Finite Group Databasebeta

← C7×#3178 #11742487

Subgroups — C7×#3178 #11742487

Order 896.

350 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#542C211yesmaximal, characteristic
2C7×#541C211yesmaximal, characteristic, complemented
3C7×#501C211yesmaximal, characteristic, complemented
4C7×#497C211yesmaximal, characteristic, complemented
5C7×#488C211yesmaximal, characteristic, complemented
6C7×#487C211yesmaximal, characteristic, complemented
7C7×#470C211yesmaximal, characteristic, complemented
8C7×#468C211yesmaximal, characteristic
9C7×#467C211yesmaximal, characteristic, complemented
10C7×#465C211yesmaximal, characteristic
11C7×#441C211yesmaximal, characteristic, complemented
12C7×#440C211yesmaximal, characteristic, complemented
13C4×C7×Q16C211yesmaximal, characteristic, complemented
14C4×C7×SD16C211yesmaximal, characteristic, complemented
15C7×#405C211yesmaximal, characteristic, complemented
16C2×C7×Q16C2211yescharacteristic
17C2×C7×Q16C2211yescharacteristic
18C2×C7×SD16C2211yescharacteristic
19C2×C7×SD16C2211yescharacteristic
20C2×C7×M16C2211yescharacteristic
21C22×C7×C8C2211yescharacteristic
22C7×C4⋊Q8C2211yescharacteristic
23C7×C4⋊Q8C2211yescharacteristic
24C7×#126—21no—
25C7×#126—21no—
26C7×#126—21no—
27C7×#125—21no—
28C7×#125—21no—
29C7×#125C2211yescharacteristic
30C7×#124—21no—
31C7×#124C2211yescharacteristic
32C7×#123—21no—
33C7×C22⋊Q8—21no—
34C7×C22⋊Q8C2211yescharacteristic
35C7×C22⋊Q8C2211yescharacteristic
36C7×C22⋊Q8C2211yescharacteristic
37C7×#121C2211yescharacteristic
38C4×C7×Q8C2211yescharacteristic
39C4×C7×Q8C2211yescharacteristic
40C4×C7×Q8C2211yescharacteristic
41C4×C7×D4C2211yescharacteristic
42C7×#117C2211yescharacteristic
43C7×#107C2211yescharacteristic
44C7×#107C2211yescharacteristic
45C7×#106C2211yescharacteristic
46C7×#106C2211yescharacteristic
47C7×#103C2211yescharacteristic
48C7×#103C2211yescharacteristic
49C7×#103C2211yescharacteristic
50C7×#103C2211yescharacteristic
51C7×#103C2211yescharacteristic
52C7×#103C2211yescharacteristic
53C7×#102C2211yescharacteristic
54C7×#102C2211yescharacteristic
55C7×#97C2211yescharacteristic
56C7×#97C2211yescharacteristic
57C4×C7×C8C2211yescharacteristic
58C4×C7×C8C2211yescharacteristic
59#3178C711yesmaximal, characteristic, direct product factor, Sylow
60C2×C7×Q8—21no—
61C2×C7×Q8C2311yescharacteristic
62C2×C7×Q8C2311yescharacteristic
63C2×C7×Q8C2311yescharacteristic
64C2×C7×D4—21no—
65C2×C7×D4C2311yescharacteristic
66C22×C4×C7—21no—
67C22×C4×C7C2311yescharacteristic
68C7×Q16—22no—
69C7×Q16—22no—
70C7×SD16—22no—
71C7×SD16—22no—
72C7×M16—22no—
73C2×C7×C8—22no—
74C2×C7×C8C2311yescharacteristic
75C2×C7×C8C2311yescharacteristic
76C2×C7×C8C2311yescharacteristic
77C2×C7×C8C2311yescharacteristic
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⋊C4—21no—
87C7×C4⋊C4—21no—
88C7×C4⋊C4—21no—
89C7×C4⋊C4D411yescharacteristic
90C7×C4⋊C4D411yescharacteristic
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⋊C4—21no—
102C7×C22⋊C4D411yescharacteristic
103C7×C22⋊C4D411yescharacteristic
104C42×C7—21no—
105C42×C7—21no—
106C42×C7—21no—
107C42×C7C2311yescharacteristic
108C42×C7C2311yescharacteristic
109#542C2×C711yescharacteristic
110#541C2×C711yescharacteristic, complemented
111#501C2×C711yescharacteristic, complemented
112#497C2×C711yescharacteristic, complemented
113#488C2×C711yescharacteristic, complemented
114#487C2×C711yescharacteristic, complemented
115#470C2×C711yescharacteristic, complemented
116#468C2×C711yescharacteristic
117#467C2×C711yescharacteristic, complemented
118#465C2×C711yescharacteristic
119#441C2×C711yescharacteristic, complemented
120#440C2×C711yescharacteristic, complemented
121C4×Q16C2×C711yescharacteristic, complemented
122C4×SD16C2×C711yescharacteristic, complemented
123#405C2×C711yescharacteristic, complemented
124C23×C7—21no—
125C23×C7C2×D411yescharacteristic
126C7×Q8—21no—
127C7×Q8—21no—
128C7×Q8—21no—
129C7×Q8—21no—
130C7×Q8—21no—
131C7×Q8—21no—
132C7×Q8—81no—
133C7×D4—21no—
134C7×D4—21no—
135C7×D4—81no—
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—81no—
146C2×C4×C7C2×D411yescharacteristic
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#4112yes—
155C7×C8#4112yes—
156C2×Q16C22×C711yescharacteristic
157C2×Q16C22×C711yescharacteristic
158C2×SD16C22×C711yescharacteristic
159C2×SD16C22×C711yescharacteristic
160C2×M16C22×C711yescharacteristic
161C22×C8C22×C711yescharacteristic
162C4⋊Q8C22×C711yescharacteristic
163C4⋊Q8C22×C711yescharacteristic
164#126—21no—
165#126—21no—
166#126—21no—
167#125—21no—
168#125—21no—
169#125C22×C711yescharacteristic
170#124—21no—
171#124C22×C711yescharacteristic
172#123—21no—
173C22⋊Q8—21no—
174C22⋊Q8C22×C711yescharacteristic
175C22⋊Q8C22×C711yescharacteristic
176C22⋊Q8C22×C711yescharacteristic
177#121C22×C711yescharacteristic
178C4×Q8C22×C711yescharacteristic
179C4×Q8C22×C711yescharacteristic
180C4×Q8C22×C711yescharacteristic
181C4×D4C22×C711yescharacteristic
182#117C22×C711yescharacteristic
183#107C22×C711yescharacteristic
184#107C22×C711yescharacteristic
185#106C22×C711yescharacteristic
186#106C22×C711yescharacteristic
187#103C22×C711yescharacteristic
188#103C22×C711yescharacteristic
189#103C22×C711yescharacteristic
190#103C22×C711yescharacteristic
191#103C22×C711yescharacteristic
192#103C22×C711yescharacteristic
193#102C22×C711yescharacteristic
194#102C22×C711yescharacteristic
195#97C22×C711yescharacteristic
196#97C22×C711yescharacteristic
197C4×C8C22×C711yescharacteristic
198C4×C8C22×C711yescharacteristic
199C22×C7—21no—
200C22×C7—21no—
201C22×C7—21no—
202C22×C7—41no—
203C22×C7—41no—
204C22×C7—41no—
205C22×C7C22×D411yescharacteristic
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×C7—41no—
217C4×C7—41no—
218C4×C7C2×#4111yescharacteristic
219C4×C7C2×#4111yescharacteristic
220C2×Q8—21no—
221C2×Q8C23×C711yescharacteristic
222C2×Q8C23×C711yescharacteristic
223C2×Q8C23×C711yescharacteristic
224C2×D4—21no—
225C2×D4C23×C711yescharacteristic
226C22×C4—21no—
227C22×C4C23×C711yescharacteristic
228Q16—22no—
229Q16—22no—
230SD16—22no—
231SD16—22no—
232M16—22no—
233C2×C8—22no—
234C2×C8C23×C711yescharacteristic
235C2×C8C23×C711yescharacteristic
236C2×C8C23×C711yescharacteristic
237C2×C8C23×C711yescharacteristic
238C4⋊C4—21no—
239C4⋊C4—21no—
240C4⋊C4—21no—
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⋊C4C7×D411yescharacteristic
250C4⋊C4C7×D411yescharacteristic
251C4⋊C4C23×C711yescharacteristic
252C4⋊C4C23×C711yescharacteristic
253C4⋊C4C23×C711yescharacteristic
254C4⋊C4C23×C711yescharacteristic
255C22⋊C4—21no—
256C22⋊C4—21no—
257C22⋊C4—21no—
258C22⋊C4—21no—
259C22⋊C4—21no—
260C22⋊C4—21no—
261C22⋊C4—21no—
262C22⋊C4C7×D411yescharacteristic
263C22⋊C4C7×D411yescharacteristic
264C42—21no—
265C42—21no—
266C42—21no—
267C42C23×C711yescharacteristic
268C42C23×C711yescharacteristic
269C2×C7—41no—
270C2×C7—81no—
271C2×C7#53211yescharacteristic
272C2×C7#57711yescharacteristic
273C2×C7#57811yescharacteristic
274C23—21no—
275C23C2×C7×D411yescharacteristic
276Q8—21no—
277Q8—21no—
278Q8—21no—
279Q8—21no—
280Q8—21no—
281Q8—21no—
282Q8—81no—
283D4—21no—
284D4—21no—
285D4—81no—
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—81no—
296C2×C4C2×C7×D411yescharacteristic
297C2×C4C2×C7×D411yescharacteristic
298C2×C4C2×C7×D411yescharacteristic
299C2×C4C2×C7×D411yescharacteristic
300C2×C4C2×C7×D411yescharacteristic
301C2×C4C24×C711yescharacteristic
302C8—21no—
303C8—21no—
304C8C7×#4112yes—
305C8C7×#4112yes—
306C7#317811yescharacteristic, direct product factor, Sylow
307C22—21no—
308C22—21no—
309C22—21no—
310C22—41no—
311C22—41no—
312C22—41no—
313C22C22×C7×D411yescharacteristic
314C4—21no—
315C4—21no—
316C4—21no—
317C4—21no—
318C4—21no—
319C4—41no—
320C4—41no—
321C4—41no—
322C4—41no—
323C4—41no—
324C4—41no—
325C4—41no—
326C4C2×C7×#4111yescharacteristic
327C4C2×C7×#4111yescharacteristic
328C2—41no—
329C2—81no—
330C2C7×#53211yescharacteristic
331C2C7×#57711yescharacteristic
332C2C7×#57811yescharacteristic
subgroupG/H
center, socleC22×C7C22×D4
commutator, Frattini, cosocleC2×C4C24×C7
prime radicalC7×#121C22
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C7 ◁ C7×#117 ◁ G

Complements

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

NH ≅ G/Ntype
2. C7×#541C2semidirect
3. C7×#501C2semidirect
4. C7×#497C2semidirect
5. C7×#488C2semidirect
6. C7×#487C2semidirect
7. C7×#470C2semidirect
9. C7×#467C2semidirect
11. C7×#441C2semidirect
12. C7×#440C2semidirect
13. C4×C7×Q16C2semidirect
14. C4×C7×SD16C2semidirect
15. C7×#405C2semidirect
59. #3178C7direct (N × H)
110. #541C2×C7semidirect
111. #501C2×C7semidirect
112. #497C2×C7semidirect
113. #488C2×C7semidirect
114. #487C2×C7semidirect
115. #470C2×C7semidirect
117. #467C2×C7semidirect
119. #441C2×C7semidirect
120. #440C2×C7semidirect
121. C4×Q16C2×C7semidirect
122. C4×SD16C2×C7semidirect
123. #405C2×C7semidirect
306. C7#3178direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#5422
C7×#5412
C7×#5012
C7×#4972
C7×#4882
C7×#4872
C7×#4702
C7×#4682
C7×#4672
C7×#4652
C7×#4412
C7×#4402
C4×C7×Q162
C4×C7×SD162
C7×#4052
#31787