Finite Group Databasebeta

← C7×#18453 #13195669

Subgroups — C7×#18453 #13195669

Order 1792.

970 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×#3536C211yesmaximal, characteristic
2C7×#3196C211yesmaximal, characteristic, complemented
3C7×#3189C211yesmaximal, characteristic, complemented
4C7×#2816C211yesmaximal, characteristic, complemented
5C7×#1618C212yesmaximal, complemented
6C7×#1614C212yesmaximal, complemented
7C7×#1606C212yesmaximal, complemented
8C7×#1602C212yesmaximal, complemented
9C7×#1516C211yesmaximal, characteristic, complemented
10C7×#1511C211yesmaximal, characteristic, complemented
11C7×#1471C211yesmaximal, characteristic, complemented
12C7×#562—22no—
13C7×#554—21no—
14C7×#548—21no—
15C7×#547—21no—
16C7×#546—22no—
17C7×#546—21no—
18C7×#543C2212yes—
19C7×#542—21no—
20C7×#539—21no—
21C7×#538C2212yes—
22C7×#537—21no—
23C7×#537—21no—
24C7×#525C2211yescharacteristic
25C2×C7×C22⋊Q8C2211yescharacteristic
26C2×C7×C22⋊Q8C2211yescharacteristic
27C2×C7×#121C2211yescharacteristic
28C7×#485—21no—
29C7×#484—21no—
30C7×#484C2212yes—
31C7×#484C2211yescharacteristic
32C7×#481C2211yescharacteristic
33C7×#470—21no—
34C7×#468—21no—
35C7×#467—22no—
36C7×#452—21no—
37C7×#451—21no—
38C7×#451C2212yes—
39C7×#451C2211yescharacteristic
40C7×#448C2211yescharacteristic
41C7×#407C2211yescharacteristic
42C2×C7×C22⋊C8C2211yescharacteristic
43C7×#399C2212yes—
44C7×#398—21no—
45C7×#394—21no—
46C7×#394—22no—
47C7×#393C2212yes—
48C7×#381C2211yescharacteristic
49C7×#328C2212yes—
50C7×#328C2212yes—
51C7×#328C2212yes—
52C7×#327C2212yes—
53C7×#323C2212yes—
54C7×#323C2212yes—
55#18453C711yesmaximal, characteristic, direct product factor, Sylow
56C2×C7×#41—22no—
57C2×C7×#41—21no—
58C2×C7×#41D411yescharacteristic
59C22×C7×Q8D411yescharacteristic
60C22×C7×Q8C2311yescharacteristic
61C22×C7×D4D412yes—
62C22×C7×D4D411yescharacteristic
63C23×C4×C7C2311yescharacteristic
64C2×C7×Q16—22no—
65C2×C7×Q16—21no—
66C2×C7×SD16—21no—
67C2×C7×M16—21no—
68C22×C7×C8—21no—
69C7×C4⋊Q8—21no—
70C7×#126—22no—
71C7×#125—22no—
72C7×#124—21no—
73C7×#124—22no—
74C7×#124—22no—
75C7×#123—22no—
76C7×#123—22no—
77C7×#123—22no—
78C7×#123—21no—
79C7×#123—22no—
80C7×C22⋊Q8—21no—
81C7×C22⋊Q8—22no—
82C7×C22⋊Q8—22no—
83C7×C22⋊Q8—22no—
84C7×C22⋊Q8—21no—
85C7×C22⋊Q8—21no—
86C7×C22⋊Q8—22no—
87C7×C22⋊Q8—22no—
88C7×C22⋊Q8—22no—
89C7×C22⋊Q8—21no—
90C7×C22⋊Q8—21no—
91C7×C22⋊Q8—21no—
92C7×C22⋊Q8—21no—
93C7×C22⋊Q8D412yes—
94C7×C22⋊Q8D411yescharacteristic
95C7×C22⋊Q8C2312yes—
96C7×C22⋊Q8C2311yescharacteristic
97C7×#121—21no—
98C7×#121—22no—
99C7×#121—22no—
100C7×#121—22no—
101C7×#121—22no—
102C7×#121—22no—
103C7×#121—21no—
104C7×#121—21no—
105C7×#121—21no—
106C7×#121—21no—
107C7×#121D411yescharacteristic
108C7×#121C2311yescharacteristic
109C7×#120—21no—
110C7×#120—22no—
111C4×C7×Q8—21no—
112C4×C7×D4—22no—
113C4×C7×D4—21no—
114C4×C7×D4—21no—
115C4×C7×D4—21no—
116C4×C7×D4—22no—
117C7×#117D411yescharacteristic
118C2×C7×C4⋊C4—21no—
119C2×C7×C4⋊C4—21no—
120C2×C7×C4⋊C4—21no—
121C2×C7×C4⋊C4—22no—
122C2×C7×C4⋊C4D411yescharacteristic
123C2×C7×C4⋊C4D411yescharacteristic
124C2×C7×C4⋊C4C2311yescharacteristic
125C2×C7×C22⋊C4—21no—
126C2×C7×C22⋊C4—22no—
127C2×C7×C22⋊C4—21no—
128C2×C7×C22⋊C4—21no—
129C2×C7×C22⋊C4—22no—
130C2×C7×C22⋊C4—22no—
131C2×C7×C22⋊C4C2312yes—
132C7×#107—22no—
133C7×#107—21no—
134C7×#106—21no—
135C7×#103—22no—
136C7×#103—21no—
137C7×#103—22no—
138C7×#103—21no—
139C7×#103—21no—
140C7×#102—21no—
141C7×C22⋊C8—21no—
142C7×C22⋊C8—21no—
143C7×C22⋊C8C2311yescharacteristic
144C7×C22⋊C8C2312yes—
145C7×C22⋊C8C2311yescharacteristic
146C7×#95—21no—
147C7×#95—22no—
148C7×#95C2312yes—
149#3536C2×C711yescharacteristic
150#3196C2×C711yescharacteristic, complemented
151#3189C2×C711yescharacteristic, complemented
152#2816C2×C711yescharacteristic, complemented
153#1618C2×C712yescomplemented
154#1614C2×C712yescomplemented
155#1606C2×C712yescomplemented
156#1602C2×C712yescomplemented
157#1516C2×C711yescharacteristic, complemented
158#1511C2×C711yescharacteristic, complemented
159#1471C2×C711yescharacteristic, complemented
160C24×C7C2×D412yes—
161C24×C7C2×D411yescharacteristic
162C7×#41—81no—
163C7×#41—84no—
164C7×#41—82no—
165C2×C7×Q8—22no—
166C2×C7×Q8—21no—
167C2×C7×Q8—21no—
168C2×C7×Q8—21no—
169C2×C7×Q8—81no—
170C2×C7×Q8—81no—
171C2×C7×Q8C2×D411yescharacteristic
172C2×C7×Q8C2×D412yes—
173C2×C7×D4—21no—
174C2×C7×D4—22no—
175C2×C7×D4—21no—
176C2×C7×D4—22no—
177C2×C7×D4—22no—
178C2×C7×D4—21no—
179C2×C7×D4—22no—
180C2×C7×D4—21no—
181C2×C7×D4—21no—
182C2×C7×D4—81no—
183C2×C7×D4—82no—
184C2×C7×D4C2×D411yescharacteristic
185C22×C4×C7—21no—
186C22×C4×C7—22no—
187C22×C4×C7—22no—
188C22×C4×C7—22no—
189C22×C4×C7—21no—
190C22×C4×C7—21no—
191C22×C4×C7—22no—
192C22×C4×C7—21no—
193C22×C4×C7—21no—
194C22×C4×C7—81no—
195C22×C4×C7C2×D412yes—
196C22×C4×C7C2×D411yescharacteristic
197C22×C4×C7C2×D411yescharacteristic
198C22×C4×C7C2×D411yescharacteristic
199C22×C4×C7C2×D411yescharacteristic
200C22×C4×C7C2×D411yescharacteristic
201C22×C4×C7C2411yescharacteristic
202C7×Q16—82no—
203C7×Q16—81no—
204C7×SD16—81no—
205C7×M16—81no—
206C2×C7×C8—21no—
207C2×C7×C8—22no—
208C2×C7×C8—21no—
209C2×C7×C8—81no—
210C7×C4⋊C4—22no—
211C7×C4⋊C4—22no—
212C7×C4⋊C4—22no—
213C7×C4⋊C4—22no—
214C7×C4⋊C4—21no—
215C7×C4⋊C4—21no—
216C7×C4⋊C4—22no—
217C7×C4⋊C4—21no—
218C7×C4⋊C4—21no—
219C7×C4⋊C4—22no—
220C7×C4⋊C4—21no—
221C7×C4⋊C4—21no—
222C7×C4⋊C4—21no—
223C7×C4⋊C4—41no—
224C7×C4⋊C4—41no—
225C7×C4⋊C4—82no—
226C7×C4⋊C4C2×D411yescharacteristic
227C7×C4⋊C4C2×D412yes—
228C7×C4⋊C4C2×D411yescharacteristic
229C7×C22⋊C4—22no—
230C7×C22⋊C4—22no—
231C7×C22⋊C4—22no—
232C7×C22⋊C4—22no—
233C7×C22⋊C4—22no—
234C7×C22⋊C4—22no—
235C7×C22⋊C4—21no—
236C7×C22⋊C4—21no—
237C7×C22⋊C4—22no—
238C7×C22⋊C4—21no—
239C7×C22⋊C4—21no—
240C7×C22⋊C4—22no—
241C7×C22⋊C4—22no—
242C7×C22⋊C4—21no—
243C7×C22⋊C4—41no—
244C7×C22⋊C4—41no—
245C7×C22⋊C4—81no—
246C7×C22⋊C4—82no—
247C7×C22⋊C4—82no—
248C42×C7—22no—
249C42×C7—21no—
250#562—22no—
251#554—21no—
252#548—21no—
253#547—21no—
254#546—22no—
255#546—21no—
256#543C22×C712yes—
257#542—21no—
258#539—21no—
259#538C22×C712yes—
260#537—21no—
261#537—21no—
262#525C22×C711yescharacteristic
263C2×C22⋊Q8C22×C711yescharacteristic
264C2×C22⋊Q8C22×C711yescharacteristic
265C2×#121C22×C711yescharacteristic
266#485—21no—
267#484—21no—
268#484C22×C711yescharacteristic
269#484C22×C712yes—
270#481C22×C711yescharacteristic
271#470—21no—
272#468—21no—
273#467—22no—
274#452—21no—
275#451—21no—
276#451C22×C711yescharacteristic
277#451C22×C712yes—
278#448C22×C711yescharacteristic
279#407C22×C711yescharacteristic
280C2×C22⋊C8C22×C711yescharacteristic
281#399C22×C712yes—
282#398—21no—
283#394—21no—
284#394—22no—
285#393C22×C712yes—
286#381C22×C711yescharacteristic
287#328C22×C712yes—
288#328C22×C712yes—
289#328C22×C712yes—
290#327C22×C712yes—
291#323C22×C712yes—
292#323C22×C712yes—
293C23×C7—22no—
294C23×C7—22no—
295C23×C7—21no—
296C23×C7—24no—
297C23×C7—41no—
298C23×C7—41no—
299C23×C7—42no—
300C23×C7—42no—
301C23×C7#12011yescharacteristic
302C23×C7#12011yescharacteristic
303C23×C7C22×D411yescharacteristic
304C7×Q8—42no—
305C7×Q8—41no—
306C7×Q8—42no—
307C7×Q8—41no—
308C7×Q8—41no—
309C7×Q8—42no—
310C7×Q8—82no—
311C7×D4—41no—
312C7×D4—42no—
313C7×D4—44no—
314C7×D4—42no—
315C7×D4—42no—
316C7×D4—81no—
317C7×D4—82no—
318C7×D4—81no—
319C7×D4—82no—
320C7×D4—82no—
321C2×C4×C7—21no—
322C2×C4×C7—22no—
323C2×C4×C7—21no—
324C2×C4×C7—22no—
325C2×C4×C7—21no—
326C2×C4×C7—22no—
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—42no—
334C2×C4×C7—42no—
335C2×C4×C7—42no—
336C2×C4×C7—42no—
337C2×C4×C7—41no—
338C2×C4×C7—42no—
339C2×C4×C7—42no—
340C2×C4×C7—41no—
341C2×C4×C7—41no—
342C2×C4×C7—41no—
343C2×C4×C7—42no—
344C2×C4×C7—81no—
345C2×C4×C7—82no—
346C2×C4×C7—82no—
347C2×C4×C7—82no—
348C2×C4×C7#12011yescharacteristic
349C2×C4×C7#12011yescharacteristic
350C2×C4×C7C22×D411yescharacteristic
351C2×C4×C7C22×D411yescharacteristic
352C7×C8—42no—
353C7×C8—41no—
354C7×C8—41no—
355C2×#41—22no—
356C2×#41—21no—
357C2×#41C7×D411yescharacteristic
358C22×Q8C7×D411yescharacteristic
359C22×Q8C23×C711yescharacteristic
360C22×D4C7×D412yes—
361C22×D4C7×D411yescharacteristic
362C23×C4C23×C711yescharacteristic
363C2×Q16—22no—
364C2×Q16—21no—
365C2×SD16—21no—
366C2×M16—21no—
367C22×C8—21no—
368C4⋊Q8—21no—
369#126—22no—
370#125—22no—
371#124—21no—
372#124—22no—
373#124—22no—
374#123—22no—
375#123—22no—
376#123—22no—
377#123—21no—
378#123—22no—
379C22⋊Q8—21no—
380C22⋊Q8—22no—
381C22⋊Q8—22no—
382C22⋊Q8—22no—
383C22⋊Q8—21no—
384C22⋊Q8—21no—
385C22⋊Q8—22no—
386C22⋊Q8—22no—
387C22⋊Q8—22no—
388C22⋊Q8—21no—
389C22⋊Q8—21no—
390C22⋊Q8—21no—
391C22⋊Q8—21no—
392C22⋊Q8C7×D412yes—
393C22⋊Q8C7×D411yescharacteristic
394C22⋊Q8C23×C712yes—
395C22⋊Q8C23×C711yescharacteristic
396#121—21no—
397#121—22no—
398#121—22no—
399#121—22no—
400#121—22no—
401#121—22no—
402#121—21no—
403#121—21no—
404#121—21no—
405#121—21no—
406#121C7×D411yescharacteristic
407#121C23×C711yescharacteristic
408#120—21no—
409#120—22no—
410C4×Q8—21no—
411C4×D4—22no—
412C4×D4—21no—
413C4×D4—21no—
414C4×D4—21no—
415C4×D4—22no—
416#117C7×D411yescharacteristic
417C2×C4⋊C4—21no—
418C2×C4⋊C4—21no—
419C2×C4⋊C4—21no—
420C2×C4⋊C4—22no—
421C2×C4⋊C4C7×D411yescharacteristic
422C2×C4⋊C4C7×D411yescharacteristic
423C2×C4⋊C4C23×C711yescharacteristic
424C2×C22⋊C4—21no—
425C2×C22⋊C4—22no—
426C2×C22⋊C4—21no—
427C2×C22⋊C4—21no—
428C2×C22⋊C4—22no—
429C2×C22⋊C4—22no—
430C2×C22⋊C4C23×C712yes—
431#107—22no—
432#107—21no—
433#106—21no—
434#103—22no—
435#103—21no—
436#103—21no—
437#103—22no—
438#103—21no—
439#102—21no—
440C22⋊C8—21no—
441C22⋊C8—21no—
442C22⋊C8C23×C712yes—
443C22⋊C8C23×C711yescharacteristic
444C22⋊C8C23×C711yescharacteristic
445#95—21no—
446#95—22no—
447#95C23×C712yes—
448C22×C7—21no—
449C22×C7—21no—
450C22×C7—21no—
451C22×C7—21no—
452C22×C7—21no—
453C22×C7—21no—
454C22×C7—21no—
455C22×C7—21no—
456C22×C7—41no—
457C22×C7—42no—
458C22×C7—41no—
459C22×C7—42no—
460C22×C7—41no—
461C22×C7—42no—
462C22×C7—82no—
463C22×C7—84no—
464C22×C7C2×#12011yescharacteristic
465C22×C7#57812yes—
466C4×C7—21no—
467C4×C7—21no—
468C4×C7—21no—
469C4×C7—21no—
470C4×C7—41no—
471C4×C7—42no—
472C4×C7—42no—
473C4×C7—41no—
474C4×C7—41no—
475C4×C7—42no—
476C4×C7—82no—
477C4×C7—81no—
478C4×C7—81no—
479C24C2×C7×D412yes—
480C24C2×C7×D411yescharacteristic
481#41—81no—
482#41—84no—
483#41—82no—
484C2×Q8—22no—
485C2×Q8—21no—
486C2×Q8—21no—
487C2×Q8—21no—
488C2×Q8—81no—
489C2×Q8—81no—
490C2×Q8C2×C7×D411yescharacteristic
491C2×Q8C2×C7×D412yes—
492C2×D4—21no—
493C2×D4—22no—
494C2×D4—21no—
495C2×D4—22no—
496C2×D4—22no—
497C2×D4—21no—
498C2×D4—22no—
499C2×D4—21no—
500C2×D4—21no—
501C2×D4—81no—
502C2×D4—82no—
503C2×D4C2×C7×D411yescharacteristic
504C22×C4—21no—
505C22×C4—22no—
506C22×C4—22no—
507C22×C4—22no—
508C22×C4—21no—
509C22×C4—21no—
510C22×C4—22no—
511C22×C4—21no—
512C22×C4—21no—
513C22×C4—81no—
514C22×C4C2×C7×D412yes—
515C22×C4C2×C7×D411yescharacteristic
516C22×C4C2×C7×D411yescharacteristic
517C22×C4C2×C7×D411yescharacteristic
518C22×C4C2×C7×D411yescharacteristic
519C22×C4C2×C7×D411yescharacteristic
520C22×C4C24×C711yescharacteristic
521Q16—82no—
522Q16—81no—
523SD16—81no—
524M16—81no—
525C2×C8—21no—
526C2×C8—22no—
527C2×C8—21no—
528C2×C8—81no—
529C4⋊C4—22no—
530C4⋊C4—22no—
531C4⋊C4—22no—
532C4⋊C4—22no—
533C4⋊C4—21no—
534C4⋊C4—21no—
535C4⋊C4—22no—
536C4⋊C4—21no—
537C4⋊C4—21no—
538C4⋊C4—22no—
539C4⋊C4—21no—
540C4⋊C4—21no—
541C4⋊C4—21no—
542C4⋊C4—41no—
543C4⋊C4—41no—
544C4⋊C4—82no—
545C4⋊C4C2×C7×D411yescharacteristic
546C4⋊C4C2×C7×D412yes—
547C4⋊C4C2×C7×D411yescharacteristic
548C22⋊C4—22no—
549C22⋊C4—22no—
550C22⋊C4—22no—
551C22⋊C4—22no—
552C22⋊C4—22no—
553C22⋊C4—22no—
554C22⋊C4—21no—
555C22⋊C4—21no—
556C22⋊C4—22no—
557C22⋊C4—21no—
558C22⋊C4—21no—
559C22⋊C4—22no—
560C22⋊C4—22no—
561C22⋊C4—21no—
562C22⋊C4—41no—
563C22⋊C4—41no—
564C22⋊C4—81no—
565C22⋊C4—82no—
566C22⋊C4—82no—
567C42—22no—
568C42—21no—
569C2×C7—22no—
570C2×C7—41no—
571C2×C7—41no—
572C2×C7—81no—
573C2×C7—82no—
574C2×C7#301111yescharacteristic
575C2×C7#301911yescharacteristic
576C2×C7#302611yescharacteristic
577C23—22no—
578C23—22no—
579C23—21no—
580C23—24no—
581C23—41no—
582C23—41no—
583C23—42no—
584C23—42no—
585C23C7×#12011yescharacteristic
586C23C7×#12011yescharacteristic
587C23C22×C7×D411yescharacteristic
588Q8—42no—
589Q8—41no—
590Q8—42no—
591Q8—41no—
592Q8—41no—
593Q8—42no—
594Q8—82no—
595D4—41no—
596D4—42no—
597D4—44no—
598D4—42no—
599D4—42no—
600D4—81no—
601D4—82no—
602D4—81no—
603D4—82no—
604D4—82no—
605C2×C4—21no—
606C2×C4—22no—
607C2×C4—21no—
608C2×C4—22no—
609C2×C4—21no—
610C2×C4—22no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—41no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—41no—
625C2×C4—41no—
626C2×C4—41no—
627C2×C4—42no—
628C2×C4—81no—
629C2×C4—82no—
630C2×C4—82no—
631C2×C4—82no—
632C2×C4C7×#12011yescharacteristic
633C2×C4C7×#12011yescharacteristic
634C2×C4C22×C7×D411yescharacteristic
635C2×C4C22×C7×D411yescharacteristic
636C8—42no—
637C8—41no—
638C8—41no—
639C7#1845311yescharacteristic, direct product factor, Sylow
640C22—21no—
641C22—21no—
642C22—21no—
643C22—21no—
644C22—21no—
645C22—21no—
646C22—21no—
647C22—21no—
648C22—41no—
649C22—42no—
650C22—41no—
651C22—42no—
652C22—41no—
653C22—42no—
654C22—82no—
655C22—84no—
656C22C2×C7×#12011yescharacteristic
657C22C7×#57812yes—
658C4—21no—
659C4—21no—
660C4—21no—
661C4—21no—
662C4—41no—
663C4—42no—
664C4—42no—
665C4—41no—
666C4—41no—
667C4—42no—
668C4—82no—
669C4—81no—
670C4—81no—
671C2—22no—
672C2—41no—
673C2—41no—
674C2—82no—
675C2—81no—
676C2C7×#301111yescharacteristic
677C2C7×#301911yescharacteristic
678C2C7×#302611yescharacteristic
subgroupG/H
center, socleC22×C7C2×#120
commutator, Frattini, cosocleC22×C4C24×C7
prime radicalC7×#3536C2
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
2. C7×#3196C2semidirect
3. C7×#3189C2semidirect
4. C7×#2816C2semidirect
5. C7×#1618C2semidirect
6. C7×#1614C2semidirect
7. C7×#1606C2semidirect
8. C7×#1602C2semidirect
9. C7×#1516C2semidirect
10. C7×#1511C2semidirect
11. C7×#1471C2semidirect
55. #18453C7direct (N × H)
150. #3196C2×C7semidirect
151. #3189C2×C7semidirect
152. #2816C2×C7semidirect
153. #1618C2×C7semidirect
154. #1614C2×C7semidirect
155. #1606C2×C7semidirect
156. #1602C2×C7semidirect
157. #1516C2×C7semidirect
158. #1511C2×C7semidirect
159. #1471C2×C7semidirect
639. C7#18453direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#35362
C7×#31962
C7×#31892
C7×#28162
C7×#16182
C7×#16142
C7×#16062
C7×#16022
C7×#15162
C7×#15112
C7×#14712
#184537