Finite Group Databasebeta

← C7×#19043 #13196259

Subgroups — C7×#19043 #13196259

Order 1792.

950 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3558C211yesmaximal, characteristic
2C7×#3215C211yesmaximal, characteristic, complemented
3C7×#3181C211yesmaximal, characteristic, complemented
4C7×#3135C211yesmaximal, characteristic, complemented
5C7×#1662C212yesmaximal, complemented
6C7×#1661C212yesmaximal, complemented
7C7×#1659C212yesmaximal, complemented
8C7×#1659C212yesmaximal, complemented
9C7×#1532C211yesmaximal, characteristic, complemented
10C7×#1530C211yesmaximal, characteristic, complemented
11C2×C7×#329C211yesmaximal, characteristic, complemented
12C7×#562C2212yes—
13C7×H(3,𝔽4)C2212yes—
14C7×#559—21no—
15C7×#548—21no—
16C7×#546—22no—
17C7×#546—21no—
18C7×D42—21no—
19C7×#542—21no—
20C7×#538—22no—
21C7×#538—21no—
22C7×#535—21no—
23C7×#534—21no—
24C7×#533C2211yescharacteristic, complemented
25C7×#532C2211yescharacteristic
26C7×#529C2211yescharacteristic
27C2×C7×#124C2211yescharacteristic
28C7×#501C2211yescharacteristic, complemented
29C7×#497C2211yescharacteristic, complemented
30C7×#497C2212yescomplemented
31C7×#491—21no—
32C7×#491C2211yescharacteristic, complemented
33C7×#490—21no—
34C7×#489C2211yescharacteristic, complemented
35C7×#489C2212yescomplemented
36C7×#489C2212yescomplemented
37C7×#489C2212yescomplemented
38C7×#467—21no—
39C7×#465—21no—
40C7×#464—22no—
41C7×#451—21no—
42C7×#450—21no—
43C7×#441—21no—
44C7×#440—21no—
45C7×#432C2211yescharacteristic, complemented
46C7×#416—22no—
47C2×C7×#97C2211yescharacteristic, complemented
48C2×C7×#97C2211yescharacteristic, complemented
49C7×#330C2212yescomplemented
50C7×#329C2212yescomplemented
51C7×#329C2212yescomplemented
52C7×#329C2212yescomplemented
53C7×#329C2212yescomplemented
54C7×#329C2212yescomplemented
55#19043C711yesmaximal, characteristic, direct product factor, Sylow
56C2×C7×#41—21no—
57C2×C7×#41D412yes—
58C22×C7×Q8D411yescharacteristic
59C22×C7×D4—22no—
60C22×C7×D4—21no—
61C22×C7×D4D411yescharacteristic
62C2×C7×Q16—21no—
63C2×C7×Q16—22no—
64C2×C7×SD16—21no—
65C2×C7×SD16—22no—
66C22×C7×C8—21no—
67C22×C7×C8—21no—
68C7×C4⋊Q8D411yescharacteristic
69C7×C4⋊Q8C2311yescharacteristic
70C7×C4⋊Q8C2311yescharacteristic
71C7×#127—21no—
72C7×#127D411yescharacteristic
73C7×#126—22no—
74C7×#126—21no—
75C7×#125—21no—
76C7×#125C2311yescharacteristic
77C7×#124—21no—
78C7×#124—22no—
79C7×#124—22no—
80C7×#124—22no—
81C7×#124—21no—
82C7×#124—21no—
83C7×#124D412yes—
84C7×#124C2312yes—
85C7×#124C2312yes—
86C7×#124C2311yescharacteristic
87C7×#123—22no—
88C7×#123—22no—
89C7×#123—21no—
90C7×#123—22no—
91C7×#123—22no—
92C7×C22⋊Q8—21no—
93C7×C22⋊Q8—22no—
94C7×C22⋊Q8—22no—
95C7×C22⋊Q8—21no—
96C7×C22⋊Q8—21no—
97C7×C22⋊Q8D411yescharacteristic
98C7×#121—22no—
99C7×#121—22no—
100C7×#121—22no—
101C7×#121—21no—
102C7×#121—22no—
103C7×#121—22no—
104C7×#121—21no—
105C7×#121—21no—
106C7×#121D411yescharacteristic
107C7×#120—22no—
108C7×#120—22no—
109C7×#120—22no—
110C7×#120—22no—
111C7×#120—22no—
112C4×C7×Q8—21no—
113C4×C7×Q8—21no—
114C4×C7×Q8—21no—
115C4×C7×Q8D411yescharacteristic
116C4×C7×D4—22no—
117C4×C7×D4—22no—
118C4×C7×D4—21no—
119C4×C7×D4—21no—
120C4×C7×D4—21no—
121C4×C7×D4—21no—
122C4×C7×D4D411yescharacteristic
123C7×#117—21no—
124C7×#117—21no—
125C2×C7×C22⋊C4—22no—
126C2×C7×C22⋊C4—21no—
127C2×C7×C22⋊C4—21no—
128C2×C42×C7C2311yescharacteristic
129C7×#107—21no—
130C7×#106—21no—
131C7×C4⋊C8—21no—
132C7×#103—21no—
133C7×#103—22no—
134C7×#103—22no—
135C7×#103—22no—
136C7×#103—21no—
137C7×#103—21no—
138C7×#102—22no—
139C7×#102—22no—
140C7×#102—21no—
141C7×#102—22no—
142C7×C22⋊C8—21no—
143C7×#97—21no—
144C7×#97—21no—
145C7×#97C2312yes—
146C7×#97C2311yescharacteristic
147C7×#97C2312yes—
148C7×#97C2311yescharacteristic
149#3558C2×C711yescharacteristic
150#3215C2×C711yescharacteristic, complemented
151#3181C2×C711yescharacteristic, complemented
152#3135C2×C711yescharacteristic, complemented
153#1662C2×C712yescomplemented
154#1661C2×C712yescomplemented
155#1659C2×C712yescomplemented
156#1659C2×C712yescomplemented
157#1532C2×C711yescharacteristic, complemented
158#1530C2×C711yescharacteristic, complemented
159C2×#329C2×C711yescharacteristic, complemented
160C24×C7—22no—
161C24×C7—21no—
162C7×#41—44no—
163C7×#41—82no—
164C2×C7×Q8—21no—
165C2×C7×Q8—21no—
166C2×C7×Q8—21no—
167C2×C7×Q8—81no—
168C2×C7×Q8C2×D412yes—
169C2×C7×Q8C2×D411yescharacteristic
170C2×C7×Q8C2×D412yes—
171C2×C7×D4—22no—
172C2×C7×D4—21no—
173C2×C7×D4—22no—
174C2×C7×D4—22no—
175C2×C7×D4—22no—
176C2×C7×D4—22no—
177C2×C7×D4—22no—
178C2×C7×D4—21no—
179C2×C7×D4—21no—
180C2×C7×D4—21no—
181C2×C7×D4—81no—
182C2×C7×D4—82no—
183C2×C7×D4—84no—
184C2×C7×D4C2×D412yes—
185C2×C7×D4C2×D411yescharacteristic
186C2×C7×D4C2×D412yes—
187C22×C4×C7—22no—
188C22×C4×C7—22no—
189C22×C4×C7—21no—
190C22×C4×C7—21no—
191C22×C4×C7—22no—
192C22×C4×C7C2×D411yescharacteristic
193C22×C4×C7C2×D411yescharacteristic
194C22×C4×C7C2×D411yescharacteristic
195C7×Q16—42no—
196C7×Q16—82no—
197C7×SD16—42no—
198C7×SD16—82no—
199C2×C7×C8—21no—
200C2×C7×C8—22no—
201C2×C7×C8—21no—
202C2×C7×C8—22no—
203C2×C7×C8—42no—
204C2×C7×C8—42no—
205C7×C4⋊C4—22no—
206C7×C4⋊C4—22no—
207C7×C4⋊C4—21no—
208C7×C4⋊C4—21no—
209C7×C4⋊C4—21no—
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—21no—
217C7×C4⋊C4—41no—
218C7×C4⋊C4—41no—
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—22no—
228C7×C22⋊C4—21no—
229C7×C22⋊C4—22no—
230C7×C22⋊C4—21no—
231C7×C22⋊C4—22no—
232C7×C22⋊C4—22no—
233C7×C22⋊C4—21no—
234C7×C22⋊C4—21no—
235C7×C22⋊C4—22no—
236C7×C22⋊C4—22no—
237C7×C22⋊C4—41no—
238C42×C7—22no—
239C42×C7—21no—
240C42×C7—41no—
241C42×C7C2×D411yescharacteristic
242C42×C7C2×D411yescharacteristic
243C42×C7C2×D411yescharacteristic
244C42×C7C2411yescharacteristic
245#562C22×C712yes—
246H(3,𝔽4)C22×C712yes—
247#559—21no—
248#548—21no—
249#546—22no—
250#546—21no—
251D42—21no—
252#542—21no—
253#538—22no—
254#538—21no—
255#535—21no—
256#534—21no—
257#533C22×C711yescharacteristic, complemented
258#532C22×C711yescharacteristic
259#529C22×C711yescharacteristic
260C2×#124C22×C711yescharacteristic
261#501C22×C711yescharacteristic, complemented
262#497C22×C711yescharacteristic, complemented
263#497C22×C712yescomplemented
264#491—21no—
265#491C22×C711yescharacteristic, complemented
266#490—21no—
267#489C22×C712yescomplemented
268#489C22×C712yescomplemented
269#489C22×C711yescharacteristic, complemented
270#489C22×C712yescomplemented
271#467—21no—
272#465—21no—
273#464—22no—
274#451—21no—
275#450—21no—
276#441—21no—
277#440—21no—
278#432C22×C711yescharacteristic, complemented
279#416—22no—
280C2×#97C22×C711yescharacteristic, complemented
281C2×#97C22×C711yescharacteristic, complemented
282#330C22×C712yescomplemented
283#329C22×C712yescomplemented
284#329C22×C712yescomplemented
285#329C22×C712yescomplemented
286#329C22×C712yescomplemented
287#329C22×C712yescomplemented
288C23×C7—22no—
289C23×C7—21no—
290C23×C7—22no—
291C23×C7—42no—
292C23×C7—81no—
293C23×C7—81no—
294C23×C7—82no—
295C23×C7—82no—
296C23×C7—82no—
297C23×C7#12011yescharacteristic
298C7×Q8—41no—
299C7×Q8—42no—
300C7×Q8—42no—
301C7×Q8—42no—
302C7×Q8—41no—
303C7×Q8—42no—
304C7×Q8—41no—
305C7×D4—41no—
306C7×D4—42no—
307C7×D4—42no—
308C7×D4—42no—
309C7×D4—42no—
310C7×D4—42no—
311C7×D4—44no—
312C7×D4—81no—
313C7×D4—82no—
314C7×D4—82no—
315C7×D4—82no—
316C7×D4—82no—
317C7×D4—82no—
318C2×C4×C7—22no—
319C2×C4×C7—21no—
320C2×C4×C7—22no—
321C2×C4×C7—22no—
322C2×C4×C7—21no—
323C2×C4×C7—21no—
324C2×C4×C7—41no—
325C2×C4×C7—41no—
326C2×C4×C7—42no—
327C2×C4×C7—41no—
328C2×C4×C7—41no—
329C2×C4×C7—82no—
330C2×C4×C7—81no—
331C2×C4×C7—82no—
332C2×C4×C7—82no—
333C2×C4×C7#12011yescharacteristic
334C2×C4×C7#12011yescharacteristic
335C2×C4×C7#12011yescharacteristic
336C2×C4×C7C22×D411yescharacteristic
337C2×C4×C7C22×D411yescharacteristic
338C2×C4×C7C22×D411yescharacteristic
339C7×C8—22no—
340C7×C8—42no—
341C7×C8—41no—
342C7×C8—42no—
343C2×#41—21no—
344C2×#41C7×D412yes—
345C22×Q8C7×D411yescharacteristic
346C22×D4—22no—
347C22×D4—21no—
348C22×D4C7×D411yescharacteristic
349C2×Q16—21no—
350C2×Q16—22no—
351C2×SD16—21no—
352C2×SD16—22no—
353C22×C8—21no—
354C22×C8—21no—
355C4⋊Q8C7×D411yescharacteristic
356C4⋊Q8C23×C711yescharacteristic
357C4⋊Q8C23×C711yescharacteristic
358#127—21no—
359#127C7×D411yescharacteristic
360#126—22no—
361#126—21no—
362#125—21no—
363#125C23×C711yescharacteristic
364#124—21no—
365#124—22no—
366#124—22no—
367#124—22no—
368#124—21no—
369#124—21no—
370#124C7×D412yes—
371#124C23×C712yes—
372#124C23×C711yescharacteristic
373#124C23×C712yes—
374#123—22no—
375#123—22no—
376#123—21no—
377#123—22no—
378#123—22no—
379C22⋊Q8—21no—
380C22⋊Q8—22no—
381C22⋊Q8—22no—
382C22⋊Q8—21no—
383C22⋊Q8—21no—
384C22⋊Q8C7×D411yescharacteristic
385#121—22no—
386#121—22no—
387#121—22no—
388#121—21no—
389#121—22no—
390#121—22no—
391#121—21no—
392#121—21no—
393#121C7×D411yescharacteristic
394#120—22no—
395#120—22no—
396#120—22no—
397#120—22no—
398#120—22no—
399C4×Q8—21no—
400C4×Q8—21no—
401C4×Q8—21no—
402C4×Q8C7×D411yescharacteristic
403C4×D4—22no—
404C4×D4—22no—
405C4×D4—21no—
406C4×D4—21no—
407C4×D4—21no—
408C4×D4—21no—
409C4×D4C7×D411yescharacteristic
410#117—21no—
411#117—21no—
412C2×C22⋊C4—22no—
413C2×C22⋊C4—21no—
414C2×C22⋊C4—21no—
415C2×C42C23×C711yescharacteristic
416#107—21no—
417#106—21no—
418C4⋊C8—21no—
419#103—22no—
420#103—22no—
421#103—21no—
422#103—22no—
423#103—21no—
424#103—21no—
425#102—22no—
426#102—22no—
427#102—21no—
428#102—22no—
429C22⋊C8—21no—
430#97—21no—
431#97—21no—
432#97C23×C711yescharacteristic
433#97C23×C711yescharacteristic
434#97C23×C712yes—
435#97C23×C712yes—
436C22×C7—21no—
437C22×C7—21no—
438C22×C7—42no—
439C22×C7—42no—
440C22×C7—41no—
441C22×C7—41no—
442C22×C7—42no—
443C22×C7—42no—
444C22×C7—41no—
445C22×C7—42no—
446C22×C7—42no—
447C22×C7—162no—
448C22×C7—164no—
449C22×C7#45512yes—
450C22×C7C2×#12011yescharacteristic
451C4×C7—21no—
452C4×C7—21no—
453C4×C7—21no—
454C4×C7—21no—
455C4×C7—21no—
456C4×C7—21no—
457C4×C7—41no—
458C4×C7—42no—
459C4×C7—41no—
460C4×C7—41no—
461C4×C7—42no—
462C4×C7—81no—
463C4×C7—81no—
464C24—22no—
465C24—21no—
466#41—44no—
467#41—82no—
468C2×Q8—21no—
469C2×Q8—21no—
470C2×Q8—21no—
471C2×Q8—81no—
472C2×Q8C2×C7×D412yes—
473C2×Q8C2×C7×D411yescharacteristic
474C2×Q8C2×C7×D412yes—
475C2×D4—22no—
476C2×D4—21no—
477C2×D4—22no—
478C2×D4—22no—
479C2×D4—22no—
480C2×D4—22no—
481C2×D4—22no—
482C2×D4—21no—
483C2×D4—21no—
484C2×D4—21no—
485C2×D4—81no—
486C2×D4—82no—
487C2×D4—84no—
488C2×D4C2×C7×D412yes—
489C2×D4C2×C7×D411yescharacteristic
490C2×D4C2×C7×D412yes—
491C22×C4—22no—
492C22×C4—22no—
493C22×C4—21no—
494C22×C4—21no—
495C22×C4—22no—
496C22×C4C2×C7×D411yescharacteristic
497C22×C4C2×C7×D411yescharacteristic
498C22×C4C2×C7×D411yescharacteristic
499Q16—42no—
500Q16—82no—
501SD16—42no—
502SD16—82no—
503C2×C8—21no—
504C2×C8—22no—
505C2×C8—21no—
506C2×C8—22no—
507C2×C8—42no—
508C2×C8—42no—
509C4⋊C4—22no—
510C4⋊C4—22no—
511C4⋊C4—21no—
512C4⋊C4—21no—
513C4⋊C4—21no—
514C4⋊C4—21no—
515C4⋊C4—21no—
516C4⋊C4—21no—
517C4⋊C4—21no—
518C4⋊C4—21no—
519C4⋊C4—21no—
520C4⋊C4—21no—
521C4⋊C4—41no—
522C4⋊C4—41no—
523C4⋊C4C2×C7×D411yescharacteristic
524C4⋊C4C2×C7×D411yescharacteristic
525C22⋊C4—22no—
526C22⋊C4—22no—
527C22⋊C4—22no—
528C22⋊C4—22no—
529C22⋊C4—22no—
530C22⋊C4—21no—
531C22⋊C4—22no—
532C22⋊C4—21no—
533C22⋊C4—22no—
534C22⋊C4—21no—
535C22⋊C4—22no—
536C22⋊C4—22no—
537C22⋊C4—21no—
538C22⋊C4—21no—
539C22⋊C4—22no—
540C22⋊C4—22no—
541C22⋊C4—41no—
542C42—22no—
543C42—21no—
544C42—41no—
545C42C2×C7×D411yescharacteristic
546C42C2×C7×D411yescharacteristic
547C42C2×C7×D411yescharacteristic
548C42C24×C711yescharacteristic
549C2×C7—22no—
550C2×C7—82no—
551C2×C7—82no—
552C2×C7—81no—
553C2×C7C2×#45511yescharacteristic
554C2×C7#301811yescharacteristic
555C2×C7#301911yescharacteristic
556C23—22no—
557C23—21no—
558C23—22no—
559C23—42no—
560C23—81no—
561C23—81no—
562C23—82no—
563C23—82no—
564C23—82no—
565C23C7×#12011yescharacteristic
566Q8—41no—
567Q8—42no—
568Q8—42no—
569Q8—42no—
570Q8—41no—
571Q8—42no—
572Q8—41no—
573D4—41no—
574D4—42no—
575D4—42no—
576D4—42no—
577D4—42no—
578D4—42no—
579D4—44no—
580D4—81no—
581D4—82no—
582D4—82no—
583D4—82no—
584D4—82no—
585D4—82no—
586C2×C4—22no—
587C2×C4—21no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—42no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—82no—
598C2×C4—81no—
599C2×C4—82no—
600C2×C4—82no—
601C2×C4C7×#12011yescharacteristic
602C2×C4C7×#12011yescharacteristic
603C2×C4C7×#12011yescharacteristic
604C2×C4C22×C7×D411yescharacteristic
605C2×C4C22×C7×D411yescharacteristic
606C2×C4C22×C7×D411yescharacteristic
607C8—22no—
608C8—42no—
609C8—41no—
610C8—42no—
611C7#1904311yescharacteristic, direct product factor, Sylow
612C22—21no—
613C22—21no—
614C22—42no—
615C22—42no—
616C22—41no—
617C22—41no—
618C22—42no—
619C22—42no—
620C22—41no—
621C22—42no—
622C22—42no—
623C22—162no—
624C22—164no—
625C22C7×#45512yes—
626C22C2×C7×#12011yescharacteristic
627C4—21no—
628C4—21no—
629C4—21no—
630C4—21no—
631C4—21no—
632C4—21no—
633C4—41no—
634C4—42no—
635C4—41no—
636C4—41no—
637C4—42no—
638C4—81no—
639C4—81no—
640C2—22no—
641C2—82no—
642C2—81no—
643C2—82no—
644C2C2×C7×#45511yescharacteristic
645C2C7×#301811yescharacteristic
646C2C7×#301911yescharacteristic
subgroupG/H
center, socleC22×C7C2×#120
commutator, Frattini, cosocleC42C24×C7
prime radicalC7×#3558C2
LCSG ▷ C42 ▷ 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×#3215C2semidirect
3. C7×#3181C2semidirect
4. C7×#3135C2semidirect
5. C7×#1662C2semidirect
6. C7×#1661C2semidirect
7. C7×#1659C2semidirect
8. C7×#1659C2semidirect
9. C7×#1532C2semidirect
10. C7×#1530C2semidirect
11. C2×C7×#329C2semidirect
24. C7×#533C22semidirect
28. C7×#501C22semidirect
29. C7×#497C22semidirect
30. C7×#497C22semidirect
32. C7×#491C22semidirect
34. C7×#489C22semidirect
35. C7×#489C22semidirect
36. C7×#489C22semidirect
37. C7×#489C22semidirect
45. C7×#432C22semidirect
47. C2×C7×#97C22semidirect
48. C2×C7×#97C22semidirect
49. C7×#330C22semidirect
50. C7×#329C22semidirect
51. C7×#329C22semidirect
52. C7×#329C22semidirect
53. C7×#329C22semidirect
54. C7×#329C22semidirect
55. #19043C7direct (N × H)
150. #3215C2×C7semidirect
151. #3181C2×C7semidirect
152. #3135C2×C7semidirect
153. #1662C2×C7semidirect
154. #1661C2×C7semidirect
155. #1659C2×C7semidirect
156. #1659C2×C7semidirect
157. #1532C2×C7semidirect
158. #1530C2×C7semidirect
159. C2×#329C2×C7semidirect
257. #533C22×C7semidirect
261. #501C22×C7semidirect
262. #497C22×C7semidirect
263. #497C22×C7semidirect
265. #491C22×C7semidirect
267. #489C22×C7semidirect
268. #489C22×C7semidirect
269. #489C22×C7semidirect
270. #489C22×C7semidirect
278. #432C22×C7semidirect
280. C2×#97C22×C7semidirect
281. C2×#97C22×C7semidirect
282. #330C22×C7semidirect
283. #329C22×C7semidirect
284. #329C22×C7semidirect
285. #329C22×C7semidirect
286. #329C22×C7semidirect
287. #329C22×C7semidirect
611. C7#19043direct (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×#32152
C7×#31812
C7×#31352
C7×#16622
C7×#16612
C7×#16592
C7×#15322
C7×#15302
C2×C7×#3292
#190437