Finite Group Databasebeta

← C7×#3121 #11742430

Subgroups — C7×#3121 #11742430

Order 896.

370 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#542C211yesmaximal, characteristic
2C7×#519C211yesmaximal, characteristic, complemented
3C7×#485C211yesmaximal, characteristic, complemented
4C7×#482C211yesmaximal, characteristic
5C7×#477C211yesmaximal, characteristic, complemented
6C7×#475C211yesmaximal, characteristic, complemented
7C7×#464C211yesmaximal, characteristic, complemented
8C7×#464C211yesmaximal, characteristic, complemented
9C7×#451C211yesmaximal, characteristic, complemented
10C7×#448C211yesmaximal, characteristic
11C7×#441C211yesmaximal, characteristic, complemented
12C7×#440C211yesmaximal, characteristic, complemented
13C4×C7×Q16C211yesmaximal, characteristic, complemented
14C4×C7×SD16C211yesmaximal, characteristic, complemented
15C7×#433C211yesmaximal, characteristic, complemented
16C22×C7×Q8C2211yescharacteristic
17C2×C7×Q16C2211yescharacteristic
18C2×C7×Q16C2211yescharacteristic
19C2×C7×SD16C2211yescharacteristic
20C2×C7×SD16C2211yescharacteristic
21C7×C4⋊Q8C2211yescharacteristic
22C7×#126—21no—
23C7×#124—21no—
24C7×#124C2211yescharacteristic
25C7×#123—21no—
26C7×C22⋊Q8—21no—
27C7×C22⋊Q8C2211yescharacteristic
28C7×#121C2211yescharacteristic
29C4×C7×Q8—21no—
30C4×C7×Q8—21no—
31C4×C7×Q8C2211yescharacteristic
32C4×C7×Q8C2211yescharacteristic
33C4×C7×Q8C2211yescharacteristic
34C4×C7×Q8C2211yescharacteristic
35C4×C7×Q8C2211yescharacteristic
36C4×C7×D4C2211yescharacteristic
37C7×#117—21no—
38C7×#117—21no—
39C7×#117C2211yescharacteristic
40C7×#117C2211yescharacteristic
41C7×#107C2211yescharacteristic
42C7×#107C2211yescharacteristic
43C7×#106C2211yescharacteristic
44C7×#106C2211yescharacteristic
45C7×C4⋊C8C2211yescharacteristic
46C7×C4⋊C8C2211yescharacteristic
47C7×#103C2211yescharacteristic
48C7×#103C2211yescharacteristic
49C7×#103C2211yescharacteristic
50C7×#103C2211yescharacteristic
51C7×#103C2211yescharacteristic
52C7×#103C2211yescharacteristic
53C7×#102C2211yescharacteristic
54C7×#102C2211yescharacteristic
55C7×C22⋊C8C2211yescharacteristic
56C7×C22⋊C8C2211yescharacteristic
57C7×#97C2211yescharacteristic
58C4×C7×C8C2211yescharacteristic
59#3121C711yesmaximal, characteristic, direct product factor, Sylow
60C2×C7×Q8—22no—
61C2×C7×Q8—22no—
62C2×C7×Q8—21no—
63C2×C7×Q8—21no—
64C2×C7×Q8C2311yescharacteristic
65C2×C7×Q8C2311yescharacteristic
66C2×C7×Q8C2311yescharacteristic
67C2×C7×D4—21no—
68C2×C7×D4C2311yescharacteristic
69C22×C4×C7—21no—
70C22×C4×C7—21no—
71C22×C4×C7C2311yescharacteristic
72C7×Q16—22no—
73C7×Q16—22no—
74C7×SD16—22no—
75C7×SD16—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⋊C4—21no—
85C7×C4⋊C4—21no—
86C7×C4⋊C4—21no—
87C7×C4⋊C4D411yescharacteristic
88C7×C4⋊C4D411yescharacteristic
89C7×C4⋊C4C2311yescharacteristic
90C7×C4⋊C4C2311yescharacteristic
91C7×C4⋊C4C2311yescharacteristic
92C7×C4⋊C4C2311yescharacteristic
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×C7—21no—
106C42×C7—21no—
107C42×C7C2311yescharacteristic
108C42×C7C2311yescharacteristic
109#542C2×C711yescharacteristic
110#519C2×C711yescharacteristic, complemented
111#485C2×C711yescharacteristic, complemented
112#482C2×C711yescharacteristic
113#477C2×C711yescharacteristic, complemented
114#475C2×C711yescharacteristic, complemented
115#464C2×C711yescharacteristic, complemented
116#464C2×C711yescharacteristic, complemented
117#451C2×C711yescharacteristic, complemented
118#448C2×C711yescharacteristic
119#441C2×C711yescharacteristic, complemented
120#440C2×C711yescharacteristic, complemented
121C4×Q16C2×C711yescharacteristic, complemented
122C4×SD16C2×C711yescharacteristic, complemented
123#433C2×C711yescharacteristic, complemented
124C23×C7—21no—
125C23×C7C2×D411yescharacteristic
126C7×Q8—24no—
127C7×Q8—21no—
128C7×Q8—21no—
129C7×Q8—21no—
130C7×Q8—21no—
131C7×Q8—81no—
132C7×Q8#4112yes—
133C7×Q8#4112yes—
134C7×D4—21no—
135C7×D4—21no—
136C7×D4—81no—
137C2×C4×C7—24no—
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—21no—
146C2×C4×C7—21no—
147C2×C4×C7—81no—
148C2×C4×C7C2×D411yescharacteristic
149C2×C4×C7C2×D411yescharacteristic
150C2×C4×C7C2×D411yescharacteristic
151C2×C4×C7C2×D411yescharacteristic
152C2×C4×C7C2×D411yescharacteristic
153C2×C4×C7C2411yescharacteristic
154C7×C8—21no—
155C7×C8—21no—
156C7×C8—21no—
157C7×C8—21no—
158C22×Q8C22×C711yescharacteristic
159C2×Q16C22×C711yescharacteristic
160C2×Q16C22×C711yescharacteristic
161C2×SD16C22×C711yescharacteristic
162C2×SD16C22×C711yescharacteristic
163C4⋊Q8C22×C711yescharacteristic
164#126—21no—
165#124—21no—
166#124C22×C711yescharacteristic
167#123—21no—
168C22⋊Q8—21no—
169C22⋊Q8C22×C711yescharacteristic
170#121C22×C711yescharacteristic
171C4×Q8—21no—
172C4×Q8—21no—
173C4×Q8C22×C711yescharacteristic
174C4×Q8C22×C711yescharacteristic
175C4×Q8C22×C711yescharacteristic
176C4×Q8C22×C711yescharacteristic
177C4×Q8C22×C711yescharacteristic
178C4×D4C22×C711yescharacteristic
179#117—21no—
180#117—21no—
181#117C22×C711yescharacteristic
182#117C22×C711yescharacteristic
183#107C22×C711yescharacteristic
184#107C22×C711yescharacteristic
185#106C22×C711yescharacteristic
186#106C22×C711yescharacteristic
187C4⋊C8C22×C711yescharacteristic
188C4⋊C8C22×C711yescharacteristic
189#103C22×C711yescharacteristic
190#103C22×C711yescharacteristic
191#103C22×C711yescharacteristic
192#103C22×C711yescharacteristic
193#103C22×C711yescharacteristic
194#103C22×C711yescharacteristic
195#102C22×C711yescharacteristic
196#102C22×C711yescharacteristic
197C22⋊C8C22×C711yescharacteristic
198C22⋊C8C22×C711yescharacteristic
199#97C22×C711yescharacteristic
200C4×C8C22×C711yescharacteristic
201C22×C7—21no—
202C22×C7—21no—
203C22×C7—21no—
204C22×C7—41no—
205C22×C7—41no—
206C22×C7—41no—
207C22×C7C22×D411yescharacteristic
208C4×C7—22no—
209C4×C7—22no—
210C4×C7—21no—
211C4×C7—21no—
212C4×C7—21no—
213C4×C7—21no—
214C4×C7—21no—
215C4×C7—41no—
216C4×C7—41no—
217C4×C7—41no—
218C4×C7—41no—
219C4×C7—41no—
220C4×C7C2×#4111yescharacteristic
221C4×C7C2×#4111yescharacteristic
222C2×Q8—22no—
223C2×Q8—22no—
224C2×Q8—21no—
225C2×Q8—21no—
226C2×Q8C23×C711yescharacteristic
227C2×Q8C23×C711yescharacteristic
228C2×Q8C23×C711yescharacteristic
229C2×D4—21no—
230C2×D4C23×C711yescharacteristic
231C22×C4—21no—
232C22×C4—21no—
233C22×C4C23×C711yescharacteristic
234Q16—22no—
235Q16—22no—
236SD16—22no—
237SD16—22no—
238C2×C8C23×C711yescharacteristic
239C2×C8C23×C711yescharacteristic
240C2×C8C23×C711yescharacteristic
241C2×C8C23×C711yescharacteristic
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⋊C4C7×D411yescharacteristic
262C22⋊C4C7×D411yescharacteristic
263C42—21no—
264C42—21no—
265C42—21no—
266C42—21no—
267C42—21no—
268C42—21no—
269C42C23×C711yescharacteristic
270C42C23×C711yescharacteristic
271C2×C7—41no—
272C2×C7—81no—
273C2×C7#52511yescharacteristic
274C2×C7#57711yescharacteristic
275C2×C7#57811yescharacteristic
276C23—21no—
277C23C2×C7×D411yescharacteristic
278Q8—24no—
279Q8—21no—
280Q8—21no—
281Q8—21no—
282Q8—21no—
283Q8—81no—
284Q8C7×#4112yes—
285Q8C7×#4112yes—
286D4—21no—
287D4—21no—
288D4—81no—
289C2×C4—24no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—81no—
300C2×C4C2×C7×D411yescharacteristic
301C2×C4C2×C7×D411yescharacteristic
302C2×C4C2×C7×D411yescharacteristic
303C2×C4C2×C7×D411yescharacteristic
304C2×C4C2×C7×D411yescharacteristic
305C2×C4C24×C711yescharacteristic
306C8—21no—
307C8—21no—
308C8—21no—
309C8—21no—
310C7#312111yescharacteristic, direct product factor, Sylow
311C22—21no—
312C22—21no—
313C22—21no—
314C22—41no—
315C22—41no—
316C22—41no—
317C22C22×C7×D411yescharacteristic
318C4—22no—
319C4—22no—
320C4—21no—
321C4—21no—
322C4—21no—
323C4—21no—
324C4—21no—
325C4—41no—
326C4—41no—
327C4—41no—
328C4—41no—
329C4—41no—
330C4C2×C7×#4111yescharacteristic
331C4C2×C7×#4111yescharacteristic
332C2—41no—
333C2—81no—
334C2C7×#52511yescharacteristic
335C2C7×#57711yescharacteristic
336C2C7×#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×#519C2semidirect
3. C7×#485C2semidirect
5. C7×#477C2semidirect
6. C7×#475C2semidirect
7. C7×#464C2semidirect
8. C7×#464C2semidirect
9. C7×#451C2semidirect
11. C7×#441C2semidirect
12. C7×#440C2semidirect
13. C4×C7×Q16C2semidirect
14. C4×C7×SD16C2semidirect
15. C7×#433C2semidirect
59. #3121C7direct (N × H)
110. #519C2×C7semidirect
111. #485C2×C7semidirect
113. #477C2×C7semidirect
114. #475C2×C7semidirect
115. #464C2×C7semidirect
116. #464C2×C7semidirect
117. #451C2×C7semidirect
119. #441C2×C7semidirect
120. #440C2×C7semidirect
121. C4×Q16C2×C7semidirect
122. C4×SD16C2×C7semidirect
123. #433C2×C7semidirect
310. C7#3121direct (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×#5192
C7×#4852
C7×#4822
C7×#4772
C7×#4752
C7×#4642
C7×#4512
C7×#4482
C7×#4412
C7×#4402
C4×C7×Q162
C4×C7×SD162
C7×#4332
#31217