Finite Group Databasebeta

← C24×C3×C4×C7×C23 #72207671

Subgroups — C24×C3×C4×C7×C23 #72207671

Order 30912.

5448 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
1C25×C3×C7×C23C211yesmaximal, characteristic
2C23×C3×C4×C7×C23C2130yesmaximal, direct product factor
3C24×C4×C7×C23C311yesmaximal, characteristic, direct product factor, Hall
4C24×C3×C7×C23C4116yesdirect product factor
5C24×C3×C7×C23C22115yes—
6C22×C3×C4×C7×C23C221140yesdirect product factor
7C23×C4×C7×C23C2×C3130yesdirect product factor
8C25×C7×C23C2×C311yescharacteristic
9C24×C3×C4×C23C711yesmaximal, characteristic, direct product factor, Hall
10C23×C3×C7×C23C2×C41120yesdirect product factor
11C23×C3×C7×C23C23135yes—
12C2×C3×C4×C7×C23C231120yesdirect product factor
13C24×C7×C23C3×C4116yesdirect product factor
14C24×C7×C23C22×C3115yes—
15C22×C4×C7×C23C22×C31140yesdirect product factor
16C25×C3×C23C2×C711yescharacteristic
17C23×C3×C4×C23C2×C7130yesdirect product factor
18C22×C3×C7×C23C22×C41140yesdirect product factor
19C22×C3×C7×C23C24115yes—
20C3×C4×C7×C23C24116yesdirect product factor
21C24×C4×C23C3×C711yescharacteristic, direct product factor, Hall
22C24×C3×C4×C7C2311yesmaximal, characteristic, direct product factor, Hall
23C23×C7×C23C2×C3×C41120yesdirect product factor
24C23×C7×C23C23×C3135yes—
25C2×C4×C7×C23C23×C31120yesdirect product factor
26C24×C3×C23C4×C7116yesdirect product factor
27C24×C3×C23C22×C7115yes—
28C22×C3×C4×C23C22×C71140yesdirect product factor
29C2×C3×C7×C23C23×C4130yesdirect product factor
30C2×C3×C7×C23C2511yescharacteristic
31C25×C23C2×C3×C711yescharacteristic
32C23×C4×C23C2×C3×C7130yesdirect product factor
33C25×C3×C7C2×C2311yescharacteristic
34C23×C3×C4×C7C2×C23130yesdirect product factor
35C22×C7×C23C22×C3×C41140yesdirect product factor
36C22×C7×C23C24×C3115yes—
37C4×C7×C23C24×C3116yesdirect product factor
38C23×C3×C23C2×C4×C71120yesdirect product factor
39C23×C3×C23C23×C7135yes—
40C2×C3×C4×C23C23×C71120yesdirect product factor
41C3×C7×C23C24×C411yescharacteristic, direct product factor, Hall
42C24×C4×C7C3×C2311yescharacteristic, direct product factor, Hall
43C24×C23C3×C4×C7116yesdirect product factor
44C24×C23C22×C3×C7115yes—
45C22×C4×C23C22×C3×C71140yesdirect product factor
46C24×C3×C7C4×C23116yesdirect product factor
47C24×C3×C7C22×C23115yes—
48C22×C3×C4×C7C22×C231140yesdirect product factor
49C2×C7×C23C23×C3×C4130yesdirect product factor
50C2×C7×C23C25×C311yescharacteristic
51C22×C3×C23C22×C4×C71140yesdirect product factor
52C22×C3×C23C24×C7115yes—
53C3×C4×C23C24×C7116yesdirect product factor
54C25×C7C2×C3×C2311yescharacteristic
55C23×C4×C7C2×C3×C23130yesdirect product factor
56C24×C3×C4C7×C2311yescharacteristic, direct product factor, Hall
57C23×C23C2×C3×C4×C71120yesdirect product factor
58C23×C23C23×C3×C7135yes—
59C2×C4×C23C23×C3×C71120yesdirect product factor
60C23×C3×C7C2×C4×C231120yesdirect product factor
61C23×C3×C7C23×C23135yes—
62C2×C3×C4×C7C23×C231120yesdirect product factor
63C7×C23C24×C3×C411yescharacteristic, direct product factor, Hall
64C2×C3×C23C23×C4×C7130yesdirect product factor
65C2×C3×C23C25×C711yescharacteristic
66C24×C7C3×C4×C23116yesdirect product factor
67C24×C7C22×C3×C23115yes—
68C22×C4×C7C22×C3×C231140yesdirect product factor
69C25×C3C2×C7×C2311yescharacteristic
70C23×C3×C4C2×C7×C23130yesdirect product factor
71C22×C23C22×C3×C4×C71140yesdirect product factor
72C22×C23C24×C3×C7115yes—
73C4×C23C24×C3×C7116yesdirect product factor
74C22×C3×C7C22×C4×C231140yesdirect product factor
75C22×C3×C7C24×C23115yes—
76C3×C4×C7C24×C23116yesdirect product factor
77C3×C23C24×C4×C711yescharacteristic, direct product factor, Hall
78C24×C4C3×C7×C2311yescharacteristic, direct product factor, Sylow
79C23×C7C2×C3×C4×C231120yesdirect product factor
80C23×C7C23×C3×C23135yes—
81C2×C4×C7C23×C3×C231120yesdirect product factor
82C24×C3C4×C7×C23116yesdirect product factor
83C24×C3C22×C7×C23115yes—
84C22×C3×C4C22×C7×C231140yesdirect product factor
85C2×C23C23×C3×C4×C7130yesdirect product factor
86C2×C23C25×C3×C711yescharacteristic
87C2×C3×C7C23×C4×C23130yesdirect product factor
88C2×C3×C7C25×C2311yescharacteristic
89C25C2×C3×C7×C2311yescharacteristic
90C23×C4C2×C3×C7×C23130yesdirect product factor
91C22×C7C22×C3×C4×C231140yesdirect product factor
92C22×C7C24×C3×C23115yes—
93C4×C7C24×C3×C23116yesdirect product factor
94C23×C3C2×C4×C7×C231120yesdirect product factor
95C23×C3C23×C7×C23135yes—
96C2×C3×C4C23×C7×C231120yesdirect product factor
97C23C24×C3×C4×C711yescharacteristic, direct product factor, Sylow
98C3×C7C24×C4×C2311yescharacteristic, direct product factor, Hall
99C24C3×C4×C7×C23116yesdirect product factor
100C24C22×C3×C7×C23115yes—
101C22×C4C22×C3×C7×C231140yesdirect product factor
102C2×C7C23×C3×C4×C23130yesdirect product factor
103C2×C7C25×C3×C2311yescharacteristic
104C22×C3C22×C4×C7×C231140yesdirect product factor
105C22×C3C24×C7×C23115yes—
106C3×C4C24×C7×C23116yesdirect product factor
107C23C2×C3×C4×C7×C231120yesdirect product factor
108C23C23×C3×C7×C23135yes—
109C2×C4C23×C3×C7×C231120yesdirect product factor
110C7C24×C3×C4×C2311yescharacteristic, direct product factor, Sylow
111C2×C3C25×C7×C2311yescharacteristic
112C2×C3C23×C4×C7×C23130yesdirect product factor
113C22C22×C3×C4×C7×C231140yesdirect product factor
114C22C24×C3×C7×C23115yes—
115C4C24×C3×C7×C23116yesdirect product factor
116C3C24×C4×C7×C2311yescharacteristic, direct product factor, Sylow
117C2C23×C3×C4×C7×C23130yesdirect product factor
118C2C25×C3×C7×C2311yescharacteristic
subgroupG/H
centerGC1
commutatorC1G
Frattini, cosocleC2C25×C3×C7×C23
socle, prime radicalC25×C3×C7×C23C2
LCSG ▷ C1
UCSC1 ◁ G

Complements

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

NH ≅ G/Ntype
2. C23×C3×C4×C7×C23C2direct (N × H)
3. C24×C4×C7×C23C3direct (N × H)
4. C24×C3×C7×C23C4direct (N × H)
6. C22×C3×C4×C7×C23C22direct (N × H)
7. C23×C4×C7×C23C2×C3direct (N × H)
9. C24×C3×C4×C23C7direct (N × H)
10. C23×C3×C7×C23C2×C4direct (N × H)
12. C2×C3×C4×C7×C23C23direct (N × H)
13. C24×C7×C23C3×C4direct (N × H)
15. C22×C4×C7×C23C22×C3direct (N × H)
17. C23×C3×C4×C23C2×C7direct (N × H)
18. C22×C3×C7×C23C22×C4direct (N × H)
20. C3×C4×C7×C23C24direct (N × H)
21. C24×C4×C23C3×C7direct (N × H)
22. C24×C3×C4×C7C23direct (N × H)
23. C23×C7×C23C2×C3×C4direct (N × H)
25. C2×C4×C7×C23C23×C3direct (N × H)
26. C24×C3×C23C4×C7direct (N × H)
28. C22×C3×C4×C23C22×C7direct (N × H)
29. C2×C3×C7×C23C23×C4direct (N × H)
32. C23×C4×C23C2×C3×C7direct (N × H)
34. C23×C3×C4×C7C2×C23direct (N × H)
35. C22×C7×C23C22×C3×C4direct (N × H)
37. C4×C7×C23C24×C3direct (N × H)
38. C23×C3×C23C2×C4×C7direct (N × H)
40. C2×C3×C4×C23C23×C7direct (N × H)
41. C3×C7×C23C24×C4direct (N × H)
42. C24×C4×C7C3×C23direct (N × H)
43. C24×C23C3×C4×C7direct (N × H)
45. C22×C4×C23C22×C3×C7direct (N × H)
46. C24×C3×C7C4×C23direct (N × H)
48. C22×C3×C4×C7C22×C23direct (N × H)
49. C2×C7×C23C23×C3×C4direct (N × H)
51. C22×C3×C23C22×C4×C7direct (N × H)
53. C3×C4×C23C24×C7direct (N × H)
55. C23×C4×C7C2×C3×C23direct (N × H)
56. C24×C3×C4C7×C23direct (N × H)
57. C23×C23C2×C3×C4×C7direct (N × H)
59. C2×C4×C23C23×C3×C7direct (N × H)
60. C23×C3×C7C2×C4×C23direct (N × H)
62. C2×C3×C4×C7C23×C23direct (N × H)
63. C7×C23C24×C3×C4direct (N × H)
64. C2×C3×C23C23×C4×C7direct (N × H)
66. C24×C7C3×C4×C23direct (N × H)
68. C22×C4×C7C22×C3×C23direct (N × H)
70. C23×C3×C4C2×C7×C23direct (N × H)
71. C22×C23C22×C3×C4×C7direct (N × H)
73. C4×C23C24×C3×C7direct (N × H)
74. C22×C3×C7C22×C4×C23direct (N × H)
76. C3×C4×C7C24×C23direct (N × H)
77. C3×C23C24×C4×C7direct (N × H)
78. C24×C4C3×C7×C23direct (N × H)
79. C23×C7C2×C3×C4×C23direct (N × H)
81. C2×C4×C7C23×C3×C23direct (N × H)
82. C24×C3C4×C7×C23direct (N × H)
84. C22×C3×C4C22×C7×C23direct (N × H)
85. C2×C23C23×C3×C4×C7direct (N × H)
87. C2×C3×C7C23×C4×C23direct (N × H)
90. C23×C4C2×C3×C7×C23direct (N × H)
91. C22×C7C22×C3×C4×C23direct (N × H)
93. C4×C7C24×C3×C23direct (N × H)
94. C23×C3C2×C4×C7×C23direct (N × H)
96. C2×C3×C4C23×C7×C23direct (N × H)
97. C23C24×C3×C4×C7direct (N × H)
98. C3×C7C24×C4×C23direct (N × H)
99. C24C3×C4×C7×C23direct (N × H)
101. C22×C4C22×C3×C7×C23direct (N × H)
102. C2×C7C23×C3×C4×C23direct (N × H)
104. C22×C3C22×C4×C7×C23direct (N × H)
106. C3×C4C24×C7×C23direct (N × H)
107. C23C2×C3×C4×C7×C23direct (N × H)
109. C2×C4C23×C3×C7×C23direct (N × H)
110. C7C24×C3×C4×C23direct (N × H)
112. C2×C3C23×C4×C7×C23direct (N × H)
113. C22C22×C3×C4×C7×C23direct (N × H)
115. C4C24×C3×C7×C23direct (N × H)
116. C3C24×C4×C7×C23direct (N × H)
117. C2C23×C3×C4×C7×C23direct (N × H)

Maximal subgroups

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

subgroupindex
C25×C3×C7×C232
C23×C3×C4×C7×C232
C24×C4×C7×C233
C24×C3×C4×C237
C24×C3×C4×C723