Finite Group Databasebeta

← C7×#9331 #13186547

Subgroups — C7×#9331 #13186547

Order 1792.

438 conjugacy classes of subgroups.

The subgroup lattice has 434 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3196C211yesmaximal, characteristic
2C7×#2978C211yesmaximal, characteristic
3C7×#1516C211yesmaximal, characteristic
4C7×#1511C211yesmaximal, characteristic, complemented
5C7×#1472C211yesmaximal, characteristic, complemented
6C7×#1306C211yesmaximal, characteristic, complemented
7C7×#1293C211yesmaximal, characteristic, complemented
8C7×#525C2211yescharacteristic
9C2×C7×C22⋊Q8C2211yescharacteristic
10C7×#485—21no—
11C7×#484C411yescharacteristic, complemented
12C7×#481C411yescharacteristic, complemented
13C7×#470—21no—
14C7×#468—21no—
15C7×#452—21no—
16C7×#451C411yescharacteristic, complemented
17C7×#448C411yescharacteristic, complemented
18C7×#436—21no—
19C7×#436—21no—
20C7×#435—21no—
21C7×#435—21no—
22C7×#433—21no—
23C7×#408—21no—
24C7×#407C2211yescharacteristic
25C7×#407C2211yescharacteristic
26C7×#407C2211yescharacteristic
27C2×C7×C22⋊C8C2211yescharacteristic
28C7×#381C2211yescharacteristic
29C7×#328—21no—
30C7×#328—21no—
31C7×#328—21no—
32C7×#327—21no—
33C7×#324—21no—
34C7×#324—21no—
35#9331C711yesmaximal, characteristic, direct product factor, Sylow
36C2×C7×#41D411yescharacteristic
37C22×C7×Q8C2×C411yescharacteristic
38C23×C4×C7C2311yescharacteristic
39C2×C7×Q16—21no—
40C2×C7×SD16—21no—
41C2×C7×M16—21no—
42C2×C7×M16—21no—
43C2×C7×M16—21no—
44C2×C7×M16—21no—
45C22×C7×C8—21no—
46C22×C7×C8—21no—
47C7×#123—21no—
48C7×C22⋊Q8—21no—
49C7×C22⋊Q8—21no—
50C7×C22⋊Q8—21no—
51C7×C22⋊Q8—21no—
52C7×C22⋊Q8C2×C411yescharacteristic
53C7×C22⋊Q8D411yescharacteristic
54C7×#121C2×C411yescharacteristic
55C7×#121D411yescharacteristic
56C7×#120—21no—
57C4×C7×D4—21no—
58C4×C7×D4—21no—
59C7×#117D411yescharacteristic
60C2×C7×C4⋊C4—21no—
61C2×C7×C4⋊C4C2×C411yescharacteristic
62C2×C7×C22⋊C4—21no—
63C2×C7×C22⋊C4—21no—
64C7×#107—21no—
65C7×#106—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⋊C8—21no—
76C7×C22⋊C8—21no—
77C7×C22⋊C8—21no—
78C7×C22⋊C8—21no—
79C7×C22⋊C8C2×C411yescharacteristic
80C7×C22⋊C8C2×C411yescharacteristic
81C7×C22⋊C8D411yescharacteristic
82C7×C22⋊C8D411yescharacteristic
83C7×C22⋊C8D411yescharacteristic
84C7×C22⋊C8D411yescharacteristic
85C7×#97—21no—
86C4×C7×C8—21no—
87C7×#95—21no—
88C7×#95—21no—
89#3196C2×C711yescharacteristic
90#2978C2×C711yescharacteristic
91#1516C2×C711yescharacteristic
92#1511C2×C711yescharacteristic, complemented
93#1472C2×C711yescharacteristic, complemented
94#1306C2×C711yescharacteristic, complemented
95#1293C2×C711yescharacteristic, complemented
96C24×C7#4111yescharacteristic
97C7×#41—81no—
98C2×C7×Q8—21no—
99C2×C7×Q8—21no—
100C2×C7×Q8—81no—
101C2×C7×Q8C22⋊C411yescharacteristic
102C2×C7×D4—21no—
103C2×C7×D4—21no—
104C2×C7×D4—21no—
105C2×C7×D4C22⋊C411yescharacteristic
106C22×C4×C7—21no—
107C22×C4×C7—21no—
108C22×C4×C7—21no—
109C22×C4×C7—41no—
110C22×C4×C7—81no—
111C22×C4×C7C22×C411yescharacteristic
112C22×C4×C7C2×D411yescharacteristic
113C22×C4×C7C2×D411yescharacteristic
114C22×C4×C7C2×D411yescharacteristic
115C22×C4×C7C2×D411yescharacteristic
116C22×C4×C7#4111yescharacteristic
117C7×Q16—81no—
118C7×SD16—81no—
119C7×M16—81no—
120C7×M16—81no—
121C7×M16—81no—
122C7×M16—81no—
123C2×C7×C8—21no—
124C2×C7×C8—21no—
125C2×C7×C8—21no—
126C2×C7×C8—21no—
127C2×C7×C8—21no—
128C2×C7×C8—21no—
129C2×C7×C8—41no—
130C2×C7×C8—81no—
131C2×C7×C8—81no—
132C7×C4⋊C4—21no—
133C7×C4⋊C4—21no—
134C7×C4⋊C4—21no—
135C7×C4⋊C4—21no—
136C7×C4⋊C4—41no—
137C7×C4⋊C4—41no—
138C7×C4⋊C4C22⋊C411yescharacteristic
139C7×C4⋊C4C22⋊C411yescharacteristic
140C7×C22⋊C4—21no—
141C7×C22⋊C4—21no—
142C7×C22⋊C4—21no—
143C7×C22⋊C4—21no—
144C7×C22⋊C4—21no—
145C7×C22⋊C4—41no—
146C7×C22⋊C4—41no—
147C7×C22⋊C4—81no—
148C42×C7—21no—
149#525C22×C711yescharacteristic
150C2×C22⋊Q8C22×C711yescharacteristic
151#485—21no—
152#484C4×C711yescharacteristic, complemented
153#481C4×C711yescharacteristic, complemented
154#470—21no—
155#468—21no—
156#452—21no—
157#451C4×C711yescharacteristic, complemented
158#448C4×C711yescharacteristic, complemented
159#436—21no—
160#436—21no—
161#435—21no—
162#435—21no—
163#433—21no—
164#408—21no—
165#407C22×C711yescharacteristic
166#407C22×C711yescharacteristic
167#407C22×C711yescharacteristic
168C2×C22⋊C8C22×C711yescharacteristic
169#381C22×C711yescharacteristic
170#328—21no—
171#328—21no—
172#328—21no—
173#327—21no—
174#324—21no—
175#324—21no—
176C23×C7—21no—
177C23×C7—41no—
178C23×C7—41no—
179C23×C7—41no—
180C23×C7C4×D411yescharacteristic
181C23×C7#12111yescharacteristic
182C23×C7#12311yescharacteristic
183C7×Q8—41no—
184C7×Q8—41no—
185C7×Q8—41no—
186C7×Q8—81no—
187C7×D4—41no—
188C7×D4—81no—
189C7×D4—81no—
190C7×D4—81no—
191C2×C4×C7—21no—
192C2×C4×C7—21no—
193C2×C4×C7—21no—
194C2×C4×C7—41no—
195C2×C4×C7—41no—
196C2×C4×C7—41no—
197C2×C4×C7—41no—
198C2×C4×C7—41no—
199C2×C4×C7—41no—
200C2×C4×C7—41no—
201C2×C4×C7—41no—
202C2×C4×C7—42no—
203C2×C4×C7—41no—
204C2×C4×C7—41no—
205C2×C4×C7—81no—
206C2×C4×C7—81no—
207C2×C4×C7—81no—
208C2×C4×C7—81no—
209C2×C4×C7C2×C22⋊C411yescharacteristic
210C2×C4×C7C4×D411yescharacteristic
211C2×C4×C7#12011yescharacteristic
212C2×C4×C7#12111yescharacteristic
213C7×C8—41no—
214C7×C8—41no—
215C7×C8—41no—
216C7×C8—41no—
217C7×C8—41no—
218C7×C8—41no—
219C7×C8—81no—
220C2×#41C7×D411yescharacteristic
221C22×Q8C2×C4×C711yescharacteristic
222C23×C4C23×C711yescharacteristic
223C2×Q16—21no—
224C2×SD16—21no—
225C2×M16—21no—
226C2×M16—21no—
227C2×M16—21no—
228C2×M16—21no—
229C22×C8—21no—
230C22×C8—21no—
231#123—21no—
232C22⋊Q8—21no—
233C22⋊Q8—21no—
234C22⋊Q8—21no—
235C22⋊Q8—21no—
236C22⋊Q8C2×C4×C711yescharacteristic
237C22⋊Q8C7×D411yescharacteristic
238#121C2×C4×C711yescharacteristic
239#121C7×D411yescharacteristic
240#120—21no—
241C4×D4—21no—
242C4×D4—21no—
243#117C7×D411yescharacteristic
244C2×C4⋊C4—21no—
245C2×C4⋊C4C2×C4×C711yescharacteristic
246C2×C22⋊C4—21no—
247C2×C22⋊C4—21no—
248#107—21no—
249#106—21no—
250C4⋊C8—21no—
251C4⋊C8—21no—
252#103—21no—
253#103—21no—
254#103—21no—
255#102—21no—
256C22⋊C8—21no—
257C22⋊C8—21no—
258C22⋊C8—21no—
259C22⋊C8—21no—
260C22⋊C8—21no—
261C22⋊C8—21no—
262C22⋊C8—21no—
263C22⋊C8C2×C4×C711yescharacteristic
264C22⋊C8C2×C4×C711yescharacteristic
265C22⋊C8C7×D411yescharacteristic
266C22⋊C8C7×D411yescharacteristic
267C22⋊C8C7×D411yescharacteristic
268C22⋊C8C7×D411yescharacteristic
269#97—21no—
270C4×C8—21no—
271#95—21no—
272#95—21no—
273C22×C7—21no—
274C22×C7—21no—
275C22×C7—21no—
276C22×C7—21no—
277C22×C7—21no—
278C22×C7—21no—
279C22×C7—21no—
280C22×C7—21no—
281C22×C7—21no—
282C22×C7—41no—
283C22×C7—41no—
284C22×C7—41no—
285C22×C7—82no—
286C22×C7#38611yescharacteristic
287C4×C7—21no—
288C4×C7—21no—
289C4×C7—21no—
290C4×C7—21no—
291C4×C7—41no—
292C4×C7—41no—
293C4×C7—41no—
294C4×C7—81no—
295C4×C7—81no—
296C4×C7—161no—
297C24C7×#4111yescharacteristic
298#41—81no—
299C2×Q8—21no—
300C2×Q8—21no—
301C2×Q8—81no—
302C2×Q8C7×C22⋊C411yescharacteristic
303C2×D4—21no—
304C2×D4—21no—
305C2×D4—21no—
306C2×D4C7×C22⋊C411yescharacteristic
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—41no—
311C22×C4—81no—
312C22×C4C22×C4×C711yescharacteristic
313C22×C4C2×C7×D411yescharacteristic
314C22×C4C2×C7×D411yescharacteristic
315C22×C4C2×C7×D411yescharacteristic
316C22×C4C2×C7×D411yescharacteristic
317C22×C4C7×#4111yescharacteristic
318Q16—81no—
319SD16—81no—
320M16—81no—
321M16—81no—
322M16—81no—
323M16—81no—
324C2×C8—21no—
325C2×C8—21no—
326C2×C8—21no—
327C2×C8—21no—
328C2×C8—21no—
329C2×C8—21no—
330C2×C8—41no—
331C2×C8—81no—
332C2×C8—81no—
333C4⋊C4—21no—
334C4⋊C4—21no—
335C4⋊C4—21no—
336C4⋊C4—21no—
337C4⋊C4—41no—
338C4⋊C4—41no—
339C4⋊C4C7×C22⋊C411yescharacteristic
340C4⋊C4C7×C22⋊C411yescharacteristic
341C22⋊C4—21no—
342C22⋊C4—21no—
343C22⋊C4—21no—
344C22⋊C4—21no—
345C22⋊C4—21no—
346C22⋊C4—41no—
347C22⋊C4—41no—
348C22⋊C4—81no—
349C42—21no—
350C2×C7—41no—
351C2×C7—41no—
352C2×C7—41no—
353C2×C7—81no—
354C2×C7#188111yescharacteristic
355C2×C7#190211yescharacteristic
356C2×C7#191311yescharacteristic
357C23—21no—
358C23—41no—
359C23—41no—
360C23—41no—
361C23C4×C7×D411yescharacteristic
362C23C7×#12111yescharacteristic
363C23C7×#12311yescharacteristic
364Q8—41no—
365Q8—41no—
366Q8—41no—
367Q8—81no—
368D4—41no—
369D4—81no—
370D4—81no—
371D4—81no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—42no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—81no—
387C2×C4—81no—
388C2×C4—81no—
389C2×C4—81no—
390C2×C4C2×C7×C22⋊C411yescharacteristic
391C2×C4C4×C7×D411yescharacteristic
392C2×C4C7×#12011yescharacteristic
393C2×C4C7×#12111yescharacteristic
394C8—41no—
395C8—41no—
396C8—41no—
397C8—41no—
398C8—41no—
399C8—41no—
400C8—81no—
401C7#933111yescharacteristic, direct product factor, Sylow
402C22—21no—
403C22—21no—
404C22—21no—
405C22—21no—
406C22—21no—
407C22—21no—
408C22—21no—
409C22—21no—
410C22—21no—
411C22—41no—
412C22—41no—
413C22—41no—
414C22—82no—
415C22C7×#38611yescharacteristic
416C4—21no—
417C4—21no—
418C4—21no—
419C4—21no—
420C4—41no—
421C4—41no—
422C4—41no—
423C4—81no—
424C4—81no—
425C4—161no—
426C2—41no—
427C2—41no—
428C2—41no—
429C2—81no—
430C2C7×#188111yescharacteristic
431C2C7×#190211yescharacteristic
432C2C7×#191311yescharacteristic
subgroupG/H
center, socleC22×C7#386
commutatorC22×C4C22×C4×C7
Frattini, cosocleC23×C4C23×C7
prime radicalC7×#525C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C7 ◁ C23×C4×C7 ◁ G

Complements

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

NH ≅ G/Ntype
4. C7×#1511C2semidirect
5. C7×#1472C2semidirect
6. C7×#1306C2semidirect
7. C7×#1293C2semidirect
11. C7×#484C4semidirect
12. C7×#481C4semidirect
16. C7×#451C4semidirect
17. C7×#448C4semidirect
35. #9331C7direct (N × H)
92. #1511C2×C7semidirect
93. #1472C2×C7semidirect
94. #1306C2×C7semidirect
95. #1293C2×C7semidirect
152. #484C4×C7semidirect
153. #481C4×C7semidirect
157. #451C4×C7semidirect
158. #448C4×C7semidirect
401. C7#9331direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#31962
C7×#29782
C7×#15162
C7×#15112
C7×#14722
C7×#13062
C7×#12932
#93317