Finite Group Databasebeta

← C7×#18460 #13195676

Subgroups — C7×#18460 #13195676

Order 1792.

936 conjugacy classes of subgroups.

The subgroup lattice has 670 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3559C211yesmaximal, characteristic, complemented
2C7×#3213C211yesmaximal, characteristic, complemented
3C7×#3211C211yesmaximal, characteristic, complemented
4C7×#2820C211yesmaximal, characteristic, complemented
5C7×#1604C212yesmaximal, complemented
6C7×#1603C212yesmaximal, complemented
7C7×#1598C212yesmaximal, complemented
8C7×#1597C212yesmaximal, complemented
9C7×#1517C211yesmaximal, characteristic, complemented
10C7×#1512C211yesmaximal, characteristic, complemented
11C7×#1470C211yesmaximal, characteristic, complemented
12C7×#563—21no—
13C7×#562—21no—
14C7×#554—21no—
15C7×#553—21no—
16C7×#553—21no—
17C7×#547—22no—
18C7×#547—21no—
19C7×#544—21no—
20C7×#540—21no—
21C7×#539—22no—
22C7×#539C2212yes—
23C7×#537C2212yes—
24C7×#532C2211yescharacteristic
25C7×#529C2211yescharacteristic
26C2×C7×#125C2211yescharacteristic
27C2×C7×#124C2211yescharacteristic
28C7×#496—21no—
29C7×#495—21no—
30C7×#489—21no—
31C7×#488—21no—
32C7×#464—21no—
33C7×#463—21no—
34C7×#463—22no—
35C7×#451C2211yescharacteristic
36C7×#450C2211yescharacteristic
37C7×#448C2211yescharacteristic
38C7×#448C2212yes—
39C7×#447C2211yescharacteristic
40C7×#447C2212yes—
41C7×#432C2211yescharacteristic
42C7×#431C2211yescharacteristic
43C7×#399—21no—
44C7×#397—21no—
45C7×#396—22no—
46C7×#395C2212yes—
47C7×#394C2212yes—
48C7×#382C2211yescharacteristic
49C7×#328C2212yes—
50C7×#327C2212yes—
51C7×#327C2212yes—
52C7×#327C2212yes—
53C7×#324C2212yes—
54C7×#324C2212yes—
55#18460C711yesmaximal, characteristic, direct product factor, Sylow
56C2×C7×#41—21no—
57C2×C7×#41—22no—
58C22×C7×Q8C2311yescharacteristic
59C22×C7×D4C2311yescharacteristic
60C2×C7×Q16—21no—
61C2×C7×SD16—21no—
62C2×C7×SD16—21no—
63C2×C7×SD16—22no—
64C2×C7×D8—21no—
65C2×C7×D8—22no—
66C7×C4⋊Q8—21no—
67C7×C4⋊Q8—22no—
68C7×C4⋊Q8D411yescharacteristic
69C7×#127D411yescharacteristic
70C7×#126—22no—
71C7×#126—22no—
72C7×#126—21no—
73C7×#125—21no—
74C7×#125—22no—
75C7×#125—21no—
76C7×#125—21no—
77C7×#125—21no—
78C7×#125—21no—
79C7×#125D411yescharacteristic
80C7×#124—21no—
81C7×#124—21no—
82C7×#124—21no—
83C7×#124—21no—
84C7×#124—21no—
85C7×#124D411yescharacteristic
86C7×#123—22no—
87C7×#123—22no—
88C7×#123—22no—
89C7×#123—22no—
90C7×#123—22no—
91C7×#123—21no—
92C7×#123—22no—
93C7×C22⋊Q8—22no—
94C7×C22⋊Q8—22no—
95C7×C22⋊Q8—22no—
96C7×C22⋊Q8—22no—
97C7×C22⋊Q8—22no—
98C7×C22⋊Q8—22no—
99C7×C22⋊Q8C2311yescharacteristic
100C7×#121—21no—
101C7×#121—22no—
102C7×#121—21no—
103C7×#121—22no—
104C7×#121—22no—
105C7×#121—21no—
106C7×#121—22no—
107C7×#121C2311yescharacteristic
108C7×#121C2312yes—
109C4×C7×Q8D411yescharacteristic
110C4×C7×D4—22no—
111C4×C7×D4—21no—
112C4×C7×D4—21no—
113C4×C7×D4—21no—
114C4×C7×D4—22no—
115C4×C7×D4—21no—
116C4×C7×D4D411yescharacteristic
117C4×C7×D4D412yes—
118C7×#117—21no—
119C7×#117—21no—
120C2×C7×C4⋊C4—22no—
121C2×C7×C4⋊C4—21no—
122C2×C7×C4⋊C4—21no—
123C2×C7×C4⋊C4—22no—
124C2×C7×C4⋊C4D412yes—
125C2×C7×C4⋊C4D411yescharacteristic
126C2×C7×C4⋊C4D411yescharacteristic
127C2×C7×C4⋊C4C2312yes—
128C2×C7×C22⋊C4—21no—
129C2×C7×C22⋊C4—21no—
130C2×C7×C22⋊C4—22no—
131C2×C7×C22⋊C4—22no—
132C2×C42×C7C2311yescharacteristic
133C7×C4⋊C8—21no—
134C7×C4⋊C8—21no—
135C7×#103—21no—
136C7×#103—21no—
137C7×#103—22no—
138C7×#102—21no—
139C7×#102—21no—
140C7×#102—22no—
141C7×C22⋊C8C2311yescharacteristic
142C7×C22⋊C8C2311yescharacteristic
143C7×C22⋊C8C2312yes—
144C7×#97—21no—
145C4×C7×C8—21no—
146C7×#95—21no—
147C7×#95—22no—
148C7×#95C2312yes—
149#3559C2×C711yescharacteristic, complemented
150#3213C2×C711yescharacteristic, complemented
151#3211C2×C711yescharacteristic, complemented
152#2820C2×C711yescharacteristic, complemented
153#1604C2×C712yescomplemented
154#1603C2×C712yescomplemented
155#1598C2×C712yescomplemented
156#1597C2×C712yescomplemented
157#1517C2×C711yescharacteristic, complemented
158#1512C2×C711yescharacteristic, complemented
159#1470C2×C711yescharacteristic, complemented
160C24×C7—21no—
161C7×#41—81no—
162C7×#41—81no—
163C7×#41—82no—
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×Q8C2×D411yescharacteristic
171C2×C7×D4—21no—
172C2×C7×D4—22no—
173C2×C7×D4—22no—
174C2×C7×D4—22no—
175C2×C7×D4—21no—
176C2×C7×D4—21no—
177C2×C7×D4—21no—
178C2×C7×D4—21no—
179C2×C7×D4—81no—
180C2×C7×D4C2×D411yescharacteristic
181C2×C7×D4C2×D412yes—
182C22×C4×C7—22no—
183C22×C4×C7—22no—
184C22×C4×C7—22no—
185C22×C4×C7—21no—
186C22×C4×C7—22no—
187C22×C4×C7—22no—
188C22×C4×C7—21no—
189C22×C4×C7C2×D412yes—
190C22×C4×C7C2×D411yescharacteristic
191C22×C4×C7C2×D412yes—
192C22×C4×C7C2×D411yescharacteristic
193C22×C4×C7C2411yescharacteristic
194C7×Q16—81no—
195C7×SD16—82no—
196C7×SD16—81no—
197C7×SD16—81no—
198C7×D8—82no—
199C7×D8—81no—
200C2×C7×C8—21no—
201C2×C7×C8—21no—
202C2×C7×C8—22no—
203C7×C4⋊C4—22no—
204C7×C4⋊C4—22no—
205C7×C4⋊C4—22no—
206C7×C4⋊C4—22no—
207C7×C4⋊C4—21no—
208C7×C4⋊C4—21no—
209C7×C4⋊C4—22no—
210C7×C4⋊C4—21no—
211C7×C4⋊C4—21no—
212C7×C4⋊C4—22no—
213C7×C4⋊C4—21no—
214C7×C4⋊C4—21no—
215C7×C4⋊C4—21no—
216C7×C4⋊C4—21no—
217C7×C4⋊C4—22no—
218C7×C4⋊C4—22no—
219C7×C4⋊C4—21no—
220C7×C4⋊C4—81no—
221C7×C4⋊C4—82no—
222C7×C4⋊C4C2×D412yes—
223C7×C4⋊C4C2×D411yescharacteristic
224C7×C4⋊C4C2×D411yescharacteristic
225C7×C22⋊C4—22no—
226C7×C22⋊C4—22no—
227C7×C22⋊C4—22no—
228C7×C22⋊C4—22no—
229C7×C22⋊C4—22no—
230C7×C22⋊C4—21no—
231C7×C22⋊C4—21no—
232C7×C22⋊C4—21no—
233C7×C22⋊C4—21no—
234C7×C22⋊C4—22no—
235C7×C22⋊C4—21no—
236C7×C22⋊C4—22no—
237C7×C22⋊C4—41no—
238C7×C22⋊C4—41no—
239C7×C22⋊C4—41no—
240C7×C22⋊C4—41no—
241C7×C22⋊C4—82no—
242C7×C22⋊C4—82no—
243C42×C7—22no—
244C42×C7—21no—
245C42×C7C2×D411yescharacteristic
246C42×C7C2×D411yescharacteristic
247C42×C7C2×D411yescharacteristic
248C42×C7C2×D411yescharacteristic
249#563—21no—
250#562—21no—
251#554—21no—
252#553—21no—
253#553—21no—
254#547—22no—
255#547—21no—
256#544—21no—
257#540—21no—
258#539—22no—
259#539C22×C712yes—
260#537C22×C712yes—
261#532C22×C711yescharacteristic
262#529C22×C711yescharacteristic
263C2×#125C22×C711yescharacteristic
264C2×#124C22×C711yescharacteristic
265#496—21no—
266#495—21no—
267#489—21no—
268#488—21no—
269#464—21no—
270#463—21no—
271#463—22no—
272#451C22×C711yescharacteristic
273#450C22×C711yescharacteristic
274#448C22×C712yes—
275#448C22×C711yescharacteristic
276#447C22×C712yes—
277#447C22×C711yescharacteristic
278#432C22×C711yescharacteristic
279#431C22×C711yescharacteristic
280#399—21no—
281#397—21no—
282#396—22no—
283#395C22×C712yes—
284#394C22×C712yes—
285#382C22×C711yescharacteristic
286#328C22×C712yes—
287#327C22×C712yes—
288#327C22×C712yes—
289#327C22×C712yes—
290#324C22×C712yes—
291#324C22×C712yes—
292C23×C7—21no—
293C23×C7—22no—
294C23×C7—42no—
295C23×C7—42no—
296C23×C7—41no—
297C23×C7—41no—
298C23×C7—41no—
299C23×C7C22×D411yescharacteristic
300C7×Q8—41no—
301C7×Q8—42no—
302C7×Q8—41no—
303C7×Q8—41no—
304C7×Q8—81no—
305C7×Q8—82no—
306C7×D4—42no—
307C7×D4—41no—
308C7×D4—42no—
309C7×D4—42no—
310C7×D4—41no—
311C7×D4—41no—
312C7×D4—42no—
313C7×D4—41no—
314C7×D4—42no—
315C7×D4—41no—
316C7×D4—81no—
317C7×D4—82no—
318C7×D4—82no—
319C2×C4×C7—21no—
320C2×C4×C7—21no—
321C2×C4×C7—21no—
322C2×C4×C7—21no—
323C2×C4×C7—22no—
324C2×C4×C7—22no—
325C2×C4×C7—21no—
326C2×C4×C7—22no—
327C2×C4×C7—22no—
328C2×C4×C7—22no—
329C2×C4×C7—41no—
330C2×C4×C7—41no—
331C2×C4×C7—42no—
332C2×C4×C7—42no—
333C2×C4×C7—42no—
334C2×C4×C7—42no—
335C2×C4×C7—42no—
336C2×C4×C7—41no—
337C2×C4×C7—42no—
338C2×C4×C7—82no—
339C2×C4×C7—82no—
340C2×C4×C7—82no—
341C2×C4×C7—82no—
342C2×C4×C7#12011yescharacteristic
343C2×C4×C7#12011yescharacteristic
344C2×C4×C7#12011yescharacteristic
345C2×C4×C7#12011yescharacteristic
346C2×C4×C7C22×D411yescharacteristic
347C2×C4×C7C22×D411yescharacteristic
348C7×C8—42no—
349C7×C8—41no—
350C7×C8—41no—
351C2×#41—21no—
352C2×#41—22no—
353C22×Q8C23×C711yescharacteristic
354C22×D4C23×C711yescharacteristic
355C2×Q16—21no—
356C2×SD16—21no—
357C2×SD16—21no—
358C2×SD16—22no—
359C2×D8—21no—
360C2×D8—22no—
361C4⋊Q8—21no—
362C4⋊Q8—22no—
363C4⋊Q8C7×D411yescharacteristic
364#127C7×D411yescharacteristic
365#126—22no—
366#126—22no—
367#126—21no—
368#125—21no—
369#125—22no—
370#125—21no—
371#125—21no—
372#125—21no—
373#125—21no—
374#125C7×D411yescharacteristic
375#124—21no—
376#124—21no—
377#124—21no—
378#124—21no—
379#124—21no—
380#124C7×D411yescharacteristic
381#123—22no—
382#123—22no—
383#123—22no—
384#123—22no—
385#123—22no—
386#123—21no—
387#123—22no—
388C22⋊Q8—22no—
389C22⋊Q8—22no—
390C22⋊Q8—22no—
391C22⋊Q8—22no—
392C22⋊Q8—22no—
393C22⋊Q8—22no—
394C22⋊Q8C23×C711yescharacteristic
395#121—21no—
396#121—22no—
397#121—21no—
398#121—22no—
399#121—22no—
400#121—21no—
401#121—22no—
402#121C23×C711yescharacteristic
403#121C23×C712yes—
404C4×Q8C7×D411yescharacteristic
405C4×D4—22no—
406C4×D4—21no—
407C4×D4—21no—
408C4×D4—21no—
409C4×D4—22no—
410C4×D4—21no—
411C4×D4C7×D411yescharacteristic
412C4×D4C7×D412yes—
413#117—21no—
414#117—21no—
415C2×C4⋊C4—22no—
416C2×C4⋊C4—21no—
417C2×C4⋊C4—21no—
418C2×C4⋊C4—22no—
419C2×C4⋊C4C7×D412yes—
420C2×C4⋊C4C7×D411yescharacteristic
421C2×C4⋊C4C7×D411yescharacteristic
422C2×C4⋊C4C23×C712yes—
423C2×C22⋊C4—21no—
424C2×C22⋊C4—21no—
425C2×C22⋊C4—22no—
426C2×C22⋊C4—22no—
427C2×C42C23×C711yescharacteristic
428C4⋊C8—21no—
429C4⋊C8—21no—
430#103—22no—
431#103—21no—
432#103—21no—
433#102—22no—
434#102—21no—
435#102—21no—
436C22⋊C8C23×C711yescharacteristic
437C22⋊C8C23×C711yescharacteristic
438C22⋊C8C23×C712yes—
439#97—21no—
440C4×C8—21no—
441#95—21no—
442#95—22no—
443#95C23×C712yes—
444C22×C7—21no—
445C22×C7—21no—
446C22×C7—42no—
447C22×C7—41no—
448C22×C7—41no—
449C22×C7—42no—
450C22×C7—42no—
451C22×C7—41no—
452C22×C7—82no—
453C22×C7—82no—
454C22×C7—81no—
455C22×C7—81no—
456C22×C7—81no—
457C22×C7C2×#12011yescharacteristic
458C22×C7#57611yescharacteristic
459C22×C7#57711yescharacteristic
460C4×C7—21no—
461C4×C7—21no—
462C4×C7—21no—
463C4×C7—21no—
464C4×C7—21no—
465C4×C7—21no—
466C4×C7—42no—
467C4×C7—42no—
468C4×C7—41no—
469C4×C7—41no—
470C4×C7—42no—
471C4×C7—41no—
472C4×C7—82no—
473C4×C7—81no—
474C24—21no—
475#41—81no—
476#41—81no—
477#41—82no—
478#41—82no—
479C2×Q8—22no—
480C2×Q8—21no—
481C2×Q8—21no—
482C2×Q8—21no—
483C2×Q8—81no—
484C2×Q8C2×C7×D411yescharacteristic
485C2×D4—21no—
486C2×D4—22no—
487C2×D4—22no—
488C2×D4—22no—
489C2×D4—21no—
490C2×D4—21no—
491C2×D4—21no—
492C2×D4—21no—
493C2×D4—81no—
494C2×D4C2×C7×D411yescharacteristic
495C2×D4C2×C7×D412yes—
496C22×C4—22no—
497C22×C4—22no—
498C22×C4—22no—
499C22×C4—21no—
500C22×C4—22no—
501C22×C4—22no—
502C22×C4—21no—
503C22×C4C2×C7×D412yes—
504C22×C4C2×C7×D411yescharacteristic
505C22×C4C2×C7×D412yes—
506C22×C4C2×C7×D411yescharacteristic
507C22×C4C24×C711yescharacteristic
508Q16—81no—
509SD16—82no—
510SD16—81no—
511SD16—81no—
512D8—82no—
513D8—81no—
514C2×C8—21no—
515C2×C8—21no—
516C2×C8—22no—
517C4⋊C4—22no—
518C4⋊C4—22no—
519C4⋊C4—22no—
520C4⋊C4—22no—
521C4⋊C4—21no—
522C4⋊C4—21no—
523C4⋊C4—22no—
524C4⋊C4—21no—
525C4⋊C4—21no—
526C4⋊C4—22no—
527C4⋊C4—21no—
528C4⋊C4—21no—
529C4⋊C4—21no—
530C4⋊C4—21no—
531C4⋊C4—22no—
532C4⋊C4—22no—
533C4⋊C4—21no—
534C4⋊C4—81no—
535C4⋊C4—82no—
536C4⋊C4C2×C7×D412yes—
537C4⋊C4C2×C7×D411yescharacteristic
538C4⋊C4C2×C7×D411yescharacteristic
539C22⋊C4—22no—
540C22⋊C4—22no—
541C22⋊C4—22no—
542C22⋊C4—22no—
543C22⋊C4—22no—
544C22⋊C4—21no—
545C22⋊C4—21no—
546C22⋊C4—21no—
547C22⋊C4—21no—
548C22⋊C4—22no—
549C22⋊C4—21no—
550C22⋊C4—22no—
551C22⋊C4—41no—
552C22⋊C4—41no—
553C22⋊C4—41no—
554C22⋊C4—41no—
555C22⋊C4—82no—
556C22⋊C4—82no—
557C42—22no—
558C42—21no—
559C42C2×C7×D411yescharacteristic
560C42C2×C7×D411yescharacteristic
561C42C2×C7×D411yescharacteristic
562C42C2×C7×D411yescharacteristic
563C2×C7—21no—
564C2×C7—21no—
565C2×C7—81no—
566C2×C7—82no—
567C2×C7—161no—
568C2×C7#301211yescharacteristic
569C2×C7#302011yescharacteristic
570C2×C7#302711yescharacteristic
571C23—21no—
572C23—22no—
573C23—42no—
574C23—42no—
575C23—41no—
576C23—41no—
577C23—41no—
578C23C22×C7×D411yescharacteristic
579Q8—41no—
580Q8—42no—
581Q8—41no—
582Q8—41no—
583Q8—81no—
584Q8—82no—
585D4—42no—
586D4—41no—
587D4—42no—
588D4—42no—
589D4—41no—
590D4—41no—
591D4—42no—
592D4—41no—
593D4—42no—
594D4—41no—
595D4—81no—
596D4—82no—
597D4—82no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—21no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—41no—
616C2×C4—42no—
617C2×C4—82no—
618C2×C4—82no—
619C2×C4—82no—
620C2×C4—82no—
621C2×C4C7×#12011yescharacteristic
622C2×C4C7×#12011yescharacteristic
623C2×C4C7×#12011yescharacteristic
624C2×C4C7×#12011yescharacteristic
625C2×C4C22×C7×D411yescharacteristic
626C2×C4C22×C7×D411yescharacteristic
627C8—42no—
628C8—41no—
629C8—41no—
630C7#1846011yescharacteristic, direct product factor, Sylow
631C22—21no—
632C22—21no—
633C22—42no—
634C22—41no—
635C22—41no—
636C22—42no—
637C22—42no—
638C22—41no—
639C22—82no—
640C22—82no—
641C22—81no—
642C22—81no—
643C22—81no—
644C22C2×C7×#12011yescharacteristic
645C22C7×#57611yescharacteristic
646C22C7×#57711yescharacteristic
647C4—21no—
648C4—21no—
649C4—21no—
650C4—21no—
651C4—21no—
652C4—21no—
653C4—42no—
654C4—42no—
655C4—41no—
656C4—41no—
657C4—42no—
658C4—41no—
659C4—82no—
660C4—81no—
661C2—21no—
662C2—21no—
663C2—81no—
664C2—82no—
665C2—161no—
666C2C7×#301211yescharacteristic
667C2C7×#302011yescharacteristic
668C2C7×#302711yescharacteristic
subgroupG/H
center, socleC22×C7C2×#120
commutator, Frattini, cosocleC22×C4C24×C7
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C7 ◁ C2×C42×C7 ◁ G

Complements

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

NH ≅ G/Ntype
1. C7×#3559C2semidirect
2. C7×#3213C2semidirect
3. C7×#3211C2semidirect
4. C7×#2820C2semidirect
5. C7×#1604C2semidirect
6. C7×#1603C2semidirect
7. C7×#1598C2semidirect
8. C7×#1597C2semidirect
9. C7×#1517C2semidirect
10. C7×#1512C2semidirect
11. C7×#1470C2semidirect
55. #18460C7direct (N × H)
149. #3559C2×C7semidirect
150. #3213C2×C7semidirect
151. #3211C2×C7semidirect
152. #2820C2×C7semidirect
153. #1604C2×C7semidirect
154. #1603C2×C7semidirect
155. #1598C2×C7semidirect
156. #1597C2×C7semidirect
157. #1517C2×C7semidirect
158. #1512C2×C7semidirect
159. #1470C2×C7semidirect
630. C7#18460direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#35592
C7×#32132
C7×#32112
C7×#28202
C7×#16042
C7×#16032
C7×#15982
C7×#15972
C7×#15172
C7×#15122
C7×#14702
#184607