Finite Group Databasebeta

← C7×#18341 #13195557

Subgroups — C7×#18341 #13195557

Order 1792.

898 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3455C211yesmaximal, characteristic
2C7×#3236C211yesmaximal, characteristic, complemented
3C7×#3234C211yesmaximal, characteristic, complemented
4C7×#2611C211yesmaximal, characteristic, complemented
5C7×#1618C212yesmaximal, complemented
6C7×#1615C212yesmaximal, complemented
7C7×#1612C212yesmaximal, complemented
8C7×#1609C212yesmaximal, complemented
9C7×#1506C211yesmaximal, characteristic, complemented
10C7×#1505C211yesmaximal, characteristic, complemented
11C7×#1464C211yesmaximal, characteristic, complemented
12C7×#542—21no—
13C7×#542—21no—
14C7×#541—21no—
15C7×#540—21no—
16C7×#539—21no—
17C7×#539C2212yes—
18C7×#538—21no—
19C7×#538C2212yes—
20C7×#533C2211yescharacteristic
21C7×#532C2211yescharacteristic
22C7×#529—22no—
23C7×#529—22no—
24C2×C7×#125C2211yescharacteristic
25C7×#525—21no—
26C4×C7×#41—21no—
27C2×C7×#117C2211yescharacteristic
28C7×#501—21no—
29C7×#501—21no—
30C7×#491—21no—
31C7×#490—21no—
32C7×#484C2212yes—
33C7×#483C2212yes—
34C7×#481C2212yes—
35C7×#480C2212yes—
36C7×#479—22no—
37C7×#478—22no—
38C7×#432C2211yescharacteristic
39C7×#432C2211yescharacteristic
40C7×#400C2211yescharacteristic
41C7×#399C2211yescharacteristic
42C7×#398C2211yescharacteristic
43C7×#397C2211yescharacteristic
44C7×#389—21no—
45C7×#388—21no—
46C7×#387—21no—
47C7×#385—21no—
48C7×#376C2211yescharacteristic
49C7×#328C2212yes—
50C7×#328C2212yes—
51C7×#327C2212yes—
52C7×#327C2212yes—
53C7×#324C2212yes—
54C7×#323C2212yes—
55#18341C711yesmaximal, characteristic, direct product factor, Sylow
56C2×C7×#41—21no—
57C23×C4×C7D411yescharacteristic
58C7×C4⋊Q8D411yescharacteristic
59C7×C4⋊Q8D411yescharacteristic
60C7×C4⋊Q8D411yescharacteristic
61C7×#127D411yescharacteristic
62C7×#126—22no—
63C7×#126—22no—
64C7×#126—22no—
65C7×#126—22no—
66C7×#125—22no—
67C7×#125—22no—
68C7×#125—21no—
69C7×#125—21no—
70C7×#125—21no—
71C7×#125—21no—
72C7×#125—21no—
73C7×#124—22no—
74C7×#124—22no—
75C7×#124—21no—
76C7×#123—22no—
77C7×#123—22no—
78C7×#123—22no—
79C7×#123—22no—
80C7×#123—22no—
81C7×C22⋊Q8—21no—
82C7×C22⋊Q8—21no—
83C7×C22⋊Q8—22no—
84C7×C22⋊Q8—22no—
85C7×C22⋊Q8—21no—
86C7×C22⋊Q8C2312yes—
87C7×#121—21no—
88C7×#121—21no—
89C7×#121—22no—
90C7×#121—22no—
91C7×#121—21no—
92C7×#121C2312yes—
93C7×#120—22no—
94C4×C7×Q8—21no—
95C4×C7×Q8—21no—
96C4×C7×Q8—21no—
97C4×C7×Q8D412yes—
98C4×C7×D4—21no—
99C4×C7×D4—21no—
100C4×C7×D4—22no—
101C4×C7×D4—22no—
102C4×C7×D4—22no—
103C4×C7×D4—22no—
104C4×C7×D4—21no—
105C4×C7×D4D412yes—
106C7×#117—22no—
107C7×#117—21no—
108C7×#117—21no—
109C7×#117—21no—
110C7×#117—21no—
111C7×#117—21no—
112C2×C7×C4⋊C4—21no—
113C2×C7×C4⋊C4—21no—
114C2×C7×C4⋊C4—21no—
115C2×C7×C4⋊C4—21no—
116C2×C7×C4⋊C4D411yescharacteristic
117C2×C7×C4⋊C4C2311yescharacteristic
118C2×C7×C4⋊C4C2311yescharacteristic
119C2×C7×C4⋊C4C2311yescharacteristic
120C2×C7×C22⋊C4—21no—
121C2×C7×C22⋊C4—21no—
122C2×C7×C22⋊C4D411yescharacteristic
123C2×C7×C22⋊C4C2311yescharacteristic
124C2×C42×C7—22no—
125C2×C42×C7—21no—
126C2×C42×C7D411yescharacteristic
127C2×C42×C7C2311yescharacteristic
128C7×#107—22no—
129C7×#107—22no—
130C7×#106—22no—
131C7×#106—22no—
132C7×C4⋊C8—21no—
133C7×C4⋊C8—21no—
134C7×#103—22no—
135C7×#103—22no—
136C7×#102—22no—
137C7×#102—22no—
138C7×C22⋊C8C2312yes—
139C7×C22⋊C8C2312yes—
140C7×#97—21no—
141C7×#97—21no—
142C7×#95—21no—
143C7×#95—21no—
144C7×#95—21no—
145C7×#95—21no—
146C7×#95—21no—
147C7×#95C2311yescharacteristic
148C7×#95C2311yescharacteristic
149#3455C2×C711yescharacteristic
150#3236C2×C711yescharacteristic, complemented
151#3234C2×C711yescharacteristic, complemented
152#2611C2×C711yescharacteristic, complemented
153#1618C2×C712yescomplemented
154#1615C2×C712yescomplemented
155#1612C2×C712yescomplemented
156#1609C2×C712yescomplemented
157#1506C2×C711yescharacteristic, complemented
158#1505C2×C711yescharacteristic, complemented
159#1464C2×C711yescharacteristic, complemented
160C24×C7C2×D411yescharacteristic
161C7×#41—42no—
162C7×#41—42no—
163C2×C7×Q8—21no—
164C2×C7×Q8—21no—
165C2×C7×Q8C2×D412yes—
166C2×C7×D4—22no—
167C2×C7×D4—22no—
168C2×C7×D4—22no—
169C2×C7×D4—21no—
170C2×C7×D4—21no—
171C2×C7×D4C2×D412yes—
172C22×C4×C7—21no—
173C22×C4×C7—22no—
174C22×C4×C7—22no—
175C22×C4×C7—22no—
176C22×C4×C7—21no—
177C22×C4×C7—21no—
178C22×C4×C7—21no—
179C22×C4×C7—21no—
180C22×C4×C7—21no—
181C22×C4×C7—41no—
182C22×C4×C7—41no—
183C22×C4×C7C2×D411yescharacteristic
184C22×C4×C7C2×D411yescharacteristic
185C22×C4×C7C2×D411yescharacteristic
186C22×C4×C7C2×D411yescharacteristic
187C22×C4×C7C2×D411yescharacteristic
188C22×C4×C7C2411yescharacteristic
189C2×C7×C8—22no—
190C2×C7×C8—22no—
191C7×C4⋊C4—22no—
192C7×C4⋊C4—22no—
193C7×C4⋊C4—22no—
194C7×C4⋊C4—22no—
195C7×C4⋊C4—21no—
196C7×C4⋊C4—21no—
197C7×C4⋊C4—21no—
198C7×C4⋊C4—21no—
199C7×C4⋊C4—21no—
200C7×C4⋊C4—21no—
201C7×C4⋊C4—22no—
202C7×C4⋊C4—21no—
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—42no—
210C7×C4⋊C4—42no—
211C7×C4⋊C4—41no—
212C7×C4⋊C4—41no—
213C7×C4⋊C4—41no—
214C7×C4⋊C4—41no—
215C7×C4⋊C4C2×D412yes—
216C7×C4⋊C4C2×D412yes—
217C7×C22⋊C4—22no—
218C7×C22⋊C4—22no—
219C7×C22⋊C4—22no—
220C7×C22⋊C4—22no—
221C7×C22⋊C4—22no—
222C7×C22⋊C4—22no—
223C7×C22⋊C4—22no—
224C7×C22⋊C4—21no—
225C7×C22⋊C4—21no—
226C7×C22⋊C4—22no—
227C7×C22⋊C4—21no—
228C7×C22⋊C4—21no—
229C7×C22⋊C4—22no—
230C7×C22⋊C4—42no—
231C7×C22⋊C4—42no—
232C42×C7—22no—
233C42×C7—22no—
234C42×C7—21no—
235C42×C7—22no—
236C42×C7—21no—
237C42×C7—42no—
238C42×C7C2×D411yescharacteristic
239C42×C7C2×D411yescharacteristic
240C42×C7C2×D411yescharacteristic
241C42×C7C2×D411yescharacteristic
242#542—21no—
243#542—21no—
244#541—21no—
245#540—21no—
246#539—21no—
247#539C22×C712yes—
248#538—21no—
249#538C22×C712yes—
250#533C22×C711yescharacteristic
251#532C22×C711yescharacteristic
252#529—22no—
253#529—22no—
254C2×#125C22×C711yescharacteristic
255#525—21no—
256C4×#41—21no—
257C2×#117C22×C711yescharacteristic
258#501—21no—
259#501—21no—
260#491—21no—
261#490—21no—
262#484C22×C712yes—
263#483C22×C712yes—
264#481C22×C712yes—
265#480C22×C712yes—
266#479—22no—
267#478—22no—
268#432C22×C711yescharacteristic
269#432C22×C711yescharacteristic
270#400C22×C711yescharacteristic
271#399C22×C711yescharacteristic
272#398C22×C711yescharacteristic
273#397C22×C711yescharacteristic
274#389—21no—
275#388—21no—
276#387—21no—
277#385—21no—
278#376C22×C711yescharacteristic
279#328C22×C712yes—
280#328C22×C712yes—
281#327C22×C712yes—
282#327C22×C712yes—
283#324C22×C712yes—
284#323C22×C712yes—
285C23×C7—21no—
286C23×C7—22no—
287C23×C7—21no—
288C23×C7—21no—
289C23×C7—41no—
290C23×C7—41no—
291C23×C7C22×D411yescharacteristic
292C7×Q8—42no—
293C7×Q8—42no—
294C7×Q8—81no—
295C7×D4—42no—
296C7×D4—42no—
297C7×D4—42no—
298C7×D4—42no—
299C7×D4—81no—
300C7×D4—82no—
301C7×D4—82no—
302C2×C4×C7—21no—
303C2×C4×C7—21no—
304C2×C4×C7—22no—
305C2×C4×C7—21no—
306C2×C4×C7—21no—
307C2×C4×C7—22no—
308C2×C4×C7—22no—
309C2×C4×C7—22no—
310C2×C4×C7—21no—
311C2×C4×C7—21no—
312C2×C4×C7—21no—
313C2×C4×C7—41no—
314C2×C4×C7—41no—
315C2×C4×C7—42no—
316C2×C4×C7—42no—
317C2×C4×C7—44no—
318C2×C4×C7—41no—
319C2×C4×C7—41no—
320C2×C4×C7—41no—
321C2×C4×C7—41no—
322C2×C4×C7—41no—
323C2×C4×C7—41no—
324C2×C4×C7—41no—
325C2×C4×C7—41no—
326C2×C4×C7—41no—
327C2×C4×C7—41no—
328C2×C4×C7—41no—
329C2×C4×C7—41no—
330C2×C4×C7—41no—
331C2×C4×C7—82no—
332C2×C4×C7—82no—
333C2×C4×C7#12011yescharacteristic
334C2×C4×C7#12011yescharacteristic
335C2×C4×C7#12011yescharacteristic
336C2×C4×C7#12011yescharacteristic
337C2×C4×C7C22×D411yescharacteristic
338C2×C4×C7C22×D411yescharacteristic
339C7×C8—42no—
340C7×C8—42no—
341C2×#41—21no—
342C23×C4C7×D411yescharacteristic
343C4⋊Q8C7×D411yescharacteristic
344C4⋊Q8C7×D411yescharacteristic
345C4⋊Q8C7×D411yescharacteristic
346#127C7×D411yescharacteristic
347#126—22no—
348#126—22no—
349#126—22no—
350#126—22no—
351#125—22no—
352#125—22no—
353#125—21no—
354#125—21no—
355#125—21no—
356#125—21no—
357#125—21no—
358#124—22no—
359#124—22no—
360#124—21no—
361#123—22no—
362#123—22no—
363#123—22no—
364#123—22no—
365#123—22no—
366C22⋊Q8—21no—
367C22⋊Q8—21no—
368C22⋊Q8—22no—
369C22⋊Q8—22no—
370C22⋊Q8—21no—
371C22⋊Q8C23×C712yes—
372#121—21no—
373#121—21no—
374#121—22no—
375#121—22no—
376#121—21no—
377#121C23×C712yes—
378#120—22no—
379C4×Q8—21no—
380C4×Q8—21no—
381C4×Q8—21no—
382C4×Q8C7×D412yes—
383C4×D4—21no—
384C4×D4—21no—
385C4×D4—22no—
386C4×D4—22no—
387C4×D4—22no—
388C4×D4—22no—
389C4×D4—21no—
390C4×D4C7×D412yes—
391#117—22no—
392#117—21no—
393#117—21no—
394#117—21no—
395#117—21no—
396#117—21no—
397C2×C4⋊C4—21no—
398C2×C4⋊C4—21no—
399C2×C4⋊C4—21no—
400C2×C4⋊C4—21no—
401C2×C4⋊C4C7×D411yescharacteristic
402C2×C4⋊C4C23×C711yescharacteristic
403C2×C4⋊C4C23×C711yescharacteristic
404C2×C4⋊C4C23×C711yescharacteristic
405C2×C22⋊C4—21no—
406C2×C22⋊C4—21no—
407C2×C22⋊C4C7×D411yescharacteristic
408C2×C22⋊C4C23×C711yescharacteristic
409C2×C42—22no—
410C2×C42—21no—
411C2×C42C7×D411yescharacteristic
412C2×C42C23×C711yescharacteristic
413#107—22no—
414#107—22no—
415#106—22no—
416#106—22no—
417C4⋊C8—21no—
418C4⋊C8—21no—
419#103—22no—
420#103—22no—
421#102—22no—
422#102—22no—
423C22⋊C8C23×C712yes—
424C22⋊C8C23×C712yes—
425#97—21no—
426#97—21no—
427#95—21no—
428#95—21no—
429#95—21no—
430#95—21no—
431#95—21no—
432#95C23×C711yescharacteristic
433#95C23×C711yescharacteristic
434C22×C7—21no—
435C22×C7—21no—
436C22×C7—42no—
437C22×C7—41no—
438C22×C7—42no—
439C22×C7—42no—
440C22×C7—41no—
441C22×C7—42no—
442C22×C7—41no—
443C22×C7—81no—
444C22×C7C2×#12011yescharacteristic
445C22×C7#57511yescharacteristic
446C22×C7#57511yescharacteristic
447C4×C7—21no—
448C4×C7—21no—
449C4×C7—21no—
450C4×C7—21no—
451C4×C7—21no—
452C4×C7—21no—
453C4×C7—42no—
454C4×C7—41no—
455C4×C7—42no—
456C4×C7—41no—
457C4×C7—42no—
458C4×C7—41no—
459C4×C7—42no—
460C4×C7—81no—
461C4×C7—81no—
462C4×C7—81no—
463C24C2×C7×D411yescharacteristic
464#41—42no—
465#41—42no—
466C2×Q8—21no—
467C2×Q8—21no—
468C2×Q8C2×C7×D412yes—
469C2×D4—22no—
470C2×D4—22no—
471C2×D4—22no—
472C2×D4—21no—
473C2×D4—21no—
474C2×D4C2×C7×D412yes—
475C22×C4—21no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—21no—
484C22×C4—41no—
485C22×C4—41no—
486C22×C4C2×C7×D411yescharacteristic
487C22×C4C2×C7×D411yescharacteristic
488C22×C4C2×C7×D411yescharacteristic
489C22×C4C2×C7×D411yescharacteristic
490C22×C4C2×C7×D411yescharacteristic
491C22×C4C24×C711yescharacteristic
492C2×C8—22no—
493C2×C8—22no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—21no—
503C4⋊C4—21no—
504C4⋊C4—22no—
505C4⋊C4—21no—
506C4⋊C4—21no—
507C4⋊C4—22no—
508C4⋊C4—21no—
509C4⋊C4—21no—
510C4⋊C4—21no—
511C4⋊C4—21no—
512C4⋊C4—42no—
513C4⋊C4—42no—
514C4⋊C4—41no—
515C4⋊C4—41no—
516C4⋊C4—41no—
517C4⋊C4—41no—
518C4⋊C4C2×C7×D412yes—
519C4⋊C4C2×C7×D412yes—
520C22⋊C4—22no—
521C22⋊C4—22no—
522C22⋊C4—22no—
523C22⋊C4—22no—
524C22⋊C4—22no—
525C22⋊C4—22no—
526C22⋊C4—22no—
527C22⋊C4—21no—
528C22⋊C4—21no—
529C22⋊C4—22no—
530C22⋊C4—21no—
531C22⋊C4—21no—
532C22⋊C4—22no—
533C22⋊C4—42no—
534C22⋊C4—42no—
535C42—22no—
536C42—22no—
537C42—21no—
538C42—22no—
539C42—21no—
540C42—42no—
541C42C2×C7×D411yescharacteristic
542C42C2×C7×D411yescharacteristic
543C42C2×C7×D411yescharacteristic
544C42C2×C7×D411yescharacteristic
545C2×C7—21no—
546C2×C7—21no—
547C2×C7—82no—
548C2×C7—81no—
549C2×C7#300511yescharacteristic
550C2×C7#301811yescharacteristic
551C2×C7#302811yescharacteristic
552C23—21no—
553C23—22no—
554C23—21no—
555C23—21no—
556C23—41no—
557C23—41no—
558C23C22×C7×D411yescharacteristic
559Q8—42no—
560Q8—42no—
561Q8—81no—
562D4—42no—
563D4—42no—
564D4—42no—
565D4—42no—
566D4—81no—
567D4—82no—
568D4—82no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—22no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—44no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—82no—
599C2×C4—82no—
600C2×C4C7×#12011yescharacteristic
601C2×C4C7×#12011yescharacteristic
602C2×C4C7×#12011yescharacteristic
603C2×C4C7×#12011yescharacteristic
604C2×C4C22×C7×D411yescharacteristic
605C2×C4C22×C7×D411yescharacteristic
606C8—42no—
607C8—42no—
608C7#1834111yescharacteristic, direct product factor, Sylow
609C22—21no—
610C22—21no—
611C22—42no—
612C22—41no—
613C22—42no—
614C22—42no—
615C22—41no—
616C22—42no—
617C22—41no—
618C22—81no—
619C22C2×C7×#12011yescharacteristic
620C22C7×#57511yescharacteristic
621C22C7×#57511yescharacteristic
622C4—21no—
623C4—21no—
624C4—21no—
625C4—21no—
626C4—21no—
627C4—21no—
628C4—42no—
629C4—41no—
630C4—42no—
631C4—41no—
632C4—42no—
633C4—41no—
634C4—42no—
635C4—81no—
636C4—81no—
637C4—81no—
638C2—21no—
639C2—21no—
640C2—81no—
641C2—82no—
642C2C7×#300511yescharacteristic
643C2C7×#301811yescharacteristic
644C2C7×#302811yescharacteristic
subgroupG/H
center, socleC22×C7C2×#120
commutator, Frattini, cosocleC22×C4C24×C7
prime radicalC7×#3455C2
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
2. C7×#3236C2semidirect
3. C7×#3234C2semidirect
4. C7×#2611C2semidirect
5. C7×#1618C2semidirect
6. C7×#1615C2semidirect
7. C7×#1612C2semidirect
8. C7×#1609C2semidirect
9. C7×#1506C2semidirect
10. C7×#1505C2semidirect
11. C7×#1464C2semidirect
55. #18341C7direct (N × H)
150. #3236C2×C7semidirect
151. #3234C2×C7semidirect
152. #2611C2×C7semidirect
153. #1618C2×C7semidirect
154. #1615C2×C7semidirect
155. #1612C2×C7semidirect
156. #1609C2×C7semidirect
157. #1506C2×C7semidirect
158. #1505C2×C7semidirect
159. #1464C2×C7semidirect
608. C7#18341direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#34552
C7×#32362
C7×#32342
C7×#26112
C7×#16182
C7×#16152
C7×#16122
C7×#16092
C7×#15062
C7×#15052
C7×#14642
#183417