hall
27C2×C3×C7×C17—28no—
28C2×C3×C7×C17D3×C718yescomplemented
29C2×C3×C7×C17C2×C3×C718yescomplemented
30C7×D51C2×C3×C7116yesdirect product factor
31C3×C7×D17—316no—
32C2×C3×D51C7211yescharacteristic, direct product factor, Hall
33C22×C3×C72—511no—
34C2×D3×C72—171no—
35C2×C7×D17—38no—
36C32×C72C2×D1711yescharacteristic, complemented, Hall
37C3×C7×C17—28no—
38C3×C7×C17C2×D3×C718yescomplemented
39C3×C7×C17C22×C3×C718yescomplemented
40C2×C32×C17C2×C7211yescharacteristic, complemented
41C3×D51C2×C7212yesdirect product factor
42C2×C3×C72—21no—
43C2×C3×C72—512no—
44C2×C3×C72C3×D1711yescharacteristic, complemented
45C2×C3×C72D5111yescharacteristic, direct product factor
46D3×C72—172no—
47C2×C3×D3×C7—178no—
48C2×C7×C17C3×D3×C718yescomplemented
49C7×D17—316no—
50C2×D51C3×C7211yescharacteristic, direct product factor
51C2×C3×D17—31no—
52C22×C72—511noHall
53C32×C17C22×C7211yescharacteristic, complemented, Hall
54C3×C72—21no—
55C3×C72C2×C3×D1711yescharacteristic, complemented
56C3×C72C2×D5111yescharacteristic, direct product factor
57C2×C32×C7C7×D1718yescomplemented
58C3×D3×C7—1716no—
59C7×C17C2×C3×D3×C718yescomplemented
60C2×C3×C17—21no—
61C2×C3×C17D3×C7211yescharacteristic, complemented
62C2×C3×C17C2×C3×C7211yescharacteristic, complemented
63D51C2×C3×C7212yesdirect product factor
64C3×D17—32no—
65C2×C72—512no—
66C2×C72C3×D5111yescharacteristic, direct product factor
67C22×C3×C7—518no—
68C2×D3×C7—178no—
69C2×D17—31noHall
70C32×C7C2×C7×D1718yescomplemented
71C3×C17—21no—
72C3×C17C2×D3×C7211yescharacteristic, complemented
73C3×C17C22×C3×C7211yescharacteristic, complemented
74C72C2×C3×D5111yescharacteristic, direct product factor, Sylow
75C2×C3×C7—28no—
76C2×C3×C7—5116no—
77C2×C3×C7C3×C7×D1718yescomplemented
78C2×C3×C7C7×D5118yesdirect product factor
79D3×C7—1716no—
80C2×C3×D3—171noHall
81C2×C17C3×D3×C7211yescharacteristic, complemented
82D17—32no—
83C22×C7—518no—
84C3×C7—28no—
85C3×C7C2×C3×C7×D1718yescomplemented
86C3×C7C2×C7×D5118yesdirect product factor
87C2×C32C72×D1711yescharacteristic, complemented
88C3×D3—172no—
89C17C2×C3×D3×C7211yescharacteristic, complemented, Sylow
90C2×C7—5116no—
91C2×C7C3×C7×D5118yesdirect product factor
92C22×C3—511no—
93C2×D3—171no—
94C32C2×C72×D1711yescharacteristic, complemented, Sylow
95C7C2×C3×C7×D5118yesdirect product factor
96C2×C3—21no—
97C2×C3—512no—
98C2×C3C3×C72×D1711yescharacteristic, complemented
99C2×C3C72×D5111yescharacteristic, direct product factor
100D3—172no—
101C22—511noSylow
102C3—21no—
103C3C2×C72×D5111yescharacteristic, direct product factor
104C3C2×C3×C72×D1711yescharacteristic, complemented
105C2—512no—
106C2C3×C72×D5111yescharacteristic, direct product factor
subgroupG/H
center, hypercenterC2×C3×C72D51
commutator, cosocle, hypocenterC3×C17C22×C3×C72
FrattiniC1G
Fitting, socleC2×C32×C72×C17C2
LCSG ▷ C3×C17
UCSC1 ◁ C2×C3×C72

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C3×C72×D51C2direct (N × H)
2. C2×C32×C72×C17C2semidirect
4. C2×C72×D51C3direct (N × H)
5. C32×C72×C17C22semidirect
6. C72×D51C2×C3direct (N × H)
9. C2×C3×C72×C17D3semidirect
10. C2×C3×C72×C17C2×C3semidirect
11. C2×C3×C7×D51C7direct (N × H)
14. C3×C72×C17C2×D3semidirect
15. C3×C72×C17C22×C3semidirect
16. C3×C7×D51C2×C7direct (N × H)
17. C2×C32×C7×C17C2×C7semidirect
19. C2×C72×C17C3×D3semidirect
21. C2×C7×D51C3×C7direct (N × H)
23. C32×C7×C17C22×C7semidirect
24. C2×C32×C72D17semidirect
26. C72×C17C2×C3×D3semidirect
28. C2×C3×C7×C17D3×C7semidirect
29. C2×C3×C7×C17C2×C3×C7semidirect
30. C7×D51C2×C3×C7direct (N × H)
32. C2×C3×D51C72direct (N × H)
36. C32×C72C2×D17semidirect
38. C3×C7×C17C2×D3×C7semidirect
39. C3×C7×C17C22×C3×C7semidirect
40. C2×C32×C17C2×C72semidirect
41. C3×D51C2×C72direct (N × H)
44. C2×C3×C72C3×D17semidirect
45. C2×C3×C72D51direct (N × H)
48. C2×C7×C17C3×D3×C7semidirect
50. C2×D51C3×C72direct (N × H)
53. C32×C17C22×C72semidirect
55. C3×C72C2×C3×D17semidirect
56. C3×C72C2×D51direct (N × H)
57. C2×C32×C7C7×D17semidirect
59. C7×C17C2×C3×D3×C7semidirect
61. C2×C3×C17D3×C72semidirect
62. C2×C3×C17C2×C3×C72semidirect
63. D51C2×C3×C72direct (N × H)
66. C2×C72C3×D51direct (N × H)
70. C32×C7C2×C7×D17semidirect
72. C3×C17C2×D3×C72semidirect
73. C3×C17C22×C3×C72semidirect
74. C72C2×C3×D51direct (N × H)
77. C2×C3×C7C3×C7×D17semidirect
78. C2×C3×C7C7×D51direct (N × H)
81. C2×C17C3×D3×C72semidirect
85. C3×C7C2×C3×C7×D17semidirect
86. C3×C7C2×C7×D51direct (N × H)
87. C2×C32C72×D17semidirect
89. C17C2×C3×D3×C72semidirect
91. C2×C7C3×C7×D51direct (N × H)
94. C32C2×C72×D17semidirect
95. C7C2×C3×C7×D51direct (N × H)
98. C2×C3C3×C72×D17semidirect
99. C2×C3C72×D51direct (N × H)
103. C3C2×C72×D51direct (N × H)
104. C3C2×C3×C72×D17semidirect
106. C2C3×C72×D51direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C3×C72×D512
C2×C32×C72×C172
C2×C3×C72×D173
C2×C72×D513
C2×C3×C7×D517
C2×C3×D3×C7217