Finite Group Databasebeta

← C7×#9265 #13186481

Subgroups — C7×#9265 #13186481

Order 1792.

438 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
1C7×#3211C211yesmaximal, characteristic
2C7×#2968C211yesmaximal, characteristic, complemented
3C7×#1532C211yesmaximal, characteristic, complemented
4C2×C7×#329C211yesmaximal, characteristic, complemented
5C7×#1469C211yesmaximal, characteristic, complemented
6C7×#1302C211yesmaximal, characteristic, complemented
7C7×#1281C211yesmaximal, characteristic, complemented
8C7×#529C2211yescharacteristic
9C2×C7×#124C2211yescharacteristic
10C7×#495—21no—
11C7×#488—21no—
12C7×#464—21no—
13C7×#463—21no—
14C7×#451C411yescharacteristic
15C7×#450C411yescharacteristic
16C7×#448C411yescharacteristic
17C7×#447C411yescharacteristic
18C7×#446—21no—
19C7×#435—21no—
20C7×#435—21no—
21C7×C8×D4—21no—
22C7×#433—21no—
23C7×#432C2211yescharacteristic
24C7×#431C2211yescharacteristic
25C7×#431C2211yescharacteristic
26C7×#405—21no—
27C2×C7×#97C2211yescharacteristic
28C2×C7×#97C2211yescharacteristic
29C7×#330—21no—
30C7×#329—21no—
31C7×#329—21no—
32C7×#329—21no—
33C7×#323—21no—
34C7×#323—21no—
35#9265C711yesmaximal, characteristic, direct product factor, Sylow
36C22×C7×Q8C2×C411yescharacteristic
37C22×C7×D4C2×C411yescharacteristic
38C2×C7×Q16—21no—
39C2×C7×SD16—21no—
40C2×C7×SD16—21no—
41C2×C7×D8—21no—
42C2×C7×M16—21no—
43C22×C7×C8—21no—
44C22×C7×C8—21no—
45C22×C7×C8—21no—
46C7×#126—21no—
47C7×#125D411yescharacteristic
48C7×#124—21no—
49C7×#124—21no—
50C7×#124—21no—
51C7×#124—21no—
52C7×#124D411yescharacteristic
53C7×#123—21no—
54C7×C22⋊Q8C2×C411yescharacteristic
55C7×#121C2×C411yescharacteristic
56C4×C7×Q8D411yescharacteristic
57C4×C7×D4—21no—
58C4×C7×D4D411yescharacteristic
59C7×#117—21no—
60C2×C7×C22⋊C4—21no—
61C2×C7×C22⋊C4—21no—
62C2×C42×C7C2311yescharacteristic
63C7×C4⋊C8—21no—
64C7×C4⋊C8—21no—
65C7×C4⋊C8—21no—
66C7×C4⋊C8—21no—
67C7×C4⋊C8—21no—
68C7×#103—21no—
69C7×#103—21no—
70C7×#102—21no—
71C7×#102—21no—
72C7×C22⋊C8—21no—
73C7×C22⋊C8—21no—
74C7×C22⋊C8—21no—
75C7×C22⋊C8C2×C411yescharacteristic
76C7×C22⋊C8C2×C411yescharacteristic
77C7×C22⋊C8D411yescharacteristic
78C7×C22⋊C8D411yescharacteristic
79C7×#97—21no—
80C7×#97—21no—
81C7×#97—21no—
82C7×#97—21no—
83C7×#97—21no—
84C7×#97D411yescharacteristic
85C7×#97D411yescharacteristic
86C4×C7×C8—21no—
87C4×C7×C8—21no—
88C4×C7×C8—21no—
89#3211C2×C711yescharacteristic
90#2968C2×C711yescharacteristic, complemented
91#1532C2×C711yescharacteristic, complemented
92C2×#329C2×C711yescharacteristic, complemented
93#1469C2×C711yescharacteristic, complemented
94#1302C2×C711yescharacteristic, complemented
95#1281C2×C711yescharacteristic, complemented
96C24×C7—21no—
97C2×C7×Q8—21no—
98C2×C7×Q8—21no—
99C2×C7×Q8—81no—
100C2×C7×Q8C22⋊C411yescharacteristic
101C2×C7×D4—21no—
102C2×C7×D4—21no—
103C2×C7×D4—21no—
104C2×C7×D4—81no—
105C2×C7×D4C22⋊C411yescharacteristic
106C22×C4×C7—21no—
107C22×C4×C7—21no—
108C22×C4×C7C22×C411yescharacteristic
109C22×C4×C7C2×D411yescharacteristic
110C22×C4×C7#4111yescharacteristic
111C7×Q16—81no—
112C7×SD16—81no—
113C7×SD16—81no—
114C7×D8—81no—
115C7×M16—42no—
116C2×C7×C8—21no—
117C2×C7×C8—22no—
118C2×C7×C8—21no—
119C2×C7×C8—21no—
120C2×C7×C8—21no—
121C2×C7×C8—21no—
122C2×C7×C8—21no—
123C2×C7×C8—42no—
124C2×C7×C8—42no—
125C2×C7×C8—41no—
126C2×C7×C8—41no—
127C2×C7×C8—41no—
128C7×C4⋊C4—21no—
129C7×C4⋊C4—21no—
130C7×C4⋊C4—21no—
131C7×C4⋊C4—21no—
132C7×C4⋊C4C22⋊C411yescharacteristic
133C7×C4⋊C4C22⋊C411yescharacteristic
134C7×C22⋊C4—21no—
135C7×C22⋊C4—21no—
136C7×C22⋊C4—21no—
137C7×C22⋊C4—21no—
138C7×C22⋊C4—21no—
139C7×C22⋊C4—41no—
140C7×C22⋊C4—41no—
141C7×C22⋊C4—41no—
142C7×C22⋊C4—41no—
143C42×C7—21no—
144C42×C7C2×D411yescharacteristic
145C42×C7C2×D411yescharacteristic
146C42×C7C2×D411yescharacteristic
147C42×C7#4111yescharacteristic
148#529C22×C711yescharacteristic
149C2×#124C22×C711yescharacteristic
150#495—21no—
151#488—21no—
152#464—21no—
153#463—21no—
154#451C4×C711yescharacteristic
155#450C4×C711yescharacteristic
156#448C4×C711yescharacteristic
157#447C4×C711yescharacteristic
158#446—21no—
159#435—21no—
160#435—21no—
161C8×D4—21no—
162#433—21no—
163#432C22×C711yescharacteristic
164#431C22×C711yescharacteristic
165#431C22×C711yescharacteristic
166#405—21no—
167C2×#97C22×C711yescharacteristic
168C2×#97C22×C711yescharacteristic
169#330—21no—
170#329—21no—
171#329—21no—
172#329—21no—
173#323—21no—
174#323—21no—
175C23×C7—21no—
176C23×C7—41no—
177C23×C7—41no—
178C23×C7—41no—
179C23×C7—81no—
180C23×C7—81no—
181C23×C7C4×D411yescharacteristic
182C7×Q8—41no—
183C7×Q8—41no—
184C7×Q8—41no—
185C7×Q8—81no—
186C7×D4—41no—
187C7×D4—41no—
188C7×D4—41no—
189C7×D4—41no—
190C7×D4—41no—
191C7×D4—81no—
192C2×C4×C7—21no—
193C2×C4×C7—21no—
194C2×C4×C7—21no—
195C2×C4×C7—21no—
196C2×C4×C7—21no—
197C2×C4×C7—41no—
198C2×C4×C7—41no—
199C2×C4×C7—41no—
200C2×C4×C7—41no—
201C2×C4×C7—41no—
202C2×C4×C7—81no—
203C2×C4×C7C2×C22⋊C411yescharacteristic
204C2×C4×C7C4×D411yescharacteristic
205C2×C4×C7#12011yescharacteristic
206C2×C4×C7#12111yescharacteristic
207C2×C4×C7#12111yescharacteristic
208C2×C4×C7#12311yescharacteristic
209C7×C8—22no—
210C7×C8—42no—
211C7×C8—41no—
212C7×C8—41no—
213C7×C8—41no—
214C7×C8—41no—
215C7×C8—41no—
216C7×C8—81no—
217C22×Q8C2×C4×C711yescharacteristic
218C22×D4C2×C4×C711yescharacteristic
219C2×Q16—21no—
220C2×SD16—21no—
221C2×SD16—21no—
222C2×D8—21no—
223C2×M16—21no—
224C22×C8—21no—
225C22×C8—21no—
226C22×C8—21no—
227#126—21no—
228#125C7×D411yescharacteristic
229#124—21no—
230#124—21no—
231#124—21no—
232#124—21no—
233#124C7×D411yescharacteristic
234#123—21no—
235C22⋊Q8C2×C4×C711yescharacteristic
236#121C2×C4×C711yescharacteristic
237C4×Q8C7×D411yescharacteristic
238C4×D4—21no—
239C4×D4C7×D411yescharacteristic
240#117—21no—
241C2×C22⋊C4—21no—
242C2×C22⋊C4—21no—
243C2×C42C23×C711yescharacteristic
244C4⋊C8—21no—
245C4⋊C8—21no—
246C4⋊C8—21no—
247C4⋊C8—21no—
248C4⋊C8—21no—
249#103—21no—
250#103—21no—
251#102—21no—
252#102—21no—
253C22⋊C8—21no—
254C22⋊C8—21no—
255C22⋊C8—21no—
256C22⋊C8C2×C4×C711yescharacteristic
257C22⋊C8C2×C4×C711yescharacteristic
258C22⋊C8C7×D411yescharacteristic
259C22⋊C8C7×D411yescharacteristic
260#97—21no—
261#97—21no—
262#97—21no—
263#97—21no—
264#97—21no—
265#97C7×D411yescharacteristic
266#97C7×D411yescharacteristic
267C4×C8—21no—
268C4×C8—21no—
269C4×C8—21no—
270C22×C7—21no—
271C22×C7—21no—
272C22×C7—41no—
273C22×C7—41no—
274C22×C7—41no—
275C22×C7—81no—
276C22×C7—81no—
277C22×C7—81no—
278C22×C7—161no—
279C22×C7—161no—
280C22×C7#38611yescharacteristic
281C22×C7#44311yescharacteristic
282C22×C7#44411yescharacteristic
283C4×C7—21no—
284C4×C7—21no—
285C4×C7—21no—
286C4×C7—21no—
287C4×C7—21no—
288C4×C7—21no—
289C4×C7—41no—
290C4×C7—41no—
291C4×C7—41no—
292C4×C7—81no—
293C24—21no—
294C2×Q8—21no—
295C2×Q8—21no—
296C2×Q8—81no—
297C2×Q8C7×C22⋊C411yescharacteristic
298C2×D4—21no—
299C2×D4—21no—
300C2×D4—21no—
301C2×D4—81no—
302C2×D4C7×C22⋊C411yescharacteristic
303C22×C4—21no—
304C22×C4—21no—
305C22×C4C22×C4×C711yescharacteristic
306C22×C4C2×C7×D411yescharacteristic
307C22×C4C7×#4111yescharacteristic
308Q16—81no—
309SD16—81no—
310SD16—81no—
311D8—81no—
312M16—42no—
313C2×C8—21no—
314C2×C8—22no—
315C2×C8—21no—
316C2×C8—21no—
317C2×C8—21no—
318C2×C8—21no—
319C2×C8—21no—
320C2×C8—42no—
321C2×C8—42no—
322C2×C8—41no—
323C2×C8—41no—
324C2×C8—41no—
325C4⋊C4—21no—
326C4⋊C4—21no—
327C4⋊C4—21no—
328C4⋊C4—21no—
329C4⋊C4C7×C22⋊C411yescharacteristic
330C4⋊C4C7×C22⋊C411yescharacteristic
331C22⋊C4—21no—
332C22⋊C4—21no—
333C22⋊C4—21no—
334C22⋊C4—21no—
335C22⋊C4—21no—
336C22⋊C4—41no—
337C22⋊C4—41no—
338C22⋊C4—41no—
339C22⋊C4—41no—
340C42—21no—
341C42C2×C7×D411yescharacteristic
342C42C2×C7×D411yescharacteristic
343C42C2×C7×D411yescharacteristic
344C42C7×#4111yescharacteristic
345C2×C7—21no—
346C2×C7—21no—
347C2×C7—81no—
348C2×C7—161no—
349C2×C7#187611yescharacteristic
350C2×C7#190611yescharacteristic
351C2×C7#191411yescharacteristic
352C23—21no—
353C23—41no—
354C23—41no—
355C23—41no—
356C23—81no—
357C23—81no—
358C23C4×C7×D411yescharacteristic
359Q8—41no—
360Q8—41no—
361Q8—41no—
362Q8—81no—
363D4—41no—
364D4—41no—
365D4—41no—
366D4—41no—
367D4—41no—
368D4—81no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—81no—
380C2×C4C2×C7×C22⋊C411yescharacteristic
381C2×C4C4×C7×D411yescharacteristic
382C2×C4C7×#12011yescharacteristic
383C2×C4C7×#12111yescharacteristic
384C2×C4C7×#12111yescharacteristic
385C2×C4C7×#12311yescharacteristic
386C8—22no—
387C8—42no—
388C8—41no—
389C8—41no—
390C8—41no—
391C8—41no—
392C8—41no—
393C8—81no—
394C7#926511yescharacteristic, direct product factor, Sylow
395C22—21no—
396C22—21no—
397C22—41no—
398C22—41no—
399C22—41no—
400C22—81no—
401C22—81no—
402C22—81no—
403C22—161no—
404C22—161no—
405C22C7×#38611yescharacteristic
406C22C7×#44311yescharacteristic
407C22C7×#44411yescharacteristic
408C4—21no—
409C4—21no—
410C4—21no—
411C4—21no—
412C4—21no—
413C4—21no—
414C4—41no—
415C4—41no—
416C4—41no—
417C4—81no—
418C2—21no—
419C2—21no—
420C2—81no—
421C2—161no—
422C2C7×#187611yescharacteristic
423C2C7×#190611yescharacteristic
424C2C7×#191411yescharacteristic
subgroupG/H
center, socleC22×C7#386
commutatorC22×C4C22×C4×C7
Frattini, cosocleC2×C42C23×C7
prime radicalC7×#447C4
LCSG ▷ C22×C4 ▷ 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×#2968C2semidirect
3. C7×#1532C2semidirect
4. C2×C7×#329C2semidirect
5. C7×#1469C2semidirect
6. C7×#1302C2semidirect
7. C7×#1281C2semidirect
35. #9265C7direct (N × H)
90. #2968C2×C7semidirect
91. #1532C2×C7semidirect
92. C2×#329C2×C7semidirect
93. #1469C2×C7semidirect
94. #1302C2×C7semidirect
95. #1281C2×C7semidirect
394. C7#9265direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#32112
C7×#29682
C7×#15322
C2×C7×#3292
C7×#14692
C7×#13022
C7×#12812
#92657