Finite Group Databasebeta

← C7×#36696 #13213912

Subgroups — C7×#36696 #13213912

Order 1792.

1908 conjugacy classes of subgroups.

The subgroup lattice has 420 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#2843C211yesmaximal, characteristic, complemented
2C7×#2830C213yesmaximal, complemented
3C7×#2829C216yesmaximal, complemented
4C7×#2828C213yesmaximal, complemented
5C7×#2823C211yesmaximal, characteristic, complemented
6C7×#2818C212yesmaximal, complemented
7C7×#2395C213yesmaximal, complemented
8C7×#2392C216yesmaximal, complemented
9C7×#2388C213yesmaximal, complemented
10C4×C7×#127C211yesmaximal, characteristic, complemented
11C4×C7×#126C212yesmaximal, complemented
12C22×C7×#41C2211yescharacteristic
13C2×C7×#127C2211yescharacteristic
14C2×C7×#126C2212yes—
15C2×C4×C7×D4C2216yes—
16C2×C4×C7×D4C2213yes—
17C2×C4×C7×D4C2213yes—
18C2×C7×#117C2213yes—
19C7×#400C2213yes—
20C7×#399C2213yes—
21C7×#398C2216yes—
22C7×#398C2213yes—
23C7×#397C2216yes—
24C7×#397C2216yes—
25C7×#397C2213yes—
26C7×#396C2216yes—
27C7×#396C2216yes—
28C7×#395C2213yes—
29C7×#394C2213yes—
30C7×#394C2216yes—
31C7×#394C2216yes—
32C7×#394C2216yes—
33C7×#394C2216yes—
34C7×#394C2216yes—
35C7×#393C2213yes—
36C7×#392C2216yes—
37C7×#392C2213yes—
38C7×#392C2216yes—
39C7×#390C2213yes—
40C7×#390C2213yes—
41C7×#388C2216yes—
42C7×#387C2216yes—
43C7×#384C2211yescharacteristic
44C7×#383C2212yes—
45C4×C7×C4⋊C4C2216yes—
46C4×C7×C22⋊C4C2216yes—
47C4×C7×C22⋊C4C2213yes—
48C4×C7×C22⋊C4C2213yes—
49C43×C7C2211yescharacteristic
50#36696C711yesmaximal, characteristic, direct product factor, Sylow
51C2×C7×#41—46no—
52C22×C7×Q8C2311yescharacteristic
53C22×C7×D4C2313yes—
54C22×C7×D4C2313yes—
55C22×C7×D4C2312yes—
56C22×C7×D4C2311yescharacteristic
57C22×C7×D4C2316yes—
58C23×C4×C7C2312yes—
59C23×C4×C7C2311yescharacteristic
60C23×C4×C7C2313yes—
61C7×#127—24no—
62C7×#126—28no—
63C4×C7×D4—412no—
64C4×C7×D4—46no—
65C4×C7×D4—46no—
66C7×#117—46no—
67C2×C7×C4⋊C4C2316yes—
68C2×C7×C4⋊C4C2313yes—
69C2×C7×C4⋊C4C2313yes—
70C2×C7×C4⋊C4C2316yes—
71C2×C7×C4⋊C4C2313yes—
72C2×C7×C4⋊C4C2313yes—
73C2×C7×C4⋊C4C2316yes—
74C2×C7×C22⋊C4C2316yes—
75C2×C7×C22⋊C4C2316yes—
76C2×C7×C22⋊C4C2316yes—
77C2×C7×C22⋊C4C2316yes—
78C2×C7×C22⋊C4C2316yes—
79C2×C7×C22⋊C4C2313yes—
80C2×C7×C22⋊C4C2316yes—
81C2×C7×C22⋊C4C2313yes—
82C2×C7×C22⋊C4C2316yes—
83C2×C7×C22⋊C4C2316yes—
84C2×C42×C7C2313yes—
85C2×C42×C7C2316yes—
86C2×C42×C7C2313yes—
87C2×C42×C7C2311yescharacteristic
88C7×#95C2316yes—
89C7×#95C2316yes—
90C7×#95C2313yes—
91C7×#95C2313yes—
92C7×#95C2313yes—
93C7×#95C2313yes—
94C7×#95C2316yes—
95C7×#95C2316yes—
96#2843C2×C711yescharacteristic, complemented
97#2830C2×C713yescomplemented
98#2829C2×C716yescomplemented
99#2828C2×C713yescomplemented
100#2823C2×C711yescharacteristic, complemented
101#2818C2×C712yescomplemented
102#2395C2×C713yescomplemented
103#2392C2×C716yescomplemented
104#2388C2×C713yescomplemented
105C4×#127C2×C711yescharacteristic, complemented
106C4×#126C2×C712yescomplemented
107C24×C7C2412yes—
108C24×C7C2413yes—
109C24×C7C2411yescharacteristic
110C7×#41—416no—
111C2×C7×Q8—43no—
112C2×C7×D4—26no—
113C2×C7×D4—26no—
114C2×C7×D4—43no—
115C2×C7×D4—43no—
116C2×C7×D4—46no—
117C2×C7×D4—43no—
118C2×C7×D4—43no—
119C2×C7×D4—46no—
120C2×C7×D4—46no—
121C2×C7×D4—46no—
122C2×C7×D4—43no—
123C22×C4×C7—26no—
124C22×C4×C7—212no—
125C22×C4×C7—26no—
126C22×C4×C7—212no—
127C22×C4×C7C2416yes—
128C22×C4×C7C2412yes—
129C22×C4×C7C2411yescharacteristic
130C22×C4×C7C2413yes—
131C22×C4×C7C2416yes—
132C22×C4×C7C2413yes—
133C22×C4×C7C2413yes—
134C22×C4×C7C2411yescharacteristic
135C7×C4⋊C4—212no—
136C7×C4⋊C4—46no—
137C7×C4⋊C4—43no—
138C7×C4⋊C4—43no—
139C7×C4⋊C4—43no—
140C7×C4⋊C4—43no—
141C7×C4⋊C4—46no—
142C7×C22⋊C4—212no—
143C7×C22⋊C4—46no—
144C7×C22⋊C4—46no—
145C7×C22⋊C4—46no—
146C7×C22⋊C4—46no—
147C7×C22⋊C4—46no—
148C7×C22⋊C4—43no—
149C7×C22⋊C4—43no—
150C7×C22⋊C4—46no—
151C7×C22⋊C4—46no—
152C42×C7—212no—
153C42×C7—26no—
154C42×C7—43no—
155C42×C7#4114yescomplemented
156C22×#41C22×C711yescharacteristic
157C2×#127C22×C711yescharacteristic
158C2×#126C22×C712yes—
159C2×C4×D4C22×C713yes—
160C2×C4×D4C22×C713yes—
161C2×C4×D4C22×C716yes—
162C2×#117C22×C713yes—
163#400C22×C713yes—
164#399C22×C713yes—
165#398C22×C713yes—
166#398C22×C716yes—
167#397C22×C716yes—
168#397C22×C713yes—
169#397C22×C716yes—
170#396C22×C716yes—
171#396C22×C716yes—
172#395C22×C713yes—
173#394C22×C716yes—
174#394C22×C716yes—
175#394C22×C713yes—
176#394C22×C716yes—
177#394C22×C716yes—
178#394C22×C716yes—
179#393C22×C713yes—
180#392C22×C716yes—
181#392C22×C716yes—
182#392C22×C713yes—
183#390C22×C713yes—
184#390C22×C713yes—
185#388C22×C716yes—
186#387C22×C716yes—
187#384C22×C711yescharacteristic
188#383C22×C712yes—
189C4×C4⋊C4C22×C716yes—
190C4×C22⋊C4C22×C716yes—
191C4×C22⋊C4C22×C713yes—
192C4×C22⋊C4C22×C713yes—
193C43C22×C711yescharacteristic
194C23×C7—22no—
195C23×C7—23no—
196C23×C7—21no—
197C23×C7—23no—
198C23×C7—26no—
199C23×C7—23no—
200C23×C7—26no—
201C23×C7—23no—
202C23×C7—26no—
203C23×C7—26no—
204C23×C7—23no—
205C23×C7C2511yescharacteristic
206C7×Q8—44no—
207C7×D4—412no—
208C7×D4—412no—
209C7×D4—424no—
210C7×D4—48no—
211C7×D4—44no—
212C2×C4×C7—23no—
213C2×C4×C7—23no—
214C2×C4×C7—26no—
215C2×C4×C7—23no—
216C2×C4×C7—26no—
217C2×C4×C7—26no—
218C2×C4×C7—26no—
219C2×C4×C7—23no—
220C2×C4×C7—23no—
221C2×C4×C7—26no—
222C2×C4×C7—26no—
223C2×C4×C7—26no—
224C2×C4×C7—26no—
225C2×C4×C7—26no—
226C2×C4×C7—412no—
227C2×C4×C7—44no—
228C2×C4×C7—48no—
229C2×C4×C7C2×#4116yes—
230C2×C4×C7H(5,𝔽2)16yes—
231C2×#41—46no—
232C22×Q8C23×C711yescharacteristic
233C22×D4C23×C713yes—
234C22×D4C23×C713yes—
235C22×D4C23×C716yes—
236C22×D4C23×C712yes—
237C22×D4C23×C711yescharacteristic
238C23×C4C23×C713yes—
239C23×C4C23×C711yescharacteristic
240C23×C4C23×C712yes—
241#127—24no—
242#126—28no—
243C4×D4—412no—
244C4×D4—46no—
245C4×D4—46no—
246#117—46no—
247C2×C4⋊C4C23×C716yes—
248C2×C4⋊C4C23×C713yes—
249C2×C4⋊C4C23×C713yes—
250C2×C4⋊C4C23×C713yes—
251C2×C4⋊C4C23×C713yes—
252C2×C4⋊C4C23×C716yes—
253C2×C4⋊C4C23×C716yes—
254C2×C22⋊C4C23×C716yes—
255C2×C22⋊C4C23×C716yes—
256C2×C22⋊C4C23×C713yes—
257C2×C22⋊C4C23×C716yes—
258C2×C22⋊C4C23×C713yes—
259C2×C22⋊C4C23×C716yes—
260C2×C22⋊C4C23×C716yes—
261C2×C22⋊C4C23×C716yes—
262C2×C22⋊C4C23×C716yes—
263C2×C22⋊C4C23×C716yes—
264C2×C42C23×C716yes—
265C2×C42C23×C713yes—
266C2×C42C23×C713yes—
267C2×C42C23×C711yescharacteristic
268#95C23×C716yes—
269#95C23×C716yes—
270#95C23×C713yes—
271#95C23×C713yes—
272#95C23×C713yes—
273#95C23×C713yes—
274#95C23×C716yes—
275#95C23×C716yes—
276C22×C7—43no—
277C22×C7—46no—
278C22×C7—43no—
279C22×C7—46no—
280C22×C7—41no—
281C22×C7—42no—
282C22×C7—43no—
283C22×C7—46no—
284C22×C7—46no—
285C22×C7—43no—
286C22×C7—43no—
287C22×C7C22×#4111yescharacteristic
288C22×C7C2×H(5,𝔽2)13yes—
289C22×C7#58513yes—
290C4×C7—26no—
291C4×C7—46no—
292C4×C7—43no—
293C4×C7—41no—
294C4×C7—42no—
295C4×C7—43no—
296C4×C7—46no—
297C4×C7#56014yes—
298C24C24×C712yes—
299C24C24×C713yes—
300C24C24×C711yescharacteristic
301#41—416no—
302C2×Q8—43no—
303C2×D4—26no—
304C2×D4—26no—
305C2×D4—43no—
306C2×D4—43no—
307C2×D4—46no—
308C2×D4—46no—
309C2×D4—43no—
310C2×D4—43no—
311C2×D4—43no—
312C2×D4—46no—
313C2×D4—46no—
314C22×C4—26no—
315C22×C4—212no—
316C22×C4—26no—
317C22×C4—212no—
318C22×C4C24×C716yes—
319C22×C4C24×C712yes—
320C22×C4C24×C711yescharacteristic
321C22×C4C24×C713yes—
322C22×C4C24×C716yes—
323C22×C4C24×C713yes—
324C22×C4C24×C713yes—
325C22×C4C24×C711yescharacteristic
326C4⋊C4—212no—
327C4⋊C4—43no—
328C4⋊C4—43no—
329C4⋊C4—43no—
330C4⋊C4—43no—
331C4⋊C4—46no—
332C4⋊C4—46no—
333C22⋊C4—212no—
334C22⋊C4—46no—
335C22⋊C4—46no—
336C22⋊C4—43no—
337C22⋊C4—43no—
338C22⋊C4—46no—
339C22⋊C4—46no—
340C22⋊C4—46no—
341C22⋊C4—46no—
342C22⋊C4—46no—
343C42—212no—
344C42—26no—
345C42—43no—
346C42C7×#4114yescomplemented
347C2×C7—81no—
348C2×C7—83no—
349C2×C7—82no—
350C2×C7#346013yes—
351C2×C7C2×#56011yescharacteristic
352C2×C7#352913yes—
353C23—22no—
354C23—23no—
355C23—21no—
356C23—23no—
357C23—26no—
358C23—23no—
359C23—26no—
360C23—23no—
361C23—26no—
362C23—26no—
363C23—23no—
364C23C25×C711yescharacteristic
365Q8—44no—
366D4—412no—
367D4—412no—
368D4—424no—
369D4—48no—
370D4—44no—
371C2×C4—23no—
372C2×C4—23no—
373C2×C4—26no—
374C2×C4—23no—
375C2×C4—26no—
376C2×C4—26no—
377C2×C4—26no—
378C2×C4—23no—
379C2×C4—23no—
380C2×C4—26no—
381C2×C4—26no—
382C2×C4—26no—
383C2×C4—26no—
384C2×C4—26no—
385C2×C4—412no—
386C2×C4—44no—
387C2×C4—48no—
388C2×C4C2×C7×#4116yes—
389C2×C4C7×H(5,𝔽2)16yes—
390C7#3669611yescharacteristic, direct product factor, Sylow
391C22—43no—
392C22—46no—
393C22—43no—
394C22—46no—
395C22—41no—
396C22—42no—
397C22—43no—
398C22—46no—
399C22—46no—
400C22—43no—
401C22—43no—
402C22C22×C7×#4111yescharacteristic
403C22C2×C7×H(5,𝔽2)13yes—
404C22C7×#58513yes—
405C4—26no—
406C4—46no—
407C4—41no—
408C4—42no—
409C4—43no—
410C4—46no—
411C4—43no—
412C4C7×#56014yes—
413C2—81no—
414C2—83no—
415C2—82no—
416C2C7×#346013yes—
417C2C2×C7×#56011yescharacteristic
418C2C7×#352913yes—
subgroupG/H
centerC22×C4×C7C24
commutator, Frattini, cosocleC23C25×C7
socleC23×C7C25
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C22×C4×C7 ◁ G

Complements

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

NH ≅ G/Ntype
1. C7×#2843C2semidirect
2. C7×#2830C2semidirect
3. C7×#2829C2semidirect
4. C7×#2828C2semidirect
5. C7×#2823C2semidirect
6. C7×#2818C2semidirect
7. C7×#2395C2semidirect
8. C7×#2392C2semidirect
9. C7×#2388C2semidirect
10. C4×C7×#127C2semidirect
11. C4×C7×#126C2semidirect
50. #36696C7direct (N × H)
96. #2843C2×C7semidirect
97. #2830C2×C7semidirect
98. #2829C2×C7semidirect
99. #2828C2×C7semidirect
100. #2823C2×C7semidirect
101. #2818C2×C7semidirect
102. #2395C2×C7semidirect
103. #2392C2×C7semidirect
104. #2388C2×C7semidirect
105. C4×#127C2×C7semidirect
106. C4×#126C2×C7semidirect
155. C42×C7#41semidirect
346. C42C7×#41semidirect
390. C7#36696direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#28432
C7×#28302
C7×#28292
C7×#28282
C7×#28232
C7×#28182
C7×#23952
C7×#23922
C7×#23882
C4×C7×#1272
C4×C7×#1262
#366967