Finite Group Databasebeta

← C7×#14589 #13191805

Subgroups — C7×#14589 #13191805

Order 1792.

1650 conjugacy classes of subgroups.

The subgroup lattice has 660 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#394C211yesmaximal, characteristic
2C2×C7×#388C212yesmaximal, complemented
3C7×#1441C211yesmaximal, characteristic, complemented
4C7×#1439C212yesmaximal, complemented
5C7×#1438C212yesmaximal
6C7×#1437C212yesmaximal, complemented
7C7×#1437C212yesmaximal, complemented
8C7×C4⋊#95C211yesmaximal, characteristic, complemented
9C7×#1434C212yesmaximal, complemented
10C23×C7×D4C2211yescharacteristic
11C22×C7×C4⋊C4C2211yescharacteristic
12C22×C7×C4⋊C4C2211yescharacteristic
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×C7×C22⋊C4C2212yes—
20C22×C42×C7C2212yes—
21C22×C42×C7C2211yescharacteristic
22C7×#394C418yescomplemented
23C7×#388—28no—
24C2×C7×#95C2212yes—
25C2×C7×#95C2212yes—
26C2×C7×#95C2212yes—
27C2×C7×#95C2212yes—
28C2×C7×#95C2212yes—
29C2×C7×#95C2212yes—
30C2×C7×#95C2212yes—
31C2×C7×#95C2211yescharacteristic
32C2×C7×#95C2211yescharacteristic
33#14589C711yesmaximal, characteristic, direct product factor, Sylow
34C25×C7C2312yes—
35C22×C7×D4—42no—
36C22×C7×D4—42no—
37C22×C7×D4—41no—
38C22×C7×D4—41no—
39C22×C7×D4C2×C414yes—
40C23×C4×C7C2312yes—
41C23×C4×C7C2311yescharacteristic
42C23×C4×C7C2312yes—
43C23×C4×C7C2311yescharacteristic
44C23×C4×C7C2312yes—
45C23×C4×C7C2312yes—
46C23×C4×C7C2312yes—
47C23×C4×C7C2311yescharacteristic
48C2×C7×C4⋊C4—41no—
49C2×C7×C4⋊C4—41no—
50C2×C7×C4⋊C4—41no—
51C2×C7×C4⋊C4—41no—
52C2×C7×C4⋊C4—42no—
53C2×C7×C4⋊C4—41no—
54C2×C7×C4⋊C4C2×C414yes—
55C2×C7×C4⋊C4D414yescomplemented
56C2×C7×C22⋊C4—24no—
57C2×C7×C22⋊C4—24no—
58C2×C7×C22⋊C4—24no—
59C2×C7×C22⋊C4—24no—
60C2×C7×C22⋊C4—24no—
61C2×C7×C22⋊C4—24no—
62C2×C7×C22⋊C4—24no—
63C2×C7×C22⋊C4—24no—
64C2×C7×C22⋊C4—42no—
65C2×C7×C22⋊C4—42no—
66C2×C7×C22⋊C4—42no—
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⋊C4C2×C418yes—
73C2×C7×C22⋊C4C2×C418yes—
74C2×C42×C7—24no—
75C2×C42×C7—24no—
76C2×C42×C7—22no—
77C2×C42×C7—42no—
78C2×C42×C7—41no—
79C2×C42×C7—41no—
80C7×#95—24no—
81C7×#95—24no—
82C7×#95—24no—
83C7×#95—42no—
84C7×#95—42no—
85C7×#95—42no—
86C7×#95—42no—
87C7×#95—41no—
88C7×#95C2×C414yes—
89C2×#394C2×C711yescharacteristic
90C2×#388C2×C712yescomplemented
91#1441C2×C711yescharacteristic, complemented
92#1439C2×C712yescomplemented
93#1438C2×C712yes—
94#1437C2×C712yescomplemented
95#1437C2×C712yescomplemented
96C4⋊#95C2×C711yescharacteristic, complemented
97#1434C2×C712yescomplemented
98C24×C7—22no—
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×C7—22no—
112C24×C7C22×C414yes—
113C24×C7C2411yescharacteristic
114C2×C7×D4—44no—
115C2×C7×D4—48no—
116C2×C7×D4—44no—
117C2×C7×D4—48no—
118C2×C7×D4—44no—
119C22×C4×C7—21no—
120C22×C4×C7—21no—
121C22×C4×C7—21no—
122C22×C4×C7—21no—
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—22no—
135C22×C4×C7—22no—
136C22×C4×C7—22no—
137C22×C4×C7—22no—
138C22×C4×C7—22no—
139C22×C4×C7—22no—
140C22×C4×C7—22no—
141C22×C4×C7—21no—
142C22×C4×C7—22no—
143C22×C4×C7—22no—
144C22×C4×C7—22no—
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—21no—
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×C7C22×C414yes—
163C22×C4×C7C22×C412yes—
164C22×C4×C7C22×C412yes—
165C22×C4×C7C22×C412yes—
166C22×C4×C7C2×D414yes—
167C22×C4×C7C2×D412yes—
168C22×C4×C7#4114yes—
169C22×C4×C7#4114yes—
170C22×C4×C7#4112yes—
171C7×C4⋊C4—44no—
172C7×C4⋊C4—44no—
173C7×C4⋊C4—44no—
174C7×C22⋊C4—216no—
175C7×C22⋊C4—48no—
176C7×C22⋊C4—48no—
177C7×C22⋊C4—48no—
178C7×C22⋊C4—48no—
179C7×C22⋊C4—48no—
180C42×C7—48no—
181C42×C7—44no—
182C23×D4C22×C711yescharacteristic
183C22×C4⋊C4C22×C711yescharacteristic
184C22×C4⋊C4C22×C711yescharacteristic
185C22×C4⋊C4C22×C711yescharacteristic
186C22×C22⋊C4C22×C712yes—
187C22×C22⋊C4C22×C712yes—
188C22×C22⋊C4C22×C712yes—
189C22×C22⋊C4C22×C712yes—
190C22×C22⋊C4C22×C712yes—
191C22×C22⋊C4C22×C712yes—
192C22×C42C22×C712yes—
193C22×C42C22×C711yescharacteristic
194#394C4×C718yescomplemented
195#388—28no—
196C2×#95C22×C712yes—
197C2×#95C22×C712yes—
198C2×#95C22×C712yes—
199C2×#95C22×C712yes—
200C2×#95C22×C712yes—
201C2×#95C22×C712yes—
202C2×#95C22×C712yes—
203C2×#95C22×C711yescharacteristic
204C2×#95C22×C711yescharacteristic
205C23×C7—24no—
206C23×C7—24no—
207C23×C7—24no—
208C23×C7—24no—
209C23×C7—24no—
210C23×C7—24no—
211C23×C7—24no—
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—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—42no—
236C23×C7—42no—
237C23×C7—42no—
238C23×C7—42no—
239C23×C7—42no—
240C23×C7C23×C411yescharacteristic
241C23×C7C22×D411yescharacteristic
242C23×C7C2×#4111yescharacteristic
243C23×C7C2×#4112yes—
244C23×C7C2×#4112yes—
245C23×C7H(5,𝔽2)11yescharacteristic
246C23×C7H(5,𝔽2)11yescharacteristic
247C23×C7H(5,𝔽2)12yes—
248C23×C7H(5,𝔽2)12yes—
249C23×C7H(5,𝔽2)11yescharacteristic
250C23×C72−1+411yescharacteristic
251C7×D4—416no—
252C2×C4×C7—22no—
253C2×C4×C7—22no—
254C2×C4×C7—22no—
255C2×C4×C7—22no—
256C2×C4×C7—22no—
257C2×C4×C7—22no—
258C2×C4×C7—24no—
259C2×C4×C7—24no—
260C2×C4×C7—24no—
261C2×C4×C7—24no—
262C2×C4×C7—22no—
263C2×C4×C7—22no—
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—24no—
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—42no—
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—41no—
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×C7C4×D414yes—
304C25C23×C712yes—
305C22×D4—42no—
306C22×D4—42no—
307C22×D4—41no—
308C22×D4—41no—
309C22×D4C2×C4×C714yes—
310C23×C4C23×C712yes—
311C23×C4C23×C711yescharacteristic
312C23×C4C23×C712yes—
313C23×C4C23×C711yescharacteristic
314C23×C4C23×C712yes—
315C23×C4C23×C712yes—
316C23×C4C23×C712yes—
317C23×C4C23×C711yescharacteristic
318C2×C4⋊C4—41no—
319C2×C4⋊C4—41no—
320C2×C4⋊C4—41no—
321C2×C4⋊C4—41no—
322C2×C4⋊C4—42no—
323C2×C4⋊C4—41no—
324C2×C4⋊C4C2×C4×C714yes—
325C2×C4⋊C4C7×D414yescomplemented
326C2×C22⋊C4—24no—
327C2×C22⋊C4—24no—
328C2×C22⋊C4—24no—
329C2×C22⋊C4—24no—
330C2×C22⋊C4—24no—
331C2×C22⋊C4—24no—
332C2×C22⋊C4—24no—
333C2×C22⋊C4—24no—
334C2×C22⋊C4—42no—
335C2×C22⋊C4—42no—
336C2×C22⋊C4—42no—
337C2×C22⋊C4—42no—
338C2×C22⋊C4—42no—
339C2×C22⋊C4—42no—
340C2×C22⋊C4—42no—
341C2×C22⋊C4—42no—
342C2×C22⋊C4C2×C4×C718yes—
343C2×C22⋊C4C2×C4×C718yes—
344C2×C42—24no—
345C2×C42—24no—
346C2×C42—22no—
347C2×C42—42no—
348C2×C42—41no—
349C2×C42—41no—
350#95—24no—
351#95—24no—
352#95—24no—
353#95—42no—
354#95—42no—
355#95—42no—
356#95—42no—
357#95—41no—
358#95C2×C4×C714yes—
359C22×C7—44no—
360C22×C7—44no—
361C22×C7—44no—
362C22×C7—44no—
363C22×C7—44no—
364C22×C7—44no—
365C22×C7—44no—
366C22×C7—82no—
367C22×C7—82no—
368C22×C7—82no—
369C22×C7—82no—
370C22×C7—82no—
371C22×C7—82no—
372C22×C7—82no—
373C22×C7—82no—
374C22×C7C2×C4×D411yescharacteristic
375C22×C7C4×#4112yes—
376C22×C7#51811yescharacteristic
377C22×C7#51812yes—
378C22×C7#52011yescharacteristic
379C22×C7#53411yescharacteristic
380C22×C7#53711yescharacteristic
381C22×C7#53811yescharacteristic
382C22×C7#53812yes—
383C22×C7#53911yescharacteristic
384C22×C7#53912yes—
385C22×C7#54012yes—
386C22×C7#54212yes—
387C22×C7#54612yes—
388C22×C7#55012yes—
389C22×C7#55112yes—
390C22×C7#55312yes—
391C22×C7#55512yes—
392C22×C7#55511yescharacteristic
393C22×C7#55511yescharacteristic
394C22×C7#56011yescharacteristic
395C22×C7#56012yes—
396C22×C7H(3,𝔽4)11yescharacteristic
397C4×C7—24no—
398C4×C7—44no—
399C4×C7—42no—
400C4×C7—42no—
401C4×C7—44no—
402C4×C7—44no—
403C4×C7—44no—
404C4×C7—82no—
405C24—22no—
406C24—22no—
407C24—22no—
408C24—22no—
409C24—22no—
410C24—22no—
411C24—22no—
412C24—22no—
413C24—22no—
414C24—22no—
415C24—22no—
416C24—22no—
417C24—22no—
418C24—22no—
419C24C22×C4×C714yes—
420C24C24×C711yescharacteristic
421C2×D4—44no—
422C2×D4—48no—
423C2×D4—44no—
424C2×D4—48no—
425C2×D4—44no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
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—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—21no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4C22×C4×C714yes—
470C22×C4C22×C4×C712yes—
471C22×C4C22×C4×C712yes—
472C22×C4C22×C4×C712yes—
473C22×C4C2×C7×D414yes—
474C22×C4C2×C7×D412yes—
475C22×C4C7×#4114yes—
476C22×C4C7×#4114yes—
477C22×C4C7×#4112yes—
478C4⋊C4—44no—
479C4⋊C4—44no—
480C4⋊C4—44no—
481C22⋊C4—216no—
482C22⋊C4—48no—
483C22⋊C4—48no—
484C22⋊C4—48no—
485C22⋊C4—48no—
486C22⋊C4—48no—
487C42—48no—
488C42—44no—
489C2×C7—84no—
490C2×C7#232511yescharacteristic
491C2×C7#233512yes—
492C2×C7#236312yes—
493C2×C7#237211yescharacteristic
494C2×C7#237511yescharacteristic
495C2×C7#269911yescharacteristic
496C2×C7#270512yes—
497C2×C7#273311yescharacteristic
498C2×C7#275612yes—
499C2×C7#278511yescharacteristic
500C2×C7#279511yescharacteristic
501C23—24no—
502C23—24no—
503C23—24no—
504C23—24no—
505C23—24no—
506C23—24no—
507C23—24no—
508C23—42no—
509C23—42no—
510C23—42no—
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—42no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23C23×C4×C711yescharacteristic
537C23C22×C7×D411yescharacteristic
538C23C2×C7×#4111yescharacteristic
539C23C2×C7×#4112yes—
540C23C2×C7×#4112yes—
541C23C7×H(5,𝔽2)11yescharacteristic
542C23C7×H(5,𝔽2)11yescharacteristic
543C23C7×H(5,𝔽2)12yes—
544C23C7×H(5,𝔽2)12yes—
545C23C7×H(5,𝔽2)11yescharacteristic
546C23C7×2−1+411yescharacteristic
547D4—416no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—41no—
587C2×C4—42no—
588C2×C4—41no—
589C2×C4—42no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4C4×C7×D414yes—
600C7#1458911yescharacteristic, direct product factor, Sylow
601C22—44no—
602C22—44no—
603C22—44no—
604C22—44no—
605C22—44no—
606C22—44no—
607C22—44no—
608C22—82no—
609C22—82no—
610C22—82no—
611C22—82no—
612C22—82no—
613C22—82no—
614C22—82no—
615C22—82no—
616C22C2×C4×C7×D411yescharacteristic
617C22C4×C7×#4112yes—
618C22C7×#51811yescharacteristic
619C22C7×#51812yes—
620C22C7×#52011yescharacteristic
621C22C7×#53411yescharacteristic
622C22C7×#53711yescharacteristic
623C22C7×#53812yes—
624C22C7×#53811yescharacteristic
625C22C7×#53912yes—
626C22C7×#53911yescharacteristic
627C22C7×#54012yes—
628C22C7×#54212yes—
629C22C7×#54612yes—
630C22C7×#55012yes—
631C22C7×#55112yes—
632C22C7×#55312yes—
633C22C7×#55511yescharacteristic
634C22C7×#55511yescharacteristic
635C22C7×#55512yes—
636C22C7×#56011yescharacteristic
637C22C7×#56012yes—
638C22C7×H(3,𝔽4)11yescharacteristic
639C4—24no—
640C4—44no—
641C4—44no—
642C4—44no—
643C4—44no—
644C4—42no—
645C4—42no—
646C4—82no—
647C2—84no—
648C2C7×#232511yescharacteristic
649C2C7×#233512yes—
650C2C7×#236312yes—
651C2C7×#237211yescharacteristic
652C2C7×#237511yescharacteristic
653C2C7×#269911yescharacteristic
654C2C7×#270512yes—
655C2C7×#273311yescharacteristic
656C2C7×#275612yes—
657C2C7×#278511yescharacteristic
658C2C7×#279511yescharacteristic
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
3. C7×#1441C2semidirect
4. C7×#1439C2semidirect
6. C7×#1437C2semidirect
7. C7×#1437C2semidirect
8. C7×C4⋊#95C2semidirect
9. C7×#1434C2semidirect
22. C7×#394C4semidirect
33. #14589C7direct (N × H)
55. C2×C7×C4⋊C4D4semidirect
90. C2×#388C2×C7semidirect
91. #1441C2×C7semidirect
92. #1439C2×C7semidirect
94. #1437C2×C7semidirect
95. #1437C2×C7semidirect
96. C4⋊#95C2×C7semidirect
97. #1434C2×C7semidirect
194. #394C4×C7semidirect
325. C2×C4⋊C4C7×D4semidirect
600. C7#14589direct (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
C7×#14412
C7×#14392
C7×#14382
C7×#14372
C7×C4⋊#952
C7×#14342
#145897