Finite Group Databasebeta

← C7×#63923 #72246834

Subgroups — C7×#63923 #72246834

Order 2016.

574 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 574 conjugacy classes of subgroups — too many to draw. Only its 108 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
1C22×D32×C7C211yesmaximal, characteristic, complemented
2C2×C7×C4⋊(C3⋊D3)C211yesmaximal, characteristic, complemented
3C2×C7×#626C212yesmaximal, complemented
4C3×C7×(C2×D3)⋊C4C212yesmaximal, complemented
5C7×#3722C211yesmaximal, characteristic, complemented
6C7×#882—32nomaximal
7C22×C7×C3⋊D3C2211yescharacteristic
8C22×C7×C3⋊D3C2211yescharacteristic, complemented
9C22×C3×D3×C7C2212yes—
10C2×D32×C7—22no—
11C2×D32×C7—24no—
12C2×C32×C4×C7C2211yescharacteristic, complemented
13C7×C4⋊(C3⋊D3)—22no—
14C2×C3×C7×Dic3C2212yescomplemented
15C7×#626—24no—
16C23×D3×C7—32no—
17C2×C7×#67—32no—
18C2×C7×D12—32no—
19C2×C7×D12—32no—
20C2×C7×D12—61no—
21C3×C7×C22⋊C4—32no—
22C7×(C2×D3)⋊C4—32no—
23C7×(C2×D3)⋊C4D312yescomplemented
24#63923C711yesmaximal, characteristic, direct product factor, Hall
25C22×C32×C7C2311yescharacteristic
26C2×C7×C3⋊D3—22no—
27C2×C7×C3⋊D3—21no—
28C2×C7×C3⋊D3—22no—
29C2×C7×C3⋊D3D412yescomplemented
30C2×C3×D3×C7—22no—
31C2×C3×D3×C7—22no—
32C2×C3×D3×C7D414yescomplemented
33D32×C7—28no—
34C32×C4×C7—21no—
35C3×C7×Dic3—22no—
36C7×#120—91noHall
37C23×C3×C7—32no—
38C22×D3×C7—32no—
39C22×D3×C7—32no—
40C22×D3×C7—61no—
41C22×D3×C7—61no—
42C22×D3×C7—64no—
43C22×D3×C7—64no—
44C22×D3×C7—64no—
45C22×D3×C7C2×D312yes—
46C2×C3×C4×C7—21no—
47C2×C3×C4×C7—32no—
48C2×C3×C4×C7C2×D312yescomplemented
49C7×#67—64no—
50C2×C7×Dic3C2×D312yescomplemented
51C7×D12—64no—
52C7×D12—62no—
53C7×D12—62no—
54C7×D12—64no—
55C22×D32C2×C711yescharacteristic, complemented
56C2×C4⋊(C3⋊D3)C2×C711yescharacteristic, complemented
57C2×#626C2×C712yescomplemented
58C3×(C2×D3)⋊C4C2×C712yescomplemented
59#3722C2×C711yescharacteristic, complemented
60C2×C32×C7C2×D412yes—
61C2×C32×C7C2×D411yescharacteristic
62C7×C3⋊D3—22no—
63C7×C3⋊D3—41no—
64C3×D3×C7—24no—
65C24×C7—91no—
66C2×C7×D4—92no—
67C2×C7×D4—91no—
68C7×C22⋊C4—32no—
69C7×C22⋊C4—91no—
70#882—32no—
71C22×C3×C7—21no—
72C22×C3×C7—34no—
73C22×C3×C7—62no—
74C22×C3×C7—62no—
75C22×C3×C7C22×D312yes—
76C2×D3×C7—22no—
77C2×D3×C7—22no—
78C2×D3×C7—34no—
79C2×D3×C7—62no—
80C2×D3×C7—616no—
81C2×D3×C7—62no—
82C2×D3×C7—62no—
83C2×D3×C7—61no—
84C2×D3×C7—61no—
85C2×D3×C7—62no—
86C2×D3×C7—62no—
87C2×D3×C7—62no—
88C2×D3×C7—122no—
89C2×D3×C7—122no—
90C2×D3×C7D1214yescomplemented
91C3×C4×C7—21no—
92C3×C4×C7—21no—
93C3×C4×C7—22no—
94C3×C4×C7—62no—
95C7×Dic3—22no—
96C22×C3⋊D3C22×C711yescharacteristic
97C22×C3⋊D3C22×C711yescharacteristic, complemented
98C22×C3×D3C22×C712yes—
99C2×D32—22no—
100C2×D32—24no—
101C2×C32×C4C22×C711yescharacteristic, complemented
102C4⋊(C3⋊D3)—22no—
103C2×C3×Dic3C22×C712yescomplemented
104#626—24no—
105C32×C7#12011yescharacteristic, complemented, Hall
106C23×C7—32no—
107C23×C7—91no—
108C23×C7—91no—
109C23×C7—184no—
110C23×C7—182no—
111C7×D4—182no—
112C7×D4—184no—
113C2×C4×C7—32no—
114C2×C4×C7D3211yescharacteristic, complemented
115C23×D3—32no—
116C2×#67—32no—
117C2×D12—32no—
118C2×D12—32no—
119C2×D12—61no—
120C3×C22⋊C4—32no—
121(C2×D3)⋊C4—32no—
122(C2×D3)⋊C4D3×C712yescomplemented
123C2×C3×C7—22no—
124C2×C3×C7—21no—
125C2×C3×C7—64no—
126C2×C3×C7C2×D1212yes—
127C2×C3×C7D3×D412yes—
128C2×C3×C7D3×D412yes—
129D3×C7—24no—
130D3×C7—64no—
131D3×C7—122no—
132D3×C7—121no—
133D3×C7—121no—
134D3×C7—122no—
135C22×C32C23×C711yescharacteristic
136C2×C3⋊D3—21no—
137C2×C3⋊D3—22no—
138C2×C3⋊D3—22no—
139C2×C3⋊D3C7×D412yescomplemented
140C2×C3×D3—22no—
141C2×C3×D3—22no—
142C2×C3×D3C7×D414yescomplemented
143D32—28no—
144C32×C4—21no—
145C3×Dic3—22no—
146#120—91noSylow
147C22×C7—34no—
148C22×C7—62no—
149C22×C7—62no—
150C22×C7—92no—
151C22×C7—188no—
152C22×C7—182no—
153C22×C7—182no—
154C22×C7—181no—
155C22×C7C2×D3211yescharacteristic
156C4×C7—21no—
157C4×C7—62no—
158C23×C3—32no—
159C22×D3—32no—
160C22×D3—32no—
161C22×D3—61no—
162C22×D3—61no—
163C22×D3—64no—
164C22×D3—64no—
165C22×D3—64no—
166C22×D3C2×D3×C712yes—
167C2×C3×C4—21no—
168C2×C3×C4—32no—
169C2×C3×C4C2×D3×C712yescomplemented
170#67—64no—
171C2×Dic3C2×D3×C712yescomplemented
172D12—64no—
173D12—62no—
174D12—62no—
175D12—64no—
176C3×C7—21no—
177C3×C7#88212yescomplemented
178C2×C32C2×C7×D412yes—
179C2×C32C2×C7×D411yescharacteristic
180C3⋊D3—22no—
181C3⋊D3—41no—
182C3×D3—24no—
183C24—91no—
184C2×D4—92no—
185C2×D4—91no—
186C22⋊C4—32no—
187C22⋊C4—91no—
188C2×C7—64no—
189C2×C7—182no—
190C2×C7—361no—
191C2×C7D3×D1212yes—
192C2×C7#381111yescharacteristic
193C22×C3—21no—
194C22×C3—34no—
195C22×C3—62no—
196C22×C3—62no—
197C22×C3C22×D3×C712yes—
198C2×D3—22no—
199C2×D3—22no—
200C2×D3—34no—
201C2×D3—62no—
202C2×D3—616no—
203C2×D3—62no—
204C2×D3—62no—
205C2×D3—61no—
206C2×D3—61no—
207C2×D3—62no—
208C2×D3—62no—
209C2×D3—62no—
210C2×D3—122no—
211C2×D3—122no—
212C2×D3C7×D1214yescomplemented
213C3×C4—21no—
214C3×C4—21no—
215C3×C4—22no—
216C3×C4—62no—
217Dic3—22no—
218C32C7×#12011yescharacteristic, complemented, Sylow
219C23—32no—
220C23—91no—
221C23—91no—
222C23—184no—
223C23—182no—
224D4—182no—
225D4—184no—
226C2×C4—32no—
227C2×C4D32×C711yescharacteristic, complemented
228C7#6392311yescharacteristic, direct product factor, Sylow
229C2×C3—22no—
230C2×C3—21no—
231C2×C3—64no—
232C2×C3C2×C7×D1212yes—
233C2×C3D3×C7×D412yes—
234C2×C3D3×C7×D412yes—
235D3—24no—
236D3—64no—
237D3—122no—
238D3—121no—
239D3—121no—
240D3—122no—
241C22—34no—
242C22—62no—
243C22—62no—
244C22—92no—
245C22—188no—
246C22—182no—
247C22—182no—
248C22—181no—
249C22C2×D32×C711yescharacteristic
250C4—21no—
251C4—62no—
252C3—21no—
253C3C7×#88212yescomplemented
254C2—64no—
255C2—182no—
256C2—361no—
257C2D3×C7×D1212yes—
258C2C7×#381111yescharacteristic
subgroupG/H
centerC22×C7C2×D32
commutator, cosocleC22×C32C23×C7
FrattiniC22C2×D32×C7
FittingC2×C32×C4×C7C22
socleC22×C32×C7C23
hypercenterC2×C4×C7D32
hypocenterC32C7×#120
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22×C7 ◁ C2×C4×C7

Complements

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

NH ≅ G/Ntype
1. C22×D32×C7C2semidirect
2. C2×C7×C4⋊(C3⋊D3)C2semidirect
3. C2×C7×#626C2semidirect
4. C3×C7×(C2×D3)⋊C4C2semidirect
5. C7×#3722C2semidirect
8. C22×C7×C3⋊D3C22semidirect
12. C2×C32×C4×C7C22semidirect
14. C2×C3×C7×Dic3C22semidirect
23. C7×(C2×D3)⋊C4D3semidirect
24. #63923C7direct (N × H)
29. C2×C7×C3⋊D3D4semidirect
32. C2×C3×D3×C7D4semidirect
48. C2×C3×C4×C7C2×D3semidirect
50. C2×C7×Dic3C2×D3semidirect
55. C22×D32C2×C7semidirect
56. C2×C4⋊(C3⋊D3)C2×C7semidirect
57. C2×#626C2×C7semidirect
58. C3×(C2×D3)⋊C4C2×C7semidirect
59. #3722C2×C7semidirect
90. C2×D3×C7D12semidirect
97. C22×C3⋊D3C22×C7semidirect
101. C2×C32×C4C22×C7semidirect
103. C2×C3×Dic3C22×C7semidirect
105. C32×C7#120semidirect
114. C2×C4×C7D32semidirect
122. (C2×D3)⋊C4D3×C7semidirect
139. C2×C3⋊D3C7×D4semidirect
142. C2×C3×D3C7×D4semidirect
169. C2×C3×C4C2×D3×C7semidirect
171. C2×Dic3C2×D3×C7semidirect
177. C3×C7#882semidirect
212. C2×D3C7×D12semidirect
218. C32C7×#120semidirect
227. C2×C4D32×C7semidirect
228. C7#63923direct (N × H)
253. C3C7×#882semidirect

Maximal subgroups

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

subgroupindex
C22×D32×C72
C2×C7×C4⋊(C3⋊D3)2
C2×C7×#6262
C3×C7×(C2×D3)⋊C42
C7×#37222
C7×#8823
#639237