Finite Group Databasebeta

← C23×C3×C53×C8 #72201281

Subgroups — C23×C3×C53×C8 #72201281

Order 24000.

21632 conjugacy classes of subgroups.

The subgroup lattice has 152 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C3×C4×C53C211yesmaximal, characteristic
2C22×C3×C53×C8C2114yesmaximal, direct product factor
3C23×C53×C8C311yesmaximal, characteristic, direct product factor, Hall
4C24×C3×C53C411yescharacteristic
5C22×C3×C4×C53C417yes—
6C22×C3×C4×C53C2217yes—
7C2×C3×C53×C8C22128yesdirect product factor
8C23×C3×C52×C8C5131yesmaximal, direct product factor
9C23×C4×C53C2×C311yescharacteristic
10C22×C53×C8C2×C3114yesdirect product factor
11C23×C3×C53C818yesdirect product factor
12C23×C3×C53C2×C417yes—
13C2×C3×C4×C53C2×C4121yes—
14C2×C3×C4×C53C2317yes—
15C3×C53×C8C2318yesdirect product factor
16C23×C3×C4×C52C2×C5131yes—
17C22×C3×C52×C8C2×C51434yesdirect product factor
18C24×C53C3×C411yescharacteristic
19C22×C4×C53C3×C417yes—
20C22×C4×C53C22×C317yes—
21C2×C53×C8C22×C3128yesdirect product factor
22C23×C52×C8C3×C5131yesdirect product factor
23C22×C3×C53C2×C8128yesdirect product factor
24C22×C3×C53C22×C417yes—
25C3×C4×C53C22×C417yes—
26C3×C4×C53C2411yescharacteristic
27C24×C3×C52C4×C5131yes—
28C22×C3×C4×C52C4×C51217yes—
29C22×C3×C4×C52C22×C51217yes—
30C2×C3×C52×C8C22×C51868yesdirect product factor
31C23×C53C3×C818yesdirect product factor
32C23×C53C2×C3×C417yes—
33C2×C4×C53C2×C3×C4121yes—
34C2×C4×C53C23×C317yes—
35C53×C8C23×C318yesdirect product factor
36C23×C3×C5×C8C52131yesdirect product factor
37C23×C4×C52C2×C3×C5131yes—
38C22×C52×C8C2×C3×C51434yesdirect product factor
39C2×C3×C53C22×C8114yesdirect product factor
40C2×C3×C53C23×C411yescharacteristic
41C23×C3×C52C5×C81248yesdirect product factor
42C23×C3×C52C2×C4×C51217yes—
43C2×C3×C4×C52C2×C4×C51651yes—
44C2×C3×C4×C52C23×C51217yes—
45C3×C52×C8C23×C51248yesdirect product factor
46C22×C53C2×C3×C8128yesdirect product factor
47C22×C53C22×C3×C417yes—
48C4×C53C22×C3×C417yes—
49C4×C53C24×C311yescharacteristic
50C23×C3×C4×C5C2×C52131yes—
51C22×C3×C5×C8C2×C521434yesdirect product factor
52C24×C52C3×C4×C5131yes—
53C22×C4×C52C3×C4×C51217yes—
54C22×C4×C52C22×C3×C51217yes—
55C2×C52×C8C22×C3×C51868yesdirect product factor
56C3×C53C23×C811yescharacteristic, direct product factor, Hall
57C23×C5×C8C3×C52131yesdirect product factor
58C22×C3×C52C2×C5×C81868yesdirect product factor
59C22×C3×C52C22×C4×C51217yes—
60C3×C4×C52C22×C4×C51217yes—
61C3×C4×C52C24×C5131yes—
62C2×C53C22×C3×C8114yesdirect product factor
63C2×C53C23×C3×C411yescharacteristic
64C24×C3×C5C4×C52131yes—
65C22×C3×C4×C5C4×C521217yes—
66C22×C3×C4×C5C22×C521217yes—
67C2×C3×C5×C8C22×C521868yesdirect product factor
68C23×C52C3×C5×C81248yesdirect product factor
69C23×C52C2×C3×C4×C51217yes—
70C2×C4×C52C2×C3×C4×C51651yes—
71C2×C4×C52C23×C3×C51217yes—
72C52×C8C23×C3×C51248yesdirect product factor
73C23×C3×C8C5311yescharacteristic, direct product factor, Hall
74C23×C4×C5C2×C3×C52131yes—
75C22×C5×C8C2×C3×C521434yesdirect product factor
76C2×C3×C52C22×C5×C81434yesdirect product factor
77C2×C3×C52C23×C4×C5131yes—
78C53C23×C3×C811yescharacteristic, direct product factor, Sylow
79C23×C3×C5C52×C81248yesdirect product factor
80C23×C3×C5C2×C4×C521217yes—
81C2×C3×C4×C5C2×C4×C521651yes—
82C2×C3×C4×C5C23×C521217yes—
83C3×C5×C8C23×C521248yesdirect product factor
84C22×C52C2×C3×C5×C81868yesdirect product factor
85C22×C52C22×C3×C4×C51217yes—
86C4×C52C22×C3×C4×C51217yes—
87C4×C52C24×C3×C5131yes—
88C23×C3×C4C2×C5311yescharacteristic
89C22×C3×C8C2×C53114yesdirect product factor
90C24×C5C3×C4×C52131yes—
91C22×C4×C5C3×C4×C521217yes—
92C22×C4×C5C22×C3×C521217yes—
93C2×C5×C8C22×C3×C521868yesdirect product factor
94C3×C52C23×C5×C8131yesdirect product factor
95C23×C8C3×C5311yescharacteristic, direct product factor, Sylow
96C22×C3×C5C2×C52×C81868yesdirect product factor
97C22×C3×C5C22×C4×C521217yes—
98C3×C4×C5C22×C4×C521217yes—
99C3×C4×C5C24×C52131yes—
100C2×C52C22×C3×C5×C81434yesdirect product factor
101C2×C52C23×C3×C4×C5131yes—
102C24×C3C4×C5311yescharacteristic
103C22×C3×C4C4×C5317yes—
104C22×C3×C4C22×C5317yes—
105C2×C3×C8C22×C53128yesdirect product factor
106C23×C5C3×C52×C81248yesdirect product factor
107C23×C5C2×C3×C4×C521217yes—
108C2×C4×C5C2×C3×C4×C521651yes—
109C2×C4×C5C23×C3×C521217yes—
110C5×C8C23×C3×C521248yesdirect product factor
111C23×C4C2×C3×C5311yescharacteristic
112C22×C8C2×C3×C53114yesdirect product factor
113C2×C3×C5C22×C52×C81434yesdirect product factor
114C2×C3×C5C23×C4×C52131yes—
115C52C23×C3×C5×C8131yesdirect product factor
116C23×C3C53×C818yesdirect product factor
117C23×C3C2×C4×C5317yes—
118C2×C3×C4C2×C4×C53121yes—
119C2×C3×C4C23×C5317yes—
120C3×C8C23×C5318yesdirect product factor
121C22×C5C2×C3×C52×C81868yesdirect product factor
122C22×C5C22×C3×C4×C521217yes—
123C4×C5C22×C3×C4×C521217yes—
124C4×C5C24×C3×C52131yes—
125C24C3×C4×C5311yescharacteristic
126C22×C4C3×C4×C5317yes—
127C22×C4C22×C3×C5317yes—
128C2×C8C22×C3×C53128yesdirect product factor
129C3×C5C23×C52×C8131yesdirect product factor
130C22×C3C2×C53×C8128yesdirect product factor
131C22×C3C22×C4×C5317yes—
132C3×C4C22×C4×C5317yes—
133C3×C4C24×C5311yescharacteristic
134C2×C5C22×C3×C52×C81434yesdirect product factor
135C2×C5C23×C3×C4×C52131yes—
136C23C3×C53×C818yesdirect product factor
137C23C2×C3×C4×C5317yes—
138C2×C4C2×C3×C4×C53121yes—
139C2×C4C23×C3×C5317yes—
140C8C23×C3×C5318yesdirect product factor
141C2×C3C22×C53×C8114yesdirect product factor
142C2×C3C23×C4×C5311yescharacteristic
143C5C23×C3×C52×C8131yesdirect product factor
144C22C2×C3×C53×C8128yesdirect product factor
145C22C22×C3×C4×C5317yes—
146C4C22×C3×C4×C5317yes—
147C4C24×C3×C5311yescharacteristic
148C3C23×C53×C811yescharacteristic, direct product factor, Sylow
149C2C22×C3×C53×C8114yesdirect product factor
150C2C23×C3×C4×C5311yescharacteristic
subgroupG/H
centerGC1
commutatorC1G
Frattini, cosocleC4C24×C3×C53
socle, prime radicalC24×C3×C53C4
LCSG ▷ C1
UCSC1 ◁ G

Complements

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

NH ≅ G/Ntype
2. C22×C3×C53×C8C2direct (N × H)
3. C23×C53×C8C3direct (N × H)
7. C2×C3×C53×C8C22direct (N × H)
8. C23×C3×C52×C8C5direct (N × H)
10. C22×C53×C8C2×C3direct (N × H)
11. C23×C3×C53C8direct (N × H)
15. C3×C53×C8C23direct (N × H)
17. C22×C3×C52×C8C2×C5direct (N × H)
21. C2×C53×C8C22×C3direct (N × H)
22. C23×C52×C8C3×C5direct (N × H)
23. C22×C3×C53C2×C8direct (N × H)
30. C2×C3×C52×C8C22×C5direct (N × H)
31. C23×C53C3×C8direct (N × H)
35. C53×C8C23×C3direct (N × H)
36. C23×C3×C5×C8C52direct (N × H)
38. C22×C52×C8C2×C3×C5direct (N × H)
39. C2×C3×C53C22×C8direct (N × H)
41. C23×C3×C52C5×C8direct (N × H)
45. C3×C52×C8C23×C5direct (N × H)
46. C22×C53C2×C3×C8direct (N × H)
51. C22×C3×C5×C8C2×C52direct (N × H)
55. C2×C52×C8C22×C3×C5direct (N × H)
56. C3×C53C23×C8direct (N × H)
57. C23×C5×C8C3×C52direct (N × H)
58. C22×C3×C52C2×C5×C8direct (N × H)
62. C2×C53C22×C3×C8direct (N × H)
67. C2×C3×C5×C8C22×C52direct (N × H)
68. C23×C52C3×C5×C8direct (N × H)
72. C52×C8C23×C3×C5direct (N × H)
73. C23×C3×C8C53direct (N × H)
75. C22×C5×C8C2×C3×C52direct (N × H)
76. C2×C3×C52C22×C5×C8direct (N × H)
78. C53C23×C3×C8direct (N × H)
79. C23×C3×C5C52×C8direct (N × H)
83. C3×C5×C8C23×C52direct (N × H)
84. C22×C52C2×C3×C5×C8direct (N × H)
89. C22×C3×C8C2×C53direct (N × H)
93. C2×C5×C8C22×C3×C52direct (N × H)
94. C3×C52C23×C5×C8direct (N × H)
95. C23×C8C3×C53direct (N × H)
96. C22×C3×C5C2×C52×C8direct (N × H)
100. C2×C52C22×C3×C5×C8direct (N × H)
105. C2×C3×C8C22×C53direct (N × H)
106. C23×C5C3×C52×C8direct (N × H)
110. C5×C8C23×C3×C52direct (N × H)
112. C22×C8C2×C3×C53direct (N × H)
113. C2×C3×C5C22×C52×C8direct (N × H)
115. C52C23×C3×C5×C8direct (N × H)
116. C23×C3C53×C8direct (N × H)
120. C3×C8C23×C53direct (N × H)
121. C22×C5C2×C3×C52×C8direct (N × H)
128. C2×C8C22×C3×C53direct (N × H)
129. C3×C5C23×C52×C8direct (N × H)
130. C22×C3C2×C53×C8direct (N × H)
134. C2×C5C22×C3×C52×C8direct (N × H)
136. C23C3×C53×C8direct (N × H)
140. C8C23×C3×C53direct (N × H)
141. C2×C3C22×C53×C8direct (N × H)
143. C5C23×C3×C52×C8direct (N × H)
144. C22C2×C3×C53×C8direct (N × H)
148. C3C23×C53×C8direct (N × H)
149. C2C22×C3×C53×C8direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C3×C4×C532
C22×C3×C53×C82
C23×C53×C83
C23×C3×C52×C85