Finite Group Databasebeta

← C7×#61887 #13239103

Subgroups — C7×#61887 #13239103

Order 1792.

1740 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3523C211yesmaximal, characteristic, complemented
2C7×#3480C211yesmaximal, characteristic, complemented
3C7×#3395C211yesmaximal, characteristic, complemented
4C7×#3394C213yesmaximal, complemented
5C7×#3393C213yesmaximal, complemented
6C7×#3392C211yesmaximal, characteristic, complemented
7C7×#3365C213yesmaximal, complemented
8C7×#3364C213yesmaximal, complemented
9C7×#3359C211yesmaximal, characteristic, complemented
10C7×#3358C211yesmaximal, characteristic, complemented
11C7×#3212C213yesmaximal, complemented
12C7×#3210C213yesmaximal, complemented
13C7×#3107C213yesmaximal, complemented
14C7×#3102C213yesmaximal, complemented
15C7×#2982C211yesmaximal, characteristic, complemented
16C7×#559C2213yes—
17C7×#557C2211yescharacteristic
18C7×#557C2211yescharacteristic
19C7×#557C2213yes—
20C7×#556C2213yes—
21C7×#556C2213yes—
22C7×#553—23no—
23C7×#552—23no—
24C7×#550C2211yescharacteristic
25C7×D4×Q8—21no—
26C7×D4×Q8C2211yescharacteristic
27C7×#548—21no—
28C7×#541—23no—
29C7×#533C2213yes—
30C7×#532C2213yes—
31C7×#529—23no—
32C7×#529C2213yes—
33C7×#519—21no—
34C4×C7×#41—21no—
35C4×C7×#41C2213yes—
36C2×C4×C7×Q8C2211yescharacteristic
37C7×#495C2216yes—
38C7×#494C2213yes—
39C7×#492C2213yes—
40C7×#488C2216yes—
41C7×#487C2213yes—
42C7×#486C2213yes—
43C7×#485C2213yes—
44C7×#482C2211yescharacteristic
45C7×#479C2213yes—
46C7×#478C2213yes—
47C7×#475C2213yes—
48C7×#474C2213yes—
49C7×#464C2216yes—
50C7×#464C2216yes—
51C7×#464C2213yes—
52C7×#463C2213yes—
53C7×#462C2213yes—
54C7×#462C2213yes—
55C7×#461C2213yes—
56C7×#461C2213yes—
57C7×#460C2213yes—
58C7×#459C2213yes—
59C7×#452C2213yes—
60C7×#452C2216yes—
61C7×#449C2213yes—
62C7×C8×Q8C2211yescharacteristic
63C7×C8×Q8C2211yescharacteristic
64C7×C8×Q8C2213yes—
65C7×C8×Q8C2213yes—
66C4×C7×Q16C2213yes—
67C4×C7×Q16C2216yes—
68C4×C7×SD16C2213yes—
69C4×C7×SD16C2213yes—
70C4×C7×SD16C2216yes—
71C4×C7×D8C2213yes—
72C7×#431C2213yes—
73C7×#431C2213yes—
74#61887C711yesmaximal, characteristic, direct product factor, Sylow
75C2×C7×#41—21no—
76C2×C7×#41C2313yes—
77C22×C7×Q8—21no—
78C22×C7×Q8D411yescharacteristic
79C2×C7×Q16C2313yes—
80C2×C7×Q16C2316yes—
81C2×C7×SD16C2313yes—
82C2×C7×SD16C2316yes—
83C2×C7×SD16C2313yes—
84C2×C7×D8C2313yes—
85C7×C4⋊Q8—23no—
86C7×C4⋊Q8C2313yes—
87C7×C4⋊Q8C2313yes—
88C7×C4⋊Q8C2313yes—
89C7×#127C2313yes—
90C7×#126—26no—
91C7×#126—26no—
92C7×#126—23no—
93C7×#125—23no—
94C7×#125—26no—
95C7×#125—23no—
96C7×#125—23no—
97C7×#125C2313yes—
98C7×#124—23no—
99C7×#124C2316yes—
100C7×#124C2313yes—
101C7×#123—23no—
102C7×#123—26no—
103C7×C22⋊Q8—23no—
104C7×C22⋊Q8—23no—
105C7×C22⋊Q8—23no—
106C7×C22⋊Q8—23no—
107C7×C22⋊Q8C2313yes—
108C7×#121—23no—
109C7×#121—23no—
110C7×#121—23no—
111C7×#121C2311yescharacteristic
112C4×C7×Q8—21no—
113C4×C7×Q8—21no—
114C4×C7×Q8—23no—
115C4×C7×Q8—23no—
116C4×C7×Q8D411yescharacteristic
117C4×C7×Q8D413yes—
118C4×C7×Q8C2313yes—
119C4×C7×Q8C2313yes—
120C4×C7×Q8C2313yes—
121C4×C7×Q8C2316yes—
122C4×C7×Q8C2311yescharacteristic
123C4×C7×Q8C2313yes—
124C4×C7×D4—23no—
125C4×C7×D4—23no—
126C4×C7×D4—23no—
127C4×C7×D4—23no—
128C4×C7×D4—23no—
129C4×C7×D4—23no—
130C4×C7×D4C2313yes—
131C4×C7×D4C2313yes—
132C4×C7×D4C2313yes—
133C7×#117—23no—
134C7×#117—23no—
135C7×#117—23no—
136C7×#117—23no—
137C7×#117—23no—
138C7×#117C2311yescharacteristic
139C2×C7×C4⋊C4D413yes—
140C2×C42×C7—23no—
141C2×C42×C7C2313yes—
142C7×#107C2313yes—
143C7×#107C2311yescharacteristic
144C7×#106C2313yes—
145C7×#106C2311yescharacteristic
146C7×C4⋊C8C2313yes—
147C7×C4⋊C8C2313yes—
148C7×C4⋊C8C2313yes—
149C7×C4⋊C8C2313yes—
150C7×#103C2316yes—
151C7×#103C2313yes—
152C7×#103C2313yes—
153C7×#103C2313yes—
154C7×#103C2313yes—
155C7×#102C2316yes—
156C7×#102C2313yes—
157C7×#102C2313yes—
158C7×#102C2311yescharacteristic
159C7×#102C2311yescharacteristic
160C7×C22⋊C8C2311yescharacteristic
161C7×C22⋊C8C2313yes—
162C4×C7×C8C2313yes—
163C4×C7×C8C2313yes—
164C4×C7×C8C2316yes—
165#3523C2×C711yescharacteristic, complemented
166#3480C2×C711yescharacteristic, complemented
167#3395C2×C711yescharacteristic, complemented
168#3394C2×C713yescomplemented
169#3393C2×C713yescomplemented
170#3392C2×C711yescharacteristic, complemented
171#3365C2×C713yescomplemented
172#3364C2×C713yescomplemented
173#3359C2×C711yescharacteristic, complemented
174#3358C2×C711yescharacteristic, complemented
175#3212C2×C713yescomplemented
176#3210C2×C713yescomplemented
177#3107C2×C713yescomplemented
178#3102C2×C713yescomplemented
179#2982C2×C711yescharacteristic, complemented
180C7×#41—43no—
181C7×#41—43no—
182C7×#41—44no—
183C2×C7×Q8—24no—
184C2×C7×Q8—23no—
185C2×C7×Q8—44no—
186C2×C7×Q8C2×D413yes—
187C2×C7×Q8C2×D411yescharacteristic
188C2×C7×Q8C2413yes—
189C2×C7×Q8C2413yes—
190C2×C7×D4—21no—
191C2×C7×D4—23no—
192C2×C7×D4—23no—
193C2×C7×D4—23no—
194C2×C7×D4C2413yes—
195C2×C7×D4C2411yescharacteristic
196C22×C4×C7—23no—
197C22×C4×C7—23no—
198C22×C4×C7—23no—
199C22×C4×C7—23no—
200C22×C4×C7C2×D413yes—
201C22×C4×C7C2×D413yes—
202C22×C4×C7C2411yescharacteristic
203C7×Q16—46no—
204C7×Q16—43no—
205C7×SD16—43no—
206C7×SD16—43no—
207C7×SD16—46no—
208C7×D8—43no—
209C2×C7×C8C2411yescharacteristic
210C2×C7×C8C2413yes—
211C2×C7×C8C2411yescharacteristic
212C2×C7×C8C2413yes—
213C7×C4⋊C4—23no—
214C7×C4⋊C4—23no—
215C7×C4⋊C4—23no—
216C7×C4⋊C4—23no—
217C7×C4⋊C4—23no—
218C7×C4⋊C4—23no—
219C7×C4⋊C4—23no—
220C7×C4⋊C4—23no—
221C7×C4⋊C4—23no—
222C7×C4⋊C4—23no—
223C7×C4⋊C4—26no—
224C7×C4⋊C4—23no—
225C7×C4⋊C4C2×D413yes—
226C7×C4⋊C4C2×D413yes—
227C7×C4⋊C4C2×D416yes—
228C7×C4⋊C4C2413yes—
229C7×C4⋊C4C2411yescharacteristic
230C7×C4⋊C4C2411yescharacteristic
231C7×C4⋊C4C2411yescharacteristic
232C7×C22⋊C4—23no—
233C7×C22⋊C4—21no—
234C7×C22⋊C4—21no—
235C7×C22⋊C4—23no—
236C7×C22⋊C4—26no—
237C7×C22⋊C4—23no—
238C7×C22⋊C4—23no—
239C7×C22⋊C4—26no—
240C7×C22⋊C4—23no—
241C7×C22⋊C4—23no—
242C42×C7—21no—
243C42×C7—23no—
244C42×C7—23no—
245C42×C7—26no—
246C42×C7—23no—
247C42×C7C2×D413yes—
248C42×C7C2×D413yes—
249C42×C7C2413yes—
250C42×C7C2413yes—
251#559C22×C713yes—
252#557C22×C711yescharacteristic
253#557C22×C711yescharacteristic
254#557C22×C713yes—
255#556C22×C713yes—
256#556C22×C713yes—
257#553—23no—
258#552—23no—
259#550C22×C711yescharacteristic
260D4×Q8—21no—
261D4×Q8C22×C711yescharacteristic
262#548—21no—
263#541—23no—
264#533C22×C713yes—
265#532C22×C713yes—
266#529—23no—
267#529C22×C713yes—
268#519—21no—
269C4×#41—21no—
270C4×#41C22×C713yes—
271C2×C4×Q8C22×C711yescharacteristic
272#495C22×C716yes—
273#494C22×C713yes—
274#492C22×C713yes—
275#488C22×C716yes—
276#487C22×C713yes—
277#486C22×C713yes—
278#485C22×C713yes—
279#482C22×C711yescharacteristic
280#479C22×C713yes—
281#478C22×C713yes—
282#475C22×C713yes—
283#474C22×C713yes—
284#464C22×C713yes—
285#464C22×C716yes—
286#464C22×C716yes—
287#463C22×C713yes—
288#462C22×C713yes—
289#462C22×C713yes—
290#461C22×C713yes—
291#461C22×C713yes—
292#460C22×C713yes—
293#459C22×C713yes—
294#452C22×C713yes—
295#452C22×C716yes—
296#449C22×C713yes—
297C8×Q8C22×C711yescharacteristic
298C8×Q8C22×C711yescharacteristic
299C8×Q8C22×C713yes—
300C8×Q8C22×C713yes—
301C4×Q16C22×C713yes—
302C4×Q16C22×C716yes—
303C4×SD16C22×C713yes—
304C4×SD16C22×C713yes—
305C4×SD16C22×C716yes—
306C4×D8C22×C713yes—
307#431C22×C713yes—
308#431C22×C713yes—
309C23×C7—21no—
310C23×C7—23no—
311C23×C7C22×D411yescharacteristic
312C7×Q8—26no—
313C7×Q8—23no—
314C7×Q8—23no—
315C7×Q8—23no—
316C7×Q8—23no—
317C7×Q8—46no—
318C7×Q8#13514yes—
319C7×D4—21no—
320C7×D4—23no—
321C7×D4—21no—
322C7×D4—23no—
323C7×D4—42no—
324C7×D4—46no—
325C7×D4—46no—
326C7×D4—83no—
327C2×C4×C7—26no—
328C2×C4×C7—21no—
329C2×C4×C7—21no—
330C2×C4×C7—21no—
331C2×C4×C7—21no—
332C2×C4×C7—23no—
333C2×C4×C7—21no—
334C2×C4×C7—23no—
335C2×C4×C7—23no—
336C2×C4×C7—43no—
337C2×C4×C7—46no—
338C2×C4×C7—46no—
339C2×C4×C7—46no—
340C2×C4×C7—83no—
341C2×C4×C7C22×D413yes—
342C2×C4×C7C22×D411yescharacteristic
343C2×C4×C7C22×D413yes—
344C2×C4×C7C22×D413yes—
345C2×C4×C7C22×D413yes—
346C2×C4×C7C2511yescharacteristic
347C7×C8—23no—
348C7×C8—21no—
349C7×C8—23no—
350C7×C8—21no—
351C2×#41—21no—
352C2×#41C23×C713yes—
353C22×Q8—21no—
354C22×Q8C7×D411yescharacteristic
355C2×Q16C23×C716yes—
356C2×Q16C23×C713yes—
357C2×SD16C23×C713yes—
358C2×SD16C23×C716yes—
359C2×SD16C23×C713yes—
360C2×D8C23×C713yes—
361C4⋊Q8—23no—
362C4⋊Q8C23×C713yes—
363C4⋊Q8C23×C713yes—
364C4⋊Q8C23×C713yes—
365#127C23×C713yes—
366#126—26no—
367#126—26no—
368#126—23no—
369#125—23no—
370#125—26no—
371#125—23no—
372#125—23no—
373#125C23×C713yes—
374#124—23no—
375#124C23×C713yes—
376#124C23×C716yes—
377#123—23no—
378#123—26no—
379C22⋊Q8—23no—
380C22⋊Q8—23no—
381C22⋊Q8—23no—
382C22⋊Q8—23no—
383C22⋊Q8C23×C713yes—
384#121—23no—
385#121—23no—
386#121—23no—
387#121C23×C711yescharacteristic
388C4×Q8—21no—
389C4×Q8—21no—
390C4×Q8—23no—
391C4×Q8—23no—
392C4×Q8C7×D411yescharacteristic
393C4×Q8C7×D413yes—
394C4×Q8C23×C713yes—
395C4×Q8C23×C713yes—
396C4×Q8C23×C713yes—
397C4×Q8C23×C716yes—
398C4×Q8C23×C711yescharacteristic
399C4×Q8C23×C713yes—
400C4×D4—23no—
401C4×D4—23no—
402C4×D4—23no—
403C4×D4—23no—
404C4×D4—23no—
405C4×D4—23no—
406C4×D4C23×C713yes—
407C4×D4C23×C713yes—
408C4×D4C23×C713yes—
409#117—23no—
410#117—23no—
411#117—23no—
412#117—23no—
413#117—23no—
414#117C23×C711yescharacteristic
415C2×C4⋊C4C7×D413yes—
416C2×C42—23no—
417C2×C42C23×C713yes—
418#107C23×C713yes—
419#107C23×C711yescharacteristic
420#106C23×C713yes—
421#106C23×C711yescharacteristic
422C4⋊C8C23×C713yes—
423C4⋊C8C23×C713yes—
424C4⋊C8C23×C713yes—
425C4⋊C8C23×C713yes—
426#103C23×C713yes—
427#103C23×C713yes—
428#103C23×C713yes—
429#103C23×C713yes—
430#103C23×C716yes—
431#102C23×C711yescharacteristic
432#102C23×C713yes—
433#102C23×C711yescharacteristic
434#102C23×C713yes—
435#102C23×C716yes—
436C22⋊C8C23×C713yes—
437C22⋊C8C23×C711yescharacteristic
438C4×C8C23×C713yes—
439C4×C8C23×C713yes—
440C4×C8C23×C716yes—
441C22×C7—21no—
442C22×C7—21no—
443C22×C7—21no—
444C22×C7—43no—
445C22×C7—41no—
446C22×C7—43no—
447C22×C7—41no—
448C22×C7—41no—
449C22×C7—43no—
450C22×C7C23×D411yescharacteristic
451C4×C7—23no—
452C4×C7—23no—
453C4×C7—21no—
454C4×C7—23no—
455C4×C7—41no—
456C4×C7—41no—
457C4×C7—43no—
458C4×C7—43no—
459C4×C7—41no—
460C4×C7—43no—
461C4×C7C2×#13516yes—
462C4×C7#58511yescharacteristic
463C4×C7#58511yescharacteristic
464#41—43no—
465#41—43no—
466#41—44no—
467C2×Q8—24no—
468C2×Q8—23no—
469C2×Q8—44no—
470C2×Q8C2×C7×D413yes—
471C2×Q8C2×C7×D411yescharacteristic
472C2×Q8C24×C713yes—
473C2×Q8C24×C713yes—
474C2×D4—21no—
475C2×D4—23no—
476C2×D4—23no—
477C2×D4—23no—
478C2×D4C24×C713yes—
479C2×D4C24×C711yescharacteristic
480C22×C4—23no—
481C22×C4—23no—
482C22×C4—23no—
483C22×C4—23no—
484C22×C4C2×C7×D413yes—
485C22×C4C2×C7×D413yes—
486C22×C4C24×C711yescharacteristic
487Q16—46no—
488Q16—43no—
489SD16—43no—
490SD16—43no—
491SD16—46no—
492D8—43no—
493C2×C8C24×C711yescharacteristic
494C2×C8C24×C713yes—
495C2×C8C24×C711yescharacteristic
496C2×C8C24×C713yes—
497C4⋊C4—23no—
498C4⋊C4—23no—
499C4⋊C4—23no—
500C4⋊C4—23no—
501C4⋊C4—23no—
502C4⋊C4—23no—
503C4⋊C4—23no—
504C4⋊C4—23no—
505C4⋊C4—23no—
506C4⋊C4—23no—
507C4⋊C4—26no—
508C4⋊C4—23no—
509C4⋊C4C2×C7×D413yes—
510C4⋊C4C2×C7×D413yes—
511C4⋊C4C2×C7×D416yes—
512C4⋊C4C24×C713yes—
513C4⋊C4C24×C711yescharacteristic
514C4⋊C4C24×C711yescharacteristic
515C4⋊C4C24×C711yescharacteristic
516C22⋊C4—23no—
517C22⋊C4—21no—
518C22⋊C4—21no—
519C22⋊C4—23no—
520C22⋊C4—26no—
521C22⋊C4—23no—
522C22⋊C4—23no—
523C22⋊C4—26no—
524C22⋊C4—23no—
525C22⋊C4—23no—
526C42—21no—
527C42—23no—
528C42—23no—
529C42—26no—
530C42—23no—
531C42C2×C7×D413yes—
532C42C2×C7×D413yes—
533C42C24×C713yes—
534C42C24×C713yes—
535C2×C7—41no—
536C2×C7—83no—
537C2×C7—81no—
538C2×C7#348711yescharacteristic
539C2×C7C22×#13511yescharacteristic
540C2×C7#358611yescharacteristic
541C23—21no—
542C23—23no—
543C23C22×C7×D411yescharacteristic
544Q8—26no—
545Q8—23no—
546Q8—23no—
547Q8—23no—
548Q8—23no—
549Q8—46no—
550Q8C7×#13514yes—
551D4—21no—
552D4—23no—
553D4—21no—
554D4—23no—
555D4—42no—
556D4—46no—
557D4—46no—
558D4—83no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—26no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—23no—
565C2×C4—21no—
566C2×C4—23no—
567C2×C4—23no—
568C2×C4—43no—
569C2×C4—46no—
570C2×C4—46no—
571C2×C4—46no—
572C2×C4—83no—
573C2×C4C22×C7×D413yes—
574C2×C4C22×C7×D411yescharacteristic
575C2×C4C22×C7×D413yes—
576C2×C4C22×C7×D413yes—
577C2×C4C22×C7×D413yes—
578C2×C4C25×C711yescharacteristic
579C8—23no—
580C8—21no—
581C8—23no—
582C8—21no—
583C7#6188711yescharacteristic, direct product factor, Sylow
584C22—21no—
585C22—21no—
586C22—21no—
587C22—41no—
588C22—43no—
589C22—43no—
590C22—41no—
591C22—43no—
592C22—41no—
593C22C23×C7×D411yescharacteristic
594C4—23no—
595C4—23no—
596C4—23no—
597C4—21no—
598C4—41no—
599C4—43no—
600C4—41no—
601C4—41no—
602C4—43no—
603C4—43no—
604C4C2×C7×#13516yes—
605C4C7×#58511yescharacteristic
606C4C7×#58511yescharacteristic
607C2—41no—
608C2—81no—
609C2—83no—
610C2C7×#348711yescharacteristic
611C2C22×C7×#13511yescharacteristic
612C2C7×#358611yescharacteristic
subgroupG/H
center, socleC22×C7C23×D4
commutator, Frattini, cosocleC2×C4C25×C7
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C7 ◁ C2×C4×C7×Q8 ◁ G

Complements

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

NH ≅ G/Ntype
1. C7×#3523C2semidirect
2. C7×#3480C2semidirect
3. C7×#3395C2semidirect
4. C7×#3394C2semidirect
5. C7×#3393C2semidirect
6. C7×#3392C2semidirect
7. C7×#3365C2semidirect
8. C7×#3364C2semidirect
9. C7×#3359C2semidirect
10. C7×#3358C2semidirect
11. C7×#3212C2semidirect
12. C7×#3210C2semidirect
13. C7×#3107C2semidirect
14. C7×#3102C2semidirect
15. C7×#2982C2semidirect
74. #61887C7direct (N × H)
165. #3523C2×C7semidirect
166. #3480C2×C7semidirect
167. #3395C2×C7semidirect
168. #3394C2×C7semidirect
169. #3393C2×C7semidirect
170. #3392C2×C7semidirect
171. #3365C2×C7semidirect
172. #3364C2×C7semidirect
173. #3359C2×C7semidirect
174. #3358C2×C7semidirect
175. #3212C2×C7semidirect
176. #3210C2×C7semidirect
177. #3107C2×C7semidirect
178. #3102C2×C7semidirect
179. #2982C2×C7semidirect
583. C7#61887direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#35232
C7×#34802
C7×#33952
C7×#33942
C7×#33932
C7×#33922
C7×#33652
C7×#33642
C7×#33592
C7×#33582
C7×#32122
C7×#32102
C7×#31072
C7×#31022
C7×#29822
#618877