Finite Group Databasebeta

← C7×#34643 #13211859

Subgroups — C7×#34643 #13211859

Order 1792.

2234 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#529C212yesmaximal, complemented
2C7×#2664C214yesmaximal, complemented
3C7×#2663C214yesmaximal, complemented
4C7×#2662C214yesmaximal, complemented
5C7×#2661C214yesmaximal, complemented
6C7×#2399C211yesmaximal, characteristic, complemented
7C7×#2398C212yesmaximal, complemented
8C7×#2397C211yesmaximal, characteristic, complemented
9C7×#2392C214yesmaximal, complemented
10C7×#2293C214yesmaximal, complemented
11C4×C7×#117C211yesmaximal, characteristic, complemented
12C7×#529—216no—
13C2×C7×#126C2214yes—
14C2×C7×#125C2212yes—
15C2×C7×#124C2212yes—
16C2×C7×#123C2214yes—
17C2×C7×C22⋊Q8C2212yes—
18C2×C7×#121C2212yes—
19C2×C4×C7×Q8C2212yes—
20C2×C4×C7×D4C2214yes—
21C2×C4×C7×D4C2214yes—
22C2×C4×C7×D4C2212yes—
23C2×C7×#117C2212yes—
24C2×C7×#117C2214yes—
25C22×C42×C7C2211yescharacteristic
26C7×#400C2214yes—
27C7×#399C2214yes—
28C7×#398C2214yes—
29C7×#398C2214yes—
30C7×#397C2214yes—
31C7×#397C2214yes—
32C7×#396C2218yes—
33C7×#395C2214yes—
34C7×#394C2218yes—
35C7×#394C2214yes—
36C7×#394C2214yes—
37C7×#393C2214yes—
38C7×#392C2214yes—
39C7×#392C2214yes—
40C7×#390C2218yes—
41C7×#389C2218yes—
42C7×#388C2218yes—
43C7×#388C2218yes—
44C4×C7×C4⋊C4C2214yes—
45C4×C7×C4⋊C4C2214yes—
46C4×C7×C22⋊C4C2218yes—
47C4×C7×C22⋊C4C2214yes—
48C7×#376C2211yescharacteristic
49C7×#376C2211yescharacteristic
50C43×C7C2212yes—
51#34643C711yesmaximal, characteristic, direct product factor, Sylow
52C22×C7×Q8C2312yes—
53C22×C7×D4C2314yes—
54C22×C7×D4C2312yes—
55C22×C7×D4C2314yes—
56C23×C4×C7C2314yes—
57C23×C4×C7C2311yescharacteristic
58C23×C4×C7C2312yes—
59C7×#126—216no—
60C7×#125—28no—
61C7×#124—28no—
62C7×#123—216no—
63C7×C22⋊Q8—28no—
64C7×#121—28no—
65C4×C7×Q8—28no—
66C4×C7×D4—216no—
67C4×C7×D4—28no—
68C4×C7×D4—216no—
69C7×#117—216no—
70C7×#117—28no—
71C2×C7×C4⋊C4C2314yes—
72C2×C7×C4⋊C4C2314yes—
73C2×C7×C4⋊C4C2314yes—
74C2×C7×C4⋊C4C2312yes—
75C2×C7×C4⋊C4C2314yes—
76C2×C7×C4⋊C4C2314yes—
77C2×C7×C4⋊C4C2312yes—
78C2×C7×C4⋊C4C2314yes—
79C2×C7×C4⋊C4C2318yes—
80C2×C7×C22⋊C4C2314yes—
81C2×C7×C22⋊C4C2314yes—
82C2×C7×C22⋊C4C2314yes—
83C2×C7×C22⋊C4C2314yes—
84C2×C7×C22⋊C4C2314yes—
85C2×C7×C22⋊C4C2318yes—
86C2×C7×C22⋊C4C2314yes—
87C2×C7×C22⋊C4C2318yes—
88C2×C7×C22⋊C4C2318yes—
89C2×C42×C7D418yescomplemented
90C2×C42×C7C2312yes—
91C2×C42×C7C2314yes—
92C2×C42×C7C2318yes—
93C2×C42×C7C2312yes—
94C2×C42×C7C2311yescharacteristic
95C2×C42×C7C2311yescharacteristic
96C2×C42×C7C2312yes—
97C7×#95C2314yes—
98C7×#95C2314yes—
99C7×#95C2318yes—
100C7×#95C2318yes—
101C7×#95C2314yes—
102C7×#95C2314yes—
103C2×#529C2×C712yescomplemented
104#2664C2×C714yescomplemented
105#2663C2×C714yescomplemented
106#2662C2×C714yescomplemented
107#2661C2×C714yescomplemented
108#2399C2×C711yescharacteristic, complemented
109#2398C2×C712yescomplemented
110#2397C2×C711yescharacteristic, complemented
111#2392C2×C714yescomplemented
112#2293C2×C714yescomplemented
113C4×#117C2×C711yescharacteristic, complemented
114C24×C7C2414yes—
115C24×C7C2411yescharacteristic
116C2×C7×Q8—24no—
117C2×C7×Q8—44no—
118C2×C7×D4—28no—
119C2×C7×D4—24no—
120C2×C7×D4—28no—
121C2×C7×D4—48no—
122C2×C7×D4—44no—
123C2×C7×D4—48no—
124C22×C4×C7—24no—
125C22×C4×C7—28no—
126C22×C4×C7—216no—
127C22×C4×C7—44no—
128C22×C4×C7C2×D414yes—
129C22×C4×C7C2×D418yes—
130C22×C4×C7C2414yes—
131C22×C4×C7C2414yes—
132C22×C4×C7C2414yes—
133C22×C4×C7C2414yes—
134C22×C4×C7C2412yes—
135C22×C4×C7C2412yes—
136C22×C4×C7C2411yescharacteristic
137C22×C4×C7C2414yes—
138C22×C4×C7C2411yescharacteristic
139C7×C4⋊C4—28no—
140C7×C4⋊C4—28no—
141C7×C4⋊C4—24no—
142C7×C4⋊C4—28no—
143C7×C4⋊C4—24no—
144C7×C4⋊C4—28no—
145C7×C4⋊C4—28no—
146C7×C4⋊C4—28no—
147C7×C4⋊C4—48no—
148C7×C22⋊C4—28no—
149C7×C22⋊C4—28no—
150C7×C22⋊C4—28no—
151C7×C22⋊C4—28no—
152C7×C22⋊C4—28no—
153C7×C22⋊C4—28no—
154C7×C22⋊C4—216no—
155C7×C22⋊C4—48no—
156C7×C22⋊C4—48no—
157C42×C7—28no—
158C42×C7—24no—
159C42×C7—48no—
160C42×C7C2×D414yes—
161C42×C7C2×D414yes—
162C42×C7C2×D418yes—
163C42×C7#4118yes—
164#529—216no—
165C2×#126C22×C714yes—
166C2×#125C22×C712yes—
167C2×#124C22×C712yes—
168C2×#123C22×C714yes—
169C2×C22⋊Q8C22×C712yes—
170C2×#121C22×C712yes—
171C2×C4×Q8C22×C712yes—
172C2×C4×D4C22×C714yes—
173C2×C4×D4C22×C714yes—
174C2×C4×D4C22×C712yes—
175C2×#117C22×C712yes—
176C2×#117C22×C714yes—
177C22×C42C22×C711yescharacteristic
178#400C22×C714yes—
179#399C22×C714yes—
180#398C22×C714yes—
181#398C22×C714yes—
182#397C22×C714yes—
183#397C22×C714yes—
184#396C22×C718yes—
185#395C22×C714yes—
186#394C22×C718yes—
187#394C22×C714yes—
188#394C22×C714yes—
189#393C22×C714yes—
190#392C22×C714yes—
191#392C22×C714yes—
192#390C22×C718yes—
193#389C22×C718yes—
194#388C22×C718yes—
195#388C22×C718yes—
196C4×C4⋊C4C22×C714yes—
197C4×C4⋊C4C22×C714yes—
198C4×C22⋊C4C22×C718yes—
199C4×C22⋊C4C22×C714yes—
200#376C22×C711yescharacteristic
201#376C22×C711yescharacteristic
202C43C22×C712yes—
203C23×C7—22no—
204C23×C7—24no—
205C23×C7—24no—
206C23×C7—28no—
207C23×C7—24no—
208C23×C7—28no—
209C23×C7—24no—
210C23×C7C22×D412yes—
211C23×C7C2511yescharacteristic
212C7×Q8—48no—
213C7×D4—48no—
214C7×D4—416no—
215C7×D4—416no—
216C2×C4×C7—22no—
217C2×C4×C7—24no—
218C2×C4×C7—28no—
219C2×C4×C7—24no—
220C2×C4×C7—28no—
221C2×C4×C7—24no—
222C2×C4×C7—24no—
223C2×C4×C7—24no—
224C2×C4×C7—24no—
225C2×C4×C7—28no—
226C2×C4×C7—28no—
227C2×C4×C7—24no—
228C2×C4×C7—28no—
229C2×C4×C7—24no—
230C2×C4×C7—48no—
231C2×C4×C7—416no—
232C2×C4×C7C22×D412yes—
233C2×C4×C7C22×D414yes—
234C2×C4×C7C22×D414yes—
235C2×C4×C7C22×D412yes—
236C2×C4×C7C2×#4118yes—
237C2×C4×C7C2×#4114yes—
238C22×Q8C23×C712yes—
239C22×D4C23×C714yes—
240C22×D4C23×C712yes—
241C22×D4C23×C714yes—
242C23×C4C23×C714yes—
243C23×C4C23×C711yescharacteristic
244C23×C4C23×C712yes—
245#126—216no—
246#125—28no—
247#124—28no—
248#123—216no—
249C22⋊Q8—28no—
250#121—28no—
251C4×Q8—28no—
252C4×D4—216no—
253C4×D4—28no—
254C4×D4—216no—
255#117—216no—
256#117—28no—
257C2×C4⋊C4C23×C714yes—
258C2×C4⋊C4C23×C714yes—
259C2×C4⋊C4C23×C714yes—
260C2×C4⋊C4C23×C712yes—
261C2×C4⋊C4C23×C714yes—
262C2×C4⋊C4C23×C714yes—
263C2×C4⋊C4C23×C712yes—
264C2×C4⋊C4C23×C714yes—
265C2×C4⋊C4C23×C718yes—
266C2×C22⋊C4C23×C714yes—
267C2×C22⋊C4C23×C714yes—
268C2×C22⋊C4C23×C714yes—
269C2×C22⋊C4C23×C714yes—
270C2×C22⋊C4C23×C714yes—
271C2×C22⋊C4C23×C714yes—
272C2×C22⋊C4C23×C718yes—
273C2×C22⋊C4C23×C718yes—
274C2×C22⋊C4C23×C718yes—
275C2×C42C7×D418yescomplemented
276C2×C42C23×C714yes—
277C2×C42C23×C712yes—
278C2×C42C23×C718yes—
279C2×C42C23×C711yescharacteristic
280C2×C42C23×C711yescharacteristic
281C2×C42C23×C712yes—
282C2×C42C23×C712yes—
283#95C23×C714yes—
284#95C23×C714yes—
285#95C23×C718yes—
286#95C23×C718yes—
287#95C23×C714yes—
288#95C23×C714yes—
289C22×C7—22no—
290C22×C7—24no—
291C22×C7—44no—
292C22×C7—44no—
293C22×C7—44no—
294C22×C7—44no—
295C22×C7—48no—
296C22×C7—48no—
297C22×C7C23×D411yescharacteristic
298C22×C7C22×#4112yes—
299C22×C7#58514yes—
300C4×C7—22no—
301C4×C7—24no—
302C4×C7—24no—
303C4×C7—28no—
304C4×C7—44no—
305C4×C7—44no—
306C4×C7—44no—
307C4×C7—44no—
308C4×C7#53214yes—
309C24C24×C714yes—
310C24C24×C711yescharacteristic
311C2×Q8—24no—
312C2×Q8—44no—
313C2×D4—28no—
314C2×D4—24no—
315C2×D4—28no—
316C2×D4—48no—
317C2×D4—44no—
318C2×D4—48no—
319C22×C4—24no—
320C22×C4—28no—
321C22×C4—216no—
322C22×C4—44no—
323C22×C4C2×C7×D414yes—
324C22×C4C2×C7×D418yes—
325C22×C4C24×C714yes—
326C22×C4C24×C714yes—
327C22×C4C24×C714yes—
328C22×C4C24×C714yes—
329C22×C4C24×C712yes—
330C22×C4C24×C712yes—
331C22×C4C24×C711yescharacteristic
332C22×C4C24×C714yes—
333C22×C4C24×C711yescharacteristic
334C4⋊C4—28no—
335C4⋊C4—28no—
336C4⋊C4—24no—
337C4⋊C4—28no—
338C4⋊C4—24no—
339C4⋊C4—28no—
340C4⋊C4—28no—
341C4⋊C4—28no—
342C4⋊C4—48no—
343C22⋊C4—28no—
344C22⋊C4—28no—
345C22⋊C4—28no—
346C22⋊C4—28no—
347C22⋊C4—28no—
348C22⋊C4—28no—
349C22⋊C4—216no—
350C22⋊C4—48no—
351C22⋊C4—48no—
352C42—28no—
353C42—24no—
354C42—48no—
355C42C2×C7×D414yes—
356C42C2×C7×D414yes—
357C42C2×C7×D418yes—
358C42C7×#4118yes—
359C2×C7—42no—
360C2×C7—84no—
361C2×C7C2×#53211yescharacteristic
362C2×C7#344912yes—
363C2×C7#345514yes—
364C23—22no—
365C23—24no—
366C23—24no—
367C23—28no—
368C23—24no—
369C23—28no—
370C23—24no—
371C23C22×C7×D412yes—
372C23C25×C711yescharacteristic
373Q8—48no—
374D4—48no—
375D4—416no—
376D4—416no—
377C2×C4—22no—
378C2×C4—24no—
379C2×C4—28no—
380C2×C4—24no—
381C2×C4—28no—
382C2×C4—24no—
383C2×C4—24no—
384C2×C4—24no—
385C2×C4—24no—
386C2×C4—28no—
387C2×C4—28no—
388C2×C4—24no—
389C2×C4—28no—
390C2×C4—24no—
391C2×C4—48no—
392C2×C4—416no—
393C2×C4C22×C7×D412yes—
394C2×C4C22×C7×D414yes—
395C2×C4C22×C7×D414yes—
396C2×C4C22×C7×D412yes—
397C2×C4C2×C7×#4118yes—
398C2×C4C2×C7×#4114yes—
399C7#3464311yescharacteristic, direct product factor, Sylow
400C22—22no—
401C22—24no—
402C22—44no—
403C22—44no—
404C22—44no—
405C22—44no—
406C22—48no—
407C22—48no—
408C22C23×C7×D411yescharacteristic
409C22C22×C7×#4112yes—
410C22C7×#58514yes—
411C4—22no—
412C4—24no—
413C4—24no—
414C4—28no—
415C4—44no—
416C4—44no—
417C4—44no—
418C4—44no—
419C4C7×#53214yes—
420C2—42no—
421C2—84no—
422C2C2×C7×#53211yescharacteristic
423C2C7×#344912yes—
424C2C7×#345514yes—
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. C2×C7×#529C2semidirect
2. C7×#2664C2semidirect
3. C7×#2663C2semidirect
4. C7×#2662C2semidirect
5. C7×#2661C2semidirect
6. C7×#2399C2semidirect
7. C7×#2398C2semidirect
8. C7×#2397C2semidirect
9. C7×#2392C2semidirect
10. C7×#2293C2semidirect
11. C4×C7×#117C2semidirect
51. #34643C7direct (N × H)
89. C2×C42×C7D4semidirect
103. C2×#529C2×C7semidirect
104. #2664C2×C7semidirect
105. #2663C2×C7semidirect
106. #2662C2×C7semidirect
107. #2661C2×C7semidirect
108. #2399C2×C7semidirect
109. #2398C2×C7semidirect
110. #2397C2×C7semidirect
111. #2392C2×C7semidirect
112. #2293C2×C7semidirect
113. C4×#117C2×C7semidirect
275. C2×C42C7×D4semidirect
399. C7#34643direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C7×#5292
C7×#26642
C7×#26632
C7×#26622
C7×#26612
C7×#23992
C7×#23982
C7×#23972
C7×#23922
C7×#22932
C4×C7×#1172
#346437