Finite Group Databasebeta

← C7×#19006 #13196222

Subgroups — C7×#19006 #13196222

Order 1792.

938 conjugacy classes of subgroups; only the 222 normal subgroups are shown.

normal subgroups only

This group has 938 conjugacy classes of subgroups — too many to draw. Only its 222 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×C7×#562C211yesmaximal, characteristic
2C7×#3215C212yesmaximal, complemented
3C7×#3166C211yesmaximal, characteristic, complemented
4C7×#1661C218yesmaximal, complemented
5C7×#1534C211yesmaximal, characteristic, complemented
6C7×#1527C212yesmaximal, complemented
7C7×#562—22no—
8C7×#562—24no—
9C7×#562C2214yes—
10C7×#533C2211yescharacteristic
11C2×C7×#125C2211yescharacteristic
12C2×C7×#124C2212yes—
13C2×C7×#123—22no—
14C2×C7×C22⋊Q8—22no—
15C2×C7×#121—22no—
16C7×#501C2214yescomplemented
17C7×#499—22no—
18C7×#497C2214yescomplemented
19C7×#489C2214yescomplemented
20C7×#464—24no—
21C7×#451—22no—
22C7×#450—22no—
23C7×#432C2212yes—
24C7×#428—22no—
25C4×C7×M16C2211yescharacteristic
26C7×#330C2214yescomplemented
27C7×#329C2218yescomplemented
28#19006C711yesmaximal, characteristic, direct product factor, Sylow
29C22×C7×Q8D412yes—
30C22×C7×D4—22no—
31C22×C7×D4D412yes—
32C23×C4×C7—22no—
33C2×C7×Q16—24no—
34C2×C7×SD16—24no—
35C2×C7×M16—21no—
36C7×C4⋊Q8C2312yes—
37C7×#125—22no—
38C7×#125—21no—
39C7×#125D412yes—
40C7×#125C2312yes—
41C7×#124—22no—
42C7×#124—22no—
43C7×#124D414yes—
44C7×#124C2314yes—
45C7×#123—28no—
46C7×#123—28no—
47C7×C22⋊Q8—28no—
48C7×C22⋊Q8—28no—
49C7×C22⋊Q8—22no—
50C7×#121—28no—
51C7×#121—28no—
52C4×C7×Q8—22no—
53C7×#117—21no—
54C2×C7×C4⋊C4—22no—
55C2×C7×C4⋊C4—22no—
56C2×C7×C4⋊C4D412yes—
57C2×C7×C22⋊C4—22no—
58C2×C7×C22⋊C4—22no—
59C2×C7×C22⋊C4—22no—
60C2×C42×C7C2311yescharacteristic
61C7×#107—24no—
62C7×#106—24no—
63C7×C4⋊C8—22no—
64C7×#103—24no—
65C7×#102—24no—
66C7×C22⋊C8—22no—
67C7×#97C2314yes—
68C7×#97C2312yes—
69C4×C7×C8—21no—
70C2×#562C2×C711yescharacteristic
71#3215C2×C712yescomplemented
72#3166C2×C711yescharacteristic, complemented
73#1661C2×C718yescomplemented
74#1534C2×C711yescharacteristic, complemented
75#1527C2×C712yescomplemented
76C24×C7—22no—
77C2×C7×Q8—22no—
78C2×C7×Q8—22no—
79C2×C7×Q8—82no—
80C2×C7×Q8C2×D414yes—
81C2×C7×D4—28no—
82C2×C7×D4—22no—
83C2×C7×D4—82no—
84C2×C7×D4—84no—
85C2×C7×D4C2×D414yes—
86C22×C4×C7—22no—
87C22×C4×C7—22no—
88C22×C4×C7—28no—
89C22×C4×C7—84no—
90C22×C4×C7C2×D411yescharacteristic
91C22×C4×C7C2×D412yes—
92C7×Q16—84no—
93C7×SD16—84no—
94C7×M16—24no—
95C2×C7×C8—22no—
96C2×C7×C8—24no—
97C7×C4⋊C4—28no—
98C7×C4⋊C4—24no—
99C7×C4⋊C4—22no—
100C7×C4⋊C4—24no—
101C7×C4⋊C4—22no—
102C7×C4⋊C4—22no—
103C7×C4⋊C4—42no—
104C7×C4⋊C4C2×D414yes—
105C7×C22⋊C4—28no—
106C7×C22⋊C4—24no—
107C7×C22⋊C4—24no—
108C7×C22⋊C4—24no—
109C7×C22⋊C4—24no—
110C7×C22⋊C4—41no—
111C42×C7—41no—
112C42×C7C2×D411yescharacteristic
113C42×C7C2×D412yes—
114C42×C7C2411yescharacteristic
115#562—22no—
116#562—24no—
117#562C22×C714yes—
118#533C22×C711yescharacteristic
119C2×#125C22×C711yescharacteristic
120C2×#124C22×C712yes—
121C2×#123—22no—
122C2×C22⋊Q8—22no—
123C2×#121—22no—
124#501C22×C714yescomplemented
125#499—22no—
126#497C22×C714yescomplemented
127#489C22×C714yescomplemented
128#464—24no—
129#451—22no—
130#450—22no—
131#432C22×C712yes—
132#428—22no—
133C4×M16C22×C711yescharacteristic
134#330C22×C714yescomplemented
135#329C22×C718yescomplemented
136C23×C7—24no—
137C23×C7—44no—
138C23×C7—44no—
139C23×C7—44no—
140C23×C7#12011yescharacteristic
141C7×Q8—44no—
142C7×Q8—44no—
143C7×Q8—82no—
144C7×D4—44no—
145C7×D4—82no—
146C7×D4—88no—
147C2×C4×C7—22no—
148C2×C4×C7—24no—
149C2×C4×C7—24no—
150C2×C4×C7—42no—
151C2×C4×C7—44no—
152C2×C4×C7—44no—
153C2×C4×C7—42no—
154C2×C4×C7—88no—
155C2×C4×C7#12011yescharacteristic
156C2×C4×C7#12012yes—
157C2×C4×C7C22×D411yescharacteristic
158C2×C4×C7C22×D412yes—
159C7×C8—24no—
160C7×C8—44no—
161C22×Q8C7×D412yes—
162C22×D4—22no—
163C22×D4C7×D412yes—
164C23×C4—22no—
165C2×Q16—24no—
166C2×SD16—24no—
167C2×M16—21no—
168C4⋊Q8C23×C712yes—
169#125—22no—
170#125—21no—
171#125C7×D412yes—
172#125C23×C712yes—
173#124—22no—
174#124—22no—
175#124C7×D414yes—
176#124C23×C714yes—
177#123—28no—
178#123—28no—
179C22⋊Q8—28no—
180C22⋊Q8—28no—
181C22⋊Q8—22no—
182#121—28no—
183#121—28no—
184C4×Q8—22no—
185#117—21no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4C7×D412yes—
189C2×C22⋊C4—22no—
190C2×C22⋊C4—22no—
191C2×C22⋊C4—22no—
192C2×C42C23×C711yescharacteristic
193#107—24no—
194#106—24no—
195C4⋊C8—22no—
196#103—24no—
197#102—24no—
198C22⋊C8—22no—
199#97C23×C712yes—
200#97C23×C714yes—
201C4×C8—21no—
202C22×C7—22no—
203C22×C7—44no—
204C22×C7—44no—
205C22×C7—44no—
206C22×C7—88no—
207C22×C7#45612yes—
208C22×C7C2×#12011yescharacteristic
209C4×C7—21no—
210C4×C7—21no—
211C4×C7—22no—
212C4×C7—22no—
213C4×C7—44no—
214C4×C7—44no—
215C4×C7—82no—
216C24—22no—
217C2×Q8—22no—
218C2×Q8—22no—
219C2×Q8—82no—
220C2×Q8C2×C7×D414yes—
221C2×D4—28no—
222C2×D4—22no—
223C2×D4—82no—
224C2×D4—84no—
225C2×D4C2×C7×D414yes—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—28no—
229C22×C4—84no—
230C22×C4C2×C7×D411yescharacteristic
231C22×C4C2×C7×D412yes—
232Q16—84no—
233SD16—84no—
234M16—24no—
235C2×C8—22no—
236C2×C8—24no—
237C4⋊C4—28no—
238C4⋊C4—24no—
239C4⋊C4—22no—
240C4⋊C4—24no—
241C4⋊C4—22no—
242C4⋊C4—22no—
243C4⋊C4—42no—
244C4⋊C4C2×C7×D414yes—
245C22⋊C4—28no—
246C22⋊C4—24no—
247C22⋊C4—24no—
248C22⋊C4—24no—
249C22⋊C4—24no—
250C22⋊C4—41no—
251C42—41no—
252C42C2×C7×D411yescharacteristic
253C42C2×C7×D412yes—
254C42C24×C711yescharacteristic
255C2×C7—22no—
256C2×C7—84no—
257C2×C7C2×#45611yescharacteristic
258C2×C7#301912yes—
259C23—24no—
260C23—44no—
261C23—44no—
262C23—44no—
263C23C7×#12011yescharacteristic
264Q8—44no—
265Q8—44no—
266Q8—82no—
267D4—44no—
268D4—82no—
269D4—88no—
270C2×C4—22no—
271C2×C4—24no—
272C2×C4—24no—
273C2×C4—42no—
274C2×C4—44no—
275C2×C4—44no—
276C2×C4—42no—
277C2×C4—88no—
278C2×C4C7×#12011yescharacteristic
279C2×C4C7×#12012yes—
280C2×C4C22×C7×D411yescharacteristic
281C2×C4C22×C7×D412yes—
282C8—24no—
283C8—44no—
284C7#1900611yescharacteristic, direct product factor, Sylow
285C22—22no—
286C22—44no—
287C22—44no—
288C22—44no—
289C22—88no—
290C22C7×#45612yes—
291C22C2×C7×#12011yescharacteristic
292C4—21no—
293C4—21no—
294C4—22no—
295C4—22no—
296C4—44no—
297C4—44no—
298C4—82no—
299C2—22no—
300C2—84no—
301C2C2×C7×#45611yescharacteristic
302C2C7×#301912yes—
subgroupG/H
center, socleC22×C7C2×#120
commutator, Frattini, cosocleC42C24×C7
prime radicalC2×C7×#562C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C7 ◁ C2×C42×C7 ◁ G

Complements

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

NH ≅ G/Ntype
2. C7×#3215C2semidirect
3. C7×#3166C2semidirect
4. C7×#1661C2semidirect
5. C7×#1534C2semidirect
6. C7×#1527C2semidirect
16. C7×#501C22semidirect
18. C7×#497C22semidirect
19. C7×#489C22semidirect
26. C7×#330C22semidirect
27. C7×#329C22semidirect
28. #19006C7direct (N × H)
71. #3215C2×C7semidirect
72. #3166C2×C7semidirect
73. #1661C2×C7semidirect
74. #1534C2×C7semidirect
75. #1527C2×C7semidirect
124. #501C22×C7semidirect
126. #497C22×C7semidirect
127. #489C22×C7semidirect
134. #330C22×C7semidirect
135. #329C22×C7semidirect
284. C7#19006direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C7×#5622
C7×#32152
C7×#31662
C7×#16612
C7×#15342
C7×#15272
#190067