Finite Group Databasebeta

← C7×#18466 #13195682

Subgroups — C7×#18466 #13195682

Order 1792.

954 conjugacy classes of subgroups.

The subgroup lattice has 668 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3558C211yesmaximal, characteristic
2C7×#3234C211yesmaximal, characteristic, complemented
3C7×#3232C211yesmaximal, characteristic, complemented
4C7×#2820C211yesmaximal, characteristic, complemented
5C7×#1618C212yesmaximal, complemented
6C7×#1613C212yesmaximal, complemented
7C7×#1612C212yesmaximal, complemented
8C7×#1607C212yesmaximal, complemented
9C7×#1517C211yesmaximal, characteristic, complemented
10C7×#1512C211yesmaximal, characteristic, complemented
11C7×#1469C211yesmaximal, characteristic, complemented
12C7×#562—21no—
13C7×H(3,𝔽4)—21no—
14C7×#559—21no—
15C7×#548—21no—
16C7×#546—21no—
17C7×#546—22no—
18C7×D42—21no—
19C7×#542—21no—
20C7×#538—22no—
21C7×#538C2212yes—
22C7×#535—21no—
23C7×#534C2212yes—
24C7×#532C2211yescharacteristic
25C7×#529C2211yescharacteristic
26C2×C7×#125C2211yescharacteristic, complemented
27C2×C7×#124C2211yescharacteristic
28C7×#501—21no—
29C7×#499—21no—
30C7×#490—21no—
31C7×#488—21no—
32C7×#484C2211yescharacteristic, complemented
33C7×#483C2211yescharacteristic, complemented
34C7×#481C2211yescharacteristic
35C7×#481C2212yescomplemented
36C7×#480C2211yescharacteristic
37C7×#480C2212yescomplemented
38C7×#479—21no—
39C7×#478—21no—
40C7×#478—22no—
41C7×#432C2211yescharacteristic, complemented
42C7×#431C2211yescharacteristic, complemented
43C7×#399C2212yescomplemented
44C7×#397C2212yescomplemented
45C7×#396—22no—
46C7×#395—21no—
47C7×#394—21no—
48C7×#382C2211yescharacteristic, complemented
49C7×#328C2212yescomplemented
50C7×#327C2212yes—
51C7×#327C2212yescomplemented
52C7×#327C2212yescomplemented
53C7×#323C2212yescomplemented
54C7×#323C2212yescomplemented
55#18466C711yesmaximal, characteristic, direct product factor, Sylow
56C2×C7×#41—21no—
57C2×C7×#41—21no—
58C22×C7×Q8D411yescharacteristic
59C22×C7×D4—21no—
60C22×C7×D4—22no—
61C22×C7×D4D411yescharacteristic
62C7×C4⋊Q8D411yescharacteristic
63C7×#127—21no—
64C7×#127D411yescharacteristic
65C7×#126—22no—
66C7×#126—21no—
67C7×#125—21no—
68C7×#125—21no—
69C7×#125—21no—
70C7×#125—21no—
71C7×#125D411yescharacteristic
72C7×#124—21no—
73C7×#124—22no—
74C7×#124—22no—
75C7×#124—22no—
76C7×#124—21no—
77C7×#124—21no—
78C7×#124—21no—
79C7×#124—21no—
80C7×#124—21no—
81C7×#124D411yescharacteristic
82C7×#123—22no—
83C7×#123—21no—
84C7×#123—22no—
85C7×#123—22no—
86C7×#123—22no—
87C7×C22⋊Q8—22no—
88C7×C22⋊Q8—21no—
89C7×C22⋊Q8—22no—
90C7×C22⋊Q8C2311yescharacteristic
91C7×#121—22no—
92C7×#121—22no—
93C7×#121—22no—
94C7×#121—21no—
95C7×#121—22no—
96C7×#121—21no—
97C7×#121—22no—
98C7×#121C2311yescharacteristic
99C7×#121C2312yes—
100C7×#120—22no—
101C7×#120—22no—
102C7×#120—22no—
103C7×#120—22no—
104C7×#120—22no—
105C4×C7×Q8—21no—
106C4×C7×Q8D411yescharacteristic
107C4×C7×D4—22no—
108C4×C7×D4—21no—
109C4×C7×D4—21no—
110C4×C7×D4—22no—
111C4×C7×D4—21no—
112C4×C7×D4D411yescharacteristic
113C4×C7×D4D412yes—
114C7×#117—21no—
115C2×C7×C4⋊C4—21no—
116C2×C7×C4⋊C4—21no—
117C2×C7×C4⋊C4—21no—
118C2×C7×C4⋊C4—22no—
119C2×C7×C4⋊C4C2311yescharacteristic
120C2×C7×C4⋊C4C2311yescharacteristic
121C2×C7×C22⋊C4—22no—
122C2×C7×C22⋊C4—22no—
123C2×C7×C22⋊C4—22no—
124C2×C7×C22⋊C4D412yes—
125C2×C7×C22⋊C4C2312yes—
126C2×C42×C7C2311yescharacteristic
127C7×#107—22no—
128C7×#107—21no—
129C7×#107—21no—
130C7×#106—22no—
131C7×#106—21no—
132C7×#106—21no—
133C7×C4⋊C8—21no—
134C7×C4⋊C8—21no—
135C7×#103—21no—
136C7×#103—21no—
137C7×#102—21no—
138C7×#102—22no—
139C7×#102—21no—
140C7×#102—22no—
141C7×C22⋊C8C2311yescharacteristic
142C7×C22⋊C8C2312yes—
143C7×C22⋊C8C2311yescharacteristic
144C7×#97—21no—
145C4×C7×C8—21no—
146C7×#95—21no—
147C7×#95—22no—
148C7×#95C2312yes—
149#3558C2×C711yescharacteristic
150#3234C2×C711yescharacteristic, complemented
151#3232C2×C711yescharacteristic, complemented
152#2820C2×C711yescharacteristic, complemented
153#1618C2×C712yescomplemented
154#1613C2×C712yescomplemented
155#1612C2×C712yescomplemented
156#1607C2×C712yescomplemented
157#1517C2×C711yescharacteristic, complemented
158#1512C2×C711yescharacteristic, complemented
159#1469C2×C711yescharacteristic, complemented
160C24×C7—22no—
161C24×C7C2×D412yes—
162C7×#41—81no—
163C7×#41—81no—
164C7×#41—82no—
165C2×C7×Q8—21no—
166C2×C7×Q8—21no—
167C2×C7×Q8—21no—
168C2×C7×Q8—81no—
169C2×C7×Q8C2×D411yescharacteristic
170C2×C7×D4—22no—
171C2×C7×D4—21no—
172C2×C7×D4—22no—
173C2×C7×D4—22no—
174C2×C7×D4—22no—
175C2×C7×D4—22no—
176C2×C7×D4—22no—
177C2×C7×D4—21no—
178C2×C7×D4—21no—
179C2×C7×D4—21no—
180C2×C7×D4—21no—
181C2×C7×D4—82no—
182C2×C7×D4—81no—
183C2×C7×D4—82no—
184C2×C7×D4—82no—
185C2×C7×D4C2×D411yescharacteristic
186C2×C7×D4C2×D412yes—
187C22×C4×C7—22no—
188C22×C4×C7—22no—
189C22×C4×C7—21no—
190C22×C4×C7—22no—
191C22×C4×C7—22no—
192C22×C4×C7—21no—
193C22×C4×C7—21no—
194C22×C4×C7C2×D411yescharacteristic
195C22×C4×C7C2×D412yes—
196C22×C4×C7C2×D411yescharacteristic
197C22×C4×C7C2411yescharacteristic
198C2×C7×C8—21no—
199C2×C7×C8—21no—
200C2×C7×C8—22no—
201C7×C4⋊C4—22no—
202C7×C4⋊C4—22no—
203C7×C4⋊C4—21no—
204C7×C4⋊C4—21no—
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—21no—
212C7×C4⋊C4—41no—
213C7×C4⋊C4—41no—
214C7×C4⋊C4—41no—
215C7×C4⋊C4—41no—
216C7×C4⋊C4—81no—
217C7×C4⋊C4—82no—
218C7×C4⋊C4C2×D412yes—
219C7×C4⋊C4C2×D411yescharacteristic
220C7×C4⋊C4C2×D411yescharacteristic
221C7×C22⋊C4—22no—
222C7×C22⋊C4—22no—
223C7×C22⋊C4—22no—
224C7×C22⋊C4—22no—
225C7×C22⋊C4—22no—
226C7×C22⋊C4—21no—
227C7×C22⋊C4—21no—
228C7×C22⋊C4—22no—
229C7×C22⋊C4—22no—
230C7×C22⋊C4—21no—
231C7×C22⋊C4—21no—
232C7×C22⋊C4—22no—
233C7×C22⋊C4—22no—
234C7×C22⋊C4—22no—
235C7×C22⋊C4—21no—
236C7×C22⋊C4—22no—
237C7×C22⋊C4—82no—
238C7×C22⋊C4—82no—
239C42×C7—22no—
240C42×C7—21no—
241C42×C7C2×D411yescharacteristic
242C42×C7C2×D411yescharacteristic
243C42×C7C2×D411yescharacteristic
244C42×C7C2×D411yescharacteristic
245#562—21no—
246H(3,𝔽4)—21no—
247#559—21no—
248#548—21no—
249#546—21no—
250#546—22no—
251D42—21no—
252#542—21no—
253#538—22no—
254#538C22×C712yes—
255#535—21no—
256#534C22×C712yes—
257#532C22×C711yescharacteristic
258#529C22×C711yescharacteristic
259C2×#125C22×C711yescharacteristic, complemented
260C2×#124C22×C711yescharacteristic
261#501—21no—
262#499—21no—
263#490—21no—
264#488—21no—
265#484C22×C711yescharacteristic, complemented
266#483C22×C711yescharacteristic, complemented
267#481C22×C711yescharacteristic
268#481C22×C712yescomplemented
269#480C22×C711yescharacteristic
270#480C22×C712yescomplemented
271#479—21no—
272#478—21no—
273#478—22no—
274#432C22×C711yescharacteristic, complemented
275#431C22×C711yescharacteristic, complemented
276#399C22×C712yescomplemented
277#397C22×C712yescomplemented
278#396—22no—
279#395—21no—
280#394—21no—
281#382C22×C711yescharacteristic, complemented
282#328C22×C712yescomplemented
283#327C22×C712yes—
284#327C22×C712yescomplemented
285#327C22×C712yescomplemented
286#323C22×C712yescomplemented
287#323C22×C712yescomplemented
288C23×C7—22no—
289C23×C7—21no—
290C23×C7—22no—
291C23×C7—22no—
292C23×C7—22no—
293C23×C7—42no—
294C23×C7—42no—
295C23×C7—82no—
296C23×C7—82no—
297C23×C7—82no—
298C23×C7C22×D411yescharacteristic
299C7×Q8—41no—
300C7×Q8—42no—
301C7×Q8—81no—
302C7×Q8—81no—
303C7×D4—42no—
304C7×D4—41no—
305C7×D4—42no—
306C7×D4—42no—
307C7×D4—41no—
308C7×D4—42no—
309C7×D4—41no—
310C7×D4—42no—
311C7×D4—81no—
312C7×D4—81no—
313C7×D4—82no—
314C7×D4—82no—
315C7×D4—82no—
316C7×D4—82no—
317C7×D4—82no—
318C2×C4×C7—21no—
319C2×C4×C7—21no—
320C2×C4×C7—21no—
321C2×C4×C7—22no—
322C2×C4×C7—22no—
323C2×C4×C7—21no—
324C2×C4×C7—21no—
325C2×C4×C7—41no—
326C2×C4×C7—41no—
327C2×C4×C7—42no—
328C2×C4×C7—42no—
329C2×C4×C7—42no—
330C2×C4×C7—42no—
331C2×C4×C7—42no—
332C2×C4×C7—41no—
333C2×C4×C7—41no—
334C2×C4×C7—41no—
335C2×C4×C7—42no—
336C2×C4×C7—41no—
337C2×C4×C7—82no—
338C2×C4×C7—82no—
339C2×C4×C7—82no—
340C2×C4×C7#12011yescharacteristic
341C2×C4×C7#12011yescharacteristic
342C2×C4×C7#12011yescharacteristic
343C2×C4×C7#12011yescharacteristic
344C2×C4×C7C22×D411yescharacteristic
345C2×C4×C7C22×D411yescharacteristic
346C7×C8—42no—
347C7×C8—41no—
348C7×C8—41no—
349C2×#41—21no—
350C2×#41—21no—
351C22×Q8C7×D411yescharacteristic
352C22×D4—21no—
353C22×D4—22no—
354C22×D4C7×D411yescharacteristic
355C4⋊Q8C7×D411yescharacteristic
356#127—21no—
357#127C7×D411yescharacteristic
358#126—22no—
359#126—21no—
360#125—21no—
361#125—21no—
362#125—21no—
363#125—21no—
364#125C7×D411yescharacteristic
365#124—21no—
366#124—22no—
367#124—22no—
368#124—22no—
369#124—21no—
370#124—21no—
371#124—21no—
372#124—21no—
373#124—21no—
374#124C7×D411yescharacteristic
375#123—22no—
376#123—21no—
377#123—22no—
378#123—22no—
379#123—22no—
380C22⋊Q8—22no—
381C22⋊Q8—21no—
382C22⋊Q8—22no—
383C22⋊Q8C23×C711yescharacteristic
384#121—22no—
385#121—22no—
386#121—22no—
387#121—21no—
388#121—22no—
389#121—21no—
390#121—22no—
391#121C23×C711yescharacteristic
392#121C23×C712yes—
393#120—22no—
394#120—22no—
395#120—22no—
396#120—22no—
397#120—22no—
398C4×Q8—21no—
399C4×Q8C7×D411yescharacteristic
400C4×D4—22no—
401C4×D4—21no—
402C4×D4—21no—
403C4×D4—22no—
404C4×D4—21no—
405C4×D4C7×D411yescharacteristic
406C4×D4C7×D412yes—
407#117—21no—
408C2×C4⋊C4—21no—
409C2×C4⋊C4—21no—
410C2×C4⋊C4—21no—
411C2×C4⋊C4—22no—
412C2×C4⋊C4C23×C711yescharacteristic
413C2×C4⋊C4C23×C711yescharacteristic
414C2×C22⋊C4—22no—
415C2×C22⋊C4—22no—
416C2×C22⋊C4—22no—
417C2×C22⋊C4C7×D412yes—
418C2×C22⋊C4C23×C712yes—
419C2×C42C23×C711yescharacteristic
420#107—22no—
421#107—21no—
422#107—21no—
423#106—22no—
424#106—21no—
425#106—21no—
426C4⋊C8—21no—
427C4⋊C8—21no—
428#103—21no—
429#103—21no—
430#102—21no—
431#102—22no—
432#102—21no—
433#102—22no—
434C22⋊C8C23×C712yes—
435C22⋊C8C23×C711yescharacteristic
436C22⋊C8C23×C711yescharacteristic
437#97—21no—
438C4×C8—21no—
439#95—21no—
440#95—22no—
441#95C23×C712yes—
442C22×C7—21no—
443C22×C7—21no—
444C22×C7—42no—
445C22×C7—42no—
446C22×C7—41no—
447C22×C7—41no—
448C22×C7—42no—
449C22×C7—42no—
450C22×C7—42no—
451C22×C7—41no—
452C22×C7—42no—
453C22×C7—82no—
454C22×C7—82no—
455C22×C7—164no—
456C22×C7C2×#12011yescharacteristic
457C22×C7#57611yescharacteristic
458C22×C7#57711yescharacteristic
459C4×C7—21no—
460C4×C7—21no—
461C4×C7—21no—
462C4×C7—21no—
463C4×C7—21no—
464C4×C7—21no—
465C4×C7—41no—
466C4×C7—42no—
467C4×C7—41no—
468C4×C7—42no—
469C4×C7—41no—
470C4×C7—82no—
471C4×C7—81no—
472C4×C7—81no—
473C24—22no—
474C24C2×C7×D412yes—
475#41—81no—
476#41—81no—
477#41—82no—
478C2×Q8—21no—
479C2×Q8—21no—
480C2×Q8—21no—
481C2×Q8—81no—
482C2×Q8C2×C7×D411yescharacteristic
483C2×D4—22no—
484C2×D4—21no—
485C2×D4—22no—
486C2×D4—22no—
487C2×D4—22no—
488C2×D4—22no—
489C2×D4—22no—
490C2×D4—21no—
491C2×D4—21no—
492C2×D4—21no—
493C2×D4—21no—
494C2×D4—82no—
495C2×D4—81no—
496C2×D4—82no—
497C2×D4—82no—
498C2×D4C2×C7×D411yescharacteristic
499C2×D4C2×C7×D412yes—
500C22×C4—22no—
501C22×C4—22no—
502C22×C4—21no—
503C22×C4—22no—
504C22×C4—22no—
505C22×C4—21no—
506C22×C4—21no—
507C22×C4C2×C7×D411yescharacteristic
508C22×C4C2×C7×D412yes—
509C22×C4C2×C7×D411yescharacteristic
510C22×C4C24×C711yescharacteristic
511C2×C8—21no—
512C2×C8—21no—
513C2×C8—22no—
514C4⋊C4—22no—
515C4⋊C4—22no—
516C4⋊C4—21no—
517C4⋊C4—21no—
518C4⋊C4—21no—
519C4⋊C4—21no—
520C4⋊C4—21no—
521C4⋊C4—21no—
522C4⋊C4—21no—
523C4⋊C4—21no—
524C4⋊C4—21no—
525C4⋊C4—41no—
526C4⋊C4—41no—
527C4⋊C4—41no—
528C4⋊C4—41no—
529C4⋊C4—81no—
530C4⋊C4—82no—
531C4⋊C4C2×C7×D412yes—
532C4⋊C4C2×C7×D411yescharacteristic
533C4⋊C4C2×C7×D411yescharacteristic
534C22⋊C4—22no—
535C22⋊C4—22no—
536C22⋊C4—22no—
537C22⋊C4—22no—
538C22⋊C4—22no—
539C22⋊C4—21no—
540C22⋊C4—21no—
541C22⋊C4—22no—
542C22⋊C4—22no—
543C22⋊C4—21no—
544C22⋊C4—21no—
545C22⋊C4—22no—
546C22⋊C4—22no—
547C22⋊C4—22no—
548C22⋊C4—21no—
549C22⋊C4—22no—
550C22⋊C4—82no—
551C22⋊C4—82no—
552C42—22no—
553C42—21no—
554C42C2×C7×D411yescharacteristic
555C42C2×C7×D411yescharacteristic
556C42C2×C7×D411yescharacteristic
557C42C2×C7×D411yescharacteristic
558C2×C7—21no—
559C2×C7—21no—
560C2×C7—82no—
561C2×C7—82no—
562C2×C7—81no—
563C2×C7#301211yescharacteristic
564C2×C7#301911yescharacteristic
565C2×C7#302811yescharacteristic
566C23—22no—
567C23—21no—
568C23—22no—
569C23—22no—
570C23—22no—
571C23—42no—
572C23—42no—
573C23—82no—
574C23—82no—
575C23—82no—
576C23C22×C7×D411yescharacteristic
577Q8—41no—
578Q8—42no—
579Q8—81no—
580Q8—81no—
581D4—42no—
582D4—41no—
583D4—42no—
584D4—42no—
585D4—41no—
586D4—42no—
587D4—41no—
588D4—42no—
589D4—81no—
590D4—81no—
591D4—82no—
592D4—82no—
593D4—82no—
594D4—82no—
595D4—82no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—42no—
614C2×C4—41no—
615C2×C4—82no—
616C2×C4—82no—
617C2×C4—82no—
618C2×C4C7×#12011yescharacteristic
619C2×C4C7×#12011yescharacteristic
620C2×C4C7×#12011yescharacteristic
621C2×C4C7×#12011yescharacteristic
622C2×C4C22×C7×D411yescharacteristic
623C2×C4C22×C7×D411yescharacteristic
624C8—42no—
625C8—41no—
626C8—41no—
627C7#1846611yescharacteristic, direct product factor, Sylow
628C22—21no—
629C22—21no—
630C22—42no—
631C22—42no—
632C22—41no—
633C22—41no—
634C22—42no—
635C22—42no—
636C22—42no—
637C22—41no—
638C22—42no—
639C22—82no—
640C22—82no—
641C22—164no—
642C22C2×C7×#12011yescharacteristic
643C22C7×#57611yescharacteristic
644C22C7×#57711yescharacteristic
645C4—21no—
646C4—21no—
647C4—21no—
648C4—21no—
649C4—21no—
650C4—21no—
651C4—41no—
652C4—42no—
653C4—41no—
654C4—42no—
655C4—41no—
656C4—82no—
657C4—81no—
658C4—81no—
659C2—21no—
660C2—21no—
661C2—82no—
662C2—81no—
663C2—82no—
664C2C7×#301211yescharacteristic
665C2C7×#301911yescharacteristic
666C2C7×#302811yescharacteristic
subgroupG/H
center, socleC22×C7C2×#120
commutator, Frattini, cosocleC22×C4C24×C7
prime radicalC7×#3558C2
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×#3234C2semidirect
3. C7×#3232C2semidirect
4. C7×#2820C2semidirect
5. C7×#1618C2semidirect
6. C7×#1613C2semidirect
7. C7×#1612C2semidirect
8. C7×#1607C2semidirect
9. C7×#1517C2semidirect
10. C7×#1512C2semidirect
11. C7×#1469C2semidirect
26. C2×C7×#125C22semidirect
32. C7×#484C22semidirect
33. C7×#483C22semidirect
35. C7×#481C22semidirect
37. C7×#480C22semidirect
41. C7×#432C22semidirect
42. C7×#431C22semidirect
43. C7×#399C22semidirect
44. C7×#397C22semidirect
48. C7×#382C22semidirect
49. C7×#328C22semidirect
51. C7×#327C22semidirect
52. C7×#327C22semidirect
53. C7×#323C22semidirect
54. C7×#323C22semidirect
55. #18466C7direct (N × H)
150. #3234C2×C7semidirect
151. #3232C2×C7semidirect
152. #2820C2×C7semidirect
153. #1618C2×C7semidirect
154. #1613C2×C7semidirect
155. #1612C2×C7semidirect
156. #1607C2×C7semidirect
157. #1517C2×C7semidirect
158. #1512C2×C7semidirect
159. #1469C2×C7semidirect
259. C2×#125C22×C7semidirect
265. #484C22×C7semidirect
266. #483C22×C7semidirect
268. #481C22×C7semidirect
270. #480C22×C7semidirect
274. #432C22×C7semidirect
275. #431C22×C7semidirect
276. #399C22×C7semidirect
277. #397C22×C7semidirect
281. #382C22×C7semidirect
282. #328C22×C7semidirect
284. #327C22×C7semidirect
285. #327C22×C7semidirect
286. #323C22×C7semidirect
287. #323C22×C7semidirect
627. C7#18466direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#35582
C7×#32342
C7×#32322
C7×#28202
C7×#16182
C7×#16132
C7×#16122
C7×#16072
C7×#15172
C7×#15122
C7×#14692
#184667