Finite Group Databasebeta

← C7×#18314 #13195530

Subgroups — C7×#18314 #13195530

Order 1792.

958 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3459C211yesmaximal, characteristic
2C7×#3232C212yesmaximal, complemented
3C7×#2633C211yesmaximal, characteristic, complemented
4C7×#1618C212yesmaximal, complemented
5C7×#1614C211yesmaximal, characteristic, complemented
6C7×#1613C211yesmaximal, characteristic, complemented
7C7×#1612C212yesmaximal, complemented
8C7×#1608C211yesmaximal, characteristic, complemented
9C7×#1607C211yesmaximal, 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×#125C2211yescharacteristic, complemented
22C7×#525—21no—
23C7×#525—22no—
24C2×C4×C7×D4—21no—
25C2×C4×C7×D4C2211yescharacteristic
26C7×#499—22no—
27C7×#488—22no—
28C7×#484C2212yescomplemented
29C7×#483C2212yescomplemented
30C7×#481C2212yes—
31C7×#480C2212yes—
32C7×#479—22no—
33C7×#478—22no—
34C7×#431C2212yescomplemented
35C7×#399C2211yescharacteristic
36C7×#399C2211yescharacteristic, complemented
37C7×#398—21no—
38C7×#397C2211yescharacteristic
39C7×#397C2211yescharacteristic, complemented
40C7×#394—21no—
41C7×#385—21no—
42C7×#385—21no—
43C7×#383C2211yescharacteristic, complemented
44C7×#328C2212yescomplemented
45C7×#328C2212yescomplemented
46C7×#327C2212yes—
47C7×#327C2212yes—
48C7×#323C2211yescharacteristic
49C7×#323C2211yescharacteristic
50C7×#323C2212yescomplemented
51#18314C711yesmaximal, characteristic, direct product factor, Sylow
52C2×C7×#41—21no—
53C22×C7×D4—21no—
54C22×C7×D4D411yescharacteristic
55C23×C4×C7—21no—
56C23×C4×C7D411yescharacteristic
57C23×C4×C7D411yescharacteristic
58C7×#126—22no—
59C7×#126—22no—
60C7×#126—22no—
61C7×#125—22no—
62C7×#125—22no—
63C7×#125—22no—
64C7×#125D412yes—
65C7×#124—22no—
66C7×#124D412yes—
67C7×#123—22no—
68C7×#123—22no—
69C7×#123—22no—
70C7×#123—22no—
71C7×#123—22no—
72C7×#123—22no—
73C7×C22⋊Q8—21no—
74C7×C22⋊Q8—22no—
75C7×C22⋊Q8—22no—
76C7×C22⋊Q8—21no—
77C7×C22⋊Q8—22no—
78C7×C22⋊Q8C2312yes—
79C7×#121—22no—
80C7×#121—21no—
81C7×#121—22no—
82C7×#121—22no—
83C7×#121—21no—
84C7×#121C2312yes—
85C7×#120—22no—
86C7×#120—22no—
87C7×#120—22no—
88C4×C7×Q8D412yes—
89C4×C7×D4—22no—
90C4×C7×D4—22no—
91C4×C7×D4—22no—
92C4×C7×D4—22no—
93C4×C7×D4—22no—
94C4×C7×D4—22no—
95C4×C7×D4—22no—
96C4×C7×D4D412yes—
97C7×#117—21no—
98C7×#117—22no—
99C2×C7×C4⋊C4—21no—
100C2×C7×C4⋊C4—21no—
101C2×C7×C4⋊C4—21no—
102C2×C7×C4⋊C4—21no—
103C2×C7×C4⋊C4—21no—
104C2×C7×C4⋊C4D411yescharacteristic
105C2×C7×C4⋊C4C2311yescharacteristic
106C2×C7×C4⋊C4C2311yescharacteristic
107C2×C7×C22⋊C4—21no—
108C2×C7×C22⋊C4—21no—
109C2×C7×C22⋊C4—21no—
110C2×C7×C22⋊C4—21no—
111C2×C7×C22⋊C4—21no—
112C2×C7×C22⋊C4—21no—
113C2×C7×C22⋊C4C2311yescharacteristic
114C2×C7×C22⋊C4C2311yescharacteristic
115C2×C42×C7—21no—
116C2×C42×C7C2311yescharacteristic
117C7×#107—22no—
118C7×#107—22no—
119C7×#106—22no—
120C7×#106—22no—
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
136#3232C2×C712yescomplemented
137#2633C2×C711yescharacteristic, complemented
138#1618C2×C712yescomplemented
139#1614C2×C711yescharacteristic, complemented
140#1613C2×C711yescharacteristic, complemented
141#1612C2×C712yescomplemented
142#1608C2×C711yescharacteristic, complemented
143#1607C2×C711yescharacteristic, complemented
144#1521C2×C712yescomplemented
145#1460C2×C711yescharacteristic, complemented
146C24×C7—21no—
147C24×C7C2×D411yescharacteristic
148C24×C7C2×D411yescharacteristic
149C7×#41—82no—
150C2×C7×Q8—21no—
151C2×C7×Q8C2×D412yes—
152C2×C7×D4—22no—
153C2×C7×D4—22no—
154C2×C7×D4—22no—
155C2×C7×D4—22no—
156C2×C7×D4—21no—
157C2×C7×D4—22no—
158C2×C7×D4—22no—
159C2×C7×D4—41no—
160C2×C7×D4—42no—
161C2×C7×D4—42no—
162C2×C7×D4C2×D412yes—
163C22×C4×C7—21no—
164C22×C4×C7—21no—
165C22×C4×C7—21no—
166C22×C4×C7—22no—
167C22×C4×C7—21no—
168C22×C4×C7—21no—
169C22×C4×C7—22no—
170C22×C4×C7—21no—
171C22×C4×C7—21no—
172C22×C4×C7—21no—
173C22×C4×C7—21no—
174C22×C4×C7—21no—
175C22×C4×C7—21no—
176C22×C4×C7—81no—
177C22×C4×C7—81no—
178C22×C4×C7—82no—
179C22×C4×C7C2×D411yescharacteristic
180C22×C4×C7C2×D411yescharacteristic
181C22×C4×C7C2×D411yescharacteristic
182C22×C4×C7C2×D411yescharacteristic
183C22×C4×C7C2411yescharacteristic
184C2×C7×C8—22no—
185C2×C7×C8—22no—
186C7×C4⋊C4—22no—
187C7×C4⋊C4—21no—
188C7×C4⋊C4—21no—
189C7×C4⋊C4—22no—
190C7×C4⋊C4—22no—
191C7×C4⋊C4—22no—
192C7×C4⋊C4—22no—
193C7×C4⋊C4—22no—
194C7×C4⋊C4—22no—
195C7×C4⋊C4—22no—
196C7×C4⋊C4—22no—
197C7×C4⋊C4—41no—
198C7×C4⋊C4—42no—
199C7×C4⋊C4—41no—
200C7×C4⋊C4—81no—
201C7×C4⋊C4—81no—
202C7×C4⋊C4C2×D412yes—
203C7×C4⋊C4C2×D412yes—
204C7×C22⋊C4—22no—
205C7×C22⋊C4—22no—
206C7×C22⋊C4—22no—
207C7×C22⋊C4—22no—
208C7×C22⋊C4—22no—
209C7×C22⋊C4—22no—
210C7×C22⋊C4—22no—
211C7×C22⋊C4—22no—
212C7×C22⋊C4—22no—
213C7×C22⋊C4—22no—
214C7×C22⋊C4—22no—
215C7×C22⋊C4—22no—
216C7×C22⋊C4—42no—
217C7×C22⋊C4—42no—
218C7×C22⋊C4—81no—
219C7×C22⋊C4—81no—
220C42×C7—22no—
221C42×C7—22no—
222C42×C7C2×D412yes—
223C42×C7C2×D412yes—
224#543C22×C711yescharacteristic
225#539—22no—
226#539—21no—
227#538—22no—
228#538C22×C712yes—
229#534—21no—
230#534C22×C711yescharacteristic
231#529—22no—
232#529C22×C712yes—
233C2×#125C22×C711yescharacteristic, complemented
234#525—21no—
235#525—22no—
236C2×C4×D4—21no—
237C2×C4×D4C22×C711yescharacteristic
238#499—22no—
239#488—22no—
240#484C22×C712yescomplemented
241#483C22×C712yescomplemented
242#481C22×C712yes—
243#480C22×C712yes—
244#479—22no—
245#478—22no—
246#431C22×C712yescomplemented
247#399C22×C711yescharacteristic
248#399C22×C711yescharacteristic, complemented
249#398—21no—
250#397C22×C711yescharacteristic
251#397C22×C711yescharacteristic, complemented
252#394—21no—
253#385—21no—
254#385—21no—
255#383C22×C711yescharacteristic, complemented
256#328C22×C712yescomplemented
257#328C22×C712yescomplemented
258#327C22×C712yes—
259#327C22×C712yes—
260#323C22×C711yescharacteristic
261#323C22×C711yescharacteristic
262#323C22×C712yescomplemented
263C23×C7—21no—
264C23×C7—21no—
265C23×C7—21no—
266C23×C7—22no—
267C23×C7—21no—
268C23×C7—21no—
269C23×C7—41no—
270C23×C7—41no—
271C23×C7—41no—
272C23×C7—41no—
273C23×C7—42no—
274C23×C7—81no—
275C23×C7—81no—
276C23×C7#13512yes—
277C23×C7C22×D411yescharacteristic
278C7×Q8—42no—
279C7×Q8—81no—
280C7×D4—44no—
281C7×D4—42no—
282C7×D4—44no—
283C7×D4—42no—
284C7×D4—42no—
285C7×D4—82no—
286C7×D4—81no—
287C7×D4—82no—
288C2×C4×C7—21no—
289C2×C4×C7—21no—
290C2×C4×C7—22no—
291C2×C4×C7—22no—
292C2×C4×C7—21no—
293C2×C4×C7—21no—
294C2×C4×C7—22no—
295C2×C4×C7—41no—
296C2×C4×C7—41no—
297C2×C4×C7—42no—
298C2×C4×C7—42no—
299C2×C4×C7—44no—
300C2×C4×C7—42no—
301C2×C4×C7—41no—
302C2×C4×C7—41no—
303C2×C4×C7—41no—
304C2×C4×C7—41no—
305C2×C4×C7—41no—
306C2×C4×C7—41no—
307C2×C4×C7—41no—
308C2×C4×C7—41no—
309C2×C4×C7—41no—
310C2×C4×C7—41no—
311C2×C4×C7—41no—
312C2×C4×C7—41no—
313C2×C4×C7—41no—
314C2×C4×C7—41no—
315C2×C4×C7—81no—
316C2×C4×C7—81no—
317C2×C4×C7—81no—
318C2×C4×C7—82no—
319C2×C4×C7—81no—
320C2×C4×C7#12011yescharacteristic
321C2×C4×C7#12012yes—
322C2×C4×C7#12011yescharacteristic
323C2×C4×C7C22×D412yes—
324C7×C8—42no—
325C7×C8—42no—
326C2×#41—21no—
327C22×D4—21no—
328C22×D4C7×D411yescharacteristic
329C23×C4—21no—
330C23×C4C7×D411yescharacteristic
331C23×C4C7×D411yescharacteristic
332#126—22no—
333#126—22no—
334#126—22no—
335#125—22no—
336#125—22no—
337#125—22no—
338#125C7×D412yes—
339#124—22no—
340#124C7×D412yes—
341#123—22no—
342#123—22no—
343#123—22no—
344#123—22no—
345#123—22no—
346#123—22no—
347C22⋊Q8—21no—
348C22⋊Q8—22no—
349C22⋊Q8—22no—
350C22⋊Q8—21no—
351C22⋊Q8—22no—
352C22⋊Q8C23×C712yes—
353#121—22no—
354#121—21no—
355#121—22no—
356#121—22no—
357#121—21no—
358#121C23×C712yes—
359#120—22no—
360#120—22no—
361#120—22no—
362C4×Q8C7×D412yes—
363C4×D4—22no—
364C4×D4—22no—
365C4×D4—22no—
366C4×D4—22no—
367C4×D4—22no—
368C4×D4—22no—
369C4×D4—22no—
370C4×D4C7×D412yes—
371#117—21no—
372#117—22no—
373C2×C4⋊C4—21no—
374C2×C4⋊C4—21no—
375C2×C4⋊C4—21no—
376C2×C4⋊C4—21no—
377C2×C4⋊C4—21no—
378C2×C4⋊C4C7×D411yescharacteristic
379C2×C4⋊C4C23×C711yescharacteristic
380C2×C4⋊C4C23×C711yescharacteristic
381C2×C22⋊C4—21no—
382C2×C22⋊C4—21no—
383C2×C22⋊C4—21no—
384C2×C22⋊C4—21no—
385C2×C22⋊C4—21no—
386C2×C22⋊C4—21no—
387C2×C22⋊C4C23×C711yescharacteristic
388C2×C22⋊C4C23×C711yescharacteristic
389C2×C42—21no—
390C2×C42C23×C711yescharacteristic
391#107—22no—
392#107—22no—
393#106—22no—
394#106—22no—
395C4⋊C8—22no—
396#103—22no—
397#103—22no—
398#102—22no—
399#102—22no—
400C22⋊C8C23×C712yes—
401C22⋊C8C23×C712yes—
402C4×C8—22no—
403#95—21no—
404#95—21no—
405#95—21no—
406#95—21no—
407#95C23×C711yescharacteristic
408#95C23×C711yescharacteristic
409C22×C7—21no—
410C22×C7—21no—
411C22×C7—22no—
412C22×C7—42no—
413C22×C7—42no—
414C22×C7—41no—
415C22×C7—41no—
416C22×C7—41no—
417C22×C7—42no—
418C22×C7—41no—
419C22×C7—42no—
420C22×C7—42no—
421C22×C7—41no—
422C22×C7—81no—
423C22×C7—81no—
424C22×C7—84no—
425C22×C7C2×#12011yescharacteristic
426C22×C7C2×#13511yescharacteristic
427C22×C7#57511yescharacteristic
428C4×C7—22no—
429C4×C7—21no—
430C4×C7—21no—
431C4×C7—22no—
432C4×C7—42no—
433C4×C7—41no—
434C4×C7—41no—
435C4×C7—42no—
436C4×C7—42no—
437C4×C7—81no—
438C4×C7—81no—
439C4×C7—81no—
440C4×C7—81no—
441C24—21no—
442C24C2×C7×D411yescharacteristic
443C24C2×C7×D411yescharacteristic
444#41—82no—
445C2×Q8—21no—
446C2×Q8C2×C7×D412yes—
447C2×D4—22no—
448C2×D4—22no—
449C2×D4—22no—
450C2×D4—22no—
451C2×D4—21no—
452C2×D4—22no—
453C2×D4—22no—
454C2×D4—41no—
455C2×D4—42no—
456C2×D4—42no—
457C2×D4C2×C7×D412yes—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—22no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—22no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—81no—
472C22×C4—81no—
473C22×C4—82no—
474C22×C4C2×C7×D411yescharacteristic
475C22×C4C2×C7×D411yescharacteristic
476C22×C4C2×C7×D411yescharacteristic
477C22×C4C2×C7×D411yescharacteristic
478C22×C4C24×C711yescharacteristic
479C2×C8—22no—
480C2×C8—22no—
481C4⋊C4—22no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—41no—
493C4⋊C4—42no—
494C4⋊C4—41no—
495C4⋊C4—81no—
496C4⋊C4—81no—
497C4⋊C4C2×C7×D412yes—
498C4⋊C4C2×C7×D412yes—
499C22⋊C4—22no—
500C22⋊C4—22no—
501C22⋊C4—22no—
502C22⋊C4—22no—
503C22⋊C4—22no—
504C22⋊C4—22no—
505C22⋊C4—22no—
506C22⋊C4—22no—
507C22⋊C4—22no—
508C22⋊C4—22no—
509C22⋊C4—22no—
510C22⋊C4—22no—
511C22⋊C4—42no—
512C22⋊C4—42no—
513C22⋊C4—81no—
514C22⋊C4—81no—
515C42—22no—
516C42—22no—
517C42C2×C7×D412yes—
518C42C2×C7×D412yes—
519C2×C7—21no—
520C2×C7—21no—
521C2×C7—42no—
522C2×C7—81no—
523C2×C7—82no—
524C2×C7#300211yescharacteristic
525C2×C7#301911yescharacteristic
526C2×C7#302811yescharacteristic
527C23—21no—
528C23—21no—
529C23—21no—
530C23—22no—
531C23—21no—
532C23—21no—
533C23—41no—
534C23—41no—
535C23—41no—
536C23—41no—
537C23—42no—
538C23—81no—
539C23—81no—
540C23C7×#13512yes—
541C23C22×C7×D411yescharacteristic
542Q8—42no—
543Q8—81no—
544D4—44no—
545D4—42no—
546D4—44no—
547D4—42no—
548D4—42no—
549D4—82no—
550D4—81no—
551D4—82no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—22no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—44no—
564C2×C4—42no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—81no—
580C2×C4—81no—
581C2×C4—81no—
582C2×C4—82no—
583C2×C4—81no—
584C2×C4C7×#12011yescharacteristic
585C2×C4C7×#12012yes—
586C2×C4C7×#12011yescharacteristic
587C2×C4C22×C7×D412yes—
588C8—42no—
589C8—42no—
590C7#1831411yescharacteristic, direct product factor, Sylow
591C22—21no—
592C22—21no—
593C22—22no—
594C22—42no—
595C22—42no—
596C22—41no—
597C22—41no—
598C22—41no—
599C22—42no—
600C22—41no—
601C22—42no—
602C22—42no—
603C22—41no—
604C22—81no—
605C22—81no—
606C22—84no—
607C22C2×C7×#12011yescharacteristic
608C22C2×C7×#13511yescharacteristic
609C22C7×#57511yescharacteristic
610C4—22no—
611C4—21no—
612C4—21no—
613C4—22no—
614C4—42no—
615C4—41no—
616C4—41no—
617C4—42no—
618C4—42no—
619C4—81no—
620C4—81no—
621C4—81no—
622C4—81no—
623C2—21no—
624C2—21no—
625C2—42no—
626C2—81no—
627C2—82no—
628C2C7×#300211yescharacteristic
629C2C7×#301911yescharacteristic
630C2C7×#302811yescharacteristic
subgroupG/H
center, socleC22×C7C2×#120
commutator, Frattini, cosocleC22×C4C24×C7
prime radicalC7×#3459C2
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×#3232C2semidirect
3. C7×#2633C2semidirect
4. C7×#1618C2semidirect
5. C7×#1614C2semidirect
6. C7×#1613C2semidirect
7. C7×#1612C2semidirect
8. C7×#1608C2semidirect
9. C7×#1607C2semidirect
10. C7×#1521C2semidirect
11. C7×#1460C2semidirect
21. C2×C7×#125C22semidirect
28. C7×#484C22semidirect
29. C7×#483C22semidirect
34. C7×#431C22semidirect
36. C7×#399C22semidirect
39. C7×#397C22semidirect
43. C7×#383C22semidirect
44. C7×#328C22semidirect
45. C7×#328C22semidirect
50. C7×#323C22semidirect
51. #18314C7direct (N × H)
136. #3232C2×C7semidirect
137. #2633C2×C7semidirect
138. #1618C2×C7semidirect
139. #1614C2×C7semidirect
140. #1613C2×C7semidirect
141. #1612C2×C7semidirect
142. #1608C2×C7semidirect
143. #1607C2×C7semidirect
144. #1521C2×C7semidirect
145. #1460C2×C7semidirect
233. C2×#125C22×C7semidirect
240. #484C22×C7semidirect
241. #483C22×C7semidirect
246. #431C22×C7semidirect
248. #399C22×C7semidirect
251. #397C22×C7semidirect
255. #383C22×C7semidirect
256. #328C22×C7semidirect
257. #328C22×C7semidirect
262. #323C22×C7semidirect
590. C7#18314direct (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×#32322
C7×#26332
C7×#16182
C7×#16142
C7×#16132
C7×#16122
C7×#16082
C7×#16072
C7×#15212
C7×#14602
#183147