Finite Group Databasebeta

← C2×C3×C7×D357 #44473017

Subgroups — C2×C3×C7×D357 #44473017

Order 29988.

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

normal subgroups only

This group has 248 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×C7×D357C212yesmaximal, direct product factor
2C2×C32×C72×C17C211yesmaximal, characteristic, complemented
3C2×C3×C7×D119—31nomaximal
4C2×C7×D357C311yesmaximal, characteristic, direct product factor
5C32×C72×C17C2211yescharacteristic, complemented, Hall
6C7×D357C2×C312yesdirect product factor
7C3×C7×D119—32no—
8C2×C3×C72×C17—21no—
9C2×C3×C72×C17D311yescharacteristic, complemented
10C2×C3×C72×C17C2×C311yescharacteristic, complemented
11C2×C3×C7×D51—71nomaximal
12C2×C3×D357C711yesmaximal, characteristic, direct product factor
13C2×C7×D119—31noHall
14C3×C72×C17—21no—
15C3×C72×C17C2×D311yescharacteristic, complemented
16C3×C72×C17C22×C311yescharacteristic, complemented
17C3×D357C2×C712yesdirect product factor
18C3×C7×D51—72no—
19C2×C32×C7×C17—23no—
20C2×C32×C7×C17D711yescharacteristic, complemented
21C2×C32×C7×C17C2×C711yescharacteristic, complemented
22C2×C3×C7×D21—171nomaximal, Hall
23C2×C72×C17C3×D311yescharacteristic, complemented
24C7×D119—32no—
25C2×D357C3×C711yescharacteristic, direct product factor
26C2×C7×D51—71no—
27C2×C3×D119—31no—
28C2×C3×C7×D17—211no—
29C32×C7×C17—23no—
30C32×C7×C17C2×D711yescharacteristic, complemented
31C32×C7×C17C22×C711yescharacteristic, complemented
32C2×C32×C72D1711yescharacteristic, complemented
33C3×C7×D21—172no—
34C72×C17C2×C3×D311yescharacteristic, complemented, Hall
35C2×C3×C7×C17—26no—
36C2×C3×C7×C17—21no—
37C2×C3×C7×C17—21no—
38C2×C3×C7×C17—23no—
39C2×C3×C7×C17—23no—
40C2×C3×C7×C17D3×C711yescharacteristic, complemented
41C2×C3×C7×C17C3×D711yescharacteristic, complemented
42C2×C3×C7×C17D2111yescharacteristic, complemented
43C2×C3×C7×C17C2×C3×C711yescharacteristic, complemented
44D357C2×C3×C712yesdirect product factor
45C7×D51—72no—
46C3×D119—32no—
47C3×C7×D17—212no—
48C2×C3×D51—71noHall
49C2×C7×D21—171no—
50C2×C3×C7×D7—511no—
51C2×D119—31no—
52C2×C7×D17—211no—
53C32×C72C2×D1711yescharacteristic, complemented, Hall
54C3×C7×C17—26no—
55C3×C7×C17—21no—
56C3×C7×C17—21no—
57C3×C7×C17—23no—
58C3×C7×C17—23no—
59C3×C7×C17C2×C3×D711yescharacteristic, complemented
60C3×C7×C17C2×D3×C711yescharacteristic, complemented
61C3×C7×C17C2×D2111yescharacteristic, complemented
62C3×C7×C17C22×C3×C711yescharacteristic, complemented
63C2×C32×C17C7×D711yescharacteristic, complemented
64C3×D51—72no—
65C2×C3×C72—21no—
66C2×C3×C72C3×D1711yescharacteristic, complemented
67C2×C3×C72D5111yescharacteristic, complemented
68C7×D21—172no—
69C3×C7×D7—512no—
70C2×C3×D21—171no—
71C2×C3×D3×C7—1191no—
72C2×C7×C17—23no—
73C2×C7×C17C3×D3×C711yescharacteristic, complemented
74C2×C7×C17C3×D2111yescharacteristic, complemented
75D119—32no—
76C7×D17—212no—
77C2×D51—71no—
78C2×C3×D17—211no—
79C2×C7×D7—511noHall
80C32×C17C2×C7×D711yescharacteristic, complemented, Hall
81C3×C72—21no—
82C3×C72C2×C3×D1711yescharacteristic, complemented
83C3×C72C2×D5111yescharacteristic, complemented
84C2×C32×C7—23no—
85C2×C32×C7C7×D1711yescharacteristic, complemented
86C2×C32×C7D11911yescharacteristic, complemented
87C3×D21—172no—
88C3×D3×C7—1192no—
89C7×C17—23no—
90C7×C17C2×C3×D3×C711yescharacteristic, complemented
91C7×C17C2×C3×D2111yescharacteristic, complemented
92C2×C3×C17—21no—
93C2×C3×C17C3×C7×D711yescharacteristic, complemented
94C2×C3×C17C7×D2111yescharacteristic, complemented
95D51—72no—
96C3×D17—212no—
97C2×C72C3×D5111yescharacteristic, complemented
98C7×D7—512no—
99C22×C3×C7—3571no—
100C2×D21—171no—
101C2×D3×C7—1191no—
102C2×C3×D7—511no—
103C2×D17—211noHall
104C32×C7—23no—
105C32×C7C2×C7×D1711yescharacteristic, complemented
106C32×C7C2×D11911yescharacteristic, complemented
107C3×C17—21no—
108C3×C17C2×C3×C7×D711yescharacteristic, complemented
109C3×C17C2×C7×D2111yescharacteristic, complemented
110C72C2×C3×D5111yescharacteristic, complemented, Sylow
111C2×C3×C7—26no—
112C2×C3×C7—21no—
113C2×C3×C7—21no—
114C2×C3×C7—23no—
115C2×C3×C7—23no—
116C2×C3×C7—3572no—
117C2×C3×C7C3×C7×D1711yescharacteristic, complemented
118C2×C3×C7C3×D11911yescharacteristic, complemented
119C2×C3×C7C7×D5111yescharacteristic, complemented
120C2×C3×C7D35711yescharacteristic, direct product factor
121D21—172no—
122C3×D7—512no—
123D3×C7—1192no—
124C2×C3×D3—1191noHall
125C2×C17C3×C7×D2111yescharacteristic, complemented
126D17—212no—
127C22×C7—3571no—
128C2×D7—511no—
129C3×C7—26no—
130C3×C7—21no—
131C3×C7—21no—
132C3×C7—23no—
133C3×C7—23no—
134C3×C7C2×C3×C7×D1711yescharacteristic, complemented
135C3×C7C2×C3×D11911yescharacteristic, complemented
136C3×C7C2×C7×D5111yescharacteristic, complemented
137C3×C7C2×D35711yescharacteristic, direct product factor
138C2×C32C7×D11911yescharacteristic, complemented
139C3×D3—1192no—
140C17C2×C3×C7×D2111yescharacteristic, complemented, Sylow
141C2×C7—23no—
142C2×C7—3572no—
143C2×C7C3×C7×D5111yescharacteristic, complemented
144C2×C7C3×D35711yescharacteristic, direct product factor
145D7—512no—
146C22×C3—3571no—
147C2×D3—1191no—
148C32C2×C7×D11911yescharacteristic, complemented, Sylow
149C7—23no—
150C7C2×C3×D35711yescharacteristic, direct product factor
151C7C2×C3×C7×D5111yescharacteristic, complemented
152C2×C3—21no—
153C2×C3—3572no—
154C2×C3C3×C7×D11911yescharacteristic, complemented
155C2×C3C7×D35711yescharacteristic, direct product factor
156D3—1192no—
157C22—3571noSylow
158C3—21no—
159C3C2×C7×D35711yescharacteristic, direct product factor
160C3C2×C3×C7×D11911yescharacteristic, complemented
161C2—3572no—
162C2C3×C7×D35711yescharacteristic, direct product factor
subgroupG/H
center, hypercenterC2×C3×C7D357
commutator, cosocle, hypocenterC3×C7×C17C22×C3×C7
FrattiniC1G
Fitting, socleC2×C32×C72×C17C2
LCSG ▷ C3×C7×C17
UCSC1 ◁ C2×C3×C7

Complements

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

NH ≅ G/Ntype
1. C3×C7×D357C2direct (N × H)
2. C2×C32×C72×C17C2semidirect
4. C2×C7×D357C3direct (N × H)
5. C32×C72×C17C22semidirect
6. C7×D357C2×C3direct (N × H)
9. C2×C3×C72×C17D3semidirect
10. C2×C3×C72×C17C2×C3semidirect
12. C2×C3×D357C7direct (N × H)
15. C3×C72×C17C2×D3semidirect
16. C3×C72×C17C22×C3semidirect
17. C3×D357C2×C7direct (N × H)
20. C2×C32×C7×C17D7semidirect
21. C2×C32×C7×C17C2×C7semidirect
23. C2×C72×C17C3×D3semidirect
25. C2×D357C3×C7direct (N × H)
30. C32×C7×C17C2×D7semidirect
31. C32×C7×C17C22×C7semidirect
32. C2×C32×C72D17semidirect
34. C72×C17C2×C3×D3semidirect
40. C2×C3×C7×C17D3×C7semidirect
41. C2×C3×C7×C17C3×D7semidirect
42. C2×C3×C7×C17D21semidirect
43. C2×C3×C7×C17C2×C3×C7semidirect
44. D357C2×C3×C7direct (N × H)
53. C32×C72C2×D17semidirect
59. C3×C7×C17C2×C3×D7semidirect
60. C3×C7×C17C2×D3×C7semidirect
61. C3×C7×C17C2×D21semidirect
62. C3×C7×C17C22×C3×C7semidirect
63. C2×C32×C17C7×D7semidirect
66. C2×C3×C72C3×D17semidirect
67. C2×C3×C72D51semidirect
73. C2×C7×C17C3×D3×C7semidirect
74. C2×C7×C17C3×D21semidirect
80. C32×C17C2×C7×D7semidirect
82. C3×C72C2×C3×D17semidirect
83. C3×C72C2×D51semidirect
85. C2×C32×C7C7×D17semidirect
86. C2×C32×C7D119semidirect
90. C7×C17C2×C3×D3×C7semidirect
91. C7×C17C2×C3×D21semidirect
93. C2×C3×C17C3×C7×D7semidirect
94. C2×C3×C17C7×D21semidirect
97. C2×C72C3×D51semidirect
105. C32×C7C2×C7×D17semidirect
106. C32×C7C2×D119semidirect
108. C3×C17C2×C3×C7×D7semidirect
109. C3×C17C2×C7×D21semidirect
110. C72C2×C3×D51semidirect
117. C2×C3×C7C3×C7×D17semidirect
118. C2×C3×C7C3×D119semidirect
119. C2×C3×C7C7×D51semidirect
120. C2×C3×C7D357direct (N × H)
125. C2×C17C3×C7×D21semidirect
134. C3×C7C2×C3×C7×D17semidirect
135. C3×C7C2×C3×D119semidirect
136. C3×C7C2×C7×D51semidirect
137. C3×C7C2×D357direct (N × H)
138. C2×C32C7×D119semidirect
140. C17C2×C3×C7×D21semidirect
143. C2×C7C3×C7×D51semidirect
144. C2×C7C3×D357direct (N × H)
148. C32C2×C7×D119semidirect
150. C7C2×C3×D357direct (N × H)
151. C7C2×C3×C7×D51semidirect
154. C2×C3C3×C7×D119semidirect
155. C2×C3C7×D357direct (N × H)
159. C3C2×C7×D357direct (N × H)
160. C3C2×C3×C7×D119semidirect
162. C2C3×C7×D357direct (N × H)

Maximal subgroups

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

subgroupindex
C3×C7×D3572
C2×C32×C72×C172
C2×C3×C7×D1193
C2×C7×D3573
C2×C3×C7×D517
C2×C3×D3577
C2×C3×C7×D2117