Finite Group Databasebeta

← C7×#61303 #13238519

Subgroups — C7×#61303 #13238519

Order 1792.

1840 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3480C211yesmaximal, characteristic, complemented
2C7×Q8×#41C211yesmaximal, characteristic, complemented
3C7×#3365C213yesmaximal, complemented
4C7×#3364C213yesmaximal, complemented
5C7×#3363C211yesmaximal, characteristic, complemented
6C7×#3362C211yesmaximal, characteristic, complemented
7C7×#3361C213yesmaximal, complemented
8C7×#3360C213yesmaximal, complemented
9C7×#3359C211yesmaximal, characteristic, complemented
10C7×#3358C211yesmaximal, characteristic, complemented
11C7×#3131C213yesmaximal, complemented
12C7×#3076C213yesmaximal, complemented
13C7×#3035C213yesmaximal, complemented
14C2×C7×C8×Q8C211yesmaximal, characteristic, complemented
15C4×C7×#135C213yesmaximal, complemented
16C2×C7×#135C2213yes—
17C7×#559C2213yes—
18C7×Q82C2211yescharacteristic
19C7×#557C2211yescharacteristic
20C7×#557C2213yes—
21C7×#556C2213yes—
22C7×#554—23no—
23C7×#554C2213yes—
24C7×#553—23no—
25C7×#550C2211yescharacteristic
26C7×D4×Q8—21no—
27C7×D4×Q8C2211yescharacteristic
28C7×#548—21no—
29C7×#533—23no—
30C7×#533C2213yes—
31C7×#532C2213yes—
32C7×#529—23no—
33C4×C7×#41—21no—
34C4×C7×#41C2213yes—
35C4×C7×#41C2213yes—
36C2×C4×C7×Q8—21no—
37C2×C4×C7×Q8C2211yescharacteristic
38C7×#488C2216yes—
39C7×#488C2216yes—
40C7×#487C2213yes—
41C7×#487C2213yes—
42C7×#486C2213yes—
43C7×#486C2213yes—
44C7×#479C2216yes—
45C7×#478C2216yes—
46C7×#477C2213yes—
47C7×#476C2213yes—
48C7×#475C2213yes—
49C7×#474C2213yes—
50C7×#464C2216yes—
51C7×#463C2216yes—
52C7×#462C2213yes—
53C7×#461C2213yes—
54C7×#460C2213yes—
55C7×#459C2213yes—
56C7×C8×Q8C2211yescharacteristic
57C7×C8×Q8C2211yescharacteristic
58C7×C8×Q8C2213yes—
59C7×C8×Q8C2213yes—
60C4×C7×Q16C2213yes—
61C4×C7×Q16C2213yes—
62C4×C7×SD16C2213yes—
63C4×C7×SD16C2213yes—
64C4×C7×SD16C2213yes—
65C4×C7×SD16C2213yes—
66C4×C7×D8C2213yes—
67C4×C7×D8C2213yes—
68C7×#427C2213yes—
69C2×C7×C4⋊C8C2213yes—
70C7×#416C2211yescharacteristic
71C7×#416C2213yes—
72C7×#416C2216yes—
73C2×C4×C7×C8C2213yes—
74#61303C711yesmaximal, characteristic, direct product factor, Sylow
75C2×C7×#41—21no—
76C2×C7×#41C2313yes—
77C2×C7×#41C2311yescharacteristic
78C22×C7×Q8—21no—
79C22×C7×Q8D411yescharacteristic
80C7×#135—212no—
81C2×C7×Q16C2313yes—
82C2×C7×SD16C2313yes—
83C2×C7×SD16C2313yes—
84C2×C7×D8C2313yes—
85C22×C7×C8C2311yescharacteristic
86C22×C7×C8C2313yes—
87C7×C4⋊Q8—23no—
88C7×C4⋊Q8—23no—
89C7×C4⋊Q8C2313yes—
90C7×C4⋊Q8C2313yes—
91C7×C4⋊Q8C2313yes—
92C7×#127C2313yes—
93C7×#126—26no—
94C7×#125—23no—
95C7×#125—26no—
96C7×#125C2316yes—
97C7×#124—23no—
98C7×#124C2316yes—
99C7×#123—26no—
100C7×C22⋊Q8—23no—
101C7×C22⋊Q8—23no—
102C7×C22⋊Q8—23no—
103C7×C22⋊Q8—23no—
104C7×C22⋊Q8—23no—
105C7×C22⋊Q8—26no—
106C7×#121—23no—
107C7×#121—23no—
108C7×#121—23no—
109C4×C7×Q8—21no—
110C4×C7×Q8—21no—
111C4×C7×Q8—23no—
112C4×C7×Q8—23no—
113C4×C7×Q8—23no—
114C4×C7×Q8D411yescharacteristic
115C4×C7×Q8D413yes—
116C4×C7×Q8C2313yes—
117C4×C7×Q8C2313yes—
118C4×C7×Q8C2313yes—
119C4×C7×Q8C2313yes—
120C4×C7×Q8C2311yescharacteristic
121C4×C7×Q8C2313yes—
122C4×C7×D4—23no—
123C4×C7×D4—23no—
124C4×C7×D4—23no—
125C4×C7×D4—23no—
126C4×C7×D4—23no—
127C4×C7×D4—23no—
128C4×C7×D4—23no—
129C4×C7×D4C2313yes—
130C4×C7×D4C2313yes—
131C4×C7×D4C2313yes—
132C4×C7×D4C2313yes—
133C7×#117—23no—
134C7×#117—23no—
135C7×#117—23no—
136C7×#117C2311yescharacteristic
137C7×#117C2313yes—
138C2×C7×C4⋊C4—23no—
139C2×C7×C4⋊C4D413yes—
140C2×C42×C7—23no—
141C2×C42×C7—23no—
142C2×C42×C7C2313yes—
143C7×#107C2313yes—
144C7×#107C2313yes—
145C7×#106C2313yes—
146C7×#106C2313yes—
147C7×C4⋊C8C2313yes—
148C7×C4⋊C8C2313yes—
149C7×C4⋊C8C2316yes—
150C7×#103C2316yes—
151C7×#103C2316yes—
152C7×#103C2313yes—
153C7×#103C2313yes—
154C7×#103C2311yescharacteristic
155C7×#103C2311yescharacteristic
156C7×#102C2316yes—
157C7×#102C2316yes—
158C7×#102C2313yes—
159C7×#102C2313yes—
160C7×#102C2311yescharacteristic
161C7×#102C2311yescharacteristic
162C4×C7×C8C2313yes—
163C4×C7×C8C2313yes—
164C4×C7×C8C2313yes—
165C4×C7×C8C2313yes—
166#3480C2×C711yescharacteristic, complemented
167Q8×#41C2×C711yescharacteristic, complemented
168#3365C2×C713yescomplemented
169#3364C2×C713yescomplemented
170#3363C2×C711yescharacteristic, complemented
171#3362C2×C711yescharacteristic, complemented
172#3361C2×C713yescomplemented
173#3360C2×C713yescomplemented
174#3359C2×C711yescharacteristic, complemented
175#3358C2×C711yescharacteristic, complemented
176#3131C2×C713yescomplemented
177#3076C2×C713yescomplemented
178#3035C2×C713yescomplemented
179C2×C8×Q8C2×C711yescharacteristic, complemented
180C4×#135C2×C713yescomplemented
181C7×#41—22no—
182C7×#41—26no—
183C7×#41—26no—
184C7×#41—44no—
185C7×#41#4114yescomplemented
186C2×C7×Q8—24no—
187C2×C7×Q8—23no—
188C2×C7×Q8—23no—
189C2×C7×Q8—44no—
190C2×C7×Q8C2×D413yes—
191C2×C7×Q8C2×D411yescharacteristic
192C2×C7×Q8C2413yes—
193C2×C7×Q8C2411yescharacteristic
194C2×C7×D4—21no—
195C2×C7×D4—23no—
196C2×C7×D4—23no—
197C2×C7×D4—23no—
198C2×C7×D4C2413yes—
199C2×C7×D4C2411yescharacteristic
200C22×C4×C7—21no—
201C22×C4×C7—23no—
202C22×C4×C7—23no—
203C22×C4×C7—23no—
204C22×C4×C7—23no—
205C22×C4×C7—23no—
206C22×C4×C7C2×D413yes—
207C22×C4×C7C2×D413yes—
208C22×C4×C7C2411yescharacteristic
209C7×Q16—26no—
210C7×SD16—26no—
211C7×SD16—26no—
212C7×D8—26no—
213C2×C7×C8—26no—
214C2×C7×C8—22no—
215C2×C7×C8C2411yescharacteristic
216C2×C7×C8C2411yescharacteristic
217C2×C7×C8C2413yes—
218C2×C7×C8C2413yes—
219C7×C4⋊C4—26no—
220C7×C4⋊C4—23no—
221C7×C4⋊C4—23no—
222C7×C4⋊C4—23no—
223C7×C4⋊C4—23no—
224C7×C4⋊C4—23no—
225C7×C4⋊C4—23no—
226C7×C4⋊C4—23no—
227C7×C4⋊C4—23no—
228C7×C4⋊C4—26no—
229C7×C4⋊C4C2×D413yes—
230C7×C4⋊C4C2×D413yes—
231C7×C4⋊C4C2×D416yes—
232C7×C4⋊C4C2413yes—
233C7×C4⋊C4C2413yes—
234C7×C4⋊C4C2411yescharacteristic
235C7×C4⋊C4C2411yescharacteristic
236C7×C22⋊C4—21no—
237C7×C22⋊C4—23no—
238C7×C22⋊C4—26no—
239C7×C22⋊C4—23no—
240C7×C22⋊C4—23no—
241C7×C22⋊C4—23no—
242C7×C22⋊C4—23no—
243C7×C22⋊C4—26no—
244C42×C7—23no—
245C42×C7—23no—
246C42×C7—23no—
247C42×C7—23no—
248C42×C7—21no—
249C42×C7—23no—
250C42×C7C2×D413yes—
251C42×C7C2×D413yes—
252C42×C7C2413yes—
253C42×C7C2413yes—
254C2×#135C22×C713yes—
255#559C22×C713yes—
256Q82C22×C711yescharacteristic
257#557C22×C711yescharacteristic
258#557C22×C713yes—
259#556C22×C713yes—
260#554—23no—
261#554C22×C713yes—
262#553—23no—
263#550C22×C711yescharacteristic
264D4×Q8—21no—
265D4×Q8C22×C711yescharacteristic
266#548—21no—
267#533—23no—
268#533C22×C713yes—
269#532C22×C713yes—
270#529—23no—
271C4×#41—21no—
272C4×#41C22×C713yes—
273C4×#41C22×C713yes—
274C2×C4×Q8—21no—
275C2×C4×Q8C22×C711yescharacteristic
276#488C22×C716yes—
277#488C22×C716yes—
278#487C22×C713yes—
279#487C22×C713yes—
280#486C22×C713yes—
281#486C22×C713yes—
282#479C22×C716yes—
283#478C22×C716yes—
284#477C22×C713yes—
285#476C22×C713yes—
286#475C22×C713yes—
287#474C22×C713yes—
288#464C22×C716yes—
289#463C22×C716yes—
290#462C22×C713yes—
291#461C22×C713yes—
292#460C22×C713yes—
293#459C22×C713yes—
294C8×Q8C22×C711yescharacteristic
295C8×Q8C22×C711yescharacteristic
296C8×Q8C22×C713yes—
297C8×Q8C22×C713yes—
298C4×Q16C22×C713yes—
299C4×Q16C22×C713yes—
300C4×SD16C22×C713yes—
301C4×SD16C22×C713yes—
302C4×SD16C22×C713yes—
303C4×SD16C22×C713yes—
304C4×D8C22×C713yes—
305C4×D8C22×C713yes—
306#427C22×C713yes—
307C2×C4⋊C8C22×C713yes—
308#416C22×C716yes—
309#416C22×C713yes—
310#416C22×C711yescharacteristic
311C2×C4×C8C22×C713yes—
312C23×C7—21no—
313C23×C7—23no—
314C23×C7C22×D411yescharacteristic
315C7×Q8—26no—
316C7×Q8—21no—
317C7×Q8—23no—
318C7×Q8—23no—
319C7×Q8—46no—
320C7×Q8—83no—
321C7×Q8#13514yescomplemented
322C7×Q8C2×#4112yes—
323C7×D4—24no—
324C7×D4—23no—
325C7×D4—21no—
326C7×D4—23no—
327C7×D4—46no—
328C7×D4—46no—
329C7×D4—83no—
330C7×D4C2×#4112yes—
331C2×C4×C7—21no—
332C2×C4×C7—24no—
333C2×C4×C7—26no—
334C2×C4×C7—21no—
335C2×C4×C7—21no—
336C2×C4×C7—23no—
337C2×C4×C7—23no—
338C2×C4×C7—23no—
339C2×C4×C7—43no—
340C2×C4×C7—46no—
341C2×C4×C7—46no—
342C2×C4×C7—46no—
343C2×C4×C7—83no—
344C2×C4×C7—83no—
345C2×C4×C7C22×D413yes—
346C2×C4×C7C22×D413yes—
347C2×C4×C7C22×D413yes—
348C2×C4×C7C22×D413yes—
349C2×C4×C7C22×D411yescharacteristic
350C2×C4×C7C2×#4112yes—
351C2×C4×C72−1+412yes—
352C2×C4×C7C2511yescharacteristic
353C7×C8—23no—
354C7×C8—23no—
355C7×C8—21no—
356C7×C8—21no—
357C2×#41—21no—
358C2×#41C23×C713yes—
359C2×#41C23×C711yescharacteristic
360C22×Q8—21no—
361C22×Q8C7×D411yescharacteristic
362#135—212no—
363C2×Q16C23×C713yes—
364C2×SD16C23×C713yes—
365C2×SD16C23×C713yes—
366C2×D8C23×C713yes—
367C22×C8C23×C713yes—
368C22×C8C23×C711yescharacteristic
369C4⋊Q8—23no—
370C4⋊Q8—23no—
371C4⋊Q8C23×C713yes—
372C4⋊Q8C23×C713yes—
373C4⋊Q8C23×C713yes—
374#127C23×C713yes—
375#126—26no—
376#125—23no—
377#125—26no—
378#125C23×C716yes—
379#124—23no—
380#124C23×C716yes—
381#123—26no—
382C22⋊Q8—23no—
383C22⋊Q8—23no—
384C22⋊Q8—23no—
385C22⋊Q8—23no—
386C22⋊Q8—23no—
387C22⋊Q8—26no—
388#121—23no—
389#121—23no—
390#121—23no—
391C4×Q8—21no—
392C4×Q8—21no—
393C4×Q8—23no—
394C4×Q8—23no—
395C4×Q8—23no—
396C4×Q8C7×D411yescharacteristic
397C4×Q8C7×D413yes—
398C4×Q8C23×C713yes—
399C4×Q8C23×C713yes—
400C4×Q8C23×C713yes—
401C4×Q8C23×C713yes—
402C4×Q8C23×C711yescharacteristic
403C4×Q8C23×C713yes—
404C4×D4—23no—
405C4×D4—23no—
406C4×D4—23no—
407C4×D4—23no—
408C4×D4—23no—
409C4×D4—23no—
410C4×D4—23no—
411C4×D4C23×C713yes—
412C4×D4C23×C713yes—
413C4×D4C23×C713yes—
414C4×D4C23×C713yes—
415#117—23no—
416#117—23no—
417#117—23no—
418#117C23×C711yescharacteristic
419#117C23×C713yes—
420C2×C4⋊C4—23no—
421C2×C4⋊C4C7×D413yes—
422C2×C42—23no—
423C2×C42—23no—
424C2×C42C23×C713yes—
425#107C23×C713yes—
426#107C23×C713yes—
427#106C23×C713yes—
428#106C23×C713yes—
429C4⋊C8C23×C713yes—
430C4⋊C8C23×C713yes—
431C4⋊C8C23×C716yes—
432#103C23×C711yescharacteristic
433#103C23×C711yescharacteristic
434#103C23×C713yes—
435#103C23×C713yes—
436#103C23×C716yes—
437#103C23×C716yes—
438#102C23×C711yescharacteristic
439#102C23×C711yescharacteristic
440#102C23×C713yes—
441#102C23×C713yes—
442#102C23×C716yes—
443#102C23×C716yes—
444C4×C8C23×C713yes—
445C4×C8C23×C713yes—
446C4×C8C23×C713yes—
447C4×C8C23×C713yes—
448C22×C7—21no—
449C22×C7—21no—
450C22×C7—22no—
451C22×C7—41no—
452C22×C7—41no—
453C22×C7—43no—
454C22×C7—43no—
455C22×C7—43no—
456C22×C7#54812yes—
457C22×C7C23×D411yescharacteristic
458C4×C7—23no—
459C4×C7—23no—
460C4×C7—22no—
461C4×C7—23no—
462C4×C7—41no—
463C4×C7—43no—
464C4×C7—43no—
465C4×C7—41no—
466C4×C7—43no—
467C4×C7#54812yes—
468C4×C7C2×#13516yes—
469C4×C7C22×#4111yescharacteristic
470C4×C7C2×2−1+411yescharacteristic
471#41—22no—
472#41—26no—
473#41—26no—
474#41—44no—
475#41C7×#4114yescomplemented
476C2×Q8—24no—
477C2×Q8—23no—
478C2×Q8—23no—
479C2×Q8—44no—
480C2×Q8C2×C7×D413yes—
481C2×Q8C2×C7×D411yescharacteristic
482C2×Q8C24×C713yes—
483C2×Q8C24×C711yescharacteristic
484C2×D4—21no—
485C2×D4—23no—
486C2×D4—23no—
487C2×D4—23no—
488C2×D4C24×C713yes—
489C2×D4C24×C711yescharacteristic
490C22×C4—21no—
491C22×C4—23no—
492C22×C4—23no—
493C22×C4—23no—
494C22×C4—23no—
495C22×C4—23no—
496C22×C4C2×C7×D413yes—
497C22×C4C2×C7×D413yes—
498C22×C4C24×C711yescharacteristic
499Q16—26no—
500SD16—26no—
501SD16—26no—
502D8—26no—
503C2×C8—26no—
504C2×C8—22no—
505C2×C8C24×C711yescharacteristic
506C2×C8C24×C711yescharacteristic
507C2×C8C24×C713yes—
508C2×C8C24×C713yes—
509C4⋊C4—26no—
510C4⋊C4—23no—
511C4⋊C4—23no—
512C4⋊C4—23no—
513C4⋊C4—23no—
514C4⋊C4—23no—
515C4⋊C4—23no—
516C4⋊C4—23no—
517C4⋊C4—23no—
518C4⋊C4—26no—
519C4⋊C4C2×C7×D413yes—
520C4⋊C4C2×C7×D413yes—
521C4⋊C4C2×C7×D416yes—
522C4⋊C4C24×C713yes—
523C4⋊C4C24×C713yes—
524C4⋊C4C24×C711yescharacteristic
525C4⋊C4C24×C711yescharacteristic
526C22⋊C4—21no—
527C22⋊C4—23no—
528C22⋊C4—26no—
529C22⋊C4—23no—
530C22⋊C4—23no—
531C22⋊C4—23no—
532C22⋊C4—23no—
533C22⋊C4—26no—
534C42—21no—
535C42—23no—
536C42—23no—
537C42—23no—
538C42—23no—
539C42—23no—
540C42C2×C7×D413yes—
541C42C2×C7×D413yes—
542C42C24×C713yes—
543C42C24×C713yes—
544C2×C7—22no—
545C2×C7—42no—
546C2×C7—83no—
547C2×C7C2×#54811yescharacteristic
548C2×C7C22×#13511yescharacteristic
549C2×C7#358411yescharacteristic
550C23—21no—
551C23—23no—
552C23C22×C7×D411yescharacteristic
553Q8—21no—
554Q8—26no—
555Q8—23no—
556Q8—23no—
557Q8—46no—
558Q8—83no—
559Q8C7×#13514yescomplemented
560Q8C2×C7×#4112yes—
561D4—24no—
562D4—23no—
563D4—21no—
564D4—23no—
565D4—46no—
566D4—46no—
567D4—83no—
568D4C2×C7×#4112yes—
569C2×C4—21no—
570C2×C4—24no—
571C2×C4—26no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—23no—
575C2×C4—23no—
576C2×C4—23no—
577C2×C4—43no—
578C2×C4—46no—
579C2×C4—46no—
580C2×C4—46no—
581C2×C4—83no—
582C2×C4—83no—
583C2×C4C22×C7×D413yes—
584C2×C4C22×C7×D413yes—
585C2×C4C22×C7×D413yes—
586C2×C4C22×C7×D411yescharacteristic
587C2×C4C22×C7×D413yes—
588C2×C4C2×C7×#4112yes—
589C2×C4C7×2−1+412yes—
590C2×C4C25×C711yescharacteristic
591C8—23no—
592C8—23no—
593C8—21no—
594C8—21no—
595C7#6130311yescharacteristic, direct product factor, Sylow
596C22—21no—
597C22—21no—
598C22—22no—
599C22—41no—
600C22—41no—
601C22—43no—
602C22—43no—
603C22—43no—
604C22C7×#54812yes—
605C22C23×C7×D411yescharacteristic
606C4—23no—
607C4—23no—
608C4—23no—
609C4—22no—
610C4—41no—
611C4—43no—
612C4—41no—
613C4—43no—
614C4—43no—
615C4C7×#54812yes—
616C4C2×C7×#13516yes—
617C4C22×C7×#4111yescharacteristic
618C4C2×C7×2−1+411yescharacteristic
619C2—22no—
620C2—42no—
621C2—83no—
622C2C2×C7×#54811yescharacteristic
623C2C22×C7×#13511yescharacteristic
624C2C7×#358411yescharacteristic
subgroupG/H
centerC2×C4×C7C22×D4
commutator, Frattini, cosocleC2×C4C25×C7
socleC22×C7C23×D4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4×C7 ◁ C2×C4×C7×Q8 ◁ G

Complements

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

NH ≅ G/Ntype
1. C7×#3480C2semidirect
2. C7×Q8×#41C2semidirect
3. C7×#3365C2semidirect
4. C7×#3364C2semidirect
5. C7×#3363C2semidirect
6. C7×#3362C2semidirect
7. C7×#3361C2semidirect
8. C7×#3360C2semidirect
9. C7×#3359C2semidirect
10. C7×#3358C2semidirect
11. C7×#3131C2semidirect
12. C7×#3076C2semidirect
13. C7×#3035C2semidirect
14. C2×C7×C8×Q8C2semidirect
15. C4×C7×#135C2semidirect
74. #61303C7direct (N × H)
166. #3480C2×C7semidirect
167. Q8×#41C2×C7semidirect
168. #3365C2×C7semidirect
169. #3364C2×C7semidirect
170. #3363C2×C7semidirect
171. #3362C2×C7semidirect
172. #3361C2×C7semidirect
173. #3360C2×C7semidirect
174. #3359C2×C7semidirect
175. #3358C2×C7semidirect
176. #3131C2×C7semidirect
177. #3076C2×C7semidirect
178. #3035C2×C7semidirect
179. C2×C8×Q8C2×C7semidirect
180. C4×#135C2×C7semidirect
185. C7×#41#41semidirect
321. C7×Q8#135semidirect
475. #41C7×#41semidirect
559. Q8C7×#135semidirect
595. C7#61303direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#34802
C7×Q8×#412
C7×#33652
C7×#33642
C7×#33632
C7×#33622
C7×#33612
C7×#33602
C7×#33592
C7×#33582
C7×#31312
C7×#30762
C7×#30352
C2×C7×C8×Q82
C4×C7×#1352
#613037