Finite Group Databasebeta

← C7×#18290 #13195506

Subgroups — C7×#18290 #13195506

Order 1792.

1034 conjugacy classes of subgroups.

The subgroup lattice has 680 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3459C211yesmaximal, characteristic, complemented
2C7×#3188C212yesmaximal, complemented
3C7×#2635C211yesmaximal, characteristic, complemented
4C7×#1615C212yesmaximal, complemented
5C7×#1614C211yesmaximal, characteristic, complemented
6C7×#1613C211yesmaximal, characteristic, complemented
7C7×#1597C212yesmaximal, complemented
8C7×#1596C211yesmaximal, characteristic, complemented
9C7×#1595C211yesmaximal, characteristic, complemented
10C7×#1516C212yesmaximal, 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×C7×#121C2211yescharacteristic, complemented
24C2×C4×C7×D4—21no—
25C2×C4×C7×D4C2211yescharacteristic
26C7×#484C2212yescomplemented
27C7×#482—22no—
28C7×#481C2212yes—
29C7×#466—22no—
30C7×#465—22no—
31C7×#450C2212yescomplemented
32C7×#449—22no—
33C7×#447C2212yes—
34C2×C7×C22⋊C8C2212yescomplemented
35C7×#400C2211yescharacteristic, complemented
36C7×#399C2211yescharacteristic
37C7×#394—21no—
38C7×#394C2211yescharacteristic, complemented
39C7×#392—21no—
40C7×#392C2211yescharacteristic
41C7×#388—21no—
42C7×#386—21no—
43C7×#381C2211yescharacteristic, complemented
44C7×#328C2212yescomplemented
45C7×#328C2212yescomplemented
46C7×#327C2212yes—
47C7×#327C2212yes—
48C7×#324C2212yescomplemented
49C7×#323C2211yescharacteristic
50C7×#323C2211yescharacteristic
51#18290C711yesmaximal, characteristic, direct product factor, Sylow
52C2×C7×#41D412yescomplemented
53C22×C7×D4—21no—
54C22×C7×D4—21no—
55C22×C7×D4—21no—
56C22×C7×D4D411yescharacteristic
57C22×C7×D4C2311yescharacteristic
58C23×C4×C7—21no—
59C23×C4×C7D411yescharacteristic
60C23×C4×C7C2311yescharacteristic
61C2×C7×SD16—22no—
62C2×C7×D8—22no—
63C22×C7×C8—22no—
64C7×#126—22no—
65C7×#126—22no—
66C7×#126—22no—
67C7×#125—21no—
68C7×#125—22no—
69C7×#124—22no—
70C7×#124—21no—
71C7×#123—22no—
72C7×#123—22no—
73C7×#123—22no—
74C7×#123—22no—
75C7×#123—22no—
76C7×#123—22no—
77C7×C22⋊Q8—22no—
78C7×C22⋊Q8—22no—
79C7×C22⋊Q8—21no—
80C7×C22⋊Q8—22no—
81C7×C22⋊Q8D412yescomplemented
82C7×C22⋊Q8C2312yes—
83C7×#121—22no—
84C7×#121—22no—
85C7×#121—22no—
86C7×#121—21no—
87C7×#121—22no—
88C7×#121—22no—
89C7×#121D412yescomplemented
90C7×#121C2312yes—
91C7×#120—22no—
92C7×#120—22no—
93C7×#120—22no—
94C4×C7×Q8—21no—
95C4×C7×D4—22no—
96C4×C7×D4—22no—
97C4×C7×D4—21no—
98C4×C7×D4—22no—
99C4×C7×D4—22no—
100C4×C7×D4—22no—
101C4×C7×D4—22no—
102C4×C7×D4—22no—
103C7×#117—22no—
104C7×#117D412yescomplemented
105C2×C7×C4⋊C4—21no—
106C2×C7×C4⋊C4—21no—
107C2×C7×C4⋊C4C2311yescharacteristic
108C2×C7×C4⋊C4C2311yescharacteristic
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⋊C4D411yescharacteristic
118C2×C7×C22⋊C4C2311yescharacteristic
119C2×C42×C7—21no—
120C2×C42×C7D411yescharacteristic
121C7×#107—22no—
122C7×#106—22no—
123C7×#103—22no—
124C7×#102—22no—
125C7×#102—22no—
126C7×#102—22no—
127C7×C22⋊C8—22no—
128C7×C22⋊C8C2312yes—
129C7×C22⋊C8C2312yes—
130C7×#95—21no—
131C7×#95—21no—
132C7×#95—21no—
133C7×#95C2311yescharacteristic
134C7×#95C2311yescharacteristic
135#3459C2×C711yescharacteristic, complemented
136#3188C2×C712yescomplemented
137#2635C2×C711yescharacteristic, complemented
138#1615C2×C712yescomplemented
139#1614C2×C711yescharacteristic, complemented
140#1613C2×C711yescharacteristic, complemented
141#1597C2×C712yescomplemented
142#1596C2×C711yescharacteristic, complemented
143#1595C2×C711yescharacteristic, complemented
144#1516C2×C712yescomplemented
145#1459C2×C711yescharacteristic, complemented
146C24×C7—21no—
147C24×C7—21no—
148C24×C7C2×D411yescharacteristic
149C24×C7C2×D411yescharacteristic
150C7×#41—82no—
151C2×C7×Q8—21no—
152C2×C7×Q8C2×D412yes—
153C2×C7×D4—22no—
154C2×C7×D4—22no—
155C2×C7×D4—22no—
156C2×C7×D4—22no—
157C2×C7×D4—21no—
158C2×C7×D4—22no—
159C2×C7×D4—22no—
160C2×C7×D4—22no—
161C2×C7×D4—41no—
162C2×C7×D4—42no—
163C2×C7×D4—42no—
164C2×C7×D4—42no—
165C2×C7×D4—42no—
166C2×C7×D4—81no—
167C2×C7×D4—81no—
168C2×C7×D4—81no—
169C2×C7×D4—81no—
170C2×C7×D4—81no—
171C2×C7×D4C2×D412yes—
172C22×C4×C7—21no—
173C22×C4×C7—21no—
174C22×C4×C7—21no—
175C22×C4×C7—22no—
176C22×C4×C7—21no—
177C22×C4×C7—22no—
178C22×C4×C7—21no—
179C22×C4×C7—21no—
180C22×C4×C7—21no—
181C22×C4×C7—21no—
182C22×C4×C7—81no—
183C22×C4×C7—81no—
184C22×C4×C7—82no—
185C22×C4×C7C2×D411yescharacteristic
186C22×C4×C7C2×D411yescharacteristic
187C22×C4×C7C2×D411yescharacteristic
188C22×C4×C7C2×D411yescharacteristic
189C22×C4×C7C2×D412yes—
190C22×C4×C7C2×D412yes—
191C22×C4×C7C2411yescharacteristic
192C7×SD16—82no—
193C7×D8—82no—
194C2×C7×C8—22no—
195C2×C7×C8—22no—
196C2×C7×C8—82no—
197C7×C4⋊C4—22no—
198C7×C4⋊C4—22no—
199C7×C4⋊C4—22no—
200C7×C4⋊C4—21no—
201C7×C4⋊C4—21no—
202C7×C4⋊C4—22no—
203C7×C4⋊C4—22no—
204C7×C4⋊C4—22no—
205C7×C4⋊C4—22no—
206C7×C4⋊C4—22no—
207C7×C4⋊C4—81no—
208C7×C4⋊C4C2×D412yes—
209C7×C4⋊C4C2×D412yes—
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—22no—
217C7×C22⋊C4—22no—
218C7×C22⋊C4—22no—
219C7×C22⋊C4—22no—
220C7×C22⋊C4—22no—
221C7×C22⋊C4—22no—
222C7×C22⋊C4—42no—
223C7×C22⋊C4—41no—
224C7×C22⋊C4—41no—
225C7×C22⋊C4—81no—
226C7×C22⋊C4—81no—
227C7×C22⋊C4—81no—
228C7×C22⋊C4—81no—
229C42×C7—22no—
230C42×C7—22no—
231C42×C7—21no—
232C42×C7—21no—
233#543C22×C711yescharacteristic
234#539—22no—
235#539C22×C712yes—
236#538—22no—
237#538—21no—
238#534—21no—
239#534C22×C711yescharacteristic
240#529—22no—
241#529—21no—
242#525—22no—
243#525C22×C712yes—
244C2×#121C22×C711yescharacteristic, complemented
245C2×C4×D4—21no—
246C2×C4×D4C22×C711yescharacteristic
247#484C22×C712yescomplemented
248#482—22no—
249#481C22×C712yes—
250#466—22no—
251#465—22no—
252#450C22×C712yescomplemented
253#449—22no—
254#447C22×C712yes—
255C2×C22⋊C8C22×C712yescomplemented
256#400C22×C711yescharacteristic, complemented
257#399C22×C711yescharacteristic
258#394—21no—
259#394C22×C711yescharacteristic, complemented
260#392—21no—
261#392C22×C711yescharacteristic
262#388—21no—
263#386—21no—
264#381C22×C711yescharacteristic, complemented
265#328C22×C712yescomplemented
266#328C22×C712yescomplemented
267#327C22×C712yes—
268#327C22×C712yes—
269#324C22×C712yescomplemented
270#323C22×C711yescharacteristic
271#323C22×C711yescharacteristic
272C23×C7—21no—
273C23×C7—21no—
274C23×C7—21no—
275C23×C7—21no—
276C23×C7—22no—
277C23×C7—41no—
278C23×C7—41no—
279C23×C7—41no—
280C23×C7—41no—
281C23×C7—42no—
282C23×C7—41no—
283C23×C7—41no—
284C23×C7—41no—
285C23×C7—41no—
286C23×C7—41no—
287C23×C7—41no—
288C23×C7—41no—
289C23×C7—81no—
290C23×C7—81no—
291C23×C7#12012yes—
292C23×C7#13512yes—
293C23×C7C22×D411yescharacteristic
294C7×Q8—42no—
295C7×Q8—81no—
296C7×D4—42no—
297C7×D4—42no—
298C7×D4—42no—
299C7×D4—44no—
300C7×D4—44no—
301C7×D4—42no—
302C7×D4—42no—
303C7×D4—82no—
304C7×D4—82no—
305C7×D4—81no—
306C7×D4—84no—
307C7×D4—84no—
308C2×C4×C7—21no—
309C2×C4×C7—22no—
310C2×C4×C7—21no—
311C2×C4×C7—22no—
312C2×C4×C7—21no—
313C2×C4×C7—22no—
314C2×C4×C7—21no—
315C2×C4×C7—21no—
316C2×C4×C7—42no—
317C2×C4×C7—41no—
318C2×C4×C7—42no—
319C2×C4×C7—41no—
320C2×C4×C7—42no—
321C2×C4×C7—44no—
322C2×C4×C7—42no—
323C2×C4×C7—41no—
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—81no—
335C2×C4×C7—81no—
336C2×C4×C7—82no—
337C2×C4×C7—81no—
338C2×C4×C7#12011yescharacteristic
339C2×C4×C7#12011yescharacteristic
340C2×C4×C7C22×D412yes—
341C7×C8—42no—
342C7×C8—42no—
343C2×#41C7×D412yescomplemented
344C22×D4—21no—
345C22×D4—21no—
346C22×D4—21no—
347C22×D4C7×D411yescharacteristic
348C22×D4C23×C711yescharacteristic
349C23×C4—21no—
350C23×C4C7×D411yescharacteristic
351C23×C4C23×C711yescharacteristic
352C2×SD16—22no—
353C2×D8—22no—
354C22×C8—22no—
355#126—22no—
356#126—22no—
357#126—22no—
358#125—21no—
359#125—22no—
360#124—22no—
361#124—21no—
362#123—22no—
363#123—22no—
364#123—22no—
365#123—22no—
366#123—22no—
367#123—22no—
368C22⋊Q8—22no—
369C22⋊Q8—22no—
370C22⋊Q8—21no—
371C22⋊Q8—22no—
372C22⋊Q8C7×D412yescomplemented
373C22⋊Q8C23×C712yes—
374#121—22no—
375#121—22no—
376#121—22no—
377#121—21no—
378#121—22no—
379#121—22no—
380#121C7×D412yescomplemented
381#121C23×C712yes—
382#120—22no—
383#120—22no—
384#120—22no—
385C4×Q8—21no—
386C4×D4—22no—
387C4×D4—22no—
388C4×D4—21no—
389C4×D4—22no—
390C4×D4—22no—
391C4×D4—22no—
392C4×D4—22no—
393C4×D4—22no—
394#117—22no—
395#117C7×D412yescomplemented
396C2×C4⋊C4—21no—
397C2×C4⋊C4—21no—
398C2×C4⋊C4C23×C711yescharacteristic
399C2×C4⋊C4C23×C711yescharacteristic
400C2×C22⋊C4—21no—
401C2×C22⋊C4—21no—
402C2×C22⋊C4—21no—
403C2×C22⋊C4—21no—
404C2×C22⋊C4—21no—
405C2×C22⋊C4—21no—
406C2×C22⋊C4—21no—
407C2×C22⋊C4—21no—
408C2×C22⋊C4C7×D411yescharacteristic
409C2×C22⋊C4C23×C711yescharacteristic
410C2×C42—21no—
411C2×C42C7×D411yescharacteristic
412#107—22no—
413#106—22no—
414#103—22no—
415#102—22no—
416#102—22no—
417#102—22no—
418C22⋊C8—22no—
419C22⋊C8C23×C712yes—
420C22⋊C8C23×C712yes—
421#95—21no—
422#95—21no—
423#95—21no—
424#95C23×C711yescharacteristic
425#95C23×C711yescharacteristic
426C22×C7—21no—
427C22×C7—21no—
428C22×C7—22no—
429C22×C7—22no—
430C22×C7—22no—
431C22×C7—22no—
432C22×C7—42no—
433C22×C7—41no—
434C22×C7—41no—
435C22×C7—42no—
436C22×C7—42no—
437C22×C7—42no—
438C22×C7—81no—
439C22×C7—81no—
440C22×C7—81no—
441C22×C7—81no—
442C22×C7—81no—
443C22×C7—81no—
444C22×C7—81no—
445C22×C7—84no—
446C22×C7—81no—
447C22×C7—81no—
448C22×C7—81no—
449C22×C7C2×#12011yescharacteristic
450C22×C7C2×#13511yescharacteristic
451C22×C7#57511yescharacteristic
452C4×C7—22no—
453C4×C7—21no—
454C4×C7—21no—
455C4×C7—41no—
456C4×C7—42no—
457C4×C7—41no—
458C4×C7—42no—
459C4×C7—41no—
460C4×C7—42no—
461C4×C7—81no—
462C4×C7—81no—
463C4×C7—81no—
464C24—21no—
465C24—21no—
466C24C2×C7×D411yescharacteristic
467C24C2×C7×D411yescharacteristic
468#41—82no—
469C2×Q8—21no—
470C2×Q8C2×C7×D412yes—
471C2×D4—22no—
472C2×D4—22no—
473C2×D4—22no—
474C2×D4—22no—
475C2×D4—21no—
476C2×D4—22no—
477C2×D4—22no—
478C2×D4—22no—
479C2×D4—41no—
480C2×D4—42no—
481C2×D4—42no—
482C2×D4—42no—
483C2×D4—42no—
484C2×D4—81no—
485C2×D4—81no—
486C2×D4—81no—
487C2×D4—81no—
488C2×D4—81no—
489C2×D4C2×C7×D412yes—
490C22×C4—21no—
491C22×C4—21no—
492C22×C4—21no—
493C22×C4—22no—
494C22×C4—21no—
495C22×C4—22no—
496C22×C4—21no—
497C22×C4—21no—
498C22×C4—21no—
499C22×C4—21no—
500C22×C4—81no—
501C22×C4—81no—
502C22×C4—82no—
503C22×C4C2×C7×D411yescharacteristic
504C22×C4C2×C7×D411yescharacteristic
505C22×C4C2×C7×D411yescharacteristic
506C22×C4C2×C7×D411yescharacteristic
507C22×C4C2×C7×D412yes—
508C22×C4C2×C7×D412yes—
509C22×C4C24×C711yescharacteristic
510SD16—82no—
511D8—82no—
512C2×C8—22no—
513C2×C8—22no—
514C2×C8—82no—
515C4⋊C4—22no—
516C4⋊C4—22no—
517C4⋊C4—22no—
518C4⋊C4—21no—
519C4⋊C4—21no—
520C4⋊C4—22no—
521C4⋊C4—22no—
522C4⋊C4—22no—
523C4⋊C4—22no—
524C4⋊C4—22no—
525C4⋊C4—81no—
526C4⋊C4C2×C7×D412yes—
527C4⋊C4C2×C7×D412yes—
528C22⋊C4—22no—
529C22⋊C4—22no—
530C22⋊C4—22no—
531C22⋊C4—22no—
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—42no—
541C22⋊C4—41no—
542C22⋊C4—41no—
543C22⋊C4—81no—
544C22⋊C4—81no—
545C22⋊C4—81no—
546C22⋊C4—81no—
547C42—22no—
548C42—22no—
549C42—21no—
550C42—21no—
551C2×C7—21no—
552C2×C7—21no—
553C2×C7—42no—
554C2×C7—42no—
555C2×C7—82no—
556C2×C7—161no—
557C2×C7#300211yescharacteristic
558C2×C7#302111yescharacteristic
559C2×C7#302611yescharacteristic
560C23—21no—
561C23—21no—
562C23—21no—
563C23—21no—
564C23—22no—
565C23—41no—
566C23—41no—
567C23—41no—
568C23—41no—
569C23—42no—
570C23—41no—
571C23—41no—
572C23—41no—
573C23—41no—
574C23—41no—
575C23—41no—
576C23—41no—
577C23—81no—
578C23—81no—
579C23C7×#12012yes—
580C23C7×#13512yes—
581C23C22×C7×D411yescharacteristic
582Q8—42no—
583Q8—81no—
584D4—42no—
585D4—42no—
586D4—42no—
587D4—44no—
588D4—44no—
589D4—42no—
590D4—42no—
591D4—82no—
592D4—82no—
593D4—81no—
594D4—84no—
595D4—84no—
596C2×C4—21no—
597C2×C4—22no—
598C2×C4—21no—
599C2×C4—22no—
600C2×C4—21no—
601C2×C4—22no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—42no—
605C2×C4—41no—
606C2×C4—42no—
607C2×C4—41no—
608C2×C4—42no—
609C2×C4—44no—
610C2×C4—42no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—81no—
623C2×C4—81no—
624C2×C4—82no—
625C2×C4—81no—
626C2×C4C7×#12011yescharacteristic
627C2×C4C7×#12011yescharacteristic
628C2×C4C22×C7×D412yes—
629C8—42no—
630C8—42no—
631C7#1829011yescharacteristic, direct product factor, Sylow
632C22—21no—
633C22—21no—
634C22—22no—
635C22—22no—
636C22—22no—
637C22—22no—
638C22—42no—
639C22—41no—
640C22—41no—
641C22—42no—
642C22—42no—
643C22—42no—
644C22—81no—
645C22—81no—
646C22—81no—
647C22—81no—
648C22—81no—
649C22—81no—
650C22—81no—
651C22—84no—
652C22—81no—
653C22—81no—
654C22—81no—
655C22C2×C7×#12011yescharacteristic
656C22C2×C7×#13511yescharacteristic
657C22C7×#57511yescharacteristic
658C4—22no—
659C4—21no—
660C4—21no—
661C4—41no—
662C4—42no—
663C4—41no—
664C4—42no—
665C4—41no—
666C4—42no—
667C4—81no—
668C4—81no—
669C4—81no—
670C2—21no—
671C2—21no—
672C2—42no—
673C2—42no—
674C2—82no—
675C2—161no—
676C2C7×#300211yescharacteristic
677C2C7×#302111yescharacteristic
678C2C7×#302611yescharacteristic
subgroupG/H
center, socleC22×C7C2×#120
commutator, Frattini, cosocleC22×C4C24×C7
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C7 ◁ C23×C4×C7 ◁ G

Complements

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

NH ≅ G/Ntype
1. C7×#3459C2semidirect
2. C7×#3188C2semidirect
3. C7×#2635C2semidirect
4. C7×#1615C2semidirect
5. C7×#1614C2semidirect
6. C7×#1613C2semidirect
7. C7×#1597C2semidirect
8. C7×#1596C2semidirect
9. C7×#1595C2semidirect
10. C7×#1516C2semidirect
11. C7×#1459C2semidirect
23. C2×C7×#121C22semidirect
26. C7×#484C22semidirect
31. C7×#450C22semidirect
34. C2×C7×C22⋊C8C22semidirect
35. C7×#400C22semidirect
38. C7×#394C22semidirect
43. C7×#381C22semidirect
44. C7×#328C22semidirect
45. C7×#328C22semidirect
48. C7×#324C22semidirect
51. #18290C7direct (N × H)
52. C2×C7×#41D4semidirect
81. C7×C22⋊Q8D4semidirect
89. C7×#121D4semidirect
104. C7×#117D4semidirect
135. #3459C2×C7semidirect
136. #3188C2×C7semidirect
137. #2635C2×C7semidirect
138. #1615C2×C7semidirect
139. #1614C2×C7semidirect
140. #1613C2×C7semidirect
141. #1597C2×C7semidirect
142. #1596C2×C7semidirect
143. #1595C2×C7semidirect
144. #1516C2×C7semidirect
145. #1459C2×C7semidirect
244. C2×#121C22×C7semidirect
247. #484C22×C7semidirect
252. #450C22×C7semidirect
255. C2×C22⋊C8C22×C7semidirect
256. #400C22×C7semidirect
259. #394C22×C7semidirect
264. #381C22×C7semidirect
265. #328C22×C7semidirect
266. #328C22×C7semidirect
269. #324C22×C7semidirect
343. C2×#41C7×D4semidirect
372. C22⋊Q8C7×D4semidirect
380. #121C7×D4semidirect
395. #117C7×D4semidirect
631. C7#18290direct (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×#31882
C7×#26352
C7×#16152
C7×#16142
C7×#16132
C7×#15972
C7×#15962
C7×#15952
C7×#15162
C7×#14592
#182907