Finite Group Databasebeta

← C3×C7×#932 #72245832

Subgroups — C3×C7×#932 #72245832

Order 2016.

326 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 326 conjugacy classes of subgroups — too many to draw. Only its 116 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
1C2×C32×C7×D4C211yesmaximal, characteristic, complemented
2C3×C7×#235C211yesmaximal, characteristic, complemented
3C3×C7×#214C212yesmaximal, complemented
4C3×C7×#213C212yesmaximal, complemented
5C3×C7×#208C211yesmaximal, characteristic, complemented
6C3×C7×AGL(1,ℤ/8)—31nomaximal
7C7×#932C311yesmaximal, characteristic, direct product factor
8C23×C32×C7—21no—
9C32×C7×D4—21no—
10C32×C7×D4C2212yes—
11C2×C32×C4×C7C2211yescharacteristic
12C3×C7×#67—21no—
13C3×C7×D12C2211yescharacteristic, complemented
14C3×C4×D3×C7—21no—
15C3×C7×Dic6C2211yescharacteristic, complemented
16C3×C7×C3⋊C8C2212yescomplemented
17C3×C7×#41—31no—
18C2×C3×C7×D4—21no—
19C2×C3×C7×D4D311yescharacteristic, complemented
20C2×C3×C7×D4C2×C311yescharacteristic, complemented
21C7×#235C2×C311yescharacteristic, complemented
22C3×C7×SD16—32no—
23C3×C7×D8—32no—
24C3×C7×M16—31no—
25C7×#214C2×C312yescomplemented
26C7×#213C2×C312yescomplemented
27C7×#208C2×C311yescharacteristic, complemented
28C3×#932C711yesmaximal, characteristic, direct product factor, Hall
29C22×C32×C7—22no—
30C22×C32×C7—42no—
31C22×C32×C7D411yescharacteristic
32C2×C3×D3×C7—21no—
33C32×C4×C7D411yescharacteristic
34C32×C4×C7C2311yescharacteristic
35C3×C7×Dic3—21no—
36C7×AGL(1,ℤ/8)—31noHall
37C23×C3×C7—21no—
38C23×C3×C7—21no—
39C23×C3×C7—21no—
40C23×C3×C7—21no—
41C3×C7×Q8—31no—
42C3×C7×D4—21no—
43C3×C7×D4—21no—
44C3×C7×D4—21no—
45C3×C7×D4—21no—
46C3×C7×D4—22no—
47C3×C7×D4—31no—
48C3×C7×D4—61no—
49C3×C7×D4C2×D312yes—
50C3×C7×D4C22×C312yes—
51C2×C3×C4×C7—21no—
52C2×C3×C4×C7—61no—
53C2×C3×C4×C7C2×D311yescharacteristic
54C2×C3×C4×C7C22×C311yescharacteristic
55C7×#67—21no—
56C7×D12C22×C311yescharacteristic, complemented
57C4×D3×C7—21no—
58C7×Dic6C22×C311yescharacteristic, complemented
59C3×C7×C8—32no—
60C7×C3⋊C8C22×C312yescomplemented
61C2×C32×D4C2×C711yescharacteristic, complemented
62C3×#235C2×C711yescharacteristic, complemented
63C3×#214C2×C712yescomplemented
64C3×#213C2×C712yescomplemented
65C3×#208C2×C711yescharacteristic, complemented
66C2×C32×C7—21no—
67C2×C32×C7—42no—
68C2×C32×C7C2×D411yescharacteristic
69C3×D3×C7—41no—
70C7×#41—31no—
71C2×C7×D4C3×D311yescharacteristic, complemented
72C7×SD16—32no—
73C7×D8—32no—
74C7×M16—31no—
75C3×AGL(1,ℤ/8)—31no—
76#932C3×C711yescharacteristic, direct product factor
77C22×C3×C7—22no—
78C22×C3×C7—22no—
79C22×C3×C7—22no—
80C22×C3×C7—22no—
81C22×C3×C7—21no—
82C22×C3×C7—42no—
83C22×C3×C7—42no—
84C22×C3×C7—42no—
85C22×C3×C7—42no—
86C22×C3×C7—61no—
87C22×C3×C7#6711yescharacteristic
88C22×C3×C7C3×D411yescharacteristic
89C2×D3×C7—21no—
90C3×C4×C7—21no—
91C3×C4×C7—21no—
92C3×C4×C7—61no—
93C3×C4×C7#6711yescharacteristic
94C3×C4×C7C3×D411yescharacteristic
95C3×C4×C7C22×D311yescharacteristic
96C3×C4×C7C23×C311yescharacteristic
97C7×Dic3—21no—
98C23×C32—21no—
99C32×D4—21no—
100C32×D4C22×C712yes—
101C2×C32×C4C22×C711yescharacteristic
102C3×#67—21no—
103C3×D12C22×C711yescharacteristic, complemented
104C3×C4×D3—21no—
105C3×Dic6C22×C711yescharacteristic, complemented
106C3×C3⋊C8C22×C712yescomplemented
107C32×C7AGL(1,ℤ/8)11yescharacteristic, complemented, Hall
108C23×C7—21no—
109C7×Q8—31no—
110C7×D4—21no—
111C7×D4—31no—
112C7×D4—61no—
113C7×D4C2×C3×D312yes—
114C2×C4×C7—61no—
115C2×C4×C7C2×C3×D311yescharacteristic
116C7×C8—32no—
117C3×#41—31no—
118C2×C3×D4—21no—
119C2×C3×D4D3×C711yescharacteristic, complemented
120C2×C3×D4C2×C3×C711yescharacteristic, complemented
121#235C2×C3×C711yescharacteristic, complemented
122C3×SD16—32no—
123C3×D8—32no—
124C3×M16—31no—
125#214C2×C3×C712yescomplemented
126#213C2×C3×C712yescomplemented
127#208C2×C3×C711yescharacteristic, complemented
128C2×C3×C7—21no—
129C2×C3×C7—21no—
130C2×C3×C7—21no—
131C2×C3×C7—21no—
132C2×C3×C7—21no—
133C2×C3×C7—42no—
134C2×C3×C7—42no—
135C2×C3×C7—42no—
136C2×C3×C7—42no—
137C2×C3×C7—121no—
138C2×C3×C7C2×#6711yescharacteristic
139C2×C3×C7C2×C3×D411yescharacteristic
140D3×C7—41no—
141C22×C32—22no—
142C22×C32—42no—
143C22×C32C7×D411yescharacteristic
144C2×C3×D3—21no—
145C32×C4C7×D411yescharacteristic
146C32×C4C23×C711yescharacteristic
147C3×Dic3—21no—
148AGL(1,ℤ/8)—31noSylow
149C22×C7—22no—
150C22×C7—42no—
151C22×C7—61no—
152C22×C7C3×#6711yescharacteristic
153C4×C7—61no—
154C4×C7C3×#6711yescharacteristic
155C4×C7C22×C3×D311yescharacteristic
156C23×C3—21no—
157C23×C3—21no—
158C23×C3—21no—
159C23×C3—21no—
160C3×Q8—31no—
161C3×D4—21no—
162C3×D4—21no—
163C3×D4—21no—
164C3×D4—21no—
165C3×D4—22no—
166C3×D4—31no—
167C3×D4—61no—
168C3×D4C2×D3×C712yes—
169C3×D4C22×C3×C712yes—
170C2×C3×C4—21no—
171C2×C3×C4—61no—
172C2×C3×C4C2×D3×C711yescharacteristic
173C2×C3×C4C22×C3×C711yescharacteristic
174#67—21no—
175D12C22×C3×C711yescharacteristic, complemented
176C4×D3—21no—
177Dic6C22×C3×C711yescharacteristic, complemented
178C3×C8—32no—
179C3⋊C8C22×C3×C712yescomplemented
180C3×C7—21no—
181C3×C7#93211yescharacteristic, direct product factor
182C3×C7C3×AGL(1,ℤ/8)11yescharacteristic, complemented
183C2×C32—21no—
184C2×C32—42no—
185C2×C32C2×C7×D411yescharacteristic
186C3×D3—41no—
187#41—31no—
188C2×D4C3×D3×C711yescharacteristic, complemented
189SD16—32no—
190D8—32no—
191M16—31no—
192C2×C7—21no—
193C2×C7—42no—
194C2×C7—121no—
195C2×C7C2×C3×#6711yescharacteristic
196C22×C3—22no—
197C22×C3—22no—
198C22×C3—22no—
199C22×C3—22no—
200C22×C3—21no—
201C22×C3—42no—
202C22×C3—42no—
203C22×C3—42no—
204C22×C3—42no—
205C22×C3—61no—
206C22×C3C7×#6711yescharacteristic
207C22×C3C3×C7×D411yescharacteristic
208C2×D3—21no—
209C3×C4—21no—
210C3×C4—21no—
211C3×C4—61no—
212C3×C4C7×#6711yescharacteristic
213C3×C4C3×C7×D411yescharacteristic
214C3×C4C22×D3×C711yescharacteristic
215C3×C4C23×C3×C711yescharacteristic
216Dic3—21no—
217C32C7×AGL(1,ℤ/8)11yescharacteristic, complemented, Sylow
218C23—21no—
219Q8—31no—
220D4—21no—
221D4—31no—
222D4—61no—
223D4C2×C3×D3×C712yes—
224C2×C4—61no—
225C2×C4C2×C3×D3×C711yescharacteristic
226C8—32no—
227C7C3×#93211yescharacteristic, direct product factor, Sylow
228C2×C3—21no—
229C2×C3—21no—
230C2×C3—21no—
231C2×C3—21no—
232C2×C3—21no—
233C2×C3—42no—
234C2×C3—42no—
235C2×C3—42no—
236C2×C3—42no—
237C2×C3—121no—
238C2×C3C2×C7×#6711yescharacteristic
239C2×C3C2×C3×C7×D411yescharacteristic
240D3—41no—
241C22—22no—
242C22—42no—
243C22—61no—
244C22C3×C7×#6711yescharacteristic
245C4—61no—
246C4C3×C7×#6711yescharacteristic
247C4C22×C3×D3×C711yescharacteristic
248C3—21no—
249C3C7×#93211yescharacteristic, direct product factor
250C3C3×C7×AGL(1,ℤ/8)11yescharacteristic, complemented
251C2—21no—
252C2—42no—
253C2—121no—
254C2C2×C3×C7×#6711yescharacteristic
subgroupG/H
centerC2×C3×C7C2×#67
commutator, cosocleC3×C4C23×C3×C7
FrattiniC4C22×C3×D3×C7
FittingC2×C32×C7×D4C2
socleC2×C32×C7C2×D4
hypercenterC2×C3×C7×D4D3
hypocenterC3C3×C7×AGL(1,ℤ/8)
LCSG ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C3×C7 ◁ C2×C3×C4×C7 ◁ C2×C3×C7×D4

Complements

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

NH ≅ G/Ntype
1. C2×C32×C7×D4C2semidirect
2. C3×C7×#235C2semidirect
3. C3×C7×#214C2semidirect
4. C3×C7×#213C2semidirect
5. C3×C7×#208C2semidirect
7. C7×#932C3direct (N × H)
13. C3×C7×D12C22semidirect
15. C3×C7×Dic6C22semidirect
16. C3×C7×C3⋊C8C22semidirect
19. C2×C3×C7×D4D3semidirect
20. C2×C3×C7×D4C2×C3semidirect
21. C7×#235C2×C3semidirect
25. C7×#214C2×C3semidirect
26. C7×#213C2×C3semidirect
27. C7×#208C2×C3semidirect
28. C3×#932C7direct (N × H)
56. C7×D12C22×C3semidirect
58. C7×Dic6C22×C3semidirect
60. C7×C3⋊C8C22×C3semidirect
61. C2×C32×D4C2×C7semidirect
62. C3×#235C2×C7semidirect
63. C3×#214C2×C7semidirect
64. C3×#213C2×C7semidirect
65. C3×#208C2×C7semidirect
71. C2×C7×D4C3×D3semidirect
76. #932C3×C7direct (N × H)
103. C3×D12C22×C7semidirect
105. C3×Dic6C22×C7semidirect
106. C3×C3⋊C8C22×C7semidirect
107. C32×C7AGL(1,ℤ/8)semidirect
119. C2×C3×D4D3×C7semidirect
120. C2×C3×D4C2×C3×C7semidirect
121. #235C2×C3×C7semidirect
125. #214C2×C3×C7semidirect
126. #213C2×C3×C7semidirect
127. #208C2×C3×C7semidirect
175. D12C22×C3×C7semidirect
177. Dic6C22×C3×C7semidirect
179. C3⋊C8C22×C3×C7semidirect
181. C3×C7#932direct (N × H)
182. C3×C7C3×AGL(1,ℤ/8)semidirect
188. C2×D4C3×D3×C7semidirect
217. C32C7×AGL(1,ℤ/8)semidirect
227. C7C3×#932direct (N × H)
249. C3C7×#932direct (N × H)
250. C3C3×C7×AGL(1,ℤ/8)semidirect

Maximal subgroups

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

subgroupindex
C2×C32×C7×D42
C3×C7×#2352
C3×C7×#2142
C3×C7×#2132
C3×C7×#2082
C3×C7×AGL(1,ℤ/8)3
C7×#9323
C3×#9327