Finite Group Databasebeta

← C7×#9206 #13186422

Subgroups — C7×#9206 #13186422

Order 1792.

578 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 578 conjugacy classes of subgroups — too many to draw. Only its 134 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×#2618C211yesmaximal, characteristic
2C7×#2313C211yesmaximal, characteristic, complemented
3C7×#1519C212yesmaximal
4C2×C7×#324C211yesmaximal, characteristic, complemented
5C7×#1289C212yesmaximal, complemented
6C2×C7×#123—21no—
7C2×C7×#121C2211yescharacteristic
8C22×C7×C4⋊C4C2211yescharacteristic
9C2×C7×C22⋊C8C2212yes—
10C7×#398C412yescomplemented
11C7×#396—22no—
12C7×#394C412yescomplemented
13C7×#387—24no—
14C7×#386—21no—
15C7×#381—21no—
16C7×#380C2211yescharacteristic
17C7×#380C2212yes—
18C4×C7×C22⋊C4—21no—
19C7×#327—24no—
20C7×#324—22no—
21#9206C711yesmaximal, characteristic, direct product factor, Sylow
22C22×C7×D4—21no—
23C22×C7×D4C2×C411yescharacteristic
24C23×C4×C7D411yescharacteristic
25C23×C4×C7D411yescharacteristic
26C23×C4×C7C2311yescharacteristic
27C22×C7×C8—22no—
28C7×#123—44no—
29C7×#121—22no—
30C7×#121—22no—
31C2×C7×C4⋊C4—22no—
32C2×C7×C4⋊C4—22no—
33C2×C7×C4⋊C4—22no—
34C2×C7×C4⋊C4—22no—
35C2×C7×C4⋊C4—21no—
36C2×C7×C4⋊C4C2×C411yescharacteristic
37C2×C7×C4⋊C4C2×C412yes—
38C2×C7×C4⋊C4D412yes—
39C2×C7×C22⋊C4—22no—
40C2×C7×C22⋊C4—21no—
41C2×C7×C22⋊C4—22no—
42C2×C7×C22⋊C4—21no—
43C2×C7×C22⋊C4—22no—
44C2×C42×C7—22no—
45C7×C22⋊C8—22no—
46C7×C22⋊C8—22no—
47C7×#95—22no—
48C7×#95—24no—
49C7×#95—22no—
50C7×#95C2×C412yes—
51C7×#95D414yes—
52#2618C2×C711yescharacteristic
53#2313C2×C711yescharacteristic, complemented
54#1519C2×C712yes—
55C2×#324C2×C711yescharacteristic, complemented
56#1289C2×C712yescomplemented
57C24×C7—21no—
58C24×C7C2×D411yescharacteristic
59C2×C7×D4—22no—
60C2×C7×D4—41no—
61C2×C7×D4—41no—
62C2×C7×D4—42no—
63C2×C7×D4—42no—
64C2×C7×D4—81no—
65C22×C4×C7—21no—
66C22×C4×C7—21no—
67C22×C4×C7—21no—
68C22×C4×C7—22no—
69C22×C4×C7—22no—
70C22×C4×C7—21no—
71C22×C4×C7—21no—
72C22×C4×C7—22no—
73C22×C4×C7—24no—
74C22×C4×C7—81no—
75C22×C4×C7—81no—
76C22×C4×C7C22⋊C412yes—
77C22×C4×C7C22⋊C412yes—
78C22×C4×C7C22×C411yescharacteristic
79C22×C4×C7C2×D412yes—
80C22×C4×C7C2×D411yescharacteristic
81C22×C4×C7#4112yes—
82C2×C7×C8—42no—
83C2×C7×C8—82no—
84C7×C4⋊C4—24no—
85C7×C4⋊C4—24no—
86C7×C4⋊C4—22no—
87C7×C4⋊C4—44no—
88C7×C4⋊C4—82no—
89C7×C4⋊C4—81no—
90C7×C22⋊C4—44no—
91C7×C22⋊C4—42no—
92C7×C22⋊C4—42no—
93C7×C22⋊C4—42no—
94C7×C22⋊C4—42no—
95C7×C22⋊C4—82no—
96C42×C7—82no—
97C2×#123—21no—
98C2×#121C22×C711yescharacteristic
99C22×C4⋊C4C22×C711yescharacteristic
100C2×C22⋊C8C22×C712yes—
101#398C4×C712yescomplemented
102#396—22no—
103#394C4×C712yescomplemented
104#387—24no—
105#386—21no—
106#381—21no—
107#380C22×C711yescharacteristic
108#380C22×C712yes—
109C4×C22⋊C4—21no—
110#327—24no—
111#324—22no—
112C23×C7—41no—
113C23×C7—41no—
114C23×C7—42no—
115C23×C7—42no—
116C23×C7—41no—
117C23×C7—41no—
118C23×C7—41no—
119C23×C7C2×C22⋊C411yescharacteristic
120C23×C7#12011yescharacteristic
121C23×C7#12311yescharacteristic
122C23×C7#13512yes—
123C23×C7AGL(1,ℤ/8)12yes—
124C7×D4—82no—
125C7×D4—82no—
126C7×D4—82no—
127C2×C4×C7—22no—
128C2×C4×C7—22no—
129C2×C4×C7—22no—
130C2×C4×C7—22no—
131C2×C4×C7—41no—
132C2×C4×C7—42no—
133C2×C4×C7—42no—
134C2×C4×C7—41no—
135C2×C4×C7—42no—
136C2×C4×C7—42no—
137C2×C4×C7—42no—
138C2×C4×C7—42no—
139C2×C4×C7—44no—
140C2×C4×C7—42no—
141C2×C4×C7—44no—
142C2×C4×C7—42no—
143C2×C4×C7—41no—
144C2×C4×C7—41no—
145C2×C4×C7—42no—
146C2×C4×C7—41no—
147C2×C4×C7—44no—
148C2×C4×C7—81no—
149C2×C4×C7C4×D412yes—
150C2×C4×C7#12112yes—
151C7×C8—82no—
152C22×D4—21no—
153C22×D4C2×C4×C711yescharacteristic
154C23×C4C7×D411yescharacteristic
155C23×C4C7×D411yescharacteristic
156C23×C4C23×C711yescharacteristic
157C22×C8—22no—
158#123—44no—
159#121—22no—
160#121—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4C2×C4×C711yescharacteristic
167C2×C4⋊C4C2×C4×C712yes—
168C2×C4⋊C4C7×D412yes—
169C2×C22⋊C4—21no—
170C2×C22⋊C4—22no—
171C2×C22⋊C4—22no—
172C2×C22⋊C4—21no—
173C2×C22⋊C4—22no—
174C2×C42—22no—
175C22⋊C8—22no—
176C22⋊C8—22no—
177#95—22no—
178#95—24no—
179#95—22no—
180#95C2×C4×C712yes—
181#95C7×D414yes—
182C22×C7—21no—
183C22×C7—21no—
184C22×C7—21no—
185C22×C7—21no—
186C22×C7—21no—
187C22×C7—21no—
188C22×C7—42no—
189C22×C7—42no—
190C22×C7—82no—
191C22×C7—82no—
192C22×C7—81no—
193C22×C7—81no—
194C22×C7—81no—
195C22×C7#38611yescharacteristic
196C22×C7#41611yescharacteristic
197C22×C7#41811yescharacteristic
198C22×C7#44912yes—
199C22×C7#48212yes—
200C4×C7—22no—
201C4×C7—22no—
202C4×C7—42no—
203C4×C7—42no—
204C4×C7—82no—
205C4×C7—84no—
206C4×C7—81no—
207C24—21no—
208C24C2×C7×D411yescharacteristic
209C2×D4—22no—
210C2×D4—41no—
211C2×D4—41no—
212C2×D4—42no—
213C2×D4—42no—
214C2×D4—81no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—22no—
223C22×C4—24no—
224C22×C4—81no—
225C22×C4—81no—
226C22×C4C7×C22⋊C412yes—
227C22×C4C7×C22⋊C412yes—
228C22×C4C22×C4×C711yescharacteristic
229C22×C4C2×C7×D412yes—
230C22×C4C2×C7×D411yescharacteristic
231C22×C4C7×#4112yes—
232C2×C8—42no—
233C2×C8—82no—
234C4⋊C4—24no—
235C4⋊C4—24no—
236C4⋊C4—22no—
237C4⋊C4—44no—
238C4⋊C4—82no—
239C4⋊C4—81no—
240C22⋊C4—44no—
241C22⋊C4—42no—
242C22⋊C4—42no—
243C22⋊C4—42no—
244C22⋊C4—42no—
245C22⋊C4—82no—
246C42—82no—
247C2×C7—21no—
248C2×C7—22no—
249C2×C7—21no—
250C2×C7—22no—
251C2×C7—161no—
252C2×C7#187511yescharacteristic
253C2×C7#191211yescharacteristic
254C2×C7#191311yescharacteristic
255C23—41no—
256C23—41no—
257C23—42no—
258C23—42no—
259C23—41no—
260C23—41no—
261C23—41no—
262C23C2×C7×C22⋊C411yescharacteristic
263C23C7×#12011yescharacteristic
264C23C7×#12311yescharacteristic
265C23C7×#13512yes—
266C23C7×AGL(1,ℤ/8)12yes—
267D4—82no—
268D4—82no—
269D4—82no—
270C2×C4—22no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—41no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—41no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—44no—
283C2×C4—42no—
284C2×C4—44no—
285C2×C4—42no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—42no—
289C2×C4—41no—
290C2×C4—44no—
291C2×C4—81no—
292C2×C4C4×C7×D412yes—
293C2×C4C7×#12112yes—
294C8—82no—
295C7#920611yescharacteristic, direct product factor, Sylow
296C22—21no—
297C22—21no—
298C22—21no—
299C22—21no—
300C22—21no—
301C22—21no—
302C22—42no—
303C22—42no—
304C22—82no—
305C22—82no—
306C22—81no—
307C22—81no—
308C22—81no—
309C22C7×#38611yescharacteristic
310C22C7×#41611yescharacteristic
311C22C7×#41811yescharacteristic
312C22C7×#44912yes—
313C22C7×#48212yes—
314C4—22no—
315C4—22no—
316C4—42no—
317C4—42no—
318C4—82no—
319C4—84no—
320C4—81no—
321C2—21no—
322C2—22no—
323C2—21no—
324C2—22no—
325C2—161no—
326C2C7×#187511yescharacteristic
327C2C7×#191211yescharacteristic
328C2C7×#191311yescharacteristic
subgroupG/H
center, socleC22×C7#386
commutatorC22×C4C22×C4×C7
Frattini, cosocleC23×C4C23×C7
prime radicalC2×C7×#121C22
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. C2×C7×#324C2semidirect
5. C7×#1289C2semidirect
10. C7×#398C4semidirect
12. C7×#394C4semidirect
21. #9206C7direct (N × H)
53. #2313C2×C7semidirect
55. C2×#324C2×C7semidirect
56. #1289C2×C7semidirect
101. #398C4×C7semidirect
103. #394C4×C7semidirect
295. C7#9206direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#26182
C7×#23132
C7×#15192
C2×C7×#3242
C7×#12892
#92067