Finite Group Databasebeta

← C7×#9326 #13186542

Subgroups — C7×#9326 #13186542

Order 1792.

574 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3112C211yesmaximal, characteristic
2C7×#2276C211yesmaximal, characteristic, complemented
3C7×#1980C211yesmaximal, characteristic
4C7×#1505C211yesmaximal, characteristic
5C7×#1491C211yesmaximal, characteristic, complemented
6C7×#1300C211yesmaximal, characteristic, complemented
7C7×#1285C211yesmaximal, characteristic, complemented
8C7×#532C2211yescharacteristic
9C2×C4×C7×Q8—21no—
10C2×C4×C7×Q8C2211yescharacteristic
11C7×#480—21no—
12C7×#477C412yescomplemented
13C7×#460C412yescomplemented
14C7×#448—21no—
15C7×#440—22no—
16C7×#432C2211yescharacteristic
17C7×#432C2211yescharacteristic
18C2×C7×#104—22no—
19C4×C7×M16C2211yescharacteristic
20C4×C7×C4⋊C4—21no—
21C4×C7×C4⋊C4—22no—
22C4×C7×C4⋊C4—21no—
23C4×C7×C4⋊C4C2211yescharacteristic
24C7×#376—21no—
25C7×#376C2211yescharacteristic
26C7×#327—21no—
27C7×#327—21no—
28C7×Q8⋊C8—22no—
29#9326C711yesmaximal, characteristic, direct product factor, Sylow
30C2×C7×#41—21no—
31C22×C7×Q8D411yescharacteristic
32C2×C7×SD16—22no—
33C2×C7×M16—21no—
34C7×C4⋊Q8C2×C411yescharacteristic
35C7×#127C2×C411yescharacteristic
36C7×#124—21no—
37C7×#121—21no—
38C7×#121—21no—
39C4×C7×Q8—28no—
40C4×C7×Q8—21no—
41C4×C7×Q8—21no—
42C4×C7×Q8C2×C412yes—
43C4×C7×Q8D412yes—
44C4×C7×D4—21no—
45C4×C7×D4—21no—
46C2×C7×C4⋊C4—22no—
47C2×C7×C4⋊C4—21no—
48C2×C7×C4⋊C4—21no—
49C2×C7×C4⋊C4D411yescharacteristic
50C2×C42×C7—22no—
51C2×C42×C7—21no—
52C2×C42×C7—21no—
53C2×C42×C7—22no—
54C2×C42×C7D412yes—
55C2×C42×C7C2311yescharacteristic
56C7×#107—22no—
57C7×C4⋊C8—21no—
58C7×C4⋊C8C2×C412yes—
59C7×#104—44no—
60C7×#103—22no—
61C7×#102—22no—
62C7×C22⋊C8—21no—
63C7×C22⋊C8—21no—
64C7×#97—21no—
65C7×#97—21no—
66C7×#97—21no—
67C4×C7×C8—21no—
68C7×#95—22no—
69C7×#95—21no—
70C7×#95—21no—
71C7×#95D411yescharacteristic
72C7×#95D411yescharacteristic
73#3112C2×C711yescharacteristic
74#2276C2×C711yescharacteristic, complemented
75#1980C2×C711yescharacteristic
76#1505C2×C711yescharacteristic
77#1491C2×C711yescharacteristic, complemented
78#1300C2×C711yescharacteristic, complemented
79#1285C2×C711yescharacteristic, complemented
80C7×#41—42no—
81C7×#41—42no—
82C2×C7×Q8—24no—
83C2×C7×Q8—21no—
84C2×C7×Q8—21no—
85C2×C7×Q8C22⋊C412yes—
86C2×C7×D4—21no—
87C2×C7×D4—21no—
88C2×C7×D4—21no—
89C2×C7×D4—41no—
90C22×C4×C7—21no—
91C22×C4×C7—21no—
92C22×C4×C7—21no—
93C22×C4×C7—22no—
94C22×C4×C7—21no—
95C22×C4×C7—41no—
96C22×C4×C7C2×D411yescharacteristic
97C22×C4×C7C2×D411yescharacteristic
98C22×C4×C7C2×D411yescharacteristic
99C7×SD16—44no—
100C7×M16—22no—
101C7×M16—41no—
102C2×C7×C8—22no—
103C2×C7×C8—41no—
104C2×C7×C8—41no—
105C7×C4⋊C4—28no—
106C7×C4⋊C4—24no—
107C7×C4⋊C4—22no—
108C7×C4⋊C4—22no—
109C7×C4⋊C4—21no—
110C7×C4⋊C4—21no—
111C7×C4⋊C4—21no—
112C7×C4⋊C4C22⋊C412yes—
113C7×C22⋊C4—41no—
114C7×C22⋊C4—41no—
115C42×C7—28no—
116C42×C7—24no—
117C42×C7—22no—
118C42×C7—22no—
119C42×C7—42no—
120C42×C7—82no—
121C42×C7C22×C411yescharacteristic
122C42×C7C2×D411yescharacteristic
123C42×C7#4111yescharacteristic
124C42×C7#4111yescharacteristic
125#532C22×C711yescharacteristic
126C2×C4×Q8—21no—
127C2×C4×Q8C22×C711yescharacteristic
128#480—21no—
129#477C4×C712yescomplemented
130#460C4×C712yescomplemented
131#448—21no—
132#440—22no—
133#432C22×C711yescharacteristic
134#432C22×C711yescharacteristic
135C2×#104—22no—
136C4×M16C22×C711yescharacteristic
137C4×C4⋊C4—21no—
138C4×C4⋊C4—22no—
139C4×C4⋊C4—21no—
140C4×C4⋊C4C22×C711yescharacteristic
141#376—21no—
142#376C22×C711yescharacteristic
143#327—21no—
144#327—21no—
145Q8⋊C8—22no—
146C23×C7—41no—
147C23×C7#12011yescharacteristic
148C7×Q8—24no—
149C7×Q8—22no—
150C7×Q8—42no—
151C7×Q8#10414yescomplemented
152C7×D4—42no—
153C7×D4—42no—
154C7×D4—81no—
155C7×D4—81no—
156C7×D4—81no—
157C2×C4×C7—24no—
158C2×C4×C7—22no—
159C2×C4×C7—21no—
160C2×C4×C7—21no—
161C2×C4×C7—22no—
162C2×C4×C7—22no—
163C2×C4×C7—24no—
164C2×C4×C7—22no—
165C2×C4×C7—22no—
166C2×C4×C7—41no—
167C2×C4×C7—41no—
168C2×C4×C7—42no—
169C2×C4×C7—41no—
170C2×C4×C7—42no—
171C2×C4×C7—41no—
172C2×C4×C7—41no—
173C2×C4×C7—41no—
174C2×C4×C7—41no—
175C2×C4×C7—41no—
176C2×C4×C7—82no—
177C2×C4×C7C2×C22⋊C411yescharacteristic
178C2×C4×C7C4×D411yescharacteristic
179C2×C4×C7C4×D411yescharacteristic
180C2×C4×C7#12111yescharacteristic
181C2×C4×C7#12111yescharacteristic
182C2×C4×C7#12311yescharacteristic
183C7×C8—42no—
184C7×C8—42no—
185C7×C8—81no—
186C2×#41—21no—
187C22×Q8C7×D411yescharacteristic
188C2×SD16—22no—
189C2×M16—21no—
190C4⋊Q8C2×C4×C711yescharacteristic
191#127C2×C4×C711yescharacteristic
192#124—21no—
193#121—21no—
194#121—21no—
195C4×Q8—28no—
196C4×Q8—21no—
197C4×Q8—21no—
198C4×Q8C2×C4×C712yes—
199C4×Q8C7×D412yes—
200C4×D4—21no—
201C4×D4—21no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4C7×D411yescharacteristic
206C2×C42—22no—
207C2×C42—21no—
208C2×C42—21no—
209C2×C42—22no—
210C2×C42C7×D412yes—
211C2×C42C23×C711yescharacteristic
212#107—22no—
213C4⋊C8—21no—
214C4⋊C8C2×C4×C712yes—
215#104—44no—
216#103—22no—
217#102—22no—
218C22⋊C8—21no—
219C22⋊C8—21no—
220#97—21no—
221#97—21no—
222#97—21no—
223C4×C8—21no—
224#95—22no—
225#95—21no—
226#95—21no—
227#95C7×D411yescharacteristic
228#95C7×D411yescharacteristic
229C22×C7—21no—
230C22×C7—21no—
231C22×C7—81no—
232C22×C7—81no—
233C22×C7—81no—
234C22×C7#38611yescharacteristic
235C22×C7#45311yescharacteristic
236C22×C7#45411yescharacteristic
237C4×C7—21no—
238C4×C7—21no—
239C4×C7—24no—
240C4×C7—21no—
241C4×C7—21no—
242C4×C7—21no—
243C4×C7—44no—
244C4×C7—42no—
245C4×C7—42no—
246C4×C7—42no—
247C4×C7—81no—
248C4×C7C2×#10412yes—
249#41—42no—
250#41—42no—
251C2×Q8—24no—
252C2×Q8—21no—
253C2×Q8—21no—
254C2×Q8C7×C22⋊C412yes—
255C2×D4—21no—
256C2×D4—21no—
257C2×D4—21no—
258C2×D4—41no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—22no—
263C22×C4—21no—
264C22×C4—41no—
265C22×C4C2×C7×D411yescharacteristic
266C22×C4C2×C7×D411yescharacteristic
267C22×C4C2×C7×D411yescharacteristic
268SD16—44no—
269M16—22no—
270M16—41no—
271C2×C8—22no—
272C2×C8—41no—
273C2×C8—41no—
274C4⋊C4—28no—
275C4⋊C4—24no—
276C4⋊C4—22no—
277C4⋊C4—22no—
278C4⋊C4—21no—
279C4⋊C4—21no—
280C4⋊C4—21no—
281C4⋊C4C7×C22⋊C412yes—
282C22⋊C4—41no—
283C22⋊C4—41no—
284C42—28no—
285C42—24no—
286C42—22no—
287C42—22no—
288C42—42no—
289C42—82no—
290C42C22×C4×C711yescharacteristic
291C42C2×C7×D411yescharacteristic
292C42C7×#4111yescharacteristic
293C42C7×#4111yescharacteristic
294C2×C7—21no—
295C2×C7—21no—
296C2×C7—161no—
297C2×C7#188311yescharacteristic
298C2×C7#189611yescharacteristic
299C2×C7#190511yescharacteristic
300C23—41no—
301C23C7×#12011yescharacteristic
302Q8—24no—
303Q8—22no—
304Q8—42no—
305Q8C7×#10414yescomplemented
306D4—42no—
307D4—42no—
308D4—81no—
309D4—81no—
310D4—81no—
311C2×C4—24no—
312C2×C4—22no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—24no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—42no—
323C2×C4—41no—
324C2×C4—42no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—82no—
331C2×C4C2×C7×C22⋊C411yescharacteristic
332C2×C4C4×C7×D411yescharacteristic
333C2×C4C4×C7×D411yescharacteristic
334C2×C4C7×#12111yescharacteristic
335C2×C4C7×#12111yescharacteristic
336C2×C4C7×#12311yescharacteristic
337C8—42no—
338C8—42no—
339C8—81no—
340C7#932611yescharacteristic, direct product factor, Sylow
341C22—21no—
342C22—21no—
343C22—81no—
344C22—81no—
345C22—81no—
346C22C7×#38611yescharacteristic
347C22C7×#45311yescharacteristic
348C22C7×#45411yescharacteristic
349C4—21no—
350C4—21no—
351C4—24no—
352C4—21no—
353C4—21no—
354C4—21no—
355C4—44no—
356C4—42no—
357C4—42no—
358C4—42no—
359C4—81no—
360C4C2×C7×#10412yes—
361C2—21no—
362C2—21no—
363C2—161no—
364C2C7×#188311yescharacteristic
365C2C7×#189611yescharacteristic
366C2C7×#190511yescharacteristic
subgroupG/H
center, socleC22×C7#386
commutatorC42C22×C4×C7
Frattini, cosocleC2×C42C23×C7
prime radicalC7×#532C22
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C7 ◁ C2×C42×C7 ◁ G

Complements

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

NH ≅ G/Ntype
2. C7×#2276C2semidirect
5. C7×#1491C2semidirect
6. C7×#1300C2semidirect
7. C7×#1285C2semidirect
12. C7×#477C4semidirect
13. C7×#460C4semidirect
29. #9326C7direct (N × H)
74. #2276C2×C7semidirect
77. #1491C2×C7semidirect
78. #1300C2×C7semidirect
79. #1285C2×C7semidirect
129. #477C4×C7semidirect
130. #460C4×C7semidirect
151. C7×Q8#104semidirect
305. Q8C7×#104semidirect
340. C7#9326direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#31122
C7×#22762
C7×#19802
C7×#15052
C7×#14912
C7×#13002
C7×#12852
#93267