Finite Group Databasebeta

← C24×C33×C7×C9 #72204309

Subgroups — C24×C33×C7×C9 #72204309

Order 27216.

53064 conjugacy classes of subgroups.

The subgroup lattice has 120 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C33×C7×C9C2115yesmaximal, direct product factor
2C24×C34×C7C311yesmaximal, characteristic
3C24×C32×C7×C9C3139yesmaximal, direct product factor
4C22×C33×C7×C9C22135yesdirect product factor
5C23×C32×C7×C9C2×C31585yesdirect product factor
6C23×C34×C7C2×C3115yes—
7C24×C33×C9C711yesmaximal, characteristic, direct product factor, Hall
8C2×C33×C7×C9C23115yesdirect product factor
9C24×C3×C7×C9C321117yesdirect product factor
10C24×C33×C7C9127yesdirect product factor
11C24×C33×C7C32113yes—
12C22×C34×C7C22×C3135yes—
13C22×C32×C7×C9C22×C311365yesdirect product factor
14C23×C33×C9C2×C7115yesdirect product factor
15C33×C7×C9C2411yescharacteristic, direct product factor, Hall
16C23×C33×C7C2×C91405yesdirect product factor
17C23×C33×C7C2×C321195yes—
18C23×C3×C7×C9C2×C3211755yesdirect product factor
19C24×C34C3×C711yescharacteristic
20C24×C32×C9C3×C7139yesdirect product factor
21C2×C34×C7C23×C3115yes—
22C2×C32×C7×C9C23×C31585yesdirect product factor
23C24×C32×C7C3×C91117yesdirect product factor
24C24×C32×C7C33113yes—
25C24×C7×C9C33127yesdirect product factor
26C22×C33×C9C22×C7135yesdirect product factor
27C22×C33×C7C22×C91945yesdirect product factor
28C22×C33×C7C22×C321455yes—
29C22×C3×C7×C9C22×C3214095yesdirect product factor
30C23×C34C2×C3×C7115yes—
31C23×C32×C9C2×C3×C71585yesdirect product factor
32C34×C7C24×C311yescharacteristic
33C32×C7×C9C24×C3139yesdirect product factor
34C23×C32×C7C2×C3×C911755yesdirect product factor
35C23×C32×C7C2×C331195yes—
36C23×C7×C9C2×C331405yesdirect product factor
37C2×C33×C9C23×C7115yesdirect product factor
38C24×C33C7×C9127yesdirect product factor
39C24×C33C32×C7113yes—
40C24×C3×C9C32×C71117yesdirect product factor
41C2×C33×C7C23×C91405yesdirect product factor
42C2×C33×C7C23×C321195yes—
43C2×C3×C7×C9C23×C3211755yesdirect product factor
44C24×C3×C7C32×C9139yesdirect product factor
45C24×C3×C7C3411yescharacteristic
46C22×C34C22×C3×C7135yes—
47C22×C32×C9C22×C3×C711365yesdirect product factor
48C22×C32×C7C22×C3×C914095yesdirect product factor
49C22×C32×C7C22×C331455yes—
50C22×C7×C9C22×C331945yesdirect product factor
51C33×C9C24×C711yescharacteristic, direct product factor, Sylow
52C23×C33C2×C7×C91405yesdirect product factor
53C23×C33C2×C32×C71195yes—
54C23×C3×C9C2×C32×C711755yesdirect product factor
55C33×C7C24×C9127yesdirect product factor
56C33×C7C24×C32113yes—
57C3×C7×C9C24×C321117yesdirect product factor
58C23×C3×C7C2×C32×C91585yesdirect product factor
59C23×C3×C7C2×C34115yes—
60C2×C34C23×C3×C7115yes—
61C2×C32×C9C23×C3×C71585yesdirect product factor
62C24×C32C3×C7×C91117yesdirect product factor
63C24×C32C33×C7113yes—
64C24×C9C33×C7127yesdirect product factor
65C2×C32×C7C23×C3×C911755yesdirect product factor
66C2×C32×C7C23×C331195yes—
67C2×C7×C9C23×C331405yesdirect product factor
68C24×C7C33×C911yescharacteristic, direct product factor, Hall
69C22×C33C22×C7×C91945yesdirect product factor
70C22×C33C22×C32×C71455yes—
71C22×C3×C9C22×C32×C714095yesdirect product factor
72C22×C3×C7C22×C32×C911365yesdirect product factor
73C22×C3×C7C22×C34135yes—
74C34C24×C3×C711yescharacteristic
75C32×C9C24×C3×C7139yesdirect product factor
76C23×C32C2×C3×C7×C911755yesdirect product factor
77C23×C32C2×C33×C71195yes—
78C23×C9C2×C33×C71405yesdirect product factor
79C32×C7C24×C3×C91117yesdirect product factor
80C32×C7C24×C33113yes—
81C7×C9C24×C33127yesdirect product factor
82C23×C7C2×C33×C9115yesdirect product factor
83C2×C33C23×C7×C91405yesdirect product factor
84C2×C33C23×C32×C71195yes—
85C2×C3×C9C23×C32×C711755yesdirect product factor
86C24×C3C32×C7×C9139yesdirect product factor
87C24×C3C34×C711yescharacteristic
88C2×C3×C7C23×C32×C91585yesdirect product factor
89C2×C3×C7C23×C34115yes—
90C22×C32C22×C3×C7×C914095yesdirect product factor
91C22×C32C22×C33×C71455yes—
92C22×C9C22×C33×C71945yesdirect product factor
93C22×C7C22×C33×C9135yesdirect product factor
94C33C24×C7×C9127yesdirect product factor
95C33C24×C32×C7113yes—
96C3×C9C24×C32×C71117yesdirect product factor
97C23×C3C2×C32×C7×C91585yesdirect product factor
98C23×C3C2×C34×C7115yes—
99C3×C7C24×C32×C9139yesdirect product factor
100C3×C7C24×C3411yescharacteristic
101C2×C32C23×C3×C7×C911755yesdirect product factor
102C2×C32C23×C33×C71195yes—
103C2×C9C23×C33×C71405yesdirect product factor
104C24C33×C7×C911yescharacteristic, direct product factor, Sylow
105C2×C7C23×C33×C9115yesdirect product factor
106C22×C3C22×C32×C7×C911365yesdirect product factor
107C22×C3C22×C34×C7135yes—
108C32C24×C33×C7113yes—
109C32C24×C3×C7×C91117yesdirect product factor
110C9C24×C33×C7127yesdirect product factor
111C23C2×C33×C7×C9115yesdirect product factor
112C7C24×C33×C911yescharacteristic, direct product factor, Sylow
113C2×C3C23×C34×C7115yes—
114C2×C3C23×C32×C7×C91585yesdirect product factor
115C22C22×C33×C7×C9135yesdirect product factor
116C3C24×C32×C7×C9139yesdirect product factor
117C3C24×C34×C711yescharacteristic
118C2C23×C33×C7×C9115yesdirect product factor
subgroupG/H
centerGC1
commutatorC1G
Frattini, cosocleC3C24×C34×C7
socle, prime radicalC24×C34×C7C3
LCSG ▷ C1
UCSC1 ◁ G

Complements

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

NH ≅ G/Ntype
1. C23×C33×C7×C9C2direct (N × H)
3. C24×C32×C7×C9C3direct (N × H)
4. C22×C33×C7×C9C22direct (N × H)
5. C23×C32×C7×C9C2×C3direct (N × H)
7. C24×C33×C9C7direct (N × H)
8. C2×C33×C7×C9C23direct (N × H)
9. C24×C3×C7×C9C32direct (N × H)
10. C24×C33×C7C9direct (N × H)
13. C22×C32×C7×C9C22×C3direct (N × H)
14. C23×C33×C9C2×C7direct (N × H)
15. C33×C7×C9C24direct (N × H)
16. C23×C33×C7C2×C9direct (N × H)
18. C23×C3×C7×C9C2×C32direct (N × H)
20. C24×C32×C9C3×C7direct (N × H)
22. C2×C32×C7×C9C23×C3direct (N × H)
23. C24×C32×C7C3×C9direct (N × H)
25. C24×C7×C9C33direct (N × H)
26. C22×C33×C9C22×C7direct (N × H)
27. C22×C33×C7C22×C9direct (N × H)
29. C22×C3×C7×C9C22×C32direct (N × H)
31. C23×C32×C9C2×C3×C7direct (N × H)
33. C32×C7×C9C24×C3direct (N × H)
34. C23×C32×C7C2×C3×C9direct (N × H)
36. C23×C7×C9C2×C33direct (N × H)
37. C2×C33×C9C23×C7direct (N × H)
38. C24×C33C7×C9direct (N × H)
40. C24×C3×C9C32×C7direct (N × H)
41. C2×C33×C7C23×C9direct (N × H)
43. C2×C3×C7×C9C23×C32direct (N × H)
44. C24×C3×C7C32×C9direct (N × H)
47. C22×C32×C9C22×C3×C7direct (N × H)
48. C22×C32×C7C22×C3×C9direct (N × H)
50. C22×C7×C9C22×C33direct (N × H)
51. C33×C9C24×C7direct (N × H)
52. C23×C33C2×C7×C9direct (N × H)
54. C23×C3×C9C2×C32×C7direct (N × H)
55. C33×C7C24×C9direct (N × H)
57. C3×C7×C9C24×C32direct (N × H)
58. C23×C3×C7C2×C32×C9direct (N × H)
61. C2×C32×C9C23×C3×C7direct (N × H)
62. C24×C32C3×C7×C9direct (N × H)
64. C24×C9C33×C7direct (N × H)
65. C2×C32×C7C23×C3×C9direct (N × H)
67. C2×C7×C9C23×C33direct (N × H)
68. C24×C7C33×C9direct (N × H)
69. C22×C33C22×C7×C9direct (N × H)
71. C22×C3×C9C22×C32×C7direct (N × H)
72. C22×C3×C7C22×C32×C9direct (N × H)
75. C32×C9C24×C3×C7direct (N × H)
76. C23×C32C2×C3×C7×C9direct (N × H)
78. C23×C9C2×C33×C7direct (N × H)
79. C32×C7C24×C3×C9direct (N × H)
81. C7×C9C24×C33direct (N × H)
82. C23×C7C2×C33×C9direct (N × H)
83. C2×C33C23×C7×C9direct (N × H)
85. C2×C3×C9C23×C32×C7direct (N × H)
86. C24×C3C32×C7×C9direct (N × H)
88. C2×C3×C7C23×C32×C9direct (N × H)
90. C22×C32C22×C3×C7×C9direct (N × H)
92. C22×C9C22×C33×C7direct (N × H)
93. C22×C7C22×C33×C9direct (N × H)
94. C33C24×C7×C9direct (N × H)
96. C3×C9C24×C32×C7direct (N × H)
97. C23×C3C2×C32×C7×C9direct (N × H)
99. C3×C7C24×C32×C9direct (N × H)
101. C2×C32C23×C3×C7×C9direct (N × H)
103. C2×C9C23×C33×C7direct (N × H)
104. C24C33×C7×C9direct (N × H)
105. C2×C7C23×C33×C9direct (N × H)
106. C22×C3C22×C32×C7×C9direct (N × H)
109. C32C24×C3×C7×C9direct (N × H)
110. C9C24×C33×C7direct (N × H)
111. C23C2×C33×C7×C9direct (N × H)
112. C7C24×C33×C9direct (N × H)
114. C2×C3C23×C32×C7×C9direct (N × H)
115. C22C22×C33×C7×C9direct (N × H)
116. C3C24×C32×C7×C9direct (N × H)
118. C2C23×C33×C7×C9direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C33×C7×C92
C24×C34×C73
C24×C32×C7×C93
C24×C33×C97