Finite Group Databasebeta

← C7×#8998 #13186214

Subgroups — C7×#8998 #13186214

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×#3172C211yesmaximal, characteristic
2C7×#2995C211yesmaximal, characteristic
3C7×#1995C211yesmaximal, characteristic, complemented
4C7×#1500C211yesmaximal, characteristic
5C7×#1497C211yesmaximal, characteristic, complemented
6C7×#1300C211yesmaximal, characteristic, complemented
7C7×#1299C211yesmaximal, characteristic, complemented
8C7×#533C2211yescharacteristic
9C7×#529C2211yescharacteristic
10C7×#501C411yescharacteristic, complemented
11C7×#497C411yescharacteristic, complemented
12C7×#491C411yescharacteristic, complemented
13C7×#489C411yescharacteristic, complemented
14C7×#470—21no—
15C7×#468—21no—
16C7×#446—21no—
17C4×C7×Q16—21no—
18C4×C7×SD16—21no—
19C7×#436—21no—
20C7×#435—21no—
21C7×#435—21no—
22C7×#433—21no—
23C7×#432C2211yescharacteristic
24C7×#431C2211yescharacteristic
25C7×#424—21no—
26C7×#423C2211yescharacteristic
27C4×C7×M16C2211yescharacteristic
28C7×#376C2211yescharacteristic
29C7×#339—21no—
30C7×#339—21no—
31C7×#328—21no—
32C7×#327—21no—
33C7×Q8⋊C8—21no—
34C7×Q8⋊C8—21no—
35#8998C711yesmaximal, characteristic, direct product factor, Sylow
36C2×C7×Q16—21no—
37C2×C7×SD16—21no—
38C2×C7×M16—21no—
39C2×C7×M16—21no—
40C2×C7×M16D411yescharacteristic
41C2×C7×M16D411yescharacteristic
42C22×C7×C8—21no—
43C7×C4⋊Q8C2×C411yescharacteristic
44C7×C4⋊Q8C2×C411yescharacteristic
45C7×#126—21no—
46C7×#125—21no—
47C7×#125C2×C411yescharacteristic
48C7×#124C2×C411yescharacteristic
49C7×#123—21no—
50C7×C22⋊Q8—21no—
51C7×C22⋊Q8—21no—
52C7×C22⋊Q8D411yescharacteristic
53C7×#121D411yescharacteristic
54C4×C7×Q8—21no—
55C4×C7×Q8—21no—
56C4×C7×Q8D411yescharacteristic
57C4×C7×D4—21no—
58C4×C7×D4D411yescharacteristic
59C7×#117—21no—
60C7×#117—21no—
61C2×C42×C7—21no—
62C2×C42×C7C2311yescharacteristic
63C7×#107—21no—
64C7×#106—21no—
65C7×C4⋊C8—21no—
66C7×C4⋊C8—21no—
67C7×C4⋊C8—21no—
68C7×C4⋊C8—21no—
69C7×C4⋊C8—21no—
70C7×C4⋊C8D411yescharacteristic
71C7×C4⋊C8D411yescharacteristic
72C7×#103—21no—
73C7×#103—21no—
74C7×#103—21no—
75C7×#102—21no—
76C7×C22⋊C8—21no—
77C7×C22⋊C8—21no—
78C7×C22⋊C8—21no—
79C7×C22⋊C8—21no—
80C7×#97—21no—
81C7×#97—21no—
82C7×#97C2×C411yescharacteristic
83C7×#97C2×C411yescharacteristic
84C4×C7×C8—21no—
85C4×C7×C8—21no—
86C4×C7×C8—21no—
87C7×#95—21no—
88C7×#95—21no—
89#3172C2×C711yescharacteristic
90#2995C2×C711yescharacteristic
91#1995C2×C711yescharacteristic, complemented
92#1500C2×C711yescharacteristic
93#1497C2×C711yescharacteristic, complemented
94#1300C2×C711yescharacteristic, complemented
95#1299C2×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—
109C7×M16—41no—
110C7×M16—41no—
111C7×M16—81no—
112C2×C7×C8—21no—
113C2×C7×C8—21no—
114C2×C7×C8—21no—
115C2×C7×C8—21no—
116C2×C7×C8—21no—
117C2×C7×C8—21no—
118C2×C7×C8—41no—
119C2×C7×C8—81no—
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—21no—
128C7×C4⋊C4—41no—
129C7×C4⋊C4—41no—
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—22no—
140C42×C7—41no—
141C42×C7—41no—
142C42×C7C22×C411yescharacteristic
143C42×C7C2×D411yescharacteristic
144C42×C7C2×D411yescharacteristic
145C42×C7#4111yescharacteristic
146#533C22×C711yescharacteristic
147#529C22×C711yescharacteristic
148#501C4×C711yescharacteristic, complemented
149#497C4×C711yescharacteristic, complemented
150#491C4×C711yescharacteristic, complemented
151#489C4×C711yescharacteristic, complemented
152#470—21no—
153#468—21no—
154#446—21no—
155C4×Q16—21no—
156C4×SD16—21no—
157#436—21no—
158#435—21no—
159#435—21no—
160#433—21no—
161#432C22×C711yescharacteristic
162#431C22×C711yescharacteristic
163#424—21no—
164#423C22×C711yescharacteristic
165C4×M16C22×C711yescharacteristic
166#376C22×C711yescharacteristic
167#339—21no—
168#339—21no—
169#328—21no—
170#327—21no—
171Q8⋊C8—21no—
172Q8⋊C8—21no—
173C23×C7—21no—
174C23×C7#12111yescharacteristic
175C7×Q8—41no—
176C7×Q8—41no—
177C7×Q8—41no—
178C7×Q8—42no—
179C7×D4—41no—
180C7×D4—81no—
181C2×C4×C7—21no—
182C2×C4×C7—21no—
183C2×C4×C7—21no—
184C2×C4×C7—21no—
185C2×C4×C7—21no—
186C2×C4×C7—41no—
187C2×C4×C7—41no—
188C2×C4×C7—41no—
189C2×C4×C7—41no—
190C2×C4×C7—42no—
191C2×C4×C7—41no—
192C2×C4×C7—41no—
193C2×C4×C7—81no—
194C2×C4×C7—81no—
195C2×C4×C7C2×C22⋊C411yescharacteristic
196C2×C4×C7C4×D411yescharacteristic
197C2×C4×C7C4×D411yescharacteristic
198C2×C4×C7#12011yescharacteristic
199C2×C4×C7#12111yescharacteristic
200C2×C4×C7#12311yescharacteristic
201C7×C8—22no—
202C7×C8—41no—
203C7×C8—41no—
204C7×C8—41no—
205C7×C8—41no—
206C7×C8—41no—
207C7×C8—81no—
208C2×Q16—21no—
209C2×SD16—21no—
210C2×M16—21no—
211C2×M16—21no—
212C2×M16C7×D411yescharacteristic
213C2×M16C7×D411yescharacteristic
214C22×C8—21no—
215C4⋊Q8C2×C4×C711yescharacteristic
216C4⋊Q8C2×C4×C711yescharacteristic
217#126—21no—
218#125—21no—
219#125C2×C4×C711yescharacteristic
220#124C2×C4×C711yescharacteristic
221#123—21no—
222C22⋊Q8—21no—
223C22⋊Q8—21no—
224C22⋊Q8C7×D411yescharacteristic
225#121C7×D411yescharacteristic
226C4×Q8—21no—
227C4×Q8—21no—
228C4×Q8C7×D411yescharacteristic
229C4×D4—21no—
230C4×D4C7×D411yescharacteristic
231#117—21no—
232#117—21no—
233C2×C42—21no—
234C2×C42C23×C711yescharacteristic
235#107—21no—
236#106—21no—
237C4⋊C8—21no—
238C4⋊C8—21no—
239C4⋊C8—21no—
240C4⋊C8—21no—
241C4⋊C8—21no—
242C4⋊C8C7×D411yescharacteristic
243C4⋊C8C7×D411yescharacteristic
244#103—21no—
245#103—21no—
246#103—21no—
247#102—21no—
248C22⋊C8—21no—
249C22⋊C8—21no—
250C22⋊C8—21no—
251C22⋊C8—21no—
252#97—21no—
253#97—21no—
254#97C2×C4×C711yescharacteristic
255#97C2×C4×C711yescharacteristic
256C4×C8—21no—
257C4×C8—21no—
258C4×C8—21no—
259#95—21no—
260#95—21no—
261C22×C7—21no—
262C22×C7—21no—
263C22×C7—21no—
264C22×C7—41no—
265C22×C7—41no—
266C22×C7—41no—
267C22×C7#38611yescharacteristic
268C4×C7—21no—
269C4×C7—21no—
270C4×C7—21no—
271C4×C7—21no—
272C4×C7—21no—
273C4×C7—41no—
274C4×C7—41no—
275C4×C7—41no—
276C4×C7—82no—
277C4×C7—81no—
278C4×C7—81no—
279C4×C7#45511yescharacteristic
280C4×C7#45611yescharacteristic
281C2×Q8—21no—
282C2×Q8—21no—
283C2×Q8C7×C22⋊C411yescharacteristic
284C2×D4—21no—
285C2×D4C7×C22⋊C411yescharacteristic
286C22×C4—21no—
287C22×C4—41no—
288C22×C4C2×C7×D411yescharacteristic
289C22×C4C2×C7×D411yescharacteristic
290C22×C4C7×#4111yescharacteristic
291Q16—42no—
292SD16—42no—
293M16—42no—
294M16—41no—
295M16—41no—
296M16—81no—
297C2×C8—21no—
298C2×C8—21no—
299C2×C8—21no—
300C2×C8—21no—
301C2×C8—21no—
302C2×C8—21no—
303C2×C8—41no—
304C2×C8—81no—
305C4⋊C4—21no—
306C4⋊C4—21no—
307C4⋊C4—21no—
308C4⋊C4—21no—
309C4⋊C4—21no—
310C4⋊C4—21no—
311C4⋊C4—21no—
312C4⋊C4—21no—
313C4⋊C4—41no—
314C4⋊C4—41no—
315C4⋊C4C7×C22⋊C411yescharacteristic
316C4⋊C4C7×C22⋊C411yescharacteristic
317C22⋊C4—21no—
318C22⋊C4—21no—
319C22⋊C4—21no—
320C22⋊C4—21no—
321C22⋊C4—21no—
322C22⋊C4—41no—
323C42—21no—
324C42—22no—
325C42—41no—
326C42—41no—
327C42C22×C4×C711yescharacteristic
328C42C2×C7×D411yescharacteristic
329C42C2×C7×D411yescharacteristic
330C42C7×#4111yescharacteristic
331C2×C7—41no—
332C2×C7—81no—
333C2×C7#188111yescharacteristic
334C2×C7#188211yescharacteristic
335C2×C7#188411yescharacteristic
336C23—21no—
337C23C7×#12111yescharacteristic
338Q8—41no—
339Q8—41no—
340Q8—41no—
341Q8—42no—
342D4—41no—
343D4—81no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—42no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—81no—
357C2×C4—81no—
358C2×C4C2×C7×C22⋊C411yescharacteristic
359C2×C4C4×C7×D411yescharacteristic
360C2×C4C4×C7×D411yescharacteristic
361C2×C4C7×#12011yescharacteristic
362C2×C4C7×#12111yescharacteristic
363C2×C4C7×#12311yescharacteristic
364C8—22no—
365C8—41no—
366C8—41no—
367C8—41no—
368C8—41no—
369C8—41no—
370C8—81no—
371C7#899811yescharacteristic, direct product factor, Sylow
372C22—21no—
373C22—21no—
374C22—21no—
375C22—41no—
376C22—41no—
377C22—41no—
378C22C7×#38611yescharacteristic
379C4—21no—
380C4—21no—
381C4—21no—
382C4—21no—
383C4—21no—
384C4—41no—
385C4—41no—
386C4—41no—
387C4—82no—
388C4—81no—
389C4—81no—
390C4C7×#45511yescharacteristic
391C4C7×#45611yescharacteristic
392C2—41no—
393C2—81no—
394C2C7×#188111yescharacteristic
395C2C7×#188211yescharacteristic
396C2C7×#188411yescharacteristic
subgroupG/H
centerC2×C4×C7#120
commutatorC42C22×C4×C7
Frattini, cosocleC2×C42C23×C7
socleC22×C7#386
prime radicalC7×#121D4
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4×C7 ◁ C2×C42×C7 ◁ G

Complements

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

NH ≅ G/Ntype
3. C7×#1995C2semidirect
5. C7×#1497C2semidirect
6. C7×#1300C2semidirect
7. C7×#1299C2semidirect
10. C7×#501C4semidirect
11. C7×#497C4semidirect
12. C7×#491C4semidirect
13. C7×#489C4semidirect
35. #8998C7direct (N × H)
91. #1995C2×C7semidirect
93. #1497C2×C7semidirect
94. #1300C2×C7semidirect
95. #1299C2×C7semidirect
148. #501C4×C7semidirect
149. #497C4×C7semidirect
150. #491C4×C7semidirect
151. #489C4×C7semidirect
371. C7#8998direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#31722
C7×#29952
C7×#19952
C7×#15002
C7×#14972
C7×#13002
C7×#12992
#89987