Finite Group Databasebeta

← C24×C32×C52×C9 #72209154

Subgroups — C24×C32×C52×C9 #72209154

Order 32400.

26800 conjugacy classes of subgroups.

The subgroup lattice has 135 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C32×C52×C9C2115yesmaximal, direct product factor
2C24×C33×C52C311yesmaximal, characteristic
3C24×C3×C52×C9C3112yesmaximal, direct product factor
4C22×C32×C52×C9C22135yesdirect product factor
5C24×C32×C5×C9C516yesmaximal, direct product factor
6C23×C33×C52C2×C3115yes—
7C23×C3×C52×C9C2×C31180yesdirect product factor
8C2×C32×C52×C9C23115yesdirect product factor
9C24×C32×C52C919yesdirect product factor
10C24×C32×C52C3214yes—
11C24×C52×C9C3219yesdirect product factor
12C23×C32×C5×C9C2×C5190yesdirect product factor
13C22×C33×C52C22×C3135yes—
14C22×C3×C52×C9C22×C31420yesdirect product factor
15C24×C33×C5C3×C516yes—
16C24×C3×C5×C9C3×C5172yesdirect product factor
17C32×C52×C9C2411yescharacteristic, direct product factor, Hall
18C23×C32×C52C2×C91135yesdirect product factor
19C23×C32×C52C2×C32160yes—
20C23×C52×C9C2×C321135yesdirect product factor
21C22×C32×C5×C9C22×C51210yesdirect product factor
22C2×C33×C52C23×C3115yes—
23C2×C3×C52×C9C23×C31180yesdirect product factor
24C24×C32×C9C5211yescharacteristic, direct product factor, Hall
25C24×C3×C52C3×C9112yesdirect product factor
26C24×C3×C52C3311yescharacteristic
27C23×C33×C5C2×C3×C5190yes—
28C23×C3×C5×C9C2×C3×C511080yesdirect product factor
29C22×C32×C52C22×C91315yesdirect product factor
30C22×C32×C52C22×C321140yes—
31C22×C52×C9C22×C321315yesdirect product factor
32C2×C32×C5×C9C23×C5190yesdirect product factor
33C24×C32×C5C5×C9154yesdirect product factor
34C24×C32×C5C32×C5124yes—
35C24×C5×C9C32×C5154yesdirect product factor
36C33×C52C24×C311yescharacteristic
37C3×C52×C9C24×C3112yesdirect product factor
38C23×C32×C9C2×C52115yesdirect product factor
39C23×C3×C52C2×C3×C91180yesdirect product factor
40C23×C3×C52C2×C33115yes—
41C22×C33×C5C22×C3×C51210yes—
42C22×C3×C5×C9C22×C3×C512520yesdirect product factor
43C2×C32×C52C23×C91135yesdirect product factor
44C2×C32×C52C23×C32160yes—
45C2×C52×C9C23×C321135yesdirect product factor
46C24×C33C3×C5211yescharacteristic
47C24×C3×C9C3×C52112yesdirect product factor
48C32×C5×C9C24×C516yesdirect product factor
49C24×C52C32×C911yescharacteristic, direct product factor, Hall
50C23×C32×C5C2×C5×C91810yesdirect product factor
51C23×C32×C5C2×C32×C51360yes—
52C23×C5×C9C2×C32×C51810yesdirect product factor
53C22×C32×C9C22×C52135yesdirect product factor
54C22×C3×C52C22×C3×C91420yesdirect product factor
55C22×C3×C52C22×C33135yes—
56C2×C33×C5C23×C3×C5190yes—
57C2×C3×C5×C9C23×C3×C511080yesdirect product factor
58C24×C3×C5C3×C5×C9172yesdirect product factor
59C24×C3×C5C33×C516yes—
60C32×C52C24×C919yesdirect product factor
61C32×C52C24×C3214yes—
62C52×C9C24×C3219yesdirect product factor
63C23×C33C2×C3×C52115yes—
64C23×C3×C9C2×C3×C521180yesdirect product factor
65C23×C52C2×C32×C9115yesdirect product factor
66C22×C32×C5C22×C5×C911890yesdirect product factor
67C22×C32×C5C22×C32×C51840yes—
68C22×C5×C9C22×C32×C511890yesdirect product factor
69C2×C32×C9C23×C52115yesdirect product factor
70C2×C3×C52C23×C3×C91180yesdirect product factor
71C2×C3×C52C23×C33115yes—
72C24×C32C52×C919yesdirect product factor
73C24×C32C32×C5214yes—
74C24×C9C32×C5219yesdirect product factor
75C33×C5C24×C3×C516yes—
76C3×C5×C9C24×C3×C5172yesdirect product factor
77C23×C3×C5C2×C3×C5×C911080yesdirect product factor
78C23×C3×C5C2×C33×C5190yes—
79C22×C33C22×C3×C52135yes—
80C22×C3×C9C22×C3×C521420yesdirect product factor
81C22×C52C22×C32×C9135yesdirect product factor
82C2×C32×C5C23×C5×C91810yesdirect product factor
83C2×C32×C5C23×C32×C51360yes—
84C2×C5×C9C23×C32×C51810yesdirect product factor
85C32×C9C24×C5211yescharacteristic, direct product factor, Sylow
86C24×C5C32×C5×C916yesdirect product factor
87C3×C52C24×C3×C9112yesdirect product factor
88C3×C52C24×C3311yescharacteristic
89C23×C32C2×C52×C91135yesdirect product factor
90C23×C32C2×C32×C52160yes—
91C23×C9C2×C32×C521135yesdirect product factor
92C22×C3×C5C22×C3×C5×C912520yesdirect product factor
93C22×C3×C5C22×C33×C51210yes—
94C2×C33C23×C3×C52115yes—
95C2×C3×C9C23×C3×C521180yesdirect product factor
96C2×C52C23×C32×C9115yesdirect product factor
97C24×C3C3×C52×C9112yesdirect product factor
98C24×C3C33×C5211yescharacteristic
99C32×C5C24×C5×C9154yesdirect product factor
100C32×C5C24×C32×C5124yes—
101C5×C9C24×C32×C5154yesdirect product factor
102C23×C5C2×C32×C5×C9190yesdirect product factor
103C22×C32C22×C52×C91315yesdirect product factor
104C22×C32C22×C32×C521140yes—
105C22×C9C22×C32×C521315yesdirect product factor
106C2×C3×C5C23×C3×C5×C911080yesdirect product factor
107C2×C3×C5C23×C33×C5190yes—
108C33C24×C3×C5211yescharacteristic
109C3×C9C24×C3×C52112yesdirect product factor
110C52C24×C32×C911yescharacteristic, direct product factor, Sylow
111C23×C3C2×C3×C52×C91180yesdirect product factor
112C23×C3C2×C33×C52115yes—
113C22×C5C22×C32×C5×C91210yesdirect product factor
114C2×C32C23×C52×C91135yesdirect product factor
115C2×C32C23×C32×C52160yes—
116C2×C9C23×C32×C521135yesdirect product factor
117C24C32×C52×C911yescharacteristic, direct product factor, Sylow
118C3×C5C24×C3×C5×C9172yesdirect product factor
119C3×C5C24×C33×C516yes—
120C22×C3C22×C3×C52×C91420yesdirect product factor
121C22×C3C22×C33×C52135yes—
122C2×C5C23×C32×C5×C9190yesdirect product factor
123C32C24×C52×C919yesdirect product factor
124C32C24×C32×C5214yes—
125C9C24×C32×C5219yesdirect product factor
126C23C2×C32×C52×C9115yesdirect product factor
127C2×C3C23×C3×C52×C91180yesdirect product factor
128C2×C3C23×C33×C52115yes—
129C5C24×C32×C5×C916yesdirect product factor
130C22C22×C32×C52×C9135yesdirect product factor
131C3C24×C3×C52×C9112yesdirect product factor
132C3C24×C33×C5211yescharacteristic
133C2C23×C32×C52×C9115yesdirect product factor
subgroupG/H
centerGC1
commutatorC1G
Frattini, cosocleC3C24×C33×C52
socle, prime radicalC24×C33×C52C3
LCSG ▷ C1
UCSC1 ◁ G

Complements

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

NH ≅ G/Ntype
1. C23×C32×C52×C9C2direct (N × H)
3. C24×C3×C52×C9C3direct (N × H)
4. C22×C32×C52×C9C22direct (N × H)
5. C24×C32×C5×C9C5direct (N × H)
7. C23×C3×C52×C9C2×C3direct (N × H)
8. C2×C32×C52×C9C23direct (N × H)
9. C24×C32×C52C9direct (N × H)
11. C24×C52×C9C32direct (N × H)
12. C23×C32×C5×C9C2×C5direct (N × H)
14. C22×C3×C52×C9C22×C3direct (N × H)
16. C24×C3×C5×C9C3×C5direct (N × H)
17. C32×C52×C9C24direct (N × H)
18. C23×C32×C52C2×C9direct (N × H)
20. C23×C52×C9C2×C32direct (N × H)
21. C22×C32×C5×C9C22×C5direct (N × H)
23. C2×C3×C52×C9C23×C3direct (N × H)
24. C24×C32×C9C52direct (N × H)
25. C24×C3×C52C3×C9direct (N × H)
28. C23×C3×C5×C9C2×C3×C5direct (N × H)
29. C22×C32×C52C22×C9direct (N × H)
31. C22×C52×C9C22×C32direct (N × H)
32. C2×C32×C5×C9C23×C5direct (N × H)
33. C24×C32×C5C5×C9direct (N × H)
35. C24×C5×C9C32×C5direct (N × H)
37. C3×C52×C9C24×C3direct (N × H)
38. C23×C32×C9C2×C52direct (N × H)
39. C23×C3×C52C2×C3×C9direct (N × H)
42. C22×C3×C5×C9C22×C3×C5direct (N × H)
43. C2×C32×C52C23×C9direct (N × H)
45. C2×C52×C9C23×C32direct (N × H)
47. C24×C3×C9C3×C52direct (N × H)
48. C32×C5×C9C24×C5direct (N × H)
49. C24×C52C32×C9direct (N × H)
50. C23×C32×C5C2×C5×C9direct (N × H)
52. C23×C5×C9C2×C32×C5direct (N × H)
53. C22×C32×C9C22×C52direct (N × H)
54. C22×C3×C52C22×C3×C9direct (N × H)
57. C2×C3×C5×C9C23×C3×C5direct (N × H)
58. C24×C3×C5C3×C5×C9direct (N × H)
60. C32×C52C24×C9direct (N × H)
62. C52×C9C24×C32direct (N × H)
64. C23×C3×C9C2×C3×C52direct (N × H)
65. C23×C52C2×C32×C9direct (N × H)
66. C22×C32×C5C22×C5×C9direct (N × H)
68. C22×C5×C9C22×C32×C5direct (N × H)
69. C2×C32×C9C23×C52direct (N × H)
70. C2×C3×C52C23×C3×C9direct (N × H)
72. C24×C32C52×C9direct (N × H)
74. C24×C9C32×C52direct (N × H)
76. C3×C5×C9C24×C3×C5direct (N × H)
77. C23×C3×C5C2×C3×C5×C9direct (N × H)
80. C22×C3×C9C22×C3×C52direct (N × H)
81. C22×C52C22×C32×C9direct (N × H)
82. C2×C32×C5C23×C5×C9direct (N × H)
84. C2×C5×C9C23×C32×C5direct (N × H)
85. C32×C9C24×C52direct (N × H)
86. C24×C5C32×C5×C9direct (N × H)
87. C3×C52C24×C3×C9direct (N × H)
89. C23×C32C2×C52×C9direct (N × H)
91. C23×C9C2×C32×C52direct (N × H)
92. C22×C3×C5C22×C3×C5×C9direct (N × H)
95. C2×C3×C9C23×C3×C52direct (N × H)
96. C2×C52C23×C32×C9direct (N × H)
97. C24×C3C3×C52×C9direct (N × H)
99. C32×C5C24×C5×C9direct (N × H)
101. C5×C9C24×C32×C5direct (N × H)
102. C23×C5C2×C32×C5×C9direct (N × H)
103. C22×C32C22×C52×C9direct (N × H)
105. C22×C9C22×C32×C52direct (N × H)
106. C2×C3×C5C23×C3×C5×C9direct (N × H)
109. C3×C9C24×C3×C52direct (N × H)
110. C52C24×C32×C9direct (N × H)
111. C23×C3C2×C3×C52×C9direct (N × H)
113. C22×C5C22×C32×C5×C9direct (N × H)
114. C2×C32C23×C52×C9direct (N × H)
116. C2×C9C23×C32×C52direct (N × H)
117. C24C32×C52×C9direct (N × H)
118. C3×C5C24×C3×C5×C9direct (N × H)
120. C22×C3C22×C3×C52×C9direct (N × H)
122. C2×C5C23×C32×C5×C9direct (N × H)
123. C32C24×C52×C9direct (N × H)
125. C9C24×C32×C52direct (N × H)
126. C23C2×C32×C52×C9direct (N × H)
127. C2×C3C23×C3×C52×C9direct (N × H)
129. C5C24×C32×C5×C9direct (N × H)
130. C22C22×C32×C52×C9direct (N × H)
131. C3C24×C3×C52×C9direct (N × H)
133. C2C23×C32×C52×C9direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C32×C52×C92
C24×C33×C523
C24×C3×C52×C93
C24×C32×C5×C95