Finite Group Databasebeta

← C2×C32×C4×C7×C49 #72202000

Subgroups — C2×C32×C4×C7×C49 #72202000

Order 24696.

864 conjugacy classes of subgroups.

The subgroup lattice has 108 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C32×C7×C49C211yesmaximal, characteristic
2C32×C4×C7×C49C212yesmaximal, direct product factor
3C2×C3×C4×C7×C49C314yesmaximal, direct product factor
4C2×C32×C7×C49C412yesdirect product factor
5C2×C32×C7×C49C2211yescharacteristic
6C22×C3×C7×C49C2×C314yes—
7C3×C4×C7×C49C2×C318yesdirect product factor
8C2×C32×C4×C72C711yesmaximal, characteristic
9C2×C32×C4×C49C717yesmaximal, direct product factor
10C32×C7×C49C2×C411yescharacteristic, direct product factor, Hall
11C2×C4×C7×C49C3211yescharacteristic, direct product factor, Hall
12C2×C3×C7×C49C3×C418yesdirect product factor
13C2×C3×C7×C49C22×C314yes—
14C22×C32×C72C2×C711yescharacteristic
15C32×C4×C72C2×C712yes—
16C22×C32×C49C2×C717yes—
17C32×C4×C49C2×C7114yesdirect product factor
18C22×C7×C49C2×C3211yescharacteristic
19C4×C7×C49C2×C3212yesdirect product factor
20C2×C3×C4×C72C3×C714yes—
21C2×C3×C4×C49C3×C7128yesdirect product factor
22C3×C7×C49C2×C3×C414yesdirect product factor
23C2×C32×C72C4×C712yes—
24C2×C32×C72C22×C711yescharacteristic
25C2×C32×C49C4×C7114yesdirect product factor
26C2×C32×C49C22×C717yes—
27C2×C7×C49C32×C412yesdirect product factor
28C2×C7×C49C22×C3211yescharacteristic
29C22×C3×C72C2×C3×C714yes—
30C3×C4×C72C2×C3×C718yes—
31C22×C3×C49C2×C3×C7128yes—
32C3×C4×C49C2×C3×C7156yesdirect product factor
33C2×C32×C4×C7C4917yesdirect product factor
34C2×C32×C4×C7C7211yescharacteristic
35C32×C72C2×C4×C711yescharacteristic
36C32×C49C2×C4×C717yesdirect product factor
37C2×C4×C72C32×C711yescharacteristic
38C2×C4×C49C32×C717yesdirect product factor
39C7×C49C2×C32×C411yescharacteristic, direct product factor, Sylow
40C2×C3×C72C3×C4×C718yes—
41C2×C3×C72C22×C3×C714yes—
42C2×C3×C49C3×C4×C7156yesdirect product factor
43C2×C3×C49C22×C3×C7128yes—
44C22×C32×C7C2×C4917yes—
45C22×C32×C7C2×C7211yescharacteristic
46C32×C4×C7C2×C49114yesdirect product factor
47C32×C4×C7C2×C7212yes—
48C22×C72C2×C32×C711yescharacteristic
49C4×C72C2×C32×C712yes—
50C22×C49C2×C32×C717yes—
51C4×C49C2×C32×C7114yesdirect product factor
52C2×C3×C4×C7C3×C49128yesdirect product factor
53C2×C3×C4×C7C3×C7214yes—
54C3×C72C2×C3×C4×C714yes—
55C3×C49C2×C3×C4×C7128yesdirect product factor
56C2×C32×C7C4×C49114yesdirect product factor
57C2×C32×C7C22×C4917yes—
58C2×C32×C7C4×C7212yes—
59C2×C32×C7C22×C7211yescharacteristic
60C2×C72C32×C4×C712yes—
61C2×C72C22×C32×C711yescharacteristic
62C2×C49C32×C4×C7114yesdirect product factor
63C2×C49C22×C32×C717yes—
64C22×C3×C7C2×C3×C49128yes—
65C22×C3×C7C2×C3×C7214yes—
66C3×C4×C7C2×C3×C49156yesdirect product factor
67C3×C4×C7C2×C3×C7218yes—
68C2×C32×C4C7×C4911yescharacteristic, direct product factor, Hall
69C32×C7C2×C4×C4917yesdirect product factor
70C32×C7C2×C4×C7211yescharacteristic
71C2×C4×C7C32×C4917yesdirect product factor
72C2×C4×C7C32×C7211yescharacteristic
73C72C2×C32×C4×C711yescharacteristic
74C49C2×C32×C4×C717yesdirect product factor
75C2×C3×C7C3×C4×C49156yesdirect product factor
76C2×C3×C7C22×C3×C49128yes—
77C2×C3×C7C3×C4×C7218yes—
78C2×C3×C7C22×C3×C7214yes—
79C22×C32C2×C7×C4911yescharacteristic
80C32×C4C2×C7×C4912yesdirect product factor
81C22×C7C2×C32×C4917yes—
82C22×C7C2×C32×C7211yescharacteristic
83C4×C7C2×C32×C49114yesdirect product factor
84C4×C7C2×C32×C7212yes—
85C2×C3×C4C3×C7×C4914yesdirect product factor
86C3×C7C2×C3×C4×C49128yesdirect product factor
87C3×C7C2×C3×C4×C7214yes—
88C2×C32C4×C7×C4912yesdirect product factor
89C2×C32C22×C7×C4911yescharacteristic
90C2×C7C32×C4×C49114yesdirect product factor
91C2×C7C22×C32×C4917yes—
92C2×C7C32×C4×C7212yes—
93C2×C7C22×C32×C7211yescharacteristic
94C22×C3C2×C3×C7×C4914yes—
95C3×C4C2×C3×C7×C4918yesdirect product factor
96C32C2×C4×C7×C4911yescharacteristic, direct product factor, Sylow
97C2×C4C32×C7×C4911yescharacteristic, direct product factor, Sylow
98C7C2×C32×C4×C4917yesdirect product factor
99C7C2×C32×C4×C7211yescharacteristic
100C2×C3C3×C4×C7×C4918yesdirect product factor
101C2×C3C22×C3×C7×C4914yes—
102C22C2×C32×C7×C4911yescharacteristic
103C4C2×C32×C7×C4912yesdirect product factor
104C3C2×C3×C4×C7×C4914yesdirect product factor
105C2C32×C4×C7×C4912yesdirect product factor
106C2C22×C32×C7×C4911yescharacteristic
subgroupG/H
centerGC1
commutatorC1G
Frattini, cosocleC2×C7C22×C32×C72
socle, prime radicalC22×C32×C72C2×C7
LCSG ▷ C1
UCSC1 ◁ G

Complements

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

NH ≅ G/Ntype
2. C32×C4×C7×C49C2direct (N × H)
3. C2×C3×C4×C7×C49C3direct (N × H)
4. C2×C32×C7×C49C4direct (N × H)
7. C3×C4×C7×C49C2×C3direct (N × H)
9. C2×C32×C4×C49C7direct (N × H)
10. C32×C7×C49C2×C4direct (N × H)
11. C2×C4×C7×C49C32direct (N × H)
12. C2×C3×C7×C49C3×C4direct (N × H)
17. C32×C4×C49C2×C7direct (N × H)
19. C4×C7×C49C2×C32direct (N × H)
21. C2×C3×C4×C49C3×C7direct (N × H)
22. C3×C7×C49C2×C3×C4direct (N × H)
25. C2×C32×C49C4×C7direct (N × H)
27. C2×C7×C49C32×C4direct (N × H)
32. C3×C4×C49C2×C3×C7direct (N × H)
33. C2×C32×C4×C7C49direct (N × H)
36. C32×C49C2×C4×C7direct (N × H)
38. C2×C4×C49C32×C7direct (N × H)
39. C7×C49C2×C32×C4direct (N × H)
42. C2×C3×C49C3×C4×C7direct (N × H)
46. C32×C4×C7C2×C49direct (N × H)
51. C4×C49C2×C32×C7direct (N × H)
52. C2×C3×C4×C7C3×C49direct (N × H)
55. C3×C49C2×C3×C4×C7direct (N × H)
56. C2×C32×C7C4×C49direct (N × H)
62. C2×C49C32×C4×C7direct (N × H)
66. C3×C4×C7C2×C3×C49direct (N × H)
68. C2×C32×C4C7×C49direct (N × H)
69. C32×C7C2×C4×C49direct (N × H)
71. C2×C4×C7C32×C49direct (N × H)
74. C49C2×C32×C4×C7direct (N × H)
75. C2×C3×C7C3×C4×C49direct (N × H)
80. C32×C4C2×C7×C49direct (N × H)
83. C4×C7C2×C32×C49direct (N × H)
85. C2×C3×C4C3×C7×C49direct (N × H)
86. C3×C7C2×C3×C4×C49direct (N × H)
88. C2×C32C4×C7×C49direct (N × H)
90. C2×C7C32×C4×C49direct (N × H)
95. C3×C4C2×C3×C7×C49direct (N × H)
96. C32C2×C4×C7×C49direct (N × H)
97. C2×C4C32×C7×C49direct (N × H)
98. C7C2×C32×C4×C49direct (N × H)
100. C2×C3C3×C4×C7×C49direct (N × H)
103. C4C2×C32×C7×C49direct (N × H)
104. C3C2×C3×C4×C7×C49direct (N × H)
105. C2C32×C4×C7×C49direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C32×C7×C492
C32×C4×C7×C492
C2×C3×C4×C7×C493
C2×C32×C4×C727
C2×C32×C4×C497