Finite Group Databasebeta

← C2×C3×C5×C7×D69 #44394670

Subgroups — C2×C3×C5×C7×D69 #44394670

Order 28980.

176 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 176 conjugacy classes of subgroups — too many to draw. Only its 88 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×C7×D69C212yesmaximal, direct product factor
2C2×C32×C5×C7×C23C211yesmaximal, characteristic, complemented
3C2×C3×C5×C7×D23—31nomaximal
4C2×C5×C7×D69C311yesmaximal, characteristic, direct product factor
5C32×C5×C7×C23C2211yescharacteristic, complemented, Hall
6C2×C3×C7×D69C511yesmaximal, characteristic, direct product factor, Hall
7C5×C7×D69C2×C312yesdirect product factor
8C3×C5×C7×D23—32no—
9C2×C3×C5×C7×C23—21no—
10C2×C3×C5×C7×C23D311yescharacteristic, complemented
11C2×C3×C5×C7×C23C2×C311yescharacteristic, complemented
12C2×C3×C5×D69C711yesmaximal, characteristic, direct product factor, Hall
13C2×C5×C7×D23—31noHall
14C3×C7×D69C2×C512yesdirect product factor
15C2×C32×C7×C23C2×C511yescharacteristic, complemented
16C3×C5×C7×C23—21no—
17C3×C5×C7×C23C2×D311yescharacteristic, complemented
18C3×C5×C7×C23C22×C311yescharacteristic, complemented
19C3×C5×D69C2×C712yesdirect product factor
20C2×C32×C5×C23C2×C711yescharacteristic, complemented
21C2×C7×D69C3×C511yescharacteristic, direct product factor
22C2×C3×C7×D23—31no—
23C2×C5×C7×C23C3×D311yescharacteristic, complemented
24C5×C7×D23—32no—
25C32×C7×C23C22×C511yescharacteristic, complemented, Hall
26C2×C5×D69C3×C711yescharacteristic, direct product factor
27C2×C3×C5×D23—31no—
28C2×C3×C5×D3×C7—231nomaximal, Hall
29C32×C5×C23C22×C711yescharacteristic, complemented, Hall
30C2×C3×C7×C23—21no—
31C2×C3×C7×C23C5×D311yescharacteristic, complemented
32C2×C3×C7×C23C2×C3×C511yescharacteristic, complemented
33C7×D69C2×C3×C512yesdirect product factor
34C3×C7×D23—32no—
35C2×C3×D69C5×C711yescharacteristic, direct product factor, Hall
36C5×C7×C23C2×C3×D311yescharacteristic, complemented, Hall
37C2×C3×C5×C23—21no—
38C2×C3×C5×C23D3×C711yescharacteristic, complemented
39C2×C3×C5×C23C2×C3×C711yescharacteristic, complemented
40C5×D69C2×C3×C712yesdirect product factor
41C3×C5×D23—32no—
42C2×C7×D23—31noHall
43C2×C32×C5×C7D2311yescharacteristic, complemented
44C3×C5×D3×C7—232no—
45C3×C7×C23—21no—
46C3×C7×C23C2×C5×D311yescharacteristic, complemented
47C3×C7×C23C22×C3×C511yescharacteristic, complemented
48C2×C5×D23—31noHall
49C22×C3×C5×C7—691no—
50C2×C5×D3×C7—231no—
51C2×C32×C23C2×C5×C711yescharacteristic, complemented
52C3×D69C2×C5×C712yesdirect product factor
53C3×C5×C23—21no—
54C3×C5×C23C2×D3×C711yescharacteristic, complemented
55C3×C5×C23C22×C3×C711yescharacteristic, complemented
56C2×C7×C23C3×C5×D311yescharacteristic, complemented
57C7×D23—32no—
58C32×C5×C7C2×D2311yescharacteristic, complemented, Hall
59C2×D69C3×C5×C711yescharacteristic, direct product factor
60C2×C3×D23—31no—
61C2×C3×D3×C7—231noHall
62C2×C5×C23C3×D3×C711yescharacteristic, complemented
63C5×D23—32no—
64C2×C3×C5×C7—21no—
65C2×C3×C5×C7—692no—
66C2×C3×C5×C7C3×D2311yescharacteristic, complemented
67C2×C3×C5×C7D6911yescharacteristic, direct product factor
68C5×D3×C7—232no—
69C32×C23C22×C5×C711yescharacteristic, complemented, Hall
70C2×C3×C5×D3—231noHall
71C7×C23C2×C3×C5×D311yescharacteristic, complemented, Hall
72C22×C5×C7—691noHall
73C2×C3×C23—21no—
74C2×C3×C23C5×D3×C711yescharacteristic, complemented
75C2×C3×C23C2×C3×C5×C711yescharacteristic, complemented
76D69C2×C3×C5×C712yesdirect product factor
77C3×D23—32no—
78C2×C32×C7C5×D2311yescharacteristic, complemented
79C3×D3×C7—232no—
80C5×C23C2×C3×D3×C711yescharacteristic, complemented, Hall
81C3×C5×C7—21no—
82C3×C5×C7C2×C3×D2311yescharacteristic, complemented
83C3×C5×C7C2×D6911yescharacteristic, direct product factor
84C2×D23—31noHall
85C2×C32×C5C7×D2311yescharacteristic, complemented
86C3×C5×D3—232no—
87C22×C3×C7—691no—
88C2×D3×C7—231no—
89C2×C5×C7—692no—
90C2×C5×C7C3×D6911yescharacteristic, direct product factor
91C3×C23—21no—
92C3×C23C2×C5×D3×C711yescharacteristic, complemented
93C3×C23C22×C3×C5×C711yescharacteristic, complemented
94C32×C7C2×C5×D2311yescharacteristic, complemented, Hall
95C22×C3×C5—691no—
96C2×C5×D3—231no—
97C2×C23C3×C5×D3×C711yescharacteristic, complemented
98D23—32no—
99C32×C5C2×C7×D2311yescharacteristic, complemented, Hall
100C2×C3×C7—21no—
101C2×C3×C7—692no—
102C2×C3×C7C3×C5×D2311yescharacteristic, complemented
103C2×C3×C7C5×D6911yescharacteristic, direct product factor
104D3×C7—232no—
105C2×C3×D3—231noHall
106C5×C7C2×C3×D6911yescharacteristic, direct product factor, Hall
107C2×C3×C5—21no—
108C2×C3×C5—692no—
109C2×C3×C5C3×C7×D2311yescharacteristic, complemented
110C2×C3×C5C7×D6911yescharacteristic, direct product factor
111C5×D3—232no—
112C22×C7—691noHall
113C23C2×C3×C5×D3×C711yescharacteristic, complemented, Sylow
114C3×C7—21no—
115C3×C7C2×C3×C5×D2311yescharacteristic, complemented
116C3×C7C2×C5×D6911yescharacteristic, direct product factor
117C22×C5—691noHall
118C2×C32C5×C7×D2311yescharacteristic, complemented
119C3×D3—232no—
120C3×C5—21no—
121C3×C5C2×C3×C7×D2311yescharacteristic, complemented
122C3×C5C2×C7×D6911yescharacteristic, direct product factor
123C2×C7—692no—
124C2×C7C3×C5×D6911yescharacteristic, direct product factor
125C22×C3—691no—
126C2×D3—231no—
127C2×C5—692no—
128C2×C5C3×C7×D6911yescharacteristic, direct product factor
129C32C2×C5×C7×D2311yescharacteristic, complemented, Sylow
130C7C2×C3×C5×D6911yescharacteristic, direct product factor, Sylow
131C2×C3—21no—
132C2×C3—692no—
133C2×C3C3×C5×C7×D2311yescharacteristic, complemented
134C2×C3C5×C7×D6911yescharacteristic, direct product factor
135D3—232no—
136C5C2×C3×C7×D6911yescharacteristic, direct product factor, Sylow
137C22—691noSylow
138C3—21no—
139C3C2×C5×C7×D6911yescharacteristic, direct product factor
140C3C2×C3×C5×C7×D2311yescharacteristic, complemented
141C2—692no—
142C2C3×C5×C7×D6911yescharacteristic, direct product factor
subgroupG/H
center, hypercenterC2×C3×C5×C7D69
commutator, cosocle, hypocenterC3×C23C22×C3×C5×C7
FrattiniC1G
Fitting, socleC2×C32×C5×C7×C23C2
LCSG ▷ C3×C23
UCSC1 ◁ C2×C3×C5×C7

Complements

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

NH ≅ G/Ntype
1. C3×C5×C7×D69C2direct (N × H)
2. C2×C32×C5×C7×C23C2semidirect
4. C2×C5×C7×D69C3direct (N × H)
5. C32×C5×C7×C23C22semidirect
6. C2×C3×C7×D69C5direct (N × H)
7. C5×C7×D69C2×C3direct (N × H)
10. C2×C3×C5×C7×C23D3semidirect
11. C2×C3×C5×C7×C23C2×C3semidirect
12. C2×C3×C5×D69C7direct (N × H)
14. C3×C7×D69C2×C5direct (N × H)
15. C2×C32×C7×C23C2×C5semidirect
17. C3×C5×C7×C23C2×D3semidirect
18. C3×C5×C7×C23C22×C3semidirect
19. C3×C5×D69C2×C7direct (N × H)
20. C2×C32×C5×C23C2×C7semidirect
21. C2×C7×D69C3×C5direct (N × H)
23. C2×C5×C7×C23C3×D3semidirect
25. C32×C7×C23C22×C5semidirect
26. C2×C5×D69C3×C7direct (N × H)
29. C32×C5×C23C22×C7semidirect
31. C2×C3×C7×C23C5×D3semidirect
32. C2×C3×C7×C23C2×C3×C5semidirect
33. C7×D69C2×C3×C5direct (N × H)
35. C2×C3×D69C5×C7direct (N × H)
36. C5×C7×C23C2×C3×D3semidirect
38. C2×C3×C5×C23D3×C7semidirect
39. C2×C3×C5×C23C2×C3×C7semidirect
40. C5×D69C2×C3×C7direct (N × H)
43. C2×C32×C5×C7D23semidirect
46. C3×C7×C23C2×C5×D3semidirect
47. C3×C7×C23C22×C3×C5semidirect
51. C2×C32×C23C2×C5×C7semidirect
52. C3×D69C2×C5×C7direct (N × H)
54. C3×C5×C23C2×D3×C7semidirect
55. C3×C5×C23C22×C3×C7semidirect
56. C2×C7×C23C3×C5×D3semidirect
58. C32×C5×C7C2×D23semidirect
59. C2×D69C3×C5×C7direct (N × H)
62. C2×C5×C23C3×D3×C7semidirect
66. C2×C3×C5×C7C3×D23semidirect
67. C2×C3×C5×C7D69direct (N × H)
69. C32×C23C22×C5×C7semidirect
71. C7×C23C2×C3×C5×D3semidirect
74. C2×C3×C23C5×D3×C7semidirect
75. C2×C3×C23C2×C3×C5×C7semidirect
76. D69C2×C3×C5×C7direct (N × H)
78. C2×C32×C7C5×D23semidirect
80. C5×C23C2×C3×D3×C7semidirect
82. C3×C5×C7C2×C3×D23semidirect
83. C3×C5×C7C2×D69direct (N × H)
85. C2×C32×C5C7×D23semidirect
90. C2×C5×C7C3×D69direct (N × H)
92. C3×C23C2×C5×D3×C7semidirect
93. C3×C23C22×C3×C5×C7semidirect
94. C32×C7C2×C5×D23semidirect
97. C2×C23C3×C5×D3×C7semidirect
99. C32×C5C2×C7×D23semidirect
102. C2×C3×C7C3×C5×D23semidirect
103. C2×C3×C7C5×D69direct (N × H)
106. C5×C7C2×C3×D69direct (N × H)
109. C2×C3×C5C3×C7×D23semidirect
110. C2×C3×C5C7×D69direct (N × H)
113. C23C2×C3×C5×D3×C7semidirect
115. C3×C7C2×C3×C5×D23semidirect
116. C3×C7C2×C5×D69direct (N × H)
118. C2×C32C5×C7×D23semidirect
121. C3×C5C2×C3×C7×D23semidirect
122. C3×C5C2×C7×D69direct (N × H)
124. C2×C7C3×C5×D69direct (N × H)
128. C2×C5C3×C7×D69direct (N × H)
129. C32C2×C5×C7×D23semidirect
130. C7C2×C3×C5×D69direct (N × H)
133. C2×C3C3×C5×C7×D23semidirect
134. C2×C3C5×C7×D69direct (N × H)
136. C5C2×C3×C7×D69direct (N × H)
139. C3C2×C5×C7×D69direct (N × H)
140. C3C2×C3×C5×C7×D23semidirect
142. C2C3×C5×C7×D69direct (N × H)

Maximal subgroups

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

subgroupindex
C3×C5×C7×D692
C2×C32×C5×C7×C232
C2×C3×C5×C7×D233
C2×C5×C7×D693
C2×C3×C7×D695
C2×C3×C5×D697
C2×C3×C5×D3×C723