Finite Group Databasebeta

← C23×C33×C4×C52 #72199113

Subgroups — C23×C33×C4×C52 #72199113

Order 21600.

26432 conjugacy classes of subgroups.

The subgroup lattice has 144 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C24×C33×C52C211yesmaximal, characteristic
2C22×C33×C4×C52C2114yesmaximal, direct product factor
3C23×C32×C4×C52C3113yesmaximal, direct product factor
4C23×C33×C52C418yesdirect product factor
5C23×C33×C52C2217yes—
6C2×C33×C4×C52C22128yesdirect product factor
7C23×C33×C4×C5C516yesmaximal, direct product factor
8C24×C32×C52C2×C3113yes—
9C22×C32×C4×C52C2×C31182yesdirect product factor
10C22×C33×C52C2×C4128yesdirect product factor
11C22×C33×C52C2317yes—
12C33×C4×C52C2318yesdirect product factor
13C23×C3×C4×C52C32113yesdirect product factor
14C24×C33×C5C2×C516yes—
15C22×C33×C4×C5C2×C5184yesdirect product factor
16C23×C32×C52C3×C41104yesdirect product factor
17C23×C32×C52C22×C3191yes—
18C2×C32×C4×C52C22×C31364yesdirect product factor
19C23×C32×C4×C5C3×C5178yesdirect product factor
20C2×C33×C52C22×C4114yesdirect product factor
21C2×C33×C52C2411yescharacteristic
22C24×C3×C52C2×C32113yes—
23C22×C3×C4×C52C2×C321182yesdirect product factor
24C23×C33×C5C4×C5148yesdirect product factor
25C23×C33×C5C22×C5142yes—
26C2×C33×C4×C5C22×C51168yesdirect product factor
27C22×C32×C52C2×C3×C41364yesdirect product factor
28C22×C32×C52C23×C3191yes—
29C32×C4×C52C23×C31104yesdirect product factor
30C23×C33×C4C5211yescharacteristic, direct product factor, Hall
31C23×C4×C52C3311yescharacteristic, direct product factor, Hall
32C24×C32×C5C2×C3×C5178yes—
33C22×C32×C4×C5C2×C3×C511092yesdirect product factor
34C33×C52C23×C411yescharacteristic, direct product factor, Hall
35C23×C3×C52C32×C41104yesdirect product factor
36C23×C3×C52C22×C32191yes—
37C2×C3×C4×C52C22×C321364yesdirect product factor
38C22×C33×C5C2×C4×C51168yesdirect product factor
39C22×C33×C5C23×C5142yes—
40C33×C4×C5C23×C5148yesdirect product factor
41C23×C3×C4×C5C32×C5178yesdirect product factor
42C2×C32×C52C22×C3×C41182yesdirect product factor
43C2×C32×C52C24×C3113yes—
44C24×C33C2×C5211yescharacteristic
45C22×C33×C4C2×C52114yesdirect product factor
46C24×C52C2×C3311yescharacteristic
47C22×C4×C52C2×C33114yesdirect product factor
48C23×C32×C5C3×C4×C51624yesdirect product factor
49C23×C32×C5C22×C3×C51546yes—
50C2×C32×C4×C5C22×C3×C512184yesdirect product factor
51C22×C3×C52C2×C32×C41364yesdirect product factor
52C22×C3×C52C23×C32191yes—
53C3×C4×C52C23×C321104yesdirect product factor
54C23×C32×C4C3×C52113yesdirect product factor
55C2×C33×C5C22×C4×C5184yesdirect product factor
56C2×C33×C5C24×C516yes—
57C24×C3×C5C2×C32×C5178yes—
58C22×C3×C4×C5C2×C32×C511092yesdirect product factor
59C32×C52C23×C3×C4113yesdirect product factor
60C23×C33C4×C5218yesdirect product factor
61C23×C33C22×C5217yes—
62C2×C33×C4C22×C52128yesdirect product factor
63C23×C52C33×C418yesdirect product factor
64C23×C52C22×C3317yes—
65C2×C4×C52C22×C33128yesdirect product factor
66C22×C32×C5C2×C3×C4×C512184yesdirect product factor
67C22×C32×C5C23×C3×C51546yes—
68C32×C4×C5C23×C3×C51624yesdirect product factor
69C23×C4×C5C33×C516yesdirect product factor
70C2×C3×C52C22×C32×C41182yesdirect product factor
71C2×C3×C52C24×C32113yes—
72C24×C32C2×C3×C52113yes—
73C22×C32×C4C2×C3×C521182yesdirect product factor
74C33×C5C23×C4×C516yesdirect product factor
75C23×C3×C5C32×C4×C51624yesdirect product factor
76C23×C3×C5C22×C32×C51546yes—
77C2×C3×C4×C5C22×C32×C512184yesdirect product factor
78C22×C33C2×C4×C52128yesdirect product factor
79C22×C33C23×C5217yes—
80C33×C4C23×C5218yesdirect product factor
81C22×C52C2×C33×C4128yesdirect product factor
82C22×C52C23×C3317yes—
83C4×C52C23×C3318yesdirect product factor
84C23×C3×C4C32×C52113yesdirect product factor
85C2×C32×C5C22×C3×C4×C511092yesdirect product factor
86C2×C32×C5C24×C3×C5178yes—
87C24×C5C2×C33×C516yes—
88C22×C4×C5C2×C33×C5184yesdirect product factor
89C3×C52C23×C32×C4113yesdirect product factor
90C23×C32C3×C4×C521104yesdirect product factor
91C23×C32C22×C3×C52191yes—
92C2×C32×C4C22×C3×C521364yesdirect product factor
93C22×C3×C5C2×C32×C4×C512184yesdirect product factor
94C22×C3×C5C23×C32×C51546yes—
95C3×C4×C5C23×C32×C51624yesdirect product factor
96C2×C33C22×C4×C52114yesdirect product factor
97C2×C33C24×C5211yescharacteristic
98C2×C52C22×C33×C4114yesdirect product factor
99C2×C52C24×C3311yescharacteristic
100C24×C3C2×C32×C52113yes—
101C22×C3×C4C2×C32×C521182yesdirect product factor
102C32×C5C23×C3×C4×C5178yesdirect product factor
103C23×C5C33×C4×C5148yesdirect product factor
104C23×C5C22×C33×C5142yes—
105C2×C4×C5C22×C33×C51168yesdirect product factor
106C22×C32C2×C3×C4×C521364yesdirect product factor
107C22×C32C23×C3×C52191yes—
108C32×C4C23×C3×C521104yesdirect product factor
109C23×C4C33×C5211yescharacteristic, direct product factor, Sylow
110C2×C3×C5C22×C32×C4×C511092yesdirect product factor
111C2×C3×C5C24×C32×C5178yes—
112C33C23×C4×C5211yescharacteristic, direct product factor, Sylow
113C52C23×C33×C411yescharacteristic, direct product factor, Sylow
114C23×C3C32×C4×C521104yesdirect product factor
115C23×C3C22×C32×C52191yes—
116C2×C3×C4C22×C32×C521364yesdirect product factor
117C22×C5C2×C33×C4×C51168yesdirect product factor
118C22×C5C23×C33×C5142yes—
119C4×C5C23×C33×C5148yesdirect product factor
120C2×C32C22×C3×C4×C521182yesdirect product factor
121C2×C32C24×C3×C52113yes—
122C24C2×C33×C5211yescharacteristic
123C22×C4C2×C33×C52114yesdirect product factor
124C3×C5C23×C32×C4×C5178yesdirect product factor
125C22×C3C2×C32×C4×C521364yesdirect product factor
126C22×C3C23×C32×C52191yes—
127C3×C4C23×C32×C521104yesdirect product factor
128C2×C5C22×C33×C4×C5184yesdirect product factor
129C2×C5C24×C33×C516yes—
130C32C23×C3×C4×C52113yesdirect product factor
131C23C33×C4×C5218yesdirect product factor
132C23C22×C33×C5217yes—
133C2×C4C22×C33×C52128yesdirect product factor
134C2×C3C22×C32×C4×C521182yesdirect product factor
135C2×C3C24×C32×C52113yes—
136C5C23×C33×C4×C516yesdirect product factor
137C22C2×C33×C4×C52128yesdirect product factor
138C22C23×C33×C5217yes—
139C4C23×C33×C5218yesdirect product factor
140C3C23×C32×C4×C52113yesdirect product factor
141C2C22×C33×C4×C52114yesdirect product factor
142C2C24×C33×C5211yescharacteristic
subgroupG/H
centerGC1
commutatorC1G
Frattini, cosocleC2C24×C33×C52
socle, prime radicalC24×C33×C52C2
LCSG ▷ C1
UCSC1 ◁ G

Complements

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

NH ≅ G/Ntype
2. C22×C33×C4×C52C2direct (N × H)
3. C23×C32×C4×C52C3direct (N × H)
4. C23×C33×C52C4direct (N × H)
6. C2×C33×C4×C52C22direct (N × H)
7. C23×C33×C4×C5C5direct (N × H)
9. C22×C32×C4×C52C2×C3direct (N × H)
10. C22×C33×C52C2×C4direct (N × H)
12. C33×C4×C52C23direct (N × H)
13. C23×C3×C4×C52C32direct (N × H)
15. C22×C33×C4×C5C2×C5direct (N × H)
16. C23×C32×C52C3×C4direct (N × H)
18. C2×C32×C4×C52C22×C3direct (N × H)
19. C23×C32×C4×C5C3×C5direct (N × H)
20. C2×C33×C52C22×C4direct (N × H)
23. C22×C3×C4×C52C2×C32direct (N × H)
24. C23×C33×C5C4×C5direct (N × H)
26. C2×C33×C4×C5C22×C5direct (N × H)
27. C22×C32×C52C2×C3×C4direct (N × H)
29. C32×C4×C52C23×C3direct (N × H)
30. C23×C33×C4C52direct (N × H)
31. C23×C4×C52C33direct (N × H)
33. C22×C32×C4×C5C2×C3×C5direct (N × H)
34. C33×C52C23×C4direct (N × H)
35. C23×C3×C52C32×C4direct (N × H)
37. C2×C3×C4×C52C22×C32direct (N × H)
38. C22×C33×C5C2×C4×C5direct (N × H)
40. C33×C4×C5C23×C5direct (N × H)
41. C23×C3×C4×C5C32×C5direct (N × H)
42. C2×C32×C52C22×C3×C4direct (N × H)
45. C22×C33×C4C2×C52direct (N × H)
47. C22×C4×C52C2×C33direct (N × H)
48. C23×C32×C5C3×C4×C5direct (N × H)
50. C2×C32×C4×C5C22×C3×C5direct (N × H)
51. C22×C3×C52C2×C32×C4direct (N × H)
53. C3×C4×C52C23×C32direct (N × H)
54. C23×C32×C4C3×C52direct (N × H)
55. C2×C33×C5C22×C4×C5direct (N × H)
58. C22×C3×C4×C5C2×C32×C5direct (N × H)
59. C32×C52C23×C3×C4direct (N × H)
60. C23×C33C4×C52direct (N × H)
62. C2×C33×C4C22×C52direct (N × H)
63. C23×C52C33×C4direct (N × H)
65. C2×C4×C52C22×C33direct (N × H)
66. C22×C32×C5C2×C3×C4×C5direct (N × H)
68. C32×C4×C5C23×C3×C5direct (N × H)
69. C23×C4×C5C33×C5direct (N × H)
70. C2×C3×C52C22×C32×C4direct (N × H)
73. C22×C32×C4C2×C3×C52direct (N × H)
74. C33×C5C23×C4×C5direct (N × H)
75. C23×C3×C5C32×C4×C5direct (N × H)
77. C2×C3×C4×C5C22×C32×C5direct (N × H)
78. C22×C33C2×C4×C52direct (N × H)
80. C33×C4C23×C52direct (N × H)
81. C22×C52C2×C33×C4direct (N × H)
83. C4×C52C23×C33direct (N × H)
84. C23×C3×C4C32×C52direct (N × H)
85. C2×C32×C5C22×C3×C4×C5direct (N × H)
88. C22×C4×C5C2×C33×C5direct (N × H)
89. C3×C52C23×C32×C4direct (N × H)
90. C23×C32C3×C4×C52direct (N × H)
92. C2×C32×C4C22×C3×C52direct (N × H)
93. C22×C3×C5C2×C32×C4×C5direct (N × H)
95. C3×C4×C5C23×C32×C5direct (N × H)
96. C2×C33C22×C4×C52direct (N × H)
98. C2×C52C22×C33×C4direct (N × H)
101. C22×C3×C4C2×C32×C52direct (N × H)
102. C32×C5C23×C3×C4×C5direct (N × H)
103. C23×C5C33×C4×C5direct (N × H)
105. C2×C4×C5C22×C33×C5direct (N × H)
106. C22×C32C2×C3×C4×C52direct (N × H)
108. C32×C4C23×C3×C52direct (N × H)
109. C23×C4C33×C52direct (N × H)
110. C2×C3×C5C22×C32×C4×C5direct (N × H)
112. C33C23×C4×C52direct (N × H)
113. C52C23×C33×C4direct (N × H)
114. C23×C3C32×C4×C52direct (N × H)
116. C2×C3×C4C22×C32×C52direct (N × H)
117. C22×C5C2×C33×C4×C5direct (N × H)
119. C4×C5C23×C33×C5direct (N × H)
120. C2×C32C22×C3×C4×C52direct (N × H)
123. C22×C4C2×C33×C52direct (N × H)
124. C3×C5C23×C32×C4×C5direct (N × H)
125. C22×C3C2×C32×C4×C52direct (N × H)
127. C3×C4C23×C32×C52direct (N × H)
128. C2×C5C22×C33×C4×C5direct (N × H)
130. C32C23×C3×C4×C52direct (N × H)
131. C23C33×C4×C52direct (N × H)
133. C2×C4C22×C33×C52direct (N × H)
134. C2×C3C22×C32×C4×C52direct (N × H)
136. C5C23×C33×C4×C5direct (N × H)
137. C22C2×C33×C4×C52direct (N × H)
139. C4C23×C33×C52direct (N × H)
140. C3C23×C32×C4×C52direct (N × H)
141. C2C22×C33×C4×C52direct (N × H)

Maximal subgroups

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

subgroupindex
C24×C33×C522
C22×C33×C4×C522
C23×C32×C4×C523
C23×C33×C4×C55