Finite Group Databasebeta

← C7×#9476 #13186692

Subgroups — C7×#9476 #13186692

Order 1792.

818 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 818 conjugacy classes of subgroups — too many to draw. Only its 130 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×#2406C211yesmaximal, characteristic
2C7×#2274C211yesmaximal, characteristic, complemented
3C7×#1789C212yesmaximal
4C7×#1463C211yesmaximal, characteristic, complemented
5C7×#1284C212yesmaximal, complemented
6C22×C7×#41C2211yescharacteristic
7C2×C7×#123—21no—
8C2×C7×#117—21no—
9C22×C7×C4⋊C4C2211yescharacteristic
10C2×C7×#104—24no—
11C7×#407C2212yes—
12C7×#395C412yescomplemented
13C7×#394C412yescomplemented
14C7×#386—21no—
15C7×#385—22no—
16C7×#381C2211yescharacteristic
17C4×C7×C4⋊C4—24no—
18C4×C7×C22⋊C4—21no—
19C4×C7×C22⋊C4C2212yes—
20C7×#324—22no—
21#9476C711yesmaximal, characteristic, direct product factor, Sylow
22C2×C7×#41—22no—
23C2×C7×#41—22no—
24C2×C7×#41—44no—
25C22×C7×Q8C2×C411yescharacteristic
26C22×C7×D4—21no—
27C22×C7×D4C2×C411yescharacteristic
28C23×C4×C7—21no—
29C23×C4×C7D411yescharacteristic
30C23×C4×C7D411yescharacteristic
31C23×C4×C7C2311yescharacteristic
32C2×C7×M16—22no—
33C7×#123—44no—
34C7×#117—28no—
35C2×C7×C4⋊C4—22no—
36C2×C7×C4⋊C4—22no—
37C2×C7×C4⋊C4—22no—
38C2×C7×C4⋊C4—22no—
39C2×C7×C4⋊C4C2×C412yes—
40C2×C7×C4⋊C4D412yes—
41C2×C7×C22⋊C4—22no—
42C2×C7×C22⋊C4—21no—
43C2×C7×C22⋊C4—22no—
44C2×C7×C22⋊C4—22no—
45C2×C42×C7—22no—
46C2×C42×C7—22no—
47C2×C42×C7D414yes—
48C7×#104—48no—
49C7×C22⋊C8—22no—
50C7×C22⋊C8—22no—
51C7×#95—22no—
52C7×#95—22no—
53C7×#95—22no—
54C7×#95—21no—
55C7×#95C2×C412yes—
56#2406C2×C711yescharacteristic
57#2274C2×C711yescharacteristic, complemented
58#1789C2×C712yes—
59#1463C2×C711yescharacteristic, complemented
60#1284C2×C712yescomplemented
61C24×C7—21no—
62C24×C7C2×D411yescharacteristic
63C7×#41—28no—
64C7×#41—44no—
65C7×#41—44no—
66C7×#41—44no—
67C2×C7×Q8—22no—
68C2×C7×Q8—42no—
69C2×C7×D4—24no—
70C2×C7×D4—22no—
71C2×C7×D4—42no—
72C2×C7×D4—42no—
73C2×C7×D4—41no—
74C2×C7×D4—41no—
75C22×C4×C7—21no—
76C22×C4×C7—21no—
77C22×C4×C7—24no—
78C22×C4×C7—21no—
79C22×C4×C7—24no—
80C22×C4×C7—21no—
81C22×C4×C7—21no—
82C22×C4×C7—22no—
83C22×C4×C7—42no—
84C22×C4×C7—42no—
85C22×C4×C7—42no—
86C22×C4×C7—42no—
87C22×C4×C7—42no—
88C22×C4×C7C22⋊C412yes—
89C22×C4×C7C22⋊C412yes—
90C22×C4×C7C22×C411yescharacteristic
91C22×C4×C7C2×D412yes—
92C22×C4×C7C2×D411yescharacteristic
93C22×C4×C7#4112yes—
94C7×M16—44no—
95C2×C7×C8—42no—
96C7×C4⋊C4—28no—
97C7×C4⋊C4—24no—
98C7×C4⋊C4—24no—
99C7×C4⋊C4—44no—
100C7×C22⋊C4—28no—
101C7×C22⋊C4—44no—
102C7×C22⋊C4—42no—
103C7×C22⋊C4—44no—
104C42×C7—28no—
105C42×C7—44no—
106C42×C7—82no—
107C22×#41C22×C711yescharacteristic
108C2×#123—21no—
109C2×#117—21no—
110C22×C4⋊C4C22×C711yescharacteristic
111C2×#104—24no—
112#407C22×C712yes—
113#395C4×C712yescomplemented
114#394C4×C712yescomplemented
115#386—21no—
116#385—22no—
117#381C22×C711yescharacteristic
118C4×C4⋊C4—24no—
119C4×C22⋊C4—21no—
120C4×C22⋊C4C22×C712yes—
121#324—22no—
122C23×C7—22no—
123C23×C7—22no—
124C23×C7—22no—
125C23×C7—41no—
126C23×C7—41no—
127C23×C7—42no—
128C23×C7—41no—
129C23×C7—42no—
130C23×C7C2×C22⋊C411yescharacteristic
131C23×C7#12011yescharacteristic
132C23×C7#12311yescharacteristic
133C7×Q8—24no—
134C7×Q8—42no—
135C7×D4—24no—
136C7×D4—42no—
137C7×D4—44no—
138C7×D4—44no—
139C2×C4×C7—22no—
140C2×C4×C7—22no—
141C2×C4×C7—24no—
142C2×C4×C7—22no—
143C2×C4×C7—22no—
144C2×C4×C7—24no—
145C2×C4×C7—28no—
146C2×C4×C7—22no—
147C2×C4×C7—22no—
148C2×C4×C7—41no—
149C2×C4×C7—41no—
150C2×C4×C7—42no—
151C2×C4×C7—48no—
152C2×C4×C7—44no—
153C2×C4×C7—41no—
154C2×C4×C7—41no—
155C2×C4×C7—44no—
156C2×C4×C7—42no—
157C2×C4×C7—42no—
158C2×C4×C7—42no—
159C2×C4×C7—42no—
160C2×C4×C7—42no—
161C2×C4×C7—81no—
162C2×C4×C7#10414yescomplemented
163C2×C4×C7C4×D412yes—
164C2×C4×C7#12112yes—
165C7×C8—82no—
166C2×#41—22no—
167C2×#41—22no—
168C2×#41—44no—
169C22×Q8C2×C4×C711yescharacteristic
170C22×D4—21no—
171C22×D4C2×C4×C711yescharacteristic
172C23×C4—21no—
173C23×C4C7×D411yescharacteristic
174C23×C4C7×D411yescharacteristic
175C23×C4C23×C711yescharacteristic
176C2×M16—22no—
177#123—44no—
178#117—28no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4C2×C4×C712yes—
184C2×C4⋊C4C7×D412yes—
185C2×C22⋊C4—22no—
186C2×C22⋊C4—21no—
187C2×C22⋊C4—22no—
188C2×C22⋊C4—22no—
189C2×C42—22no—
190C2×C42—22no—
191C2×C42C7×D414yes—
192#104—48no—
193C22⋊C8—22no—
194C22⋊C8—22no—
195#95—22no—
196#95—22no—
197#95—22no—
198#95—21no—
199#95C2×C4×C712yes—
200C22×C7—21no—
201C22×C7—21no—
202C22×C7—21no—
203C22×C7—21no—
204C22×C7—21no—
205C22×C7—21no—
206C22×C7—21no—
207C22×C7—44no—
208C22×C7—44no—
209C22×C7—42no—
210C22×C7—82no—
211C22×C7—81no—
212C22×C7—81no—
213C22×C7#38611yescharacteristic
214C22×C7C2×#10412yes—
215C22×C7B(4,𝔽2)12yes—
216C4×C7—22no—
217C4×C7—22no—
218C4×C7—24no—
219C4×C7—48no—
220C4×C7—42no—
221C4×C7—42no—
222C4×C7—82no—
223C24—21no—
224C24C2×C7×D411yescharacteristic
225#41—28no—
226#41—44no—
227#41—44no—
228#41—44no—
229C2×Q8—22no—
230C2×Q8—42no—
231C2×D4—24no—
232C2×D4—22no—
233C2×D4—42no—
234C2×D4—42no—
235C2×D4—41no—
236C2×D4—41no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—24no—
240C22×C4—21no—
241C22×C4—24no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—22no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4C7×C22⋊C412yes—
251C22×C4C7×C22⋊C412yes—
252C22×C4C22×C4×C711yescharacteristic
253C22×C4C2×C7×D412yes—
254C22×C4C2×C7×D411yescharacteristic
255C22×C4C7×#4112yes—
256M16—44no—
257C2×C8—42no—
258C4⋊C4—28no—
259C4⋊C4—24no—
260C4⋊C4—24no—
261C4⋊C4—44no—
262C22⋊C4—28no—
263C22⋊C4—44no—
264C22⋊C4—42no—
265C22⋊C4—44no—
266C42—28no—
267C42—44no—
268C42—82no—
269C2×C7—22no—
270C2×C7—22no—
271C2×C7—41no—
272C2×C7—82no—
273C2×C7#188911yescharacteristic
274C2×C7#189611yescharacteristic
275C2×C7#191211yescharacteristic
276C23—22no—
277C23—22no—
278C23—22no—
279C23—41no—
280C23—41no—
281C23—42no—
282C23—41no—
283C23—42no—
284C23C2×C7×C22⋊C411yescharacteristic
285C23C7×#12011yescharacteristic
286C23C7×#12311yescharacteristic
287Q8—24no—
288Q8—42no—
289D4—24no—
290D4—42no—
291D4—44no—
292D4—44no—
293C2×C4—22no—
294C2×C4—22no—
295C2×C4—24no—
296C2×C4—22no—
297C2×C4—22no—
298C2×C4—24no—
299C2×C4—28no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—42no—
305C2×C4—48no—
306C2×C4—44no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—44no—
310C2×C4—42no—
311C2×C4—42no—
312C2×C4—42no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—81no—
316C2×C4C7×#10414yescomplemented
317C2×C4C4×C7×D412yes—
318C2×C4C7×#12112yes—
319C8—82no—
320C7#947611yescharacteristic, direct product factor, Sylow
321C22—21no—
322C22—21no—
323C22—21no—
324C22—21no—
325C22—21no—
326C22—21no—
327C22—21no—
328C22—44no—
329C22—44no—
330C22—42no—
331C22—82no—
332C22—81no—
333C22—81no—
334C22C7×#38611yescharacteristic
335C22C2×C7×#10412yes—
336C22C7×B(4,𝔽2)12yes—
337C4—22no—
338C4—22no—
339C4—24no—
340C4—48no—
341C4—42no—
342C4—42no—
343C4—82no—
344C2—22no—
345C2—22no—
346C2—41no—
347C2—82no—
348C2C7×#188911yescharacteristic
349C2C7×#189611yescharacteristic
350C2C7×#191211yescharacteristic
subgroupG/H
center, socleC22×C7#386
commutatorC22×C4C22×C4×C7
Frattini, cosocleC23×C4C23×C7
prime radicalC22×C7×#41C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C7 ◁ C23×C4×C7 ◁ G

Complements

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

NH ≅ G/Ntype
2. C7×#2274C2semidirect
4. C7×#1463C2semidirect
5. C7×#1284C2semidirect
12. C7×#395C4semidirect
13. C7×#394C4semidirect
21. #9476C7direct (N × H)
57. #2274C2×C7semidirect
59. #1463C2×C7semidirect
60. #1284C2×C7semidirect
113. #395C4×C7semidirect
114. #394C4×C7semidirect
162. C2×C4×C7#104semidirect
316. C2×C4C7×#104semidirect
320. C7#9476direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#24062
C7×#22742
C7×#17892
C7×#14632
C7×#12842
#94767