Finite Group Databasebeta

← C7×#9341 #13186557

Subgroups — C7×#9341 #13186557

Order 1792.

534 conjugacy classes of subgroups.

The subgroup lattice has 460 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3193C211yesmaximal, characteristic
2C7×#2377C211yesmaximal, characteristic
3C7×#1996C211yesmaximal, characteristic, complemented
4C7×#1511C211yesmaximal, characteristic, complemented
5C7×#1507C211yesmaximal, characteristic
6C7×#1305C211yesmaximal, characteristic, complemented
7C7×#1284C211yesmaximal, characteristic, complemented
8C2×C7×C22⋊Q8C2211yescharacteristic
9C2×C7×#121C2211yescharacteristic
10C7×#484—21no—
11C7×#480C412yescomplemented
12C7×#468—22no—
13C7×#450—21no—
14C7×#448C412yescomplemented
15C7×#407C2211yescharacteristic
16C7×#407C2211yescharacteristic
17C7×#390—21no—
18C7×#388—21no—
19C7×#387—21no—
20C7×#386—21no—
21C7×#386—21no—
22C7×#385—21no—
23C7×#381C2211yescharacteristic
24C7×#380C2211yescharacteristic
25C4×C7×C22⋊C4C2211yescharacteristic
26C7×#342—21no—
27C7×#342—21no—
28C7×#328—21no—
29C7×#328—21no—
30C7×#327—22no—
31#9341C711yesmaximal, characteristic, direct product factor, Sylow
32C22×C7×Q8C2×C411yescharacteristic, complemented
33C22×C7×D4—21no—
34C22×C7×D4D411yescharacteristic
35C23×C4×C7—21no—
36C23×C4×C7C2311yescharacteristic
37C2×C7×SD16—22no—
38C2×C7×M16—21no—
39C2×C7×M16—21no—
40C7×C22⋊Q8—21no—
41C7×C22⋊Q8—21no—
42C7×C22⋊Q8—21no—
43C7×C22⋊Q8—21no—
44C7×#121—21no—
45C7×#121—21no—
46C7×#121C2×C412yes—
47C7×#121D412yes—
48C2×C7×C4⋊C4—21no—
49C2×C7×C4⋊C4—21no—
50C2×C7×C4⋊C4—21no—
51C2×C7×C4⋊C4C2×C411yescharacteristic, complemented
52C2×C7×C4⋊C4D411yescharacteristic
53C2×C7×C22⋊C4—21no—
54C2×C7×C22⋊C4—21no—
55C2×C7×C22⋊C4—21no—
56C2×C7×C22⋊C4—21no—
57C2×C7×C22⋊C4—21no—
58C2×C7×C22⋊C4—21no—
59C2×C7×C22⋊C4—21no—
60C2×C7×C22⋊C4D411yescharacteristic
61C2×C7×C22⋊C4D411yescharacteristic
62C2×C42×C7—21no—
63C2×C42×C7—21no—
64C7×#107—22no—
65C7×#103—22no—
66C7×#102—22no—
67C7×C22⋊C8—21no—
68C7×C22⋊C8—21no—
69C7×C22⋊C8—21no—
70C7×C22⋊C8C2×C412yescomplemented
71C7×#95—21no—
72C7×#95—21no—
73C7×#95—21no—
74C7×#95—21no—
75C7×#95—21no—
76C7×#95D411yescharacteristic
77C7×#95D411yescharacteristic
78#3193C2×C711yescharacteristic
79#2377C2×C711yescharacteristic
80#1996C2×C711yescharacteristic, complemented
81#1511C2×C711yescharacteristic, complemented
82#1507C2×C711yescharacteristic
83#1305C2×C711yescharacteristic, complemented
84#1284C2×C711yescharacteristic, complemented
85C24×C7—21no—
86C24×C7C2×D411yescharacteristic
87C2×C7×Q8—21no—
88C2×C7×Q8—21no—
89C2×C7×Q8—81no—
90C2×C7×D4—22no—
91C2×C7×D4—22no—
92C2×C7×D4—21no—
93C2×C7×D4—81no—
94C2×C7×D4—81no—
95C2×C7×D4—81no—
96C2×C7×D4C22⋊C412yes—
97C22×C4×C7—21no—
98C22×C4×C7—21no—
99C22×C4×C7—21no—
100C22×C4×C7—21no—
101C22×C4×C7—21no—
102C22×C4×C7—21no—
103C22×C4×C7—21no—
104C22×C4×C7—41no—
105C22×C4×C7—81no—
106C22×C4×C7—81no—
107C22×C4×C7—81no—
108C22×C4×C7—81no—
109C22×C4×C7C22×C411yescharacteristic
110C22×C4×C7C2×D411yescharacteristic
111C22×C4×C7C2×D411yescharacteristic
112C22×C4×C7C2×D411yescharacteristic
113C22×C4×C7#4111yescharacteristic
114C22×C4×C7#4111yescharacteristic
115C7×SD16—82no—
116C7×M16—81no—
117C7×M16—81no—
118C2×C7×C8—22no—
119C2×C7×C8—41no—
120C7×C4⋊C4—21no—
121C7×C4⋊C4—21no—
122C7×C4⋊C4—21no—
123C7×C4⋊C4—41no—
124C7×C4⋊C4—41no—
125C7×C4⋊C4—81no—
126C7×C4⋊C4—81no—
127C7×C4⋊C4C22⋊C412yes—
128C7×C22⋊C4—22no—
129C7×C22⋊C4—22no—
130C7×C22⋊C4—22no—
131C7×C22⋊C4—41no—
132C7×C22⋊C4—41no—
133C7×C22⋊C4—41no—
134C7×C22⋊C4—41no—
135C7×C22⋊C4—41no—
136C7×C22⋊C4—81no—
137C7×C22⋊C4—81no—
138C7×C22⋊C4—81no—
139C7×C22⋊C4—81no—
140C42×C7—81no—
141C42×C7—81no—
142C2×C22⋊Q8C22×C711yescharacteristic
143C2×#121C22×C711yescharacteristic
144#484—21no—
145#480C4×C712yescomplemented
146#468—22no—
147#450—21no—
148#448C4×C712yescomplemented
149#407C22×C711yescharacteristic
150#407C22×C711yescharacteristic
151#390—21no—
152#388—21no—
153#387—21no—
154#386—21no—
155#386—21no—
156#385—21no—
157#381C22×C711yescharacteristic
158#380C22×C711yescharacteristic
159C4×C22⋊C4C22×C711yescharacteristic
160#342—21no—
161#342—21no—
162#328—21no—
163#328—21no—
164#327—22no—
165C23×C7—21no—
166C23×C7—21no—
167C23×C7—22no—
168C23×C7—41no—
169C23×C7—41no—
170C23×C7—41no—
171C23×C7—41no—
172C23×C7—41no—
173C23×C7—41no—
174C23×C7—41no—
175C23×C7—41no—
176C23×C7C4×D411yescharacteristic
177C23×C7#12011yescharacteristic
178C23×C7#12111yescharacteristic
179C7×Q8—42no—
180C7×Q8—81no—
181C7×D4—42no—
182C7×D4—81no—
183C7×D4—82no—
184C7×D4—82no—
185C2×C4×C7—22no—
186C2×C4×C7—41no—
187C2×C4×C7—41no—
188C2×C4×C7—41no—
189C2×C4×C7—41no—
190C2×C4×C7—41no—
191C2×C4×C7—41no—
192C2×C4×C7—41no—
193C2×C4×C7—41no—
194C2×C4×C7—41no—
195C2×C4×C7—41no—
196C2×C4×C7—42no—
197C2×C4×C7—41no—
198C2×C4×C7—41no—
199C2×C4×C7—41no—
200C2×C4×C7—41no—
201C2×C4×C7—41no—
202C2×C4×C7—41no—
203C2×C4×C7—41no—
204C2×C4×C7—41no—
205C2×C4×C7—41no—
206C2×C4×C7—41no—
207C2×C4×C7—41no—
208C2×C4×C7—41no—
209C2×C4×C7—81no—
210C2×C4×C7—81no—
211C2×C4×C7—84no—
212C2×C4×C7—81no—
213C2×C4×C7—81no—
214C2×C4×C7C2×C22⋊C411yescharacteristic
215C2×C4×C7C4×D411yescharacteristic
216C2×C4×C7#12111yescharacteristic
217C2×C4×C7#12311yescharacteristic
218C7×C8—42no—
219C7×C8—81no—
220C22×Q8C2×C4×C711yescharacteristic, complemented
221C22×D4—21no—
222C22×D4C7×D411yescharacteristic
223C23×C4—21no—
224C23×C4C23×C711yescharacteristic
225C2×SD16—22no—
226C2×M16—21no—
227C2×M16—21no—
228C22⋊Q8—21no—
229C22⋊Q8—21no—
230C22⋊Q8—21no—
231C22⋊Q8—21no—
232#121—21no—
233#121—21no—
234#121C2×C4×C712yes—
235#121C7×D412yes—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4C2×C4×C711yescharacteristic, complemented
240C2×C4⋊C4C7×D411yescharacteristic
241C2×C22⋊C4—21no—
242C2×C22⋊C4—21no—
243C2×C22⋊C4—21no—
244C2×C22⋊C4—21no—
245C2×C22⋊C4—21no—
246C2×C22⋊C4—21no—
247C2×C22⋊C4—21no—
248C2×C22⋊C4C7×D411yescharacteristic
249C2×C22⋊C4C7×D411yescharacteristic
250C2×C42—21no—
251C2×C42—21no—
252#107—22no—
253#103—22no—
254#102—22no—
255C22⋊C8—21no—
256C22⋊C8—21no—
257C22⋊C8—21no—
258C22⋊C8C2×C4×C712yescomplemented
259#95—21no—
260#95—21no—
261#95—21no—
262#95—21no—
263#95—21no—
264#95C7×D411yescharacteristic
265#95C7×D411yescharacteristic
266C22×C7—21no—
267C22×C7—21no—
268C22×C7—21no—
269C22×C7—21no—
270C22×C7—21no—
271C22×C7—21no—
272C22×C7—21no—
273C22×C7—21no—
274C22×C7—42no—
275C22×C7—42no—
276C22×C7—42no—
277C22×C7—81no—
278C22×C7—81no—
279C22×C7—81no—
280C22×C7—81no—
281C22×C7—81no—
282C22×C7—81no—
283C22×C7#38611yescharacteristic
284C22×C7#45511yescharacteristic
285C22×C7#45611yescharacteristic
286C4×C7—21no—
287C4×C7—21no—
288C4×C7—21no—
289C4×C7—21no—
290C4×C7—42no—
291C4×C7—81no—
292C4×C7—81no—
293C4×C7—81no—
294C4×C7—81no—
295C4×C7—82no—
296C4×C7—81no—
297C4×C7—81no—
298C24—21no—
299C24C2×C7×D411yescharacteristic
300C2×Q8—21no—
301C2×Q8—21no—
302C2×Q8—81no—
303C2×D4—22no—
304C2×D4—22no—
305C2×D4—21no—
306C2×D4—81no—
307C2×D4—81no—
308C2×D4—81no—
309C2×D4C7×C22⋊C412yes—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—41no—
318C22×C4—81no—
319C22×C4—81no—
320C22×C4—81no—
321C22×C4—81no—
322C22×C4C22×C4×C711yescharacteristic
323C22×C4C2×C7×D411yescharacteristic
324C22×C4C2×C7×D411yescharacteristic
325C22×C4C2×C7×D411yescharacteristic
326C22×C4C7×#4111yescharacteristic
327C22×C4C7×#4111yescharacteristic
328SD16—82no—
329M16—81no—
330M16—81no—
331C2×C8—22no—
332C2×C8—41no—
333C4⋊C4—21no—
334C4⋊C4—21no—
335C4⋊C4—21no—
336C4⋊C4—41no—
337C4⋊C4—41no—
338C4⋊C4—81no—
339C4⋊C4—81no—
340C4⋊C4C7×C22⋊C412yes—
341C22⋊C4—22no—
342C22⋊C4—22no—
343C22⋊C4—22no—
344C22⋊C4—41no—
345C22⋊C4—41no—
346C22⋊C4—41no—
347C22⋊C4—41no—
348C22⋊C4—41no—
349C22⋊C4—81no—
350C22⋊C4—81no—
351C22⋊C4—81no—
352C22⋊C4—81no—
353C42—81no—
354C42—81no—
355C2×C7—21no—
356C2×C7—21no—
357C2×C7—41no—
358C2×C7—41no—
359C2×C7—82no—
360C2×C7#188211yescharacteristic
361C2×C7#190411yescharacteristic
362C2×C7#191311yescharacteristic
363C23—21no—
364C23—21no—
365C23—22no—
366C23—41no—
367C23—41no—
368C23—41no—
369C23—41no—
370C23—41no—
371C23—41no—
372C23—41no—
373C23—41no—
374C23C4×C7×D411yescharacteristic
375C23C7×#12011yescharacteristic
376C23C7×#12111yescharacteristic
377Q8—42no—
378Q8—81no—
379D4—42no—
380D4—81no—
381D4—82no—
382D4—82no—
383C2×C4—22no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—42no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—81no—
408C2×C4—81no—
409C2×C4—84no—
410C2×C4—81no—
411C2×C4—81no—
412C2×C4C2×C7×C22⋊C411yescharacteristic
413C2×C4C4×C7×D411yescharacteristic
414C2×C4C7×#12111yescharacteristic
415C2×C4C7×#12311yescharacteristic
416C8—42no—
417C8—81no—
418C7#934111yescharacteristic, direct product factor, Sylow
419C22—21no—
420C22—21no—
421C22—21no—
422C22—21no—
423C22—21no—
424C22—21no—
425C22—21no—
426C22—21no—
427C22—42no—
428C22—42no—
429C22—42no—
430C22—81no—
431C22—81no—
432C22—81no—
433C22—81no—
434C22—81no—
435C22—81no—
436C22C7×#38611yescharacteristic
437C22C7×#45511yescharacteristic
438C22C7×#45611yescharacteristic
439C4—21no—
440C4—21no—
441C4—21no—
442C4—21no—
443C4—42no—
444C4—81no—
445C4—81no—
446C4—81no—
447C4—81no—
448C4—82no—
449C4—81no—
450C4—81no—
451C2—21no—
452C2—21no—
453C2—41no—
454C2—41no—
455C2—82no—
456C2C7×#188211yescharacteristic
457C2C7×#190411yescharacteristic
458C2C7×#191311yescharacteristic
subgroupG/H
center, socleC22×C7#386
commutatorC22×C4C22×C4×C7
Frattini, cosocleC23×C4C23×C7
prime radicalC2×C7×#121C22
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
3. C7×#1996C2semidirect
4. C7×#1511C2semidirect
6. C7×#1305C2semidirect
7. C7×#1284C2semidirect
11. C7×#480C4semidirect
14. C7×#448C4semidirect
31. #9341C7direct (N × H)
32. C22×C7×Q8C2×C4semidirect
51. C2×C7×C4⋊C4C2×C4semidirect
70. C7×C22⋊C8C2×C4semidirect
80. #1996C2×C7semidirect
81. #1511C2×C7semidirect
83. #1305C2×C7semidirect
84. #1284C2×C7semidirect
145. #480C4×C7semidirect
148. #448C4×C7semidirect
220. C22×Q8C2×C4×C7semidirect
239. C2×C4⋊C4C2×C4×C7semidirect
258. C22⋊C8C2×C4×C7semidirect
418. C7#9341direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#31932
C7×#23772
C7×#19962
C7×#15112
C7×#15072
C7×#13052
C7×#12842
#93417