Finite Group Databasebeta

← C7×#18306 #13195522

Subgroups — C7×#18306 #13195522

Order 1792.

1040 conjugacy classes of subgroups.

The subgroup lattice has 686 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3459C211yesmaximal, characteristic, complemented
2C7×#3211C212yesmaximal, complemented
3C7×#2631C211yesmaximal, characteristic, complemented
4C7×#1606C212yesmaximal, complemented
5C7×#1602C211yesmaximal, characteristic, complemented
6C7×#1601C211yesmaximal, characteristic, complemented
7C7×#1600C212yesmaximal, complemented
8C7×#1596C211yesmaximal, characteristic, complemented
9C7×#1595C211yesmaximal, characteristic, complemented
10C7×#1521C212yesmaximal, complemented
11C7×#1460C211yesmaximal, characteristic, complemented
12C7×#543C2211yescharacteristic
13C7×#539—22no—
14C7×#539—21no—
15C7×#538—22no—
16C7×#538C2212yes—
17C7×#534—21no—
18C7×#534C2211yescharacteristic
19C7×#529—22no—
20C7×#529C2212yes—
21C2×C7×#124C2211yescharacteristic, complemented
22C7×#525—21no—
23C7×#525—22no—
24C2×C4×C7×D4—21no—
25C2×C4×C7×D4C2211yescharacteristic
26C7×#495—22no—
27C7×#488—22no—
28C7×#464—22no—
29C7×#463—22no—
30C7×#451C2212yescomplemented
31C7×#450C2212yescomplemented
32C7×#448C2212yes—
33C7×#447C2212yes—
34C7×#431C2212yescomplemented
35C7×#397—21no—
36C7×#393C2211yescharacteristic
37C7×#393C2211yescharacteristic, complemented
38C7×#392—21no—
39C7×#392C2211yescharacteristic
40C7×#392C2211yescharacteristic, complemented
41C7×#386—21no—
42C7×#386—21no—
43C7×#383C2211yescharacteristic, complemented
44C7×#328C2212yescomplemented
45C7×#328C2212yescomplemented
46C7×#327C2212yes—
47C7×#327C2212yes—
48C7×#323C2211yescharacteristic
49C7×#323C2211yescharacteristic
50C7×#323C2212yescomplemented
51#18306C711yesmaximal, characteristic, direct product factor, Sylow
52C2×C7×#41—21no—
53C22×C7×Q8C2311yescharacteristic
54C22×C7×D4—21no—
55C22×C7×D4—21no—
56C22×C7×D4—21no—
57C22×C7×D4D411yescharacteristic
58C22×C7×D4C2311yescharacteristic
59C23×C4×C7—21no—
60C23×C4×C7D411yescharacteristic
61C23×C4×C7D411yescharacteristic
62C2×C7×Q16—22no—
63C2×C7×SD16—22no—
64C2×C7×SD16—22no—
65C2×C7×D8—22no—
66C7×#126—22no—
67C7×#126—22no—
68C7×#126—22no—
69C7×#125—22no—
70C7×#125D412yescomplemented
71C7×#124—22no—
72C7×#124—22no—
73C7×#124—22no—
74C7×#124D412yescomplemented
75C7×#123—22no—
76C7×#123—22no—
77C7×#123—22no—
78C7×#123—22no—
79C7×#123—22no—
80C7×#123—22no—
81C7×C22⋊Q8—21no—
82C7×C22⋊Q8—22no—
83C7×C22⋊Q8—22no—
84C7×C22⋊Q8—21no—
85C7×C22⋊Q8—22no—
86C7×C22⋊Q8C2312yes—
87C7×#121—22no—
88C7×#121—21no—
89C7×#121—22no—
90C7×#121—22no—
91C7×#121—21no—
92C7×#121C2312yes—
93C7×#120—22no—
94C7×#120—22no—
95C7×#120—22no—
96C4×C7×Q8D412yescomplemented
97C4×C7×D4—22no—
98C4×C7×D4—22no—
99C4×C7×D4—22no—
100C4×C7×D4—22no—
101C4×C7×D4—22no—
102C4×C7×D4—22no—
103C4×C7×D4—22no—
104C4×C7×D4D412yescomplemented
105C7×#117—21no—
106C7×#117—22no—
107C2×C7×C4⋊C4—21no—
108C2×C7×C4⋊C4D411yescharacteristic
109C2×C7×C22⋊C4—21no—
110C2×C7×C22⋊C4—21no—
111C2×C7×C22⋊C4—21no—
112C2×C7×C22⋊C4—21no—
113C2×C7×C22⋊C4—21no—
114C2×C7×C22⋊C4—21no—
115C2×C7×C22⋊C4—21no—
116C2×C7×C22⋊C4—21no—
117C2×C7×C22⋊C4C2311yescharacteristic
118C2×C7×C22⋊C4C2311yescharacteristic
119C2×C42×C7—21no—
120C2×C42×C7C2311yescharacteristic
121C7×C4⋊C8—22no—
122C7×#103—22no—
123C7×#103—22no—
124C7×#102—22no—
125C7×#102—22no—
126C7×C22⋊C8C2312yes—
127C7×C22⋊C8C2312yes—
128C4×C7×C8—22no—
129C7×#95—21no—
130C7×#95—21no—
131C7×#95—21no—
132C7×#95—21no—
133C7×#95C2311yescharacteristic
134C7×#95C2311yescharacteristic
135#3459C2×C711yescharacteristic, complemented
136#3211C2×C712yescomplemented
137#2631C2×C711yescharacteristic, complemented
138#1606C2×C712yescomplemented
139#1602C2×C711yescharacteristic, complemented
140#1601C2×C711yescharacteristic, complemented
141#1600C2×C712yescomplemented
142#1596C2×C711yescharacteristic, complemented
143#1595C2×C711yescharacteristic, complemented
144#1521C2×C712yescomplemented
145#1460C2×C711yescharacteristic, complemented
146C24×C7—21no—
147C24×C7—21no—
148C24×C7C2×D411yescharacteristic
149C24×C7C2×D411yescharacteristic
150C7×#41—82no—
151C2×C7×Q8—21no—
152C2×C7×Q8—22no—
153C2×C7×Q8—81no—
154C2×C7×Q8C2×D412yes—
155C2×C7×D4—22no—
156C2×C7×D4—22no—
157C2×C7×D4—22no—
158C2×C7×D4—22no—
159C2×C7×D4—21no—
160C2×C7×D4—22no—
161C2×C7×D4—22no—
162C2×C7×D4—22no—
163C2×C7×D4—41no—
164C2×C7×D4—42no—
165C2×C7×D4—42no—
166C2×C7×D4—42no—
167C2×C7×D4—42no—
168C2×C7×D4—81no—
169C2×C7×D4—81no—
170C2×C7×D4—81no—
171C2×C7×D4—81no—
172C2×C7×D4—81no—
173C2×C7×D4C2×D412yes—
174C22×C4×C7—21no—
175C22×C4×C7—21no—
176C22×C4×C7—21no—
177C22×C4×C7—22no—
178C22×C4×C7—21no—
179C22×C4×C7—21no—
180C22×C4×C7—22no—
181C22×C4×C7—21no—
182C22×C4×C7—21no—
183C22×C4×C7—21no—
184C22×C4×C7—21no—
185C22×C4×C7—21no—
186C22×C4×C7—81no—
187C22×C4×C7—81no—
188C22×C4×C7—82no—
189C22×C4×C7C2×D411yescharacteristic
190C22×C4×C7C2×D411yescharacteristic
191C22×C4×C7C2×D411yescharacteristic
192C22×C4×C7C2×D411yescharacteristic
193C22×C4×C7C2411yescharacteristic
194C7×Q16—82no—
195C7×SD16—82no—
196C7×SD16—82no—
197C7×D8—82no—
198C2×C7×C8—22no—
199C2×C7×C8—22no—
200C7×C4⋊C4—22no—
201C7×C4⋊C4—21no—
202C7×C4⋊C4—21no—
203C7×C4⋊C4—22no—
204C7×C4⋊C4—22no—
205C7×C4⋊C4—22no—
206C7×C4⋊C4—22no—
207C7×C4⋊C4—22no—
208C7×C4⋊C4—22no—
209C7×C4⋊C4C2×D412yes—
210C7×C4⋊C4C2×D412yes—
211C7×C22⋊C4—22no—
212C7×C22⋊C4—22no—
213C7×C22⋊C4—22no—
214C7×C22⋊C4—22no—
215C7×C22⋊C4—22no—
216C7×C22⋊C4—22no—
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—41no—
224C7×C22⋊C4—42no—
225C7×C22⋊C4—41no—
226C7×C22⋊C4—81no—
227C7×C22⋊C4—81no—
228C7×C22⋊C4—81no—
229C7×C22⋊C4—81no—
230C42×C7—22no—
231C42×C7—22no—
232C42×C7C2×D412yes—
233C42×C7C2×D412yes—
234#543C22×C711yescharacteristic
235#539—22no—
236#539—21no—
237#538—22no—
238#538C22×C712yes—
239#534—21no—
240#534C22×C711yescharacteristic
241#529—22no—
242#529C22×C712yes—
243C2×#124C22×C711yescharacteristic, complemented
244#525—21no—
245#525—22no—
246C2×C4×D4—21no—
247C2×C4×D4C22×C711yescharacteristic
248#495—22no—
249#488—22no—
250#464—22no—
251#463—22no—
252#451C22×C712yescomplemented
253#450C22×C712yescomplemented
254#448C22×C712yes—
255#447C22×C712yes—
256#431C22×C712yescomplemented
257#397—21no—
258#393C22×C711yescharacteristic
259#393C22×C711yescharacteristic, complemented
260#392—21no—
261#392C22×C711yescharacteristic
262#392C22×C711yescharacteristic, complemented
263#386—21no—
264#386—21no—
265#383C22×C711yescharacteristic, complemented
266#328C22×C712yescomplemented
267#328C22×C712yescomplemented
268#327C22×C712yes—
269#327C22×C712yes—
270#323C22×C711yescharacteristic
271#323C22×C711yescharacteristic
272#323C22×C712yescomplemented
273C23×C7—21no—
274C23×C7—21no—
275C23×C7—21no—
276C23×C7—22no—
277C23×C7—21no—
278C23×C7—21no—
279C23×C7—41no—
280C23×C7—41no—
281C23×C7—41no—
282C23×C7—41no—
283C23×C7—42no—
284C23×C7—41no—
285C23×C7—41no—
286C23×C7—41no—
287C23×C7—41no—
288C23×C7—41no—
289C23×C7—41no—
290C23×C7—41no—
291C23×C7—81no—
292C23×C7—81no—
293C23×C7#13512yes—
294C23×C7C22×D411yescharacteristic
295C7×Q8—42no—
296C7×Q8—42no—
297C7×Q8—42no—
298C7×Q8—81no—
299C7×D4—44no—
300C7×D4—42no—
301C7×D4—42no—
302C7×D4—42no—
303C7×D4—44no—
304C7×D4—42no—
305C7×D4—42no—
306C7×D4—82no—
307C7×D4—81no—
308C7×D4—82no—
309C7×D4—84no—
310C7×D4—84no—
311C2×C4×C7—21no—
312C2×C4×C7—21no—
313C2×C4×C7—22no—
314C2×C4×C7—22no—
315C2×C4×C7—21no—
316C2×C4×C7—21no—
317C2×C4×C7—22no—
318C2×C4×C7—41no—
319C2×C4×C7—41no—
320C2×C4×C7—42no—
321C2×C4×C7—42no—
322C2×C4×C7—44no—
323C2×C4×C7—42no—
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—41no—
332C2×C4×C7—41no—
333C2×C4×C7—41no—
334C2×C4×C7—41no—
335C2×C4×C7—81no—
336C2×C4×C7—81no—
337C2×C4×C7—81no—
338C2×C4×C7—82no—
339C2×C4×C7—81no—
340C2×C4×C7#12011yescharacteristic
341C2×C4×C7#12012yes—
342C2×C4×C7#12011yescharacteristic
343C2×C4×C7C22×D412yes—
344C7×C8—42no—
345C7×C8—42no—
346C2×#41—21no—
347C22×Q8C23×C711yescharacteristic
348C22×D4—21no—
349C22×D4—21no—
350C22×D4—21no—
351C22×D4C7×D411yescharacteristic
352C22×D4C23×C711yescharacteristic
353C23×C4—21no—
354C23×C4C7×D411yescharacteristic
355C23×C4C7×D411yescharacteristic
356C2×Q16—22no—
357C2×SD16—22no—
358C2×SD16—22no—
359C2×D8—22no—
360#126—22no—
361#126—22no—
362#126—22no—
363#125—22no—
364#125C7×D412yescomplemented
365#124—22no—
366#124—22no—
367#124—22no—
368#124C7×D412yescomplemented
369#123—22no—
370#123—22no—
371#123—22no—
372#123—22no—
373#123—22no—
374#123—22no—
375C22⋊Q8—21no—
376C22⋊Q8—22no—
377C22⋊Q8—22no—
378C22⋊Q8—21no—
379C22⋊Q8—22no—
380C22⋊Q8C23×C712yes—
381#121—22no—
382#121—21no—
383#121—22no—
384#121—22no—
385#121—21no—
386#121C23×C712yes—
387#120—22no—
388#120—22no—
389#120—22no—
390C4×Q8C7×D412yescomplemented
391C4×D4—22no—
392C4×D4—22no—
393C4×D4—22no—
394C4×D4—22no—
395C4×D4—22no—
396C4×D4—22no—
397C4×D4—22no—
398C4×D4C7×D412yescomplemented
399#117—21no—
400#117—22no—
401C2×C4⋊C4—21no—
402C2×C4⋊C4C7×D411yescharacteristic
403C2×C22⋊C4—21no—
404C2×C22⋊C4—21no—
405C2×C22⋊C4—21no—
406C2×C22⋊C4—21no—
407C2×C22⋊C4—21no—
408C2×C22⋊C4—21no—
409C2×C22⋊C4—21no—
410C2×C22⋊C4—21no—
411C2×C22⋊C4C23×C711yescharacteristic
412C2×C22⋊C4C23×C711yescharacteristic
413C2×C42—21no—
414C2×C42C23×C711yescharacteristic
415C4⋊C8—22no—
416#103—22no—
417#103—22no—
418#102—22no—
419#102—22no—
420C22⋊C8C23×C712yes—
421C22⋊C8C23×C712yes—
422C4×C8—22no—
423#95—21no—
424#95—21no—
425#95—21no—
426#95—21no—
427#95C23×C711yescharacteristic
428#95C23×C711yescharacteristic
429C22×C7—21no—
430C22×C7—21no—
431C22×C7—22no—
432C22×C7—42no—
433C22×C7—42no—
434C22×C7—41no—
435C22×C7—41no—
436C22×C7—41no—
437C22×C7—42no—
438C22×C7—41no—
439C22×C7—42no—
440C22×C7—42no—
441C22×C7—41no—
442C22×C7—81no—
443C22×C7—81no—
444C22×C7—81no—
445C22×C7—81no—
446C22×C7—81no—
447C22×C7—81no—
448C22×C7—84no—
449C22×C7—81no—
450C22×C7—81no—
451C22×C7—81no—
452C22×C7C2×#12011yescharacteristic
453C22×C7C2×#13511yescharacteristic
454C22×C7#57511yescharacteristic
455C4×C7—22no—
456C4×C7—21no—
457C4×C7—21no—
458C4×C7—22no—
459C4×C7—42no—
460C4×C7—41no—
461C4×C7—41no—
462C4×C7—42no—
463C4×C7—42no—
464C4×C7—81no—
465C4×C7—81no—
466C4×C7—81no—
467C24—21no—
468C24—21no—
469C24C2×C7×D411yescharacteristic
470C24C2×C7×D411yescharacteristic
471#41—82no—
472C2×Q8—21no—
473C2×Q8—22no—
474C2×Q8—81no—
475C2×Q8C2×C7×D412yes—
476C2×D4—22no—
477C2×D4—22no—
478C2×D4—22no—
479C2×D4—22no—
480C2×D4—21no—
481C2×D4—22no—
482C2×D4—22no—
483C2×D4—22no—
484C2×D4—41no—
485C2×D4—42no—
486C2×D4—42no—
487C2×D4—42no—
488C2×D4—42no—
489C2×D4—81no—
490C2×D4—81no—
491C2×D4—81no—
492C2×D4—81no—
493C2×D4—81no—
494C2×D4C2×C7×D412yes—
495C22×C4—21no—
496C22×C4—21no—
497C22×C4—21no—
498C22×C4—22no—
499C22×C4—21no—
500C22×C4—21no—
501C22×C4—22no—
502C22×C4—21no—
503C22×C4—21no—
504C22×C4—21no—
505C22×C4—21no—
506C22×C4—21no—
507C22×C4—81no—
508C22×C4—81no—
509C22×C4—82no—
510C22×C4C2×C7×D411yescharacteristic
511C22×C4C2×C7×D411yescharacteristic
512C22×C4C2×C7×D411yescharacteristic
513C22×C4C2×C7×D411yescharacteristic
514C22×C4C24×C711yescharacteristic
515Q16—82no—
516SD16—82no—
517SD16—82no—
518D8—82no—
519C2×C8—22no—
520C2×C8—22no—
521C4⋊C4—22no—
522C4⋊C4—21no—
523C4⋊C4—21no—
524C4⋊C4—22no—
525C4⋊C4—22no—
526C4⋊C4—22no—
527C4⋊C4—22no—
528C4⋊C4—22no—
529C4⋊C4—22no—
530C4⋊C4C2×C7×D412yes—
531C4⋊C4C2×C7×D412yes—
532C22⋊C4—22no—
533C22⋊C4—22no—
534C22⋊C4—22no—
535C22⋊C4—22no—
536C22⋊C4—22no—
537C22⋊C4—22no—
538C22⋊C4—22no—
539C22⋊C4—22no—
540C22⋊C4—22no—
541C22⋊C4—22no—
542C22⋊C4—22no—
543C22⋊C4—22no—
544C22⋊C4—41no—
545C22⋊C4—42no—
546C22⋊C4—41no—
547C22⋊C4—81no—
548C22⋊C4—81no—
549C22⋊C4—81no—
550C22⋊C4—81no—
551C42—22no—
552C42—22no—
553C42C2×C7×D412yes—
554C42C2×C7×D412yes—
555C2×C7—21no—
556C2×C7—21no—
557C2×C7—42no—
558C2×C7—81no—
559C2×C7—82no—
560C2×C7—161no—
561C2×C7#300211yescharacteristic
562C2×C7#301911yescharacteristic
563C2×C7#302711yescharacteristic
564C23—21no—
565C23—21no—
566C23—21no—
567C23—22no—
568C23—21no—
569C23—21no—
570C23—41no—
571C23—41no—
572C23—41no—
573C23—41no—
574C23—42no—
575C23—41no—
576C23—41no—
577C23—41no—
578C23—41no—
579C23—41no—
580C23—41no—
581C23—41no—
582C23—81no—
583C23—81no—
584C23C7×#13512yes—
585C23C22×C7×D411yescharacteristic
586Q8—42no—
587Q8—42no—
588Q8—42no—
589Q8—81no—
590D4—44no—
591D4—42no—
592D4—42no—
593D4—42no—
594D4—44no—
595D4—42no—
596D4—42no—
597D4—82no—
598D4—81no—
599D4—82no—
600D4—84no—
601D4—84no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—22no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—44no—
614C2×C4—42no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—41no—
623C2×C4—41no—
624C2×C4—41no—
625C2×C4—41no—
626C2×C4—81no—
627C2×C4—81no—
628C2×C4—81no—
629C2×C4—82no—
630C2×C4—81no—
631C2×C4C7×#12011yescharacteristic
632C2×C4C7×#12012yes—
633C2×C4C7×#12011yescharacteristic
634C2×C4C22×C7×D412yes—
635C8—42no—
636C8—42no—
637C7#1830611yescharacteristic, direct product factor, Sylow
638C22—21no—
639C22—21no—
640C22—22no—
641C22—42no—
642C22—42no—
643C22—41no—
644C22—41no—
645C22—41no—
646C22—42no—
647C22—41no—
648C22—42no—
649C22—42no—
650C22—41no—
651C22—81no—
652C22—81no—
653C22—81no—
654C22—81no—
655C22—81no—
656C22—81no—
657C22—84no—
658C22—81no—
659C22—81no—
660C22—81no—
661C22C2×C7×#12011yescharacteristic
662C22C2×C7×#13511yescharacteristic
663C22C7×#57511yescharacteristic
664C4—22no—
665C4—21no—
666C4—21no—
667C4—22no—
668C4—42no—
669C4—41no—
670C4—41no—
671C4—42no—
672C4—42no—
673C4—81no—
674C4—81no—
675C4—81no—
676C2—21no—
677C2—21no—
678C2—42no—
679C2—81no—
680C2—82no—
681C2—161no—
682C2C7×#300211yescharacteristic
683C2C7×#301911yescharacteristic
684C2C7×#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×#3459C2semidirect
2. C7×#3211C2semidirect
3. C7×#2631C2semidirect
4. C7×#1606C2semidirect
5. C7×#1602C2semidirect
6. C7×#1601C2semidirect
7. C7×#1600C2semidirect
8. C7×#1596C2semidirect
9. C7×#1595C2semidirect
10. C7×#1521C2semidirect
11. C7×#1460C2semidirect
21. C2×C7×#124C22semidirect
30. C7×#451C22semidirect
31. C7×#450C22semidirect
34. C7×#431C22semidirect
37. C7×#393C22semidirect
40. C7×#392C22semidirect
43. C7×#383C22semidirect
44. C7×#328C22semidirect
45. C7×#328C22semidirect
50. C7×#323C22semidirect
51. #18306C7direct (N × H)
70. C7×#125D4semidirect
74. C7×#124D4semidirect
96. C4×C7×Q8D4semidirect
104. C4×C7×D4D4semidirect
135. #3459C2×C7semidirect
136. #3211C2×C7semidirect
137. #2631C2×C7semidirect
138. #1606C2×C7semidirect
139. #1602C2×C7semidirect
140. #1601C2×C7semidirect
141. #1600C2×C7semidirect
142. #1596C2×C7semidirect
143. #1595C2×C7semidirect
144. #1521C2×C7semidirect
145. #1460C2×C7semidirect
243. C2×#124C22×C7semidirect
252. #451C22×C7semidirect
253. #450C22×C7semidirect
256. #431C22×C7semidirect
259. #393C22×C7semidirect
262. #392C22×C7semidirect
265. #383C22×C7semidirect
266. #328C22×C7semidirect
267. #328C22×C7semidirect
272. #323C22×C7semidirect
364. #125C7×D4semidirect
368. #124C7×D4semidirect
390. C4×Q8C7×D4semidirect
398. C4×D4C7×D4semidirect
637. C7#18306direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#34592
C7×#32112
C7×#26312
C7×#16062
C7×#16022
C7×#16012
C7×#16002
C7×#15962
C7×#15952
C7×#15212
C7×#14602
#183067