Finite Group Databasebeta

← C7×#18334 #13195550

Subgroups — C7×#18334 #13195550

Order 1792.

972 conjugacy classes of subgroups.

The subgroup lattice has 698 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3455C211yesmaximal, characteristic, complemented
2C7×#3215C211yesmaximal, characteristic, complemented
3C7×#3213C211yesmaximal, characteristic, complemented
4C7×#2610C211yesmaximal, characteristic, complemented
5C7×#1606C212yesmaximal, complemented
6C7×#1603C212yesmaximal, complemented
7C7×#1600C212yesmaximal, complemented
8C7×#1597C212yesmaximal, 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×#124C2211yescharacteristic
25C7×#525—21no—
26C4×C7×#41—21no—
27C2×C7×#117C2211yescharacteristic
28C7×#497—21no—
29C7×#496—21no—
30C7×#489—21no—
31C7×#489—21no—
32C7×#464—22no—
33C7×#463—22no—
34C7×#451C2212yes—
35C7×#450C2212yes—
36C7×#448C2212yes—
37C7×#447C2212yes—
38C7×#432C2211yescharacteristic
39C7×#432C2211yescharacteristic
40C7×#395C2211yescharacteristic
41C7×#394C2211yescharacteristic
42C7×#393C2211yescharacteristic
43C7×#392C2211yescharacteristic
44C7×#390—21no—
45C7×#388—21no—
46C7×#388—21no—
47C7×#386—21no—
48C7×#376C2211yescharacteristic
49C7×#328C2212yes—
50C7×#328C2212yes—
51C7×#327C2212yes—
52C7×#327C2212yes—
53C7×#324C2212yes—
54C7×#323C2212yes—
55#18334C711yesmaximal, characteristic, direct product factor, Sylow
56C2×C7×#41—21no—
57C22×C7×Q8C2311yescharacteristic
58C22×C7×D4—21no—
59C22×C7×D4C2311yescharacteristic
60C23×C4×C7D411yescharacteristic
61C2×C7×Q16—22no—
62C2×C7×SD16—22no—
63C2×C7×SD16—22no—
64C2×C7×D8—22no—
65C7×C4⋊Q8D411yescharacteristic, complemented
66C7×C4⋊Q8D411yescharacteristic, complemented
67C7×C4⋊Q8D411yescharacteristic, complemented
68C7×#127D411yescharacteristic, complemented
69C7×#126—22no—
70C7×#126—22no—
71C7×#126—22no—
72C7×#126—22no—
73C7×#125—22no—
74C7×#125—22no—
75C7×#125—21no—
76C7×#124—22no—
77C7×#124—22no—
78C7×#124—21no—
79C7×#124—21no—
80C7×#124—21no—
81C7×#124—21no—
82C7×#124—21no—
83C7×#123—22no—
84C7×#123—22no—
85C7×#123—22no—
86C7×#123—22no—
87C7×#123—22no—
88C7×C22⋊Q8—21no—
89C7×C22⋊Q8—21no—
90C7×C22⋊Q8—22no—
91C7×C22⋊Q8—22no—
92C7×C22⋊Q8—21no—
93C7×C22⋊Q8C2312yes—
94C7×#121—21no—
95C7×#121—21no—
96C7×#121—22no—
97C7×#121—22no—
98C7×#121—21no—
99C7×#121C2312yes—
100C7×#120—22no—
101C4×C7×Q8—21no—
102C4×C7×Q8—21no—
103C4×C7×Q8—21no—
104C4×C7×Q8D412yescomplemented
105C4×C7×D4—21no—
106C4×C7×D4—21no—
107C4×C7×D4—22no—
108C4×C7×D4—22no—
109C4×C7×D4—22no—
110C4×C7×D4—22no—
111C4×C7×D4—21no—
112C4×C7×D4D412yescomplemented
113C7×#117—22no—
114C7×#117—21no—
115C7×#117—21no—
116C7×#117—21no—
117C7×#117—21no—
118C7×#117—21no—
119C2×C7×C4⋊C4—21no—
120C2×C7×C4⋊C4D411yescharacteristic
121C2×C7×C4⋊C4C2311yescharacteristic
122C2×C7×C22⋊C4—21no—
123C2×C7×C22⋊C4—21no—
124C2×C7×C22⋊C4—21no—
125C2×C7×C22⋊C4—21no—
126C2×C7×C22⋊C4—21no—
127C2×C7×C22⋊C4—21no—
128C2×C7×C22⋊C4D411yescharacteristic
129C2×C7×C22⋊C4C2311yescharacteristic
130C2×C42×C7—22no—
131C2×C42×C7—21no—
132C2×C42×C7D411yescharacteristic
133C2×C42×C7C2311yescharacteristic
134C7×C4⋊C8—21no—
135C7×C4⋊C8—21no—
136C7×#103—22no—
137C7×#103—22no—
138C7×#102—22no—
139C7×#102—22no—
140C7×C22⋊C8C2312yes—
141C7×C22⋊C8C2312yes—
142C7×#97—21no—
143C7×#97—21no—
144C7×#95—21no—
145C7×#95—21no—
146C7×#95—21no—
147C7×#95C2311yescharacteristic
148C7×#95C2311yescharacteristic
149#3455C2×C711yescharacteristic, complemented
150#3215C2×C711yescharacteristic, complemented
151#3213C2×C711yescharacteristic, complemented
152#2610C2×C711yescharacteristic, complemented
153#1606C2×C712yescomplemented
154#1603C2×C712yescomplemented
155#1600C2×C712yescomplemented
156#1597C2×C712yescomplemented
157#1506C2×C711yescharacteristic, complemented
158#1505C2×C711yescharacteristic, complemented
159#1464C2×C711yescharacteristic, complemented
160C24×C7—21no—
161C24×C7C2×D411yescharacteristic
162C7×#41—42no—
163C7×#41—42no—
164C2×C7×Q8—21no—
165C2×C7×Q8—21no—
166C2×C7×Q8—21no—
167C2×C7×Q8—21no—
168C2×C7×Q8—81no—
169C2×C7×Q8C2×D412yes—
170C2×C7×D4—22no—
171C2×C7×D4—22no—
172C2×C7×D4—22no—
173C2×C7×D4—21no—
174C2×C7×D4—21no—
175C2×C7×D4—21no—
176C2×C7×D4—21no—
177C2×C7×D4—42no—
178C2×C7×D4—42no—
179C2×C7×D4—81no—
180C2×C7×D4—81no—
181C2×C7×D4C2×D412yes—
182C22×C4×C7—21no—
183C22×C4×C7—22no—
184C22×C4×C7—22no—
185C22×C4×C7—22no—
186C22×C4×C7—21no—
187C22×C4×C7—21no—
188C22×C4×C7—21no—
189C22×C4×C7—21no—
190C22×C4×C7—41no—
191C22×C4×C7—41no—
192C22×C4×C7C2×D411yescharacteristic
193C22×C4×C7C2×D411yescharacteristic
194C22×C4×C7C2×D411yescharacteristic
195C22×C4×C7C2×D411yescharacteristic
196C22×C4×C7C2×D411yescharacteristic
197C22×C4×C7C2411yescharacteristic
198C7×Q16—82no—
199C7×SD16—82no—
200C7×SD16—82no—
201C7×D8—82no—
202C2×C7×C8—22no—
203C2×C7×C8—22no—
204C7×C4⋊C4—22no—
205C7×C4⋊C4—22no—
206C7×C4⋊C4—22no—
207C7×C4⋊C4—22no—
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—22no—
215C7×C4⋊C4—21no—
216C7×C4⋊C4—21no—
217C7×C4⋊C4—22no—
218C7×C4⋊C4—81no—
219C7×C4⋊C4C2×D412yes—
220C7×C4⋊C4C2×D412yes—
221C7×C22⋊C4—22no—
222C7×C22⋊C4—22no—
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—21no—
230C7×C22⋊C4—22no—
231C7×C22⋊C4—21no—
232C7×C22⋊C4—21no—
233C7×C22⋊C4—22no—
234C7×C22⋊C4—42no—
235C7×C22⋊C4—42no—
236C7×C22⋊C4—41no—
237C7×C22⋊C4—41no—
238C7×C22⋊C4—41no—
239C7×C22⋊C4—41no—
240C7×C22⋊C4—81no—
241C7×C22⋊C4—81no—
242C42×C7—22no—
243C42×C7—22no—
244C42×C7—21no—
245C42×C7—22no—
246C42×C7—21no—
247C42×C7—42no—
248C42×C7C2×D411yescharacteristic
249C42×C7C2×D411yescharacteristic
250C42×C7C2×D411yescharacteristic
251C42×C7C2×D411yescharacteristic
252#542—21no—
253#542—21no—
254#541—21no—
255#540—21no—
256#539—21no—
257#539C22×C712yes—
258#538—21no—
259#538C22×C712yes—
260#533C22×C711yescharacteristic
261#532C22×C711yescharacteristic
262#529—22no—
263#529—22no—
264C2×#124C22×C711yescharacteristic
265#525—21no—
266C4×#41—21no—
267C2×#117C22×C711yescharacteristic
268#497—21no—
269#496—21no—
270#489—21no—
271#489—21no—
272#464—22no—
273#463—22no—
274#451C22×C712yes—
275#450C22×C712yes—
276#448C22×C712yes—
277#447C22×C712yes—
278#432C22×C711yescharacteristic
279#432C22×C711yescharacteristic
280#395C22×C711yescharacteristic
281#394C22×C711yescharacteristic
282#393C22×C711yescharacteristic
283#392C22×C711yescharacteristic
284#390—21no—
285#388—21no—
286#388—21no—
287#386—21no—
288#376C22×C711yescharacteristic
289#328C22×C712yes—
290#328C22×C712yes—
291#327C22×C712yes—
292#327C22×C712yes—
293#324C22×C712yes—
294#323C22×C712yes—
295C23×C7—21no—
296C23×C7—22no—
297C23×C7—21no—
298C23×C7—21no—
299C23×C7—41no—
300C23×C7—41no—
301C23×C7—41no—
302C23×C7—41no—
303C23×C7—41no—
304C23×C7—41no—
305C23×C7—41no—
306C23×C7—41no—
307C23×C7—41no—
308C23×C7C22×D411yescharacteristic
309C7×Q8—42no—
310C7×Q8—42no—
311C7×Q8—42no—
312C7×Q8—42no—
313C7×Q8—81no—
314C7×D4—42no—
315C7×D4—42no—
316C7×D4—42no—
317C7×D4—42no—
318C7×D4—42no—
319C7×D4—42no—
320C7×D4—81no—
321C7×D4—82no—
322C7×D4—82no—
323C7×D4—84no—
324C2×C4×C7—21no—
325C2×C4×C7—21no—
326C2×C4×C7—22no—
327C2×C4×C7—21no—
328C2×C4×C7—21no—
329C2×C4×C7—22no—
330C2×C4×C7—22no—
331C2×C4×C7—22no—
332C2×C4×C7—21no—
333C2×C4×C7—21no—
334C2×C4×C7—21no—
335C2×C4×C7—41no—
336C2×C4×C7—41no—
337C2×C4×C7—42no—
338C2×C4×C7—42no—
339C2×C4×C7—44no—
340C2×C4×C7—41no—
341C2×C4×C7—41no—
342C2×C4×C7—41no—
343C2×C4×C7—41no—
344C2×C4×C7—41no—
345C2×C4×C7—41no—
346C2×C4×C7—41no—
347C2×C4×C7—41no—
348C2×C4×C7—41no—
349C2×C4×C7—41no—
350C2×C4×C7—82no—
351C2×C4×C7—82no—
352C2×C4×C7#12011yescharacteristic
353C2×C4×C7#12011yescharacteristic
354C2×C4×C7#12011yescharacteristic
355C2×C4×C7#12011yescharacteristic
356C2×C4×C7C22×D411yescharacteristic
357C2×C4×C7C22×D411yescharacteristic
358C7×C8—42no—
359C7×C8—42no—
360C2×#41—21no—
361C22×Q8C23×C711yescharacteristic
362C22×D4—21no—
363C22×D4C23×C711yescharacteristic
364C23×C4C7×D411yescharacteristic
365C2×Q16—22no—
366C2×SD16—22no—
367C2×SD16—22no—
368C2×D8—22no—
369C4⋊Q8C7×D411yescharacteristic, complemented
370C4⋊Q8C7×D411yescharacteristic, complemented
371C4⋊Q8C7×D411yescharacteristic, complemented
372#127C7×D411yescharacteristic, complemented
373#126—22no—
374#126—22no—
375#126—22no—
376#126—22no—
377#125—22no—
378#125—22no—
379#125—21no—
380#124—22no—
381#124—22no—
382#124—21no—
383#124—21no—
384#124—21no—
385#124—21no—
386#124—21no—
387#123—22no—
388#123—22no—
389#123—22no—
390#123—22no—
391#123—22no—
392C22⋊Q8—21no—
393C22⋊Q8—21no—
394C22⋊Q8—22no—
395C22⋊Q8—22no—
396C22⋊Q8—21no—
397C22⋊Q8C23×C712yes—
398#121—21no—
399#121—21no—
400#121—22no—
401#121—22no—
402#121—21no—
403#121C23×C712yes—
404#120—22no—
405C4×Q8—21no—
406C4×Q8—21no—
407C4×Q8—21no—
408C4×Q8C7×D412yescomplemented
409C4×D4—21no—
410C4×D4—21no—
411C4×D4—22no—
412C4×D4—22no—
413C4×D4—22no—
414C4×D4—22no—
415C4×D4—21no—
416C4×D4C7×D412yescomplemented
417#117—22no—
418#117—21no—
419#117—21no—
420#117—21no—
421#117—21no—
422#117—21no—
423C2×C4⋊C4—21no—
424C2×C4⋊C4C7×D411yescharacteristic
425C2×C4⋊C4C23×C711yescharacteristic
426C2×C22⋊C4—21no—
427C2×C22⋊C4—21no—
428C2×C22⋊C4—21no—
429C2×C22⋊C4—21no—
430C2×C22⋊C4—21no—
431C2×C22⋊C4—21no—
432C2×C22⋊C4C7×D411yescharacteristic
433C2×C22⋊C4C23×C711yescharacteristic
434C2×C42—22no—
435C2×C42—21no—
436C2×C42C7×D411yescharacteristic
437C2×C42C23×C711yescharacteristic
438C4⋊C8—21no—
439C4⋊C8—21no—
440#103—22no—
441#103—22no—
442#102—22no—
443#102—22no—
444C22⋊C8C23×C712yes—
445C22⋊C8C23×C712yes—
446#97—21no—
447#97—21no—
448#95—21no—
449#95—21no—
450#95—21no—
451#95C23×C711yescharacteristic
452#95C23×C711yescharacteristic
453C22×C7—21no—
454C22×C7—21no—
455C22×C7—42no—
456C22×C7—41no—
457C22×C7—42no—
458C22×C7—42no—
459C22×C7—41no—
460C22×C7—42no—
461C22×C7—41no—
462C22×C7—81no—
463C22×C7—81no—
464C22×C7—81no—
465C22×C7—81no—
466C22×C7—81no—
467C22×C7—81no—
468C22×C7—81no—
469C22×C7—81no—
470C22×C7C2×#12011yescharacteristic
471C22×C7#57511yescharacteristic
472C22×C7#57511yescharacteristic
473C4×C7—21no—
474C4×C7—21no—
475C4×C7—21no—
476C4×C7—21no—
477C4×C7—21no—
478C4×C7—21no—
479C4×C7—42no—
480C4×C7—41no—
481C4×C7—42no—
482C4×C7—41no—
483C4×C7—42no—
484C4×C7—41no—
485C4×C7—42no—
486C4×C7—81no—
487C4×C7—81no—
488C24—21no—
489C24C2×C7×D411yescharacteristic
490#41—42no—
491#41—42no—
492C2×Q8—21no—
493C2×Q8—21no—
494C2×Q8—21no—
495C2×Q8—21no—
496C2×Q8—81no—
497C2×Q8C2×C7×D412yes—
498C2×D4—22no—
499C2×D4—22no—
500C2×D4—22no—
501C2×D4—21no—
502C2×D4—21no—
503C2×D4—21no—
504C2×D4—21no—
505C2×D4—42no—
506C2×D4—42no—
507C2×D4—81no—
508C2×D4—81no—
509C2×D4C2×C7×D412yes—
510C22×C4—21no—
511C22×C4—22no—
512C22×C4—22no—
513C22×C4—22no—
514C22×C4—21no—
515C22×C4—21no—
516C22×C4—21no—
517C22×C4—21no—
518C22×C4—41no—
519C22×C4—41no—
520C22×C4C2×C7×D411yescharacteristic
521C22×C4C2×C7×D411yescharacteristic
522C22×C4C2×C7×D411yescharacteristic
523C22×C4C2×C7×D411yescharacteristic
524C22×C4C2×C7×D411yescharacteristic
525C22×C4C24×C711yescharacteristic
526Q16—82no—
527SD16—82no—
528SD16—82no—
529D8—82no—
530C2×C8—22no—
531C2×C8—22no—
532C4⋊C4—22no—
533C4⋊C4—22no—
534C4⋊C4—22no—
535C4⋊C4—22no—
536C4⋊C4—21no—
537C4⋊C4—21no—
538C4⋊C4—21no—
539C4⋊C4—21no—
540C4⋊C4—21no—
541C4⋊C4—21no—
542C4⋊C4—22no—
543C4⋊C4—21no—
544C4⋊C4—21no—
545C4⋊C4—22no—
546C4⋊C4—81no—
547C4⋊C4C2×C7×D412yes—
548C4⋊C4C2×C7×D412yes—
549C22⋊C4—22no—
550C22⋊C4—22no—
551C22⋊C4—22no—
552C22⋊C4—22no—
553C22⋊C4—22no—
554C22⋊C4—22no—
555C22⋊C4—22no—
556C22⋊C4—21no—
557C22⋊C4—21no—
558C22⋊C4—22no—
559C22⋊C4—21no—
560C22⋊C4—21no—
561C22⋊C4—22no—
562C22⋊C4—42no—
563C22⋊C4—42no—
564C22⋊C4—41no—
565C22⋊C4—41no—
566C22⋊C4—41no—
567C22⋊C4—41no—
568C22⋊C4—81no—
569C22⋊C4—81no—
570C42—22no—
571C42—22no—
572C42—21no—
573C42—22no—
574C42—21no—
575C42—42no—
576C42C2×C7×D411yescharacteristic
577C42C2×C7×D411yescharacteristic
578C42C2×C7×D411yescharacteristic
579C42C2×C7×D411yescharacteristic
580C2×C7—21no—
581C2×C7—21no—
582C2×C7—82no—
583C2×C7—81no—
584C2×C7—161no—
585C2×C7#300511yescharacteristic
586C2×C7#301811yescharacteristic
587C2×C7#302711yescharacteristic
588C23—21no—
589C23—22no—
590C23—21no—
591C23—21no—
592C23—41no—
593C23—41no—
594C23—41no—
595C23—41no—
596C23—41no—
597C23—41no—
598C23—41no—
599C23—41no—
600C23—41no—
601C23C22×C7×D411yescharacteristic
602Q8—42no—
603Q8—42no—
604Q8—42no—
605Q8—42no—
606Q8—81no—
607D4—42no—
608D4—42no—
609D4—42no—
610D4—42no—
611D4—42no—
612D4—42no—
613D4—81no—
614D4—82no—
615D4—82no—
616D4—84no—
617C2×C4—21no—
618C2×C4—21no—
619C2×C4—22no—
620C2×C4—21no—
621C2×C4—21no—
622C2×C4—22no—
623C2×C4—22no—
624C2×C4—22no—
625C2×C4—21no—
626C2×C4—21no—
627C2×C4—21no—
628C2×C4—41no—
629C2×C4—41no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—44no—
633C2×C4—41no—
634C2×C4—41no—
635C2×C4—41no—
636C2×C4—41no—
637C2×C4—41no—
638C2×C4—41no—
639C2×C4—41no—
640C2×C4—41no—
641C2×C4—41no—
642C2×C4—41no—
643C2×C4—82no—
644C2×C4—82no—
645C2×C4C7×#12011yescharacteristic
646C2×C4C7×#12011yescharacteristic
647C2×C4C7×#12011yescharacteristic
648C2×C4C7×#12011yescharacteristic
649C2×C4C22×C7×D411yescharacteristic
650C2×C4C22×C7×D411yescharacteristic
651C8—42no—
652C8—42no—
653C7#1833411yescharacteristic, direct product factor, Sylow
654C22—21no—
655C22—21no—
656C22—42no—
657C22—41no—
658C22—42no—
659C22—42no—
660C22—41no—
661C22—42no—
662C22—41no—
663C22—81no—
664C22—81no—
665C22—81no—
666C22—81no—
667C22—81no—
668C22—81no—
669C22—81no—
670C22—81no—
671C22C2×C7×#12011yescharacteristic
672C22C7×#57511yescharacteristic
673C22C7×#57511yescharacteristic
674C4—21no—
675C4—21no—
676C4—21no—
677C4—21no—
678C4—21no—
679C4—21no—
680C4—42no—
681C4—41no—
682C4—42no—
683C4—41no—
684C4—42no—
685C4—41no—
686C4—42no—
687C4—81no—
688C4—81no—
689C2—21no—
690C2—21no—
691C2—81no—
692C2—82no—
693C2—161no—
694C2C7×#300511yescharacteristic
695C2C7×#301811yescharacteristic
696C2C7×#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×#3455C2semidirect
2. C7×#3215C2semidirect
3. C7×#3213C2semidirect
4. C7×#2610C2semidirect
5. C7×#1606C2semidirect
6. C7×#1603C2semidirect
7. C7×#1600C2semidirect
8. C7×#1597C2semidirect
9. C7×#1506C2semidirect
10. C7×#1505C2semidirect
11. C7×#1464C2semidirect
55. #18334C7direct (N × H)
65. C7×C4⋊Q8D4semidirect
66. C7×C4⋊Q8D4semidirect
67. C7×C4⋊Q8D4semidirect
68. C7×#127D4semidirect
104. C4×C7×Q8D4semidirect
112. C4×C7×D4D4semidirect
149. #3455C2×C7semidirect
150. #3215C2×C7semidirect
151. #3213C2×C7semidirect
152. #2610C2×C7semidirect
153. #1606C2×C7semidirect
154. #1603C2×C7semidirect
155. #1600C2×C7semidirect
156. #1597C2×C7semidirect
157. #1506C2×C7semidirect
158. #1505C2×C7semidirect
159. #1464C2×C7semidirect
369. C4⋊Q8C7×D4semidirect
370. C4⋊Q8C7×D4semidirect
371. C4⋊Q8C7×D4semidirect
372. #127C7×D4semidirect
408. C4×Q8C7×D4semidirect
416. C4×D4C7×D4semidirect
653. C7#18334direct (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×#32152
C7×#32132
C7×#26102
C7×#16062
C7×#16032
C7×#16002
C7×#15972
C7×#15062
C7×#15052
C7×#14642
#183347