Finite Group Databasebeta

← C7×#24241 #13201457

Subgroups — C7×#24241 #13201457

Order 1792.

1438 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#547C211yesmaximal, characteristic
2C2×C7×#462C211yesmaximal, characteristic, complemented
3C2×C7×#452C212yesmaximal, complemented
4C2×C4×C7×SD16C211yesmaximal, characteristic, complemented
5C7×#2538C211yesmaximal, characteristic, complemented
6C7×#2035C212yesmaximal, complemented
7C7×#2001C212yesmaximal, complemented
8C7×#1956C211yesmaximal, characteristic, complemented
9C7×#1954C211yesmaximal, characteristic, complemented
10C7×#1802C211yesmaximal, characteristic, complemented
11C7×#1788C212yesmaximal, complemented
12C22×C7×#41C2212yes—
13C22×C7×Q16C2212yes—
14C22×C7×SD16C2211yescharacteristic
15C7×#547—24no—
16C7×#547—24no—
17C2×C7×C4⋊Q8C2211yescharacteristic
18C2×C7×#123—22no—
19C2×C7×C22⋊Q8C2212yes—
20C2×C7×C22⋊Q8C2212yes—
21C2×C7×#121—21no—
22C2×C4×C7×Q8C2211yescharacteristic
23C2×C4×C7×D4C2211yescharacteristic
24C2×C4×C7×D4C2211yescharacteristic
25C22×C7×C4⋊C4—21no—
26C7×#462—24no—
27C7×#452—28no—
28C4×C7×SD16—24no—
29C2×C7×#106C2211yescharacteristic
30C2×C7×C4⋊C8C2211yescharacteristic
31C2×C7×#103C2211yescharacteristic
32C2×C7×#103C2212yes—
33C2×C7×#103C2212yes—
34C2×C7×#103C2212yes—
35C2×C7×#102C2211yescharacteristic
36C2×C7×#102C2212yes—
37C2×C7×C22⋊C8C2212yes—
38C2×C7×C22⋊C8C2212yes—
39C2×C4×C7×C8C2211yescharacteristic
40C7×#396—22no—
41C7×#390—21no—
42C7×#389—21no—
43C7×#384C2211yescharacteristic
44C7×#380C2212yes—
45C7×#340C2212yes—
46#24241C711yesmaximal, characteristic, direct product factor, Sylow
47C2×C7×#41—24no—
48C2×C7×#41—24no—
49C2×C7×#41—24no—
50C2×C7×#41—24no—
51C2×C7×#41—42no—
52C2×C7×#41—42no—
53C22×C7×Q8C2312yes—
54C22×C7×Q8C2311yescharacteristic
55C22×C7×D4—22no—
56C22×C7×D4D411yescharacteristic
57C22×C7×D4C2311yescharacteristic
58C23×C4×C7—22no—
59C23×C4×C7—21no—
60C23×C4×C7C2312yes—
61C2×C7×Q16—24no—
62C2×C7×Q16—24no—
63C2×C7×Q16—42no—
64C2×C7×SD16—22no—
65C2×C7×SD16—22no—
66C2×C7×SD16D414yescomplemented
67C22×C7×C8C2312yes—
68C22×C7×C8C2311yescharacteristic
69C22×C7×C8C2311yescharacteristic
70C7×C4⋊Q8—22no—
71C7×C4⋊Q8D414yes—
72C7×#123—216no—
73C7×C22⋊Q8—24no—
74C7×C22⋊Q8—24no—
75C7×C22⋊Q8—24no—
76C7×C22⋊Q8—42no—
77C7×#121—28no—
78C4×C7×Q8—22no—
79C4×C7×Q8—22no—
80C4×C7×D4—22no—
81C4×C7×D4—22no—
82C4×C7×D4—22no—
83C4×C7×D4D414yes—
84C2×C7×C4⋊C4—28no—
85C2×C7×C4⋊C4—21no—
86C2×C7×C4⋊C4—22no—
87C2×C7×C4⋊C4—21no—
88C2×C7×C4⋊C4—21no—
89C2×C7×C4⋊C4—21no—
90C2×C7×C4⋊C4—22no—
91C2×C7×C4⋊C4—21no—
92C2×C7×C4⋊C4D411yescharacteristic
93C2×C7×C4⋊C4C2312yes—
94C2×C7×C4⋊C4C2311yescharacteristic
95C2×C7×C4⋊C4C2311yescharacteristic
96C2×C7×C22⋊C4—22no—
97C2×C7×C22⋊C4—21no—
98C2×C7×C22⋊C4—22no—
99C2×C7×C22⋊C4—22no—
100C2×C7×C22⋊C4D412yes—
101C2×C42×C7—21no—
102C2×C42×C7C2311yescharacteristic
103C7×#106—22no—
104C7×C4⋊C8—22no—
105C7×#103—24no—
106C7×#103—24no—
107C7×#103—22no—
108C7×#103—42no—
109C7×#102—22no—
110C7×#102—24no—
111C7×C22⋊C8—24no—
112C7×C22⋊C8—42no—
113C4×C7×C8—22no—
114C7×#95—22no—
115C2×#547C2×C711yescharacteristic
116C2×#462C2×C711yescharacteristic, complemented
117C2×#452C2×C712yescomplemented
118C2×C4×SD16C2×C711yescharacteristic, complemented
119#2538C2×C711yescharacteristic, complemented
120#2035C2×C712yescomplemented
121#2001C2×C712yescomplemented
122#1956C2×C711yescharacteristic, complemented
123#1954C2×C711yescharacteristic, complemented
124#1802C2×C711yescharacteristic, complemented
125#1788C2×C712yescomplemented
126C24×C7—21no—
127C24×C7C2×D412yes—
128C7×#41—48no—
129C7×#41—48no—
130C7×#41—48no—
131C7×#41—48no—
132C2×C7×Q8—22no—
133C2×C7×Q8—21no—
134C2×C7×Q8—21no—
135C2×C7×Q8—22no—
136C2×C7×Q8—21no—
137C2×C7×Q8—21no—
138C2×C7×Q8—22no—
139C2×C7×Q8—22no—
140C2×C7×Q8—21no—
141C2×C7×Q8—22no—
142C2×C7×Q8C2×D414yes—
143C2×C7×Q8C2×D412yes—
144C2×C7×D4—28no—
145C2×C7×D4—22no—
146C2×C7×D4—22no—
147C2×C7×D4—22no—
148C2×C7×D4—21no—
149C2×C7×D4—21no—
150C2×C7×D4—21no—
151C2×C7×D4—44no—
152C2×C7×D4—44no—
153C2×C7×D4C2×D412yes—
154C2×C7×D4C2×D412yes—
155C2×C7×D4#4112yes—
156C22×C4×C7—28no—
157C22×C4×C7—21no—
158C22×C4×C7—21no—
159C22×C4×C7—22no—
160C22×C4×C7—24no—
161C22×C4×C7—22no—
162C22×C4×C7—22no—
163C22×C4×C7—22no—
164C22×C4×C7—22no—
165C22×C4×C7—22no—
166C22×C4×C7—22no—
167C22×C4×C7—21no—
168C22×C4×C7—22no—
169C22×C4×C7—44no—
170C22×C4×C7—44no—
171C22×C4×C7—42no—
172C22×C4×C7—42no—
173C22×C4×C7C2×D412yes—
174C22×C4×C7C2×D411yescharacteristic
175C22×C4×C7C2×D411yescharacteristic
176C22×C4×C7C2411yescharacteristic
177C7×Q16—48no—
178C7×SD16—28no—
179C2×C7×C8—21no—
180C2×C7×C8—21no—
181C2×C7×C8—21no—
182C2×C7×C8—22no—
183C2×C7×C8—22no—
184C2×C7×C8—21no—
185C2×C7×C8—22no—
186C2×C7×C8—21no—
187C2×C7×C8C2×D412yes—
188C7×C4⋊C4—28no—
189C7×C4⋊C4—22no—
190C7×C4⋊C4—24no—
191C7×C4⋊C4—24no—
192C7×C4⋊C4—21no—
193C7×C4⋊C4—22no—
194C7×C4⋊C4—21no—
195C7×C4⋊C4—21no—
196C7×C4⋊C4—22no—
197C7×C4⋊C4—44no—
198C7×C4⋊C4—42no—
199C7×C4⋊C4—42no—
200C7×C4⋊C4—81no—
201C7×C4⋊C4C2×D412yes—
202C7×C22⋊C4—28no—
203C7×C22⋊C4—24no—
204C7×C22⋊C4—24no—
205C7×C22⋊C4—44no—
206C7×C22⋊C4—82no—
207C42×C7—42no—
208C42×C7C2×D412yes—
209C42×C7C2×D412yes—
210C22×#41C22×C712yes—
211C22×Q16C22×C712yes—
212C22×SD16C22×C711yescharacteristic
213#547—24no—
214#547—24no—
215C2×C4⋊Q8C22×C711yescharacteristic
216C2×#123—22no—
217C2×C22⋊Q8C22×C712yes—
218C2×C22⋊Q8C22×C712yes—
219C2×#121—21no—
220C2×C4×Q8C22×C711yescharacteristic
221C2×C4×D4C22×C711yescharacteristic
222C2×C4×D4C22×C711yescharacteristic
223C22×C4⋊C4—21no—
224#462—24no—
225#452—28no—
226C4×SD16—24no—
227C2×#106C22×C711yescharacteristic
228C2×C4⋊C8C22×C711yescharacteristic
229C2×#103C22×C711yescharacteristic
230C2×#103C22×C712yes—
231C2×#103C22×C712yes—
232C2×#103C22×C712yes—
233C2×#102C22×C712yes—
234C2×#102C22×C711yescharacteristic
235C2×C22⋊C8C22×C712yes—
236C2×C22⋊C8C22×C712yes—
237C2×C4×C8C22×C711yescharacteristic
238#396—22no—
239#390—21no—
240#389—21no—
241#384C22×C711yescharacteristic
242#380C22×C712yes—
243#340C22×C712yes—
244C23×C7—22no—
245C23×C7—22no—
246C23×C7—22no—
247C23×C7—22no—
248C23×C7—22no—
249C23×C7—22no—
250C23×C7—22no—
251C23×C7—22no—
252C23×C7—22no—
253C23×C7—22no—
254C23×C7—41no—
255C23×C7—41no—
256C23×C7—41no—
257C23×C7—41no—
258C23×C7C22×D411yescharacteristic
259C7×Q8—22no—
260C7×Q8—22no—
261C7×Q8—24no—
262C7×Q8—24no—
263C7×Q8—42no—
264C7×Q8—41no—
265C7×Q8—42no—
266C7×Q8—41no—
267C7×D4—22no—
268C7×D4—22no—
269C7×D4—22no—
270C7×D4—416no—
271C7×D4—44no—
272C7×D4#12114yes—
273C2×C4×C7—21no—
274C2×C4×C7—21no—
275C2×C4×C7—22no—
276C2×C4×C7—22no—
277C2×C4×C7—24no—
278C2×C4×C7—22no—
279C2×C4×C7—22no—
280C2×C4×C7—22no—
281C2×C4×C7—416no—
282C2×C4×C7—41no—
283C2×C4×C7—42no—
284C2×C4×C7—42no—
285C2×C4×C7—41no—
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—48no—
293C2×C4×C7—41no—
294C2×C4×C7—42no—
295C2×C4×C7—42no—
296C2×C4×C7—41no—
297C2×C4×C7—42no—
298C2×C4×C7—41no—
299C2×C4×C7#12012yes—
300C2×C4×C7#12012yes—
301C2×C4×C7#13512yes—
302C2×C4×C7#13712yes—
303C2×C4×C7C22×D411yescharacteristic
304C2×C4×C7C22×D411yescharacteristic
305C2×C4×C7C22×D411yescharacteristic
306C2×C4×C7C2×#4111yescharacteristic
307C2×C4×C7H(5,𝔽2)11yescharacteristic
308C2×C4×C7H(5,𝔽2)11yescharacteristic
309C7×C8—22no—
310C7×C8—41no—
311C7×C8—42no—
312C2×#41—24no—
313C2×#41—24no—
314C2×#41—24no—
315C2×#41—24no—
316C2×#41—42no—
317C2×#41—42no—
318C22×Q8C23×C711yescharacteristic
319C22×Q8C23×C712yes—
320C22×D4—22no—
321C22×D4C7×D411yescharacteristic
322C22×D4C23×C711yescharacteristic
323C23×C4—21no—
324C23×C4—22no—
325C23×C4C23×C712yes—
326C2×Q16—24no—
327C2×Q16—24no—
328C2×Q16—42no—
329C2×SD16—22no—
330C2×SD16—22no—
331C2×SD16C7×D414yescomplemented
332C22×C8C23×C712yes—
333C22×C8C23×C711yescharacteristic
334C22×C8C23×C711yescharacteristic
335C4⋊Q8—22no—
336C4⋊Q8C7×D414yes—
337#123—216no—
338C22⋊Q8—24no—
339C22⋊Q8—24no—
340C22⋊Q8—24no—
341C22⋊Q8—42no—
342#121—28no—
343C4×Q8—22no—
344C4×Q8—22no—
345C4×D4—22no—
346C4×D4—22no—
347C4×D4—22no—
348C4×D4C7×D414yes—
349C2×C4⋊C4—28no—
350C2×C4⋊C4—21no—
351C2×C4⋊C4—22no—
352C2×C4⋊C4—21no—
353C2×C4⋊C4—21no—
354C2×C4⋊C4—22no—
355C2×C4⋊C4—21no—
356C2×C4⋊C4—21no—
357C2×C4⋊C4C7×D411yescharacteristic
358C2×C4⋊C4C23×C711yescharacteristic
359C2×C4⋊C4C23×C712yes—
360C2×C4⋊C4C23×C711yescharacteristic
361C2×C22⋊C4—22no—
362C2×C22⋊C4—21no—
363C2×C22⋊C4—22no—
364C2×C22⋊C4—22no—
365C2×C22⋊C4C7×D412yes—
366C2×C42—21no—
367C2×C42C23×C711yescharacteristic
368#106—22no—
369C4⋊C8—22no—
370#103—24no—
371#103—24no—
372#103—22no—
373#103—42no—
374#102—22no—
375#102—24no—
376C22⋊C8—24no—
377C22⋊C8—42no—
378C4×C8—22no—
379#95—22no—
380C22×C7—24no—
381C22×C7—42no—
382C22×C7—42no—
383C22×C7—42no—
384C22×C7—42no—
385C22×C7—42no—
386C22×C7—42no—
387C22×C7—42no—
388C22×C7—42no—
389C22×C7—42no—
390C22×C7—42no—
391C22×C7—42no—
392C22×C7—42no—
393C22×C7—42no—
394C22×C7C2×#12011yescharacteristic
395C22×C7D4211yescharacteristic
396C22×C7#54611yescharacteristic
397C22×C7C2×#13511yescharacteristic
398C22×C7C2×#13711yescharacteristic
399C22×C7#57711yescharacteristic
400C22×C7#57811yescharacteristic
401C4×C7—22no—
402C4×C7—42no—
403C4×C7—44no—
404C4×C7—42no—
405C4×C7—42no—
406C4×C7—81no—
407C4×C7—82no—
408C4×C7#45214yescomplemented
409C4×C7C2×#12111yescharacteristic
410C4×C7#53511yescharacteristic
411C4×C7D4211yescharacteristic
412C4×C7#54611yescharacteristic
413C24—21no—
414C24C2×C7×D412yes—
415#41—48no—
416#41—48no—
417#41—48no—
418#41—48no—
419C2×Q8—22no—
420C2×Q8—21no—
421C2×Q8—21no—
422C2×Q8—22no—
423C2×Q8—21no—
424C2×Q8—21no—
425C2×Q8—22no—
426C2×Q8—22no—
427C2×Q8—21no—
428C2×Q8—22no—
429C2×Q8C2×C7×D414yes—
430C2×Q8C2×C7×D412yes—
431C2×D4—28no—
432C2×D4—22no—
433C2×D4—22no—
434C2×D4—22no—
435C2×D4—21no—
436C2×D4—21no—
437C2×D4—21no—
438C2×D4—44no—
439C2×D4—44no—
440C2×D4C2×C7×D412yes—
441C2×D4C2×C7×D412yes—
442C2×D4C7×#4112yes—
443C22×C4—28no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—22no—
447C22×C4—24no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—21no—
455C22×C4—22no—
456C22×C4—44no—
457C22×C4—44no—
458C22×C4—42no—
459C22×C4—42no—
460C22×C4C2×C7×D412yes—
461C22×C4C2×C7×D411yescharacteristic
462C22×C4C2×C7×D411yescharacteristic
463C22×C4C24×C711yescharacteristic
464Q16—48no—
465SD16—28no—
466C2×C8—21no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—22no—
470C2×C8—22no—
471C2×C8—21no—
472C2×C8—22no—
473C2×C8—21no—
474C2×C8C2×C7×D412yes—
475C4⋊C4—28no—
476C4⋊C4—22no—
477C4⋊C4—24no—
478C4⋊C4—24no—
479C4⋊C4—21no—
480C4⋊C4—22no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—22no—
484C4⋊C4—44no—
485C4⋊C4—42no—
486C4⋊C4—42no—
487C4⋊C4—81no—
488C4⋊C4C2×C7×D412yes—
489C22⋊C4—28no—
490C22⋊C4—24no—
491C22⋊C4—24no—
492C22⋊C4—44no—
493C22⋊C4—82no—
494C42—42no—
495C42C2×C7×D412yes—
496C42C2×C7×D412yes—
497C2×C7—44no—
498C2×C7—82no—
499C2×C7#240811yescharacteristic
500C2×C7C2×#45211yescharacteristic
501C2×C7#301311yescharacteristic
502C2×C7#327611yescharacteristic
503C2×C7#328411yescharacteristic
504C2×C7#330211yescharacteristic
505C2×C7#333111yescharacteristic
506C23—22no—
507C23—22no—
508C23—22no—
509C23—22no—
510C23—22no—
511C23—22no—
512C23—22no—
513C23—22no—
514C23—22no—
515C23—22no—
516C23—41no—
517C23—41no—
518C23—41no—
519C23—41no—
520C23C22×C7×D411yescharacteristic
521Q8—22no—
522Q8—22no—
523Q8—24no—
524Q8—24no—
525Q8—42no—
526Q8—41no—
527Q8—42no—
528Q8—41no—
529D4—22no—
530D4—22no—
531D4—22no—
532D4—416no—
533D4—44no—
534D4C7×#12114yes—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—24no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—416no—
544C2×C4—41no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—48no—
555C2×C4—41no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—41no—
559C2×C4—42no—
560C2×C4—41no—
561C2×C4C7×#12012yes—
562C2×C4C7×#12012yes—
563C2×C4C7×#13512yes—
564C2×C4C7×#13712yes—
565C2×C4C22×C7×D411yescharacteristic
566C2×C4C22×C7×D411yescharacteristic
567C2×C4C22×C7×D411yescharacteristic
568C2×C4C2×C7×#4111yescharacteristic
569C2×C4C7×H(5,𝔽2)11yescharacteristic
570C2×C4C7×H(5,𝔽2)11yescharacteristic
571C8—22no—
572C8—41no—
573C8—42no—
574C7#2424111yescharacteristic, direct product factor, Sylow
575C22—24no—
576C22—42no—
577C22—42no—
578C22—42no—
579C22—42no—
580C22—42no—
581C22—42no—
582C22—42no—
583C22—42no—
584C22—42no—
585C22—42no—
586C22—42no—
587C22—42no—
588C22—42no—
589C22C2×C7×#12011yescharacteristic
590C22C7×D4211yescharacteristic
591C22C7×#54611yescharacteristic
592C22C2×C7×#13511yescharacteristic
593C22C2×C7×#13711yescharacteristic
594C22C7×#57711yescharacteristic
595C22C7×#57811yescharacteristic
596C4—22no—
597C4—42no—
598C4—44no—
599C4—42no—
600C4—42no—
601C4—81no—
602C4—82no—
603C4C7×#45214yescomplemented
604C4C2×C7×#12111yescharacteristic
605C4C7×#53511yescharacteristic
606C4C7×D4211yescharacteristic
607C4C7×#54611yescharacteristic
608C2—44no—
609C2—82no—
610C2C7×#240811yescharacteristic
611C2C2×C7×#45211yescharacteristic
612C2C7×#301311yescharacteristic
613C2C7×#327611yescharacteristic
614C2C7×#328411yescharacteristic
615C2C7×#330211yescharacteristic
616C2C7×#333111yescharacteristic
subgroupG/H
center, socleC23×C7C22×D4
commutator, Frattini, cosocleC22×C4C24×C7
prime radicalC2×C7×#547C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C7 ◁ C2×C4×C7×D4 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C7×#462C2semidirect
3. C2×C7×#452C2semidirect
4. C2×C4×C7×SD16C2semidirect
5. C7×#2538C2semidirect
6. C7×#2035C2semidirect
7. C7×#2001C2semidirect
8. C7×#1956C2semidirect
9. C7×#1954C2semidirect
10. C7×#1802C2semidirect
11. C7×#1788C2semidirect
46. #24241C7direct (N × H)
66. C2×C7×SD16D4semidirect
116. C2×#462C2×C7semidirect
117. C2×#452C2×C7semidirect
118. C2×C4×SD16C2×C7semidirect
119. #2538C2×C7semidirect
120. #2035C2×C7semidirect
121. #2001C2×C7semidirect
122. #1956C2×C7semidirect
123. #1954C2×C7semidirect
124. #1802C2×C7semidirect
125. #1788C2×C7semidirect
331. C2×SD16C7×D4semidirect
408. C4×C7#452semidirect
574. C7#24241direct (N × H)
603. C4C7×#452semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#5472
C2×C7×#4622
C2×C7×#4522
C2×C4×C7×SD162
C7×#25382
C7×#20352
C7×#20012
C7×#19562
C7×#19542
C7×#18022
C7×#17882
#242417