Finite Group Databasebeta

← C7×#14571 #13191787

Subgroups — C7×#14571 #13191787

Order 1792.

1966 conjugacy classes of subgroups.

The subgroup lattice has 652 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#394C211yesmaximal, characteristic
2C2×C7×#388C212yesmaximal, complemented
3C2×C7×#386C212yesmaximal
4C7×#1445C211yesmaximal, characteristic, complemented
5C7×#1442C212yesmaximal, complemented
6C7×#1439C212yesmaximal, complemented
7C7×#1436C211yesmaximal, characteristic, complemented
8C7×C22⋊#95C212yesmaximal, complemented
9C7×C22⋊#95C212yesmaximal, complemented
10C23×C7×D4C2211yescharacteristic
11C24×C4×C7C2212yes—
12C22×C7×C4⋊C4C2212yes—
13C22×C7×C4⋊C4C2211yescharacteristic
14C22×C7×C22⋊C4C2212yes—
15C22×C7×C22⋊C4C2212yes—
16C22×C7×C22⋊C4C2212yes—
17C22×C7×C22⋊C4C2212yes—
18C22×C7×C22⋊C4C2212yes—
19C22×C42×C7C2211yescharacteristic
20C7×#394C418yescomplemented
21C7×#388—28no—
22C7×#386—28no—
23C2×C7×#95C2212yes—
24C2×C7×#95C2212yes—
25C2×C7×#95C2212yes—
26C2×C7×#95C2212yes—
27C2×C7×#95C2212yes—
28C2×C7×#95C2211yescharacteristic
29C2×C7×#95C2212yes—
30C2×C7×#95C2211yescharacteristic
31C2×C7×#95C2212yes—
32C2×C7×#95C2211yescharacteristic
33C2×C7×#95C2211yescharacteristic
34#14571C711yesmaximal, characteristic, direct product factor, Sylow
35C25×C7C2312yes—
36C22×C7×D4—24no—
37C22×C7×D4—24no—
38C22×C7×D4—41no—
39C22×C7×D4—41no—
40C22×C7×D4C2×C414yes—
41C23×C4×C7—24no—
42C23×C4×C7—24no—
43C23×C4×C7—24no—
44C23×C4×C7—42no—
45C23×C4×C7—42no—
46C23×C4×C7—42no—
47C23×C4×C7—42no—
48C23×C4×C7C2312yes—
49C23×C4×C7C2311yescharacteristic
50C23×C4×C7C2311yescharacteristic
51C23×C4×C7C2312yes—
52C23×C4×C7C2311yescharacteristic
53C23×C4×C7C2312yes—
54C23×C4×C7C2312yes—
55C23×C4×C7C2312yes—
56C2×C7×C4⋊C4—24no—
57C2×C7×C4⋊C4—41no—
58C2×C7×C4⋊C4—41no—
59C2×C7×C4⋊C4—42no—
60C2×C7×C4⋊C4—42no—
61C2×C7×C4⋊C4C2×C414yescomplemented
62C2×C7×C22⋊C4—24no—
63C2×C7×C22⋊C4—24no—
64C2×C7×C22⋊C4—24no—
65C2×C7×C22⋊C4—24no—
66C2×C7×C22⋊C4—24no—
67C2×C7×C22⋊C4—24no—
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×C7×C22⋊C4—42no—
74C2×C7×C22⋊C4C2×C418yescomplemented
75C2×C7×C22⋊C4C2×C418yescomplemented
76C2×C7×C22⋊C4D418yes—
77C2×C42×C7—24no—
78C2×C42×C7—41no—
79C7×#95—24no—
80C7×#95—24no—
81C7×#95—24no—
82C7×#95—24no—
83C7×#95—24no—
84C7×#95—42no—
85C7×#95—41no—
86C7×#95—42no—
87C7×#95—41no—
88C7×#95—41no—
89C7×#95C2×C414yescomplemented
90C2×#394C2×C711yescharacteristic
91C2×#388C2×C712yescomplemented
92C2×#386C2×C712yes—
93#1445C2×C711yescharacteristic, complemented
94#1442C2×C712yescomplemented
95#1439C2×C712yescomplemented
96#1436C2×C711yescharacteristic, complemented
97C22⋊#95C2×C712yescomplemented
98C22⋊#95C2×C712yescomplemented
99C24×C7—22no—
100C24×C7—22no—
101C24×C7—22no—
102C24×C7—22no—
103C24×C7—22no—
104C24×C7—22no—
105C24×C7—22no—
106C24×C7—22no—
107C24×C7—22no—
108C24×C7—22no—
109C24×C7—22no—
110C24×C7—22no—
111C24×C7C22×C414yes—
112C24×C7C2×D414yes—
113C24×C7#4114yes—
114C24×C7C2411yescharacteristic
115C2×C7×D4—216no—
116C2×C7×D4—44no—
117C2×C7×D4—44no—
118C2×C7×D4—48no—
119C2×C7×D4—44no—
120C22×C4×C7—22no—
121C22×C4×C7—22no—
122C22×C4×C7—22no—
123C22×C4×C7—22no—
124C22×C4×C7—22no—
125C22×C4×C7—22no—
126C22×C4×C7—22no—
127C22×C4×C7—22no—
128C22×C4×C7—22no—
129C22×C4×C7—22no—
130C22×C4×C7—22no—
131C22×C4×C7—22no—
132C22×C4×C7—22no—
133C22×C4×C7—22no—
134C22×C4×C7—21no—
135C22×C4×C7—21no—
136C22×C4×C7—21no—
137C22×C4×C7—22no—
138C22×C4×C7—22no—
139C22×C4×C7—22no—
140C22×C4×C7—22no—
141C22×C4×C7—22no—
142C22×C4×C7—22no—
143C22×C4×C7—22no—
144C22×C4×C7—21no—
145C22×C4×C7—22no—
146C22×C4×C7—22no—
147C22×C4×C7—22no—
148C22×C4×C7—22no—
149C22×C4×C7—22no—
150C22×C4×C7—22no—
151C22×C4×C7—22no—
152C22×C4×C7—21no—
153C22×C4×C7—21no—
154C22×C4×C7—22no—
155C22×C4×C7—22no—
156C22×C4×C7—22no—
157C22×C4×C7—22no—
158C22×C4×C7—22no—
159C22×C4×C7—22no—
160C22×C4×C7—22no—
161C22×C4×C7—22no—
162C22×C4×C7—22no—
163C22×C4×C7—48no—
164C22×C4×C7—48no—
165C22×C4×C7—48no—
166C22×C4×C7—48no—
167C22×C4×C7—48no—
168C22×C4×C7—48no—
169C22×C4×C7—48no—
170C22×C4×C7C22×C414yes—
171C22×C4×C7C22×C412yes—
172C22×C4×C7C22×C412yes—
173C22×C4×C7C22×C412yes—
174C22×C4×C7C2×D414yes—
175C22×C4×C7C2×D414yes—
176C22×C4×C7#4114yes—
177C7×C4⋊C4—44no—
178C7×C4⋊C4—48no—
179C7×C22⋊C4—216no—
180C7×C22⋊C4—48no—
181C7×C22⋊C4—48no—
182C7×C22⋊C4—48no—
183C7×C22⋊C4—48no—
184C42×C7—44no—
185C23×D4C22×C711yescharacteristic
186C24×C4C22×C712yes—
187C22×C4⋊C4C22×C712yes—
188C22×C4⋊C4C22×C711yescharacteristic
189C22×C22⋊C4C22×C712yes—
190C22×C22⋊C4C22×C712yes—
191C22×C22⋊C4C22×C712yes—
192C22×C22⋊C4C22×C712yes—
193C22×C22⋊C4C22×C712yes—
194C22×C42C22×C711yescharacteristic
195#394C4×C718yescomplemented
196#388—28no—
197#386—28no—
198C2×#95C22×C712yes—
199C2×#95C22×C712yes—
200C2×#95C22×C712yes—
201C2×#95C22×C712yes—
202C2×#95C22×C712yes—
203C2×#95C22×C711yescharacteristic
204C2×#95C22×C712yes—
205C2×#95C22×C711yescharacteristic
206C2×#95C22×C712yes—
207C2×#95C22×C711yescharacteristic
208C2×#95C22×C711yescharacteristic
209C23×C7—24no—
210C23×C7—24no—
211C23×C7—24no—
212C23×C7—24no—
213C23×C7—24no—
214C23×C7—24no—
215C23×C7—24no—
216C23×C7—24no—
217C23×C7—24no—
218C23×C7—24no—
219C23×C7—24no—
220C23×C7—24no—
221C23×C7—24no—
222C23×C7—24no—
223C23×C7—24no—
224C23×C7—24no—
225C23×C7—24no—
226C23×C7—24no—
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—42no—
236C23×C7—42no—
237C23×C7—42no—
238C23×C7—42no—
239C23×C7—42no—
240C23×C7—42no—
241C23×C7—42no—
242C23×C7—42no—
243C23×C7C4×D418yes—
244C23×C7C23×C411yescharacteristic
245C23×C7C22×D412yes—
246C23×C7C2×#4112yes—
247C23×C7C2×#4112yes—
248C23×C7H(5,𝔽2)12yes—
249C23×C7H(5,𝔽2)11yescharacteristic
250C23×C7H(5,𝔽2)11yescharacteristic
251C23×C7H(5,𝔽2)12yes—
252C23×C72−1+412yes—
253C7×D4—416no—
254C2×C4×C7—24no—
255C2×C4×C7—24no—
256C2×C4×C7—24no—
257C2×C4×C7—22no—
258C2×C4×C7—22no—
259C2×C4×C7—24no—
260C2×C4×C7—22no—
261C2×C4×C7—22no—
262C2×C4×C7—22no—
263C2×C4×C7—24no—
264C2×C4×C7—24no—
265C2×C4×C7—24no—
266C2×C4×C7—24no—
267C2×C4×C7—24no—
268C2×C4×C7—24no—
269C2×C4×C7—24no—
270C2×C4×C7—24no—
271C2×C4×C7—24no—
272C2×C4×C7—24no—
273C2×C4×C7—432no—
274C2×C4×C7—42no—
275C2×C4×C7—42no—
276C2×C4×C7—42no—
277C2×C4×C7—42no—
278C2×C4×C7—42no—
279C2×C4×C7—42no—
280C2×C4×C7—42no—
281C2×C4×C7—42no—
282C2×C4×C7—42no—
283C2×C4×C7—42no—
284C2×C4×C7—42no—
285C2×C4×C7—42no—
286C2×C4×C7—41no—
287C2×C4×C7—41no—
288C2×C4×C7—42no—
289C2×C4×C7—42no—
290C2×C4×C7—42no—
291C2×C4×C7—42no—
292C2×C4×C7—41no—
293C2×C4×C7—42no—
294C2×C4×C7—41no—
295C2×C4×C7—42no—
296C2×C4×C7—42no—
297C2×C4×C7—42no—
298C2×C4×C7—42no—
299C2×C4×C7—42no—
300C2×C4×C7—42no—
301C2×C4×C7—42no—
302C2×C4×C7—42no—
303C2×C4×C7—42no—
304C2×C4×C7—42no—
305C25C23×C712yes—
306C22×D4—24no—
307C22×D4—24no—
308C22×D4—41no—
309C22×D4—41no—
310C22×D4C2×C4×C714yes—
311C23×C4—24no—
312C23×C4—24no—
313C23×C4—24no—
314C23×C4—42no—
315C23×C4—42no—
316C23×C4—42no—
317C23×C4—42no—
318C23×C4C23×C712yes—
319C23×C4C23×C712yes—
320C23×C4C23×C711yescharacteristic
321C23×C4C23×C711yescharacteristic
322C23×C4C23×C711yescharacteristic
323C23×C4C23×C712yes—
324C23×C4C23×C712yes—
325C23×C4C23×C712yes—
326C2×C4⋊C4—24no—
327C2×C4⋊C4—41no—
328C2×C4⋊C4—41no—
329C2×C4⋊C4—42no—
330C2×C4⋊C4—42no—
331C2×C4⋊C4C2×C4×C714yescomplemented
332C2×C22⋊C4—24no—
333C2×C22⋊C4—24no—
334C2×C22⋊C4—24no—
335C2×C22⋊C4—24no—
336C2×C22⋊C4—24no—
337C2×C22⋊C4—24no—
338C2×C22⋊C4—42no—
339C2×C22⋊C4—42no—
340C2×C22⋊C4—42no—
341C2×C22⋊C4—42no—
342C2×C22⋊C4—42no—
343C2×C22⋊C4—42no—
344C2×C22⋊C4C2×C4×C718yescomplemented
345C2×C22⋊C4C2×C4×C718yescomplemented
346C2×C22⋊C4C7×D418yes—
347C2×C42—24no—
348C2×C42—41no—
349#95—24no—
350#95—24no—
351#95—24no—
352#95—24no—
353#95—24no—
354#95—42no—
355#95—41no—
356#95—42no—
357#95—41no—
358#95—41no—
359#95C2×C4×C714yescomplemented
360C22×C7—28no—
361C22×C7—28no—
362C22×C7—28no—
363C22×C7—44no—
364C22×C7—44no—
365C22×C7—44no—
366C22×C7—44no—
367C22×C7—44no—
368C22×C7—44no—
369C22×C7—44no—
370C22×C7—44no—
371C22×C7—44no—
372C22×C7—44no—
373C22×C7—44no—
374C22×C7—44no—
375C22×C7C2×C4×D412yes—
376C22×C7C4×#4111yescharacteristic
377C22×C7#51811yescharacteristic
378C22×C7#51811yescharacteristic
379C22×C7#52012yes—
380C22×C7#53412yes—
381C22×C7#53712yes—
382C22×C7#53812yes—
383C22×C7#53812yes—
384C22×C7#53912yes—
385C22×C7#54012yes—
386C22×C7D4211yescharacteristic
387C22×C7#54612yes—
388C22×C7#54612yes—
389C22×C7#54712yes—
390C22×C7#55111yescharacteristic
391C22×C7#55112yes—
392C22×C7#55212yes—
393C22×C7H(3,𝔽4)11yescharacteristic
394C22×C7#56212yes—
395C22×C7#56311yescharacteristic
396C4×C7—44no—
397C4×C7—42no—
398C4×C7—42no—
399C4×C7—42no—
400C4×C7—44no—
401C4×C7—44no—
402C4×C7—44no—
403C4×C7—82no—
404C24—22no—
405C24—22no—
406C24—22no—
407C24—22no—
408C24—22no—
409C24—22no—
410C24—22no—
411C24—22no—
412C24—22no—
413C24—22no—
414C24—22no—
415C24—22no—
416C24C22×C4×C714yes—
417C24C2×C7×D414yes—
418C24C7×#4114yes—
419C24C24×C711yescharacteristic
420C2×D4—216no—
421C2×D4—44no—
422C2×D4—44no—
423C2×D4—48no—
424C2×D4—44no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—21no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—48no—
469C22×C4—48no—
470C22×C4—48no—
471C22×C4—48no—
472C22×C4—48no—
473C22×C4—48no—
474C22×C4—48no—
475C22×C4C22×C4×C714yes—
476C22×C4C22×C4×C712yes—
477C22×C4C22×C4×C712yes—
478C22×C4C22×C4×C712yes—
479C22×C4C2×C7×D414yes—
480C22×C4C2×C7×D414yes—
481C22×C4C7×#4114yes—
482C4⋊C4—44no—
483C4⋊C4—48no—
484C22⋊C4—216no—
485C22⋊C4—48no—
486C22⋊C4—48no—
487C22⋊C4—48no—
488C22⋊C4—48no—
489C42—44no—
490C2×C7—48no—
491C2×C7#232512yes—
492C2×C7#233311yescharacteristic
493C2×C7#235811yescharacteristic
494C2×C7#235912yes—
495C2×C7#237911yescharacteristic
496C2×C7#267112yes—
497C2×C7#268412yes—
498C2×C7#269512yes—
499C2×C7#274312yes—
500C23—24no—
501C23—24no—
502C23—24no—
503C23—24no—
504C23—24no—
505C23—24no—
506C23—24no—
507C23—24no—
508C23—24no—
509C23—24no—
510C23—24no—
511C23—24no—
512C23—24no—
513C23—24no—
514C23—24no—
515C23—24no—
516C23—24no—
517C23—24no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23—42no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23C4×C7×D418yes—
535C23C23×C4×C711yescharacteristic
536C23C22×C7×D412yes—
537C23C2×C7×#4112yes—
538C23C2×C7×#4112yes—
539C23C7×H(5,𝔽2)12yes—
540C23C7×H(5,𝔽2)11yescharacteristic
541C23C7×H(5,𝔽2)11yescharacteristic
542C23C7×H(5,𝔽2)12yes—
543C23C7×2−1+412yes—
544D4—416no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—24no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—432no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—41no—
584C2×C4—42no—
585C2×C4—41no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C7#1457111yescharacteristic, direct product factor, Sylow
597C22—28no—
598C22—28no—
599C22—28no—
600C22—44no—
601C22—44no—
602C22—44no—
603C22—44no—
604C22—44no—
605C22—44no—
606C22—44no—
607C22—44no—
608C22—44no—
609C22—44no—
610C22—44no—
611C22—44no—
612C22C2×C4×C7×D412yes—
613C22C4×C7×#4111yescharacteristic
614C22C7×#51811yescharacteristic
615C22C7×#51811yescharacteristic
616C22C7×#52012yes—
617C22C7×#53412yes—
618C22C7×#53712yes—
619C22C7×#53812yes—
620C22C7×#53812yes—
621C22C7×#53912yes—
622C22C7×#54012yes—
623C22C7×D4211yescharacteristic
624C22C7×#54612yes—
625C22C7×#54612yes—
626C22C7×#54712yes—
627C22C7×#55112yes—
628C22C7×#55111yescharacteristic
629C22C7×#55212yes—
630C22C7×H(3,𝔽4)11yescharacteristic
631C22C7×#56212yes—
632C22C7×#56311yescharacteristic
633C4—44no—
634C4—44no—
635C4—44no—
636C4—44no—
637C4—42no—
638C4—42no—
639C4—42no—
640C4—82no—
641C2—48no—
642C2C7×#232512yes—
643C2C7×#233311yescharacteristic
644C2C7×#235811yescharacteristic
645C2C7×#235912yes—
646C2C7×#237911yescharacteristic
647C2C7×#267112yes—
648C2C7×#268412yes—
649C2C7×#269512yes—
650C2C7×#274312yes—
subgroupG/H
center, socleC24×C7C24
commutatorC23C23×C4×C7
Frattini, cosocleC24C24×C7
prime radicalC23×C7×D4C22
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24×C7 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C7×#388C2semidirect
4. C7×#1445C2semidirect
5. C7×#1442C2semidirect
6. C7×#1439C2semidirect
7. C7×#1436C2semidirect
8. C7×C22⋊#95C2semidirect
9. C7×C22⋊#95C2semidirect
20. C7×#394C4semidirect
34. #14571C7direct (N × H)
61. C2×C7×C4⋊C4C2×C4semidirect
74. C2×C7×C22⋊C4C2×C4semidirect
75. C2×C7×C22⋊C4C2×C4semidirect
89. C7×#95C2×C4semidirect
91. C2×#388C2×C7semidirect
93. #1445C2×C7semidirect
94. #1442C2×C7semidirect
95. #1439C2×C7semidirect
96. #1436C2×C7semidirect
97. C22⋊#95C2×C7semidirect
98. C22⋊#95C2×C7semidirect
195. #394C4×C7semidirect
331. C2×C4⋊C4C2×C4×C7semidirect
344. C2×C22⋊C4C2×C4×C7semidirect
345. C2×C22⋊C4C2×C4×C7semidirect
359. #95C2×C4×C7semidirect
596. C7#14571direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C7×#3942
C2×C7×#3882
C2×C7×#3862
C7×#14452
C7×#14422
C7×#14392
C7×#14362
C7×C22⋊#952
#145717