Finite Group Databasebeta

← C7×#18803 #13196019

Subgroups — C7×#18803 #13196019

Order 1792.

958 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3480C211yesmaximal, characteristic
2C7×#3227C211yesmaximal, characteristic, complemented
3C7×#3079C211yesmaximal, characteristic, complemented
4C2×C7×#462C211yesmaximal, characteristic, complemented
5C7×#1653C212yesmaximal, complemented
6C7×#1652C212yesmaximal, complemented
7C7×#1641C212yesmaximal, complemented
8C7×#1640C212yesmaximal, complemented
9C7×#1547C211yesmaximal, characteristic, complemented
10C7×#1487C211yesmaximal, characteristic, complemented
11C7×#1477C211yesmaximal, characteristic, complemented
12C22×C7×Q16—21no—
13C7×#559—21no—
14C7×#559C2211yescharacteristic
15C7×#557C2211yescharacteristic
16C7×#556—21no—
17C7×#556C2211yescharacteristic
18C7×#553—22no—
19C7×#553—21no—
20C7×#550C2211yescharacteristic
21C7×#548—21no—
22C7×#532—21no—
23C7×#532C2211yescharacteristic
24C2×C7×C4⋊Q8C2211yescharacteristic
25C7×#529—21no—
26C7×#529—22no—
27C4×C7×#41—21no—
28C4×C7×#41—21no—
29C4×C7×#41C2211yescharacteristic
30C2×C4×C7×Q8C2211yescharacteristic
31C7×#498C2212yes—
32C7×#486C2212yes—
33C7×#481—22no—
34C7×#477—21no—
35C7×#476—22no—
36C7×#476—21no—
37C7×#475C2212yes—
38C7×#474C2212yes—
39C7×#462—21no—
40C7×#462—21no—
41C7×#462C2212yes—
42C7×#462C2212yes—
43C7×#431C2211yescharacteristic
44C7×#428—21no—
45C7×#423C2211yescharacteristic
46C2×C7×C4⋊C8C2211yescharacteristic
47C7×#417—21no—
48C2×C7×#103—21no—
49C7×#332C2212yes—
50C7×#331C2212yes—
51C7×Q8⋊C8C2212yes—
52C7×Q8⋊C8C2212yes—
53C7×Q8⋊C8C2212yes—
54C7×D4⋊C8C2212yes—
55#18803C711yesmaximal, characteristic, direct product factor, Sylow
56C2×C7×#41—21no—
57C2×C7×#41—21no—
58C2×C7×#41D411yescharacteristic
59C22×C7×Q8—21no—
60C22×C7×Q8D411yescharacteristic
61C2×C7×Q16—22no—
62C2×C7×Q16—22no—
63C2×C7×Q16—44no—
64C2×C7×M16—21no—
65C22×C7×C8—21no—
66C7×C4⋊Q8—21no—
67C7×C4⋊Q8—21no—
68C7×C4⋊Q8—21no—
69C7×C4⋊Q8—21no—
70C7×C4⋊Q8C2312yes—
71C7×C4⋊Q8C2311yescharacteristic
72C7×#127—21no—
73C7×#127C2311yescharacteristic
74C7×#126—22no—
75C7×#126—22no—
76C7×#126—22no—
77C7×#125—22no—
78C7×#125—21no—
79C7×#124—22no—
80C7×#124—21no—
81C7×#124—21no—
82C7×#124—21no—
83C7×#124—21no—
84C7×#123—22no—
85C7×#123—22no—
86C7×#123—22no—
87C7×C22⋊Q8—22no—
88C7×C22⋊Q8—21no—
89C7×#121—22no—
90C7×#121—22no—
91C7×#121—22no—
92C7×#121—21no—
93C7×#121—21no—
94C7×#121D412yes—
95C4×C7×Q8—21no—
96C4×C7×Q8—21no—
97C4×C7×Q8—21no—
98C4×C7×Q8—21no—
99C4×C7×Q8—21no—
100C4×C7×Q8D411yescharacteristic
101C4×C7×Q8D411yescharacteristic
102C4×C7×Q8D411yescharacteristic
103C4×C7×Q8C2311yescharacteristic
104C4×C7×Q8C2311yescharacteristic
105C4×C7×Q8C2311yescharacteristic
106C4×C7×D4—22no—
107C4×C7×D4—22no—
108C4×C7×D4—22no—
109C4×C7×D4—22no—
110C4×C7×D4—22no—
111C4×C7×D4—21no—
112C4×C7×D4—21no—
113C4×C7×D4—21no—
114C4×C7×D4—21no—
115C4×C7×D4—21no—
116C4×C7×D4—21no—
117C4×C7×D4D411yescharacteristic
118C4×C7×D4D412yes—
119C4×C7×D4C2311yescharacteristic
120C7×#117—22no—
121C7×#117—22no—
122C7×#117—21no—
123C7×#117—21no—
124C7×#117D411yescharacteristic
125C2×C7×C4⋊C4—21no—
126C2×C7×C4⋊C4—21no—
127C2×C7×C4⋊C4—21no—
128C2×C7×C4⋊C4D411yescharacteristic
129C2×C42×C7—22no—
130C2×C42×C7—21no—
131C2×C42×C7—21no—
132C2×C42×C7C2311yescharacteristic
133C7×#107—22no—
134C7×#106—22no—
135C7×#106—24no—
136C7×C4⋊C8—21no—
137C7×C4⋊C8—21no—
138C7×C4⋊C8—21no—
139C7×C4⋊C8C2312yes—
140C7×C4⋊C8C2312yes—
141C7×#103—22no—
142C7×#103—22no—
143C7×#103—22no—
144C7×#102—22no—
145C7×#102—24no—
146C7×C22⋊C8—21no—
147C4×C7×C8C2312yes—
148#3480C2×C711yescharacteristic
149#3227C2×C711yescharacteristic, complemented
150#3079C2×C711yescharacteristic, complemented
151C2×#462C2×C711yescharacteristic, complemented
152#1653C2×C712yescomplemented
153#1652C2×C712yescomplemented
154#1641C2×C712yescomplemented
155#1640C2×C712yescomplemented
156#1547C2×C711yescharacteristic, complemented
157#1487C2×C711yescharacteristic, complemented
158#1477C2×C711yescharacteristic, complemented
159C7×#41—22no—
160C7×#41—22no—
161C7×#41—42no—
162C7×#41—42no—
163C7×#41—42no—
164C7×#41—42no—
165C2×C7×Q8—21no—
166C2×C7×Q8—21no—
167C2×C7×Q8—22no—
168C2×C7×Q8—22no—
169C2×C7×Q8—41no—
170C2×C7×Q8—42no—
171C2×C7×Q8—42no—
172C2×C7×Q8—41no—
173C2×C7×Q8C2×D411yescharacteristic
174C2×C7×Q8C2×D411yescharacteristic
175C2×C7×Q8C2×D411yescharacteristic
176C2×C7×D4—22no—
177C2×C7×D4—21no—
178C2×C7×D4—22no—
179C2×C7×D4—22no—
180C2×C7×D4—21no—
181C2×C7×D4—21no—
182C2×C7×D4C2×D411yescharacteristic
183C2×C7×D4C2×D412yes—
184C22×C4×C7—22no—
185C22×C4×C7—21no—
186C22×C4×C7—22no—
187C22×C4×C7—21no—
188C22×C4×C7—21no—
189C22×C4×C7—22no—
190C22×C4×C7—21no—
191C22×C4×C7—21no—
192C22×C4×C7—41no—
193C22×C4×C7C2×D411yescharacteristic
194C22×C4×C7C2×D411yescharacteristic
195C22×C4×C7C2×D411yescharacteristic
196C7×Q16—44no—
197C7×Q16—44no—
198C7×M16—42no—
199C2×C7×C8—22no—
200C2×C7×C8—22no—
201C2×C7×C8—22no—
202C2×C7×C8—42no—
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—21no—
210C7×C4⋊C4—21no—
211C7×C4⋊C4—21no—
212C7×C4⋊C4—21no—
213C7×C4⋊C4—21no—
214C7×C4⋊C4—21no—
215C7×C4⋊C4—22no—
216C7×C4⋊C4—22no—
217C7×C4⋊C4—41no—
218C7×C4⋊C4—41no—
219C7×C4⋊C4C2×D411yescharacteristic
220C7×C4⋊C4C2×D411yescharacteristic
221C7×C4⋊C4C2×D411yescharacteristic
222C7×C4⋊C4C2×D412yes—
223C7×C4⋊C4C2×D411yescharacteristic
224C7×C22⋊C4—22no—
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—22no—
232C7×C22⋊C4—21no—
233C7×C22⋊C4—22no—
234C7×C22⋊C4—21no—
235C42×C7—22no—
236C42×C7—22no—
237C42×C7—21no—
238C42×C7—21no—
239C42×C7—21no—
240C42×C7—21no—
241C42×C7—21no—
242C42×C7—21no—
243C42×C7—21no—
244C42×C7C2×D411yescharacteristic
245C42×C7C2×D411yescharacteristic
246C42×C7C2×D411yescharacteristic
247C42×C7C2411yescharacteristic
248C22×Q16—21no—
249#559—21no—
250#559C22×C711yescharacteristic
251#557C22×C711yescharacteristic
252#556—21no—
253#556C22×C711yescharacteristic
254#553—22no—
255#553—21no—
256#550C22×C711yescharacteristic
257#548—21no—
258#532—21no—
259#532C22×C711yescharacteristic
260C2×C4⋊Q8C22×C711yescharacteristic
261#529—21no—
262#529—22no—
263C4×#41—21no—
264C4×#41—21no—
265C4×#41C22×C711yescharacteristic
266C2×C4×Q8C22×C711yescharacteristic
267#498C22×C712yes—
268#486C22×C712yes—
269#481—22no—
270#477—21no—
271#476—21no—
272#476—22no—
273#475C22×C712yes—
274#474C22×C712yes—
275#462—21no—
276#462—21no—
277#462C22×C712yes—
278#462C22×C712yes—
279#431C22×C711yescharacteristic
280#428—21no—
281#423C22×C711yescharacteristic
282C2×C4⋊C8C22×C711yescharacteristic
283#417—21no—
284C2×#103—21no—
285#332C22×C712yes—
286#331C22×C712yes—
287Q8⋊C8C22×C712yes—
288Q8⋊C8C22×C712yes—
289Q8⋊C8C22×C712yes—
290D4⋊C8C22×C712yes—
291C23×C7—21no—
292C23×C7—22no—
293C23×C7#12011yescharacteristic
294C7×Q8—21no—
295C7×Q8—21no—
296C7×Q8—21no—
297C7×Q8—21no—
298C7×Q8—21no—
299C7×Q8—42no—
300C7×Q8—41no—
301C7×Q8—41no—
302C7×Q8—42no—
303C7×Q8—42no—
304C7×Q8—82no—
305C7×Q8AGL(1,ℤ/8)12yes—
306C7×D4—21no—
307C7×D4—21no—
308C7×D4—42no—
309C7×D4—44no—
310C7×D4—42no—
311C7×D4—44no—
312C7×D4—41no—
313C7×D4—41no—
314C7×D4—41no—
315C7×D4—41no—
316C7×D4—82no—
317C2×C4×C7—21no—
318C2×C4×C7—21no—
319C2×C4×C7—21no—
320C2×C4×C7—21no—
321C2×C4×C7—21no—
322C2×C4×C7—21no—
323C2×C4×C7—21no—
324C2×C4×C7—21no—
325C2×C4×C7—22no—
326C2×C4×C7—21no—
327C2×C4×C7—41no—
328C2×C4×C7—42no—
329C2×C4×C7—44no—
330C2×C4×C7—41no—
331C2×C4×C7—41no—
332C2×C4×C7—42no—
333C2×C4×C7—44no—
334C2×C4×C7—42no—
335C2×C4×C7—81no—
336C2×C4×C7—81no—
337C2×C4×C7—82no—
338C2×C4×C7—82no—
339C2×C4×C7#12011yescharacteristic
340C2×C4×C7#12011yescharacteristic
341C2×C4×C7#12011yescharacteristic
342C2×C4×C7#13512yes—
343C2×C4×C7#13712yes—
344C2×C4×C7C22×D411yescharacteristic
345C2×C4×C7C22×D411yescharacteristic
346C2×C4×C7C22×D411yescharacteristic
347C7×C8—42no—
348C7×C8—42no—
349C7×C8—42no—
350C2×#41—21no—
351C2×#41—21no—
352C2×#41C7×D411yescharacteristic
353C22×Q8—21no—
354C22×Q8C7×D411yescharacteristic
355C2×Q16—22no—
356C2×Q16—22no—
357C2×Q16—44no—
358C2×M16—21no—
359C22×C8—21no—
360C4⋊Q8—21no—
361C4⋊Q8—21no—
362C4⋊Q8—21no—
363C4⋊Q8—21no—
364C4⋊Q8C23×C712yes—
365C4⋊Q8C23×C711yescharacteristic
366#127—21no—
367#127C23×C711yescharacteristic
368#126—22no—
369#126—22no—
370#126—22no—
371#125—22no—
372#125—21no—
373#124—22no—
374#124—21no—
375#124—21no—
376#124—21no—
377#124—21no—
378#123—22no—
379#123—22no—
380#123—22no—
381C22⋊Q8—22no—
382C22⋊Q8—21no—
383#121—22no—
384#121—22no—
385#121—22no—
386#121—21no—
387#121—21no—
388#121C7×D412yes—
389C4×Q8—21no—
390C4×Q8—21no—
391C4×Q8—21no—
392C4×Q8—21no—
393C4×Q8—21no—
394C4×Q8C7×D411yescharacteristic
395C4×Q8C7×D411yescharacteristic
396C4×Q8C7×D411yescharacteristic
397C4×Q8C23×C711yescharacteristic
398C4×Q8C23×C711yescharacteristic
399C4×Q8C23×C711yescharacteristic
400C4×D4—22no—
401C4×D4—22no—
402C4×D4—22no—
403C4×D4—22no—
404C4×D4—22no—
405C4×D4—21no—
406C4×D4—21no—
407C4×D4—21no—
408C4×D4—21no—
409C4×D4—21no—
410C4×D4—21no—
411C4×D4C7×D411yescharacteristic
412C4×D4C7×D412yes—
413C4×D4C23×C711yescharacteristic
414#117—22no—
415#117—22no—
416#117—21no—
417#117—21no—
418#117C7×D411yescharacteristic
419C2×C4⋊C4—21no—
420C2×C4⋊C4—21no—
421C2×C4⋊C4—21no—
422C2×C4⋊C4C7×D411yescharacteristic
423C2×C42—22no—
424C2×C42—21no—
425C2×C42—21no—
426C2×C42C23×C711yescharacteristic
427#107—22no—
428#106—22no—
429#106—24no—
430C4⋊C8—21no—
431C4⋊C8—21no—
432C4⋊C8—21no—
433C4⋊C8C23×C712yes—
434C4⋊C8C23×C712yes—
435#103—22no—
436#103—22no—
437#103—22no—
438#102—22no—
439#102—24no—
440C22⋊C8—21no—
441C4×C8C23×C712yes—
442C22×C7—21no—
443C22×C7—21no—
444C22×C7—41no—
445C22×C7—42no—
446C22×C7—42no—
447C22×C7—41no—
448C22×C7—42no—
449C22×C7—41no—
450C22×C7#45212yes—
451C22×C7C2×#12011yescharacteristic
452C4×C7—21no—
453C4×C7—22no—
454C4×C7—21no—
455C4×C7—21no—
456C4×C7—41no—
457C4×C7—42no—
458C4×C7—41no—
459C4×C7—41no—
460C4×C7—41no—
461C4×C7—41no—
462C4×C7—41no—
463C4×C7—82no—
464C4×C7#45212yes—
465C4×C7C2×#13511yescharacteristic
466C4×C7C2×AGL(1,ℤ/8)11yescharacteristic
467C4×C7C2×#13711yescharacteristic
468C4×C7#57511yescharacteristic
469#41—22no—
470#41—22no—
471#41—42no—
472#41—42no—
473#41—42no—
474#41—42no—
475C2×Q8—21no—
476C2×Q8—21no—
477C2×Q8—22no—
478C2×Q8—22no—
479C2×Q8—41no—
480C2×Q8—42no—
481C2×Q8—42no—
482C2×Q8—41no—
483C2×Q8C2×C7×D411yescharacteristic
484C2×Q8C2×C7×D411yescharacteristic
485C2×Q8C2×C7×D411yescharacteristic
486C2×D4—22no—
487C2×D4—21no—
488C2×D4—22no—
489C2×D4—22no—
490C2×D4—21no—
491C2×D4—21no—
492C2×D4C2×C7×D411yescharacteristic
493C2×D4C2×C7×D412yes—
494C22×C4—22no—
495C22×C4—21no—
496C22×C4—22no—
497C22×C4—21no—
498C22×C4—21no—
499C22×C4—22no—
500C22×C4—21no—
501C22×C4—21no—
502C22×C4—41no—
503C22×C4C2×C7×D411yescharacteristic
504C22×C4C2×C7×D411yescharacteristic
505C22×C4C2×C7×D411yescharacteristic
506Q16—44no—
507Q16—44no—
508M16—42no—
509C2×C8—22no—
510C2×C8—22no—
511C2×C8—22no—
512C2×C8—42no—
513C4⋊C4—22no—
514C4⋊C4—22no—
515C4⋊C4—22no—
516C4⋊C4—22no—
517C4⋊C4—21no—
518C4⋊C4—21no—
519C4⋊C4—21no—
520C4⋊C4—21no—
521C4⋊C4—21no—
522C4⋊C4—21no—
523C4⋊C4—21no—
524C4⋊C4—21no—
525C4⋊C4—22no—
526C4⋊C4—22no—
527C4⋊C4—41no—
528C4⋊C4—41no—
529C4⋊C4C2×C7×D411yescharacteristic
530C4⋊C4C2×C7×D411yescharacteristic
531C4⋊C4C2×C7×D411yescharacteristic
532C4⋊C4C2×C7×D412yes—
533C4⋊C4C2×C7×D411yescharacteristic
534C22⋊C4—22no—
535C22⋊C4—22no—
536C22⋊C4—22no—
537C22⋊C4—22no—
538C22⋊C4—22no—
539C22⋊C4—22no—
540C22⋊C4—21no—
541C22⋊C4—22no—
542C22⋊C4—21no—
543C22⋊C4—22no—
544C22⋊C4—21no—
545C42—22no—
546C42—22no—
547C42—21no—
548C42—21no—
549C42—21no—
550C42—21no—
551C42—21no—
552C42—21no—
553C42—21no—
554C42C2×C7×D411yescharacteristic
555C42C2×C7×D411yescharacteristic
556C42C2×C7×D411yescharacteristic
557C42C24×C711yescharacteristic
558C2×C7—22no—
559C2×C7—81no—
560C2×C7—82no—
561C2×C7C2×#45211yescharacteristic
562C2×C7#300611yescharacteristic
563C2×C7#302211yescharacteristic
564C23—21no—
565C23—22no—
566C23C7×#12011yescharacteristic
567Q8—21no—
568Q8—21no—
569Q8—21no—
570Q8—21no—
571Q8—21no—
572Q8—42no—
573Q8—41no—
574Q8—41no—
575Q8—42no—
576Q8—42no—
577Q8—82no—
578Q8C7×AGL(1,ℤ/8)12yes—
579D4—21no—
580D4—21no—
581D4—42no—
582D4—44no—
583D4—42no—
584D4—44no—
585D4—41no—
586D4—41no—
587D4—41no—
588D4—41no—
589D4—82no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—22no—
599C2×C4—21no—
600C2×C4—41no—
601C2×C4—42no—
602C2×C4—44no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—42no—
606C2×C4—44no—
607C2×C4—42no—
608C2×C4—81no—
609C2×C4—81no—
610C2×C4—82no—
611C2×C4—82no—
612C2×C4C7×#12011yescharacteristic
613C2×C4C7×#12011yescharacteristic
614C2×C4C7×#12011yescharacteristic
615C2×C4C7×#13512yes—
616C2×C4C7×#13712yes—
617C2×C4C22×C7×D411yescharacteristic
618C2×C4C22×C7×D411yescharacteristic
619C2×C4C22×C7×D411yescharacteristic
620C8—42no—
621C8—42no—
622C8—42no—
623C7#1880311yescharacteristic, direct product factor, Sylow
624C22—21no—
625C22—21no—
626C22—41no—
627C22—42no—
628C22—42no—
629C22—41no—
630C22—42no—
631C22—41no—
632C22C7×#45212yes—
633C22C2×C7×#12011yescharacteristic
634C4—21no—
635C4—22no—
636C4—21no—
637C4—21no—
638C4—41no—
639C4—42no—
640C4—41no—
641C4—41no—
642C4—41no—
643C4—41no—
644C4—41no—
645C4—82no—
646C4C7×#45212yes—
647C4C2×C7×#13511yescharacteristic
648C4C2×C7×AGL(1,ℤ/8)11yescharacteristic
649C4C2×C7×#13711yescharacteristic
650C4C7×#57511yescharacteristic
651C2—22no—
652C2—81no—
653C2—82no—
654C2C2×C7×#45211yescharacteristic
655C2C7×#300611yescharacteristic
656C2C7×#302211yescharacteristic
subgroupG/H
center, socleC22×C7C2×#120
commutator, Frattini, cosocleC42C24×C7
prime radicalC7×#3480C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C7 ◁ C2×C42×C7 ◁ G

Complements

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

NH ≅ G/Ntype
2. C7×#3227C2semidirect
3. C7×#3079C2semidirect
4. C2×C7×#462C2semidirect
5. C7×#1653C2semidirect
6. C7×#1652C2semidirect
7. C7×#1641C2semidirect
8. C7×#1640C2semidirect
9. C7×#1547C2semidirect
10. C7×#1487C2semidirect
11. C7×#1477C2semidirect
55. #18803C7direct (N × H)
149. #3227C2×C7semidirect
150. #3079C2×C7semidirect
151. C2×#462C2×C7semidirect
152. #1653C2×C7semidirect
153. #1652C2×C7semidirect
154. #1641C2×C7semidirect
155. #1640C2×C7semidirect
156. #1547C2×C7semidirect
157. #1487C2×C7semidirect
158. #1477C2×C7semidirect
623. C7#18803direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#34802
C7×#32272
C7×#30792
C2×C7×#4622
C7×#16532
C7×#16522
C7×#16412
C7×#16402
C7×#15472
C7×#14872
C7×#14772
#188037