Finite Group Databasebeta

← C7×#3371 #11742680

Subgroups — C7×#3371 #11742680

Order 896.

344 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#559C211yesmaximal, characteristic
2C7×#557C211yesmaximal, characteristic, complemented
3C7×#490C211yesmaximal, characteristic
4C7×#489C211yesmaximal, characteristic, complemented
5C7×#488C211yesmaximal, characteristic, complemented
6C7×#479C211yesmaximal, characteristic, complemented
7C7×#478C211yesmaximal, characteristic, complemented
8C7×#476C211yesmaximal, characteristic, complemented
9C7×#464C211yesmaximal, characteristic, complemented
10C7×#464C211yesmaximal, characteristic, complemented
11C7×#460C211yesmaximal, characteristic
12C7×#446C211yesmaximal, characteristic, complemented
13C7×#441C211yesmaximal, characteristic, complemented
14C7×#440C211yesmaximal, characteristic, complemented
15C4×C7×SD16C211yesmaximal, characteristic, complemented
16C2×C7×Q16C2211yescharacteristic
17C2×C7×SD16C2211yescharacteristic
18C2×C7×SD16C2211yescharacteristic
19C7×C4⋊Q8—21no—
20C7×C4⋊Q8C2211yescharacteristic
21C7×#127C2211yescharacteristic
22C7×#125—21no—
23C7×#125—21no—
24C7×#125C2211yescharacteristic
25C7×#125C2211yescharacteristic
26C7×#124—21no—
27C7×#124C2211yescharacteristic
28C7×#124C2211yescharacteristic
29C7×#123—21no—
30C7×#123—21no—
31C4×C7×Q8—21no—
32C4×C7×Q8C2211yescharacteristic
33C4×C7×Q8C2211yescharacteristic
34C4×C7×Q8C2211yescharacteristic
35C4×C7×Q8C2211yescharacteristic
36C4×C7×Q8C2211yescharacteristic
37C4×C7×D4—21no—
38C4×C7×D4C2211yescharacteristic
39C4×C7×D4C2211yescharacteristic
40C7×#107C2211yescharacteristic
41C7×#106C2211yescharacteristic
42C7×#106C2211yescharacteristic
43C7×C4⋊C8C2211yescharacteristic
44C7×C4⋊C8C2211yescharacteristic
45C7×C4⋊C8C2211yescharacteristic
46C7×#103C2211yescharacteristic
47C7×#103C2211yescharacteristic
48C7×#103C2211yescharacteristic
49C7×#103C2211yescharacteristic
50C7×#102C2211yescharacteristic
51C7×#102C2211yescharacteristic
52C7×#102C2211yescharacteristic
53C7×#102C2211yescharacteristic
54C7×#102C2211yescharacteristic
55C7×#102C2211yescharacteristic
56C7×#97C2211yescharacteristic
57C7×#97C2211yescharacteristic
58C4×C7×C8C2211yescharacteristic
59#3371C711yesmaximal, characteristic, direct product factor, Sylow
60C2×C7×Q8—21no—
61C2×C7×Q8D411yescharacteristic
62C2×C7×Q8C2311yescharacteristic
63C2×C7×Q8C2311yescharacteristic
64C2×C7×D4—21no—
65C2×C7×D4—21no—
66C2×C7×D4C2311yescharacteristic
67C2×C7×D4C2311yescharacteristic
68C22×C4×C7—21no—
69C22×C4×C7—21no—
70C7×Q16—22no—
71C7×SD16—22no—
72C7×SD16—22no—
73C2×C7×C8C2311yescharacteristic
74C2×C7×C8C2311yescharacteristic
75C2×C7×C8C2311yescharacteristic
76C2×C7×C8C2311yescharacteristic
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⋊C4—21no—
86C7×C4⋊C4D411yescharacteristic
87C7×C4⋊C4D411yescharacteristic
88C7×C4⋊C4D411yescharacteristic
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⋊C4—21no—
99C42×C7—21no—
100C42×C7—21no—
101C42×C7—21no—
102C42×C7—21no—
103C42×C7C2311yescharacteristic
104C42×C7C2311yescharacteristic
105C42×C7C2311yescharacteristic
106#559C2×C711yescharacteristic
107#557C2×C711yescharacteristic, complemented
108#490C2×C711yescharacteristic
109#489C2×C711yescharacteristic, complemented
110#488C2×C711yescharacteristic, complemented
111#479C2×C711yescharacteristic, complemented
112#478C2×C711yescharacteristic, complemented
113#476C2×C711yescharacteristic, complemented
114#464C2×C711yescharacteristic, complemented
115#464C2×C711yescharacteristic, complemented
116#460C2×C711yescharacteristic
117#446C2×C711yescharacteristic, complemented
118#441C2×C711yescharacteristic, complemented
119#440C2×C711yescharacteristic, complemented
120C4×SD16C2×C711yescharacteristic, complemented
121C23×C7—21no—
122C23×C7—21no—
123C7×Q8—21no—
124C7×Q8—21no—
125C7×Q8—21no—
126C7×Q8—41no—
127C7×Q8—42no—
128C7×Q8#4112yes—
129C7×D4—21no—
130C7×D4—21no—
131C7×D4—21no—
132C7×D4—21no—
133C7×D4—42no—
134C7×D4—81no—
135C2×C4×C7—21no—
136C2×C4×C7—21no—
137C2×C4×C7—21no—
138C2×C4×C7—21no—
139C2×C4×C7—21no—
140C2×C4×C7—21no—
141C2×C4×C7—81no—
142C2×C4×C7—81no—
143C2×C4×C7C2×D411yescharacteristic
144C2×C4×C7C2×D411yescharacteristic
145C2×C4×C7C2×D411yescharacteristic
146C2×C4×C7C2×D411yescharacteristic
147C2×C4×C7C2×D411yescharacteristic
148C2×C4×C7C2×D411yescharacteristic
149C2×C4×C7C2411yescharacteristic
150C7×C8—21no—
151C7×C8—21no—
152C7×C8—21no—
153C7×C8—21no—
154C2×Q16C22×C711yescharacteristic
155C2×SD16C22×C711yescharacteristic
156C2×SD16C22×C711yescharacteristic
157C4⋊Q8—21no—
158C4⋊Q8C22×C711yescharacteristic
159#127C22×C711yescharacteristic
160#125—21no—
161#125—21no—
162#125C22×C711yescharacteristic
163#125C22×C711yescharacteristic
164#124—21no—
165#124C22×C711yescharacteristic
166#124C22×C711yescharacteristic
167#123—21no—
168#123—21no—
169C4×Q8—21no—
170C4×Q8C22×C711yescharacteristic
171C4×Q8C22×C711yescharacteristic
172C4×Q8C22×C711yescharacteristic
173C4×Q8C22×C711yescharacteristic
174C4×Q8C22×C711yescharacteristic
175C4×D4—21no—
176C4×D4C22×C711yescharacteristic
177C4×D4C22×C711yescharacteristic
178#107C22×C711yescharacteristic
179#106C22×C711yescharacteristic
180#106C22×C711yescharacteristic
181C4⋊C8C22×C711yescharacteristic
182C4⋊C8C22×C711yescharacteristic
183C4⋊C8C22×C711yescharacteristic
184#103C22×C711yescharacteristic
185#103C22×C711yescharacteristic
186#103C22×C711yescharacteristic
187#103C22×C711yescharacteristic
188#102C22×C711yescharacteristic
189#102C22×C711yescharacteristic
190#102C22×C711yescharacteristic
191#102C22×C711yescharacteristic
192#102C22×C711yescharacteristic
193#102C22×C711yescharacteristic
194#97C22×C711yescharacteristic
195#97C22×C711yescharacteristic
196C4×C8C22×C711yescharacteristic
197C22×C7—41no—
198C22×C7—41no—
199C22×C7—41no—
200C22×C7—41no—
201C22×C7—41no—
202C22×C7—41no—
203C22×C7C22×D411yescharacteristic
204C4×C7—22no—
205C4×C7—21no—
206C4×C7—21no—
207C4×C7—21no—
208C4×C7—21no—
209C4×C7—21no—
210C4×C7—21no—
211C4×C7—41no—
212C4×C7—41no—
213C4×C7—41no—
214C4×C7—41no—
215C4×C7—41no—
216C4×C7C2×#4111yescharacteristic
217C4×C72−1+411yescharacteristic
218C2×Q8—21no—
219C2×Q8C7×D411yescharacteristic
220C2×Q8C23×C711yescharacteristic
221C2×Q8C23×C711yescharacteristic
222C2×D4—21no—
223C2×D4—21no—
224C2×D4C23×C711yescharacteristic
225C2×D4C23×C711yescharacteristic
226C22×C4—21no—
227C22×C4—21no—
228Q16—22no—
229SD16—22no—
230SD16—22no—
231C2×C8C23×C711yescharacteristic
232C2×C8C23×C711yescharacteristic
233C2×C8C23×C711yescharacteristic
234C2×C8C23×C711yescharacteristic
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⋊C4—21no—
244C4⋊C4C7×D411yescharacteristic
245C4⋊C4C7×D411yescharacteristic
246C4⋊C4C7×D411yescharacteristic
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⋊C4—21no—
257C42—21no—
258C42—21no—
259C42—21no—
260C42—21no—
261C42C23×C711yescharacteristic
262C42C23×C711yescharacteristic
263C42C23×C711yescharacteristic
264C2×C7—81no—
265C2×C7—81no—
266C2×C7#54811yescharacteristic
267C2×C7#57511yescharacteristic
268C2×C7#57711yescharacteristic
269C23—21no—
270C23—21no—
271Q8—21no—
272Q8—21no—
273Q8—21no—
274Q8—41no—
275Q8—42no—
276Q8C7×#4112yes—
277D4—21no—
278D4—21no—
279D4—21no—
280D4—21no—
281D4—42no—
282D4—81no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
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×C4C2×C7×D411yescharacteristic
297C2×C4C24×C711yescharacteristic
298C8—21no—
299C8—21no—
300C8—21no—
301C8—21no—
302C7#337111yescharacteristic, direct product factor, Sylow
303C22—41no—
304C22—41no—
305C22—41no—
306C22—41no—
307C22—41no—
308C22—41no—
309C22C22×C7×D411yescharacteristic
310C4—22no—
311C4—21no—
312C4—21no—
313C4—21no—
314C4—21no—
315C4—21no—
316C4—21no—
317C4—41no—
318C4—41no—
319C4—41no—
320C4—41no—
321C4—41no—
322C4C2×C7×#4111yescharacteristic
323C4C7×2−1+411yescharacteristic
324C2—81no—
325C2—81no—
326C2C7×#54811yescharacteristic
327C2C7×#57511yescharacteristic
328C2C7×#57711yescharacteristic
subgroupG/H
center, socleC22×C7C22×D4
commutator, Frattini, cosocleC2×C4C24×C7
prime radicalC7×#127C22
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C7 ◁ C4×C7×Q8 ◁ G

Complements

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

NH ≅ G/Ntype
2. C7×#557C2semidirect
4. C7×#489C2semidirect
5. C7×#488C2semidirect
6. C7×#479C2semidirect
7. C7×#478C2semidirect
8. C7×#476C2semidirect
9. C7×#464C2semidirect
10. C7×#464C2semidirect
12. C7×#446C2semidirect
13. C7×#441C2semidirect
14. C7×#440C2semidirect
15. C4×C7×SD16C2semidirect
59. #3371C7direct (N × H)
107. #557C2×C7semidirect
109. #489C2×C7semidirect
110. #488C2×C7semidirect
111. #479C2×C7semidirect
112. #478C2×C7semidirect
113. #476C2×C7semidirect
114. #464C2×C7semidirect
115. #464C2×C7semidirect
117. #446C2×C7semidirect
118. #441C2×C7semidirect
119. #440C2×C7semidirect
120. C4×SD16C2×C7semidirect
302. C7#3371direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#5592
C7×#5572
C7×#4902
C7×#4892
C7×#4882
C7×#4792
C7×#4782
C7×#4762
C7×#4642
C7×#4602
C7×#4462
C7×#4412
C7×#4402
C4×C7×SD162
#33717