Finite Group Databasebeta

← C7×#3350 #11742659

Subgroups — C7×#3350 #11742659

Order 896.

348 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#556C211yesmaximal, characteristic, complemented
2C7×#553C211yesmaximal, characteristic
3C7×#501C211yesmaximal, characteristic, complemented
4C7×#485C211yesmaximal, characteristic, complemented
5C7×#483C211yesmaximal, characteristic, complemented
6C7×#482C211yesmaximal, characteristic, complemented
7C7×#482C211yesmaximal, characteristic, complemented
8C7×#479C211yesmaximal, characteristic, complemented
9C7×#476C211yesmaximal, characteristic, complemented
10C7×#468C211yesmaximal, characteristic, complemented
11C7×#465C211yesmaximal, characteristic, complemented
12C7×#440C211yesmaximal, characteristic, complemented
13C7×#435C211yesmaximal, characteristic, complemented
14C7×#428C211yesmaximal, characteristic, complemented
15C7×#427C211yesmaximal, characteristic, complemented
16C2×C7×SD16C2211yescharacteristic
17C2×C7×M16C2211yescharacteristic
18C22×C7×C8C2211yescharacteristic
19C7×C4⋊Q8C2211yescharacteristic
20C7×#126—21no—
21C7×#126—21no—
22C7×#126—21no—
23C7×#125C2211yescharacteristic
24C7×#124—21no—
25C7×#123—21no—
26C7×C22⋊Q8C2211yescharacteristic
27C7×C22⋊Q8C2211yescharacteristic
28C7×#121—21no—
29C7×#121C2211yescharacteristic
30C7×#121C2211yescharacteristic
31C4×C7×Q8—21no—
32C4×C7×Q8C2211yescharacteristic
33C4×C7×D4—21no—
34C4×C7×D4C2211yescharacteristic
35C4×C7×D4C2211yescharacteristic
36C7×#117C2211yescharacteristic
37C7×#117C2211yescharacteristic
38C7×#117C2211yescharacteristic
39C2×C7×C4⋊C4C2211yescharacteristic
40C7×#107C2211yescharacteristic
41C7×#107C2211yescharacteristic
42C7×#107C2211yescharacteristic
43C7×#107C2211yescharacteristic
44C7×#106C2211yescharacteristic
45C7×#106C2211yescharacteristic
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#3350C711yesmaximal, 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×C7C2311yescharacteristic
69C22×C4×C7C2311yescharacteristic
70C7×SD16—22no—
71C7×M16—22no—
72C2×C7×C8—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⋊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⋊C4D411yescharacteristic
100C7×C22⋊C4D411yescharacteristic
101C42×C7—21no—
102C42×C7—21no—
103C42×C7—21no—
104C42×C7—21no—
105C42×C7C2311yescharacteristic
106#556C2×C711yescharacteristic, complemented
107#553C2×C711yescharacteristic
108#501C2×C711yescharacteristic, complemented
109#485C2×C711yescharacteristic, complemented
110#483C2×C711yescharacteristic, complemented
111#482C2×C711yescharacteristic, complemented
112#482C2×C711yescharacteristic, complemented
113#479C2×C711yescharacteristic, complemented
114#476C2×C711yescharacteristic, complemented
115#468C2×C711yescharacteristic, complemented
116#465C2×C711yescharacteristic, complemented
117#440C2×C711yescharacteristic, complemented
118#435C2×C711yescharacteristic, complemented
119#428C2×C711yescharacteristic, complemented
120#427C2×C711yescharacteristic, complemented
121C23×C7—21no—
122C23×C7C2×D411yescharacteristic
123C23×C7C2×D411yescharacteristic
124C7×Q8—21no—
125C7×Q8—21no—
126C7×Q8—42no—
127C7×D4—21no—
128C7×D4—21no—
129C7×D4—41no—
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×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#4112yes—
154C2×SD16C22×C711yescharacteristic
155C2×M16C22×C711yescharacteristic
156C22×C8C22×C711yescharacteristic
157C4⋊Q8C22×C711yescharacteristic
158#126—21no—
159#126—21no—
160#126—21no—
161#125C22×C711yescharacteristic
162#124—21no—
163#123—21no—
164C22⋊Q8C22×C711yescharacteristic
165C22⋊Q8C22×C711yescharacteristic
166#121—21no—
167#121C22×C711yescharacteristic
168#121C22×C711yescharacteristic
169C4×Q8—21no—
170C4×Q8C22×C711yescharacteristic
171C4×D4—21no—
172C4×D4C22×C711yescharacteristic
173C4×D4C22×C711yescharacteristic
174#117C22×C711yescharacteristic
175#117C22×C711yescharacteristic
176#117C22×C711yescharacteristic
177C2×C4⋊C4C22×C711yescharacteristic
178#107C22×C711yescharacteristic
179#107C22×C711yescharacteristic
180#107C22×C711yescharacteristic
181#107C22×C711yescharacteristic
182#106C22×C711yescharacteristic
183#106C22×C711yescharacteristic
184#106C22×C711yescharacteristic
185#106C22×C711yescharacteristic
186#106C22×C711yescharacteristic
187C4⋊C8C22×C711yescharacteristic
188#103C22×C711yescharacteristic
189#103C22×C711yescharacteristic
190#103C22×C711yescharacteristic
191#102C22×C711yescharacteristic
192#102C22×C711yescharacteristic
193#102C22×C711yescharacteristic
194C22⋊C8C22×C711yescharacteristic
195C22⋊C8C22×C711yescharacteristic
196#97C22×C711yescharacteristic
197C22×C7—21no—
198C22×C7—21no—
199C22×C7—21no—
200C22×C7—21no—
201C22×C7—21no—
202C22×C7—21no—
203C22×C7—41no—
204C22×C7—41no—
205C22×C7—41no—
206C22×C7C22×D411yescharacteristic
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×C7—41no—
217C4×C7—41no—
218C4×C7C2×#4111yescharacteristic
219C4×C72−1+411yescharacteristic
220C2×Q8—21no—
221C2×Q8C23×C711yescharacteristic
222C2×D4—21no—
223C2×D4—21no—
224C2×D4C7×D411yescharacteristic
225C2×D4C23×C711yescharacteristic
226C22×C4—21no—
227C22×C4—21no—
228C22×C4C23×C711yescharacteristic
229C22×C4C23×C711yescharacteristic
230SD16—22no—
231M16—22no—
232C2×C8—22no—
233C2×C8C23×C711yescharacteristic
234C2×C8C23×C711yescharacteristic
235C2×C8C23×C711yescharacteristic
236C2×C8C23×C711yescharacteristic
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⋊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⋊C4—21no—
257C22⋊C4—21no—
258C22⋊C4—21no—
259C22⋊C4C7×D411yescharacteristic
260C22⋊C4C7×D411yescharacteristic
261C42—21no—
262C42—21no—
263C42—21no—
264C42—21no—
265C42C23×C711yescharacteristic
266C2×C7—41no—
267C2×C7—41no—
268C2×C7—81no—
269C2×C7#54711yescharacteristic
270C2×C7#57511yescharacteristic
271C2×C7#57711yescharacteristic
272C23—21no—
273C23C2×C7×D411yescharacteristic
274C23C2×C7×D411yescharacteristic
275Q8—21no—
276Q8—21no—
277Q8—42no—
278D4—21no—
279D4—21no—
280D4—41no—
281D4—42no—
282D4—81no—
283C2×C4—21no—
284C2×C4—21no—
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—81no—
295C2×C4—81no—
296C2×C4C2×C7×D411yescharacteristic
297C2×C4C2×C7×D411yescharacteristic
298C2×C4C2×C7×D411yescharacteristic
299C2×C4C2×C7×D411yescharacteristic
300C2×C4C24×C711yescharacteristic
301C8—21no—
302C8—21no—
303C8—21no—
304C8C7×#4112yes—
305C7#335011yescharacteristic, direct product factor, Sylow
306C22—21no—
307C22—21no—
308C22—21no—
309C22—21no—
310C22—21no—
311C22—21no—
312C22—41no—
313C22—41no—
314C22—41no—
315C22C22×C7×D411yescharacteristic
316C4—21no—
317C4—21no—
318C4—21no—
319C4—21no—
320C4—41no—
321C4—41no—
322C4—41no—
323C4—41no—
324C4—41no—
325C4—41no—
326C4—41no—
327C4C2×C7×#4111yescharacteristic
328C4C7×2−1+411yescharacteristic
329C2—41no—
330C2—41no—
331C2—81no—
332C2C7×#54711yescharacteristic
333C2C7×#57511yescharacteristic
334C2C7×#57711yescharacteristic
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×#556C2semidirect
3. C7×#501C2semidirect
4. C7×#485C2semidirect
5. C7×#483C2semidirect
6. C7×#482C2semidirect
7. C7×#482C2semidirect
8. C7×#479C2semidirect
9. C7×#476C2semidirect
10. C7×#468C2semidirect
11. C7×#465C2semidirect
12. C7×#440C2semidirect
13. C7×#435C2semidirect
14. C7×#428C2semidirect
15. C7×#427C2semidirect
59. #3350C7direct (N × H)
106. #556C2×C7semidirect
108. #501C2×C7semidirect
109. #485C2×C7semidirect
110. #483C2×C7semidirect
111. #482C2×C7semidirect
112. #482C2×C7semidirect
113. #479C2×C7semidirect
114. #476C2×C7semidirect
115. #468C2×C7semidirect
116. #465C2×C7semidirect
117. #440C2×C7semidirect
118. #435C2×C7semidirect
119. #428C2×C7semidirect
120. #427C2×C7semidirect
305. C7#3350direct (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×#5532
C7×#5012
C7×#4852
C7×#4832
C7×#4822
C7×#4792
C7×#4762
C7×#4682
C7×#4652
C7×#4402
C7×#4352
C7×#4282
C7×#4272
#33507