Finite Group Databasebeta

← C7×#18107 #13195323

Subgroups — C7×#18107 #13195323

Order 1792.

1000 conjugacy classes of subgroups.

The subgroup lattice has 634 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3459C211yesmaximal, characteristic
2C7×#3091C212yesmaximal, complemented
3C7×#2395C211yesmaximal, characteristic, complemented
4C7×#1615C212yesmaximal, complemented
5C7×#1614C211yesmaximal, characteristic, complemented
6C7×#1613C211yesmaximal, characteristic, complemented
7C7×#1609C212yesmaximal, complemented
8C7×#1608C211yesmaximal, characteristic, complemented
9C7×#1607C211yesmaximal, characteristic, complemented
10C7×#1501C212yesmaximal, complemented
11C7×#1459C211yesmaximal, characteristic, complemented
12C7×#543C2211yescharacteristic
13C7×#539—22no—
14C7×#539C2212yes—
15C7×#538—22no—
16C7×#538—21no—
17C7×#534—21no—
18C7×#534C2211yescharacteristic
19C7×#529—22no—
20C7×#529—21no—
21C7×#525—22no—
22C7×#525C2212yes—
23C2×C4×C7×D4—21no—
24C2×C4×C7×D4C2211yescharacteristic
25C2×C7×#117C2211yescharacteristic, complemented
26C7×#485—22no—
27C7×#484C2212yescomplemented
28C7×#483C2212yescomplemented
29C7×#482—22no—
30C7×#481C2212yes—
31C7×#480C2212yes—
32C7×#427—22no—
33C7×#416—22no—
34C2×C7×C22⋊C8C2212yescomplemented
35C7×#400C2211yescharacteristic, complemented
36C7×#399C2211yescharacteristic
37C7×#398C2211yescharacteristic, complemented
38C7×#397—21no—
39C7×#397C2211yescharacteristic
40C7×#394—21no—
41C4×C7×C4⋊C4—21no—
42C4×C7×C22⋊C4—21no—
43C4×C7×C22⋊C4C2211yescharacteristic, complemented
44C7×#328C2212yescomplemented
45C7×#328C2212yescomplemented
46C7×#327C2212yes—
47C7×#327C2212yes—
48C7×#324C2212yescomplemented
49C7×#323C2211yescharacteristic
50C7×#323C2211yescharacteristic
51#18107C711yesmaximal, characteristic, direct product factor, Sylow
52C2×C7×#41D412yes—
53C22×C7×D4—21no—
54C22×C7×D4D411yescharacteristic
55C23×C4×C7—21no—
56C23×C4×C7D411yescharacteristic
57C23×C4×C7C2311yescharacteristic
58C22×C7×C8—22no—
59C7×#126—22no—
60C7×#126—22no—
61C7×#126—22no—
62C7×#125—21no—
63C7×#125—22no—
64C7×#124—22no—
65C7×#124—21no—
66C7×#123—22no—
67C7×#123—22no—
68C7×#123—22no—
69C7×#123—22no—
70C7×#123—22no—
71C7×#123—22no—
72C7×C22⋊Q8—22no—
73C7×C22⋊Q8—22no—
74C7×C22⋊Q8—21no—
75C7×C22⋊Q8—22no—
76C7×C22⋊Q8D412yes—
77C7×C22⋊Q8C2312yes—
78C7×#121—22no—
79C7×#121—22no—
80C7×#121—22no—
81C7×#121—21no—
82C7×#121D412yes—
83C7×#121C2312yes—
84C7×#120—22no—
85C7×#120—22no—
86C7×#120—22no—
87C4×C7×Q8—21no—
88C4×C7×D4—22no—
89C4×C7×D4—22no—
90C4×C7×D4—21no—
91C4×C7×D4—22no—
92C4×C7×D4—22no—
93C4×C7×D4—22no—
94C4×C7×D4—22no—
95C4×C7×D4—22no—
96C7×#117—22no—
97C7×#117—22no—
98C7×#117—22no—
99C7×#117D412yes—
100C2×C7×C4⋊C4—21no—
101C2×C7×C4⋊C4—21no—
102C2×C7×C4⋊C4—21no—
103C2×C7×C4⋊C4C2311yescharacteristic
104C2×C7×C4⋊C4C2311yescharacteristic
105C2×C7×C4⋊C4C2311yescharacteristic
106C2×C7×C22⋊C4—21no—
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⋊C4D411yescharacteristic
113C2×C7×C22⋊C4C2311yescharacteristic
114C2×C42×C7—21no—
115C2×C42×C7—21no—
116C2×C42×C7—21no—
117C2×C42×C7D411yescharacteristic
118C7×#107—22no—
119C7×#107—22no—
120C7×#106—22no—
121C7×#106—22no—
122C7×#103—22no—
123C7×#103—22no—
124C7×#102—22no—
125C7×#102—22no—
126C7×C22⋊C8—22no—
127C7×C22⋊C8C2312yes—
128C7×C22⋊C8C2312yes—
129C7×#95—21no—
130C7×#95—21no—
131C7×#95—21no—
132C7×#95—21no—
133C7×#95C2311yescharacteristic
134C7×#95C2311yescharacteristic
135#3459C2×C711yescharacteristic
136#3091C2×C712yescomplemented
137#2395C2×C711yescharacteristic, complemented
138#1615C2×C712yescomplemented
139#1614C2×C711yescharacteristic, complemented
140#1613C2×C711yescharacteristic, complemented
141#1609C2×C712yescomplemented
142#1608C2×C711yescharacteristic, complemented
143#1607C2×C711yescharacteristic, complemented
144#1501C2×C712yescomplemented
145#1459C2×C711yescharacteristic, complemented
146C24×C7—21no—
147C24×C7C2×D411yescharacteristic
148C24×C7C2×D411yescharacteristic
149C7×#41—44no—
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—21no—
167C22×C4×C7—22no—
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—42no—
175C22×C4×C7—42no—
176C22×C4×C7—82no—
177C22×C4×C7C2×D411yescharacteristic
178C22×C4×C7C2×D411yescharacteristic
179C22×C4×C7C2×D412yes—
180C22×C4×C7C2×D411yescharacteristic
181C22×C4×C7C2×D411yescharacteristic
182C22×C4×C7C2×D412yes—
183C22×C4×C7C2411yescharacteristic
184C2×C7×C8—22no—
185C2×C7×C8—22no—
186C2×C7×C8—44no—
187C7×C4⋊C4—22no—
188C7×C4⋊C4—22no—
189C7×C4⋊C4—22no—
190C7×C4⋊C4—21no—
191C7×C4⋊C4—21no—
192C7×C4⋊C4—22no—
193C7×C4⋊C4—22no—
194C7×C4⋊C4—22no—
195C7×C4⋊C4—22no—
196C7×C4⋊C4—22no—
197C7×C4⋊C4—22no—
198C7×C4⋊C4—42no—
199C7×C4⋊C4—42no—
200C7×C4⋊C4C2×D412yes—
201C7×C4⋊C4C2×D412yes—
202C7×C22⋊C4—22no—
203C7×C22⋊C4—22no—
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—42no—
215C7×C22⋊C4—42no—
216C7×C22⋊C4—42no—
217C7×C22⋊C4—42no—
218C42×C7—22no—
219C42×C7—21no—
220C42×C7—22no—
221C42×C7—21no—
222C42×C7—42no—
223C42×C7—41no—
224C42×C7—41no—
225#543C22×C711yescharacteristic
226#539—22no—
227#539C22×C712yes—
228#538—22no—
229#538—21no—
230#534—21no—
231#534C22×C711yescharacteristic
232#529—22no—
233#529—21no—
234#525—22no—
235#525C22×C712yes—
236C2×C4×D4—21no—
237C2×C4×D4C22×C711yescharacteristic
238C2×#117C22×C711yescharacteristic, complemented
239#485—22no—
240#484C22×C712yescomplemented
241#483C22×C712yescomplemented
242#482—22no—
243#481C22×C712yes—
244#480C22×C712yes—
245#427—22no—
246#416—22no—
247C2×C22⋊C8C22×C712yescomplemented
248#400C22×C711yescharacteristic, complemented
249#399C22×C711yescharacteristic
250#398C22×C711yescharacteristic, complemented
251#397—21no—
252#397C22×C711yescharacteristic
253#394—21no—
254C4×C4⋊C4—21no—
255C4×C22⋊C4—21no—
256C4×C22⋊C4C22×C711yescharacteristic, complemented
257#328C22×C712yescomplemented
258#328C22×C712yescomplemented
259#327C22×C712yes—
260#327C22×C712yes—
261#324C22×C712yescomplemented
262#323C22×C711yescharacteristic
263#323C22×C711yescharacteristic
264C23×C7—21no—
265C23×C7—21no—
266C23×C7—21no—
267C23×C7—21no—
268C23×C7—22no—
269C23×C7—41no—
270C23×C7—41no—
271C23×C7—41no—
272C23×C7—41no—
273C23×C7—42no—
274C23×C7—81no—
275C23×C7—81no—
276C23×C7#12012yes—
277C23×C7#13512yes—
278C23×C7C22×D411yescharacteristic
279C7×Q8—42no—
280C7×Q8—81no—
281C7×D4—42no—
282C7×D4—44no—
283C7×D4—44no—
284C7×D4—42no—
285C7×D4—42no—
286C7×D4—82no—
287C7×D4—82no—
288C7×D4—81no—
289C2×C4×C7—22no—
290C2×C4×C7—24no—
291C2×C4×C7—22no—
292C2×C4×C7—21no—
293C2×C4×C7—22no—
294C2×C4×C7—21no—
295C2×C4×C7—22no—
296C2×C4×C7—21no—
297C2×C4×C7—22no—
298C2×C4×C7—21no—
299C2×C4×C7—21no—
300C2×C4×C7—42no—
301C2×C4×C7—41no—
302C2×C4×C7—41no—
303C2×C4×C7—42no—
304C2×C4×C7—41no—
305C2×C4×C7—42no—
306C2×C4×C7—44no—
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—41no—
316C2×C4×C7—41no—
317C2×C4×C7—41no—
318C2×C4×C7—81no—
319C2×C4×C7—81no—
320C2×C4×C7—82no—
321C2×C4×C7—81no—
322C2×C4×C7#12011yescharacteristic
323C2×C4×C7#12011yescharacteristic
324C2×C4×C7C22×D412yes—
325C7×C8—42no—
326C7×C8—42no—
327C2×#41C7×D412yes—
328C22×D4—21no—
329C22×D4C7×D411yescharacteristic
330C23×C4—21no—
331C23×C4C7×D411yescharacteristic
332C23×C4C23×C711yescharacteristic
333C22×C8—22no—
334#126—22no—
335#126—22no—
336#126—22no—
337#125—21no—
338#125—22no—
339#124—22no—
340#124—21no—
341#123—22no—
342#123—22no—
343#123—22no—
344#123—22no—
345#123—22no—
346#123—22no—
347C22⋊Q8—22no—
348C22⋊Q8—22no—
349C22⋊Q8—21no—
350C22⋊Q8—22no—
351C22⋊Q8C7×D412yes—
352C22⋊Q8C23×C712yes—
353#121—22no—
354#121—22no—
355#121—22no—
356#121—21no—
357#121C7×D412yes—
358#121C23×C712yes—
359#120—22no—
360#120—22no—
361#120—22no—
362C4×Q8—21no—
363C4×D4—22no—
364C4×D4—22no—
365C4×D4—21no—
366C4×D4—22no—
367C4×D4—22no—
368C4×D4—22no—
369C4×D4—22no—
370C4×D4—22no—
371#117—22no—
372#117—22no—
373#117—22no—
374#117C7×D412yes—
375C2×C4⋊C4—21no—
376C2×C4⋊C4—21no—
377C2×C4⋊C4—21no—
378C2×C4⋊C4C23×C711yescharacteristic
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⋊C4C7×D411yescharacteristic
388C2×C22⋊C4C23×C711yescharacteristic
389C2×C42—21no—
390C2×C42—21no—
391C2×C42—21no—
392C2×C42C7×D411yescharacteristic
393#107—22no—
394#107—22no—
395#106—22no—
396#106—22no—
397#103—22no—
398#103—22no—
399#102—22no—
400#102—22no—
401C22⋊C8—22no—
402C22⋊C8C23×C712yes—
403C22⋊C8C23×C712yes—
404#95—21no—
405#95—21no—
406#95—21no—
407#95—21no—
408#95C23×C711yescharacteristic
409#95C23×C711yescharacteristic
410C22×C7—21no—
411C22×C7—21no—
412C22×C7—22no—
413C22×C7—22no—
414C22×C7—22no—
415C22×C7—22no—
416C22×C7—42no—
417C22×C7—41no—
418C22×C7—41no—
419C22×C7—42no—
420C22×C7—42no—
421C22×C7—42no—
422C22×C7—81no—
423C22×C7—81no—
424C22×C7—81no—
425C22×C7—84no—
426C22×C7C2×#12011yescharacteristic
427C22×C7C2×#13511yescharacteristic
428C22×C7#57511yescharacteristic
429C4×C7—22no—
430C4×C7—21no—
431C4×C7—42no—
432C4×C7—41no—
433C4×C7—42no—
434C4×C7—41no—
435C4×C7—42no—
436C4×C7—41no—
437C4×C7—42no—
438C4×C7—81no—
439C4×C7—81no—
440C4×C7—81no—
441C4×C7#45812yes—
442C24—21no—
443C24C2×C7×D411yescharacteristic
444C24C2×C7×D411yescharacteristic
445#41—44no—
446C2×Q8—21no—
447C2×Q8C2×C7×D412yes—
448C2×D4—22no—
449C2×D4—22no—
450C2×D4—22no—
451C2×D4—22no—
452C2×D4—21no—
453C2×D4—22no—
454C2×D4—22no—
455C2×D4—41no—
456C2×D4—42no—
457C2×D4—42no—
458C2×D4C2×C7×D412yes—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—22no—
464C22×C4—21no—
465C22×C4—22no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—42no—
471C22×C4—42no—
472C22×C4—82no—
473C22×C4C2×C7×D411yescharacteristic
474C22×C4C2×C7×D411yescharacteristic
475C22×C4C2×C7×D412yes—
476C22×C4C2×C7×D411yescharacteristic
477C22×C4C2×C7×D411yescharacteristic
478C22×C4C2×C7×D412yes—
479C22×C4C24×C711yescharacteristic
480C2×C8—22no—
481C2×C8—22no—
482C2×C8—44no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—42no—
495C4⋊C4—42no—
496C4⋊C4C2×C7×D412yes—
497C4⋊C4C2×C7×D412yes—
498C22⋊C4—22no—
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—42no—
511C22⋊C4—42no—
512C22⋊C4—42no—
513C22⋊C4—42no—
514C42—22no—
515C42—22no—
516C42—21no—
517C42—21no—
518C42—42no—
519C42—41no—
520C42—41no—
521C2×C7—21no—
522C2×C7—21no—
523C2×C7—42no—
524C2×C7—42no—
525C2×C7—82no—
526C2×C7#300211yescharacteristic
527C2×C7#301811yescharacteristic
528C2×C7C2×#45811yescharacteristic
529C23—21no—
530C23—21no—
531C23—21no—
532C23—21no—
533C23—22no—
534C23—41no—
535C23—41no—
536C23—41no—
537C23—41no—
538C23—42no—
539C23—81no—
540C23—81no—
541C23C7×#12012yes—
542C23C7×#13512yes—
543C23C22×C7×D411yescharacteristic
544Q8—42no—
545Q8—81no—
546D4—42no—
547D4—44no—
548D4—44no—
549D4—42no—
550D4—42no—
551D4—82no—
552D4—82no—
553D4—81no—
554C2×C4—22no—
555C2×C4—24no—
556C2×C4—22no—
557C2×C4—21no—
558C2×C4—22no—
559C2×C4—21no—
560C2×C4—22no—
561C2×C4—21no—
562C2×C4—22no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—42no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—42no—
569C2×C4—41no—
570C2×C4—42no—
571C2×C4—44no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—81no—
584C2×C4—81no—
585C2×C4—82no—
586C2×C4—81no—
587C2×C4C7×#12011yescharacteristic
588C2×C4C7×#12011yescharacteristic
589C2×C4C22×C7×D412yes—
590C8—42no—
591C8—42no—
592C7#1810711yescharacteristic, direct product factor, Sylow
593C22—21no—
594C22—21no—
595C22—22no—
596C22—22no—
597C22—22no—
598C22—22no—
599C22—42no—
600C22—41no—
601C22—41no—
602C22—42no—
603C22—42no—
604C22—42no—
605C22—81no—
606C22—81no—
607C22—81no—
608C22—84no—
609C22C2×C7×#12011yescharacteristic
610C22C2×C7×#13511yescharacteristic
611C22C7×#57511yescharacteristic
612C4—22no—
613C4—21no—
614C4—42no—
615C4—41no—
616C4—42no—
617C4—41no—
618C4—42no—
619C4—41no—
620C4—42no—
621C4—81no—
622C4—81no—
623C4—81no—
624C4C7×#45812yes—
625C2—21no—
626C2—21no—
627C2—42no—
628C2—42no—
629C2—82no—
630C2C7×#300211yescharacteristic
631C2C7×#301811yescharacteristic
632C2C2×C7×#45811yescharacteristic
subgroupG/H
centerC2×C4×C7#120
commutator, Frattini, cosocleC22×C4C24×C7
socleC22×C7C2×#120
prime radicalC7×#3459C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4×C7 ◁ C23×C4×C7 ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C7×#3091C2semidirect
3. C7×#2395C2semidirect
4. C7×#1615C2semidirect
5. C7×#1614C2semidirect
6. C7×#1613C2semidirect
7. C7×#1609C2semidirect
8. C7×#1608C2semidirect
9. C7×#1607C2semidirect
10. C7×#1501C2semidirect
11. C7×#1459C2semidirect
25. C2×C7×#117C22semidirect
27. C7×#484C22semidirect
28. C7×#483C22semidirect
34. C2×C7×C22⋊C8C22semidirect
35. C7×#400C22semidirect
37. C7×#398C22semidirect
43. C4×C7×C22⋊C4C22semidirect
44. C7×#328C22semidirect
45. C7×#328C22semidirect
48. C7×#324C22semidirect
51. #18107C7direct (N × H)
136. #3091C2×C7semidirect
137. #2395C2×C7semidirect
138. #1615C2×C7semidirect
139. #1614C2×C7semidirect
140. #1613C2×C7semidirect
141. #1609C2×C7semidirect
142. #1608C2×C7semidirect
143. #1607C2×C7semidirect
144. #1501C2×C7semidirect
145. #1459C2×C7semidirect
238. C2×#117C22×C7semidirect
240. #484C22×C7semidirect
241. #483C22×C7semidirect
247. C2×C22⋊C8C22×C7semidirect
248. #400C22×C7semidirect
250. #398C22×C7semidirect
256. C4×C22⋊C4C22×C7semidirect
257. #328C22×C7semidirect
258. #328C22×C7semidirect
261. #324C22×C7semidirect
592. C7#18107direct (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×#30912
C7×#23952
C7×#16152
C7×#16142
C7×#16132
C7×#16092
C7×#16082
C7×#16072
C7×#15012
C7×#14592
#181077