Finite Group Databasebeta

← C28×C3×C4×C7 #72199039

Subgroups — C28×C3×C4×C7 #72199039

Order 21504.

64598868 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
1C27×C3×C4×C7C21510yesmaximal, direct product factor
2C29×C3×C7C211yesmaximal, characteristic
3C28×C4×C7C311yesmaximal, characteristic, direct product factor, Hall
4C28×C3×C7C41256yesdirect product factor
5C28×C3×C7C221255yes—
6C26×C3×C4×C7C22143180yesdirect product factor
7C27×C4×C7C2×C31510yesdirect product factor
8C29×C7C2×C311yescharacteristic
9C28×C3×C4C711yesmaximal, characteristic, direct product factor, Hall
10C25×C3×C4×C7C231777240yesdirect product factor
11C27×C3×C7C2×C4132640yesdirect product factor
12C27×C3×C7C23110795yes—
13C27×C3×C4C2×C71510yesdirect product factor
14C29×C3C2×C711yescharacteristic
15C28×C4C3×C711yescharacteristic, direct product factor, Sylow
16C28×C7C3×C41256yesdirect product factor
17C28×C7C22×C31255yes—
18C26×C4×C7C22×C3143180yesdirect product factor
19C26×C3×C7C22×C41690880yesdirect product factor
20C26×C3×C7C24197155yes—
21C24×C3×C4×C7C2413212592yesdirect product factor
22C27×C7C2×C3×C4132640yesdirect product factor
23C27×C7C23×C3110795yes—
24C25×C4×C7C23×C31777240yesdirect product factor
25C28×C3C4×C71256yesdirect product factor
26C28×C3C22×C71255yes—
27C26×C3×C4C22×C7143180yesdirect product factor
28C25×C3×C7C23×C413108960yesdirect product factor
29C25×C3×C7C251200787yes—
30C23×C3×C4×C7C2513108960yesdirect product factor
31C29C2×C3×C711yescharacteristic
32C27×C4C2×C3×C71510yesdirect product factor
33C26×C7C22×C3×C41690880yesdirect product factor
34C26×C7C24×C3197155yes—
35C24×C4×C7C24×C313212592yesdirect product factor
36C27×C3C2×C4×C7132640yesdirect product factor
37C27×C3C23×C7110795yes—
38C25×C3×C4C23×C71777240yesdirect product factor
39C24×C3×C7C24×C413212592yesdirect product factor
40C24×C3×C7C26197155yes—
41C22×C3×C4×C7C261690880yesdirect product factor
42C28C3×C4×C71256yesdirect product factor
43C28C22×C3×C71255yes—
44C26×C4C22×C3×C7143180yesdirect product factor
45C25×C7C23×C3×C413108960yesdirect product factor
46C25×C7C25×C31200787yes—
47C23×C4×C7C25×C313108960yesdirect product factor
48C26×C3C22×C4×C71690880yesdirect product factor
49C26×C3C24×C7197155yes—
50C24×C3×C4C24×C713212592yesdirect product factor
51C23×C3×C7C25×C41777240yesdirect product factor
52C23×C3×C7C27110795yes—
53C2×C3×C4×C7C27132640yesdirect product factor
54C27C2×C3×C4×C7132640yesdirect product factor
55C27C23×C3×C7110795yes—
56C25×C4C23×C3×C71777240yesdirect product factor
57C24×C7C24×C3×C413212592yesdirect product factor
58C24×C7C26×C3197155yes—
59C22×C4×C7C26×C31690880yesdirect product factor
60C25×C3C23×C4×C713108960yesdirect product factor
61C25×C3C25×C71200787yes—
62C23×C3×C4C25×C713108960yesdirect product factor
63C22×C3×C7C26×C4143180yesdirect product factor
64C22×C3×C7C281255yes—
65C3×C4×C7C281256yesdirect product factor
66C26C22×C3×C4×C71690880yesdirect product factor
67C26C24×C3×C7197155yes—
68C24×C4C24×C3×C713212592yesdirect product factor
69C23×C7C25×C3×C41777240yesdirect product factor
70C23×C7C27×C3110795yes—
71C2×C4×C7C27×C3132640yesdirect product factor
72C24×C3C24×C4×C713212592yesdirect product factor
73C24×C3C26×C7197155yes—
74C22×C3×C4C26×C71690880yesdirect product factor
75C2×C3×C7C27×C41510yesdirect product factor
76C2×C3×C7C2911yescharacteristic
77C25C23×C3×C4×C713108960yesdirect product factor
78C25C25×C3×C71200787yes—
79C23×C4C25×C3×C713108960yesdirect product factor
80C22×C7C26×C3×C4143180yesdirect product factor
81C22×C7C28×C31255yes—
82C4×C7C28×C31256yesdirect product factor
83C23×C3C25×C4×C71777240yesdirect product factor
84C23×C3C27×C7110795yes—
85C2×C3×C4C27×C7132640yesdirect product factor
86C3×C7C28×C411yescharacteristic, direct product factor, Hall
87C24C24×C3×C4×C713212592yesdirect product factor
88C24C26×C3×C7197155yes—
89C22×C4C26×C3×C71690880yesdirect product factor
90C2×C7C29×C311yescharacteristic
91C2×C7C27×C3×C41510yesdirect product factor
92C22×C3C26×C4×C7143180yesdirect product factor
93C22×C3C28×C71255yes—
94C3×C4C28×C71256yesdirect product factor
95C23C27×C3×C7110795yes—
96C23C25×C3×C4×C71777240yesdirect product factor
97C2×C4C27×C3×C7132640yesdirect product factor
98C7C28×C3×C411yescharacteristic, direct product factor, Sylow
99C2×C3C29×C711yescharacteristic
100C2×C3C27×C4×C71510yesdirect product factor
101C22C26×C3×C4×C7143180yesdirect product factor
102C22C28×C3×C71255yes—
103C4C28×C3×C71256yesdirect product factor
104C3C28×C4×C711yescharacteristic, direct product factor, Sylow
105C2C29×C3×C711yescharacteristic
106C2C27×C3×C4×C71510yesdirect product factor
subgroupG/H
centerGC1
commutatorC1G
Frattini, cosocleC2C29×C3×C7
socle, prime radicalC29×C3×C7C2
LCSG ▷ C1
UCSC1 ◁ G

Complements

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

NH ≅ G/Ntype
1. C27×C3×C4×C7C2direct (N × H)
3. C28×C4×C7C3direct (N × H)
4. C28×C3×C7C4direct (N × H)
6. C26×C3×C4×C7C22direct (N × H)
7. C27×C4×C7C2×C3direct (N × H)
9. C28×C3×C4C7direct (N × H)
10. C25×C3×C4×C7C23direct (N × H)
11. C27×C3×C7C2×C4direct (N × H)
13. C27×C3×C4C2×C7direct (N × H)
15. C28×C4C3×C7direct (N × H)
16. C28×C7C3×C4direct (N × H)
18. C26×C4×C7C22×C3direct (N × H)
19. C26×C3×C7C22×C4direct (N × H)
21. C24×C3×C4×C7C24direct (N × H)
22. C27×C7C2×C3×C4direct (N × H)
24. C25×C4×C7C23×C3direct (N × H)
25. C28×C3C4×C7direct (N × H)
27. C26×C3×C4C22×C7direct (N × H)
28. C25×C3×C7C23×C4direct (N × H)
30. C23×C3×C4×C7C25direct (N × H)
32. C27×C4C2×C3×C7direct (N × H)
33. C26×C7C22×C3×C4direct (N × H)
35. C24×C4×C7C24×C3direct (N × H)
36. C27×C3C2×C4×C7direct (N × H)
38. C25×C3×C4C23×C7direct (N × H)
39. C24×C3×C7C24×C4direct (N × H)
41. C22×C3×C4×C7C26direct (N × H)
42. C28C3×C4×C7direct (N × H)
44. C26×C4C22×C3×C7direct (N × H)
45. C25×C7C23×C3×C4direct (N × H)
47. C23×C4×C7C25×C3direct (N × H)
48. C26×C3C22×C4×C7direct (N × H)
50. C24×C3×C4C24×C7direct (N × H)
51. C23×C3×C7C25×C4direct (N × H)
53. C2×C3×C4×C7C27direct (N × H)
54. C27C2×C3×C4×C7direct (N × H)
56. C25×C4C23×C3×C7direct (N × H)
57. C24×C7C24×C3×C4direct (N × H)
59. C22×C4×C7C26×C3direct (N × H)
60. C25×C3C23×C4×C7direct (N × H)
62. C23×C3×C4C25×C7direct (N × H)
63. C22×C3×C7C26×C4direct (N × H)
65. C3×C4×C7C28direct (N × H)
66. C26C22×C3×C4×C7direct (N × H)
68. C24×C4C24×C3×C7direct (N × H)
69. C23×C7C25×C3×C4direct (N × H)
71. C2×C4×C7C27×C3direct (N × H)
72. C24×C3C24×C4×C7direct (N × H)
74. C22×C3×C4C26×C7direct (N × H)
75. C2×C3×C7C27×C4direct (N × H)
77. C25C23×C3×C4×C7direct (N × H)
79. C23×C4C25×C3×C7direct (N × H)
80. C22×C7C26×C3×C4direct (N × H)
82. C4×C7C28×C3direct (N × H)
83. C23×C3C25×C4×C7direct (N × H)
85. C2×C3×C4C27×C7direct (N × H)
86. C3×C7C28×C4direct (N × H)
87. C24C24×C3×C4×C7direct (N × H)
89. C22×C4C26×C3×C7direct (N × H)
91. C2×C7C27×C3×C4direct (N × H)
92. C22×C3C26×C4×C7direct (N × H)
94. C3×C4C28×C7direct (N × H)
96. C23C25×C3×C4×C7direct (N × H)
97. C2×C4C27×C3×C7direct (N × H)
98. C7C28×C3×C4direct (N × H)
100. C2×C3C27×C4×C7direct (N × H)
101. C22C26×C3×C4×C7direct (N × H)
103. C4C28×C3×C7direct (N × H)
104. C3C28×C4×C7direct (N × H)
106. C2C27×C3×C4×C7direct (N × H)

Maximal subgroups

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

subgroupindex
C27×C3×C4×C72
C29×C3×C72
C28×C4×C73
C28×C3×C47