Finite Group Databasebeta

← C7×#19019 #13196235

Subgroups — C7×#19019 #13196235

Order 1792.

958 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3561C211yesmaximal, characteristic, complemented
2C7×#3235C211yesmaximal, characteristic, complemented
3C7×#3171C211yesmaximal, characteristic, complemented
4C7×#3153C211yesmaximal, characteristic, complemented
5C7×#1661C212yesmaximal, complemented
6C7×#1660C212yesmaximal, complemented
7C7×#1657C212yesmaximal, complemented
8C7×#1656C212yesmaximal, complemented
9C7×#1529C211yesmaximal, characteristic, complemented
10C7×#1527C211yesmaximal, characteristic, complemented
11C7×#1526C211yesmaximal, characteristic, complemented
12C2×C7×#137—21no—
13C2×C7×AGL(1,ℤ/8)—21no—
14C7×#563C2212yes—
15C7×#562C2212yes—
16C7×#556—21no—
17C7×#551—22no—
18C7×D4×Q8—21no—
19C7×#548—21no—
20C7×#546—21no—
21C7×#543—21no—
22C7×#542—21no—
23C7×#542—22no—
24C7×#538—21no—
25C7×#536—21no—
26C7×#533C2211yescharacteristic
27C7×#532C2211yescharacteristic
28C7×#529C2211yescharacteristic
29C2×C7×#124C2211yescharacteristic
30C7×#501C2211yescharacteristic, complemented
31C7×#501C2212yescomplemented
32C7×#497C2212yes—
33C7×#496C2212yes—
34C7×#496C2211yescharacteristic, complemented
35C7×#495—22no—
36C7×#490C2211yescharacteristic, complemented
37C7×#490C2212yescomplemented
38C7×#489C2211yescharacteristic, complemented
39C7×#482—22no—
40C7×#476—21no—
41C7×#474—21no—
42C7×#469—21no—
43C7×#468—21no—
44C4×C7×SD16—21no—
45C4×C7×D8—21no—
46C7×#432C2211yescharacteristic
47C4×C7×M16C2211yescharacteristic
48C4×C7×M16C2211yescharacteristic
49C7×#330C2212yescomplemented
50C7×#330C2212yescomplemented
51C7×#330C2212yescomplemented
52C7×#329C2212yes—
53C7×#329C2212yes—
54C7×#329C2212yescomplemented
55#19019C711yesmaximal, characteristic, direct product factor, Sylow
56C2×C7×#41—21no—
57C2×C7×#41—21no—
58C22×C7×Q8—21no—
59C22×C7×Q8D411yescharacteristic
60C22×C7×D4D411yescharacteristic
61C7×#137—44no—
62C7×AGL(1,ℤ/8)—44no—
63C2×C7×Q16—22no—
64C2×C7×SD16—21no—
65C2×C7×SD16—22no—
66C2×C7×SD16—22no—
67C2×C7×D8—21no—
68C2×C7×D8—22no—
69C2×C7×M16—21no—
70C2×C7×M16—21no—
71C7×C4⋊Q8—21no—
72C7×C4⋊Q8—22no—
73C7×C4⋊Q8D411yescharacteristic
74C7×C4⋊Q8D411yescharacteristic
75C7×C4⋊Q8C2311yescharacteristic
76C7×#127C2311yescharacteristic
77C7×#126—22no—
78C7×#126—22no—
79C7×#126—22no—
80C7×#126—21no—
81C7×#125C2311yescharacteristic
82C7×#125C2312yes—
83C7×#124—21no—
84C7×#124—22no—
85C7×#124—22no—
86C7×#124—21no—
87C7×#124—21no—
88C7×#124—21no—
89C7×#124D412yes—
90C7×#124C2311yescharacteristic
91C7×#124C2312yes—
92C7×#123—22no—
93C7×#123—22no—
94C7×#123—21no—
95C7×#123—22no—
96C7×#123—22no—
97C7×C22⋊Q8—22no—
98C7×C22⋊Q8—22no—
99C7×C22⋊Q8—21no—
100C7×C22⋊Q8—22no—
101C7×C22⋊Q8—22no—
102C7×C22⋊Q8—22no—
103C7×C22⋊Q8—21no—
104C7×C22⋊Q8—21no—
105C7×C22⋊Q8—22no—
106C7×C22⋊Q8—22no—
107C7×C22⋊Q8D411yescharacteristic
108C7×#121—21no—
109C7×#121—22no—
110C7×#121—22no—
111C7×#121—21no—
112C7×#121—21no—
113C7×#121D411yescharacteristic
114C7×#120—22no—
115C4×C7×Q8—21no—
116C4×C7×Q8—21no—
117C4×C7×Q8—21no—
118C4×C7×Q8D411yescharacteristic
119C4×C7×D4—22no—
120C4×C7×D4—21no—
121C4×C7×D4—22no—
122C4×C7×D4—21no—
123C4×C7×D4—21no—
124C4×C7×D4—21no—
125C4×C7×D4D411yescharacteristic
126C7×#117—22no—
127C7×#117—21no—
128C7×#117D412yes—
129C2×C7×C22⋊C4—22no—
130C2×C7×C22⋊C4—21no—
131C2×C7×C22⋊C4—21no—
132C2×C42×C7C2311yescharacteristic
133C7×#107—21no—
134C7×#107—22no—
135C7×#106—21no—
136C7×#106—22no—
137C7×C4⋊C8—21no—
138C7×#103—21no—
139C7×#102—21no—
140C7×#102—21no—
141C7×#102—21no—
142C7×#102—22no—
143C7×#102—22no—
144C7×C22⋊C8—21no—
145C7×#97C2311yescharacteristic
146C7×#97C2312yes—
147C7×#97C2311yescharacteristic
148C7×#97C2312yes—
149C4×C7×C8—21no—
150C4×C7×C8—21no—
151#3561C2×C711yescharacteristic, complemented
152#3235C2×C711yescharacteristic, complemented
153#3171C2×C711yescharacteristic, complemented
154#3153C2×C711yescharacteristic, complemented
155#1661C2×C712yescomplemented
156#1660C2×C712yescomplemented
157#1657C2×C712yescomplemented
158#1656C2×C712yescomplemented
159#1529C2×C711yescharacteristic, complemented
160#1527C2×C711yescharacteristic, complemented
161#1526C2×C711yescharacteristic, complemented
162C24×C7—21no—
163C7×#41—42no—
164C7×#41—42no—
165C7×#41—82no—
166C2×C7×Q8—22no—
167C2×C7×Q8—21no—
168C2×C7×Q8—21no—
169C2×C7×Q8—21no—
170C2×C7×Q8—21no—
171C2×C7×Q8—42no—
172C2×C7×Q8—82no—
173C2×C7×Q8C2×D411yescharacteristic
174C2×C7×Q8C2×D412yes—
175C2×C7×D4—21no—
176C2×C7×D4—22no—
177C2×C7×D4—22no—
178C2×C7×D4—21no—
179C2×C7×D4—21no—
180C2×C7×D4—21no—
181C2×C7×D4—21no—
182C2×C7×D4—42no—
183C2×C7×D4—41no—
184C2×C7×D4C2×D411yescharacteristic
185C2×C7×D4C2×D412yes—
186C22×C4×C7—22no—
187C22×C4×C7—22no—
188C22×C4×C7—22no—
189C22×C4×C7—21no—
190C22×C4×C7—21no—
191C22×C4×C7—41no—
192C22×C4×C7C2×D411yescharacteristic
193C22×C4×C7C2×D411yescharacteristic
194C22×C4×C7C2×D411yescharacteristic
195C7×Q16—44no—
196C7×SD16—42no—
197C7×SD16—44no—
198C7×SD16—44no—
199C7×D8—42no—
200C7×D8—44no—
201C7×M16—24no—
202C7×M16—42no—
203C2×C7×C8—21no—
204C2×C7×C8—22no—
205C2×C7×C8—21no—
206C2×C7×C8—22no—
207C7×C4⋊C4—22no—
208C7×C4⋊C4—22no—
209C7×C4⋊C4—22no—
210C7×C4⋊C4—22no—
211C7×C4⋊C4—21no—
212C7×C4⋊C4—21no—
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—21no—
221C7×C4⋊C4C2×D412yes—
222C7×C4⋊C4C2×D411yescharacteristic
223C7×C4⋊C4C2×D412yes—
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—22no—
232C7×C22⋊C4—22no—
233C7×C22⋊C4—22no—
234C7×C22⋊C4—21no—
235C7×C22⋊C4—21no—
236C7×C22⋊C4—22no—
237C7×C22⋊C4—22no—
238C7×C22⋊C4—21no—
239C7×C22⋊C4—21no—
240C7×C22⋊C4—41no—
241C7×C22⋊C4—41no—
242C42×C7—22no—
243C42×C7—22no—
244C42×C7—21no—
245C42×C7C2×D411yescharacteristic
246C42×C7C2×D411yescharacteristic
247C42×C7C2×D411yescharacteristic
248C42×C7C2411yescharacteristic
249C2×#137—21no—
250C2×AGL(1,ℤ/8)—21no—
251#563C22×C712yes—
252#562C22×C712yes—
253#556—21no—
254#551—22no—
255D4×Q8—21no—
256#548—21no—
257#546—21no—
258#543—21no—
259#542—21no—
260#542—22no—
261#538—21no—
262#536—21no—
263#533C22×C711yescharacteristic
264#532C22×C711yescharacteristic
265#529C22×C711yescharacteristic
266C2×#124C22×C711yescharacteristic
267#501C22×C711yescharacteristic, complemented
268#501C22×C712yescomplemented
269#497C22×C712yes—
270#496C22×C712yes—
271#496C22×C711yescharacteristic, complemented
272#495—22no—
273#490C22×C711yescharacteristic, complemented
274#490C22×C712yescomplemented
275#489C22×C711yescharacteristic, complemented
276#482—22no—
277#476—21no—
278#474—21no—
279#469—21no—
280#468—21no—
281C4×SD16—21no—
282C4×D8—21no—
283#432C22×C711yescharacteristic
284C4×M16C22×C711yescharacteristic
285C4×M16C22×C711yescharacteristic
286#330C22×C712yescomplemented
287#330C22×C712yescomplemented
288#330C22×C712yescomplemented
289#329C22×C712yes—
290#329C22×C712yes—
291#329C22×C712yescomplemented
292C23×C7—22no—
293C23×C7—21no—
294C23×C7—42no—
295C23×C7—41no—
296C23×C7—81no—
297C23×C7—81no—
298C23×C7#12011yescharacteristic
299C7×Q8—42no—
300C7×Q8—41no—
301C7×Q8—41no—
302C7×Q8—41no—
303C7×Q8—42no—
304C7×Q8—42no—
305C7×Q8—81no—
306C7×Q8—82no—
307C7×D4—42no—
308C7×D4—41no—
309C7×D4—41no—
310C7×D4—41no—
311C7×D4—41no—
312C7×D4—41no—
313C7×D4—42no—
314C7×D4—42no—
315C7×D4—42no—
316C7×D4—82no—
317C7×D4—82no—
318C7×D4—82no—
319C2×C4×C7—22no—
320C2×C4×C7—21no—
321C2×C4×C7—22no—
322C2×C4×C7—21no—
323C2×C4×C7—22no—
324C2×C4×C7—22no—
325C2×C4×C7—21no—
326C2×C4×C7—41no—
327C2×C4×C7—41no—
328C2×C4×C7—42no—
329C2×C4×C7—41no—
330C2×C4×C7—81no—
331C2×C4×C7—82no—
332C2×C4×C7—82no—
333C2×C4×C7—82no—
334C2×C4×C7—82no—
335C2×C4×C7#12011yescharacteristic
336C2×C4×C7#12011yescharacteristic
337C2×C4×C7#12011yescharacteristic
338C2×C4×C7C22×D411yescharacteristic
339C2×C4×C7C22×D411yescharacteristic
340C2×C4×C7C22×D411yescharacteristic
341C7×C8—24no—
342C7×C8—22no—
343C7×C8—42no—
344C7×C8—41no—
345C2×#41—21no—
346C2×#41—21no—
347C22×Q8—21no—
348C22×Q8C7×D411yescharacteristic
349C22×D4C7×D411yescharacteristic
350#137—44no—
351AGL(1,ℤ/8)—44no—
352C2×Q16—22no—
353C2×SD16—22no—
354C2×SD16—21no—
355C2×SD16—22no—
356C2×D8—22no—
357C2×D8—21no—
358C2×M16—21no—
359C2×M16—21no—
360C4⋊Q8—21no—
361C4⋊Q8—22no—
362C4⋊Q8C7×D411yescharacteristic
363C4⋊Q8C7×D411yescharacteristic
364C4⋊Q8C23×C711yescharacteristic
365#127C23×C711yescharacteristic
366#126—22no—
367#126—22no—
368#126—22no—
369#126—21no—
370#125C23×C711yescharacteristic
371#125C23×C712yes—
372#124—21no—
373#124—22no—
374#124—22no—
375#124—21no—
376#124—21no—
377#124—21no—
378#124C7×D412yes—
379#124C23×C712yes—
380#124C23×C711yescharacteristic
381#123—22no—
382#123—22no—
383#123—21no—
384#123—22no—
385#123—22no—
386C22⋊Q8—22no—
387C22⋊Q8—22no—
388C22⋊Q8—21no—
389C22⋊Q8—22no—
390C22⋊Q8—22no—
391C22⋊Q8—22no—
392C22⋊Q8—21no—
393C22⋊Q8—21no—
394C22⋊Q8—22no—
395C22⋊Q8—22no—
396C22⋊Q8C7×D411yescharacteristic
397#121—21no—
398#121—22no—
399#121—22no—
400#121—21no—
401#121—21no—
402#121C7×D411yescharacteristic
403#120—22no—
404C4×Q8—21no—
405C4×Q8—21no—
406C4×Q8—21no—
407C4×Q8C7×D411yescharacteristic
408C4×D4—22no—
409C4×D4—21no—
410C4×D4—22no—
411C4×D4—21no—
412C4×D4—21no—
413C4×D4—21no—
414C4×D4C7×D411yescharacteristic
415#117—22no—
416#117—21no—
417#117C7×D412yes—
418C2×C22⋊C4—22no—
419C2×C22⋊C4—21no—
420C2×C22⋊C4—21no—
421C2×C42C23×C711yescharacteristic
422#107—21no—
423#107—22no—
424#106—21no—
425#106—22no—
426C4⋊C8—21no—
427#103—21no—
428#102—21no—
429#102—21no—
430#102—21no—
431#102—22no—
432#102—22no—
433C22⋊C8—21no—
434#97C23×C712yes—
435#97C23×C711yescharacteristic
436#97C23×C711yescharacteristic
437#97C23×C712yes—
438C4×C8—21no—
439C4×C8—21no—
440C22×C7—21no—
441C22×C7—21no—
442C22×C7—21no—
443C22×C7—41no—
444C22×C7—42no—
445C22×C7—42no—
446C22×C7—41no—
447C22×C7—41no—
448C22×C7—42no—
449C22×C7—81no—
450C22×C7—81no—
451C22×C7—81no—
452C22×C7—162no—
453C22×C7C2×#12011yescharacteristic
454C4×C7—21no—
455C4×C7—21no—
456C4×C7—21no—
457C4×C7—21no—
458C4×C7—21no—
459C4×C7—41no—
460C4×C7—42no—
461C4×C7—42no—
462C4×C7—41no—
463C4×C7—42no—
464C4×C7—41no—
465C4×C7—81no—
466C4×C7#45312yes—
467C24—21no—
468#41—42no—
469#41—42no—
470#41—82no—
471C2×Q8—22no—
472C2×Q8—21no—
473C2×Q8—21no—
474C2×Q8—21no—
475C2×Q8—21no—
476C2×Q8—42no—
477C2×Q8—82no—
478C2×Q8C2×C7×D411yescharacteristic
479C2×Q8C2×C7×D412yes—
480C2×D4—21no—
481C2×D4—22no—
482C2×D4—22no—
483C2×D4—21no—
484C2×D4—21no—
485C2×D4—21no—
486C2×D4—21no—
487C2×D4—42no—
488C2×D4—41no—
489C2×D4C2×C7×D411yescharacteristic
490C2×D4C2×C7×D412yes—
491C22×C4—22no—
492C22×C4—22no—
493C22×C4—22no—
494C22×C4—21no—
495C22×C4—21no—
496C22×C4—41no—
497C22×C4C2×C7×D411yescharacteristic
498C22×C4C2×C7×D411yescharacteristic
499C22×C4C2×C7×D411yescharacteristic
500Q16—44no—
501SD16—42no—
502SD16—44no—
503SD16—44no—
504D8—42no—
505D8—44no—
506M16—24no—
507M16—42no—
508C2×C8—21no—
509C2×C8—22no—
510C2×C8—21no—
511C2×C8—22no—
512C4⋊C4—22no—
513C4⋊C4—22no—
514C4⋊C4—22no—
515C4⋊C4—22no—
516C4⋊C4—21no—
517C4⋊C4—21no—
518C4⋊C4—21no—
519C4⋊C4—21no—
520C4⋊C4—21no—
521C4⋊C4—21no—
522C4⋊C4—22no—
523C4⋊C4—22no—
524C4⋊C4—21no—
525C4⋊C4—21no—
526C4⋊C4C2×C7×D412yes—
527C4⋊C4C2×C7×D411yescharacteristic
528C4⋊C4C2×C7×D412yes—
529C4⋊C4C2×C7×D411yescharacteristic
530C22⋊C4—22no—
531C22⋊C4—22no—
532C22⋊C4—22no—
533C22⋊C4—22no—
534C22⋊C4—22no—
535C22⋊C4—21no—
536C22⋊C4—22no—
537C22⋊C4—22no—
538C22⋊C4—22no—
539C22⋊C4—21no—
540C22⋊C4—21no—
541C22⋊C4—22no—
542C22⋊C4—22no—
543C22⋊C4—21no—
544C22⋊C4—21no—
545C22⋊C4—41no—
546C22⋊C4—41no—
547C42—22no—
548C42—22no—
549C42—21no—
550C42C2×C7×D411yescharacteristic
551C42C2×C7×D411yescharacteristic
552C42C2×C7×D411yescharacteristic
553C42C24×C711yescharacteristic
554C2×C7—41no—
555C2×C7—81no—
556C2×C7—82no—
557C2×C7—161no—
558C2×C7C2×#45311yescharacteristic
559C2×C7#301811yescharacteristic
560C2×C7#302011yescharacteristic
561C23—22no—
562C23—21no—
563C23—42no—
564C23—41no—
565C23—81no—
566C23—81no—
567C23C7×#12011yescharacteristic
568Q8—42no—
569Q8—41no—
570Q8—41no—
571Q8—41no—
572Q8—42no—
573Q8—42no—
574Q8—81no—
575Q8—82no—
576D4—42no—
577D4—41no—
578D4—41no—
579D4—41no—
580D4—41no—
581D4—41no—
582D4—42no—
583D4—42no—
584D4—42no—
585D4—82no—
586D4—82no—
587D4—82no—
588C2×C4—22no—
589C2×C4—21no—
590C2×C4—22no—
591C2×C4—21no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—21no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—42no—
598C2×C4—41no—
599C2×C4—81no—
600C2×C4—82no—
601C2×C4—82no—
602C2×C4—82no—
603C2×C4—82no—
604C2×C4C7×#12011yescharacteristic
605C2×C4C7×#12011yescharacteristic
606C2×C4C7×#12011yescharacteristic
607C2×C4C22×C7×D411yescharacteristic
608C2×C4C22×C7×D411yescharacteristic
609C2×C4C22×C7×D411yescharacteristic
610C8—24no—
611C8—22no—
612C8—42no—
613C8—41no—
614C7#1901911yescharacteristic, direct product factor, Sylow
615C22—21no—
616C22—21no—
617C22—21no—
618C22—41no—
619C22—42no—
620C22—42no—
621C22—41no—
622C22—41no—
623C22—42no—
624C22—81no—
625C22—81no—
626C22—81no—
627C22—162no—
628C22C2×C7×#12011yescharacteristic
629C4—21no—
630C4—21no—
631C4—21no—
632C4—21no—
633C4—21no—
634C4—41no—
635C4—42no—
636C4—42no—
637C4—41no—
638C4—42no—
639C4—41no—
640C4—81no—
641C4C7×#45312yes—
642C2—41no—
643C2—82no—
644C2—81no—
645C2—161no—
646C2C2×C7×#45311yescharacteristic
647C2C7×#301811yescharacteristic
648C2C7×#302011yescharacteristic
subgroupG/H
center, socleC22×C7C2×#120
commutator, Frattini, cosocleC42C24×C7
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
1. C7×#3561C2semidirect
2. C7×#3235C2semidirect
3. C7×#3171C2semidirect
4. C7×#3153C2semidirect
5. C7×#1661C2semidirect
6. C7×#1660C2semidirect
7. C7×#1657C2semidirect
8. C7×#1656C2semidirect
9. C7×#1529C2semidirect
10. C7×#1527C2semidirect
11. C7×#1526C2semidirect
30. C7×#501C22semidirect
31. C7×#501C22semidirect
34. C7×#496C22semidirect
36. C7×#490C22semidirect
37. C7×#490C22semidirect
38. C7×#489C22semidirect
49. C7×#330C22semidirect
50. C7×#330C22semidirect
51. C7×#330C22semidirect
54. C7×#329C22semidirect
55. #19019C7direct (N × H)
151. #3561C2×C7semidirect
152. #3235C2×C7semidirect
153. #3171C2×C7semidirect
154. #3153C2×C7semidirect
155. #1661C2×C7semidirect
156. #1660C2×C7semidirect
157. #1657C2×C7semidirect
158. #1656C2×C7semidirect
159. #1529C2×C7semidirect
160. #1527C2×C7semidirect
161. #1526C2×C7semidirect
267. #501C22×C7semidirect
268. #501C22×C7semidirect
271. #496C22×C7semidirect
273. #490C22×C7semidirect
274. #490C22×C7semidirect
275. #489C22×C7semidirect
286. #330C22×C7semidirect
287. #330C22×C7semidirect
288. #330C22×C7semidirect
291. #329C22×C7semidirect
614. C7#19019direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#35612
C7×#32352
C7×#31712
C7×#31532
C7×#16612
C7×#16602
C7×#16572
C7×#16562
C7×#15292
C7×#15272
C7×#15262
#190197