Finite Group Databasebeta

← C7×#9036 #13186252

Subgroups — C7×#9036 #13186252

Order 1792.

566 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#2921C211yesmaximal, characteristic, complemented
2C7×#2392C211yesmaximal, characteristic
3C7×#1486C212yesmaximal, complemented
4C7×#1458C211yesmaximal, characteristic, complemented
5C7×#1303C212yesmaximal, complemented
6C2×C4×C7×D4—21no—
7C2×C4×C7×D4C2211yescharacteristic
8C2×C7×#117C2211yescharacteristic
9C7×#433—24no—
10C7×#431C2212yes—
11C7×#423C2212yes—
12C7×#407—21no—
13C4×C7×M16—21no—
14C7×#398C411yescharacteristic
15C7×#397C411yescharacteristic
16C7×#396—21no—
17C7×#394—21no—
18C7×#394C411yescharacteristic
19C7×#392C411yescharacteristic
20C4×C7×C4⋊C4C2211yescharacteristic
21C4×C7×C22⋊C4—21no—
22C7×D4⋊C8—24no—
23C7×#324—21no—
24C7×#323—21no—
25#9036C711yesmaximal, characteristic, direct product factor, Sylow
26C22×C7×D4—21no—
27C22×C7×D4C2×C411yescharacteristic
28C23×C4×C7—21no—
29C23×C4×C7D411yescharacteristic
30C2×C7×M16—22no—
31C4×C7×D4—22no—
32C4×C7×D4—22no—
33C4×C7×D4—44no—
34C7×#117—22no—
35C7×#117—22no—
36C2×C7×C4⋊C4—21no—
37C2×C7×C4⋊C4—21no—
38C2×C7×C4⋊C4C2×C411yescharacteristic
39C2×C7×C4⋊C4C2×C411yescharacteristic
40C2×C7×C4⋊C4D411yescharacteristic
41C2×C7×C22⋊C4—21no—
42C2×C7×C22⋊C4—21no—
43C2×C7×C22⋊C4—21no—
44C2×C7×C22⋊C4—21no—
45C2×C7×C22⋊C4—21no—
46C2×C7×C22⋊C4C2×C411yescharacteristic
47C2×C7×C22⋊C4D411yescharacteristic
48C2×C42×C7—21no—
49C2×C42×C7D411yescharacteristic
50C2×C42×C7C2311yescharacteristic
51C7×C4⋊C8—22no—
52C7×C4⋊C8—22no—
53C7×C4⋊C8D414yescomplemented
54C7×C22⋊C8—24no—
55C7×C22⋊C8—22no—
56C7×#97—22no—
57C4×C7×C8—22no—
58C4×C7×C8—22no—
59C7×#95—21no—
60C7×#95—21no—
61C7×#95C2×C411yescharacteristic
62C7×#95C2×C411yescharacteristic
63#2921C2×C711yescharacteristic, complemented
64#2392C2×C711yescharacteristic
65#1486C2×C712yescomplemented
66#1458C2×C711yescharacteristic, complemented
67#1303C2×C712yescomplemented
68C24×C7—21no—
69C24×C7C22⋊C411yescharacteristic
70C2×C7×D4—22no—
71C2×C7×D4—42no—
72C2×C7×D4—42no—
73C2×C7×D4—42no—
74C2×C7×D4—42no—
75C22×C4×C7—21no—
76C22×C4×C7—21no—
77C22×C4×C7—21no—
78C22×C4×C7—22no—
79C22×C4×C7—22no—
80C22×C4×C7—21no—
81C22×C4×C7—21no—
82C22×C4×C7—41no—
83C22×C4×C7—41no—
84C22×C4×C7—42no—
85C22×C4×C7—42no—
86C22×C4×C7C22⋊C411yescharacteristic
87C22×C4×C7C22⋊C411yescharacteristic
88C22×C4×C7C22⋊C411yescharacteristic
89C22×C4×C7C22×C411yescharacteristic
90C22×C4×C7C2×D411yescharacteristic
91C22×C4×C7C2×D411yescharacteristic
92C7×M16—42no—
93C7×M16—42no—
94C2×C7×C8—24no—
95C2×C7×C8—42no—
96C7×C4⋊C4—22no—
97C7×C4⋊C4—22no—
98C7×C4⋊C4—22no—
99C7×C4⋊C4—42no—
100C7×C4⋊C4—42no—
101C7×C22⋊C4—22no—
102C7×C22⋊C4—22no—
103C7×C22⋊C4—42no—
104C7×C22⋊C4—42no—
105C7×C22⋊C4—42no—
106C7×C22⋊C4—42no—
107C7×C22⋊C4—42no—
108C42×C7—21no—
109C42×C7—21no—
110C42×C7—42no—
111C42×C7C2×D412yes—
112C42×C7#4112yes—
113C2×C4×D4—21no—
114C2×C4×D4C22×C711yescharacteristic
115C2×#117C22×C711yescharacteristic
116#433—24no—
117#431C22×C712yes—
118#423C22×C712yes—
119#407—21no—
120C4×M16—21no—
121#398C4×C711yescharacteristic
122#397C4×C711yescharacteristic
123#396—21no—
124#394—21no—
125#394C4×C711yescharacteristic
126#392C4×C711yescharacteristic
127C4×C4⋊C4C22×C711yescharacteristic
128C4×C22⋊C4—21no—
129D4⋊C8—24no—
130#324—21no—
131#323—21no—
132C23×C7—22no—
133C23×C7—21no—
134C23×C7—21no—
135C23×C7—21no—
136C23×C7—41no—
137C23×C7—41no—
138C23×C7—41no—
139C23×C7—41no—
140C23×C7—41no—
141C23×C7—41no—
142C23×C7—41no—
143C23×C7—41no—
144C23×C7C2×C22⋊C411yescharacteristic
145C7×D4—24no—
146C7×D4—42no—
147C7×D4—48no—
148C2×C4×C7—22no—
149C2×C4×C7—21no—
150C2×C4×C7—21no—
151C2×C4×C7—21no—
152C2×C4×C7—21no—
153C2×C4×C7—21no—
154C2×C4×C7—41no—
155C2×C4×C7—41no—
156C2×C4×C7—41no—
157C2×C4×C7—41no—
158C2×C4×C7—42no—
159C2×C4×C7—42no—
160C2×C4×C7—41no—
161C2×C4×C7—41no—
162C2×C4×C7—41no—
163C2×C4×C7—41no—
164C2×C4×C7—41no—
165C2×C4×C7—41no—
166C2×C4×C7—41no—
167C2×C4×C7—41no—
168C2×C4×C7—82no—
169C2×C4×C7—82no—
170C2×C4×C7C4×D412yes—
171C2×C4×C7#12011yescharacteristic
172C2×C4×C7#12112yes—
173C2×C4×C7#12311yescharacteristic
174C2×C4×C7#13512yes—
175C2×C4×C7AGL(1,ℤ/8)12yes—
176C7×C8—44no—
177C7×C8—82no—
178C22×D4—21no—
179C22×D4C2×C4×C711yescharacteristic
180C23×C4—21no—
181C23×C4C7×D411yescharacteristic
182C2×M16—22no—
183C4×D4—22no—
184C4×D4—22no—
185C4×D4—44no—
186#117—22no—
187#117—22no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4C2×C4×C711yescharacteristic
191C2×C4⋊C4C2×C4×C711yescharacteristic
192C2×C4⋊C4C7×D411yescharacteristic
193C2×C22⋊C4—21no—
194C2×C22⋊C4—21no—
195C2×C22⋊C4—21no—
196C2×C22⋊C4—21no—
197C2×C22⋊C4—21no—
198C2×C22⋊C4C2×C4×C711yescharacteristic
199C2×C22⋊C4C7×D411yescharacteristic
200C2×C42—21no—
201C2×C42C7×D411yescharacteristic
202C2×C42C23×C711yescharacteristic
203C4⋊C8—22no—
204C4⋊C8—22no—
205C4⋊C8C7×D414yescomplemented
206C22⋊C8—24no—
207C22⋊C8—22no—
208#97—22no—
209C4×C8—22no—
210C4×C8—22no—
211#95—21no—
212#95—21no—
213#95C2×C4×C711yescharacteristic
214#95C2×C4×C711yescharacteristic
215C22×C7—21no—
216C22×C7—21no—
217C22×C7—42no—
218C22×C7—42no—
219C22×C7—42no—
220C22×C7—41no—
221C22×C7—41no—
222C22×C7—42no—
223C22×C7—41no—
224C22×C7—81no—
225C22×C7—81no—
226C22×C7—81no—
227C22×C7—81no—
228C22×C7—81no—
229C22×C7#38611yescharacteristic
230C22×C7#41611yescharacteristic
231C22×C7#41811yescharacteristic
232C4×C7—22no—
233C4×C7—22no—
234C4×C7—42no—
235C4×C7—41no—
236C4×C7—41no—
237C4×C7—41no—
238C4×C7—81no—
239C4×C7—81no—
240C4×C7#44912yes—
241C4×C7#48212yes—
242C24—21no—
243C24C7×C22⋊C411yescharacteristic
244C2×D4—22no—
245C2×D4—42no—
246C2×D4—42no—
247C2×D4—42no—
248C2×D4—42no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4C7×C22⋊C411yescharacteristic
261C22×C4C7×C22⋊C411yescharacteristic
262C22×C4C7×C22⋊C411yescharacteristic
263C22×C4C22×C4×C711yescharacteristic
264C22×C4C2×C7×D411yescharacteristic
265C22×C4C2×C7×D411yescharacteristic
266M16—42no—
267M16—42no—
268C2×C8—24no—
269C2×C8—42no—
270C4⋊C4—22no—
271C4⋊C4—22no—
272C4⋊C4—22no—
273C4⋊C4—42no—
274C4⋊C4—42no—
275C22⋊C4—22no—
276C22⋊C4—22no—
277C22⋊C4—42no—
278C22⋊C4—42no—
279C22⋊C4—42no—
280C22⋊C4—42no—
281C22⋊C4—42no—
282C42—21no—
283C42—21no—
284C42—42no—
285C42C2×C7×D412yes—
286C42C7×#4112yes—
287C2×C7—21no—
288C2×C7—21no—
289C2×C7—81no—
290C2×C7—82no—
291C2×C7#187511yescharacteristic
292C2×C7#188411yescharacteristic
293C2×C7#189111yescharacteristic
294C23—22no—
295C23—21no—
296C23—21no—
297C23—21no—
298C23—41no—
299C23—41no—
300C23—41no—
301C23—41no—
302C23—41no—
303C23—41no—
304C23—41no—
305C23—41no—
306C23C2×C7×C22⋊C411yescharacteristic
307D4—24no—
308D4—42no—
309D4—48no—
310C2×C4—22no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—82no—
331C2×C4—82no—
332C2×C4C4×C7×D412yes—
333C2×C4C7×#12011yescharacteristic
334C2×C4C7×#12112yes—
335C2×C4C7×#12311yescharacteristic
336C2×C4C7×#13512yes—
337C2×C4C7×AGL(1,ℤ/8)12yes—
338C8—44no—
339C8—82no—
340C7#903611yescharacteristic, direct product factor, Sylow
341C22—21no—
342C22—21no—
343C22—42no—
344C22—42no—
345C22—42no—
346C22—41no—
347C22—41no—
348C22—42no—
349C22—41no—
350C22—81no—
351C22—81no—
352C22—81no—
353C22—81no—
354C22—81no—
355C22C7×#38611yescharacteristic
356C22C7×#41611yescharacteristic
357C22C7×#41811yescharacteristic
358C4—22no—
359C4—22no—
360C4—42no—
361C4—41no—
362C4—41no—
363C4—41no—
364C4—81no—
365C4—81no—
366C4C7×#44912yes—
367C4C7×#48212yes—
368C2—21no—
369C2—21no—
370C2—81no—
371C2—82no—
372C2C7×#187511yescharacteristic
373C2C7×#188411yescharacteristic
374C2C7×#189111yescharacteristic
subgroupG/H
centerC2×C4×C7#120
commutatorC22×C4C22×C4×C7
Frattini, cosocleC2×C42C23×C7
socleC22×C7#386
prime radicalC7×#392C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4×C7 ◁ C2×C42×C7 ◁ G

Complements

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

NH ≅ G/Ntype
1. C7×#2921C2semidirect
3. C7×#1486C2semidirect
4. C7×#1458C2semidirect
5. C7×#1303C2semidirect
25. #9036C7direct (N × H)
53. C7×C4⋊C8D4semidirect
63. #2921C2×C7semidirect
65. #1486C2×C7semidirect
66. #1458C2×C7semidirect
67. #1303C2×C7semidirect
205. C4⋊C8C7×D4semidirect
340. C7#9036direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#29212
C7×#23922
C7×#14862
C7×#14582
C7×#13032
#90367