Finite Group Databasebeta

← C7×#3241 #11742550

Subgroups — C7×#3241 #11742550

Order 896.

332 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#544C211yesmaximal, characteristic, complemented
2C7×#540C211yesmaximal, characteristic
3C7×#501C211yesmaximal, characteristic, complemented
4C7×#499C211yesmaximal, characteristic, complemented
5C7×#490C211yesmaximal, characteristic, complemented
6C7×#486C211yesmaximal, characteristic, complemented
7C7×#482C211yesmaximal, characteristic, complemented
8C7×#482C211yesmaximal, characteristic, complemented
9C7×#481C211yesmaximal, characteristic, complemented
10C7×#480C211yesmaximal, characteristic, complemented
11C7×#478C211yesmaximal, characteristic, complemented
12C7×#478C211yesmaximal, characteristic, complemented
13C7×#476C211yesmaximal, characteristic, complemented
14C7×#474C211yesmaximal, characteristic, complemented
15C7×#433C211yesmaximal, characteristic, complemented
16C7×C4⋊Q8—21no—
17C7×C4⋊Q8C2211yescharacteristic
18C7×C4⋊Q8C2211yescharacteristic
19C7×#127C2211yescharacteristic
20C7×#125—21no—
21C7×#125C2211yescharacteristic
22C7×#125C2211yescharacteristic
23C7×#125C2211yescharacteristic
24C7×#123—21no—
25C7×#123—21no—
26C7×C22⋊Q8—21no—
27C7×C22⋊Q8—21no—
28C7×#121—21no—
29C7×#121—21no—
30C7×#121C2211yescharacteristic
31C7×#121C2211yescharacteristic
32C4×C7×D4C2211yescharacteristic
33C4×C7×D4C2211yescharacteristic
34C7×#117C2211yescharacteristic
35C7×#117C2211yescharacteristic
36C2×C7×C4⋊C4C2211yescharacteristic
37C7×#107C2211yescharacteristic
38C7×#107C2211yescharacteristic
39C7×#107C2211yescharacteristic
40C7×#107C2211yescharacteristic
41C7×#106C2211yescharacteristic
42C7×#106C2211yescharacteristic
43C7×#106C2211yescharacteristic
44C7×#106C2211yescharacteristic
45C7×C4⋊C8C2211yescharacteristic
46C7×C4⋊C8C2211yescharacteristic
47C7×#102C2211yescharacteristic
48C7×#102C2211yescharacteristic
49C7×#102C2211yescharacteristic
50C7×#102C2211yescharacteristic
51C7×#102C2211yescharacteristic
52C7×#102C2211yescharacteristic
53C7×#102C2211yescharacteristic
54C7×#102C2211yescharacteristic
55C7×C22⋊C8C2211yescharacteristic
56C7×C22⋊C8C2211yescharacteristic
57C7×#97C2211yescharacteristic
58C4×C7×C8C2211yescharacteristic
59#3241C711yesmaximal, characteristic, direct product factor, Sylow
60C2×C7×Q8—21no—
61C2×C7×Q8—21no—
62C2×C7×D4—21no—
63C2×C7×D4—21no—
64C2×C7×D4—21no—
65C2×C7×D4—21no—
66C2×C7×D4C2311yescharacteristic
67C2×C7×D4C2311yescharacteristic
68C22×C4×C7—21no—
69C22×C4×C7—21no—
70C22×C4×C7—21no—
71C22×C4×C7C2311yescharacteristic
72C2×C7×C8C2311yescharacteristic
73C2×C7×C8C2311yescharacteristic
74C2×C7×C8C2311yescharacteristic
75C2×C7×C8C2311yescharacteristic
76C7×C4⋊C4—21no—
77C7×C4⋊C4—21no—
78C7×C4⋊C4—21no—
79C7×C4⋊C4—21no—
80C7×C4⋊C4—21no—
81C7×C4⋊C4—21no—
82C7×C4⋊C4—21no—
83C7×C4⋊C4—21no—
84C7×C4⋊C4—21no—
85C7×C4⋊C4D411yescharacteristic
86C7×C4⋊C4D411yescharacteristic
87C7×C4⋊C4C2311yescharacteristic
88C7×C4⋊C4C2311yescharacteristic
89C7×C4⋊C4C2311yescharacteristic
90C7×C4⋊C4C2311yescharacteristic
91C7×C4⋊C4C2311yescharacteristic
92C7×C4⋊C4C2311yescharacteristic
93C7×C22⋊C4—21no—
94C7×C22⋊C4—21no—
95C7×C22⋊C4—21no—
96C7×C22⋊C4—21no—
97C7×C22⋊C4—21no—
98C7×C22⋊C4D411yescharacteristic
99C7×C22⋊C4D411yescharacteristic
100C42×C7—21no—
101C42×C7C2311yescharacteristic
102C42×C7C2311yescharacteristic
103#544C2×C711yescharacteristic, complemented
104#540C2×C711yescharacteristic
105#501C2×C711yescharacteristic, complemented
106#499C2×C711yescharacteristic, complemented
107#490C2×C711yescharacteristic, complemented
108#486C2×C711yescharacteristic, complemented
109#482C2×C711yescharacteristic, complemented
110#482C2×C711yescharacteristic, complemented
111#481C2×C711yescharacteristic, complemented
112#480C2×C711yescharacteristic, complemented
113#478C2×C711yescharacteristic, complemented
114#478C2×C711yescharacteristic, complemented
115#476C2×C711yescharacteristic, complemented
116#474C2×C711yescharacteristic, complemented
117#433C2×C711yescharacteristic, complemented
118C23×C7—21no—
119C23×C7—21no—
120C23×C7C2×D411yescharacteristic
121C7×Q8—81no—
122C7×Q8—81no—
123C7×D4—21no—
124C7×D4—21no—
125C7×D4—21no—
126C7×D4—21no—
127C7×D4—81no—
128C7×D4—81no—
129C7×D4—81no—
130C7×D4—81no—
131C2×C4×C7—21no—
132C2×C4×C7—21no—
133C2×C4×C7—21no—
134C2×C4×C7—21no—
135C2×C4×C7—21no—
136C2×C4×C7—21no—
137C2×C4×C7—21no—
138C2×C4×C7—21no—
139C2×C4×C7—81no—
140C2×C4×C7—81no—
141C2×C4×C7—81no—
142C2×C4×C7C2×D411yescharacteristic
143C2×C4×C7C2×D411yescharacteristic
144C2×C4×C7C2×D411yescharacteristic
145C2×C4×C7C2×D411yescharacteristic
146C2×C4×C7C2×D411yescharacteristic
147C2×C4×C7C2411yescharacteristic
148C7×C8—21no—
149C7×C8—21no—
150C7×C8—21no—
151C7×C8—21no—
152C4⋊Q8—21no—
153C4⋊Q8C22×C711yescharacteristic
154C4⋊Q8C22×C711yescharacteristic
155#127C22×C711yescharacteristic
156#125—21no—
157#125C22×C711yescharacteristic
158#125C22×C711yescharacteristic
159#125C22×C711yescharacteristic
160#123—21no—
161#123—21no—
162C22⋊Q8—21no—
163C22⋊Q8—21no—
164#121—21no—
165#121—21no—
166#121C22×C711yescharacteristic
167#121C22×C711yescharacteristic
168C4×D4C22×C711yescharacteristic
169C4×D4C22×C711yescharacteristic
170#117C22×C711yescharacteristic
171#117C22×C711yescharacteristic
172C2×C4⋊C4C22×C711yescharacteristic
173#107C22×C711yescharacteristic
174#107C22×C711yescharacteristic
175#107C22×C711yescharacteristic
176#107C22×C711yescharacteristic
177#106C22×C711yescharacteristic
178#106C22×C711yescharacteristic
179#106C22×C711yescharacteristic
180#106C22×C711yescharacteristic
181C4⋊C8C22×C711yescharacteristic
182C4⋊C8C22×C711yescharacteristic
183#102C22×C711yescharacteristic
184#102C22×C711yescharacteristic
185#102C22×C711yescharacteristic
186#102C22×C711yescharacteristic
187#102C22×C711yescharacteristic
188#102C22×C711yescharacteristic
189#102C22×C711yescharacteristic
190#102C22×C711yescharacteristic
191C22⋊C8C22×C711yescharacteristic
192C22⋊C8C22×C711yescharacteristic
193#97C22×C711yescharacteristic
194C4×C8C22×C711yescharacteristic
195C22×C7—21no—
196C22×C7—21no—
197C22×C7—21no—
198C22×C7—41no—
199C22×C7—41no—
200C22×C7—41no—
201C22×C7—41no—
202C22×C7—41no—
203C22×C7—41no—
204C22×C7C22×D411yescharacteristic
205C4×C7—21no—
206C4×C7—21no—
207C4×C7—21no—
208C4×C7—21no—
209C4×C7—21no—
210C4×C7—41no—
211C4×C7—41no—
212C4×C7—41no—
213C4×C7—41no—
214C4×C7—41no—
215C4×C7—41no—
216C4×C72−1+411yescharacteristic
217C4×C72−1+411yescharacteristic
218C2×Q8—21no—
219C2×Q8—21no—
220C2×D4—21no—
221C2×D4—21no—
222C2×D4—21no—
223C2×D4—21no—
224C2×D4C23×C711yescharacteristic
225C2×D4C23×C711yescharacteristic
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4C23×C711yescharacteristic
230C2×C8C23×C711yescharacteristic
231C2×C8C23×C711yescharacteristic
232C2×C8C23×C711yescharacteristic
233C2×C8C23×C711yescharacteristic
234C4⋊C4—21no—
235C4⋊C4—21no—
236C4⋊C4—21no—
237C4⋊C4—21no—
238C4⋊C4—21no—
239C4⋊C4—21no—
240C4⋊C4—21no—
241C4⋊C4—21no—
242C4⋊C4—21no—
243C4⋊C4C7×D411yescharacteristic
244C4⋊C4C7×D411yescharacteristic
245C4⋊C4C23×C711yescharacteristic
246C4⋊C4C23×C711yescharacteristic
247C4⋊C4C23×C711yescharacteristic
248C4⋊C4C23×C711yescharacteristic
249C4⋊C4C23×C711yescharacteristic
250C4⋊C4C23×C711yescharacteristic
251C22⋊C4—21no—
252C22⋊C4—21no—
253C22⋊C4—21no—
254C22⋊C4—21no—
255C22⋊C4—21no—
256C22⋊C4C7×D411yescharacteristic
257C22⋊C4C7×D411yescharacteristic
258C42—21no—
259C42C23×C711yescharacteristic
260C42C23×C711yescharacteristic
261C2×C7—41no—
262C2×C7—81no—
263C2×C7—81no—
264C2×C7#53611yescharacteristic
265C2×C7#57611yescharacteristic
266C2×C7#57711yescharacteristic
267C23—21no—
268C23—21no—
269C23C2×C7×D411yescharacteristic
270Q8—81no—
271Q8—81no—
272D4—21no—
273D4—21no—
274D4—21no—
275D4—21no—
276D4—81no—
277D4—81no—
278D4—81no—
279D4—81no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—81no—
289C2×C4—81no—
290C2×C4—81no—
291C2×C4C2×C7×D411yescharacteristic
292C2×C4C2×C7×D411yescharacteristic
293C2×C4C2×C7×D411yescharacteristic
294C2×C4C2×C7×D411yescharacteristic
295C2×C4C2×C7×D411yescharacteristic
296C2×C4C24×C711yescharacteristic
297C8—21no—
298C8—21no—
299C8—21no—
300C8—21no—
301C7#324111yescharacteristic, direct product factor, Sylow
302C22—21no—
303C22—21no—
304C22—21no—
305C22—41no—
306C22—41no—
307C22—41no—
308C22—41no—
309C22—41no—
310C22—41no—
311C22C22×C7×D411yescharacteristic
312C4—21no—
313C4—21no—
314C4—21no—
315C4—21no—
316C4—21no—
317C4—41no—
318C4—41no—
319C4—41no—
320C4—41no—
321C4—41no—
322C4—41no—
323C4C7×2−1+411yescharacteristic
324C4C7×2−1+411yescharacteristic
325C2—41no—
326C2—81no—
327C2—81no—
328C2C7×#53611yescharacteristic
329C2C7×#57611yescharacteristic
330C2C7×#57711yescharacteristic
subgroupG/H
center, socleC22×C7C22×D4
commutator, Frattini, cosocleC2×C4C24×C7
prime radicalC7×#540C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C7 ◁ C7×#117 ◁ G

Complements

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

NH ≅ G/Ntype
1. C7×#544C2semidirect
3. C7×#501C2semidirect
4. C7×#499C2semidirect
5. C7×#490C2semidirect
6. C7×#486C2semidirect
7. C7×#482C2semidirect
8. C7×#482C2semidirect
9. C7×#481C2semidirect
10. C7×#480C2semidirect
11. C7×#478C2semidirect
12. C7×#478C2semidirect
13. C7×#476C2semidirect
14. C7×#474C2semidirect
15. C7×#433C2semidirect
59. #3241C7direct (N × H)
103. #544C2×C7semidirect
105. #501C2×C7semidirect
106. #499C2×C7semidirect
107. #490C2×C7semidirect
108. #486C2×C7semidirect
109. #482C2×C7semidirect
110. #482C2×C7semidirect
111. #481C2×C7semidirect
112. #480C2×C7semidirect
113. #478C2×C7semidirect
114. #478C2×C7semidirect
115. #476C2×C7semidirect
116. #474C2×C7semidirect
117. #433C2×C7semidirect
301. C7#3241direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#5442
C7×#5402
C7×#5012
C7×#4992
C7×#4902
C7×#4862
C7×#4822
C7×#4812
C7×#4802
C7×#4782
C7×#4762
C7×#4742
C7×#4332
#32417