Finite Group Databasebeta

← C7×#3349 #11742658

Subgroups — C7×#3349 #11742658

Order 896.

336 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#556C211yesmaximal, characteristic
2C7×#552C211yesmaximal, characteristic
3C7×#501C211yesmaximal, characteristic, complemented
4C7×#485C211yesmaximal, characteristic, complemented
5C7×#485C211yesmaximal, characteristic, complemented
6C7×#485C211yesmaximal, characteristic, complemented
7C7×#484C211yesmaximal, characteristic, complemented
8C7×#479C211yesmaximal, characteristic, complemented
9C7×#477C211yesmaximal, characteristic, complemented
10C7×#470C211yesmaximal, characteristic, complemented
11C7×#467C211yesmaximal, characteristic, complemented
12C7×#441C211yesmaximal, characteristic, complemented
13C7×#435C211yesmaximal, characteristic
14C7×#428C211yesmaximal, characteristic, complemented
15C7×#427C211yesmaximal, characteristic, complemented
16C2×C7×Q16C2211yescharacteristic
17C2×C7×M16C2211yescharacteristic
18C22×C7×C8C2211yescharacteristic
19C7×C4⋊Q8C2211yescharacteristic
20C7×#126—21no—
21C7×#126—21no—
22C7×#126—21no—
23C7×#125—21no—
24C7×#125C2211yescharacteristic
25C7×#124—21no—
26C7×#123—21no—
27C7×C22⋊Q8C2211yescharacteristic
28C7×C22⋊Q8C2211yescharacteristic
29C7×C22⋊Q8C2211yescharacteristic
30C7×C22⋊Q8C2211yescharacteristic
31C4×C7×Q8—21no—
32C4×C7×Q8C2211yescharacteristic
33C4×C7×Q8C2211yescharacteristic
34C4×C7×D4C2211yescharacteristic
35C7×#117—21no—
36C7×#117C2211yescharacteristic
37C7×#117C2211yescharacteristic
38C7×#117C2211yescharacteristic
39C2×C7×C4⋊C4C2211yescharacteristic
40C7×#107C2211yescharacteristic
41C7×#107C2211yescharacteristic
42C7×#107C2211yescharacteristic
43C7×#107C2211yescharacteristic
44C7×#107C2211yescharacteristic
45C7×#106C2211yescharacteristic
46C7×#106C2211yescharacteristic
47C7×#106C2211yescharacteristic
48C7×#106C2211yescharacteristic
49C7×C4⋊C8C2211yescharacteristic
50C7×#103C2211yescharacteristic
51C7×#103C2211yescharacteristic
52C7×#103C2211yescharacteristic
53C7×#103C2211yescharacteristic
54C7×#103C2211yescharacteristic
55C7×#103C2211yescharacteristic
56C7×C22⋊C8C2211yescharacteristic
57C7×C22⋊C8C2211yescharacteristic
58C7×#97C2211yescharacteristic
59#3349C711yesmaximal, characteristic, direct product factor, Sylow
60C2×C7×Q8—21no—
61C2×C7×Q8C2311yescharacteristic
62C2×C7×Q8C2311yescharacteristic
63C2×C7×D4D411yescharacteristic
64C22×C4×C7—21no—
65C22×C4×C7C2311yescharacteristic
66C22×C4×C7C2311yescharacteristic
67C7×Q16—22no—
68C7×M16—22no—
69C2×C7×C8—22no—
70C2×C7×C8C2311yescharacteristic
71C2×C7×C8C2311yescharacteristic
72C2×C7×C8C2311yescharacteristic
73C2×C7×C8C2311yescharacteristic
74C7×C4⋊C4—21no—
75C7×C4⋊C4—21no—
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⋊C4D411yescharacteristic
85C7×C4⋊C4C2311yescharacteristic
86C7×C4⋊C4C2311yescharacteristic
87C7×C4⋊C4C2311yescharacteristic
88C7×C4⋊C4C2311yescharacteristic
89C7×C4⋊C4C2311yescharacteristic
90C7×C4⋊C4C2311yescharacteristic
91C7×C22⋊C4—21no—
92C7×C22⋊C4—21no—
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×C7—21no—
102C42×C7—21no—
103C42×C7—21no—
104C42×C7—21no—
105C42×C7C2311yescharacteristic
106#556C2×C711yescharacteristic
107#552C2×C711yescharacteristic
108#501C2×C711yescharacteristic, complemented
109#485C2×C711yescharacteristic, complemented
110#485C2×C711yescharacteristic, complemented
111#485C2×C711yescharacteristic, complemented
112#484C2×C711yescharacteristic, complemented
113#479C2×C711yescharacteristic, complemented
114#477C2×C711yescharacteristic, complemented
115#470C2×C711yescharacteristic, complemented
116#467C2×C711yescharacteristic, complemented
117#441C2×C711yescharacteristic, complemented
118#435C2×C711yescharacteristic
119#428C2×C711yescharacteristic, complemented
120#427C2×C711yescharacteristic, complemented
121C23×C7C2×D411yescharacteristic
122C23×C7C2×D411yescharacteristic
123C7×Q8—21no—
124C7×Q8—21no—
125C7×Q8—21no—
126C7×Q8—21no—
127C7×Q8—42no—
128C7×D4—41no—
129C2×C4×C7—21no—
130C2×C4×C7—21no—
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—21no—
140C2×C4×C7—21no—
141C2×C4×C7—42no—
142C2×C4×C7C2×D411yescharacteristic
143C2×C4×C7C2×D411yescharacteristic
144C2×C4×C7C2×D411yescharacteristic
145C2×C4×C7C2×D411yescharacteristic
146C2×C4×C7C2411yescharacteristic
147C7×C8—21no—
148C7×C8—21no—
149C7×C8—21no—
150C7×C8#4112yes—
151C2×Q16C22×C711yescharacteristic
152C2×M16C22×C711yescharacteristic
153C22×C8C22×C711yescharacteristic
154C4⋊Q8C22×C711yescharacteristic
155#126—21no—
156#126—21no—
157#126—21no—
158#125—21no—
159#125C22×C711yescharacteristic
160#124—21no—
161#123—21no—
162C22⋊Q8C22×C711yescharacteristic
163C22⋊Q8C22×C711yescharacteristic
164C22⋊Q8C22×C711yescharacteristic
165C22⋊Q8C22×C711yescharacteristic
166C4×Q8—21no—
167C4×Q8C22×C711yescharacteristic
168C4×Q8C22×C711yescharacteristic
169C4×D4C22×C711yescharacteristic
170#117—21no—
171#117C22×C711yescharacteristic
172#117C22×C711yescharacteristic
173#117C22×C711yescharacteristic
174C2×C4⋊C4C22×C711yescharacteristic
175#107C22×C711yescharacteristic
176#107C22×C711yescharacteristic
177#107C22×C711yescharacteristic
178#107C22×C711yescharacteristic
179#107C22×C711yescharacteristic
180#106C22×C711yescharacteristic
181#106C22×C711yescharacteristic
182#106C22×C711yescharacteristic
183#106C22×C711yescharacteristic
184C4⋊C8C22×C711yescharacteristic
185#103C22×C711yescharacteristic
186#103C22×C711yescharacteristic
187#103C22×C711yescharacteristic
188#103C22×C711yescharacteristic
189#103C22×C711yescharacteristic
190#103C22×C711yescharacteristic
191C22⋊C8C22×C711yescharacteristic
192C22⋊C8C22×C711yescharacteristic
193#97C22×C711yescharacteristic
194C22×C7—21no—
195C22×C7—21no—
196C22×C7—21no—
197C22×C7—21no—
198C22×C7—21no—
199C22×C7—21no—
200C22×C7C22×D411yescharacteristic
201C4×C7—21no—
202C4×C7—21no—
203C4×C7—21no—
204C4×C7—21no—
205C4×C7—41no—
206C4×C7—41no—
207C4×C7—41no—
208C4×C7—41no—
209C4×C7—41no—
210C4×C7—41no—
211C4×C7—41no—
212C4×C7—41no—
213C4×C7C2×#4111yescharacteristic
214C4×C72−1+411yescharacteristic
215C2×Q8—21no—
216C2×Q8C23×C711yescharacteristic
217C2×Q8C23×C711yescharacteristic
218C2×D4C7×D411yescharacteristic
219C22×C4—21no—
220C22×C4C23×C711yescharacteristic
221C22×C4C23×C711yescharacteristic
222Q16—22no—
223M16—22no—
224C2×C8—22no—
225C2×C8C23×C711yescharacteristic
226C2×C8C23×C711yescharacteristic
227C2×C8C23×C711yescharacteristic
228C2×C8C23×C711yescharacteristic
229C4⋊C4—21no—
230C4⋊C4—21no—
231C4⋊C4—21no—
232C4⋊C4—21no—
233C4⋊C4—21no—
234C4⋊C4—21no—
235C4⋊C4—21no—
236C4⋊C4—21no—
237C4⋊C4—21no—
238C4⋊C4—21no—
239C4⋊C4C7×D411yescharacteristic
240C4⋊C4C23×C711yescharacteristic
241C4⋊C4C23×C711yescharacteristic
242C4⋊C4C23×C711yescharacteristic
243C4⋊C4C23×C711yescharacteristic
244C4⋊C4C23×C711yescharacteristic
245C4⋊C4C23×C711yescharacteristic
246C22⋊C4—21no—
247C22⋊C4—21no—
248C22⋊C4—21no—
249C22⋊C4—21no—
250C22⋊C4—21no—
251C22⋊C4—21no—
252C22⋊C4—21no—
253C22⋊C4C7×D411yescharacteristic
254C22⋊C4C7×D411yescharacteristic
255C42—21no—
256C42—21no—
257C42—21no—
258C42—21no—
259C42—21no—
260C42C23×C711yescharacteristic
261C2×C7—41no—
262C2×C7—41no—
263C2×C7#54711yescharacteristic
264C2×C7#57511yescharacteristic
265C2×C7#57811yescharacteristic
266C23C2×C7×D411yescharacteristic
267C23C2×C7×D411yescharacteristic
268Q8—21no—
269Q8—21no—
270Q8—21no—
271Q8—21no—
272Q8—42no—
273D4—41no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—42no—
287C2×C4C2×C7×D411yescharacteristic
288C2×C4C2×C7×D411yescharacteristic
289C2×C4C2×C7×D411yescharacteristic
290C2×C4C2×C7×D411yescharacteristic
291C2×C4C24×C711yescharacteristic
292C8—21no—
293C8—21no—
294C8—21no—
295C8C7×#4112yes—
296C7#334911yescharacteristic, direct product factor, Sylow
297C22—21no—
298C22—21no—
299C22—21no—
300C22—21no—
301C22—21no—
302C22—21no—
303C22C22×C7×D411yescharacteristic
304C4—21no—
305C4—21no—
306C4—21no—
307C4—21no—
308C4—41no—
309C4—41no—
310C4—41no—
311C4—41no—
312C4—41no—
313C4—41no—
314C4—41no—
315C4—41no—
316C4C2×C7×#4111yescharacteristic
317C4C7×2−1+411yescharacteristic
318C2—41no—
319C2—41no—
320C2C7×#54711yescharacteristic
321C2C7×#57511yescharacteristic
322C2C7×#57811yescharacteristic
subgroupG/H
center, socleC22×C7C22×D4
commutator, Frattini, cosocleC2×C4C24×C7
prime radicalC2×C7×D4D4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C7 ◁ C4×C7×D4 ◁ G

Complements

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

NH ≅ G/Ntype
3. C7×#501C2semidirect
4. C7×#485C2semidirect
5. C7×#485C2semidirect
6. C7×#485C2semidirect
7. C7×#484C2semidirect
8. C7×#479C2semidirect
9. C7×#477C2semidirect
10. C7×#470C2semidirect
11. C7×#467C2semidirect
12. C7×#441C2semidirect
14. C7×#428C2semidirect
15. C7×#427C2semidirect
59. #3349C7direct (N × H)
108. #501C2×C7semidirect
109. #485C2×C7semidirect
110. #485C2×C7semidirect
111. #485C2×C7semidirect
112. #484C2×C7semidirect
113. #479C2×C7semidirect
114. #477C2×C7semidirect
115. #470C2×C7semidirect
116. #467C2×C7semidirect
117. #441C2×C7semidirect
119. #428C2×C7semidirect
120. #427C2×C7semidirect
296. C7#3349direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#5562
C7×#5522
C7×#5012
C7×#4852
C7×#4842
C7×#4792
C7×#4772
C7×#4702
C7×#4672
C7×#4412
C7×#4352
C7×#4282
C7×#4272
#33497