Finite Group Databasebeta

← C7×#3347 #11742656

Subgroups — C7×#3347 #11742656

Order 896.

360 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#555C211yesmaximal, characteristic, complemented
2C7×#553C211yesmaximal, characteristic, complemented
3C7×#501C211yesmaximal, characteristic, complemented
4C7×#482C211yesmaximal, characteristic, complemented
5C7×#482C211yesmaximal, characteristic, complemented
6C7×#482C211yesmaximal, characteristic, complemented
7C7×#480C211yesmaximal, characteristic, complemented
8C7×#478C211yesmaximal, characteristic, complemented
9C7×#474C211yesmaximal, characteristic, complemented
10C7×#469C211yesmaximal, characteristic, complemented
11C7×#466C211yesmaximal, characteristic, complemented
12C7×#442C211yesmaximal, characteristic, complemented
13C7×#435C211yesmaximal, characteristic, complemented
14C7×#428C211yesmaximal, characteristic, complemented
15C7×#427C211yesmaximal, characteristic, complemented
16C2×C7×D8C2211yescharacteristic
17C2×C7×M16C2211yescharacteristic
18C22×C7×C8C2211yescharacteristic
19C7×C4⋊Q8C2211yescharacteristic
20C7×#126—21no—
21C7×#125C2211yescharacteristic
22C7×#124—21no—
23C7×#124—21no—
24C7×#123—21no—
25C7×#121—21no—
26C7×#121—21no—
27C7×#121C2211yescharacteristic
28C7×#121C2211yescharacteristic
29C7×#121C2211yescharacteristic
30C7×#121C2211yescharacteristic
31C4×C7×D4—21no—
32C4×C7×D4C2211yescharacteristic
33C4×C7×D4C2211yescharacteristic
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×#102C2211yescharacteristic
51C7×#102C2211yescharacteristic
52C7×#102C2211yescharacteristic
53C7×#102C2211yescharacteristic
54C7×#102C2211yescharacteristic
55C7×#102C2211yescharacteristic
56C7×C22⋊C8C2211yescharacteristic
57C7×C22⋊C8C2211yescharacteristic
58C7×#97C2211yescharacteristic
59#3347C711yesmaximal, characteristic, direct product factor, Sylow
60C2×C7×Q8—21no—
61C2×C7×D4—21no—
62C2×C7×D4—21no—
63C2×C7×D4—21no—
64C2×C7×D4—21no—
65C2×C7×D4D411yescharacteristic
66C2×C7×D4C2311yescharacteristic
67C2×C7×D4C2311yescharacteristic
68C22×C4×C7—21no—
69C22×C4×C7—21no—
70C22×C4×C7—21no—
71C22×C4×C7C2311yescharacteristic
72C22×C4×C7C2311yescharacteristic
73C7×D8—22no—
74C7×M16—22no—
75C2×C7×C8—22no—
76C2×C7×C8C2311yescharacteristic
77C2×C7×C8C2311yescharacteristic
78C2×C7×C8C2311yescharacteristic
79C2×C7×C8C2311yescharacteristic
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⋊C4—21no—
99C7×C22⋊C4—21no—
100C7×C22⋊C4D411yescharacteristic
101C7×C22⋊C4D411yescharacteristic
102C42×C7—21no—
103C42×C7—21no—
104C42×C7—21no—
105C42×C7C2311yescharacteristic
106#555C2×C711yescharacteristic, complemented
107#553C2×C711yescharacteristic, complemented
108#501C2×C711yescharacteristic, complemented
109#482C2×C711yescharacteristic, complemented
110#482C2×C711yescharacteristic, complemented
111#482C2×C711yescharacteristic, complemented
112#480C2×C711yescharacteristic, complemented
113#478C2×C711yescharacteristic, complemented
114#474C2×C711yescharacteristic, complemented
115#469C2×C711yescharacteristic, complemented
116#466C2×C711yescharacteristic, complemented
117#442C2×C711yescharacteristic, complemented
118#435C2×C711yescharacteristic, complemented
119#428C2×C711yescharacteristic, complemented
120#427C2×C711yescharacteristic, complemented
121C23×C7—21no—
122C23×C7—21no—
123C23×C7C2×D411yescharacteristic
124C23×C7C2×D411yescharacteristic
125C7×Q8—81no—
126C7×D4—21no—
127C7×D4—21no—
128C7×D4—21no—
129C7×D4—21no—
130C7×D4—41no—
131C7×D4—42no—
132C7×D4—81no—
133C7×D4—81no—
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—21no—
142C2×C4×C7—21no—
143C2×C4×C7—21no—
144C2×C4×C7—21no—
145C2×C4×C7—42no—
146C2×C4×C7—81no—
147C2×C4×C7—81no—
148C2×C4×C7C2×D411yescharacteristic
149C2×C4×C7C2×D411yescharacteristic
150C2×C4×C7C2×D411yescharacteristic
151C2×C4×C7C2×D411yescharacteristic
152C2×C4×C7C2411yescharacteristic
153C7×C8—21no—
154C7×C8—21no—
155C7×C8—21no—
156C7×C8#4112yes—
157C2×D8C22×C711yescharacteristic
158C2×M16C22×C711yescharacteristic
159C22×C8C22×C711yescharacteristic
160C4⋊Q8C22×C711yescharacteristic
161#126—21no—
162#125C22×C711yescharacteristic
163#124—21no—
164#124—21no—
165#123—21no—
166#121—21no—
167#121—21no—
168#121C22×C711yescharacteristic
169#121C22×C711yescharacteristic
170#121C22×C711yescharacteristic
171#121C22×C711yescharacteristic
172C4×D4—21no—
173C4×D4C22×C711yescharacteristic
174C4×D4C22×C711yescharacteristic
175C4×D4C22×C711yescharacteristic
176#117—21no—
177#117C22×C711yescharacteristic
178#117C22×C711yescharacteristic
179#117C22×C711yescharacteristic
180C2×C4⋊C4C22×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
189#106C22×C711yescharacteristic
190C4⋊C8C22×C711yescharacteristic
191#102C22×C711yescharacteristic
192#102C22×C711yescharacteristic
193#102C22×C711yescharacteristic
194#102C22×C711yescharacteristic
195#102C22×C711yescharacteristic
196#102C22×C711yescharacteristic
197C22⋊C8C22×C711yescharacteristic
198C22⋊C8C22×C711yescharacteristic
199#97C22×C711yescharacteristic
200C22×C7—21no—
201C22×C7—21no—
202C22×C7—21no—
203C22×C7—21no—
204C22×C7—21no—
205C22×C7—21no—
206C22×C7—41no—
207C22×C7—41no—
208C22×C7—41no—
209C22×C7—41no—
210C22×C7—41no—
211C22×C7—41no—
212C22×C7C22×D411yescharacteristic
213C4×C7—21no—
214C4×C7—21no—
215C4×C7—21no—
216C4×C7—21no—
217C4×C7—41no—
218C4×C7—41no—
219C4×C7—41no—
220C4×C7—41no—
221C4×C7—41no—
222C4×C7—41no—
223C4×C7C2×#4111yescharacteristic
224C4×C72−1+411yescharacteristic
225C2×Q8—21no—
226C2×D4—21no—
227C2×D4—21no—
228C2×D4—21no—
229C2×D4—21no—
230C2×D4C7×D411yescharacteristic
231C2×D4C23×C711yescharacteristic
232C2×D4C23×C711yescharacteristic
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4C23×C711yescharacteristic
237C22×C4C23×C711yescharacteristic
238D8—22no—
239M16—22no—
240C2×C8—22no—
241C2×C8C23×C711yescharacteristic
242C2×C8C23×C711yescharacteristic
243C2×C8C23×C711yescharacteristic
244C2×C8C23×C711yescharacteristic
245C4⋊C4—21no—
246C4⋊C4—21no—
247C4⋊C4—21no—
248C4⋊C4—21no—
249C4⋊C4C7×D411yescharacteristic
250C4⋊C4C23×C711yescharacteristic
251C4⋊C4C23×C711yescharacteristic
252C4⋊C4C23×C711yescharacteristic
253C4⋊C4C23×C711yescharacteristic
254C4⋊C4C23×C711yescharacteristic
255C4⋊C4C23×C711yescharacteristic
256C22⋊C4—21no—
257C22⋊C4—21no—
258C22⋊C4—21no—
259C22⋊C4—21no—
260C22⋊C4—21no—
261C22⋊C4—21no—
262C22⋊C4—21no—
263C22⋊C4—21no—
264C22⋊C4—21no—
265C22⋊C4C7×D411yescharacteristic
266C22⋊C4C7×D411yescharacteristic
267C42—21no—
268C42—21no—
269C42—21no—
270C42C23×C711yescharacteristic
271C2×C7—41no—
272C2×C7—41no—
273C2×C7—81no—
274C2×C7—81no—
275C2×C7#54711yescharacteristic
276C2×C7#57511yescharacteristic
277C2×C7#57611yescharacteristic
278C23—21no—
279C23—21no—
280C23C2×C7×D411yescharacteristic
281C23C2×C7×D411yescharacteristic
282Q8—81no—
283D4—21no—
284D4—21no—
285D4—21no—
286D4—21no—
287D4—41no—
288D4—42no—
289D4—81no—
290D4—81no—
291D4—81no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—42no—
303C2×C4—81no—
304C2×C4—81no—
305C2×C4C2×C7×D411yescharacteristic
306C2×C4C2×C7×D411yescharacteristic
307C2×C4C2×C7×D411yescharacteristic
308C2×C4C2×C7×D411yescharacteristic
309C2×C4C24×C711yescharacteristic
310C8—21no—
311C8—21no—
312C8—21no—
313C8C7×#4112yes—
314C7#334711yescharacteristic, direct product factor, Sylow
315C22—21no—
316C22—21no—
317C22—21no—
318C22—21no—
319C22—21no—
320C22—21no—
321C22—41no—
322C22—41no—
323C22—41no—
324C22—41no—
325C22—41no—
326C22—41no—
327C22C22×C7×D411yescharacteristic
328C4—21no—
329C4—21no—
330C4—21no—
331C4—21no—
332C4—41no—
333C4—41no—
334C4—41no—
335C4—41no—
336C4—41no—
337C4—41no—
338C4C2×C7×#4111yescharacteristic
339C4C7×2−1+411yescharacteristic
340C2—41no—
341C2—41no—
342C2—81no—
343C2—81no—
344C2C7×#54711yescharacteristic
345C2C7×#57511yescharacteristic
346C2C7×#57611yescharacteristic
subgroupG/H
center, socleC22×C7C22×D4
commutator, Frattini, cosocleC2×C4C24×C7
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×#555C2semidirect
2. C7×#553C2semidirect
3. C7×#501C2semidirect
4. C7×#482C2semidirect
5. C7×#482C2semidirect
6. C7×#482C2semidirect
7. C7×#480C2semidirect
8. C7×#478C2semidirect
9. C7×#474C2semidirect
10. C7×#469C2semidirect
11. C7×#466C2semidirect
12. C7×#442C2semidirect
13. C7×#435C2semidirect
14. C7×#428C2semidirect
15. C7×#427C2semidirect
59. #3347C7direct (N × H)
106. #555C2×C7semidirect
107. #553C2×C7semidirect
108. #501C2×C7semidirect
109. #482C2×C7semidirect
110. #482C2×C7semidirect
111. #482C2×C7semidirect
112. #480C2×C7semidirect
113. #478C2×C7semidirect
114. #474C2×C7semidirect
115. #469C2×C7semidirect
116. #466C2×C7semidirect
117. #442C2×C7semidirect
118. #435C2×C7semidirect
119. #428C2×C7semidirect
120. #427C2×C7semidirect
314. C7#3347direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#5552
C7×#5532
C7×#5012
C7×#4822
C7×#4802
C7×#4782
C7×#4742
C7×#4692
C7×#4662
C7×#4422
C7×#4352
C7×#4282
C7×#4272
#33477