Finite Group Databasebeta

← C7×#18461 #13195677

Subgroups — C7×#18461 #13195677

Order 1792.

880 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3563C211yesmaximal, characteristic, complemented
2C7×#3215C211yesmaximal, characteristic, complemented
3C7×#3211C211yesmaximal, characteristic
4C7×#2820C211yesmaximal, characteristic, complemented
5C7×#1605C212yesmaximal, complemented
6C7×#1603C212yesmaximal, complemented
7C7×#1599C212yesmaximal, complemented
8C7×#1597C212yesmaximal
9C7×#1518C211yesmaximal, characteristic, complemented
10C7×#1512C211yesmaximal, characteristic, complemented
11C7×#1470C211yesmaximal, characteristic, complemented
12C7×#564—21no—
13C7×#563—21no—
14C7×#557—21no—
15C7×#554—21no—
16C7×#552—21no—
17C7×#552—22no—
18C7×#552—21no—
19C7×#544—21no—
20C7×#544C2212yes—
21C7×#541—21no—
22C7×#541—22no—
23C7×#539C2212yes—
24C7×#533C2211yescharacteristic
25C7×#529C2211yescharacteristic
26C2×C7×#125C2211yescharacteristic
27C2×C7×#124C2211yescharacteristic
28C7×#497—21no—
29C7×#495—21no—
30C7×#489—21no—
31C7×#488—21no—
32C7×#464—21no—
33C7×#464—22no—
34C7×#463—21no—
35C7×#451C2211yescharacteristic
36C7×#451C2212yes—
37C7×#450C2211yescharacteristic
38C7×#450C2212yes—
39C7×#448C2211yescharacteristic
40C7×#447C2211yescharacteristic
41C7×#432C2211yescharacteristic
42C7×#431C2211yescharacteristic
43C7×#399—21no—
44C7×#397—21no—
45C7×#396—22no—
46C7×#395C2212yes—
47C7×#394C2212yes—
48C7×#382C2211yescharacteristic
49C7×#328C2212yes—
50C7×#328C2212yes—
51C7×#328C2212yes—
52C7×#327C2212yes—
53C7×#324C2212yes—
54C7×#324C2212yes—
55#18461C711yesmaximal, characteristic, direct product factor, Sylow
56C22×C7×Q8C2311yescharacteristic
57C22×C7×D4C2311yescharacteristic
58C2×C7×Q16—21no—
59C2×C7×Q16—22no—
60C2×C7×SD16—21no—
61C2×C7×SD16—22no—
62C2×C7×SD16—21no—
63C2×C7×D8—21no—
64C7×C4⋊Q8—21no—
65C7×C4⋊Q8—22no—
66C7×C4⋊Q8—22no—
67C7×C4⋊Q8D411yescharacteristic
68C7×C4⋊Q8D411yescharacteristic
69C7×#126—22no—
70C7×#126—22no—
71C7×#126—22no—
72C7×#126—22no—
73C7×#126—21no—
74C7×#125—22no—
75C7×#125—22no—
76C7×#125—22no—
77C7×#125—22no—
78C7×#125—21no—
79C7×#125—22no—
80C7×#125—22no—
81C7×#125—22no—
82C7×#125—22no—
83C7×#125—21no—
84C7×#125—21no—
85C7×#125—21no—
86C7×#125—21no—
87C7×#125—21no—
88C7×#125D411yescharacteristic
89C7×#124—21no—
90C7×#124—21no—
91C7×#124—21no—
92C7×#124—21no—
93C7×#124D411yescharacteristic
94C7×#123—22no—
95C7×#123—21no—
96C7×#123—22no—
97C7×C22⋊Q8—21no—
98C7×C22⋊Q8—21no—
99C7×C22⋊Q8—22no—
100C7×C22⋊Q8—22no—
101C7×C22⋊Q8—22no—
102C7×C22⋊Q8—21no—
103C7×C22⋊Q8—22no—
104C7×C22⋊Q8C2311yescharacteristic
105C7×C22⋊Q8C2312yes—
106C7×#121C2311yescharacteristic
107C4×C7×Q8—21no—
108C4×C7×Q8—22no—
109C4×C7×Q8—21no—
110C4×C7×Q8—21no—
111C4×C7×Q8D411yescharacteristic
112C4×C7×Q8D412yes—
113C4×C7×D4—21no—
114C4×C7×D4—22no—
115C4×C7×D4D411yescharacteristic
116C7×#117—22no—
117C7×#117—21no—
118C7×#117—21no—
119C7×#117—22no—
120C7×#117—21no—
121C2×C7×C4⋊C4—21no—
122C2×C7×C4⋊C4—21no—
123C2×C7×C4⋊C4—22no—
124C2×C7×C4⋊C4D412yes—
125C2×C7×C4⋊C4D411yescharacteristic
126C2×C7×C4⋊C4D411yescharacteristic
127C2×C7×C4⋊C4C2312yes—
128C2×C7×C22⋊C4—21no—
129C2×C7×C22⋊C4—21no—
130C2×C7×C22⋊C4—22no—
131C2×C7×C22⋊C4—22no—
132C2×C42×C7C2311yescharacteristic
133C7×C4⋊C8—21no—
134C7×C4⋊C8—21no—
135C7×#103—21no—
136C7×#103—21no—
137C7×#103—22no—
138C7×#102—21no—
139C7×#102—21no—
140C7×#102—22no—
141C7×C22⋊C8C2312yes—
142C7×C22⋊C8C2311yescharacteristic
143C7×C22⋊C8C2311yescharacteristic
144C7×#97—21no—
145C4×C7×C8—21no—
146C7×#95—21no—
147C7×#95—22no—
148C7×#95C2312yes—
149#3563C2×C711yescharacteristic, complemented
150#3215C2×C711yescharacteristic, complemented
151#3211C2×C711yescharacteristic
152#2820C2×C711yescharacteristic, complemented
153#1605C2×C712yescomplemented
154#1603C2×C712yescomplemented
155#1599C2×C712yescomplemented
156#1597C2×C712yes—
157#1518C2×C711yescharacteristic, complemented
158#1512C2×C711yescharacteristic, complemented
159#1470C2×C711yescharacteristic, complemented
160C24×C7—21no—
161C2×C7×Q8—22no—
162C2×C7×Q8—21no—
163C2×C7×Q8—21no—
164C2×C7×Q8—21no—
165C2×C7×Q8—81no—
166C2×C7×Q8C2×D412yes—
167C2×C7×Q8C2×D411yescharacteristic
168C2×C7×D4—21no—
169C2×C7×D4—21no—
170C2×C7×D4—21no—
171C2×C7×D4—81no—
172C2×C7×D4C2×D411yescharacteristic
173C22×C4×C7—22no—
174C22×C4×C7—21no—
175C22×C4×C7—22no—
176C22×C4×C7—21no—
177C22×C4×C7C2×D412yes—
178C22×C4×C7C2×D411yescharacteristic
179C22×C4×C7C2×D412yes—
180C22×C4×C7C2×D411yescharacteristic
181C22×C4×C7C2411yescharacteristic
182C7×Q16—81no—
183C7×Q16—82no—
184C7×SD16—81no—
185C7×SD16—81no—
186C7×SD16—82no—
187C7×D8—81no—
188C2×C7×C8—21no—
189C2×C7×C8—21no—
190C2×C7×C8—22no—
191C7×C4⋊C4—22no—
192C7×C4⋊C4—22no—
193C7×C4⋊C4—22no—
194C7×C4⋊C4—22no—
195C7×C4⋊C4—22no—
196C7×C4⋊C4—22no—
197C7×C4⋊C4—22no—
198C7×C4⋊C4—22no—
199C7×C4⋊C4—21no—
200C7×C4⋊C4—21no—
201C7×C4⋊C4—21no—
202C7×C4⋊C4—22no—
203C7×C4⋊C4—21no—
204C7×C4⋊C4—22no—
205C7×C4⋊C4—21no—
206C7×C4⋊C4—21no—
207C7×C4⋊C4—21no—
208C7×C4⋊C4—21no—
209C7×C4⋊C4—21no—
210C7×C4⋊C4—21no—
211C7×C4⋊C4—22no—
212C7×C4⋊C4—22no—
213C7×C4⋊C4—22no—
214C7×C4⋊C4—21no—
215C7×C4⋊C4—22no—
216C7×C4⋊C4—81no—
217C7×C4⋊C4—82no—
218C7×C4⋊C4C2×D411yescharacteristic
219C7×C4⋊C4C2×D411yescharacteristic
220C7×C4⋊C4C2×D412yes—
221C7×C22⋊C4—22no—
222C7×C22⋊C4—22no—
223C7×C22⋊C4—22no—
224C7×C22⋊C4—22no—
225C7×C22⋊C4—21no—
226C7×C22⋊C4—21no—
227C7×C22⋊C4—21no—
228C7×C22⋊C4—21no—
229C7×C22⋊C4—21no—
230C7×C22⋊C4—41no—
231C7×C22⋊C4—41no—
232C7×C22⋊C4—41no—
233C7×C22⋊C4—41no—
234C7×C22⋊C4—82no—
235C7×C22⋊C4—82no—
236C42×C7—22no—
237C42×C7—22no—
238C42×C7—22no—
239C42×C7—22no—
240C42×C7—21no—
241C42×C7C2×D411yescharacteristic
242C42×C7C2×D411yescharacteristic
243C42×C7C2×D411yescharacteristic
244C42×C7C2×D411yescharacteristic
245#564—21no—
246#563—21no—
247#557—21no—
248#554—21no—
249#552—21no—
250#552—22no—
251#552—21no—
252#544—21no—
253#544C22×C712yes—
254#541—21no—
255#541—22no—
256#539C22×C712yes—
257#533C22×C711yescharacteristic
258#529C22×C711yescharacteristic
259C2×#125C22×C711yescharacteristic
260C2×#124C22×C711yescharacteristic
261#497—21no—
262#495—21no—
263#489—21no—
264#488—21no—
265#464—21no—
266#464—22no—
267#463—21no—
268#451C22×C712yes—
269#451C22×C711yescharacteristic
270#450C22×C712yes—
271#450C22×C711yescharacteristic
272#448C22×C711yescharacteristic
273#447C22×C711yescharacteristic
274#432C22×C711yescharacteristic
275#431C22×C711yescharacteristic
276#399—21no—
277#397—21no—
278#396—22no—
279#395C22×C712yes—
280#394C22×C712yes—
281#382C22×C711yescharacteristic
282#328C22×C712yes—
283#328C22×C712yes—
284#328C22×C712yes—
285#327C22×C712yes—
286#324C22×C712yes—
287#324C22×C712yes—
288C23×C7—21no—
289C23×C7—42no—
290C23×C7—42no—
291C23×C7—41no—
292C23×C7—41no—
293C23×C7—41no—
294C23×C7C22×D411yescharacteristic
295C7×Q8—41no—
296C7×Q8—42no—
297C7×Q8—42no—
298C7×Q8—41no—
299C7×Q8—41no—
300C7×Q8—42no—
301C7×Q8—82no—
302C7×D4—41no—
303C7×D4—41no—
304C7×D4—41no—
305C7×D4—42no—
306C7×D4—41no—
307C7×D4—41no—
308C2×C4×C7—21no—
309C2×C4×C7—21no—
310C2×C4×C7—22no—
311C2×C4×C7—21no—
312C2×C4×C7—21no—
313C2×C4×C7—22no—
314C2×C4×C7—21no—
315C2×C4×C7—22no—
316C2×C4×C7—22no—
317C2×C4×C7—22no—
318C2×C4×C7—22no—
319C2×C4×C7—41no—
320C2×C4×C7—41no—
321C2×C4×C7—42no—
322C2×C4×C7—42no—
323C2×C4×C7—42no—
324C2×C4×C7—42no—
325C2×C4×C7—42no—
326C2×C4×C7—41no—
327C2×C4×C7—42no—
328C2×C4×C7—82no—
329C2×C4×C7#12011yescharacteristic
330C2×C4×C7#12011yescharacteristic
331C2×C4×C7#12011yescharacteristic
332C2×C4×C7#12011yescharacteristic
333C2×C4×C7C22×D411yescharacteristic
334C2×C4×C7C22×D411yescharacteristic
335C7×C8—41no—
336C7×C8—41no—
337C7×C8—42no—
338C22×Q8C23×C711yescharacteristic
339C22×D4C23×C711yescharacteristic
340C2×Q16—21no—
341C2×Q16—22no—
342C2×SD16—21no—
343C2×SD16—22no—
344C2×SD16—21no—
345C2×D8—21no—
346C4⋊Q8—21no—
347C4⋊Q8—22no—
348C4⋊Q8—22no—
349C4⋊Q8C7×D411yescharacteristic
350C4⋊Q8C7×D411yescharacteristic
351#126—22no—
352#126—22no—
353#126—22no—
354#126—22no—
355#126—21no—
356#125—22no—
357#125—22no—
358#125—22no—
359#125—22no—
360#125—21no—
361#125—22no—
362#125—22no—
363#125—22no—
364#125—22no—
365#125—21no—
366#125—21no—
367#125—21no—
368#125—21no—
369#125—21no—
370#125C7×D411yescharacteristic
371#124—21no—
372#124—21no—
373#124—21no—
374#124—21no—
375#124C7×D411yescharacteristic
376#123—22no—
377#123—21no—
378#123—22no—
379C22⋊Q8—21no—
380C22⋊Q8—21no—
381C22⋊Q8—22no—
382C22⋊Q8—22no—
383C22⋊Q8—22no—
384C22⋊Q8—21no—
385C22⋊Q8—22no—
386C22⋊Q8C23×C711yescharacteristic
387C22⋊Q8C23×C712yes—
388#121C23×C711yescharacteristic
389C4×Q8—21no—
390C4×Q8—22no—
391C4×Q8—21no—
392C4×Q8—21no—
393C4×Q8C7×D411yescharacteristic
394C4×Q8C7×D412yes—
395C4×D4—21no—
396C4×D4—22no—
397C4×D4C7×D411yescharacteristic
398#117—22no—
399#117—21no—
400#117—21no—
401#117—22no—
402#117—21no—
403C2×C4⋊C4—21no—
404C2×C4⋊C4—21no—
405C2×C4⋊C4—22no—
406C2×C4⋊C4C7×D412yes—
407C2×C4⋊C4C7×D411yescharacteristic
408C2×C4⋊C4C7×D411yescharacteristic
409C2×C4⋊C4C23×C712yes—
410C2×C22⋊C4—21no—
411C2×C22⋊C4—21no—
412C2×C22⋊C4—22no—
413C2×C22⋊C4—22no—
414C2×C42C23×C711yescharacteristic
415C4⋊C8—21no—
416C4⋊C8—21no—
417#103—21no—
418#103—21no—
419#103—22no—
420#102—21no—
421#102—21no—
422#102—22no—
423C22⋊C8C23×C712yes—
424C22⋊C8C23×C711yescharacteristic
425C22⋊C8C23×C711yescharacteristic
426#97—21no—
427C4×C8—21no—
428#95—21no—
429#95—22no—
430#95C23×C712yes—
431C22×C7—21no—
432C22×C7—21no—
433C22×C7—41no—
434C22×C7—41no—
435C22×C7—41no—
436C22×C7—82no—
437C22×C7—82no—
438C22×C7—81no—
439C22×C7—81no—
440C22×C7—81no—
441C22×C7C2×#12011yescharacteristic
442C22×C7#57711yescharacteristic
443C22×C7#57811yescharacteristic
444C4×C7—21no—
445C4×C7—21no—
446C4×C7—21no—
447C4×C7—21no—
448C4×C7—21no—
449C4×C7—21no—
450C4×C7—42no—
451C4×C7—41no—
452C4×C7—41no—
453C4×C7—42no—
454C4×C7—42no—
455C4×C7—42no—
456C4×C7—41no—
457C4×C7—82no—
458C4×C7—81no—
459C24—21no—
460C2×Q8—22no—
461C2×Q8—21no—
462C2×Q8—21no—
463C2×Q8—21no—
464C2×Q8—81no—
465C2×Q8C2×C7×D412yes—
466C2×Q8C2×C7×D411yescharacteristic
467C2×D4—21no—
468C2×D4—21no—
469C2×D4—21no—
470C2×D4—81no—
471C2×D4C2×C7×D411yescharacteristic
472C22×C4—22no—
473C22×C4—21no—
474C22×C4—22no—
475C22×C4—21no—
476C22×C4C2×C7×D412yes—
477C22×C4C2×C7×D411yescharacteristic
478C22×C4C2×C7×D412yes—
479C22×C4C2×C7×D411yescharacteristic
480C22×C4C24×C711yescharacteristic
481Q16—81no—
482Q16—82no—
483SD16—81no—
484SD16—81no—
485SD16—82no—
486D8—81no—
487C2×C8—21no—
488C2×C8—21no—
489C2×C8—22no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—22no—
502C4⋊C4—21no—
503C4⋊C4—22no—
504C4⋊C4—21no—
505C4⋊C4—21no—
506C4⋊C4—21no—
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—21no—
510C4⋊C4—22no—
511C4⋊C4—22no—
512C4⋊C4—22no—
513C4⋊C4—21no—
514C4⋊C4—22no—
515C4⋊C4—81no—
516C4⋊C4—82no—
517C4⋊C4C2×C7×D411yescharacteristic
518C4⋊C4C2×C7×D411yescharacteristic
519C4⋊C4C2×C7×D412yes—
520C22⋊C4—22no—
521C22⋊C4—22no—
522C22⋊C4—22no—
523C22⋊C4—22no—
524C22⋊C4—21no—
525C22⋊C4—21no—
526C22⋊C4—21no—
527C22⋊C4—21no—
528C22⋊C4—21no—
529C22⋊C4—41no—
530C22⋊C4—41no—
531C22⋊C4—41no—
532C22⋊C4—41no—
533C22⋊C4—82no—
534C22⋊C4—82no—
535C42—22no—
536C42—22no—
537C42—22no—
538C42—22no—
539C42—21no—
540C42C2×C7×D411yescharacteristic
541C42C2×C7×D411yescharacteristic
542C42C2×C7×D411yescharacteristic
543C42C2×C7×D411yescharacteristic
544C2×C7—21no—
545C2×C7—21no—
546C2×C7—81no—
547C2×C7—161no—
548C2×C7#301311yescharacteristic
549C2×C7#302011yescharacteristic
550C2×C7#302711yescharacteristic
551C23—21no—
552C23—42no—
553C23—42no—
554C23—41no—
555C23—41no—
556C23—41no—
557C23C22×C7×D411yescharacteristic
558Q8—41no—
559Q8—42no—
560Q8—42no—
561Q8—41no—
562Q8—41no—
563Q8—42no—
564Q8—82no—
565D4—41no—
566D4—41no—
567D4—41no—
568D4—42no—
569D4—41no—
570D4—41no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—22no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—22no—
577C2×C4—21no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—41no—
590C2×C4—42no—
591C2×C4—82no—
592C2×C4C7×#12011yescharacteristic
593C2×C4C7×#12011yescharacteristic
594C2×C4C7×#12011yescharacteristic
595C2×C4C7×#12011yescharacteristic
596C2×C4C22×C7×D411yescharacteristic
597C2×C4C22×C7×D411yescharacteristic
598C8—41no—
599C8—41no—
600C8—42no—
601C7#1846111yescharacteristic, direct product factor, Sylow
602C22—21no—
603C22—21no—
604C22—41no—
605C22—41no—
606C22—41no—
607C22—82no—
608C22—82no—
609C22—81no—
610C22—81no—
611C22—81no—
612C22C2×C7×#12011yescharacteristic
613C22C7×#57711yescharacteristic
614C22C7×#57811yescharacteristic
615C4—21no—
616C4—21no—
617C4—21no—
618C4—21no—
619C4—21no—
620C4—21no—
621C4—42no—
622C4—41no—
623C4—41no—
624C4—42no—
625C4—42no—
626C4—42no—
627C4—41no—
628C4—82no—
629C4—81no—
630C2—21no—
631C2—21no—
632C2—81no—
633C2—161no—
634C2C7×#301311yescharacteristic
635C2C7×#302011yescharacteristic
636C2C7×#302711yescharacteristic
subgroupG/H
center, socleC22×C7C2×#120
commutator, Frattini, cosocleC22×C4C24×C7
prime radicalC7×#447C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C7 ◁ C2×C42×C7 ◁ G

Complements

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

NH ≅ G/Ntype
1. C7×#3563C2semidirect
2. C7×#3215C2semidirect
4. C7×#2820C2semidirect
5. C7×#1605C2semidirect
6. C7×#1603C2semidirect
7. C7×#1599C2semidirect
9. C7×#1518C2semidirect
10. C7×#1512C2semidirect
11. C7×#1470C2semidirect
55. #18461C7direct (N × H)
149. #3563C2×C7semidirect
150. #3215C2×C7semidirect
152. #2820C2×C7semidirect
153. #1605C2×C7semidirect
154. #1603C2×C7semidirect
155. #1599C2×C7semidirect
157. #1518C2×C7semidirect
158. #1512C2×C7semidirect
159. #1470C2×C7semidirect
601. C7#18461direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#35632
C7×#32152
C7×#32112
C7×#28202
C7×#16052
C7×#16032
C7×#15992
C7×#15972
C7×#15182
C7×#15122
C7×#14702
#184617