Finite Group Databasebeta

← C7×#18459 #13195675

Subgroups — C7×#18459 #13195675

Order 1792.

972 conjugacy classes of subgroups.

The subgroup lattice has 682 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3561C211yesmaximal, characteristic, complemented
2C7×#3215C211yesmaximal, characteristic, complemented
3C7×#3211C211yesmaximal, characteristic, complemented
4C7×#2819C211yesmaximal, characteristic, complemented
5C7×#1606C212yesmaximal, complemented
6C7×#1602C212yesmaximal, complemented
7C7×#1600C212yesmaximal, complemented
8C7×#1596C212yesmaximal, complemented
9C7×#1518C211yesmaximal, characteristic, complemented
10C7×#1512C211yesmaximal, characteristic, complemented
11C7×#1469C211yesmaximal, characteristic, complemented
12C7×#563—21no—
13C7×#562—21no—
14C7×#556—21no—
15C7×#551—22no—
16C7×D4×Q8—21no—
17C7×#548—21no—
18C7×#546—21no—
19C7×#543C2212yes—
20C7×#542—21no—
21C7×#542—22no—
22C7×#538C2212yes—
23C7×#536—21no—
24C7×#533C2211yescharacteristic
25C7×#529C2211yescharacteristic
26C2×C7×#124C2211yescharacteristic
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×#394—22no—
44C7×#393—21no—
45C7×#393C2212yes—
46C7×#392—21no—
47C7×#392C2212yes—
48C7×#382C2211yescharacteristic
49C7×#328C2212yes—
50C7×#328C2212yes—
51C7×#328C2212yes—
52C7×#327C2212yes—
53C7×#323C2212yes—
54C7×#323C2212yes—
55#18459C711yesmaximal, characteristic, direct product factor, Sylow
56C2×C7×#41—21no—
57C22×C7×Q8—21no—
58C22×C7×Q8D411yescharacteristic
59C22×C7×Q8C2311yescharacteristic
60C22×C7×D4—22no—
61C22×C7×D4D411yescharacteristic
62C22×C7×D4C2311yescharacteristic
63C2×C7×Q16—21no—
64C2×C7×Q16—22no—
65C2×C7×SD16—21no—
66C2×C7×SD16—22no—
67C2×C7×SD16—21no—
68C2×C7×D8—21no—
69C7×C4⋊Q8—21no—
70C7×C4⋊Q8—22no—
71C7×C4⋊Q8D411yescharacteristic, complemented
72C7×C4⋊Q8D411yescharacteristic, complemented
73C7×#126—22no—
74C7×#126—22no—
75C7×#126—22no—
76C7×#126—21no—
77C7×#125—21no—
78C7×#125D411yescharacteristic, complemented
79C7×#124—21no—
80C7×#124—22no—
81C7×#124—22no—
82C7×#124—21no—
83C7×#124—21no—
84C7×#124—21no—
85C7×#124—21no—
86C7×#124—21no—
87C7×#124—21no—
88C7×#124—21no—
89C7×#124—21no—
90C7×#124D411yescharacteristic, complemented
91C7×#123—22no—
92C7×#123—22no—
93C7×#123—22no—
94C7×#123—21no—
95C7×#123—22no—
96C7×C22⋊Q8—22no—
97C7×C22⋊Q8—21no—
98C7×C22⋊Q8—22no—
99C7×C22⋊Q8—22no—
100C7×C22⋊Q8—22no—
101C7×C22⋊Q8—21no—
102C7×C22⋊Q8—22no—
103C7×C22⋊Q8—22no—
104C7×C22⋊Q8—22no—
105C7×C22⋊Q8C2311yescharacteristic
106C7×C22⋊Q8C2312yes—
107C7×#121—21no—
108C7×#121—22no—
109C7×#121—22no—
110C7×#121C2311yescharacteristic
111C7×#120—22no—
112C4×C7×Q8—21no—
113C4×C7×Q8—21no—
114C4×C7×Q8D411yescharacteristic, complemented
115C4×C7×Q8D412yescomplemented
116C4×C7×D4—21no—
117C4×C7×D4—22no—
118C4×C7×D4—21no—
119C4×C7×D4—22no—
120C4×C7×D4D411yescharacteristic, complemented
121C7×#117—22no—
122C7×#117—21no—
123C7×#117—21no—
124C2×C7×C4⋊C4—21no—
125C2×C7×C22⋊C4—22no—
126C2×C7×C22⋊C4—21no—
127C2×C7×C22⋊C4—21no—
128C2×C7×C22⋊C4—22no—
129C2×C7×C22⋊C4—22no—
130C2×C7×C22⋊C4D412yes—
131C2×C7×C22⋊C4C2312yes—
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#3561C2×C711yescharacteristic, complemented
150#3215C2×C711yescharacteristic, complemented
151#3211C2×C711yescharacteristic, complemented
152#2819C2×C711yescharacteristic, complemented
153#1606C2×C712yescomplemented
154#1602C2×C712yescomplemented
155#1600C2×C712yescomplemented
156#1596C2×C712yescomplemented
157#1518C2×C711yescharacteristic, complemented
158#1512C2×C711yescharacteristic, complemented
159#1469C2×C711yescharacteristic, complemented
160C24×C7—21no—
161C24×C7C2×D412yes—
162C7×#41—82no—
163C2×C7×Q8—22no—
164C2×C7×Q8—21no—
165C2×C7×Q8—21no—
166C2×C7×Q8—21no—
167C2×C7×Q8—21no—
168C2×C7×Q8—21no—
169C2×C7×Q8—81no—
170C2×C7×Q8—81no—
171C2×C7×Q8—82no—
172C2×C7×Q8C2×D411yescharacteristic
173C2×C7×Q8C2×D412yes—
174C2×C7×D4—21no—
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—81no—
182C2×C7×D4—81no—
183C2×C7×D4—82no—
184C2×C7×D4—82no—
185C2×C7×D4—82no—
186C2×C7×D4C2×D411yescharacteristic
187C22×C4×C7—22no—
188C22×C4×C7—22no—
189C22×C4×C7—22no—
190C22×C4×C7—21no—
191C22×C4×C7—22no—
192C22×C4×C7—21no—
193C22×C4×C7C2×D411yescharacteristic
194C22×C4×C7C2×D412yes—
195C22×C4×C7C2×D411yescharacteristic
196C22×C4×C7C2411yescharacteristic
197C7×Q16—81no—
198C7×Q16—82no—
199C7×SD16—81no—
200C7×SD16—81no—
201C7×SD16—82no—
202C7×D8—81no—
203C2×C7×C8—21no—
204C2×C7×C8—21no—
205C2×C7×C8—22no—
206C7×C4⋊C4—22no—
207C7×C4⋊C4—22no—
208C7×C4⋊C4—22no—
209C7×C4⋊C4—22no—
210C7×C4⋊C4—21no—
211C7×C4⋊C4—21no—
212C7×C4⋊C4—21no—
213C7×C4⋊C4—21no—
214C7×C4⋊C4—21no—
215C7×C4⋊C4—21no—
216C7×C4⋊C4—22no—
217C7×C4⋊C4—21no—
218C7×C4⋊C4—22no—
219C7×C4⋊C4—81no—
220C7×C4⋊C4C2×D411yescharacteristic
221C7×C4⋊C4C2×D411yescharacteristic
222C7×C4⋊C4C2×D412yes—
223C7×C22⋊C4—22no—
224C7×C22⋊C4—22no—
225C7×C22⋊C4—22no—
226C7×C22⋊C4—22no—
227C7×C22⋊C4—22no—
228C7×C22⋊C4—21no—
229C7×C22⋊C4—22no—
230C7×C22⋊C4—22no—
231C7×C22⋊C4—22no—
232C7×C22⋊C4—21no—
233C7×C22⋊C4—21no—
234C7×C22⋊C4—21no—
235C7×C22⋊C4—22no—
236C7×C22⋊C4—22no—
237C7×C22⋊C4—21no—
238C7×C22⋊C4—41no—
239C7×C22⋊C4—41no—
240C7×C22⋊C4—41no—
241C7×C22⋊C4—41no—
242C7×C22⋊C4—82no—
243C7×C22⋊C4—82no—
244C42×C7—22no—
245C42×C7—22no—
246C42×C7—21no—
247C42×C7C2×D411yescharacteristic
248C42×C7C2×D411yescharacteristic
249C42×C7C2×D411yescharacteristic
250C42×C7C2×D411yescharacteristic
251#563—21no—
252#562—21no—
253#556—21no—
254#551—22no—
255D4×Q8—21no—
256#548—21no—
257#546—21no—
258#543C22×C712yes—
259#542—21no—
260#542—22no—
261#538C22×C712yes—
262#536—21no—
263#533C22×C711yescharacteristic
264#529C22×C711yescharacteristic
265C2×#124C22×C711yescharacteristic
266C2×#124C22×C711yescharacteristic
267#497—21no—
268#495—21no—
269#489—21no—
270#488—21no—
271#464—21no—
272#464—22no—
273#463—21no—
274#451C22×C711yescharacteristic
275#451C22×C712yes—
276#450C22×C711yescharacteristic
277#450C22×C712yes—
278#448C22×C711yescharacteristic
279#447C22×C711yescharacteristic
280#432C22×C711yescharacteristic
281#431C22×C711yescharacteristic
282#394—22no—
283#393—21no—
284#393C22×C712yes—
285#392—21no—
286#392C22×C712yes—
287#382C22×C711yescharacteristic
288#328C22×C712yes—
289#328C22×C712yes—
290#328C22×C712yes—
291#327C22×C712yes—
292#323C22×C712yes—
293#323C22×C712yes—
294C23×C7—22no—
295C23×C7—21no—
296C23×C7—22no—
297C23×C7—22no—
298C23×C7—42no—
299C23×C7—42no—
300C23×C7—42no—
301C23×C7—42no—
302C23×C7—41no—
303C23×C7—41no—
304C23×C7—41no—
305C23×C7C22×D411yescharacteristic
306C7×Q8—41no—
307C7×Q8—42no—
308C7×Q8—42no—
309C7×Q8—42no—
310C7×Q8—41no—
311C7×Q8—41no—
312C7×Q8—42no—
313C7×Q8—81no—
314C7×Q8—82no—
315C7×D4—41no—
316C7×D4—42no—
317C7×D4—41no—
318C7×D4—41no—
319C7×D4—42no—
320C7×D4—41no—
321C7×D4—41no—
322C7×D4—81no—
323C7×D4—82no—
324C7×D4—82no—
325C7×D4—88no—
326C2×C4×C7—21no—
327C2×C4×C7—21no—
328C2×C4×C7—21no—
329C2×C4×C7—22no—
330C2×C4×C7—21no—
331C2×C4×C7—22no—
332C2×C4×C7—22no—
333C2×C4×C7—21no—
334C2×C4×C7—41no—
335C2×C4×C7—41no—
336C2×C4×C7—42no—
337C2×C4×C7—42no—
338C2×C4×C7—42no—
339C2×C4×C7—42no—
340C2×C4×C7—42no—
341C2×C4×C7—41no—
342C2×C4×C7—42no—
343C2×C4×C7—82no—
344C2×C4×C7—82no—
345C2×C4×C7—82no—
346C2×C4×C7#12011yescharacteristic
347C2×C4×C7#12011yescharacteristic
348C2×C4×C7#12011yescharacteristic
349C2×C4×C7#12011yescharacteristic
350C2×C4×C7C22×D411yescharacteristic
351C2×C4×C7C22×D411yescharacteristic
352C7×C8—41no—
353C7×C8—41no—
354C7×C8—42no—
355C2×#41—21no—
356C22×Q8—21no—
357C22×Q8C7×D411yescharacteristic
358C22×Q8C23×C711yescharacteristic
359C22×D4—22no—
360C22×D4C7×D411yescharacteristic
361C22×D4C23×C711yescharacteristic
362C2×Q16—21no—
363C2×Q16—22no—
364C2×SD16—21no—
365C2×SD16—22no—
366C2×SD16—21no—
367C2×D8—21no—
368C4⋊Q8—21no—
369C4⋊Q8—22no—
370C4⋊Q8C7×D411yescharacteristic, complemented
371C4⋊Q8C7×D411yescharacteristic, complemented
372#126—22no—
373#126—22no—
374#126—22no—
375#126—21no—
376#125—21no—
377#125C7×D411yescharacteristic, complemented
378#124—21no—
379#124—22no—
380#124—22no—
381#124—21no—
382#124—21no—
383#124—21no—
384#124—21no—
385#124—21no—
386#124—21no—
387#124—21no—
388#124—21no—
389#124C7×D411yescharacteristic, complemented
390#123—22no—
391#123—22no—
392#123—22no—
393#123—21no—
394#123—22no—
395C22⋊Q8—22no—
396C22⋊Q8—21no—
397C22⋊Q8—22no—
398C22⋊Q8—22no—
399C22⋊Q8—22no—
400C22⋊Q8—21no—
401C22⋊Q8—22no—
402C22⋊Q8—22no—
403C22⋊Q8—22no—
404C22⋊Q8C23×C711yescharacteristic
405C22⋊Q8C23×C712yes—
406#121—21no—
407#121—22no—
408#121—22no—
409#121C23×C711yescharacteristic
410#120—22no—
411C4×Q8—21no—
412C4×Q8—21no—
413C4×Q8C7×D411yescharacteristic, complemented
414C4×Q8C7×D412yescomplemented
415C4×D4—21no—
416C4×D4—22no—
417C4×D4—21no—
418C4×D4—22no—
419C4×D4C7×D411yescharacteristic, complemented
420#117—22no—
421#117—21no—
422#117—21no—
423C2×C4⋊C4—21no—
424C2×C22⋊C4—22no—
425C2×C22⋊C4—21no—
426C2×C22⋊C4—21no—
427C2×C22⋊C4—22no—
428C2×C22⋊C4—22no—
429C2×C22⋊C4C7×D412yes—
430C2×C22⋊C4C23×C712yes—
431C2×C42C23×C711yescharacteristic
432C4⋊C8—21no—
433C4⋊C8—21no—
434#103—21no—
435#103—21no—
436#103—22no—
437#102—21no—
438#102—21no—
439#102—22no—
440C22⋊C8C23×C712yes—
441C22⋊C8C23×C711yescharacteristic
442C22⋊C8C23×C711yescharacteristic
443#97—21no—
444C4×C8—21no—
445#95—21no—
446#95—22no—
447#95C23×C712yes—
448C22×C7—21no—
449C22×C7—21no—
450C22×C7—41no—
451C22×C7—42no—
452C22×C7—41no—
453C22×C7—42no—
454C22×C7—42no—
455C22×C7—41no—
456C22×C7—82no—
457C22×C7—82no—
458C22×C7—82no—
459C22×C7—82no—
460C22×C7—81no—
461C22×C7—81no—
462C22×C7—81no—
463C22×C7C2×#12011yescharacteristic
464C22×C7#57711yescharacteristic
465C22×C7#57811yescharacteristic
466C4×C7—21no—
467C4×C7—21no—
468C4×C7—21no—
469C4×C7—21no—
470C4×C7—21no—
471C4×C7—21no—
472C4×C7—41no—
473C4×C7—41no—
474C4×C7—42no—
475C4×C7—42no—
476C4×C7—41no—
477C4×C7—42no—
478C4×C7—82no—
479C4×C7—81no—
480C24—21no—
481C24C2×C7×D412yes—
482#41—82no—
483C2×Q8—22no—
484C2×Q8—21no—
485C2×Q8—21no—
486C2×Q8—21no—
487C2×Q8—21no—
488C2×Q8—21no—
489C2×Q8—81no—
490C2×Q8—81no—
491C2×Q8—82no—
492C2×Q8C2×C7×D411yescharacteristic
493C2×Q8C2×C7×D412yes—
494C2×D4—21no—
495C2×D4—22no—
496C2×D4—22no—
497C2×D4—21no—
498C2×D4—21no—
499C2×D4—21no—
500C2×D4—21no—
501C2×D4—81no—
502C2×D4—81no—
503C2×D4—82no—
504C2×D4—82no—
505C2×D4—82no—
506C2×D4C2×C7×D411yescharacteristic
507C22×C4—22no—
508C22×C4—22no—
509C22×C4—22no—
510C22×C4—21no—
511C22×C4—22no—
512C22×C4—21no—
513C22×C4C2×C7×D411yescharacteristic
514C22×C4C2×C7×D412yes—
515C22×C4C2×C7×D411yescharacteristic
516C22×C4C24×C711yescharacteristic
517Q16—81no—
518Q16—82no—
519SD16—81no—
520SD16—81no—
521SD16—82no—
522D8—81no—
523C2×C8—21no—
524C2×C8—21no—
525C2×C8—22no—
526C4⋊C4—22no—
527C4⋊C4—22no—
528C4⋊C4—22no—
529C4⋊C4—22no—
530C4⋊C4—21no—
531C4⋊C4—21no—
532C4⋊C4—21no—
533C4⋊C4—21no—
534C4⋊C4—21no—
535C4⋊C4—21no—
536C4⋊C4—22no—
537C4⋊C4—21no—
538C4⋊C4—22no—
539C4⋊C4—81no—
540C4⋊C4C2×C7×D411yescharacteristic
541C4⋊C4C2×C7×D411yescharacteristic
542C4⋊C4C2×C7×D412yes—
543C22⋊C4—22no—
544C22⋊C4—22no—
545C22⋊C4—22no—
546C22⋊C4—22no—
547C22⋊C4—22no—
548C22⋊C4—21no—
549C22⋊C4—22no—
550C22⋊C4—22no—
551C22⋊C4—22no—
552C22⋊C4—21no—
553C22⋊C4—21no—
554C22⋊C4—21no—
555C22⋊C4—22no—
556C22⋊C4—22no—
557C22⋊C4—21no—
558C22⋊C4—41no—
559C22⋊C4—41no—
560C22⋊C4—41no—
561C22⋊C4—41no—
562C22⋊C4—82no—
563C22⋊C4—82no—
564C42—22no—
565C42—22no—
566C42—21no—
567C42C2×C7×D411yescharacteristic
568C42C2×C7×D411yescharacteristic
569C42C2×C7×D411yescharacteristic
570C42C2×C7×D411yescharacteristic
571C2×C7—21no—
572C2×C7—21no—
573C2×C7—81no—
574C2×C7—82no—
575C2×C7—161no—
576C2×C7#301311yescharacteristic
577C2×C7#301911yescharacteristic
578C2×C7#302711yescharacteristic
579C23—22no—
580C23—21no—
581C23—22no—
582C23—22no—
583C23—42no—
584C23—42no—
585C23—42no—
586C23—42no—
587C23—41no—
588C23—41no—
589C23—41no—
590C23C22×C7×D411yescharacteristic
591Q8—41no—
592Q8—42no—
593Q8—42no—
594Q8—42no—
595Q8—41no—
596Q8—41no—
597Q8—42no—
598Q8—81no—
599Q8—82no—
600D4—41no—
601D4—42no—
602D4—41no—
603D4—41no—
604D4—42no—
605D4—41no—
606D4—41no—
607D4—81no—
608D4—82no—
609D4—82no—
610D4—88no—
611C2×C4—21no—
612C2×C4—21no—
613C2×C4—21no—
614C2×C4—22no—
615C2×C4—21no—
616C2×C4—22no—
617C2×C4—22no—
618C2×C4—21no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—41no—
627C2×C4—42no—
628C2×C4—82no—
629C2×C4—82no—
630C2×C4—82no—
631C2×C4C7×#12011yescharacteristic
632C2×C4C7×#12011yescharacteristic
633C2×C4C7×#12011yescharacteristic
634C2×C4C7×#12011yescharacteristic
635C2×C4C22×C7×D411yescharacteristic
636C2×C4C22×C7×D411yescharacteristic
637C8—41no—
638C8—41no—
639C8—42no—
640C7#1845911yescharacteristic, direct product factor, Sylow
641C22—21no—
642C22—21no—
643C22—41no—
644C22—42no—
645C22—41no—
646C22—42no—
647C22—42no—
648C22—41no—
649C22—82no—
650C22—82no—
651C22—82no—
652C22—82no—
653C22—81no—
654C22—81no—
655C22—81no—
656C22C2×C7×#12011yescharacteristic
657C22C7×#57711yescharacteristic
658C22C7×#57811yescharacteristic
659C4—21no—
660C4—21no—
661C4—21no—
662C4—21no—
663C4—21no—
664C4—21no—
665C4—41no—
666C4—41no—
667C4—42no—
668C4—42no—
669C4—41no—
670C4—42no—
671C4—82no—
672C4—81no—
673C2—21no—
674C2—21no—
675C2—82no—
676C2—81no—
677C2—161no—
678C2C7×#301311yescharacteristic
679C2C7×#301911yescharacteristic
680C2C7×#302711yescharacteristic
subgroupG/H
center, socleC22×C7C2×#120
commutator, Frattini, cosocleC22×C4C24×C7
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×#3561C2semidirect
2. C7×#3215C2semidirect
3. C7×#3211C2semidirect
4. C7×#2819C2semidirect
5. C7×#1606C2semidirect
6. C7×#1602C2semidirect
7. C7×#1600C2semidirect
8. C7×#1596C2semidirect
9. C7×#1518C2semidirect
10. C7×#1512C2semidirect
11. C7×#1469C2semidirect
55. #18459C7direct (N × H)
71. C7×C4⋊Q8D4semidirect
72. C7×C4⋊Q8D4semidirect
78. C7×#125D4semidirect
90. C7×#124D4semidirect
114. C4×C7×Q8D4semidirect
115. C4×C7×Q8D4semidirect
120. C4×C7×D4D4semidirect
149. #3561C2×C7semidirect
150. #3215C2×C7semidirect
151. #3211C2×C7semidirect
152. #2819C2×C7semidirect
153. #1606C2×C7semidirect
154. #1602C2×C7semidirect
155. #1600C2×C7semidirect
156. #1596C2×C7semidirect
157. #1518C2×C7semidirect
158. #1512C2×C7semidirect
159. #1469C2×C7semidirect
370. C4⋊Q8C7×D4semidirect
371. C4⋊Q8C7×D4semidirect
377. #125C7×D4semidirect
389. #124C7×D4semidirect
413. C4×Q8C7×D4semidirect
414. C4×Q8C7×D4semidirect
419. C4×D4C7×D4semidirect
640. C7#18459direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#35612
C7×#32152
C7×#32112
C7×#28192
C7×#16062
C7×#16022
C7×#16002
C7×#15962
C7×#15182
C7×#15122
C7×#14692
#184597