Finite Group Databasebeta

← C24×C4×C72×C9 #72205243

Subgroups — C24×C4×C72×C9 #72205243

Order 28224.

20430 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
1C25×C72×C9C211yesmaximal, characteristic
2C23×C4×C72×C9C2130yesmaximal, direct product factor
3C24×C3×C4×C72C311yesmaximal, characteristic
4C24×C72×C9C4116yesdirect product factor
5C24×C72×C9C22115yes—
6C22×C4×C72×C9C221140yesdirect product factor
7C25×C3×C72C2×C311yescharacteristic
8C23×C3×C4×C72C2×C3130yes—
9C24×C4×C7×C9C718yesmaximal, direct product factor
10C23×C72×C9C2×C41120yesdirect product factor
11C23×C72×C9C23135yes—
12C2×C4×C72×C9C231120yesdirect product factor
13C24×C4×C72C911yescharacteristic, direct product factor, Hall
14C24×C3×C72C3×C4116yes—
15C24×C3×C72C22×C3115yes—
16C22×C3×C4×C72C22×C31140yes—
17C25×C7×C9C2×C718yes—
18C23×C4×C7×C9C2×C71240yesdirect product factor
19C22×C72×C9C22×C41140yesdirect product factor
20C22×C72×C9C24115yes—
21C4×C72×C9C24116yesdirect product factor
22C25×C72C2×C911yescharacteristic
23C23×C4×C72C2×C9130yesdirect product factor
24C24×C3×C4×C7C3×C718yes—
25C23×C3×C72C2×C3×C41120yes—
26C23×C3×C72C23×C3135yes—
27C2×C3×C4×C72C23×C31120yes—
28C24×C7×C9C4×C71128yesdirect product factor
29C24×C7×C9C22×C71120yes—
30C22×C4×C7×C9C22×C711120yesdirect product factor
31C2×C72×C9C23×C4130yesdirect product factor
32C2×C72×C9C2511yescharacteristic
33C24×C72C4×C9116yesdirect product factor
34C24×C72C22×C9115yes—
35C22×C4×C72C22×C91140yesdirect product factor
36C25×C3×C7C2×C3×C718yes—
37C23×C3×C4×C7C2×C3×C71240yes—
38C22×C3×C72C22×C3×C41140yes—
39C22×C3×C72C24×C3115yes—
40C3×C4×C72C24×C3116yes—
41C24×C4×C9C7211yescharacteristic, direct product factor, Hall
42C23×C7×C9C2×C4×C71960yesdirect product factor
43C23×C7×C9C23×C71280yes—
44C2×C4×C7×C9C23×C71960yesdirect product factor
45C24×C4×C7C7×C918yesdirect product factor
46C72×C9C24×C411yescharacteristic, direct product factor, Hall
47C23×C72C2×C4×C91120yesdirect product factor
48C23×C72C23×C9135yes—
49C2×C4×C72C23×C91120yesdirect product factor
50C24×C3×C7C3×C4×C71128yes—
51C24×C3×C7C22×C3×C71120yes—
52C22×C3×C4×C7C22×C3×C711120yes—
53C2×C3×C72C23×C3×C4130yes—
54C2×C3×C72C25×C311yescharacteristic
55C25×C9C2×C7211yescharacteristic
56C23×C4×C9C2×C72130yesdirect product factor
57C22×C7×C9C22×C4×C711120yesdirect product factor
58C22×C7×C9C24×C71120yes—
59C4×C7×C9C24×C71128yesdirect product factor
60C25×C7C2×C7×C918yes—
61C23×C4×C7C2×C7×C91240yesdirect product factor
62C22×C72C22×C4×C91140yesdirect product factor
63C22×C72C24×C9115yes—
64C4×C72C24×C9116yesdirect product factor
65C24×C3×C4C3×C7211yescharacteristic
66C23×C3×C7C2×C3×C4×C71960yes—
67C23×C3×C7C23×C3×C71280yes—
68C2×C3×C4×C7C23×C3×C71960yes—
69C3×C72C24×C3×C411yescharacteristic
70C24×C9C4×C72116yesdirect product factor
71C24×C9C22×C72115yes—
72C22×C4×C9C22×C721140yesdirect product factor
73C2×C7×C9C23×C4×C71240yesdirect product factor
74C2×C7×C9C25×C718yes—
75C24×C7C4×C7×C91128yesdirect product factor
76C24×C7C22×C7×C91120yes—
77C22×C4×C7C22×C7×C911120yesdirect product factor
78C2×C72C23×C4×C9130yesdirect product factor
79C2×C72C25×C911yescharacteristic
80C25×C3C2×C3×C7211yescharacteristic
81C23×C3×C4C2×C3×C72130yes—
82C22×C3×C7C22×C3×C4×C711120yes—
83C22×C3×C7C24×C3×C71120yes—
84C3×C4×C7C24×C3×C71128yes—
85C23×C9C2×C4×C721120yesdirect product factor
86C23×C9C23×C72135yes—
87C2×C4×C9C23×C721120yesdirect product factor
88C24×C4C72×C911yescharacteristic, direct product factor, Sylow
89C7×C9C24×C4×C718yesdirect product factor
90C23×C7C2×C4×C7×C91960yesdirect product factor
91C23×C7C23×C7×C91280yes—
92C2×C4×C7C23×C7×C91960yesdirect product factor
93C72C24×C4×C911yescharacteristic, direct product factor, Sylow
94C24×C3C3×C4×C72116yes—
95C24×C3C22×C3×C72115yes—
96C22×C3×C4C22×C3×C721140yes—
97C2×C3×C7C23×C3×C4×C71240yes—
98C2×C3×C7C25×C3×C718yes—
99C22×C9C22×C4×C721140yesdirect product factor
100C22×C9C24×C72115yes—
101C4×C9C24×C72116yesdirect product factor
102C25C2×C72×C911yescharacteristic
103C23×C4C2×C72×C9130yesdirect product factor
104C22×C7C22×C4×C7×C911120yesdirect product factor
105C22×C7C24×C7×C91120yes—
106C4×C7C24×C7×C91128yesdirect product factor
107C23×C3C2×C3×C4×C721120yes—
108C23×C3C23×C3×C72135yes—
109C2×C3×C4C23×C3×C721120yes—
110C3×C7C24×C3×C4×C718yes—
111C2×C9C23×C4×C72130yesdirect product factor
112C2×C9C25×C7211yescharacteristic
113C24C4×C72×C9116yesdirect product factor
114C24C22×C72×C9115yes—
115C22×C4C22×C72×C91140yesdirect product factor
116C2×C7C23×C4×C7×C91240yesdirect product factor
117C2×C7C25×C7×C918yes—
118C22×C3C22×C3×C4×C721140yes—
119C22×C3C24×C3×C72115yes—
120C3×C4C24×C3×C72116yes—
121C9C24×C4×C7211yescharacteristic, direct product factor, Sylow
122C23C2×C4×C72×C91120yesdirect product factor
123C23C23×C72×C9135yes—
124C2×C4C23×C72×C91120yesdirect product factor
125C7C24×C4×C7×C918yesdirect product factor
126C2×C3C23×C3×C4×C72130yes—
127C2×C3C25×C3×C7211yescharacteristic
128C22C22×C4×C72×C91140yesdirect product factor
129C22C24×C72×C9115yes—
130C4C24×C72×C9116yesdirect product factor
131C3C24×C3×C4×C7211yescharacteristic
132C2C23×C4×C72×C9130yesdirect product factor
133C2C25×C72×C911yescharacteristic
subgroupG/H
centerGC1
commutatorC1G
Frattini, cosocleC2×C3C25×C3×C72
socle, prime radicalC25×C3×C72C2×C3
LCSG ▷ C1
UCSC1 ◁ G

Complements

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

NH ≅ G/Ntype
2. C23×C4×C72×C9C2direct (N × H)
4. C24×C72×C9C4direct (N × H)
6. C22×C4×C72×C9C22direct (N × H)
9. C24×C4×C7×C9C7direct (N × H)
10. C23×C72×C9C2×C4direct (N × H)
12. C2×C4×C72×C9C23direct (N × H)
13. C24×C4×C72C9direct (N × H)
18. C23×C4×C7×C9C2×C7direct (N × H)
19. C22×C72×C9C22×C4direct (N × H)
21. C4×C72×C9C24direct (N × H)
23. C23×C4×C72C2×C9direct (N × H)
28. C24×C7×C9C4×C7direct (N × H)
30. C22×C4×C7×C9C22×C7direct (N × H)
31. C2×C72×C9C23×C4direct (N × H)
33. C24×C72C4×C9direct (N × H)
35. C22×C4×C72C22×C9direct (N × H)
41. C24×C4×C9C72direct (N × H)
42. C23×C7×C9C2×C4×C7direct (N × H)
44. C2×C4×C7×C9C23×C7direct (N × H)
45. C24×C4×C7C7×C9direct (N × H)
46. C72×C9C24×C4direct (N × H)
47. C23×C72C2×C4×C9direct (N × H)
49. C2×C4×C72C23×C9direct (N × H)
56. C23×C4×C9C2×C72direct (N × H)
57. C22×C7×C9C22×C4×C7direct (N × H)
59. C4×C7×C9C24×C7direct (N × H)
61. C23×C4×C7C2×C7×C9direct (N × H)
62. C22×C72C22×C4×C9direct (N × H)
64. C4×C72C24×C9direct (N × H)
70. C24×C9C4×C72direct (N × H)
72. C22×C4×C9C22×C72direct (N × H)
73. C2×C7×C9C23×C4×C7direct (N × H)
75. C24×C7C4×C7×C9direct (N × H)
77. C22×C4×C7C22×C7×C9direct (N × H)
78. C2×C72C23×C4×C9direct (N × H)
85. C23×C9C2×C4×C72direct (N × H)
87. C2×C4×C9C23×C72direct (N × H)
88. C24×C4C72×C9direct (N × H)
89. C7×C9C24×C4×C7direct (N × H)
90. C23×C7C2×C4×C7×C9direct (N × H)
92. C2×C4×C7C23×C7×C9direct (N × H)
93. C72C24×C4×C9direct (N × H)
99. C22×C9C22×C4×C72direct (N × H)
101. C4×C9C24×C72direct (N × H)
103. C23×C4C2×C72×C9direct (N × H)
104. C22×C7C22×C4×C7×C9direct (N × H)
106. C4×C7C24×C7×C9direct (N × H)
111. C2×C9C23×C4×C72direct (N × H)
113. C24C4×C72×C9direct (N × H)
115. C22×C4C22×C72×C9direct (N × H)
116. C2×C7C23×C4×C7×C9direct (N × H)
121. C9C24×C4×C72direct (N × H)
122. C23C2×C4×C72×C9direct (N × H)
124. C2×C4C23×C72×C9direct (N × H)
125. C7C24×C4×C7×C9direct (N × H)
128. C22C22×C4×C72×C9direct (N × H)
130. C4C24×C72×C9direct (N × H)
132. C2C23×C4×C72×C9direct (N × H)

Maximal subgroups

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

subgroupindex
C25×C72×C92
C23×C4×C72×C92
C24×C3×C4×C723
C24×C4×C7×C97