Finite Group Databasebeta

← C7×#24419 #13201635

Subgroups — C7×#24419 #13201635

Order 1792.

1360 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3486C211yesmaximal, characteristic
2C7×#3056C212yesmaximal, complemented
3C2×C7×#456C211yesmaximal, characteristic, complemented
4C22×C7×#101C211yesmaximal, characteristic, complemented
5C7×#2434C211yesmaximal, characteristic, complemented
6C7×#2038C212yesmaximal, complemented
7C7×#2005C212yesmaximal, complemented
8C7×#1886C211yesmaximal, characteristic, complemented
9C7×#1871C212yesmaximal, complemented
10C7×#1857C211yesmaximal, characteristic, complemented
11C7×#1793C211yesmaximal, characteristic, complemented
12C2×C7×2−1+4C2211yescharacteristic
13C23×C7×Q8C2211yescharacteristic
14C2×C7×#137C2212yes—
15C22×C7×M16C2212yes—
16C7×D4×Q8—22no—
17C7×D4×Q8—22no—
18C7×#548—22no—
19C7×#548—22no—
20C7×#536—22no—
21C7×#536C2212yes—
22C2×C7×C4⋊Q8C2211yescharacteristic
23C2×C7×C4⋊Q8C2211yescharacteristic
24C7×#525—22no—
25C7×#525C2212yes—
26C2×C7×C22⋊Q8C2211yescharacteristic
27C7×#519C2211yescharacteristic
28C7×#470—22no—
29C7×#468—22no—
30C7×#467—22no—
31C7×#465—22no—
32C7×#456—24no—
33C7×#428C2212yes—
34C2×C7×#104C2212yes—
35C2×C7×#104C2212yes—
36C7×#419C2212yes—
37C7×#417C2212yes—
38C2×C7×#101—24no—
39C2×C7×#101C2212yes—
40C2×C7×#101C2211yescharacteristic
41C2×C7×#101C2211yescharacteristic
42C7×#395—22no—
43C7×#391C2211yescharacteristic
44C7×#391C2211yescharacteristic
45C7×#385—22no—
46C4×C7×C22⋊C4C2211yescharacteristic
47C7×#339C2212yes—
48C7×#339C2212yes—
49#24419C711yesmaximal, characteristic, direct product factor, Sylow
50C7×2−1+4—22no—
51C7×2−1+4—22no—
52C7×2−1+4—42no—
53C2×C7×#41—22no—
54C2×C7×#41—22no—
55C2×C7×#41C2312yes—
56C22×C7×Q8—22no—
57C22×C7×Q8—22no—
58C22×C7×Q8—22no—
59C22×C7×Q8—22no—
60C22×C7×Q8—22no—
61C22×C7×Q8—42no—
62C22×C7×Q8D412yes—
63C22×C7×Q8C2311yescharacteristic
64C22×C7×Q8C2311yescharacteristic
65C22×C7×Q8C2311yescharacteristic
66C23×C4×C7D412yes—
67C23×C4×C7C2311yescharacteristic
68C7×#137—24no—
69C7×#137—42no—
70C2×C7×Q16—22no—
71C2×C7×SD16—22no—
72C2×C7×M16—22no—
73C2×C7×M16—24no—
74C2×C7×M16—42no—
75C2×C7×M16C2312yes—
76C2×C7×M16C2312yes—
77C22×C7×C8—22no—
78C7×C4⋊Q8—24no—
79C7×C4⋊Q8—22no—
80C7×C4⋊Q8—22no—
81C7×C4⋊Q8—22no—
82C7×C4⋊Q8—42no—
83C7×#124—24no—
84C7×#124—22no—
85C7×#123—24no—
86C7×#123—22no—
87C7×C22⋊Q8—22no—
88C7×C22⋊Q8—22no—
89C7×C22⋊Q8—24no—
90C7×C22⋊Q8—24no—
91C7×C22⋊Q8—22no—
92C7×C22⋊Q8—22no—
93C7×C22⋊Q8—22no—
94C7×C22⋊Q8—22no—
95C7×C22⋊Q8D412yes—
96C7×C22⋊Q8D412yes—
97C7×#121—22no—
98C7×#121—22no—
99C7×#121D412yes—
100C7×#121D412yes—
101C7×#120—22no—
102C4×C7×Q8—22no—
103C4×C7×Q8—22no—
104C4×C7×D4—24no—
105C4×C7×D4—24no—
106C4×C7×D4—22no—
107C4×C7×D4—22no—
108C7×#117—22no—
109C7×#117C2312yes—
110C2×C7×C4⋊C4—22no—
111C2×C7×C4⋊C4—22no—
112C2×C7×C4⋊C4—21no—
113C2×C7×C4⋊C4C2311yescharacteristic
114C2×C7×C22⋊C4—21no—
115C2×C7×C22⋊C4—21no—
116C2×C42×C7C2311yescharacteristic
117C2×C42×C7C2311yescharacteristic
118C7×#107—22no—
119C7×#106—22no—
120C7×#104—24no—
121C7×#104—42no—
122C7×#103—22no—
123C7×#103—22no—
124C7×#103—22no—
125C7×#102—22no—
126C7×#101—22no—
127C7×#101—24no—
128C7×#101D414yescomplemented
129C7×#95—21no—
130C7×#95—22no—
131C7×#95—22no—
132#3486C2×C711yescharacteristic
133#3056C2×C712yescomplemented
134C2×#456C2×C711yescharacteristic, complemented
135C22×#101C2×C711yescharacteristic, complemented
136#2434C2×C711yescharacteristic, complemented
137#2038C2×C712yescomplemented
138#2005C2×C712yescomplemented
139#1886C2×C711yescharacteristic, complemented
140#1871C2×C712yescomplemented
141#1857C2×C711yescharacteristic, complemented
142#1793C2×C711yescharacteristic, complemented
143C24×C7C2×D411yescharacteristic
144C7×#41—22no—
145C7×#41—22no—
146C7×#41—22no—
147C7×#41—22no—
148C7×#41—44no—
149C7×#41—44no—
150C7×#41—44no—
151C7×#41—44no—
152C2×C7×Q8—22no—
153C2×C7×Q8—24no—
154C2×C7×Q8—24no—
155C2×C7×Q8—22no—
156C2×C7×Q8—21no—
157C2×C7×Q8—22no—
158C2×C7×Q8—21no—
159C2×C7×Q8—22no—
160C2×C7×Q8—22no—
161C2×C7×Q8—22no—
162C2×C7×Q8—21no—
163C2×C7×Q8—21no—
164C2×C7×Q8—22no—
165C2×C7×Q8—42no—
166C2×C7×Q8—44no—
167C2×C7×Q8—44no—
168C2×C7×Q8—44no—
169C2×C7×Q8—44no—
170C2×C7×Q8—44no—
171C2×C7×Q8—44no—
172C2×C7×Q8C2×D412yes—
173C2×C7×Q8C2×D412yes—
174C2×C7×Q8#4112yes—
175C2×C7×D4—22no—
176C2×C7×D4—22no—
177C2×C7×D4—22no—
178C2×C7×D4—22no—
179C2×C7×D4C2×D412yes—
180C22×C4×C7—21no—
181C22×C4×C7—21no—
182C22×C4×C7—22no—
183C22×C4×C7—24no—
184C22×C4×C7—22no—
185C22×C4×C7—22no—
186C22×C4×C7—21no—
187C22×C4×C7—21no—
188C22×C4×C7—24no—
189C22×C4×C7—22no—
190C22×C4×C7—21no—
191C22×C4×C7—21no—
192C22×C4×C7—42no—
193C22×C4×C7C2×D412yes—
194C22×C4×C7C2×D411yescharacteristic
195C22×C4×C7C2×D412yes—
196C22×C4×C7C2×D412yes—
197C22×C4×C7C2×D412yes—
198C22×C4×C7C2411yescharacteristic
199C7×Q16—44no—
200C7×SD16—44no—
201C7×M16—22no—
202C7×M16—22no—
203C7×M16—22no—
204C7×M16—44no—
205C7×M16C2×D414yes—
206C2×C7×C8—22no—
207C2×C7×C8—22no—
208C2×C7×C8—44no—
209C7×C4⋊C4—22no—
210C7×C4⋊C4—22no—
211C7×C4⋊C4—24no—
212C7×C4⋊C4—24no—
213C7×C4⋊C4—22no—
214C7×C4⋊C4—22no—
215C7×C4⋊C4—22no—
216C7×C4⋊C4—44no—
217C7×C4⋊C4—42no—
218C7×C4⋊C4—44no—
219C7×C4⋊C4C2×D412yes—
220C7×C4⋊C4C2×D412yes—
221C7×C22⋊C4—24no—
222C7×C22⋊C4—24no—
223C7×C22⋊C4—24no—
224C7×C22⋊C4—22no—
225C7×C22⋊C4—22no—
226C7×C22⋊C4—22no—
227C7×C22⋊C4—22no—
228C7×C22⋊C4—22no—
229C7×C22⋊C4—41no—
230C7×C22⋊C4—41no—
231C42×C7—24no—
232C42×C7—22no—
233C42×C7—21no—
234C42×C7—21no—
235C42×C7—22no—
236C2×2−1+4C22×C711yescharacteristic
237C23×Q8C22×C711yescharacteristic
238C2×#137C22×C712yes—
239C22×M16C22×C712yes—
240D4×Q8—22no—
241D4×Q8—22no—
242#548—22no—
243#548—22no—
244#536—22no—
245#536C22×C712yes—
246C2×C4⋊Q8C22×C711yescharacteristic
247C2×C4⋊Q8C22×C711yescharacteristic
248#525—22no—
249#525C22×C712yes—
250C2×C22⋊Q8C22×C711yescharacteristic
251#519C22×C711yescharacteristic
252#470—22no—
253#468—22no—
254#467—22no—
255#465—22no—
256#456—24no—
257#428C22×C712yes—
258C2×#104C22×C712yes—
259C2×#104C22×C712yes—
260#419C22×C712yes—
261#417C22×C712yes—
262C2×#101—24no—
263C2×#101C22×C711yescharacteristic
264C2×#101C22×C712yes—
265C2×#101C22×C711yescharacteristic
266#395—22no—
267#391C22×C711yescharacteristic
268#391C22×C711yescharacteristic
269#385—22no—
270C4×C22⋊C4C22×C711yescharacteristic
271#339C22×C712yes—
272#339C22×C712yes—
273C23×C7—21no—
274C23×C7—21no—
275C23×C7—22no—
276C23×C7—22no—
277C23×C7—41no—
278C23×C7#12011yescharacteristic
279C23×C7#12011yescharacteristic
280C23×C7C22×D411yescharacteristic
281C7×Q8—24no—
282C7×Q8—24no—
283C7×Q8—22no—
284C7×Q8—22no—
285C7×Q8—48no—
286C7×Q8—44no—
287C7×Q8—42no—
288C7×Q8—42no—
289C7×Q8—42no—
290C7×Q8—42no—
291C7×D4—22no—
292C7×D4—22no—
293C7×D4—44no—
294C7×D4—44no—
295C7×D4—42no—
296C7×D4—42no—
297C7×D4—44no—
298C2×C4×C7—22no—
299C2×C4×C7—21no—
300C2×C4×C7—21no—
301C2×C4×C7—22no—
302C2×C4×C7—22no—
303C2×C4×C7—22no—
304C2×C4×C7—22no—
305C2×C4×C7—22no—
306C2×C4×C7—22no—
307C2×C4×C7—22no—
308C2×C4×C7—22no—
309C2×C4×C7—22no—
310C2×C4×C7—22no—
311C2×C4×C7—22no—
312C2×C4×C7—22no—
313C2×C4×C7—22no—
314C2×C4×C7—44no—
315C2×C4×C7—44no—
316C2×C4×C7—48no—
317C2×C4×C7—41no—
318C2×C4×C7—41no—
319C2×C4×C7—41no—
320C2×C4×C7—41no—
321C2×C4×C7—42no—
322C2×C4×C7—42no—
323C2×C4×C7—42no—
324C2×C4×C7—41no—
325C2×C4×C7—41no—
326C2×C4×C7—41no—
327C2×C4×C7—41no—
328C2×C4×C7#12012yes—
329C2×C4×C7#12114yes—
330C2×C4×C7C22×D411yescharacteristic
331C2×C4×C7C22×D412yes—
332C2×C4×C7C2×#4111yescharacteristic
333C2×C4×C7H(5,𝔽2)12yes—
334C7×C8—24no—
335C7×C8—42no—
3362−1+4—22no—
3372−1+4—22no—
3382−1+4—42no—
339C2×#41—22no—
340C2×#41—22no—
341C2×#41C23×C712yes—
342C22×Q8—22no—
343C22×Q8—22no—
344C22×Q8—22no—
345C22×Q8—22no—
346C22×Q8—22no—
347C22×Q8—42no—
348C22×Q8C7×D412yes—
349C22×Q8C23×C711yescharacteristic
350C22×Q8C23×C711yescharacteristic
351C22×Q8C23×C711yescharacteristic
352C23×C4C7×D412yes—
353C23×C4C23×C711yescharacteristic
354#137—24no—
355#137—42no—
356C2×Q16—22no—
357C2×SD16—22no—
358C2×M16—22no—
359C2×M16—24no—
360C2×M16—42no—
361C2×M16C23×C712yes—
362C2×M16C23×C712yes—
363C22×C8—22no—
364C4⋊Q8—24no—
365C4⋊Q8—22no—
366C4⋊Q8—22no—
367C4⋊Q8—22no—
368C4⋊Q8—42no—
369#124—24no—
370#124—22no—
371#123—24no—
372#123—22no—
373C22⋊Q8—22no—
374C22⋊Q8—22no—
375C22⋊Q8—24no—
376C22⋊Q8—24no—
377C22⋊Q8—22no—
378C22⋊Q8—22no—
379C22⋊Q8—22no—
380C22⋊Q8—22no—
381C22⋊Q8C7×D412yes—
382C22⋊Q8C7×D412yes—
383#121—22no—
384#121—22no—
385#121C7×D412yes—
386#121C7×D412yes—
387#120—22no—
388C4×Q8—22no—
389C4×Q8—22no—
390C4×D4—24no—
391C4×D4—24no—
392C4×D4—22no—
393C4×D4—22no—
394#117—22no—
395#117C23×C712yes—
396C2×C4⋊C4—21no—
397C2×C4⋊C4—22no—
398C2×C4⋊C4—22no—
399C2×C4⋊C4C23×C711yescharacteristic
400C2×C22⋊C4—21no—
401C2×C22⋊C4—21no—
402C2×C42C23×C711yescharacteristic
403C2×C42C23×C711yescharacteristic
404#107—22no—
405#106—22no—
406#104—24no—
407#104—42no—
408#103—22no—
409#103—22no—
410#103—22no—
411#102—22no—
412#101—22no—
413#101—24no—
414#101C7×D414yescomplemented
415#95—21no—
416#95—22no—
417#95—22no—
418C22×C7—21no—
419C22×C7—21no—
420C22×C7—21no—
421C22×C7—21no—
422C22×C7—21no—
423C22×C7—21no—
424C22×C7—21no—
425C22×C7—42no—
426C22×C7—42no—
427C22×C7—42no—
428C22×C7—42no—
429C22×C7—42no—
430C22×C7#45612yescomplemented
431C22×C7C2×#12011yescharacteristic
432C22×C7C2×#12111yescharacteristic
433C22×C7#53411yescharacteristic
434C4×C7—22no—
435C4×C7—24no—
436C4×C7—42no—
437C4×C7—42no—
438C4×C7—42no—
439C4×C7—42no—
440C4×C7—42no—
441C4×C7—42no—
442C4×C7—81no—
443C4×C7—81no—
444C4×C7D4212yes—
445C4×C7#54612yes—
446C24C2×C7×D411yescharacteristic
447#41—22no—
448#41—22no—
449#41—22no—
450#41—22no—
451#41—44no—
452#41—44no—
453#41—44no—
454#41—44no—
455C2×Q8—22no—
456C2×Q8—24no—
457C2×Q8—24no—
458C2×Q8—22no—
459C2×Q8—21no—
460C2×Q8—22no—
461C2×Q8—21no—
462C2×Q8—22no—
463C2×Q8—22no—
464C2×Q8—22no—
465C2×Q8—21no—
466C2×Q8—21no—
467C2×Q8—22no—
468C2×Q8—42no—
469C2×Q8—44no—
470C2×Q8—44no—
471C2×Q8—44no—
472C2×Q8—44no—
473C2×Q8—44no—
474C2×Q8—44no—
475C2×Q8C2×C7×D412yes—
476C2×Q8C2×C7×D412yes—
477C2×Q8C7×#4112yes—
478C2×D4—22no—
479C2×D4—22no—
480C2×D4—22no—
481C2×D4—22no—
482C2×D4C2×C7×D412yes—
483C22×C4—21no—
484C22×C4—21no—
485C22×C4—22no—
486C22×C4—24no—
487C22×C4—22no—
488C22×C4—22no—
489C22×C4—21no—
490C22×C4—21no—
491C22×C4—24no—
492C22×C4—22no—
493C22×C4—21no—
494C22×C4—21no—
495C22×C4—42no—
496C22×C4C2×C7×D412yes—
497C22×C4C2×C7×D411yescharacteristic
498C22×C4C2×C7×D412yes—
499C22×C4C2×C7×D412yes—
500C22×C4C2×C7×D412yes—
501C22×C4C24×C711yescharacteristic
502Q16—44no—
503SD16—44no—
504M16—22no—
505M16—22no—
506M16—22no—
507M16—44no—
508M16C2×C7×D414yes—
509C2×C8—22no—
510C2×C8—22no—
511C2×C8—44no—
512C4⋊C4—22no—
513C4⋊C4—22no—
514C4⋊C4—24no—
515C4⋊C4—24no—
516C4⋊C4—22no—
517C4⋊C4—22no—
518C4⋊C4—22no—
519C4⋊C4—44no—
520C4⋊C4—42no—
521C4⋊C4—44no—
522C4⋊C4C2×C7×D412yes—
523C4⋊C4C2×C7×D412yes—
524C22⋊C4—24no—
525C22⋊C4—24no—
526C22⋊C4—24no—
527C22⋊C4—22no—
528C22⋊C4—22no—
529C22⋊C4—22no—
530C22⋊C4—22no—
531C22⋊C4—22no—
532C22⋊C4—41no—
533C22⋊C4—41no—
534C42—24no—
535C42—22no—
536C42—21no—
537C42—21no—
538C42—22no—
539C2×C7—21no—
540C2×C7—21no—
541C2×C7—22no—
542C2×C7—41no—
543C2×C7—82no—
544C2×C7#241011yescharacteristic
545C2×C7C2×#45611yescharacteristic
546C2×C7#302011yescharacteristic
547C23—21no—
548C23—21no—
549C23—22no—
550C23—22no—
551C23—41no—
552C23C7×#12011yescharacteristic
553C23C7×#12011yescharacteristic
554C23C22×C7×D411yescharacteristic
555Q8—24no—
556Q8—24no—
557Q8—22no—
558Q8—22no—
559Q8—48no—
560Q8—44no—
561Q8—42no—
562Q8—42no—
563Q8—42no—
564Q8—42no—
565D4—22no—
566D4—22no—
567D4—44no—
568D4—44no—
569D4—42no—
570D4—42no—
571D4—44no—
572C2×C4—22no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—48no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4C7×#12012yes—
603C2×C4C7×#12114yes—
604C2×C4C22×C7×D411yescharacteristic
605C2×C4C22×C7×D412yes—
606C2×C4C2×C7×#4111yescharacteristic
607C2×C4C7×H(5,𝔽2)12yes—
608C8—24no—
609C8—42no—
610C7#2441911yescharacteristic, direct product factor, Sylow
611C22—21no—
612C22—21no—
613C22—21no—
614C22—21no—
615C22—21no—
616C22—21no—
617C22—21no—
618C22—42no—
619C22—42no—
620C22—42no—
621C22—42no—
622C22—42no—
623C22C7×#45612yescomplemented
624C22C2×C7×#12011yescharacteristic
625C22C2×C7×#12111yescharacteristic
626C22C7×#53411yescharacteristic
627C4—22no—
628C4—24no—
629C4—42no—
630C4—42no—
631C4—42no—
632C4—42no—
633C4—42no—
634C4—42no—
635C4—81no—
636C4—81no—
637C4C7×D4212yes—
638C4C7×#54612yes—
639C2—21no—
640C2—21no—
641C2—22no—
642C2—41no—
643C2—82no—
644C2C7×#241011yescharacteristic
645C2C2×C7×#45611yescharacteristic
646C2C7×#302011yescharacteristic
subgroupG/H
center, socleC22×C7C2×#120
commutator, Frattini, cosocleC22×C4C24×C7
prime radicalC7×#3486C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C7 ◁ C23×C4×C7 ◁ G

Complements

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

NH ≅ G/Ntype
2. C7×#3056C2semidirect
3. C2×C7×#456C2semidirect
4. C22×C7×#101C2semidirect
5. C7×#2434C2semidirect
6. C7×#2038C2semidirect
7. C7×#2005C2semidirect
8. C7×#1886C2semidirect
9. C7×#1871C2semidirect
10. C7×#1857C2semidirect
11. C7×#1793C2semidirect
49. #24419C7direct (N × H)
128. C7×#101D4semidirect
133. #3056C2×C7semidirect
134. C2×#456C2×C7semidirect
135. C22×#101C2×C7semidirect
136. #2434C2×C7semidirect
137. #2038C2×C7semidirect
138. #2005C2×C7semidirect
139. #1886C2×C7semidirect
140. #1871C2×C7semidirect
141. #1857C2×C7semidirect
142. #1793C2×C7semidirect
414. #101C7×D4semidirect
430. C22×C7#456semidirect
610. C7#24419direct (N × H)
623. C22C7×#456semidirect

Maximal subgroups

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

subgroupindex
C7×#34862
C7×#30562
C2×C7×#4562
C22×C7×#1012
C7×#24342
C7×#20382
C7×#20052
C7×#18862
C7×#18712
C7×#18572
C7×#17932
#244197