Finite Group Databasebeta

← C7×#9041 #13186257

Subgroups — C7×#9041 #13186257

Order 1792.

1050 conjugacy classes of subgroups.

The subgroup lattice has 720 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#447C211yesmaximal, characteristic
2C7×#2325C211yesmaximal, characteristic, complemented
3C7×#1786C211yesmaximal, characteristic, complemented
4C2×C7×#327C211yesmaximal, characteristic, complemented
5C7×#1438C211yesmaximal, characteristic, complemented
6C7×#1294C211yesmaximal, characteristic, complemented
7C7×#1280C211yesmaximal, characteristic, complemented
8C23×C7×D4C2211yescharacteristic
9C22×C7×D8—21no—
10C2×C7×#121C2211yescharacteristic
11C2×C4×C7×D4—21no—
12C22×C7×C22⋊C4—21no—
13C22×C7×C22⋊C4—21no—
14C7×#447—22no—
15C7×#447C414yescomplemented
16C2×C7×#102—21no—
17C2×C7×#102—21no—
18C2×C7×#102—21no—
19C2×C7×C22⋊C8C2211yescharacteristic
20C2×C7×C22⋊C8C2211yescharacteristic
21C7×#388—21no—
22C7×#387—21no—
23C7×#386—21no—
24C7×#386—21no—
25C7×#385—21no—
26C7×#380C2211yescharacteristic
27C2×C7×#95C2211yescharacteristic
28C2×C7×#95C2211yescharacteristic
29C7×#327—22no—
30#9041C711yesmaximal, characteristic, direct product factor, Sylow
31C25×C7—21no—
32C22×C7×D4—21no—
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×C412yescomplemented
41C22×C7×D4D411yescharacteristic
42C23×C4×C7—21no—
43C23×C4×C7—21no—
44C23×C4×C7D411yescharacteristic
45C23×C4×C7D411yescharacteristic
46C23×C4×C7C2311yescharacteristic
47C2×C7×D8—24no—
48C2×C7×D8—42no—
49C2×C7×D8—42no—
50C22×C7×C8—21no—
51C22×C7×C8—21no—
52C7×#121—21no—
53C7×#121—21no—
54C7×#121C2×C412yes—
55C7×#121D412yes—
56C4×C7×D4—42no—
57C4×C7×D4—42no—
58C2×C7×C4⋊C4—21no—
59C2×C7×C4⋊C4D411yescharacteristic
60C2×C7×C4⋊C4D411yescharacteristic
61C2×C7×C4⋊C4D411yescharacteristic
62C2×C7×C22⋊C4—21no—
63C2×C7×C22⋊C4—21no—
64C2×C7×C22⋊C4—21no—
65C2×C7×C22⋊C4—21no—
66C2×C7×C22⋊C4—21no—
67C2×C7×C22⋊C4—42no—
68C2×C7×C22⋊C4—42no—
69C2×C7×C22⋊C4—42no—
70C2×C7×C22⋊C4—42no—
71C2×C7×C22⋊C4—42no—
72C2×C7×C22⋊C4—42no—
73C2×C42×C7—21no—
74C7×#102—24no—
75C7×#102—42no—
76C7×#102—42no—
77C7×C22⋊C8—21no—
78C7×C22⋊C8—21no—
79C7×C22⋊C8—21no—
80C7×C22⋊C8C2×C412yescomplemented
81C7×#95—21no—
82C7×#95—21no—
83C7×#95—21no—
84C7×#95—21no—
85C7×#95—21no—
86C7×#95—21no—
87C7×#95—21no—
88C2×#447C2×C711yescharacteristic
89#2325C2×C711yescharacteristic, complemented
90#1786C2×C711yescharacteristic, complemented
91C2×#327C2×C711yescharacteristic, complemented
92#1438C2×C711yescharacteristic, complemented
93#1294C2×C711yescharacteristic, complemented
94#1280C2×C711yescharacteristic, complemented
95C24×C7—22no—
96C24×C7—22no—
97C24×C7—22no—
98C24×C7—21no—
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×C7—41no—
109C24×C7—41no—
110C24×C7—41no—
111C24×C7C2×D411yescharacteristic
112C2×C7×D4—22no—
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—22no—
120C2×C7×D4—21no—
121C2×C7×D4—41no—
122C2×C7×D4—41no—
123C2×C7×D4—41no—
124C2×C7×D4—41no—
125C2×C7×D4—41no—
126C2×C7×D4—41no—
127C2×C7×D4—44no—
128C2×C7×D4—44no—
129C2×C7×D4—44no—
130C2×C7×D4—44no—
131C2×C7×D4—41no—
132C2×C7×D4—41no—
133C2×C7×D4—41no—
134C2×C7×D4—41no—
135C2×C7×D4—41no—
136C2×C7×D4—41no—
137C2×C7×D4C22⋊C412yescomplemented
138C22×C4×C7—21no—
139C22×C4×C7—21no—
140C22×C4×C7—21no—
141C22×C4×C7—22no—
142C22×C4×C7—21no—
143C22×C4×C7—21no—
144C22×C4×C7—21no—
145C22×C4×C7—21no—
146C22×C4×C7—21no—
147C22×C4×C7—41no—
148C22×C4×C7—41no—
149C22×C4×C7—42no—
150C22×C4×C7—41no—
151C22×C4×C7—41no—
152C22×C4×C7—41no—
153C22×C4×C7—41no—
154C22×C4×C7—41no—
155C22×C4×C7—41no—
156C22×C4×C7—41no—
157C22×C4×C7—41no—
158C22×C4×C7—41no—
159C22×C4×C7—41no—
160C22×C4×C7—81no—
161C22×C4×C7—81no—
162C22×C4×C7D812yescomplemented
163C22×C4×C7C22×C411yescharacteristic
164C22×C4×C7C2×D411yescharacteristic
165C22×C4×C7C2×D411yescharacteristic
166C22×C4×C7C2×D411yescharacteristic
167C22×C4×C7#4111yescharacteristic
168C22×C4×C7#4111yescharacteristic
169C7×D8—48no—
170C2×C7×C8—22no—
171C2×C7×C8—41no—
172C2×C7×C8—41no—
173C2×C7×C8—41no—
174C2×C7×C8—41no—
175C2×C7×C8—41no—
176C7×C4⋊C4—21no—
177C7×C4⋊C4—41no—
178C7×C4⋊C4—41no—
179C7×C4⋊C4—42no—
180C7×C4⋊C4C22⋊C412yescomplemented
181C7×C22⋊C4—22no—
182C7×C22⋊C4—22no—
183C7×C22⋊C4—42no—
184C7×C22⋊C4—48no—
185C7×C22⋊C4—48no—
186C7×C22⋊C4—41no—
187C7×C22⋊C4—41no—
188C7×C22⋊C4—81no—
189C7×C22⋊C4—81no—
190C42×C7—42no—
191C23×D4C22×C711yescharacteristic
192C22×D8—21no—
193C2×#121C22×C711yescharacteristic
194C2×C4×D4—21no—
195C22×C22⋊C4—21no—
196C22×C22⋊C4—21no—
197#447—22no—
198#447C4×C714yescomplemented
199C2×#102—21no—
200C2×#102—21no—
201C2×#102—21no—
202C2×C22⋊C8C22×C711yescharacteristic
203C2×C22⋊C8C22×C711yescharacteristic
204#388—21no—
205#387—21no—
206#386—21no—
207#386—21no—
208#385—21no—
209#380C22×C711yescharacteristic
210C2×#95C22×C711yescharacteristic
211C2×#95C22×C711yescharacteristic
212#327—22no—
213C23×C7—21no—
214C23×C7—21no—
215C23×C7—21no—
216C23×C7—21no—
217C23×C7—24no—
218C23×C7—22no—
219C23×C7—42no—
220C23×C7—42no—
221C23×C7—42no—
222C23×C7—42no—
223C23×C7—42no—
224C23×C7—42no—
225C23×C7—42no—
226C23×C7—42no—
227C23×C7—42no—
228C23×C7—42no—
229C23×C7—42no—
230C23×C7—42no—
231C23×C7—42no—
232C23×C7—42no—
233C23×C7—42no—
234C23×C7—42no—
235C23×C7—41no—
236C23×C7—41no—
237C23×C7—41no—
238C23×C7—41no—
239C23×C7—41no—
240C23×C7—41no—
241C23×C7—42no—
242C23×C7—42no—
243C23×C7—81no—
244C23×C7—81no—
245C23×C7—81no—
246C23×C7—81no—
247C23×C7—81no—
248C23×C7—81no—
249C23×C7—81no—
250C23×C7—81no—
251C23×C7—81no—
252C23×C7—81no—
253C23×C7—81no—
254C23×C7—81no—
255C23×C7—81no—
256C23×C7—81no—
257C23×C7—81no—
258C23×C7—81no—
259C23×C7C4×D411yescharacteristic
260C23×C7#12011yescharacteristic
261C23×C7#12111yescharacteristic
262C23×C7C2×D811yescharacteristic
263C23×C7#13511yescharacteristic
264C23×C7AGL(1,ℤ/8)11yescharacteristic
265C23×C7AGL(1,ℤ/8)11yescharacteristic
266C7×D4—24no—
267C7×D4—24no—
268C7×D4—42no—
269C7×D4—42no—
270C7×D4—42no—
271C7×D4—42no—
272C7×D4—42no—
273C7×D4—42no—
274C7×D4—42no—
275C7×D4—42no—
276C7×D4—42no—
277C7×D4—42no—
278C7×D4—42no—
279C7×D4—81no—
280C2×C4×C7—21no—
281C2×C4×C7—21no—
282C2×C4×C7—21no—
283C2×C4×C7—21no—
284C2×C4×C7—22no—
285C2×C4×C7—22no—
286C2×C4×C7—41no—
287C2×C4×C7—41no—
288C2×C4×C7—41no—
289C2×C4×C7—41no—
290C2×C4×C7—41no—
291C2×C4×C7—41no—
292C2×C4×C7—41no—
293C2×C4×C7—41no—
294C2×C4×C7—42no—
295C2×C4×C7—42no—
296C2×C4×C7—42no—
297C2×C4×C7—42no—
298C2×C4×C7—42no—
299C2×C4×C7—41no—
300C2×C4×C7—41no—
301C2×C4×C7—41no—
302C2×C4×C7—41no—
303C2×C4×C7—41no—
304C2×C4×C7—41no—
305C2×C4×C7—41no—
306C2×C4×C7—41no—
307C2×C4×C7—41no—
308C2×C4×C7—81no—
309C2×C4×C7—84no—
310C2×C4×C7—81no—
311C2×C4×C7—81no—
312C2×C4×C7—81no—
313C2×C4×C7—81no—
314C2×C4×C7—81no—
315C2×C4×C7C2×C22⋊C411yescharacteristic
316C2×C4×C7C4×D411yescharacteristic
317C2×C4×C7#12111yescharacteristic
318C2×C4×C7#12311yescharacteristic
319C7×C8—42no—
320C7×C8—81no—
321C25—21no—
322C22×D4—21no—
323C22×D4—21no—
324C22×D4—21no—
325C22×D4—21no—
326C22×D4—21no—
327C22×D4—21no—
328C22×D4—42no—
329C22×D4—42no—
330C22×D4C2×C4×C712yescomplemented
331C22×D4C7×D411yescharacteristic
332C23×C4—21no—
333C23×C4—21no—
334C23×C4C7×D411yescharacteristic
335C23×C4C7×D411yescharacteristic
336C23×C4C23×C711yescharacteristic
337C2×D8—24no—
338C2×D8—42no—
339C2×D8—42no—
340C22×C8—21no—
341C22×C8—21no—
342#121—21no—
343#121—21no—
344#121C2×C4×C712yes—
345#121C7×D412yes—
346C4×D4—42no—
347C4×D4—42no—
348C2×C4⋊C4—21no—
349C2×C4⋊C4C7×D411yescharacteristic
350C2×C4⋊C4C7×D411yescharacteristic
351C2×C4⋊C4C7×D411yescharacteristic
352C2×C22⋊C4—21no—
353C2×C22⋊C4—21no—
354C2×C22⋊C4—21no—
355C2×C22⋊C4—21no—
356C2×C22⋊C4—21no—
357C2×C22⋊C4—42no—
358C2×C22⋊C4—42no—
359C2×C22⋊C4—42no—
360C2×C22⋊C4—42no—
361C2×C22⋊C4—42no—
362C2×C22⋊C4—42no—
363C2×C42—21no—
364#102—24no—
365#102—42no—
366#102—42no—
367C22⋊C8—21no—
368C22⋊C8—21no—
369C22⋊C8—21no—
370C22⋊C8C2×C4×C712yescomplemented
371#95—21no—
372#95—21no—
373#95—21no—
374#95—21no—
375#95—21no—
376#95—21no—
377#95—21no—
378C22×C7—21no—
379C22×C7—21no—
380C22×C7—21no—
381C22×C7—21no—
382C22×C7—21no—
383C22×C7—21no—
384C22×C7—21no—
385C22×C7—21no—
386C22×C7—21no—
387C22×C7—21no—
388C22×C7—21no—
389C22×C7—21no—
390C22×C7—44no—
391C22×C7—44no—
392C22×C7—42no—
393C22×C7—42no—
394C22×C7—44no—
395C22×C7—42no—
396C22×C7—82no—
397C22×C7—82no—
398C22×C7—82no—
399C22×C7—82no—
400C22×C7—82no—
401C22×C7—82no—
402C22×C7—81no—
403C22×C7—82no—
404C22×C7—81no—
405C22×C7—82no—
406C22×C7—81no—
407C22×C7—82no—
408C22×C7—81no—
409C22×C7—82no—
410C22×C7—82no—
411C22×C7—82no—
412C22×C7#38611yescharacteristic
413C22×C7C4×D811yescharacteristic
414C22×C7#44211yescharacteristic
415C22×C7#44711yescharacteristic
416C22×C7#44911yescharacteristic
417C22×C7#45311yescharacteristic
418C22×C7#45411yescharacteristic
419C22×C7B(4,𝔽2)11yescharacteristic
420C22×C7B(4,𝔽2)11yescharacteristic
421C22×C7#45911yescharacteristic
422C22×C7#46311yescharacteristic
423C4×C7—21no—
424C4×C7—21no—
425C4×C7—21no—
426C4×C7—21no—
427C4×C7—42no—
428C4×C7—42no—
429C4×C7—82no—
430C4×C7—81no—
431C4×C7—81no—
432C4×C7—81no—
433C24—22no—
434C24—22no—
435C24—22no—
436C24—21no—
437C24—41no—
438C24—41no—
439C24—41no—
440C24—41no—
441C24—41no—
442C24—41no—
443C24—41no—
444C24—41no—
445C24—41no—
446C24—41no—
447C24—41no—
448C24—41no—
449C24C2×C7×D411yescharacteristic
450C2×D4—22no—
451C2×D4—22no—
452C2×D4—22no—
453C2×D4—22no—
454C2×D4—22no—
455C2×D4—22no—
456C2×D4—22no—
457C2×D4—22no—
458C2×D4—21no—
459C2×D4—41no—
460C2×D4—41no—
461C2×D4—41no—
462C2×D4—41no—
463C2×D4—41no—
464C2×D4—41no—
465C2×D4—44no—
466C2×D4—44no—
467C2×D4—44no—
468C2×D4—44no—
469C2×D4—41no—
470C2×D4—41no—
471C2×D4—41no—
472C2×D4—41no—
473C2×D4—41no—
474C2×D4—41no—
475C2×D4C7×C22⋊C412yescomplemented
476C22×C4—21no—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—22no—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—21no—
484C22×C4—21no—
485C22×C4—41no—
486C22×C4—41no—
487C22×C4—42no—
488C22×C4—41no—
489C22×C4—41no—
490C22×C4—41no—
491C22×C4—41no—
492C22×C4—41no—
493C22×C4—41no—
494C22×C4—41no—
495C22×C4—41no—
496C22×C4—41no—
497C22×C4—41no—
498C22×C4—81no—
499C22×C4—81no—
500C22×C4C7×D812yescomplemented
501C22×C4C22×C4×C711yescharacteristic
502C22×C4C2×C7×D411yescharacteristic
503C22×C4C2×C7×D411yescharacteristic
504C22×C4C2×C7×D411yescharacteristic
505C22×C4C7×#4111yescharacteristic
506C22×C4C7×#4111yescharacteristic
507D8—48no—
508C2×C8—22no—
509C2×C8—41no—
510C2×C8—41no—
511C2×C8—41no—
512C2×C8—41no—
513C2×C8—41no—
514C4⋊C4—21no—
515C4⋊C4—41no—
516C4⋊C4—41no—
517C4⋊C4—42no—
518C4⋊C4C7×C22⋊C412yescomplemented
519C22⋊C4—22no—
520C22⋊C4—22no—
521C22⋊C4—42no—
522C22⋊C4—48no—
523C22⋊C4—48no—
524C22⋊C4—41no—
525C22⋊C4—41no—
526C22⋊C4—81no—
527C22⋊C4—81no—
528C42—42no—
529C2×C7—21no—
530C2×C7—21no—
531C2×C7—21no—
532C2×C7—21no—
533C2×C7—82no—
534C2×C7—84no—
535C2×C7#159511yescharacteristic
536C2×C7#159711yescharacteristic
537C2×C7#159811yescharacteristic
538C2×C7#160011yescharacteristic
539C2×C7#187911yescharacteristic
540C2×C7#188311yescharacteristic
541C2×C7#188911yescharacteristic
542C23—21no—
543C23—21no—
544C23—21no—
545C23—21no—
546C23—24no—
547C23—22no—
548C23—42no—
549C23—42no—
550C23—42no—
551C23—42no—
552C23—42no—
553C23—42no—
554C23—42no—
555C23—42no—
556C23—42no—
557C23—42no—
558C23—42no—
559C23—42no—
560C23—42no—
561C23—42no—
562C23—42no—
563C23—42no—
564C23—41no—
565C23—41no—
566C23—41no—
567C23—41no—
568C23—41no—
569C23—41no—
570C23—42no—
571C23—42no—
572C23—81no—
573C23—81no—
574C23—81no—
575C23—81no—
576C23—81no—
577C23—81no—
578C23—81no—
579C23—81no—
580C23—81no—
581C23—81no—
582C23—81no—
583C23—81no—
584C23—81no—
585C23—81no—
586C23—81no—
587C23—81no—
588C23C4×C7×D411yescharacteristic
589C23C7×#12011yescharacteristic
590C23C7×#12111yescharacteristic
591C23C2×C7×D811yescharacteristic
592C23C7×#13511yescharacteristic
593C23C7×AGL(1,ℤ/8)11yescharacteristic
594C23C7×AGL(1,ℤ/8)11yescharacteristic
595D4—24no—
596D4—24no—
597D4—42no—
598D4—42no—
599D4—42no—
600D4—42no—
601D4—42no—
602D4—42no—
603D4—42no—
604D4—42no—
605D4—42no—
606D4—42no—
607D4—42no—
608D4—81no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—21no—
612C2×C4—21no—
613C2×C4—22no—
614C2×C4—22no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—41no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—41no—
629C2×C4—41no—
630C2×C4—41no—
631C2×C4—41no—
632C2×C4—41no—
633C2×C4—41no—
634C2×C4—41no—
635C2×C4—41no—
636C2×C4—41no—
637C2×C4—81no—
638C2×C4—84no—
639C2×C4—81no—
640C2×C4—81no—
641C2×C4—81no—
642C2×C4—81no—
643C2×C4—81no—
644C2×C4C2×C7×C22⋊C411yescharacteristic
645C2×C4C4×C7×D411yescharacteristic
646C2×C4C7×#12111yescharacteristic
647C2×C4C7×#12311yescharacteristic
648C8—42no—
649C8—81no—
650C7#904111yescharacteristic, direct product factor, Sylow
651C22—21no—
652C22—21no—
653C22—21no—
654C22—21no—
655C22—21no—
656C22—21no—
657C22—21no—
658C22—21no—
659C22—21no—
660C22—21no—
661C22—21no—
662C22—21no—
663C22—44no—
664C22—44no—
665C22—42no—
666C22—42no—
667C22—44no—
668C22—42no—
669C22—82no—
670C22—82no—
671C22—82no—
672C22—82no—
673C22—82no—
674C22—82no—
675C22—81no—
676C22—82no—
677C22—82no—
678C22—81no—
679C22—82no—
680C22—81no—
681C22—82no—
682C22—81no—
683C22—82no—
684C22—82no—
685C22C7×#38611yescharacteristic
686C22C4×C7×D811yescharacteristic
687C22C7×#44211yescharacteristic
688C22C7×#44711yescharacteristic
689C22C7×#44911yescharacteristic
690C22C7×#45311yescharacteristic
691C22C7×#45411yescharacteristic
692C22C7×B(4,𝔽2)11yescharacteristic
693C22C7×B(4,𝔽2)11yescharacteristic
694C22C7×#45911yescharacteristic
695C22C7×#46311yescharacteristic
696C4—21no—
697C4—21no—
698C4—21no—
699C4—21no—
700C4—42no—
701C4—42no—
702C4—82no—
703C4—81no—
704C4—81no—
705C4—81no—
706C2—21no—
707C2—21no—
708C2—21no—
709C2—21no—
710C2—84no—
711C2—82no—
712C2C7×#159511yescharacteristic
713C2C7×#159711yescharacteristic
714C2C7×#159811yescharacteristic
715C2C7×#160011yescharacteristic
716C2C7×#187911yescharacteristic
717C2C7×#188311yescharacteristic
718C2C7×#188911yescharacteristic
subgroupG/H
center, socleC23×C7#120
commutatorC22×C4C22×C4×C7
Frattini, cosocleC23×C4C23×C7
prime radicalC2×C7×#447C2
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×#2325C2semidirect
3. C7×#1786C2semidirect
4. C2×C7×#327C2semidirect
5. C7×#1438C2semidirect
6. C7×#1294C2semidirect
7. C7×#1280C2semidirect
15. C7×#447C4semidirect
30. #9041C7direct (N × H)
40. C22×C7×D4C2×C4semidirect
80. C7×C22⋊C8C2×C4semidirect
89. #2325C2×C7semidirect
90. #1786C2×C7semidirect
91. C2×#327C2×C7semidirect
92. #1438C2×C7semidirect
93. #1294C2×C7semidirect
94. #1280C2×C7semidirect
137. C2×C7×D4C22⋊C4semidirect
162. C22×C4×C7D8semidirect
180. C7×C4⋊C4C22⋊C4semidirect
198. #447C4×C7semidirect
330. C22×D4C2×C4×C7semidirect
370. C22⋊C8C2×C4×C7semidirect
475. C2×D4C7×C22⋊C4semidirect
500. C22×C4C7×D8semidirect
518. C4⋊C4C7×C22⋊C4semidirect
650. C7#9041direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C7×#4472
C7×#23252
C7×#17862
C2×C7×#3272
C7×#14382
C7×#12942
C7×#12802
#90417