Finite Group Databasebeta

← C7×#8762 #13185978

Subgroups — C7×#8762 #13185978

Order 1792.

472 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#2353C211yesmaximal, characteristic
2C7×#1799C211yesmaximal, characteristic
3C7×#1459C211yesmaximal, characteristic
4C7×#1294C212yesmaximal, complemented
5C7×#1289C212yesmaximal, complemented
6C2×C4×C7×D4C2211yescharacteristic
7C2×C7×#99—21no—
8C2×C7×C22⋊C8C2212yes—
9C7×#388—21no—
10C7×#387—21no—
11C7×#386—21no—
12C7×#383C411yescharacteristic, complemented
13C7×#381C411yescharacteristic, complemented
14C7×#380C2212yes—
15C7×#380C2211yescharacteristic
16C4×C7×C22⋊C4—21no—
17C4×C7×C22⋊C4C411yescharacteristic, complemented
18C7×#376C411yescharacteristic, complemented
19C2×C7×#95C2211yescharacteristic
20C7×#342—21no—
21C7×#324—21no—
22C7×#323—21no—
23#8762C711yesmaximal, characteristic, direct product factor, Sylow
24C22×C7×D4D411yescharacteristic
25C23×C4×C7—21no—
26C23×C4×C7C2×C411yescharacteristic
27C23×C4×C7C2311yescharacteristic
28C22×C7×C8—22no—
29C4×C7×D4—22no—
30C4×C7×D4—22no—
31C2×C7×C4⋊C4—21no—
32C2×C7×C4⋊C4—22no—
33C2×C7×C4⋊C4D411yescharacteristic
34C2×C7×C22⋊C4—21no—
35C2×C7×C22⋊C4—21no—
36C2×C7×C22⋊C4—22no—
37C2×C7×C22⋊C4—21no—
38C2×C7×C22⋊C4D411yescharacteristic
39C2×C7×C22⋊C4D411yescharacteristic
40C2×C42×C7—21no—
41C2×C42×C7—21no—
42C2×C42×C7C2×C411yescharacteristic
43C7×#99—42no—
44C7×C22⋊C8—22no—
45C7×C22⋊C8—22no—
46C7×#95—21no—
47C7×#95—21no—
48C7×#95—21no—
49C7×#95—21no—
50C7×#95—21no—
51C7×#95—22no—
52C7×#95C2×C411yescharacteristic
53C7×#95C2×C411yescharacteristic
54C7×#95C2×C411yescharacteristic
55C7×#95C2×C411yescharacteristic
56#2353C2×C711yescharacteristic
57#1799C2×C711yescharacteristic
58#1459C2×C711yescharacteristic
59#1294C2×C712yescomplemented
60#1289C2×C712yescomplemented
61C24×C7C22⋊C411yescharacteristic
62C24×C7C2×D411yescharacteristic
63C2×C7×D4—22no—
64C2×C7×D4—22no—
65C2×C7×D4—21no—
66C2×C7×D4—21no—
67C22×C4×C7—21no—
68C22×C4×C7—21no—
69C22×C4×C7—21no—
70C22×C4×C7—21no—
71C22×C4×C7—21no—
72C22×C4×C7—21no—
73C22×C4×C7—42no—
74C22×C4×C7—41no—
75C22×C4×C7—41no—
76C22×C4×C7—42no—
77C22×C4×C7—81no—
78C22×C4×C7—81no—
79C22×C4×C7—81no—
80C22×C4×C7C22⋊C411yescharacteristic
81C22×C4×C7C22⋊C411yescharacteristic
82C22×C4×C7C22⋊C411yescharacteristic
83C22×C4×C7C22×C411yescharacteristic
84C22×C4×C7C2×D411yescharacteristic
85C22×C4×C7#4112yes—
86C22×C4×C7#4112yes—
87C2×C7×C8—42no—
88C2×C7×C8—82no—
89C7×C4⋊C4—22no—
90C7×C4⋊C4—81no—
91C7×C4⋊C4—82no—
92C7×C22⋊C4—22no—
93C7×C22⋊C4—22no—
94C7×C22⋊C4—42no—
95C7×C22⋊C4—42no—
96C7×C22⋊C4—42no—
97C7×C22⋊C4—42no—
98C7×C22⋊C4—81no—
99C42×C7—21no—
100C42×C7—21no—
101C42×C7—42no—
102C42×C7—81no—
103C2×C4×D4C22×C711yescharacteristic
104C2×#99—21no—
105C2×C22⋊C8C22×C712yes—
106#388—21no—
107#387—21no—
108#386—21no—
109#383C4×C711yescharacteristic, complemented
110#381C4×C711yescharacteristic, complemented
111#380C22×C712yes—
112#380C22×C711yescharacteristic
113C4×C22⋊C4—21no—
114C4×C22⋊C4C4×C711yescharacteristic, complemented
115#376C4×C711yescharacteristic, complemented
116C2×#95C22×C711yescharacteristic
117#342—21no—
118#324—21no—
119#323—21no—
120C23×C7—21no—
121C23×C7—21no—
122C23×C7—21no—
123C23×C7—41no—
124C23×C7—41no—
125C23×C7—41no—
126C23×C7—41no—
127C23×C7C2×C22⋊C411yescharacteristic
128C23×C7#12312yes—
129C23×C7#13512yes—
130C23×C7AGL(1,ℤ/8)11yescharacteristic
131C23×C7#13711yescharacteristic
132C7×D4—42no—
133C7×D4—42no—
134C2×C4×C7—22no—
135C2×C4×C7—21no—
136C2×C4×C7—21no—
137C2×C4×C7—21no—
138C2×C4×C7—21no—
139C2×C4×C7—21no—
140C2×C4×C7—41no—
141C2×C4×C7—42no—
142C2×C4×C7—42no—
143C2×C4×C7—41no—
144C2×C4×C7—41no—
145C2×C4×C7—41no—
146C2×C4×C7—41no—
147C2×C4×C7—41no—
148C2×C4×C7—41no—
149C2×C4×C7—41no—
150C2×C4×C7—41no—
151C2×C4×C7—41no—
152C2×C4×C7—41no—
153C2×C4×C7—41no—
154C2×C4×C7—41no—
155C2×C4×C7—41no—
156C2×C4×C7—41no—
157C2×C4×C7—81no—
158C2×C4×C7—81no—
159C2×C4×C7—82no—
160C2×C4×C7—82no—
161C2×C4×C7—82no—
162C2×C4×C7—82no—
163C2×C4×C7—82no—
164C2×C4×C7#11711yescharacteristic
165C2×C4×C7#11711yescharacteristic
166C2×C4×C7#12312yes—
167C7×C8—82no—
168C22×D4C7×D411yescharacteristic
169C23×C4—21no—
170C23×C4C2×C4×C711yescharacteristic
171C23×C4C23×C711yescharacteristic
172C22×C8—22no—
173C4×D4—22no—
174C4×D4—22no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4C7×D411yescharacteristic
178C2×C22⋊C4—21no—
179C2×C22⋊C4—21no—
180C2×C22⋊C4—22no—
181C2×C22⋊C4—21no—
182C2×C22⋊C4C7×D411yescharacteristic
183C2×C22⋊C4C7×D411yescharacteristic
184C2×C42—21no—
185C2×C42—21no—
186C2×C42C2×C4×C711yescharacteristic
187#99—42no—
188C22⋊C8—22no—
189C22⋊C8—22no—
190#95—21no—
191#95—21no—
192#95—21no—
193#95—21no—
194#95—21no—
195#95—22no—
196#95C2×C4×C711yescharacteristic
197#95C2×C4×C711yescharacteristic
198#95C2×C4×C711yescharacteristic
199#95C2×C4×C711yescharacteristic
200C22×C7—21no—
201C22×C7—22no—
202C22×C7—21no—
203C22×C7—22no—
204C22×C7—41no—
205C22×C7—41no—
206C22×C7—42no—
207C22×C7—41no—
208C22×C7—41no—
209C22×C7—41no—
210C22×C7—81no—
211C22×C7—81no—
212C22×C7#38111yescharacteristic
213C22×C7#41611yescharacteristic
214C22×C7#41711yescharacteristic
215C22×C7#48212yes—
216C22×C7#48512yes—
217C4×C7—22no—
218C4×C7—21no—
219C4×C7—21no—
220C4×C7—42no—
221C4×C7—41no—
222C4×C7—41no—
223C4×C7—41no—
224C4×C7—81no—
225C4×C7—81no—
226C4×C7—81no—
227C4×C7—162no—
228C24C7×C22⋊C411yescharacteristic
229C24C2×C7×D411yescharacteristic
230C2×D4—22no—
231C2×D4—22no—
232C2×D4—21no—
233C2×D4—21no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—42no—
241C22×C4—41no—
242C22×C4—41no—
243C22×C4—42no—
244C22×C4—81no—
245C22×C4—81no—
246C22×C4—81no—
247C22×C4C7×C22⋊C411yescharacteristic
248C22×C4C7×C22⋊C411yescharacteristic
249C22×C4C7×C22⋊C411yescharacteristic
250C22×C4C22×C4×C711yescharacteristic
251C22×C4C2×C7×D411yescharacteristic
252C22×C4C7×#4112yes—
253C22×C4C7×#4112yes—
254C2×C8—42no—
255C2×C8—82no—
256C4⋊C4—22no—
257C4⋊C4—81no—
258C4⋊C4—82no—
259C22⋊C4—22no—
260C22⋊C4—22no—
261C22⋊C4—42no—
262C22⋊C4—42no—
263C22⋊C4—42no—
264C22⋊C4—42no—
265C22⋊C4—81no—
266C42—21no—
267C42—21no—
268C42—42no—
269C42—81no—
270C2×C7—21no—
271C2×C7—21no—
272C2×C7—22no—
273C2×C7—22no—
274C2×C7—81no—
275C2×C7#182511yescharacteristic
276C2×C7#182711yescharacteristic
277C2×C7#182811yescharacteristic
278C23—21no—
279C23—21no—
280C23—21no—
281C23—41no—
282C23—41no—
283C23—41no—
284C23—41no—
285C23C2×C7×C22⋊C411yescharacteristic
286C23C7×#12312yes—
287C23C7×#13512yes—
288C23C7×AGL(1,ℤ/8)11yescharacteristic
289C23C7×#13711yescharacteristic
290D4—42no—
291D4—42no—
292C2×C4—22no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—41no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—81no—
316C2×C4—81no—
317C2×C4—82no—
318C2×C4—82no—
319C2×C4—82no—
320C2×C4—82no—
321C2×C4—82no—
322C2×C4C7×#11711yescharacteristic
323C2×C4C7×#11711yescharacteristic
324C2×C4C7×#12312yes—
325C8—82no—
326C7#876211yescharacteristic, direct product factor, Sylow
327C22—21no—
328C22—22no—
329C22—21no—
330C22—22no—
331C22—41no—
332C22—41no—
333C22—42no—
334C22—41no—
335C22—41no—
336C22—41no—
337C22—81no—
338C22—81no—
339C22C7×#38111yescharacteristic
340C22C7×#41611yescharacteristic
341C22C7×#41711yescharacteristic
342C22C7×#48212yes—
343C22C7×#48512yes—
344C4—22no—
345C4—21no—
346C4—21no—
347C4—42no—
348C4—41no—
349C4—41no—
350C4—41no—
351C4—81no—
352C4—81no—
353C4—81no—
354C4—162no—
355C2—21no—
356C2—21no—
357C2—22no—
358C2—22no—
359C2—81no—
360C2C7×#182511yescharacteristic
361C2C7×#182711yescharacteristic
362C2C7×#182811yescharacteristic
subgroupG/H
center, socleC22×C7#381
commutatorC22×C4C22×C4×C7
Frattini, cosocleC23×C4C23×C7
prime radicalC22×C7×D4D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C7 ◁ C23×C4×C7 ◁ G

Complements

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

NH ≅ G/Ntype
4. C7×#1294C2semidirect
5. C7×#1289C2semidirect
12. C7×#383C4semidirect
13. C7×#381C4semidirect
17. C4×C7×C22⋊C4C4semidirect
18. C7×#376C4semidirect
23. #8762C7direct (N × H)
59. #1294C2×C7semidirect
60. #1289C2×C7semidirect
109. #383C4×C7semidirect
110. #381C4×C7semidirect
114. C4×C22⋊C4C4×C7semidirect
115. #376C4×C7semidirect
326. C7#8762direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#23532
C7×#17992
C7×#14592
C7×#12942
C7×#12892
#87627