Finite Group Databasebeta

← C7×#9042 #13186258

Subgroups — C7×#9042 #13186258

Order 1792.

966 conjugacy classes of subgroups.

The subgroup lattice has 668 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#450C211yesmaximal, characteristic
2C7×#2329C211yesmaximal, characteristic, complemented
3C7×#1786C211yesmaximal, characteristic
4C2×C7×#328C211yesmaximal, characteristic, complemented
5C7×#1438C211yesmaximal, characteristic
6C7×#1294C211yesmaximal, characteristic, complemented
7C7×#1280C211yesmaximal, characteristic, complemented
8C23×C7×D4C2211yescharacteristic
9C22×C7×SD16—21no—
10C2×C7×C22⋊Q8C2211yescharacteristic
11C22×C7×C22⋊C4—21no—
12C22×C7×C22⋊C4—21no—
13C7×#450—22no—
14C7×#450C414yescomplemented
15C2×C7×#102—21no—
16C2×C7×#102—21no—
17C2×C7×#102—21no—
18C2×C7×C22⋊C8C2211yescharacteristic
19C2×C7×C22⋊C8C2211yescharacteristic
20C7×#391—21no—
21C7×#389—21no—
22C7×#387—21no—
23C7×#387—21no—
24C7×#385—21no—
25C7×#380C2211yescharacteristic
26C4×C7×C22⋊C4—21no—
27C2×C7×#95C2211yescharacteristic
28C2×C7×#95C2211yescharacteristic
29C7×#328—22no—
30#9042C711yesmaximal, characteristic, direct product factor, Sylow
31C25×C7—21no—
32C22×C7×Q8D411yescharacteristic
33C22×C7×D4—21no—
34C22×C7×D4—21no—
35C22×C7×D4—21no—
36C22×C7×D4—21no—
37C22×C7×D4—21no—
38C22×C7×D4—42no—
39C22×C7×D4—42no—
40C22×C7×D4C2×C412yes—
41C23×C4×C7—21no—
42C23×C4×C7D411yescharacteristic
43C23×C4×C7D411yescharacteristic
44C23×C4×C7C2311yescharacteristic
45C2×C7×SD16—24no—
46C2×C7×SD16—42no—
47C2×C7×SD16—42no—
48C22×C7×C8—21no—
49C22×C7×C8—21no—
50C7×C22⋊Q8—21no—
51C7×C22⋊Q8—21no—
52C7×C22⋊Q8C2×C412yes—
53C7×C22⋊Q8D412yes—
54C2×C7×C4⋊C4—21no—
55C2×C7×C4⋊C4—21no—
56C2×C7×C4⋊C4—21no—
57C2×C7×C4⋊C4D411yescharacteristic
58C2×C7×C4⋊C4D411yescharacteristic
59C2×C7×C4⋊C4D411yescharacteristic
60C2×C7×C22⋊C4—21no—
61C2×C7×C22⋊C4—21no—
62C2×C7×C22⋊C4—42no—
63C2×C7×C22⋊C4—42no—
64C2×C7×C22⋊C4—42no—
65C2×C7×C22⋊C4—42no—
66C2×C7×C22⋊C4—42no—
67C2×C7×C22⋊C4—42no—
68C2×C42×C7—21no—
69C2×C42×C7—21no—
70C7×#102—24no—
71C7×#102—42no—
72C7×#102—42no—
73C7×C22⋊C8—21no—
74C7×C22⋊C8—21no—
75C7×C22⋊C8—21no—
76C7×C22⋊C8C2×C412yes—
77C7×#95—21no—
78C7×#95—21no—
79C7×#95—21no—
80C7×#95—21no—
81C7×#95—21no—
82C7×#95—21no—
83C7×#95—21no—
84C7×#95—21no—
85C7×#95—21no—
86C2×#450C2×C711yescharacteristic
87#2329C2×C711yescharacteristic, complemented
88#1786C2×C711yescharacteristic
89C2×#328C2×C711yescharacteristic, complemented
90#1438C2×C711yescharacteristic
91#1294C2×C711yescharacteristic, complemented
92#1280C2×C711yescharacteristic, complemented
93C24×C7—22no—
94C24×C7—22no—
95C24×C7—22no—
96C24×C7—41no—
97C24×C7—41no—
98C24×C7—41no—
99C24×C7—41no—
100C24×C7—41no—
101C24×C7—41no—
102C24×C7—41no—
103C24×C7—41no—
104C24×C7—41no—
105C24×C7—41no—
106C24×C7—41no—
107C24×C7—41no—
108C24×C7C2×D411yescharacteristic
109C2×C7×Q8—21no—
110C2×C7×Q8—41no—
111C2×C7×Q8—41no—
112C2×C7×Q8C22⋊C412yescomplemented
113C2×C7×D4—22no—
114C2×C7×D4—22no—
115C2×C7×D4—22no—
116C2×C7×D4—22no—
117C2×C7×D4—22no—
118C2×C7×D4—22no—
119C2×C7×D4—41no—
120C2×C7×D4—41no—
121C2×C7×D4—41no—
122C2×C7×D4—41no—
123C2×C7×D4—44no—
124C2×C7×D4—44no—
125C2×C7×D4—44no—
126C2×C7×D4—44no—
127C2×C7×D4—41no—
128C2×C7×D4—41no—
129C22×C4×C7—21no—
130C22×C4×C7—21no—
131C22×C4×C7—21no—
132C22×C4×C7—21no—
133C22×C4×C7—22no—
134C22×C4×C7—21no—
135C22×C4×C7—21no—
136C22×C4×C7—21no—
137C22×C4×C7—21no—
138C22×C4×C7—21no—
139C22×C4×C7—41no—
140C22×C4×C7—41no—
141C22×C4×C7—41no—
142C22×C4×C7—41no—
143C22×C4×C7—41no—
144C22×C4×C7—41no—
145C22×C4×C7—41no—
146C22×C4×C7—41no—
147C22×C4×C7—41no—
148C22×C4×C7—41no—
149C22×C4×C7—41no—
150C22×C4×C7—41no—
151C22×C4×C7SD1612yescomplemented
152C22×C4×C7C22×C411yescharacteristic
153C22×C4×C7C2×D411yescharacteristic
154C22×C4×C7C2×D411yescharacteristic
155C22×C4×C7C2×D411yescharacteristic
156C22×C4×C7#4111yescharacteristic
157C22×C4×C7#4111yescharacteristic
158C7×SD16—48no—
159C2×C7×C8—22no—
160C2×C7×C8—41no—
161C2×C7×C8—41no—
162C2×C7×C8—41no—
163C2×C7×C8—41no—
164C2×C7×C8—41no—
165C7×C4⋊C4—22no—
166C7×C4⋊C4—22no—
167C7×C4⋊C4—21no—
168C7×C4⋊C4—41no—
169C7×C4⋊C4—41no—
170C7×C4⋊C4—81no—
171C7×C4⋊C4—81no—
172C7×C4⋊C4C22⋊C412yescomplemented
173C7×C22⋊C4—22no—
174C7×C22⋊C4—22no—
175C7×C22⋊C4—48no—
176C7×C22⋊C4—48no—
177C7×C22⋊C4—41no—
178C7×C22⋊C4—41no—
179C42×C7—81no—
180C42×C7—81no—
181C23×D4C22×C711yescharacteristic
182C22×SD16—21no—
183C2×C22⋊Q8C22×C711yescharacteristic
184C22×C22⋊C4—21no—
185C22×C22⋊C4—21no—
186#450—22no—
187#450C4×C714yescomplemented
188C2×#102—21no—
189C2×#102—21no—
190C2×#102—21no—
191C2×C22⋊C8C22×C711yescharacteristic
192C2×C22⋊C8C22×C711yescharacteristic
193#391—21no—
194#389—21no—
195#387—21no—
196#387—21no—
197#385—21no—
198#380C22×C711yescharacteristic
199C4×C22⋊C4—21no—
200C2×#95C22×C711yescharacteristic
201C2×#95C22×C711yescharacteristic
202#328—22no—
203C23×C7—21no—
204C23×C7—21no—
205C23×C7—21no—
206C23×C7—21no—
207C23×C7—24no—
208C23×C7—42no—
209C23×C7—42no—
210C23×C7—42no—
211C23×C7—42no—
212C23×C7—42no—
213C23×C7—42no—
214C23×C7—42no—
215C23×C7—42no—
216C23×C7—42no—
217C23×C7—42no—
218C23×C7—42no—
219C23×C7—42no—
220C23×C7—42no—
221C23×C7—42no—
222C23×C7—42no—
223C23×C7—42no—
224C23×C7—42no—
225C23×C7—42no—
226C23×C7—81no—
227C23×C7—81no—
228C23×C7—81no—
229C23×C7—81no—
230C23×C7—81no—
231C23×C7—81no—
232C23×C7—81no—
233C23×C7—81no—
234C23×C7—81no—
235C23×C7—81no—
236C23×C7—81no—
237C23×C7—81no—
238C23×C7—81no—
239C23×C7—81no—
240C23×C7—81no—
241C23×C7—81no—
242C23×C7C4×D411yescharacteristic
243C23×C7#12011yescharacteristic
244C23×C7#12111yescharacteristic
245C23×C7C2×SD1611yescharacteristic
246C23×C7#13511yescharacteristic
247C23×C7AGL(1,ℤ/8)11yescharacteristic
248C23×C7#13711yescharacteristic
249C7×Q8—42no—
250C7×Q8—81no—
251C7×D4—24no—
252C7×D4—24no—
253C7×D4—42no—
254C7×D4—42no—
255C7×D4—42no—
256C7×D4—42no—
257C7×D4—42no—
258C7×D4—42no—
259C2×C4×C7—21no—
260C2×C4×C7—21no—
261C2×C4×C7—21no—
262C2×C4×C7—21no—
263C2×C4×C7—22no—
264C2×C4×C7—22no—
265C2×C4×C7—22no—
266C2×C4×C7—41no—
267C2×C4×C7—41no—
268C2×C4×C7—41no—
269C2×C4×C7—41no—
270C2×C4×C7—41no—
271C2×C4×C7—41no—
272C2×C4×C7—41no—
273C2×C4×C7—41no—
274C2×C4×C7—41no—
275C2×C4×C7—41no—
276C2×C4×C7—42no—
277C2×C4×C7—42no—
278C2×C4×C7—42no—
279C2×C4×C7—42no—
280C2×C4×C7—42no—
281C2×C4×C7—41no—
282C2×C4×C7—41no—
283C2×C4×C7—41no—
284C2×C4×C7—41no—
285C2×C4×C7—41no—
286C2×C4×C7—41no—
287C2×C4×C7—41no—
288C2×C4×C7—41no—
289C2×C4×C7—41no—
290C2×C4×C7—81no—
291C2×C4×C7—81no—
292C2×C4×C7—81no—
293C2×C4×C7—81no—
294C2×C4×C7—81no—
295C2×C4×C7—81no—
296C2×C4×C7C2×C22⋊C411yescharacteristic
297C2×C4×C7C4×D411yescharacteristic
298C2×C4×C7#12111yescharacteristic
299C2×C4×C7#12311yescharacteristic
300C7×C8—42no—
301C7×C8—81no—
302C25—21no—
303C22×Q8C7×D411yescharacteristic
304C22×D4—21no—
305C22×D4—21no—
306C22×D4—21no—
307C22×D4—21no—
308C22×D4—21no—
309C22×D4—42no—
310C22×D4—42no—
311C22×D4C2×C4×C712yes—
312C23×C4—21no—
313C23×C4C7×D411yescharacteristic
314C23×C4C7×D411yescharacteristic
315C23×C4C23×C711yescharacteristic
316C2×SD16—24no—
317C2×SD16—42no—
318C2×SD16—42no—
319C22×C8—21no—
320C22×C8—21no—
321C22⋊Q8—21no—
322C22⋊Q8—21no—
323C22⋊Q8C2×C4×C712yes—
324C22⋊Q8C7×D412yes—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4C7×D411yescharacteristic
329C2×C4⋊C4C7×D411yescharacteristic
330C2×C4⋊C4C7×D411yescharacteristic
331C2×C22⋊C4—21no—
332C2×C22⋊C4—21no—
333C2×C22⋊C4—42no—
334C2×C22⋊C4—42no—
335C2×C22⋊C4—42no—
336C2×C22⋊C4—42no—
337C2×C22⋊C4—42no—
338C2×C22⋊C4—42no—
339C2×C42—21no—
340C2×C42—21no—
341#102—24no—
342#102—42no—
343#102—42no—
344C22⋊C8—21no—
345C22⋊C8—21no—
346C22⋊C8—21no—
347C22⋊C8C2×C4×C712yes—
348#95—21no—
349#95—21no—
350#95—21no—
351#95—21no—
352#95—21no—
353#95—21no—
354#95—21no—
355#95—21no—
356#95—21no—
357C22×C7—21no—
358C22×C7—21no—
359C22×C7—21no—
360C22×C7—21no—
361C22×C7—21no—
362C22×C7—21no—
363C22×C7—21no—
364C22×C7—21no—
365C22×C7—21no—
366C22×C7—21no—
367C22×C7—21no—
368C22×C7—21no—
369C22×C7—44no—
370C22×C7—44no—
371C22×C7—44no—
372C22×C7—82no—
373C22×C7—82no—
374C22×C7—82no—
375C22×C7—82no—
376C22×C7—82no—
377C22×C7—82no—
378C22×C7—82no—
379C22×C7—82no—
380C22×C7—82no—
381C22×C7—82no—
382C22×C7—82no—
383C22×C7—82no—
384C22×C7#38611yescharacteristic
385C22×C7C4×SD1611yescharacteristic
386C22×C7#44011yescharacteristic
387C22×C7#44811yescharacteristic
388C22×C7#44911yescharacteristic
389C22×C7#45311yescharacteristic
390C22×C7#45411yescharacteristic
391C22×C7B(4,𝔽2)11yescharacteristic
392C22×C7B(4,𝔽2)11yescharacteristic
393C22×C7#46011yescharacteristic
394C22×C7#46411yescharacteristic
395C4×C7—21no—
396C4×C7—21no—
397C4×C7—21no—
398C4×C7—21no—
399C4×C7—42no—
400C4×C7—42no—
401C4×C7—42no—
402C4×C7—82no—
403C4×C7—81no—
404C4×C7—81no—
405C4×C7—81no—
406C24—22no—
407C24—22no—
408C24—22no—
409C24—41no—
410C24—41no—
411C24—41no—
412C24—41no—
413C24—41no—
414C24—41no—
415C24—41no—
416C24—41no—
417C24—41no—
418C24—41no—
419C24—41no—
420C24—41no—
421C24C2×C7×D411yescharacteristic
422C2×Q8—21no—
423C2×Q8—41no—
424C2×Q8—41no—
425C2×Q8C7×C22⋊C412yescomplemented
426C2×D4—22no—
427C2×D4—22no—
428C2×D4—22no—
429C2×D4—22no—
430C2×D4—22no—
431C2×D4—22no—
432C2×D4—41no—
433C2×D4—41no—
434C2×D4—41no—
435C2×D4—41no—
436C2×D4—44no—
437C2×D4—44no—
438C2×D4—44no—
439C2×D4—44no—
440C2×D4—41no—
441C2×D4—41no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—22no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—41no—
453C22×C4—41no—
454C22×C4—41no—
455C22×C4—41no—
456C22×C4—41no—
457C22×C4—41no—
458C22×C4—41no—
459C22×C4—41no—
460C22×C4—41no—
461C22×C4—41no—
462C22×C4—41no—
463C22×C4—41no—
464C22×C4C7×SD1612yescomplemented
465C22×C4C22×C4×C711yescharacteristic
466C22×C4C2×C7×D411yescharacteristic
467C22×C4C2×C7×D411yescharacteristic
468C22×C4C2×C7×D411yescharacteristic
469C22×C4C7×#4111yescharacteristic
470C22×C4C7×#4111yescharacteristic
471SD16—48no—
472C2×C8—22no—
473C2×C8—41no—
474C2×C8—41no—
475C2×C8—41no—
476C2×C8—41no—
477C2×C8—41no—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—21no—
481C4⋊C4—41no—
482C4⋊C4—41no—
483C4⋊C4—81no—
484C4⋊C4—81no—
485C4⋊C4C7×C22⋊C412yescomplemented
486C22⋊C4—22no—
487C22⋊C4—22no—
488C22⋊C4—48no—
489C22⋊C4—48no—
490C22⋊C4—41no—
491C22⋊C4—41no—
492C42—81no—
493C42—81no—
494C2×C7—21no—
495C2×C7—21no—
496C2×C7—21no—
497C2×C7—21no—
498C2×C7—84no—
499C2×C7#159611yescharacteristic
500C2×C7#159711yescharacteristic
501C2×C7#159911yescharacteristic
502C2×C7#160011yescharacteristic
503C2×C7#188011yescharacteristic
504C2×C7#188311yescharacteristic
505C2×C7#188911yescharacteristic
506C23—21no—
507C23—21no—
508C23—21no—
509C23—21no—
510C23—24no—
511C23—42no—
512C23—42no—
513C23—42no—
514C23—42no—
515C23—42no—
516C23—42no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23—42no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—81no—
530C23—81no—
531C23—81no—
532C23—81no—
533C23—81no—
534C23—81no—
535C23—81no—
536C23—81no—
537C23—81no—
538C23—81no—
539C23—81no—
540C23—81no—
541C23—81no—
542C23—81no—
543C23—81no—
544C23—81no—
545C23C4×C7×D411yescharacteristic
546C23C7×#12011yescharacteristic
547C23C7×#12111yescharacteristic
548C23C2×C7×SD1611yescharacteristic
549C23C7×#13511yescharacteristic
550C23C7×AGL(1,ℤ/8)11yescharacteristic
551C23C7×#13711yescharacteristic
552Q8—42no—
553Q8—81no—
554D4—24no—
555D4—24no—
556D4—42no—
557D4—42no—
558D4—42no—
559D4—42no—
560D4—42no—
561D4—42no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—81no—
594C2×C4—81no—
595C2×C4—81no—
596C2×C4—81no—
597C2×C4—81no—
598C2×C4—81no—
599C2×C4C2×C7×C22⋊C411yescharacteristic
600C2×C4C4×C7×D411yescharacteristic
601C2×C4C7×#12111yescharacteristic
602C2×C4C7×#12311yescharacteristic
603C8—42no—
604C8—81no—
605C7#904211yescharacteristic, direct product factor, Sylow
606C22—21no—
607C22—21no—
608C22—21no—
609C22—21no—
610C22—21no—
611C22—21no—
612C22—21no—
613C22—21no—
614C22—21no—
615C22—21no—
616C22—21no—
617C22—21no—
618C22—44no—
619C22—44no—
620C22—44no—
621C22—82no—
622C22—82no—
623C22—82no—
624C22—82no—
625C22—82no—
626C22—82no—
627C22—82no—
628C22—82no—
629C22—82no—
630C22—82no—
631C22—82no—
632C22—82no—
633C22C7×#38611yescharacteristic
634C22C4×C7×SD1611yescharacteristic
635C22C7×#44011yescharacteristic
636C22C7×#44811yescharacteristic
637C22C7×#44911yescharacteristic
638C22C7×#45311yescharacteristic
639C22C7×#45411yescharacteristic
640C22C7×B(4,𝔽2)11yescharacteristic
641C22C7×B(4,𝔽2)11yescharacteristic
642C22C7×#46011yescharacteristic
643C22C7×#46411yescharacteristic
644C4—21no—
645C4—21no—
646C4—21no—
647C4—21no—
648C4—42no—
649C4—42no—
650C4—42no—
651C4—82no—
652C4—81no—
653C4—81no—
654C4—81no—
655C2—21no—
656C2—21no—
657C2—21no—
658C2—21no—
659C2—84no—
660C2C7×#159611yescharacteristic
661C2C7×#159711yescharacteristic
662C2C7×#159911yescharacteristic
663C2C7×#160011yescharacteristic
664C2C7×#188011yescharacteristic
665C2C7×#188311yescharacteristic
666C2C7×#188911yescharacteristic
subgroupG/H
center, socleC23×C7#120
commutatorC22×C4C22×C4×C7
Frattini, cosocleC23×C4C23×C7
prime radicalC23×C7×D4C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23×C7 ◁ C23×C4×C7 ◁ G

Complements

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

NH ≅ G/Ntype
2. C7×#2329C2semidirect
4. C2×C7×#328C2semidirect
6. C7×#1294C2semidirect
7. C7×#1280C2semidirect
14. C7×#450C4semidirect
30. #9042C7direct (N × H)
87. #2329C2×C7semidirect
89. C2×#328C2×C7semidirect
91. #1294C2×C7semidirect
92. #1280C2×C7semidirect
112. C2×C7×Q8C22⋊C4semidirect
151. C22×C4×C7SD16semidirect
172. C7×C4⋊C4C22⋊C4semidirect
187. #450C4×C7semidirect
425. C2×Q8C7×C22⋊C4semidirect
464. C22×C4C7×SD16semidirect
485. C4⋊C4C7×C22⋊C4semidirect
605. C7#9042direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C7×#4502
C7×#23292
C7×#17862
C2×C7×#3282
C7×#14382
C7×#12942
C7×#12802
#90427