Finite Group Databasebeta

← C7×#9480 #13186696

Subgroups — C7×#9480 #13186696

Order 1792.

726 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 726 conjugacy classes of subgroups — too many to draw. Only its 122 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×#2313C211yesmaximal, characteristic, complemented
3C7×#1789C212yesmaximal
4C7×#1463C211yesmaximal, characteristic, complemented
5C7×#1284C212yesmaximal, complemented
6C22×C7×#41C2211yescharacteristic
7C2×C7×#123—21no—
8C22×C7×C4⋊C4C2211yescharacteristic
9C2×C7×#104—24no—
10C7×#407C2212yes—
11C7×#395C412yescomplemented
12C7×#394C412yescomplemented
13C7×#387—24no—
14C7×#386—21no—
15C7×#385—22no—
16C7×#381—21no—
17C7×#381C2211yescharacteristic
18C7×#380—21no—
19C4×C7×C22⋊C4C2212yes—
20C7×#324—22no—
21#9480C711yesmaximal, 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—
34C2×C7×C4⋊C4—22no—
35C2×C7×C4⋊C4—22no—
36C2×C7×C4⋊C4—22no—
37C2×C7×C4⋊C4—22no—
38C2×C7×C4⋊C4C2×C412yes—
39C2×C7×C4⋊C4D412yes—
40C2×C7×C22⋊C4—22no—
41C2×C7×C22⋊C4—21no—
42C2×C7×C22⋊C4—22no—
43C2×C7×C22⋊C4—22no—
44C2×C42×C7D414yes—
45C7×#104—48no—
46C7×C22⋊C8—22no—
47C7×C22⋊C8—22no—
48C7×#95—22no—
49C7×#95—24no—
50C7×#95—22no—
51C7×#95—22no—
52C7×#95—21no—
53C7×#95C2×C412yes—
54#2406C2×C711yescharacteristic
55#2313C2×C711yescharacteristic, complemented
56#1789C2×C712yes—
57#1463C2×C711yescharacteristic, complemented
58#1284C2×C712yescomplemented
59C24×C7—21no—
60C24×C7C2×D411yescharacteristic
61C7×#41—28no—
62C7×#41—44no—
63C7×#41—44no—
64C7×#41—44no—
65C2×C7×Q8—22no—
66C2×C7×Q8—42no—
67C2×C7×D4—24no—
68C2×C7×D4—22no—
69C2×C7×D4—42no—
70C2×C7×D4—42no—
71C2×C7×D4—41no—
72C2×C7×D4—41no—
73C22×C4×C7—21no—
74C22×C4×C7—21no—
75C22×C4×C7—21no—
76C22×C4×C7—21no—
77C22×C4×C7—21no—
78C22×C4×C7—22no—
79C22×C4×C7—24no—
80C22×C4×C7—42no—
81C22×C4×C7—42no—
82C22×C4×C7—42no—
83C22×C4×C7—42no—
84C22×C4×C7—42no—
85C22×C4×C7—42no—
86C22×C4×C7C22⋊C412yes—
87C22×C4×C7C22⋊C412yes—
88C22×C4×C7C22×C411yescharacteristic
89C22×C4×C7C2×D412yes—
90C22×C4×C7C2×D411yescharacteristic
91C22×C4×C7#4112yes—
92C7×M16—44no—
93C2×C7×C8—42no—
94C7×C4⋊C4—24no—
95C7×C4⋊C4—24no—
96C7×C4⋊C4—44no—
97C7×C4⋊C4—82no—
98C7×C22⋊C4—44no—
99C7×C22⋊C4—44no—
100C7×C22⋊C4—44no—
101C7×C22⋊C4—42no—
102C42×C7—44no—
103C22×#41C22×C711yescharacteristic
104C2×#123—21no—
105C22×C4⋊C4C22×C711yescharacteristic
106C2×#104—24no—
107#407C22×C712yes—
108#395C4×C712yescomplemented
109#394C4×C712yescomplemented
110#387—24no—
111#386—21no—
112#385—22no—
113#381—21no—
114#381C22×C711yescharacteristic
115#380—21no—
116C4×C22⋊C4C22×C712yes—
117#324—22no—
118C23×C7—22no—
119C23×C7—22no—
120C23×C7—22no—
121C23×C7—41no—
122C23×C7—41no—
123C23×C7—42no—
124C23×C7—41no—
125C23×C7—42no—
126C23×C7C2×C22⋊C411yescharacteristic
127C23×C7#12011yescharacteristic
128C23×C7#12311yescharacteristic
129C7×Q8—24no—
130C7×Q8—42no—
131C7×D4—24no—
132C7×D4—42no—
133C7×D4—44no—
134C7×D4—44no—
135C2×C4×C7—24no—
136C2×C4×C7—22no—
137C2×C4×C7—22no—
138C2×C4×C7—22no—
139C2×C4×C7—22no—
140C2×C4×C7—22no—
141C2×C4×C7—41no—
142C2×C4×C7—41no—
143C2×C4×C7—42no—
144C2×C4×C7—48no—
145C2×C4×C7—42no—
146C2×C4×C7—42no—
147C2×C4×C7—41no—
148C2×C4×C7—41no—
149C2×C4×C7—42no—
150C2×C4×C7—42no—
151C2×C4×C7—42no—
152C2×C4×C7—44no—
153C2×C4×C7—44no—
154C2×C4×C7—42no—
155C2×C4×C7—42no—
156C2×C4×C7—44no—
157C2×C4×C7—81no—
158C2×C4×C7C4×D412yes—
159C2×C4×C7#12112yes—
160C7×C8—82no—
161C2×#41—22no—
162C2×#41—22no—
163C2×#41—44no—
164C22×Q8C2×C4×C711yescharacteristic
165C22×D4—21no—
166C22×D4C2×C4×C711yescharacteristic
167C23×C4—21no—
168C23×C4C7×D411yescharacteristic
169C23×C4C7×D411yescharacteristic
170C23×C4C23×C711yescharacteristic
171C2×M16—22no—
172#123—44no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4C2×C4×C712yes—
178C2×C4⋊C4C7×D412yes—
179C2×C22⋊C4—21no—
180C2×C22⋊C4—22no—
181C2×C22⋊C4—22no—
182C2×C22⋊C4—22no—
183C2×C42C7×D414yes—
184#104—48no—
185C22⋊C8—22no—
186C22⋊C8—22no—
187#95—22no—
188#95—24no—
189#95—22no—
190#95—22no—
191#95—21no—
192#95C2×C4×C712yes—
193C22×C7—21no—
194C22×C7—21no—
195C22×C7—21no—
196C22×C7—21no—
197C22×C7—21no—
198C22×C7—21no—
199C22×C7—21no—
200C22×C7—44no—
201C22×C7—44no—
202C22×C7—42no—
203C22×C7—82no—
204C22×C7—81no—
205C22×C7—81no—
206C22×C7#38611yescharacteristic
207C22×C7#42112yes—
208C22×C7B(4,𝔽2)12yes—
209C4×C7—22no—
210C4×C7—22no—
211C4×C7—42no—
212C4×C7—42no—
213C4×C7—42no—
214C4×C7—84no—
215C4×C7—82no—
216C24—21no—
217C24C2×C7×D411yescharacteristic
218#41—28no—
219#41—44no—
220#41—44no—
221#41—44no—
222C2×Q8—22no—
223C2×Q8—42no—
224C2×D4—24no—
225C2×D4—22no—
226C2×D4—42no—
227C2×D4—42no—
228C2×D4—41no—
229C2×D4—41no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—22no—
236C22×C4—24no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4C7×C22⋊C412yes—
244C22×C4C7×C22⋊C412yes—
245C22×C4C22×C4×C711yescharacteristic
246C22×C4C2×C7×D412yes—
247C22×C4C2×C7×D411yescharacteristic
248C22×C4C7×#4112yes—
249M16—44no—
250C2×C8—42no—
251C4⋊C4—24no—
252C4⋊C4—24no—
253C4⋊C4—44no—
254C4⋊C4—82no—
255C22⋊C4—44no—
256C22⋊C4—44no—
257C22⋊C4—42no—
258C22⋊C4—44no—
259C42—44no—
260C2×C7—22no—
261C2×C7—22no—
262C2×C7—41no—
263C2×C7—82no—
264C2×C7#188911yescharacteristic
265C2×C7#189711yescharacteristic
266C2×C7#191211yescharacteristic
267C23—22no—
268C23—22no—
269C23—22no—
270C23—41no—
271C23—41no—
272C23—42no—
273C23—41no—
274C23—42no—
275C23C2×C7×C22⋊C411yescharacteristic
276C23C7×#12011yescharacteristic
277C23C7×#12311yescharacteristic
278Q8—24no—
279Q8—42no—
280D4—24no—
281D4—42no—
282D4—44no—
283D4—44no—
284C2×C4—24no—
285C2×C4—22no—
286C2×C4—22no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—42no—
293C2×C4—48no—
294C2×C4—42no—
295C2×C4—42no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—42no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4—44no—
302C2×C4—44no—
303C2×C4—42no—
304C2×C4—42no—
305C2×C4—44no—
306C2×C4—81no—
307C2×C4C4×C7×D412yes—
308C2×C4C7×#12112yes—
309C8—82no—
310C7#948011yescharacteristic, direct product factor, Sylow
311C22—21no—
312C22—21no—
313C22—21no—
314C22—21no—
315C22—21no—
316C22—21no—
317C22—21no—
318C22—44no—
319C22—44no—
320C22—42no—
321C22—82no—
322C22—81no—
323C22—81no—
324C22C7×#38611yescharacteristic
325C22C7×#42112yes—
326C22C7×B(4,𝔽2)12yes—
327C4—22no—
328C4—22no—
329C4—42no—
330C4—42no—
331C4—42no—
332C4—84no—
333C4—82no—
334C2—22no—
335C2—22no—
336C2—41no—
337C2—82no—
338C2C7×#188911yescharacteristic
339C2C7×#189711yescharacteristic
340C2C7×#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×#2313C2semidirect
4. C7×#1463C2semidirect
5. C7×#1284C2semidirect
11. C7×#395C4semidirect
12. C7×#394C4semidirect
21. #9480C7direct (N × H)
55. #2313C2×C7semidirect
57. #1463C2×C7semidirect
58. #1284C2×C7semidirect
108. #395C4×C7semidirect
109. #394C4×C7semidirect
310. C7#9480direct (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×#23132
C7×#17892
C7×#14632
C7×#12842
#94807