Finite Group Databasebeta

← C7×#9431 #13186647

Subgroups — C7×#9431 #13186647

Order 1792.

414 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3181C211yesmaximal, characteristic
2C7×#2968C211yesmaximal, characteristic
3C7×#1531C211yesmaximal, characteristic, complemented
4C7×#1518C211yesmaximal, characteristic, complemented
5C7×#1512C211yesmaximal, characteristic
6C7×#1302C211yesmaximal, characteristic, complemented
7C7×#1295C211yesmaximal, characteristic, complemented
8C7×#533C2211yescharacteristic
9C7×#529C2211yescharacteristic
10C7×#501C411yescharacteristic, complemented
11C7×#497C411yescharacteristic, complemented
12C7×#491C411yescharacteristic, complemented
13C7×#489C411yescharacteristic, complemented
14C7×#467—21no—
15C7×#465—21no—
16C7×#446—21no—
17C7×#441—21no—
18C7×#440—21no—
19C7×#435—21no—
20C7×#435—21no—
21C7×C8×D4—21no—
22C7×#433—21no—
23C7×#432C2211yescharacteristic
24C7×#431C2211yescharacteristic
25C7×#405—21no—
26C2×C7×#97C2211yescharacteristic
27C2×C7×#97C2211yescharacteristic
28C7×#382C2211yescharacteristic
29C7×#330—21no—
30C7×#330—21no—
31C7×#328—21no—
32C7×#328—21no—
33C7×#328—21no—
34C7×#327—21no—
35#9431C711yesmaximal, characteristic, direct product factor, Sylow
36C2×C7×Q16—21no—
37C2×C7×SD16—21no—
38C2×C7×M16—21no—
39C22×C7×C8—21no—
40C22×C7×C8—21no—
41C22×C7×C8—21no—
42C7×C4⋊Q8C2×C411yescharacteristic
43C7×C4⋊Q8C2×C411yescharacteristic
44C7×#126—21no—
45C7×#125—21no—
46C7×#125C2×C411yescharacteristic
47C7×#124C2×C411yescharacteristic
48C7×#123—21no—
49C7×C22⋊Q8—21no—
50C7×C22⋊Q8—21no—
51C7×C22⋊Q8D411yescharacteristic
52C7×#121D411yescharacteristic
53C4×C7×Q8—21no—
54C4×C7×Q8—21no—
55C4×C7×Q8D411yescharacteristic
56C4×C7×D4—21no—
57C4×C7×D4D411yescharacteristic
58C7×#117—21no—
59C7×#117—21no—
60C2×C7×C4⋊C4—21no—
61C2×C42×C7C2311yescharacteristic
62C7×#107—21no—
63C7×#106—21no—
64C7×C4⋊C8—21no—
65C7×C4⋊C8—21no—
66C7×C4⋊C8—21no—
67C7×C4⋊C8—21no—
68C7×#103—21no—
69C7×#103—21no—
70C7×#103—21no—
71C7×#102—21no—
72C7×C22⋊C8—21no—
73C7×C22⋊C8—21no—
74C7×C22⋊C8—21no—
75C7×C22⋊C8D411yescharacteristic
76C7×C22⋊C8D411yescharacteristic
77C7×#97—21no—
78C7×#97—21no—
79C7×#97—21no—
80C7×#97—21no—
81C7×#97C2×C411yescharacteristic
82C7×#97C2×C411yescharacteristic
83C7×#97D411yescharacteristic
84C7×#97D411yescharacteristic
85C4×C7×C8—21no—
86C4×C7×C8—21no—
87C7×#95—21no—
88C7×#95—21no—
89#3181C2×C711yescharacteristic
90#2968C2×C711yescharacteristic
91#1531C2×C711yescharacteristic, complemented
92#1518C2×C711yescharacteristic, complemented
93#1512C2×C711yescharacteristic
94#1302C2×C711yescharacteristic, complemented
95#1295C2×C711yescharacteristic, complemented
96C2×C7×Q8—21no—
97C2×C7×Q8—21no—
98C2×C7×Q8C22⋊C411yescharacteristic
99C2×C7×D4—21no—
100C2×C7×D4C22⋊C411yescharacteristic
101C22×C4×C7—21no—
102C22×C4×C7—41no—
103C22×C4×C7C2×D411yescharacteristic
104C22×C4×C7C2×D411yescharacteristic
105C22×C4×C7#4111yescharacteristic
106C7×Q16—42no—
107C7×SD16—42no—
108C7×M16—42no—
109C2×C7×C8—21no—
110C2×C7×C8—21no—
111C2×C7×C8—21no—
112C2×C7×C8—21no—
113C2×C7×C8—21no—
114C2×C7×C8—21no—
115C2×C7×C8—42no—
116C2×C7×C8—42no—
117C2×C7×C8—42no—
118C2×C7×C8—41no—
119C7×C4⋊C4—21no—
120C7×C4⋊C4—21no—
121C7×C4⋊C4—21no—
122C7×C4⋊C4—21no—
123C7×C4⋊C4—21no—
124C7×C4⋊C4—21no—
125C7×C4⋊C4—21no—
126C7×C4⋊C4—21no—
127C7×C4⋊C4—41no—
128C7×C4⋊C4—41no—
129C7×C4⋊C4—81no—
130C7×C4⋊C4C22⋊C411yescharacteristic
131C7×C4⋊C4C22⋊C411yescharacteristic
132C7×C22⋊C4—21no—
133C7×C22⋊C4—21no—
134C7×C22⋊C4—21no—
135C7×C22⋊C4—21no—
136C7×C22⋊C4—21no—
137C7×C22⋊C4—41no—
138C42×C7—21no—
139C42×C7—41no—
140C42×C7C22×C411yescharacteristic
141C42×C7C2×D411yescharacteristic
142C42×C7C2×D411yescharacteristic
143C42×C7#4111yescharacteristic
144#533C22×C711yescharacteristic
145#529C22×C711yescharacteristic
146#501C4×C711yescharacteristic, complemented
147#497C4×C711yescharacteristic, complemented
148#491C4×C711yescharacteristic, complemented
149#489C4×C711yescharacteristic, complemented
150#467—21no—
151#465—21no—
152#446—21no—
153#441—21no—
154#440—21no—
155#435—21no—
156#435—21no—
157C8×D4—21no—
158#433—21no—
159#432C22×C711yescharacteristic
160#431C22×C711yescharacteristic
161#405—21no—
162C2×#97C22×C711yescharacteristic
163C2×#97C22×C711yescharacteristic
164#382C22×C711yescharacteristic
165#330—21no—
166#330—21no—
167#328—21no—
168#328—21no—
169#328—21no—
170#327—21no—
171C23×C7—21no—
172C23×C7#12111yescharacteristic
173C7×Q8—41no—
174C7×Q8—41no—
175C7×Q8—41no—
176C7×Q8—81no—
177C7×D4—41no—
178C7×D4—41no—
179C7×D4—41no—
180C2×C4×C7—21no—
181C2×C4×C7—21no—
182C2×C4×C7—21no—
183C2×C4×C7—21no—
184C2×C4×C7—21no—
185C2×C4×C7—41no—
186C2×C4×C7—41no—
187C2×C4×C7—41no—
188C2×C4×C7—41no—
189C2×C4×C7—41no—
190C2×C4×C7—41no—
191C2×C4×C7—81no—
192C2×C4×C7—81no—
193C2×C4×C7—81no—
194C2×C4×C7C2×C22⋊C411yescharacteristic
195C2×C4×C7C4×D411yescharacteristic
196C2×C4×C7C4×D411yescharacteristic
197C2×C4×C7#12011yescharacteristic
198C2×C4×C7#12111yescharacteristic
199C2×C4×C7#12311yescharacteristic
200C7×C8—22no—
201C7×C8—22no—
202C7×C8—41no—
203C7×C8—41no—
204C7×C8—41no—
205C7×C8—41no—
206C7×C8—81no—
207C2×Q16—21no—
208C2×SD16—21no—
209C2×M16—21no—
210C22×C8—21no—
211C22×C8—21no—
212C22×C8—21no—
213C4⋊Q8C2×C4×C711yescharacteristic
214C4⋊Q8C2×C4×C711yescharacteristic
215#126—21no—
216#125—21no—
217#125C2×C4×C711yescharacteristic
218#124C2×C4×C711yescharacteristic
219#123—21no—
220C22⋊Q8—21no—
221C22⋊Q8—21no—
222C22⋊Q8C7×D411yescharacteristic
223#121C7×D411yescharacteristic
224C4×Q8—21no—
225C4×Q8—21no—
226C4×Q8C7×D411yescharacteristic
227C4×D4—21no—
228C4×D4C7×D411yescharacteristic
229#117—21no—
230#117—21no—
231C2×C4⋊C4—21no—
232C2×C42C23×C711yescharacteristic
233#107—21no—
234#106—21no—
235C4⋊C8—21no—
236C4⋊C8—21no—
237C4⋊C8—21no—
238C4⋊C8—21no—
239#103—21no—
240#103—21no—
241#103—21no—
242#102—21no—
243C22⋊C8—21no—
244C22⋊C8—21no—
245C22⋊C8—21no—
246C22⋊C8C7×D411yescharacteristic
247C22⋊C8C7×D411yescharacteristic
248#97—21no—
249#97—21no—
250#97—21no—
251#97—21no—
252#97C2×C4×C711yescharacteristic
253#97C2×C4×C711yescharacteristic
254#97C7×D411yescharacteristic
255#97C7×D411yescharacteristic
256C4×C8—21no—
257C4×C8—21no—
258#95—21no—
259#95—21no—
260C22×C7—21no—
261C22×C7—21no—
262C22×C7—41no—
263C22×C7—41no—
264C22×C7—41no—
265C22×C7#38611yescharacteristic
266C22×C7#47111yescharacteristic
267C22×C7#47311yescharacteristic
268C4×C7—21no—
269C4×C7—21no—
270C4×C7—21no—
271C4×C7—21no—
272C4×C7—21no—
273C4×C7—21no—
274C4×C7—41no—
275C4×C7—41no—
276C4×C7—41no—
277C4×C7—81no—
278C4×C7—81no—
279C4×C7—161no—
280C2×Q8—21no—
281C2×Q8—21no—
282C2×Q8C7×C22⋊C411yescharacteristic
283C2×D4—21no—
284C2×D4C7×C22⋊C411yescharacteristic
285C22×C4—21no—
286C22×C4—41no—
287C22×C4C2×C7×D411yescharacteristic
288C22×C4C2×C7×D411yescharacteristic
289C22×C4C7×#4111yescharacteristic
290Q16—42no—
291SD16—42no—
292M16—42no—
293C2×C8—21no—
294C2×C8—21no—
295C2×C8—21no—
296C2×C8—21no—
297C2×C8—21no—
298C2×C8—21no—
299C2×C8—42no—
300C2×C8—42no—
301C2×C8—42no—
302C2×C8—41no—
303C4⋊C4—21no—
304C4⋊C4—21no—
305C4⋊C4—21no—
306C4⋊C4—21no—
307C4⋊C4—21no—
308C4⋊C4—21no—
309C4⋊C4—21no—
310C4⋊C4—21no—
311C4⋊C4—41no—
312C4⋊C4—41no—
313C4⋊C4—81no—
314C4⋊C4C7×C22⋊C411yescharacteristic
315C4⋊C4C7×C22⋊C411yescharacteristic
316C22⋊C4—21no—
317C22⋊C4—21no—
318C22⋊C4—21no—
319C22⋊C4—21no—
320C22⋊C4—21no—
321C22⋊C4—41no—
322C42—21no—
323C42—41no—
324C42C22×C4×C711yescharacteristic
325C42C2×C7×D411yescharacteristic
326C42C2×C7×D411yescharacteristic
327C42C7×#4111yescharacteristic
328C2×C7—21no—
329C2×C7—21no—
330C2×C7—81no—
331C2×C7#188411yescharacteristic
332C2×C7#190611yescharacteristic
333C2×C7#190811yescharacteristic
334C23—21no—
335C23C7×#12111yescharacteristic
336Q8—41no—
337Q8—41no—
338Q8—41no—
339Q8—81no—
340D4—41no—
341D4—41no—
342D4—41no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—81no—
355C2×C4—81no—
356C2×C4—81no—
357C2×C4C2×C7×C22⋊C411yescharacteristic
358C2×C4C4×C7×D411yescharacteristic
359C2×C4C4×C7×D411yescharacteristic
360C2×C4C7×#12011yescharacteristic
361C2×C4C7×#12111yescharacteristic
362C2×C4C7×#12311yescharacteristic
363C8—22no—
364C8—22no—
365C8—41no—
366C8—41no—
367C8—41no—
368C8—41no—
369C8—81no—
370C7#943111yescharacteristic, direct product factor, Sylow
371C22—21no—
372C22—21no—
373C22—41no—
374C22—41no—
375C22—41no—
376C22C7×#38611yescharacteristic
377C22C7×#47111yescharacteristic
378C22C7×#47311yescharacteristic
379C4—21no—
380C4—21no—
381C4—21no—
382C4—21no—
383C4—21no—
384C4—21no—
385C4—41no—
386C4—41no—
387C4—41no—
388C4—81no—
389C4—81no—
390C4—161no—
391C2—21no—
392C2—21no—
393C2—81no—
394C2C7×#188411yescharacteristic
395C2C7×#190611yescharacteristic
396C2C7×#190811yescharacteristic
subgroupG/H
center, socleC22×C7#386
commutatorC42C22×C4×C7
Frattini, cosocleC2×C42C23×C7
prime radicalC7×#121D4
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
3. C7×#1531C2semidirect
4. C7×#1518C2semidirect
6. C7×#1302C2semidirect
7. C7×#1295C2semidirect
10. C7×#501C4semidirect
11. C7×#497C4semidirect
12. C7×#491C4semidirect
13. C7×#489C4semidirect
35. #9431C7direct (N × H)
91. #1531C2×C7semidirect
92. #1518C2×C7semidirect
94. #1302C2×C7semidirect
95. #1295C2×C7semidirect
146. #501C4×C7semidirect
147. #497C4×C7semidirect
148. #491C4×C7semidirect
149. #489C4×C7semidirect
370. C7#9431direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#31812
C7×#29682
C7×#15312
C7×#15182
C7×#15122
C7×#13022
C7×#12952
#94317