Finite Group Databasebeta

← C7×#3346 #11742655

Subgroups — C7×#3346 #11742655

Order 896.

356 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#554C211yesmaximal, characteristic, complemented
2C7×#553C211yesmaximal, characteristic
3C7×#501C211yesmaximal, characteristic, complemented
4C7×#485C211yesmaximal, characteristic, complemented
5C7×#484C211yesmaximal, characteristic, complemented
6C7×#481C211yesmaximal, characteristic, complemented
7C7×#480C211yesmaximal, characteristic, complemented
8C7×#479C211yesmaximal, characteristic, complemented
9C7×#474C211yesmaximal, characteristic, complemented
10C7×#468C211yesmaximal, characteristic, complemented
11C7×#465C211yesmaximal, characteristic, complemented
12C7×#440C211yesmaximal, characteristic, complemented
13C7×#435C211yesmaximal, characteristic, complemented
14C7×#428C211yesmaximal, characteristic, complemented
15C2×C7×#107C211yesmaximal, characteristic, complemented
16C2×C7×SD16C2211yescharacteristic
17C2×C7×M16C2211yescharacteristic
18C22×C7×C8C2211yescharacteristic
19C7×C4⋊Q8C2211yescharacteristic
20C7×#126—21no—
21C7×#125—21no—
22C7×#125C2211yescharacteristic
23C7×#123—21no—
24C7×C22⋊Q8—21no—
25C7×C22⋊Q8—21no—
26C7×C22⋊Q8C2211yescharacteristic
27C7×C22⋊Q8C2211yescharacteristic
28C7×#121—21no—
29C7×#121C2211yescharacteristic
30C7×#121C2211yescharacteristic
31C4×C7×Q8C2211yescharacteristic
32C4×C7×D4—21no—
33C4×C7×D4—21no—
34C4×C7×D4C2211yescharacteristic
35C4×C7×D4C2211yescharacteristic
36C7×#117C2211yescharacteristic
37C2×C7×C4⋊C4C2211yescharacteristic
38C2×C7×C4⋊C4C2211yescharacteristic
39C2×C7×C4⋊C4C2211yescharacteristic
40C7×#107C2211yescharacteristic
41C7×#107C2211yescharacteristic
42C7×#107C2211yescharacteristic
43C7×#107C2211yescharacteristic
44C7×#107C2211yescharacteristic
45C7×#107C2211yescharacteristic
46C7×#106C2211yescharacteristic
47C7×#106C2211yescharacteristic
48C7×#106C2211yescharacteristic
49C7×C4⋊C8C2211yescharacteristic
50C7×#103C2211yescharacteristic
51C7×#103C2211yescharacteristic
52C7×#103C2211yescharacteristic
53C7×#102C2211yescharacteristic
54C7×#102C2211yescharacteristic
55C7×#102C2211yescharacteristic
56C7×C22⋊C8C2211yescharacteristic
57C7×C22⋊C8C2211yescharacteristic
58C7×#97C2211yescharacteristic
59#3346C711yesmaximal, characteristic, direct product factor, Sylow
60C2×C7×Q8—21no—
61C2×C7×Q8C2311yescharacteristic
62C2×C7×D4—21no—
63C2×C7×D4—21no—
64C2×C7×D4D411yescharacteristic
65C2×C7×D4C2311yescharacteristic
66C22×C4×C7—21no—
67C22×C4×C7—21no—
68C22×C4×C7—21no—
69C22×C4×C7—21no—
70C22×C4×C7C2311yescharacteristic
71C22×C4×C7C2311yescharacteristic
72C7×SD16—22no—
73C7×M16—22no—
74C2×C7×C8—22no—
75C2×C7×C8C2311yescharacteristic
76C2×C7×C8C2311yescharacteristic
77C2×C7×C8C2311yescharacteristic
78C2×C7×C8C2311yescharacteristic
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⋊C4—21no—
87C7×C4⋊C4—21no—
88C7×C4⋊C4D411yescharacteristic
89C7×C4⋊C4C2311yescharacteristic
90C7×C4⋊C4C2311yescharacteristic
91C7×C4⋊C4C2311yescharacteristic
92C7×C4⋊C4C2311yescharacteristic
93C7×C4⋊C4C2311yescharacteristic
94C7×C4⋊C4C2311yescharacteristic
95C7×C22⋊C4—21no—
96C7×C22⋊C4—21no—
97C7×C22⋊C4—21no—
98C7×C22⋊C4—21no—
99C7×C22⋊C4—21no—
100C7×C22⋊C4—21no—
101C7×C22⋊C4D411yescharacteristic
102C7×C22⋊C4D411yescharacteristic
103C42×C7—21no—
104C42×C7—21no—
105C42×C7C2311yescharacteristic
106#554C2×C711yescharacteristic, complemented
107#553C2×C711yescharacteristic
108#501C2×C711yescharacteristic, complemented
109#485C2×C711yescharacteristic, complemented
110#484C2×C711yescharacteristic, complemented
111#481C2×C711yescharacteristic, complemented
112#480C2×C711yescharacteristic, complemented
113#479C2×C711yescharacteristic, complemented
114#474C2×C711yescharacteristic, complemented
115#468C2×C711yescharacteristic, complemented
116#465C2×C711yescharacteristic, complemented
117#440C2×C711yescharacteristic, complemented
118#435C2×C711yescharacteristic, complemented
119#428C2×C711yescharacteristic, complemented
120C2×#107C2×C711yescharacteristic, complemented
121C23×C7—21no—
122C23×C7C2×D411yescharacteristic
123C23×C7C2×D411yescharacteristic
124C7×Q8—21no—
125C7×Q8—21no—
126C7×Q8—81no—
127C7×D4—22no—
128C7×D4—21no—
129C7×D4—21no—
130C7×D4—42no—
131C7×D4—81no—
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—21no—
142C2×C4×C7—21no—
143C2×C4×C7—81no—
144C2×C4×C7—81no—
145C2×C4×C7—81no—
146C2×C4×C7—81no—
147C2×C4×C7C2×D411yescharacteristic
148C2×C4×C7C2×D411yescharacteristic
149C2×C4×C7C2×D411yescharacteristic
150C2×C4×C7C2×D411yescharacteristic
151C2×C4×C7C2411yescharacteristic
152C7×C8—21no—
153C7×C8—21no—
154C7×C8—21no—
155C7×C8#4112yes—
156C2×SD16C22×C711yescharacteristic
157C2×M16C22×C711yescharacteristic
158C22×C8C22×C711yescharacteristic
159C4⋊Q8C22×C711yescharacteristic
160#126—21no—
161#125—21no—
162#125C22×C711yescharacteristic
163#123—21no—
164C22⋊Q8—21no—
165C22⋊Q8—21no—
166C22⋊Q8C22×C711yescharacteristic
167C22⋊Q8C22×C711yescharacteristic
168#121—21no—
169#121C22×C711yescharacteristic
170#121C22×C711yescharacteristic
171C4×Q8C22×C711yescharacteristic
172C4×D4—21no—
173C4×D4—21no—
174C4×D4C22×C711yescharacteristic
175C4×D4C22×C711yescharacteristic
176#117C22×C711yescharacteristic
177C2×C4⋊C4C22×C711yescharacteristic
178C2×C4⋊C4C22×C711yescharacteristic
179C2×C4⋊C4C22×C711yescharacteristic
180#107C22×C711yescharacteristic
181#107C22×C711yescharacteristic
182#107C22×C711yescharacteristic
183#107C22×C711yescharacteristic
184#107C22×C711yescharacteristic
185#107C22×C711yescharacteristic
186#106C22×C711yescharacteristic
187#106C22×C711yescharacteristic
188#106C22×C711yescharacteristic
189C4⋊C8C22×C711yescharacteristic
190#103C22×C711yescharacteristic
191#103C22×C711yescharacteristic
192#103C22×C711yescharacteristic
193#102C22×C711yescharacteristic
194#102C22×C711yescharacteristic
195#102C22×C711yescharacteristic
196C22⋊C8C22×C711yescharacteristic
197C22⋊C8C22×C711yescharacteristic
198#97C22×C711yescharacteristic
199C22×C7—21no—
200C22×C7—21no—
201C22×C7—21no—
202C22×C7—21no—
203C22×C7—21no—
204C22×C7—41no—
205C22×C7—41no—
206C22×C7—41no—
207C22×C7#13712yes—
208C22×C7C22×D411yescharacteristic
209C4×C7—21no—
210C4×C7—21no—
211C4×C7—21no—
212C4×C7—21no—
213C4×C7—41no—
214C4×C7—41no—
215C4×C7—41no—
216C4×C7—41no—
217C4×C7—41no—
218C4×C7—41no—
219C4×C7—41no—
220C4×C7C2×#4111yescharacteristic
221C4×C72−1+411yescharacteristic
222C2×Q8—21no—
223C2×Q8C23×C711yescharacteristic
224C2×D4—21no—
225C2×D4—21no—
226C2×D4C7×D411yescharacteristic
227C2×D4C23×C711yescharacteristic
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4C23×C711yescharacteristic
233C22×C4C23×C711yescharacteristic
234SD16—22no—
235M16—22no—
236C2×C8—22no—
237C2×C8C23×C711yescharacteristic
238C2×C8C23×C711yescharacteristic
239C2×C8C23×C711yescharacteristic
240C2×C8C23×C711yescharacteristic
241C4⋊C4—21no—
242C4⋊C4—21no—
243C4⋊C4—21no—
244C4⋊C4—21no—
245C4⋊C4—21no—
246C4⋊C4—21no—
247C4⋊C4—21no—
248C4⋊C4—21no—
249C4⋊C4—21no—
250C4⋊C4C7×D411yescharacteristic
251C4⋊C4C23×C711yescharacteristic
252C4⋊C4C23×C711yescharacteristic
253C4⋊C4C23×C711yescharacteristic
254C4⋊C4C23×C711yescharacteristic
255C4⋊C4C23×C711yescharacteristic
256C4⋊C4C23×C711yescharacteristic
257C22⋊C4—21no—
258C22⋊C4—21no—
259C22⋊C4—21no—
260C22⋊C4—21no—
261C22⋊C4—21no—
262C22⋊C4—21no—
263C22⋊C4C7×D411yescharacteristic
264C22⋊C4C7×D411yescharacteristic
265C42—21no—
266C42—21no—
267C42C23×C711yescharacteristic
268C2×C7—22no—
269C2×C7—41no—
270C2×C7—81no—
271C2×C7#54711yescharacteristic
272C2×C7C2×#13711yescharacteristic
273C2×C7#57611yescharacteristic
274C23—21no—
275C23C2×C7×D411yescharacteristic
276C23C2×C7×D411yescharacteristic
277Q8—21no—
278Q8—21no—
279Q8—81no—
280D4—22no—
281D4—21no—
282D4—21no—
283D4—42no—
284D4—81no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—81no—
297C2×C4—81no—
298C2×C4—81no—
299C2×C4—81no—
300C2×C4C2×C7×D411yescharacteristic
301C2×C4C2×C7×D411yescharacteristic
302C2×C4C2×C7×D411yescharacteristic
303C2×C4C2×C7×D411yescharacteristic
304C2×C4C24×C711yescharacteristic
305C8—21no—
306C8—21no—
307C8—21no—
308C8C7×#4112yes—
309C7#334611yescharacteristic, direct product factor, Sylow
310C22—21no—
311C22—21no—
312C22—21no—
313C22—21no—
314C22—21no—
315C22—41no—
316C22—41no—
317C22—41no—
318C22C7×#13712yes—
319C22C22×C7×D411yescharacteristic
320C4—21no—
321C4—21no—
322C4—21no—
323C4—21no—
324C4—41no—
325C4—41no—
326C4—41no—
327C4—41no—
328C4—41no—
329C4—41no—
330C4—41no—
331C4C2×C7×#4111yescharacteristic
332C4C7×2−1+411yescharacteristic
333C2—22no—
334C2—41no—
335C2—81no—
336C2C7×#54711yescharacteristic
337C2C2×C7×#13711yescharacteristic
338C2C7×#57611yescharacteristic
subgroupG/H
center, socleC22×C7C22×D4
commutator, Frattini, cosocleC2×C4C24×C7
prime radicalC7×#553C2
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
1. C7×#554C2semidirect
3. C7×#501C2semidirect
4. C7×#485C2semidirect
5. C7×#484C2semidirect
6. C7×#481C2semidirect
7. C7×#480C2semidirect
8. C7×#479C2semidirect
9. C7×#474C2semidirect
10. C7×#468C2semidirect
11. C7×#465C2semidirect
12. C7×#440C2semidirect
13. C7×#435C2semidirect
14. C7×#428C2semidirect
15. C2×C7×#107C2semidirect
59. #3346C7direct (N × H)
106. #554C2×C7semidirect
108. #501C2×C7semidirect
109. #485C2×C7semidirect
110. #484C2×C7semidirect
111. #481C2×C7semidirect
112. #480C2×C7semidirect
113. #479C2×C7semidirect
114. #474C2×C7semidirect
115. #468C2×C7semidirect
116. #465C2×C7semidirect
117. #440C2×C7semidirect
118. #435C2×C7semidirect
119. #428C2×C7semidirect
120. C2×#107C2×C7semidirect
309. C7#3346direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#5542
C7×#5532
C7×#5012
C7×#4852
C7×#4842
C7×#4812
C7×#4802
C7×#4792
C7×#4742
C7×#4682
C7×#4652
C7×#4402
C7×#4352
C7×#4282
C2×C7×#1072
#33467