Finite Group Databasebeta

← C7×#9278 #13186494

Subgroups — C7×#9278 #13186494

Order 1792.

564 conjugacy classes of subgroups.

The subgroup lattice has 488 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3187C211yesmaximal, characteristic
2C7×#2368C211yesmaximal, characteristic, complemented
3C7×#1964C211yesmaximal, characteristic
4C7×#1520C211yesmaximal, characteristic, complemented
5C2×C7×#327C211yesmaximal, characteristic
6C7×#1294C211yesmaximal, characteristic, complemented
7C7×#1282C211yesmaximal, characteristic, complemented
8C2×C7×C22⋊Q8C2211yescharacteristic
9C2×C7×#121C2211yescharacteristic
10C7×#483C412yescomplemented
11C7×#481—21no—
12C7×#465—22no—
13C7×#450C412yescomplemented
14C7×#448—21no—
15C2×C7×#99—21no—
16C2×C7×#99—21no—
17C2×C7×C22⋊C8C2211yescharacteristic
18C2×C7×C22⋊C8C2211yescharacteristic
19C7×#391—21no—
20C7×#387—21no—
21C7×#387—21no—
22C7×#385—21no—
23C7×#380C2211yescharacteristic
24C4×C7×C4⋊C4—21no—
25C4×C7×C22⋊C4—21no—
26C4×C7×C22⋊C4C2211yescharacteristic
27C2×C7×#95C2211yescharacteristic
28C7×#328—22no—
29C7×#327—21no—
30C7×#327—21no—
31#9278C711yesmaximal, characteristic, direct product factor, Sylow
32C22×C7×Q8D411yescharacteristic
33C22×C7×D4—21no—
34C22×C7×D4C2×C411yescharacteristic
35C23×C4×C7—21no—
36C23×C4×C7C2311yescharacteristic
37C2×C7×SD16—22no—
38C22×C7×C8—21no—
39C22×C7×C8—21no—
40C7×C22⋊Q8—21no—
41C7×C22⋊Q8—21no—
42C7×C22⋊Q8C2×C412yes—
43C7×C22⋊Q8D412yes—
44C7×#121—21no—
45C7×#121—21no—
46C7×#121—21no—
47C7×#121—21no—
48C2×C7×C4⋊C4—21no—
49C2×C7×C4⋊C4—21no—
50C2×C7×C4⋊C4—21no—
51C2×C7×C4⋊C4—21no—
52C2×C7×C4⋊C4C2×C411yescharacteristic
53C2×C7×C4⋊C4D411yescharacteristic
54C2×C7×C22⋊C4—21no—
55C2×C7×C22⋊C4—21no—
56C2×C7×C22⋊C4—21no—
57C2×C7×C22⋊C4D411yescharacteristic
58C2×C7×C22⋊C4D411yescharacteristic
59C2×C42×C7—21no—
60C2×C42×C7—21no—
61C2×C42×C7—21no—
62C7×#106—22no—
63C7×#103—22no—
64C7×#102—22no—
65C7×#99—42no—
66C7×#99—42no—
67C7×C22⋊C8—21no—
68C7×C22⋊C8—21no—
69C7×C22⋊C8—21no—
70C7×C22⋊C8C2×C412yes—
71C7×#95—21no—
72C7×#95—21no—
73C7×#95—21no—
74C7×#95—21no—
75C7×#95—21no—
76C7×#95—21no—
77C7×#95—21no—
78C7×#95D411yescharacteristic
79C7×#95D411yescharacteristic
80#3187C2×C711yescharacteristic
81#2368C2×C711yescharacteristic, complemented
82#1964C2×C711yescharacteristic
83#1520C2×C711yescharacteristic, complemented
84C2×#327C2×C711yescharacteristic
85#1294C2×C711yescharacteristic, complemented
86#1282C2×C711yescharacteristic, complemented
87C24×C7—21no—
88C24×C7C2×D411yescharacteristic
89C2×C7×Q8—21no—
90C2×C7×Q8—81no—
91C2×C7×Q8C22⋊C412yes—
92C2×C7×D4—21no—
93C2×C7×D4—21no—
94C2×C7×D4—41no—
95C2×C7×D4—41no—
96C2×C7×D4—41no—
97C2×C7×D4—41no—
98C2×C7×D4—41no—
99C2×C7×D4—41no—
100C2×C7×D4—81no—
101C22×C4×C7—21no—
102C22×C4×C7—21no—
103C22×C4×C7—21no—
104C22×C4×C7—21no—
105C22×C4×C7—21no—
106C22×C4×C7—21no—
107C22×C4×C7—21no—
108C22×C4×C7—42no—
109C22×C4×C7—41no—
110C22×C4×C7—41no—
111C22×C4×C7—41no—
112C22×C4×C7—81no—
113C22×C4×C7—81no—
114C22×C4×C7C22×C411yescharacteristic
115C22×C4×C7C2×D411yescharacteristic
116C22×C4×C7C2×D411yescharacteristic
117C22×C4×C7C2×D411yescharacteristic
118C22×C4×C7#4111yescharacteristic
119C22×C4×C7#4111yescharacteristic
120C7×SD16—82no—
121C2×C7×C8—22no—
122C2×C7×C8—41no—
123C2×C7×C8—81no—
124C2×C7×C8—81no—
125C7×C4⋊C4—22no—
126C7×C4⋊C4—22no—
127C7×C4⋊C4—21no—
128C7×C4⋊C4—21no—
129C7×C4⋊C4—21no—
130C7×C4⋊C4—42no—
131C7×C4⋊C4—81no—
132C7×C4⋊C4—81no—
133C7×C4⋊C4C22⋊C412yes—
134C7×C22⋊C4—22no—
135C7×C22⋊C4—22no—
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—
143C7×C22⋊C4SD1612yescomplemented
144C42×C7—42no—
145C42×C7—81no—
146C42×C7—81no—
147C2×C22⋊Q8C22×C711yescharacteristic
148C2×#121C22×C711yescharacteristic
149#483C4×C712yescomplemented
150#481—21no—
151#465—22no—
152#450C4×C712yescomplemented
153#448—21no—
154C2×#99—21no—
155C2×#99—21no—
156C2×C22⋊C8C22×C711yescharacteristic
157C2×C22⋊C8C22×C711yescharacteristic
158#391—21no—
159#387—21no—
160#387—21no—
161#385—21no—
162#380C22×C711yescharacteristic
163C4×C4⋊C4—21no—
164C4×C22⋊C4—21no—
165C4×C22⋊C4C22×C711yescharacteristic
166C2×#95C22×C711yescharacteristic
167#328—22no—
168#327—21no—
169#327—21no—
170C23×C7—41no—
171C23×C7—41no—
172C23×C7—41no—
173C23×C7—41no—
174C23×C7—41no—
175C23×C7—81no—
176C23×C7—81no—
177C23×C7C4×D411yescharacteristic
178C23×C7#12011yescharacteristic
179C23×C7#12111yescharacteristic
180C23×C7C2×SD1611yescharacteristic
181C23×C7#13511yescharacteristic
182C23×C7AGL(1,ℤ/8)11yescharacteristic
183C23×C7#13711yescharacteristic
184C7×Q8—42no—
185C7×Q8—81no—
186C7×D4—42no—
187C7×D4—81no—
188C7×D4—81no—
189C7×D4—81no—
190C7×D4—81no—
191C7×D4—81no—
192C2×C4×C7—22no—
193C2×C4×C7—22no—
194C2×C4×C7—22no—
195C2×C4×C7—22no—
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—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—41no—
210C2×C4×C7—41no—
211C2×C4×C7—41no—
212C2×C4×C7—41no—
213C2×C4×C7—41no—
214C2×C4×C7—41no—
215C2×C4×C7—41no—
216C2×C4×C7—41no—
217C2×C4×C7—41no—
218C2×C4×C7—41no—
219C2×C4×C7—82no—
220C2×C4×C7—82no—
221C2×C4×C7—81no—
222C2×C4×C7—81no—
223C2×C4×C7—81no—
224C2×C4×C7C2×C22⋊C411yescharacteristic
225C2×C4×C7C4×D411yescharacteristic
226C2×C4×C7#12111yescharacteristic
227C2×C4×C7#12311yescharacteristic
228C7×C8—42no—
229C7×C8—81no—
230C22×Q8C7×D411yescharacteristic
231C22×D4—21no—
232C22×D4C2×C4×C711yescharacteristic
233C23×C4—21no—
234C23×C4C23×C711yescharacteristic
235C2×SD16—22no—
236C22×C8—21no—
237C22×C8—21no—
238C22⋊Q8—21no—
239C22⋊Q8—21no—
240C22⋊Q8C2×C4×C712yes—
241C22⋊Q8C7×D412yes—
242#121—21no—
243#121—21no—
244#121—21no—
245#121—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4C2×C4×C711yescharacteristic
251C2×C4⋊C4C7×D411yescharacteristic
252C2×C22⋊C4—21no—
253C2×C22⋊C4—21no—
254C2×C22⋊C4—21no—
255C2×C22⋊C4C7×D411yescharacteristic
256C2×C22⋊C4C7×D411yescharacteristic
257C2×C42—21no—
258C2×C42—21no—
259C2×C42—21no—
260#106—22no—
261#103—22no—
262#102—22no—
263#99—42no—
264#99—42no—
265C22⋊C8—21no—
266C22⋊C8—21no—
267C22⋊C8—21no—
268C22⋊C8C2×C4×C712yes—
269#95—21no—
270#95—21no—
271#95—21no—
272#95—21no—
273#95—21no—
274#95—21no—
275#95—21no—
276#95C7×D411yescharacteristic
277#95C7×D411yescharacteristic
278C22×C7—21no—
279C22×C7—21no—
280C22×C7—21no—
281C22×C7—21no—
282C22×C7—21no—
283C22×C7—21no—
284C22×C7—41no—
285C22×C7—41no—
286C22×C7—41no—
287C22×C7—41no—
288C22×C7—81no—
289C22×C7—81no—
290C22×C7—81no—
291C22×C7—161no—
292C22×C7—161no—
293C22×C7#38611yescharacteristic
294C22×C7C4×SD1611yescharacteristic
295C22×C7#44011yescharacteristic
296C22×C7#44811yescharacteristic
297C22×C7#44911yescharacteristic
298C22×C7#46011yescharacteristic
299C22×C7#46411yescharacteristic
300C4×C7—21no—
301C4×C7—21no—
302C4×C7—21no—
303C4×C7—21no—
304C4×C7—42no—
305C4×C7—42no—
306C4×C7—42no—
307C4×C7—42no—
308C4×C7—81no—
309C4×C7—81no—
310C4×C7—81no—
311C4×C7—161no—
312C24—21no—
313C24C2×C7×D411yescharacteristic
314C2×Q8—21no—
315C2×Q8—81no—
316C2×Q8C7×C22⋊C412yes—
317C2×D4—21no—
318C2×D4—21no—
319C2×D4—41no—
320C2×D4—41no—
321C2×D4—41no—
322C2×D4—41no—
323C2×D4—41no—
324C2×D4—41no—
325C2×D4—81no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—42no—
334C22×C4—41no—
335C22×C4—41no—
336C22×C4—41no—
337C22×C4—81no—
338C22×C4—81no—
339C22×C4C22×C4×C711yescharacteristic
340C22×C4C2×C7×D411yescharacteristic
341C22×C4C2×C7×D411yescharacteristic
342C22×C4C2×C7×D411yescharacteristic
343C22×C4C7×#4111yescharacteristic
344C22×C4C7×#4111yescharacteristic
345SD16—82no—
346C2×C8—22no—
347C2×C8—41no—
348C2×C8—81no—
349C2×C8—81no—
350C4⋊C4—22no—
351C4⋊C4—22no—
352C4⋊C4—21no—
353C4⋊C4—21no—
354C4⋊C4—21no—
355C4⋊C4—42no—
356C4⋊C4—81no—
357C4⋊C4—81no—
358C4⋊C4C7×C22⋊C412yes—
359C22⋊C4—22no—
360C22⋊C4—22no—
361C22⋊C4—21no—
362C22⋊C4—21no—
363C22⋊C4—21no—
364C22⋊C4—41no—
365C22⋊C4—41no—
366C22⋊C4—41no—
367C22⋊C4—41no—
368C22⋊C4C7×SD1612yescomplemented
369C42—42no—
370C42—81no—
371C42—81no—
372C2×C7—21no—
373C2×C7—21no—
374C2×C7—21no—
375C2×C7—21no—
376C2×C7—21no—
377C2×C7—21no—
378C2×C7—161no—
379C2×C7#188011yescharacteristic
380C2×C7#190611yescharacteristic
381C2×C7#191311yescharacteristic
382C23—41no—
383C23—41no—
384C23—41no—
385C23—41no—
386C23—41no—
387C23—81no—
388C23—81no—
389C23C4×C7×D411yescharacteristic
390C23C7×#12011yescharacteristic
391C23C7×#12111yescharacteristic
392C23C2×C7×SD1611yescharacteristic
393C23C7×#13511yescharacteristic
394C23C7×AGL(1,ℤ/8)11yescharacteristic
395C23C7×#13711yescharacteristic
396Q8—42no—
397Q8—81no—
398D4—42no—
399D4—81no—
400D4—81no—
401D4—81no—
402D4—81no—
403D4—81no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—82no—
432C2×C4—82no—
433C2×C4—81no—
434C2×C4—81no—
435C2×C4—81no—
436C2×C4C2×C7×C22⋊C411yescharacteristic
437C2×C4C4×C7×D411yescharacteristic
438C2×C4C7×#12111yescharacteristic
439C2×C4C7×#12311yescharacteristic
440C8—42no—
441C8—81no—
442C7#927811yescharacteristic, direct product factor, Sylow
443C22—21no—
444C22—21no—
445C22—21no—
446C22—21no—
447C22—21no—
448C22—21no—
449C22—41no—
450C22—41no—
451C22—41no—
452C22—41no—
453C22—81no—
454C22—81no—
455C22—81no—
456C22—161no—
457C22—161no—
458C22C7×#38611yescharacteristic
459C22C4×C7×SD1611yescharacteristic
460C22C7×#44011yescharacteristic
461C22C7×#44811yescharacteristic
462C22C7×#44911yescharacteristic
463C22C7×#46011yescharacteristic
464C22C7×#46411yescharacteristic
465C4—21no—
466C4—21no—
467C4—21no—
468C4—21no—
469C4—42no—
470C4—42no—
471C4—42no—
472C4—42no—
473C4—81no—
474C4—81no—
475C4—81no—
476C4—161no—
477C2—21no—
478C2—21no—
479C2—21no—
480C2—21no—
481C2—21no—
482C2—21no—
483C2—161no—
484C2C7×#188011yescharacteristic
485C2C7×#190611yescharacteristic
486C2C7×#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
2. C7×#2368C2semidirect
4. C7×#1520C2semidirect
6. C7×#1294C2semidirect
7. C7×#1282C2semidirect
10. C7×#483C4semidirect
13. C7×#450C4semidirect
31. #9278C7direct (N × H)
81. #2368C2×C7semidirect
83. #1520C2×C7semidirect
85. #1294C2×C7semidirect
86. #1282C2×C7semidirect
143. C7×C22⋊C4SD16semidirect
149. #483C4×C7semidirect
152. #450C4×C7semidirect
368. C22⋊C4C7×SD16semidirect
442. C7#9278direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#31872
C7×#23682
C7×#19642
C7×#15202
C2×C7×#3272
C7×#12942
C7×#12822
#92787