Finite Group Databasebeta

← C3×C5×C3⋊(C5⋊(C7⋊D7)) #36348073

Subgroups — C3×C5×C3⋊(C5⋊(C7⋊D7)) #36348073

Order 22050.

480 conjugacy classes of subgroups; only the 164 normal subgroups are shown.

normal subgroups only

This group has 480 conjugacy classes of subgroups — too many to draw. Only its 164 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
1C32×C52×C72C211yesmaximal, characteristic, complemented, Hall
2C5×C3⋊(C5⋊(C7⋊D7))C311yesmaximal, characteristic, direct product factor
3C3×C5×C5⋊(C7⋊D7)—31nomaximal
4C3×C3⋊(C5⋊(C7⋊D7))C511yesmaximal, characteristic, direct product factor
5C3×C5×C3⋊(C7⋊D7)—51nomaximal
6C3×C52×C72—21no—
7C3×C52×C72D311yescharacteristic, complemented
8C3×C52×C72C2×C311yescharacteristic, complemented
9C3×C5×D105—78nomaximal
10C5×C5⋊(C7⋊D7)—31noHall
11C32×C5×C72—22no—
12C32×C5×C72D511yescharacteristic, complemented
13C32×C5×C72C2×C511yescharacteristic, complemented
14C32×C52×C7D718yescomplemented
15C3⋊(C5⋊(C7⋊D7))C3×C511yescharacteristic, direct product factor
16C5×C3⋊(C7⋊D7)—51no—
17C3×C5⋊(C7⋊D7)—31no—
18C3×C5×C7⋊D7—151no—
19C52×C72C3×D311yescharacteristic, complemented, Hall
20C5×D105—78no—
21C3×C5×D35—218no—
22C3×C3⋊(C7⋊D7)—51noHall
23C3×C5×C72—24no—
24C3×C5×C72—22no—
25C3×C5×C72—22no—
26C3×C5×C72—21no—
27C3×C5×C72—21no—
28C3×C5×C72C5×D311yescharacteristic, complemented
29C3×C5×C72C3×D511yescharacteristic, complemented
30C3×C5×C72D1511yescharacteristic, complemented
31C3×C5×C72C2×C3×C511yescharacteristic, complemented
32C3×D105—78no—
33C3×C5×D21—358no—
34C3×C52×C7—28no—
35C3×C52×C7C3×D718yescomplemented
36C3×C52×C7D2118yescomplemented
37C5⋊(C7⋊D7)—31no—
38C5×C7⋊D7—151no—
39C3×C5×D15—491noHall
40C32×C72C5×D511yescharacteristic, complemented, Hall
41C5×D35—218no—
42C32×C5×C7—216no—
43C32×C5×C7C5×D718yescomplemented
44C32×C5×C7D3518yescomplemented
45C3⋊(C7⋊D7)—51no—
46C3×C7⋊D7—151no—
47C5×C72—22no—
48C5×C72C3×C5×D311yescharacteristic, complemented
49C5×C72C3×D1511yescharacteristic, complemented
50C32×C52C7⋊D711yescharacteristic, complemented, Hall
51D105—78no—
52C5×D21—358no—
53C3×D35—218no—
54C3×C5×D7—1058no—
55C52×C7C3×D2118yescomplemented
56C5×D15—491no—
57C3×C5×D5—1471no—
58C3×C72—21no—
59C3×C72C3×C5×D511yescharacteristic, complemented
60C3×C72C5×D1511yescharacteristic, complemented
61C3×D21—358no—
62C3×C5×C7—232no—
63C3×C5×C7—28no—
64C3×C5×C7—28no—
65C3×C5×C7—216no—
66C3×C5×C7—216no—
67C3×C5×C7C3×C5×D718yescomplemented
68C3×C5×C7C3×D3518yescomplemented
69C3×C5×C7C5×D2118yescomplemented
70C3×C5×C7D10518yescomplemented
71C7⋊D7—151noHall
72C3×D15—491no—
73C3×C5×D3—2451no—
74C3×C52—21no—
75C3×C52C3×C7⋊D711yescharacteristic, complemented
76C3×C52C3⋊(C7⋊D7)11yescharacteristic, complemented
77D35—218no—
78C5×D7—1058no—
79C32×C7C5×D3518yescomplemented
80C5×D5—1471noHall
81C72C3×C5×D1511yescharacteristic, complemented, Sylow
82C32×C5—22no—
83C32×C5C5×C7⋊D711yescharacteristic, complemented
84C32×C5C5⋊(C7⋊D7)11yescharacteristic, complemented
85D21—358no—
86C3×D7—1058no—
87C5×C7—216no—
88C5×C7C3×C5×D2118yescomplemented
89C5×C7C3×D10518yescomplemented
90C2×C3×C5—7351no—
91D15—491no—
92C3×D5—1471no—
93C5×D3—2451no—
94C52C3×C3⋊(C7⋊D7)11yescharacteristic, complemented, Sylow
95C3×C7—28no—
96C3×C7C3×C5×D3518yescomplemented
97C3×C7C5×D10518yescomplemented
98C3×D3—2451noHall
99C3×C5—24no—
100C3×C5—21no—
101C3×C5—21no—
102C3×C5—22no—
103C3×C5—22no—
104C3×C5C3×C5×C7⋊D711yescharacteristic, complemented
105C3×C5C3×C5⋊(C7⋊D7)11yescharacteristic, complemented
106C3×C5C5×C3⋊(C7⋊D7)11yescharacteristic, complemented
107C3×C5C3⋊(C5⋊(C7⋊D7))11yescharacteristic, direct product factor
108D7—1058no—
109C2×C5—7351no—
110D5—1471no—
111C32C5×C5⋊(C7⋊D7)11yescharacteristic, complemented, Sylow
112C7C3×C5×D10518yescomplemented
113C2×C3—7351no—
114D3—2451no—
115C5—22no—
116C5C3×C5×C3⋊(C7⋊D7)11yescharacteristic, complemented
117C5C3×C3⋊(C5⋊(C7⋊D7))11yescharacteristic, direct product factor
118C3—21no—
119C3C3×C5×C5⋊(C7⋊D7)11yescharacteristic, complemented
120C3C5×C3⋊(C5⋊(C7⋊D7))11yescharacteristic, direct product factor
121C2—7351noSylow
subgroupG/H
center, hypercenterC3×C5C3⋊(C5⋊(C7⋊D7))
commutator, cosocle, hypocenterC3×C5×C72C2×C3×C5
FrattiniC1G
Fitting, socleC32×C52×C72C2
LCSG ▷ C3×C5×C72
UCSC1 ◁ C3×C5

Complements

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

NH ≅ G/Ntype
1. C32×C52×C72C2semidirect
2. C5×C3⋊(C5⋊(C7⋊D7))C3direct (N × H)
4. C3×C3⋊(C5⋊(C7⋊D7))C5direct (N × H)
7. C3×C52×C72D3semidirect
8. C3×C52×C72C2×C3semidirect
12. C32×C5×C72D5semidirect
13. C32×C5×C72C2×C5semidirect
14. C32×C52×C7D7semidirect
15. C3⋊(C5⋊(C7⋊D7))C3×C5direct (N × H)
19. C52×C72C3×D3semidirect
28. C3×C5×C72C5×D3semidirect
29. C3×C5×C72C3×D5semidirect
30. C3×C5×C72D15semidirect
31. C3×C5×C72C2×C3×C5semidirect
35. C3×C52×C7C3×D7semidirect
36. C3×C52×C7D21semidirect
40. C32×C72C5×D5semidirect
43. C32×C5×C7C5×D7semidirect
44. C32×C5×C7D35semidirect
48. C5×C72C3×C5×D3semidirect
49. C5×C72C3×D15semidirect
50. C32×C52C7⋊D7semidirect
55. C52×C7C3×D21semidirect
59. C3×C72C3×C5×D5semidirect
60. C3×C72C5×D15semidirect
67. C3×C5×C7C3×C5×D7semidirect
68. C3×C5×C7C3×D35semidirect
69. C3×C5×C7C5×D21semidirect
70. C3×C5×C7D105semidirect
75. C3×C52C3×C7⋊D7semidirect
76. C3×C52C3⋊(C7⋊D7)semidirect
79. C32×C7C5×D35semidirect
81. C72C3×C5×D15semidirect
83. C32×C5C5×C7⋊D7semidirect
84. C32×C5C5⋊(C7⋊D7)semidirect
88. C5×C7C3×C5×D21semidirect
89. C5×C7C3×D105semidirect
94. C52C3×C3⋊(C7⋊D7)semidirect
96. C3×C7C3×C5×D35semidirect
97. C3×C7C5×D105semidirect
104. C3×C5C3×C5×C7⋊D7semidirect
105. C3×C5C3×C5⋊(C7⋊D7)semidirect
106. C3×C5C5×C3⋊(C7⋊D7)semidirect
107. C3×C5C3⋊(C5⋊(C7⋊D7))direct (N × H)
111. C32C5×C5⋊(C7⋊D7)semidirect
112. C7C3×C5×D105semidirect
116. C5C3×C5×C3⋊(C7⋊D7)semidirect
117. C5C3×C3⋊(C5⋊(C7⋊D7))direct (N × H)
119. C3C3×C5×C5⋊(C7⋊D7)semidirect
120. C3C5×C3⋊(C5⋊(C7⋊D7))direct (N × H)

Maximal subgroups

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

subgroupindex
C32×C52×C722
C5×C3⋊(C5⋊(C7⋊D7))3
C3×C5×C5⋊(C7⋊D7)3
C3×C3⋊(C5⋊(C7⋊D7))5
C3×C5×C3⋊(C7⋊D7)5
C3×C5×D1057