Finite Group Databasebeta

← C2×C3×C5×D3×C7×C23 #44394669

Subgroups — C2×C3×C5×D3×C7×C23 #44394669

Order 28980.

176 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 176 conjugacy classes of subgroups — too many to draw. Only its 112 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C5×D3×C7×C23C212yesmaximal, direct product factor
2C2×C32×C5×C7×C23C211yesmaximal, characteristic, complemented
3C2×C5×D3×C7×C23C311yesmaximal, characteristic, direct product factor
4C22×C3×C5×C7×C23—31nomaximal
5C32×C5×C7×C23C2211yescharacteristic, complemented, Hall
6C2×C3×D3×C7×C23C511yesmaximal, characteristic, direct product factor, Hall
7C5×D3×C7×C23C2×C312yesdirect product factor
8C2×C3×C5×C7×C23—21no—
9C2×C3×C5×C7×C23—32no—
10C2×C3×C5×C7×C23D311yescharacteristic, direct product factor
11C2×C3×C5×C7×C23C2×C311yescharacteristic, complemented
12C2×C3×C5×D3×C23C711yesmaximal, characteristic, direct product factor, Hall
13C22×C5×C7×C23—31noHall
14C3×D3×C7×C23C2×C512yesdirect product factor
15C2×C32×C7×C23C2×C511yescharacteristic, complemented
16C3×C5×C7×C23—21no—
17C3×C5×C7×C23C2×D311yescharacteristic, direct product factor
18C3×C5×C7×C23C22×C311yescharacteristic, complemented
19C3×C5×D3×C23C2×C712yesdirect product factor
20C2×C32×C5×C23C2×C711yescharacteristic, complemented
21C22×C3×C7×C23—31no—
22C2×D3×C7×C23C3×C511yescharacteristic, direct product factor
23C2×C5×C7×C23—32no—
24C2×C5×C7×C23C3×D311yescharacteristic, direct product factor
25C32×C7×C23C22×C511yescharacteristic, complemented, Hall
26C22×C3×C5×C23—31no—
27C2×C5×D3×C23C3×C711yescharacteristic, direct product factor
28C2×C3×C5×D3×C7C2311yesmaximal, characteristic, direct product factor, Hall
29C32×C5×C23C22×C711yescharacteristic, complemented, Hall
30C2×C3×C7×C23—21no—
31C2×C3×C7×C23—32no—
32C2×C3×C7×C23C5×D311yescharacteristic, direct product factor
33C2×C3×C7×C23C2×C3×C511yescharacteristic, complemented
34D3×C7×C23C2×C3×C512yesdirect product factor
35C2×C3×D3×C23C5×C711yescharacteristic, direct product factor, Hall
36C5×C7×C23C2×C3×D311yescharacteristic, direct product factor, Hall
37C2×C3×C5×C23—21no—
38C2×C3×C5×C23—32no—
39C2×C3×C5×C23D3×C711yescharacteristic, direct product factor
40C2×C3×C5×C23C2×C3×C711yescharacteristic, complemented
41C5×D3×C23C2×C3×C712yesdirect product factor
42C22×C7×C23—31noHall
43C2×C32×C5×C7C2×C2311yescharacteristic, complemented
44C3×C5×D3×C7C2×C2312yesdirect product factor
45C3×C7×C23—21no—
46C3×C7×C23C2×C5×D311yescharacteristic, direct product factor
47C3×C7×C23C22×C3×C511yescharacteristic, complemented
48C22×C5×C23—31noHall
49C22×C3×C5×C7—31no—
50C2×C5×D3×C7C3×C2311yescharacteristic, direct product factor
51C2×C32×C23C2×C5×C711yescharacteristic, complemented
52C3×D3×C23C2×C5×C712yesdirect product factor
53C3×C5×C23—21no—
54C3×C5×C23C2×D3×C711yescharacteristic, direct product factor
55C3×C5×C23C22×C3×C711yescharacteristic, complemented
56C2×C7×C23—32no—
57C2×C7×C23C3×C5×D311yescharacteristic, direct product factor
58C32×C5×C7C22×C2311yescharacteristic, complemented, Hall
59C22×C3×C23—31no—
60C2×D3×C23C3×C5×C711yescharacteristic, direct product factor
61C2×C3×D3×C7C5×C2311yescharacteristic, direct product factor, Hall
62C2×C5×C23—32no—
63C2×C5×C23C3×D3×C711yescharacteristic, direct product factor
64C2×C3×C5×C7—21no—
65C2×C3×C5×C7—32no—
66C2×C3×C5×C7D3×C2311yescharacteristic, direct product factor
67C2×C3×C5×C7C2×C3×C2311yescharacteristic, complemented
68C5×D3×C7C2×C3×C2312yesdirect product factor
69C32×C23C22×C5×C711yescharacteristic, complemented, Hall
70C2×C3×C5×D3C7×C2311yescharacteristic, direct product factor, Hall
71C7×C23C2×C3×C5×D311yescharacteristic, direct product factor, Hall
72C22×C5×C7—31noHall
73C2×C3×C23—21no—
74C2×C3×C23—32no—
75C2×C3×C23C5×D3×C711yescharacteristic, direct product factor
76C2×C3×C23C2×C3×C5×C711yescharacteristic, complemented
77D3×C23C2×C3×C5×C712yesdirect product factor
78C2×C32×C7C2×C5×C2311yescharacteristic, complemented
79C3×D3×C7C2×C5×C2312yesdirect product factor
80C5×C23C2×C3×D3×C711yescharacteristic, direct product factor, Hall
81C3×C5×C7—21no—
82C3×C5×C7C2×D3×C2311yescharacteristic, direct product factor
83C3×C5×C7C22×C3×C2311yescharacteristic, complemented
84C22×C23—31noHall
85C2×C32×C5C2×C7×C2311yescharacteristic, complemented
86C3×C5×D3C2×C7×C2312yesdirect product factor
87C22×C3×C7—31no—
88C2×D3×C7C3×C5×C2311yescharacteristic, direct product factor
89C2×C5×C7—32no—
90C2×C5×C7C3×D3×C2311yescharacteristic, direct product factor
91C3×C23—21no—
92C3×C23C2×C5×D3×C711yescharacteristic, direct product factor
93C3×C23C22×C3×C5×C711yescharacteristic, complemented
94C32×C7C22×C5×C2311yescharacteristic, complemented, Hall
95C22×C3×C5—31no—
96C2×C5×D3C3×C7×C2311yescharacteristic, direct product factor
97C2×C23—32no—
98C2×C23C3×C5×D3×C711yescharacteristic, direct product factor
99C32×C5C22×C7×C2311yescharacteristic, complemented, Hall
100C2×C3×C7—21no—
101C2×C3×C7—32no—
102C2×C3×C7C5×D3×C2311yescharacteristic, direct product factor
103C2×C3×C7C2×C3×C5×C2311yescharacteristic, complemented
104D3×C7C2×C3×C5×C2312yesdirect product factor
105C2×C3×D3C5×C7×C2311yescharacteristic, direct product factor, Hall
106C5×C7C2×C3×D3×C2311yescharacteristic, direct product factor, Hall
107C2×C3×C5—21no—
108C2×C3×C5—32no—
109C2×C3×C5D3×C7×C2311yescharacteristic, direct product factor
110C2×C3×C5C2×C3×C7×C2311yescharacteristic, complemented
111C5×D3C2×C3×C7×C2312yesdirect product factor
112C22×C7—31noHall
113C23C2×C3×C5×D3×C711yescharacteristic, direct product factor, Sylow
114C3×C7—21no—
115C3×C7C2×C5×D3×C2311yescharacteristic, direct product factor
116C3×C7C22×C3×C5×C2311yescharacteristic, complemented
117C22×C5—31noHall
118C2×C32C2×C5×C7×C2311yescharacteristic, complemented
119C3×D3C2×C5×C7×C2312yesdirect product factor
120C3×C5—21no—
121C3×C5C2×D3×C7×C2311yescharacteristic, direct product factor
122C3×C5C22×C3×C7×C2311yescharacteristic, complemented
123C2×C7—32no—
124C2×C7C3×C5×D3×C2311yescharacteristic, direct product factor
125C22×C3—31no—
126C2×D3C3×C5×C7×C2311yescharacteristic, direct product factor
127C2×C5—32no—
128C2×C5C3×D3×C7×C2311yescharacteristic, direct product factor
129C32C22×C5×C7×C2311yescharacteristic, complemented, Sylow
130C7C2×C3×C5×D3×C2311yescharacteristic, direct product factor, Sylow
131C2×C3—21no—
132C2×C3—32no—
133C2×C3C2×C3×C5×C7×C2311yescharacteristic, complemented
134C2×C3C5×D3×C7×C2311yescharacteristic, direct product factor
135D3C2×C3×C5×C7×C2312yesdirect product factor
136C5C2×C3×D3×C7×C2311yescharacteristic, direct product factor, Sylow
137C22—31noSylow
138C3—21no—
139C3C22×C3×C5×C7×C2311yescharacteristic, complemented
140C3C2×C5×D3×C7×C2311yescharacteristic, direct product factor
141C2—32no—
142C2C3×C5×D3×C7×C2311yescharacteristic, direct product factor
subgroupG/H
center, hypercenterC2×C3×C5×C7×C23D3
commutator, cosocle, hypocenterC3C22×C3×C5×C7×C23
FrattiniC1G
Fitting, socleC2×C32×C5×C7×C23C2
LCSG ▷ C3
UCSC1 ◁ C2×C3×C5×C7×C23

Complements

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

NH ≅ G/Ntype
1. C3×C5×D3×C7×C23C2direct (N × H)
2. C2×C32×C5×C7×C23C2semidirect
3. C2×C5×D3×C7×C23C3direct (N × H)
5. C32×C5×C7×C23C22semidirect
6. C2×C3×D3×C7×C23C5direct (N × H)
7. C5×D3×C7×C23C2×C3direct (N × H)
10. C2×C3×C5×C7×C23D3direct (N × H)
11. C2×C3×C5×C7×C23C2×C3semidirect
12. C2×C3×C5×D3×C23C7direct (N × H)
14. C3×D3×C7×C23C2×C5direct (N × H)
15. C2×C32×C7×C23C2×C5semidirect
17. C3×C5×C7×C23C2×D3direct (N × H)
18. C3×C5×C7×C23C22×C3semidirect
19. C3×C5×D3×C23C2×C7direct (N × H)
20. C2×C32×C5×C23C2×C7semidirect
22. C2×D3×C7×C23C3×C5direct (N × H)
24. C2×C5×C7×C23C3×D3direct (N × H)
25. C32×C7×C23C22×C5semidirect
27. C2×C5×D3×C23C3×C7direct (N × H)
28. C2×C3×C5×D3×C7C23direct (N × H)
29. C32×C5×C23C22×C7semidirect
32. C2×C3×C7×C23C5×D3direct (N × H)
33. C2×C3×C7×C23C2×C3×C5semidirect
34. D3×C7×C23C2×C3×C5direct (N × H)
35. C2×C3×D3×C23C5×C7direct (N × H)
36. C5×C7×C23C2×C3×D3direct (N × H)
39. C2×C3×C5×C23D3×C7direct (N × H)
40. C2×C3×C5×C23C2×C3×C7semidirect
41. C5×D3×C23C2×C3×C7direct (N × H)
43. C2×C32×C5×C7C2×C23semidirect
44. C3×C5×D3×C7C2×C23direct (N × H)
46. C3×C7×C23C2×C5×D3direct (N × H)
47. C3×C7×C23C22×C3×C5semidirect
50. C2×C5×D3×C7C3×C23direct (N × H)
51. C2×C32×C23C2×C5×C7semidirect
52. C3×D3×C23C2×C5×C7direct (N × H)
54. C3×C5×C23C2×D3×C7direct (N × H)
55. C3×C5×C23C22×C3×C7semidirect
57. C2×C7×C23C3×C5×D3direct (N × H)
58. C32×C5×C7C22×C23semidirect
60. C2×D3×C23C3×C5×C7direct (N × H)
61. C2×C3×D3×C7C5×C23direct (N × H)
63. C2×C5×C23C3×D3×C7direct (N × H)
66. C2×C3×C5×C7D3×C23direct (N × H)
67. C2×C3×C5×C7C2×C3×C23semidirect
68. C5×D3×C7C2×C3×C23direct (N × H)
69. C32×C23C22×C5×C7semidirect
70. C2×C3×C5×D3C7×C23direct (N × H)
71. C7×C23C2×C3×C5×D3direct (N × H)
75. C2×C3×C23C5×D3×C7direct (N × H)
76. C2×C3×C23C2×C3×C5×C7semidirect
77. D3×C23C2×C3×C5×C7direct (N × H)
78. C2×C32×C7C2×C5×C23semidirect
79. C3×D3×C7C2×C5×C23direct (N × H)
80. C5×C23C2×C3×D3×C7direct (N × H)
82. C3×C5×C7C2×D3×C23direct (N × H)
83. C3×C5×C7C22×C3×C23semidirect
85. C2×C32×C5C2×C7×C23semidirect
86. C3×C5×D3C2×C7×C23direct (N × H)
88. C2×D3×C7C3×C5×C23direct (N × H)
90. C2×C5×C7C3×D3×C23direct (N × H)
92. C3×C23C2×C5×D3×C7direct (N × H)
93. C3×C23C22×C3×C5×C7semidirect
94. C32×C7C22×C5×C23semidirect
96. C2×C5×D3C3×C7×C23direct (N × H)
98. C2×C23C3×C5×D3×C7direct (N × H)
99. C32×C5C22×C7×C23semidirect
102. C2×C3×C7C5×D3×C23direct (N × H)
103. C2×C3×C7C2×C3×C5×C23semidirect
104. D3×C7C2×C3×C5×C23direct (N × H)
105. C2×C3×D3C5×C7×C23direct (N × H)
106. C5×C7C2×C3×D3×C23direct (N × H)
109. C2×C3×C5D3×C7×C23direct (N × H)
110. C2×C3×C5C2×C3×C7×C23semidirect
111. C5×D3C2×C3×C7×C23direct (N × H)
113. C23C2×C3×C5×D3×C7direct (N × H)
115. C3×C7C2×C5×D3×C23direct (N × H)
116. C3×C7C22×C3×C5×C23semidirect
118. C2×C32C2×C5×C7×C23semidirect
119. C3×D3C2×C5×C7×C23direct (N × H)
121. C3×C5C2×D3×C7×C23direct (N × H)
122. C3×C5C22×C3×C7×C23semidirect
124. C2×C7C3×C5×D3×C23direct (N × H)
126. C2×D3C3×C5×C7×C23direct (N × H)
128. C2×C5C3×D3×C7×C23direct (N × H)
129. C32C22×C5×C7×C23semidirect
130. C7C2×C3×C5×D3×C23direct (N × H)
133. C2×C3C2×C3×C5×C7×C23semidirect
134. C2×C3C5×D3×C7×C23direct (N × H)
135. D3C2×C3×C5×C7×C23direct (N × H)
136. C5C2×C3×D3×C7×C23direct (N × H)
139. C3C22×C3×C5×C7×C23semidirect
140. C3C2×C5×D3×C7×C23direct (N × H)
142. C2C3×C5×D3×C7×C23direct (N × H)

Maximal subgroups

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

subgroupindex
C3×C5×D3×C7×C232
C2×C32×C5×C7×C232
C2×C5×D3×C7×C233
C22×C3×C5×C7×C233
C2×C3×D3×C7×C235
C2×C3×C5×D3×C237
C2×C3×C5×D3×C723