Finite Group Databasebeta

← C7×#61470 #13238686

Subgroups — C7×#61470 #13238686

Order 1792.

1812 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×#3482C211yesmaximal, characteristic
2C7×Q8×#41C211yesmaximal, characteristic, complemented
3C7×#3395C211yesmaximal, characteristic, complemented
4C7×Q8×SD16C211yesmaximal, characteristic, complemented
5C7×#3363C211yesmaximal, characteristic, complemented
6C7×#3359C211yesmaximal, characteristic, complemented
7C7×#3356C213yesmaximal, complemented
8C7×#3331C213yesmaximal, complemented
9C7×#3328C213yesmaximal, complemented
10C7×#3276C213yesmaximal, complemented
11C7×#3185C213yesmaximal, complemented
12C7×#3116C213yesmaximal, complemented
13C7×#3113C213yesmaximal, complemented
14C7×#2971C211yesmaximal, characteristic, complemented
15C4×C7×#137C213yesmaximal, complemented
16C2×C7×#137C2213yes—
17C7×Q82C2211yescharacteristic
18C7×#557—21no—
19C7×#557C2211yescharacteristic
20C7×#557C2211yescharacteristic
21C7×#556C2213yes—
22C7×#554—23no—
23C7×#554C2213yes—
24C7×#552—23no—
25C7×D4×Q8C2213yes—
26C7×D4×Q8C2211yescharacteristic
27C7×#548—21no—
28C7×#547C2213yes—
29C7×#533—23no—
30C7×#533C2213yes—
31C7×#533C2213yes—
32C7×#529—23no—
33C4×C7×#41—21no—
34C4×C7×#41C2213yes—
35C4×C7×#41C2211yescharacteristic
36C2×C4×C7×Q8—21no—
37C2×C4×C7×Q8C2213yes—
38C7×#498C2213yes—
39C7×#494C2213yes—
40C7×#487C2213yes—
41C7×#487C2213yes—
42C7×#485C2216yes—
43C7×#483C2216yes—
44C7×#477C2213yes—
45C7×#476C2213yes—
46C7×#475C2213yes—
47C7×#475C2213yes—
48C7×#470C2216yes—
49C7×#470C2216yes—
50C7×#462C2213yes—
51C7×#462C2213yes—
52C7×#462C2213yes—
53C7×#461C2213yes—
54C7×#452C2216yes—
55C7×#451C2216yes—
56C7×C8×Q8C2211yescharacteristic
57C7×C8×Q8C2211yescharacteristic
58C7×#441C2213yes—
59C7×#441C2216yes—
60C7×#441C2213yes—
61C7×#440C2211yescharacteristic
62C7×#440C2213yes—
63C4×C7×Q16C2213yes—
64C4×C7×Q16C2213yes—
65C4×C7×SD16C2213yes—
66C4×C7×SD16C2213yes—
67C7×#436C2213yes—
68C7×#436C2213yes—
69C7×#432C2213yes—
70C7×#428C2213yes—
71C7×#419C2213yes—
72C7×#417C2213yes—
73C4×C7×M16C2213yes—
74#61470C711yesmaximal, characteristic, direct product factor, Sylow
75C2×C7×#41D411yescharacteristic
76C2×C7×#41C2313yes—
77C22×C7×Q8—21no—
78C22×C7×Q8C2313yes—
79C7×#137—212no—
80C2×C7×Q16C2313yes—
81C2×C7×Q16C2313yes—
82C2×C7×Q16C2316yes—
83C2×C7×SD16C2313yes—
84C2×C7×SD16C2311yescharacteristic
85C2×C7×M16C2313yes—
86C2×C7×M16C2313yes—
87C7×C4⋊Q8—23no—
88C7×C4⋊Q8—23no—
89C7×C4⋊Q8C2313yes—
90C7×C4⋊Q8C2313yes—
91C7×C4⋊Q8C2313yes—
92C7×C4⋊Q8C2313yes—
93C7×#126—26no—
94C7×#125—23no—
95C7×#125—23no—
96C7×#125—23no—
97C7×#125—23no—
98C7×#125—23no—
99C7×#124—23no—
100C7×#123—26no—
101C7×C22⋊Q8—23no—
102C7×C22⋊Q8—23no—
103C7×C22⋊Q8—23no—
104C7×C22⋊Q8—26no—
105C7×C22⋊Q8—26no—
106C7×C22⋊Q8C2316yes—
107C7×C22⋊Q8C2316yes—
108C7×#121—23no—
109C4×C7×Q8—21no—
110C4×C7×Q8—21no—
111C4×C7×Q8—21no—
112C4×C7×Q8—23no—
113C4×C7×Q8—23no—
114C4×C7×Q8—23no—
115C4×C7×Q8—23no—
116C4×C7×Q8—23no—
117C4×C7×Q8D411yescharacteristic
118C4×C7×Q8C2313yes—
119C4×C7×Q8C2313yes—
120C4×C7×Q8C2313yes—
121C4×C7×Q8C2311yescharacteristic
122C4×C7×Q8C2313yes—
123C4×C7×Q8C2313yes—
124C4×C7×Q8C2311yescharacteristic
125C4×C7×D4—23no—
126C4×C7×D4—23no—
127C4×C7×D4—23no—
128C4×C7×D4—23no—
129C4×C7×D4D413yes—
130C4×C7×D4C2313yes—
131C4×C7×D4C2311yescharacteristic
132C4×C7×D4C2313yes—
133C7×#117—23no—
134C7×#117—23no—
135C7×#117—23no—
136C7×#117D413yes—
137C7×#117C2313yes—
138C2×C7×C4⋊C4—23no—
139C2×C7×C4⋊C4—23no—
140C2×C7×C4⋊C4C2313yes—
141C2×C42×C7—23no—
142C2×C42×C7—23no—
143C2×C42×C7C2313yes—
144C7×#107C2313yes—
145C7×#107C2311yescharacteristic
146C7×#106C2313yes—
147C7×#106C2316yes—
148C7×#106C2313yes—
149C7×C4⋊C8C2313yes—
150C7×C4⋊C8C2313yes—
151C7×#103C2313yes—
152C7×#103C2313yes—
153C7×#103C2316yes—
154C7×#103C2316yes—
155C7×#103C2313yes—
156C7×#103C2313yes—
157C7×#103C2313yes—
158C7×#103C2311yescharacteristic
159C7×#102C2313yes—
160C7×#102C2311yescharacteristic
161C7×C22⋊C8C2316yes—
162C7×#97C2311yescharacteristic
163C7×#97C2313yes—
164C4×C7×C8C2313yes—
165C4×C7×C8C2313yes—
166#3482C2×C711yescharacteristic
167Q8×#41C2×C711yescharacteristic, complemented
168#3395C2×C711yescharacteristic, complemented
169Q8×SD16C2×C711yescharacteristic, complemented
170#3363C2×C711yescharacteristic, complemented
171#3359C2×C711yescharacteristic, complemented
172#3356C2×C713yescomplemented
173#3331C2×C713yescomplemented
174#3328C2×C713yescomplemented
175#3276C2×C713yescomplemented
176#3185C2×C713yescomplemented
177#3116C2×C713yescomplemented
178#3113C2×C713yescomplemented
179#2971C2×C711yescharacteristic, complemented
180C4×#137C2×C713yescomplemented
181C7×#41—24no—
182C7×#41—26no—
183C7×#41—43no—
184C2×C7×Q8—23no—
185C2×C7×Q8—23no—
186C2×C7×Q8—23no—
187C2×C7×Q8—26no—
188C2×C7×Q8—43no—
189C2×C7×Q8—44no—
190C2×C7×Q8C2×D411yescharacteristic
191C2×C7×Q8C2413yes—
192C2×C7×Q8C2413yes—
193C2×C7×Q8C2411yescharacteristic
194C2×C7×D4—23no—
195C2×C7×D4C2×D413yes—
196C2×C7×D4C2411yescharacteristic
197C22×C4×C7—23no—
198C22×C4×C7—23no—
199C22×C4×C7—21no—
200C22×C4×C7—23no—
201C22×C4×C7—23no—
202C22×C4×C7C2×D413yes—
203C22×C4×C7C2×D413yes—
204C22×C4×C7C2413yes—
205C7×Q16—26no—
206C7×Q16—26no—
207C7×Q16—46no—
208C7×SD16—26no—
209C7×SD16#4114yescomplemented
210C7×M16—26no—
211C7×M16—43no—
212C2×C7×C8C2411yescharacteristic
213C2×C7×C8C2413yes—
214C2×C7×C8C2413yes—
215C2×C7×C8C2411yescharacteristic
216C7×C4⋊C4—21no—
217C7×C4⋊C4—23no—
218C7×C4⋊C4—26no—
219C7×C4⋊C4—23no—
220C7×C4⋊C4—23no—
221C7×C4⋊C4—23no—
222C7×C4⋊C4—23no—
223C7×C4⋊C4—23no—
224C7×C4⋊C4—23no—
225C7×C4⋊C4—23no—
226C7×C4⋊C4—23no—
227C7×C4⋊C4—23no—
228C7×C4⋊C4—23no—
229C7×C4⋊C4—23no—
230C7×C4⋊C4—26no—
231C7×C4⋊C4C2×D413yes—
232C7×C4⋊C4C2×D413yes—
233C7×C4⋊C4C2411yescharacteristic
234C7×C4⋊C4C2413yes—
235C7×C4⋊C4C2413yes—
236C7×C4⋊C4C2411yescharacteristic
237C7×C22⋊C4—26no—
238C7×C22⋊C4—26no—
239C7×C22⋊C4—23no—
240C7×C22⋊C4—21no—
241C7×C22⋊C4—23no—
242C7×C22⋊C4C2×D416yes—
243C42×C7—21no—
244C42×C7—23no—
245C42×C7—23no—
246C42×C7—23no—
247C42×C7—23no—
248C42×C7—23no—
249C42×C7—23no—
250C42×C7C2×D413yes—
251C42×C7C2×D413yes—
252C42×C7C2413yes—
253C42×C7C2411yescharacteristic
254C2×#137C22×C713yes—
255Q82C22×C711yescharacteristic
256#557—21no—
257#557C22×C711yescharacteristic
258#557C22×C711yescharacteristic
259#556C22×C713yes—
260#554—23no—
261#554C22×C713yes—
262#552—23no—
263D4×Q8C22×C713yes—
264D4×Q8C22×C711yescharacteristic
265#548—21no—
266#547C22×C713yes—
267#533—23no—
268#533C22×C713yes—
269#533C22×C713yes—
270#529—23no—
271C4×#41—21no—
272C4×#41C22×C713yes—
273C4×#41C22×C711yescharacteristic
274C2×C4×Q8—21no—
275C2×C4×Q8C22×C713yes—
276#498C22×C713yes—
277#494C22×C713yes—
278#487C22×C713yes—
279#487C22×C713yes—
280#485C22×C716yes—
281#483C22×C716yes—
282#477C22×C713yes—
283#476C22×C713yes—
284#475C22×C713yes—
285#475C22×C713yes—
286#470C22×C716yes—
287#470C22×C716yes—
288#462C22×C713yes—
289#462C22×C713yes—
290#462C22×C713yes—
291#461C22×C713yes—
292#452C22×C716yes—
293#451C22×C716yes—
294C8×Q8C22×C711yescharacteristic
295C8×Q8C22×C711yescharacteristic
296#441C22×C713yes—
297#441C22×C716yes—
298#441C22×C713yes—
299#440C22×C713yes—
300#440C22×C711yescharacteristic
301C4×Q16C22×C713yes—
302C4×Q16C22×C713yes—
303C4×SD16C22×C713yes—
304C4×SD16C22×C713yes—
305#436C22×C713yes—
306#436C22×C713yes—
307#432C22×C713yes—
308#428C22×C713yes—
309#419C22×C713yes—
310#417C22×C713yes—
311C4×M16C22×C713yes—
312C23×C7—21no—
313C23×C7C22×D413yes—
314C7×Q8—21no—
315C7×Q8—23no—
316C7×Q8—23no—
317C7×Q8—23no—
318C7×Q8—23no—
319C7×Q8—46no—
320C7×Q8—46no—
321C7×Q8—83no—
322C7×Q8#13714yescomplemented
323C7×Q8C2×#4112yes—
324C7×D4—26no—
325C7×D4—21no—
326C7×D4—46no—
327C7×D4C2×#4112yes—
328C2×C4×C7—21no—
329C2×C4×C7—26no—
330C2×C4×C7—23no—
331C2×C4×C7—23no—
332C2×C4×C7—21no—
333C2×C4×C7—21no—
334C2×C4×C7—23no—
335C2×C4×C7—23no—
336C2×C4×C7—23no—
337C2×C4×C7—23no—
338C2×C4×C7—43no—
339C2×C4×C7—46no—
340C2×C4×C7—46no—
341C2×C4×C7—42no—
342C2×C4×C7—46no—
343C2×C4×C7—83no—
344C2×C4×C7C22×D411yescharacteristic
345C2×C4×C7C22×D411yescharacteristic
346C2×C4×C7C22×D413yes—
347C2×C4×C7C22×D413yes—
348C2×C4×C7C22×D413yes—
349C2×C4×C7C2511yescharacteristic
350C7×C8—23no—
351C7×C8—21no—
352C7×C8—23no—
353C7×C8C2×#4112yes—
354C2×#41C7×D411yescharacteristic
355C2×#41C23×C713yes—
356C22×Q8—21no—
357C22×Q8C23×C713yes—
358#137—212no—
359C2×Q16C23×C713yes—
360C2×Q16C23×C713yes—
361C2×Q16C23×C716yes—
362C2×SD16C23×C713yes—
363C2×SD16C23×C711yescharacteristic
364C2×M16C23×C713yes—
365C2×M16C23×C713yes—
366C4⋊Q8—23no—
367C4⋊Q8—23no—
368C4⋊Q8C23×C713yes—
369C4⋊Q8C23×C713yes—
370C4⋊Q8C23×C713yes—
371C4⋊Q8C23×C713yes—
372#126—26no—
373#125—23no—
374#125—23no—
375#125—23no—
376#125—23no—
377#125—23no—
378#124—23no—
379#123—26no—
380C22⋊Q8—23no—
381C22⋊Q8—23no—
382C22⋊Q8—23no—
383C22⋊Q8—26no—
384C22⋊Q8—26no—
385C22⋊Q8C23×C716yes—
386C22⋊Q8C23×C716yes—
387#121—23no—
388C4×Q8—21no—
389C4×Q8—21no—
390C4×Q8—21no—
391C4×Q8—23no—
392C4×Q8—23no—
393C4×Q8—23no—
394C4×Q8—23no—
395C4×Q8—23no—
396C4×Q8C7×D411yescharacteristic
397C4×Q8C23×C711yescharacteristic
398C4×Q8C23×C713yes—
399C4×Q8C23×C713yes—
400C4×Q8C23×C713yes—
401C4×Q8C23×C713yes—
402C4×Q8C23×C713yes—
403C4×Q8C23×C711yescharacteristic
404C4×D4—23no—
405C4×D4—23no—
406C4×D4—23no—
407C4×D4—23no—
408C4×D4C7×D413yes—
409C4×D4C23×C711yescharacteristic
410C4×D4C23×C713yes—
411C4×D4C23×C713yes—
412#117—23no—
413#117—23no—
414#117—23no—
415#117C7×D413yes—
416#117C23×C713yes—
417C2×C4⋊C4—23no—
418C2×C4⋊C4—23no—
419C2×C4⋊C4C23×C713yes—
420C2×C42—23no—
421C2×C42—23no—
422C2×C42C23×C713yes—
423#107C23×C711yescharacteristic
424#107C23×C713yes—
425#106C23×C713yes—
426#106C23×C713yes—
427#106C23×C716yes—
428C4⋊C8C23×C713yes—
429C4⋊C8C23×C713yes—
430#103C23×C713yes—
431#103C23×C713yes—
432#103C23×C711yescharacteristic
433#103C23×C713yes—
434#103C23×C713yes—
435#103C23×C713yes—
436#103C23×C716yes—
437#103C23×C716yes—
438#102C23×C711yescharacteristic
439#102C23×C713yes—
440C22⋊C8C23×C716yes—
441#97C23×C711yescharacteristic
442#97C23×C713yes—
443C4×C8C23×C713yes—
444C4×C8C23×C713yes—
445C22×C7—23no—
446C22×C7—23no—
447C22×C7—22no—
448C22×C7—23no—
449C22×C7—41no—
450C22×C7—41no—
451C22×C7C23×D411yescharacteristic
452C4×C7—21no—
453C4×C7—23no—
454C4×C7—23no—
455C4×C7—22no—
456C4×C7—21no—
457C4×C7—41no—
458C4×C7—41no—
459C4×C7—43no—
460C4×C7—43no—
461C4×C7—43no—
462C4×C7—43no—
463C4×C7C2×#13716yes—
464C4×C7C22×#4111yescharacteristic
465C4×C7#58511yescharacteristic
466#41—24no—
467#41—26no—
468#41—43no—
469C2×Q8—23no—
470C2×Q8—23no—
471C2×Q8—23no—
472C2×Q8—26no—
473C2×Q8—43no—
474C2×Q8—44no—
475C2×Q8C2×C7×D411yescharacteristic
476C2×Q8C24×C713yes—
477C2×Q8C24×C713yes—
478C2×Q8C24×C711yescharacteristic
479C2×D4—23no—
480C2×D4C2×C7×D413yes—
481C2×D4C24×C711yescharacteristic
482C22×C4—23no—
483C22×C4—23no—
484C22×C4—21no—
485C22×C4—23no—
486C22×C4—23no—
487C22×C4C2×C7×D413yes—
488C22×C4C2×C7×D413yes—
489C22×C4C24×C713yes—
490Q16—26no—
491Q16—26no—
492Q16—46no—
493SD16—26no—
494SD16C7×#4114yescomplemented
495M16—26no—
496M16—43no—
497C2×C8C24×C711yescharacteristic
498C2×C8C24×C713yes—
499C2×C8C24×C713yes—
500C2×C8C24×C711yescharacteristic
501C4⋊C4—21no—
502C4⋊C4—23no—
503C4⋊C4—26no—
504C4⋊C4—23no—
505C4⋊C4—23no—
506C4⋊C4—23no—
507C4⋊C4—23no—
508C4⋊C4—23no—
509C4⋊C4—23no—
510C4⋊C4—23no—
511C4⋊C4—23no—
512C4⋊C4—23no—
513C4⋊C4—23no—
514C4⋊C4—23no—
515C4⋊C4—26no—
516C4⋊C4C2×C7×D413yes—
517C4⋊C4C2×C7×D413yes—
518C4⋊C4C24×C711yescharacteristic
519C4⋊C4C24×C713yes—
520C4⋊C4C24×C713yes—
521C4⋊C4C24×C711yescharacteristic
522C22⋊C4—26no—
523C22⋊C4—26no—
524C22⋊C4—23no—
525C22⋊C4—21no—
526C22⋊C4—23no—
527C22⋊C4C2×C7×D416yes—
528C42—21no—
529C42—23no—
530C42—23no—
531C42—23no—
532C42—23no—
533C42—23no—
534C42—23no—
535C42C2×C7×D413yes—
536C42C2×C7×D413yes—
537C42C24×C713yes—
538C42C24×C711yescharacteristic
539C2×C7—43no—
540C2×C7—42no—
541C2×C7D4×#4111yescharacteristic
542C2×C7C22×#13711yescharacteristic
543C2×C7#358611yescharacteristic
544C23—21no—
545C23C22×C7×D413yes—
546Q8—21no—
547Q8—23no—
548Q8—23no—
549Q8—23no—
550Q8—23no—
551Q8—46no—
552Q8—46no—
553Q8—83no—
554Q8C7×#13714yescomplemented
555Q8C2×C7×#4112yes—
556D4—21no—
557D4—26no—
558D4—46no—
559D4C2×C7×#4112yes—
560C2×C4—21no—
561C2×C4—23no—
562C2×C4—23no—
563C2×C4—26no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—23no—
567C2×C4—23no—
568C2×C4—23no—
569C2×C4—23no—
570C2×C4—43no—
571C2×C4—46no—
572C2×C4—46no—
573C2×C4—42no—
574C2×C4—46no—
575C2×C4—83no—
576C2×C4C22×C7×D411yescharacteristic
577C2×C4C22×C7×D411yescharacteristic
578C2×C4C22×C7×D413yes—
579C2×C4C22×C7×D413yes—
580C2×C4C22×C7×D413yes—
581C2×C4C25×C711yescharacteristic
582C8—23no—
583C8—21no—
584C8—23no—
585C8C2×C7×#4112yes—
586C7#6147011yescharacteristic, direct product factor, Sylow
587C22—23no—
588C22—23no—
589C22—23no—
590C22—22no—
591C22—41no—
592C22—41no—
593C22C23×C7×D411yescharacteristic
594C4—21no—
595C4—21no—
596C4—23no—
597C4—23no—
598C4—22no—
599C4—43no—
600C4—41no—
601C4—41no—
602C4—43no—
603C4—43no—
604C4—43no—
605C4C2×C7×#13716yes—
606C4C22×C7×#4111yescharacteristic
607C4C7×#58511yescharacteristic
608C2—43no—
609C2—42no—
610C2C7×D4×#4111yescharacteristic
611C2C22×C7×#13711yescharacteristic
612C2C7×#358611yescharacteristic
subgroupG/H
center, socleC22×C7C23×D4
commutator, Frattini, cosocleC2×C4C25×C7
prime radicalC7×#3482C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C7 ◁ C4×C7×#41 ◁ G

Complements

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

NH ≅ G/Ntype
2. C7×Q8×#41C2semidirect
3. C7×#3395C2semidirect
4. C7×Q8×SD16C2semidirect
5. C7×#3363C2semidirect
6. C7×#3359C2semidirect
7. C7×#3356C2semidirect
8. C7×#3331C2semidirect
9. C7×#3328C2semidirect
10. C7×#3276C2semidirect
11. C7×#3185C2semidirect
12. C7×#3116C2semidirect
13. C7×#3113C2semidirect
14. C7×#2971C2semidirect
15. C4×C7×#137C2semidirect
74. #61470C7direct (N × H)
167. Q8×#41C2×C7semidirect
168. #3395C2×C7semidirect
169. Q8×SD16C2×C7semidirect
170. #3363C2×C7semidirect
171. #3359C2×C7semidirect
172. #3356C2×C7semidirect
173. #3331C2×C7semidirect
174. #3328C2×C7semidirect
175. #3276C2×C7semidirect
176. #3185C2×C7semidirect
177. #3116C2×C7semidirect
178. #3113C2×C7semidirect
179. #2971C2×C7semidirect
180. C4×#137C2×C7semidirect
209. C7×SD16#41semidirect
322. C7×Q8#137semidirect
494. SD16C7×#41semidirect
554. Q8C7×#137semidirect
586. C7#61470direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#34822
C7×Q8×#412
C7×#33952
C7×Q8×SD162
C7×#33632
C7×#33592
C7×#33562
C7×#33312
C7×#33282
C7×#32762
C7×#31852
C7×#31162
C7×#31132
C7×#29712
C4×C7×#1372
#614707