Finite Group Databasebeta

← C2×C3×C5×D609 #49141683

Subgroups — C2×C3×C5×D609 #49141683

Order 36540.

176 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 176 conjugacy classes of subgroups — too many to draw. Only its 76 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×D609C212yesmaximal, direct product factor
2C2×C32×C5×C7×C29C211yesmaximal, characteristic, complemented
3C2×C3×C5×D203—31nomaximal
4C2×C5×D609C311yesmaximal, characteristic, direct product factor
5C32×C5×C7×C29C2211yescharacteristic, complemented, Hall
6C2×C3×D609C511yesmaximal, characteristic, direct product factor, Hall
7C5×D609C2×C312yesdirect product factor
8C3×C5×D203—32no—
9C2×C3×C5×C7×C29—21no—
10C2×C3×C5×C7×C29D311yescharacteristic, complemented
11C2×C3×C5×C7×C29C2×C311yescharacteristic, complemented
12C2×C3×C5×D87—71nomaximal, Hall
13C2×C5×D203—31noHall
14C3×D609C2×C512yesdirect product factor
15C2×C32×C7×C29C2×C511yescharacteristic, complemented
16C3×C5×C7×C29—21no—
17C3×C5×C7×C29C2×D311yescharacteristic, complemented
18C3×C5×C7×C29C22×C311yescharacteristic, complemented
19C3×C5×D87—72no—
20C2×C32×C5×C29D711yescharacteristic, complemented
21C2×D609C3×C511yescharacteristic, direct product factor
22C2×C3×D203—31no—
23C5×D203—32no—
24C2×C5×C7×C29C3×D311yescharacteristic, complemented
25C32×C7×C29C22×C511yescharacteristic, complemented, Hall
26C2×C5×D87—71no—
27C2×C3×C5×D29—211no—
28C32×C5×C29C2×D711yescharacteristic, complemented, Hall
29C2×C3×C5×D21—291nomaximal, Hall
30C2×C3×C7×C29—21no—
31C2×C3×C7×C29C5×D311yescharacteristic, complemented
32C2×C3×C7×C29C2×C3×C511yescharacteristic, complemented
33D609C2×C3×C512yesdirect product factor
34C3×D203—32no—
35C2×C3×D87—71noHall
36C5×C7×C29C2×C3×D311yescharacteristic, complemented, Hall
37C2×C3×C5×C29—21no—
38C2×C3×C5×C29C3×D711yescharacteristic, complemented
39C2×C3×C5×C29D2111yescharacteristic, complemented
40C5×D87—72no—
41C3×C5×D29—212no—
42C2×D203—31noHall
43C2×C32×C5×C7D2911yescharacteristic, complemented
44C3×C5×D21—292no—
45C3×C7×C29—21no—
46C3×C7×C29C2×C5×D311yescharacteristic, complemented
47C3×C7×C29C22×C3×C511yescharacteristic, complemented
48C2×C5×D29—211noHall
49C2×C32×C29C5×D711yescharacteristic, complemented
50C3×D87—72no—
51C3×C5×C29—21no—
52C3×C5×C29C2×C3×D711yescharacteristic, complemented
53C3×C5×C29C2×D2111yescharacteristic, complemented
54C2×C5×D21—291no—
55C2×C3×C5×D7—871no—
56C2×C7×C29C3×C5×D311yescharacteristic, complemented
57D203—32no—
58C2×D87—71no—
59C2×C3×D29—211no—
60C32×C5×C7C2×D2911yescharacteristic, complemented, Hall
61C2×C5×C29C3×D2111yescharacteristic, complemented
62C5×D29—212no—
63C32×C29C2×C5×D711yescharacteristic, complemented, Hall
64C2×C3×D21—291noHall
65C2×C3×C5×C7—21no—
66C2×C3×C5×C7C3×D2911yescharacteristic, complemented
67C2×C3×C5×C7D8711yescharacteristic, complemented
68C5×D21—292no—
69C3×C5×D7—872no—
70C7×C29C2×C3×C5×D311yescharacteristic, complemented, Hall
71C2×C3×C5×D3—2031noHall
72C2×C3×C29—21no—
73C2×C3×C29C3×C5×D711yescharacteristic, complemented
74C2×C3×C29C5×D2111yescharacteristic, complemented
75D87—72no—
76C3×D29—212no—
77C5×C29C2×C3×D2111yescharacteristic, complemented, Hall
78C2×C5×D7—871noHall
79C2×C32×C7C5×D2911yescharacteristic, complemented
80C3×D21—292no—
81C2×D29—211noHall
82C3×C5×C7—21no—
83C3×C5×C7C2×C3×D2911yescharacteristic, complemented
84C3×C5×C7C2×D8711yescharacteristic, complemented
85C2×C32×C5D20311yescharacteristic, complemented
86C3×C5×D3—2032no—
87C3×C29—21no—
88C3×C29C2×C3×C5×D711yescharacteristic, complemented
89C3×C29C2×C5×D2111yescharacteristic, complemented
90C2×D21—291no—
91C2×C3×D7—871no—
92C2×C5×C7C3×D8711yescharacteristic, complemented
93C5×D7—872no—
94C32×C7C2×C5×D2911yescharacteristic, complemented, Hall
95C22×C3×C5—6091no—
96C2×C5×D3—2031no—
97C2×C29C3×C5×D2111yescharacteristic, complemented
98D29—212no—
99C32×C5C2×D20311yescharacteristic, complemented, Hall
100C2×C3×C7—21no—
101C2×C3×C7C3×C5×D2911yescharacteristic, complemented
102C2×C3×C7C5×D8711yescharacteristic, complemented
103D21—292no—
104C3×D7—872no—
105C2×C3×D3—2031noHall
106C5×C7C2×C3×D8711yescharacteristic, complemented, Hall
107C2×C3×C5—21no—
108C2×C3×C5—6092no—
109C2×C3×C5C3×D20311yescharacteristic, complemented
110C2×C3×C5D60911yescharacteristic, direct product factor
111C5×D3—2032no—
112C29C2×C3×C5×D2111yescharacteristic, complemented, Sylow
113C2×D7—871noHall
114C3×C7—21no—
115C3×C7C2×C3×C5×D2911yescharacteristic, complemented
116C3×C7C2×C5×D8711yescharacteristic, complemented
117C22×C5—6091noHall
118C2×C32C5×D20311yescharacteristic, complemented
119C3×D3—2032no—
120C3×C5—21no—
121C3×C5C2×C3×D20311yescharacteristic, complemented
122C3×C5C2×D60911yescharacteristic, direct product factor
123C2×C7C3×C5×D8711yescharacteristic, complemented
124D7—872no—
125C22×C3—6091no—
126C2×D3—2031no—
127C2×C5—6092no—
128C2×C5C3×D60911yescharacteristic, direct product factor
129C32C2×C5×D20311yescharacteristic, complemented, Sylow
130C7C2×C3×C5×D8711yescharacteristic, complemented, Sylow
131C2×C3—21no—
132C2×C3—6092no—
133C2×C3C3×C5×D20311yescharacteristic, complemented
134C2×C3C5×D60911yescharacteristic, direct product factor
135D3—2032no—
136C5C2×C3×D60911yescharacteristic, direct product factor, Sylow
137C22—6091noSylow
138C3—21no—
139C3C2×C5×D60911yescharacteristic, direct product factor
140C3C2×C3×C5×D20311yescharacteristic, complemented
141C2—6092no—
142C2C3×C5×D60911yescharacteristic, direct product factor
subgroupG/H
center, hypercenterC2×C3×C5D609
commutator, cosocle, hypocenterC3×C7×C29C22×C3×C5
FrattiniC1G
Fitting, socleC2×C32×C5×C7×C29C2
LCSG ▷ C3×C7×C29
UCSC1 ◁ C2×C3×C5

Complements

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

NH ≅ G/Ntype
1. C3×C5×D609C2direct (N × H)
2. C2×C32×C5×C7×C29C2semidirect
4. C2×C5×D609C3direct (N × H)
5. C32×C5×C7×C29C22semidirect
6. C2×C3×D609C5direct (N × H)
7. C5×D609C2×C3direct (N × H)
10. C2×C3×C5×C7×C29D3semidirect
11. C2×C3×C5×C7×C29C2×C3semidirect
14. C3×D609C2×C5direct (N × H)
15. C2×C32×C7×C29C2×C5semidirect
17. C3×C5×C7×C29C2×D3semidirect
18. C3×C5×C7×C29C22×C3semidirect
20. C2×C32×C5×C29D7semidirect
21. C2×D609C3×C5direct (N × H)
24. C2×C5×C7×C29C3×D3semidirect
25. C32×C7×C29C22×C5semidirect
28. C32×C5×C29C2×D7semidirect
31. C2×C3×C7×C29C5×D3semidirect
32. C2×C3×C7×C29C2×C3×C5semidirect
33. D609C2×C3×C5direct (N × H)
36. C5×C7×C29C2×C3×D3semidirect
38. C2×C3×C5×C29C3×D7semidirect
39. C2×C3×C5×C29D21semidirect
43. C2×C32×C5×C7D29semidirect
46. C3×C7×C29C2×C5×D3semidirect
47. C3×C7×C29C22×C3×C5semidirect
49. C2×C32×C29C5×D7semidirect
52. C3×C5×C29C2×C3×D7semidirect
53. C3×C5×C29C2×D21semidirect
56. C2×C7×C29C3×C5×D3semidirect
60. C32×C5×C7C2×D29semidirect
61. C2×C5×C29C3×D21semidirect
63. C32×C29C2×C5×D7semidirect
66. C2×C3×C5×C7C3×D29semidirect
67. C2×C3×C5×C7D87semidirect
70. C7×C29C2×C3×C5×D3semidirect
73. C2×C3×C29C3×C5×D7semidirect
74. C2×C3×C29C5×D21semidirect
77. C5×C29C2×C3×D21semidirect
79. C2×C32×C7C5×D29semidirect
83. C3×C5×C7C2×C3×D29semidirect
84. C3×C5×C7C2×D87semidirect
85. C2×C32×C5D203semidirect
88. C3×C29C2×C3×C5×D7semidirect
89. C3×C29C2×C5×D21semidirect
92. C2×C5×C7C3×D87semidirect
94. C32×C7C2×C5×D29semidirect
97. C2×C29C3×C5×D21semidirect
99. C32×C5C2×D203semidirect
101. C2×C3×C7C3×C5×D29semidirect
102. C2×C3×C7C5×D87semidirect
106. C5×C7C2×C3×D87semidirect
109. C2×C3×C5C3×D203semidirect
110. C2×C3×C5D609direct (N × H)
112. C29C2×C3×C5×D21semidirect
115. C3×C7C2×C3×C5×D29semidirect
116. C3×C7C2×C5×D87semidirect
118. C2×C32C5×D203semidirect
121. C3×C5C2×C3×D203semidirect
122. C3×C5C2×D609direct (N × H)
123. C2×C7C3×C5×D87semidirect
128. C2×C5C3×D609direct (N × H)
129. C32C2×C5×D203semidirect
130. C7C2×C3×C5×D87semidirect
133. C2×C3C3×C5×D203semidirect
134. C2×C3C5×D609direct (N × H)
136. C5C2×C3×D609direct (N × H)
139. C3C2×C5×D609direct (N × H)
140. C3C2×C3×C5×D203semidirect
142. C2C3×C5×D609direct (N × H)

Maximal subgroups

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

subgroupindex
C3×C5×D6092
C2×C32×C5×C7×C292
C2×C3×C5×D2033
C2×C5×D6093
C2×C3×D6095
C2×C3×C5×D877
C2×C3×C5×D2129