Finite Group Databasebeta

← C7×#11535 #13188751

Subgroups — C7×#11535 #13188751

Order 1792.

440 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 106 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
1C7×#2326C211yesmaximal, characteristic
2C7×#1465C212yesmaximal
3C7×#1325C212yesmaximal, complemented
4C7×#1319C212yesmaximal, complemented
5C2×C7×#124C411yescharacteristic
6C2×C4×C7×Q8C411yescharacteristic
7C2×C4×C7×D4C2211yescharacteristic
8C7×#407—22no—
9C7×#391C2211yescharacteristic
10C7×#390C411yescharacteristic
11C7×#388—22no—
12C7×#385—22no—
13C7×#384C2211yescharacteristic
14C7×#376C411yescharacteristic
15C7×#324C2214yes—
16C7×#323—22no—
17#11535C711yesmaximal, characteristic, direct product factor, Sylow
18C22×C7×Q8C2×C411yescharacteristic
19C22×C7×D4C2×C411yescharacteristic
20C23×C4×C7D412yes—
21C2×C7×M16—42no—
22C7×#124—22no—
23C7×#124—22no—
24C4×C7×Q8—42no—
25C4×C7×D4—42no—
26C2×C7×C4⋊C4—21no—
27C2×C7×C4⋊C4—22no—
28C2×C7×C4⋊C4C2×C411yescharacteristic
29C2×C7×C4⋊C4C2311yescharacteristic
30C2×C7×C22⋊C4—22no—
31C2×C7×C22⋊C4—22no—
32C2×C7×C22⋊C4D412yes—
33C2×C42×C7—21no—
34C2×C42×C7C2×C411yescharacteristic
35C2×C42×C7C2×C411yescharacteristic
36C2×C42×C7C2×C411yescharacteristic
37C7×C22⋊C8—24no—
38C7×C22⋊C8—42no—
39C7×#95—21no—
40C7×#95—21no—
41C7×#95—22no—
42#2326C2×C711yescharacteristic
43#1465C2×C712yes—
44#1325C2×C712yescomplemented
45#1319C2×C712yescomplemented
46C24×C7C22⋊C412yes—
47C2×C7×Q8—21no—
48C2×C7×Q8—41no—
49C2×C7×Q8—41no—
50C2×C7×Q8M1612yescomplemented
51C2×C7×D4—21no—
52C2×C7×D4—21no—
53C2×C7×D4—41no—
54C2×C7×D4—41no—
55C22×C4×C7—21no—
56C22×C4×C7—21no—
57C22×C4×C7—22no—
58C22×C4×C7—22no—
59C22×C4×C7—21no—
60C22×C4×C7—21no—
61C22×C4×C7—82no—
62C22×C4×C7C22⋊C411yescharacteristic
63C22×C4×C7C22⋊C411yescharacteristic
64C22×C4×C7C22×C411yescharacteristic
65C22×C4×C7C2×D412yes—
66C7×M16—84no—
67C2×C7×C8—44no—
68C7×C4⋊C4—41no—
69C7×C4⋊C4—41no—
70C7×C4⋊C4—42no—
71C7×C4⋊C4—82no—
72C7×C22⋊C4—28no—
73C7×C22⋊C4—42no—
74C7×C22⋊C4—82no—
75C42×C7—22no—
76C42×C7—21no—
77C42×C7—41no—
78C42×C7—42no—
79C42×C7M1612yescomplemented
80C2×#124C4×C711yescharacteristic
81C2×C4×Q8C4×C711yescharacteristic
82C2×C4×D4C22×C711yescharacteristic
83#407—22no—
84#391C22×C711yescharacteristic
85#390C4×C711yescharacteristic
86#388—22no—
87#385—22no—
88#384C22×C711yescharacteristic
89#376C4×C711yescharacteristic
90#324C22×C714yes—
91#323—22no—
92C23×C7—22no—
93C23×C7—22no—
94C23×C7—22no—
95C23×C7—42no—
96C23×C7—42no—
97C23×C7C2×C22⋊C411yescharacteristic
98C7×Q8—42no—
99C7×Q8—42no—
100C7×D4—42no—
101C7×D4—42no—
102C2×C4×C7—21no—
103C2×C4×C7—21no—
104C2×C4×C7—22no—
105C2×C4×C7—22no—
106C2×C4×C7—21no—
107C2×C4×C7—22no—
108C2×C4×C7—41no—
109C2×C4×C7—44no—
110C2×C4×C7—41no—
111C2×C4×C7—41no—
112C2×C4×C7—41no—
113C2×C4×C7—41no—
114C2×C4×C7—42no—
115C2×C4×C7—41no—
116C2×C4×C7—41no—
117C2×C4×C7—42no—
118C2×C4×C7—41no—
119C2×C4×C7—41no—
120C2×C4×C7—41no—
121C2×C4×C7—82no—
122C2×C4×C7#9912yes—
123C2×C4×C7C2×M1611yescharacteristic
124C2×C4×C7C2×M1611yescharacteristic
125C7×C8—84no—
126C22×Q8C2×C4×C711yescharacteristic
127C22×D4C2×C4×C711yescharacteristic
128C23×C4C7×D412yes—
129C2×M16—42no—
130#124—22no—
131#124—22no—
132C4×Q8—42no—
133C4×D4—42no—
134C2×C4⋊C4—21no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4C2×C4×C711yescharacteristic
137C2×C4⋊C4C23×C711yescharacteristic
138C2×C22⋊C4—22no—
139C2×C22⋊C4—22no—
140C2×C22⋊C4C7×D412yes—
141C2×C42—21no—
142C2×C42C2×C4×C711yescharacteristic
143C2×C42C2×C4×C711yescharacteristic
144C2×C42C2×C4×C711yescharacteristic
145C22⋊C8—24no—
146C22⋊C8—42no—
147#95—21no—
148#95—21no—
149#95—22no—
150C22×C7—21no—
151C22×C7—21no—
152C22×C7—44no—
153C22×C7—42no—
154C22×C7—42no—
155C22×C7—42no—
156C22×C7—82no—
157C22×C7#40711yescharacteristic
158C22×C7C2×#9911yescharacteristic
159C22×C7#41311yescharacteristic
160C4×C7—22no—
161C4×C7—21no—
162C4×C7—21no—
163C4×C7—42no—
164C4×C7—42no—
165C4×C7—41no—
166C4×C7—42no—
167C4×C7—81no—
168C4×C7—81no—
169C24C7×C22⋊C412yes—
170C2×Q8—21no—
171C2×Q8—41no—
172C2×Q8—41no—
173C2×Q8C7×M1612yescomplemented
174C2×D4—21no—
175C2×D4—21no—
176C2×D4—41no—
177C2×D4—41no—
178C22×C4—21no—
179C22×C4—21no—
180C22×C4—22no—
181C22×C4—22no—
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—82no—
185C22×C4C7×C22⋊C411yescharacteristic
186C22×C4C7×C22⋊C411yescharacteristic
187C22×C4C22×C4×C711yescharacteristic
188C22×C4C2×C7×D412yes—
189M16—84no—
190C2×C8—44no—
191C4⋊C4—41no—
192C4⋊C4—41no—
193C4⋊C4—42no—
194C4⋊C4—82no—
195C22⋊C4—28no—
196C22⋊C4—42no—
197C22⋊C4—82no—
198C42—22no—
199C42—21no—
200C42—41no—
201C42—42no—
202C42C7×M1612yescomplemented
203C2×C7—21no—
204C2×C7—21no—
205C2×C7—82no—
206C2×C7#146511yescharacteristic
207C2×C7#212711yescharacteristic
208C2×C7#213511yescharacteristic
209C23—22no—
210C23—22no—
211C23—22no—
212C23—42no—
213C23—42no—
214C23C2×C7×C22⋊C411yescharacteristic
215Q8—42no—
216Q8—42no—
217D4—42no—
218D4—42no—
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—21no—
224C2×C4—22no—
225C2×C4—41no—
226C2×C4—44no—
227C2×C4—41no—
228C2×C4—41no—
229C2×C4—41no—
230C2×C4—41no—
231C2×C4—42no—
232C2×C4—41no—
233C2×C4—41no—
234C2×C4—42no—
235C2×C4—41no—
236C2×C4—41no—
237C2×C4—41no—
238C2×C4—82no—
239C2×C4C7×#9912yes—
240C2×C4C2×C7×M1611yescharacteristic
241C2×C4C2×C7×M1611yescharacteristic
242C8—84no—
243C7#1153511yescharacteristic, direct product factor, Sylow
244C22—21no—
245C22—21no—
246C22—44no—
247C22—42no—
248C22—42no—
249C22—42no—
250C22—82no—
251C22C7×#40711yescharacteristic
252C22C2×C7×#9911yescharacteristic
253C22C7×#41311yescharacteristic
254C4—21no—
255C4—21no—
256C4—22no—
257C4—42no—
258C4—42no—
259C4—42no—
260C4—41no—
261C4—81no—
262C4—81no—
263C2—21no—
264C2—21no—
265C2—82no—
266C2C7×#146511yescharacteristic
267C2C7×#212711yescharacteristic
268C2C7×#213511yescharacteristic
subgroupG/H
center, socleC22×C7C2×#99
commutatorC22×C4C22×C4×C7
Frattini, cosocleC2×C4⋊C4C23×C7
prime radicalC22×C7×D4C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C7 ◁ C22×C4×C7 ◁ C2×C4×C7×D4 ◁ G

Complements

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

NH ≅ G/Ntype
3. C7×#1325C2semidirect
4. C7×#1319C2semidirect
17. #11535C7direct (N × H)
44. #1325C2×C7semidirect
45. #1319C2×C7semidirect
50. C2×C7×Q8M16semidirect
79. C42×C7M16semidirect
173. C2×Q8C7×M16semidirect
202. C42C7×M16semidirect
243. C7#11535direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#23262
C7×#14652
C7×#13252
C7×#13192
#115357