Finite Group Databasebeta

← Dic7×#485 #13360965

Subgroups — Dic7×#485 #13360965

Order 1792.

888 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#485C211yesmaximal, characteristic
2Dic7×C22⋊Q8C211yesmaximal, characteristic
3Dic7×#117C211yesmaximal, characteristic
4#11729447C211yesmaximal, characteristic
5#11729329C211yesmaximal, characteristic
6#11728363C211yesmaximal, characteristic, complemented
7Dic7×#107C211yesmaximal, characteristic, complemented
8#11728311C211yesmaximal, characteristic, complemented
9Dic7×#106C211yesmaximal, characteristic, complemented
10#11727978C211yesmaximal, characteristic, complemented
11#11727977C211yesmaximal, characteristic, complemented
12Dic7×#103C211yesmaximal, characteristic, complemented
13Dic7×#103C211yesmaximal, characteristic, complemented
14#11727683C211yesmaximal, characteristic
15Dic7×C22⋊C8C211yesmaximal, characteristic
16C2×C7×C22⋊Q8C2211yescharacteristic
17C2×C7×#117C2211yescharacteristic
18C2×Q8×Dic7C2211yescharacteristic
19C2×#6489C2211yescharacteristic
20C22×C4×Dic7C2211yescharacteristic
21C7×#485C418yescomplemented
22C2×C7×#107C2211yescharacteristic
23C2×C7×#106C2211yescharacteristic
24C2×C7×#103C2211yescharacteristic
25C2×C7×#103C2211yescharacteristic
26C2×C7×C22⋊C8C2211yescharacteristic
27#90002C2211yescharacteristic
28C2×#6409C2211yescharacteristic
29C2×#6409C2211yescharacteristic
30#89984C2211yescharacteristic
31C2×C22⋊(C7⋊C8)C2211yescharacteristic
32C2×#6392C2211yescharacteristic
33C2×#6391C2211yescharacteristic
34C2×C8×Dic7C2211yescharacteristic
35C2×C8×Dic7C2211yescharacteristic
36#89757C2211yescharacteristic
37#89753—21no—
38C4⋊C4×Dic7—21no—
39C4⋊C4×Dic7C2211yescharacteristic
40C4⋊C4×Dic7C2211yescharacteristic
41C4⋊C4×Dic7C2211yescharacteristic
42C4⋊C4×Dic7C2211yescharacteristic
43C4⋊C4×Dic7C2211yescharacteristic
44#89722—21no—
45C22⋊C4×Dic7—21no—
46C22⋊C4×Dic7—21no—
47C22⋊C4×Dic7—21no—
48#89704—21no—
49C42×Dic7—21no—
50#89346C2211yescharacteristic
51#89346C2211yescharacteristic
52#89345C2211yescharacteristic
53#89345C2211yescharacteristic
54#89324C2211yescharacteristic
55#89324C2211yescharacteristic
56#89324C2211yescharacteristic
57#89324C2211yescharacteristic
58#89324C2211yescharacteristic
59#89324C2211yescharacteristic
60C4×#485—71nomaximal, Sylow
61C22×C7×Q8C2311yescharacteristic
62C23×C4×C7C2311yescharacteristic
63C23×Dic7D411yescharacteristic
64C23×Dic7D411yescharacteristic
65C22×C7×C8C2311yescharacteristic
66C22×C7×C8C2311yescharacteristic
67C7×C22⋊Q8—22no—
68C7×C22⋊Q8C2×C414yes—
69C7×#117—22no—
70C7×#117C2×C414yes—
71C2×C7×C4⋊C4—21no—
72C2×C7×C4⋊C4C2311yescharacteristic
73C2×C7×C4⋊C4C2311yescharacteristic
74C2×C7×C4⋊C4C2311yescharacteristic
75C2×C7×C22⋊C4—21no—
76C2×C7×C22⋊C4—21no—
77C2×C42×C7—21no—
78C2×C22⋊Dic7—21no—
79C2×C22⋊Dic7—21no—
80Q8×Dic7—22no—
81Q8×Dic7—22no—
82#6489—22no—
83#6489—22no—
84C2×C4⋊Dic7—21no—
85C2×C4⋊Dic7—21no—
86C2×C4⋊Dic7C2311yescharacteristic
87C2×C4⋊Dic7C2311yescharacteristic
88C2×C4⋊Dic7C2311yescharacteristic
89C2×C4⋊Dic7C2311yescharacteristic
90C2×C4×Dic7—21no—
91C2×C4×Dic7—22no—
92C2×C4×Dic7—22no—
93C2×C4×Dic7—21no—
94C2×C4×Dic7—21no—
95C2×C4×Dic7—21no—
96C2×C4×Dic7—21no—
97C2×C4×Dic7—21no—
98C2×C4×Dic7—21no—
99C2×C4×Dic7D411yescharacteristic
100C2×C4×Dic7D411yescharacteristic
101C2×C4×Dic7C2311yescharacteristic
102C2×C4×Dic7C2311yescharacteristic
103C22×C7⋊C8C2311yescharacteristic
104C22×C7⋊C8C2311yescharacteristic
105C7×#107C2×C414yescomplemented
106C7×#106C2×C414yescomplemented
107C7×#103C2×C414yescomplemented
108C7×#103C2×C414yescomplemented
109C7×C22⋊C8C2×C414yes—
110#6409—22no—
111#6409—22no—
112C7⋊#95—21no—
113C7⋊#95—21no—
114C7⋊#95—21no—
115C7⋊#95—21no—
116C7⋊#95—21no—
117C7⋊#95—21no—
118C7⋊#95—21no—
119C7⋊#95—21no—
120C7⋊#95—21no—
121C22⋊(C7⋊C8)—22no—
122#6392—22no—
123#6391—22no—
124C8×Dic7—22no—
125C8×Dic7—22no—
126C2×#485D711yescharacteristic
127C4×C22⋊Q8—71no—
128C4×#117—71no—
129#1928—71no—
130#1927—71no—
131#1919—71no—
132#1918—71no—
133#1873—71no—
134#1854—71no—
135#1825—71no—
136C4×#107—71no—
137C4×#106—71no—
138C4×#103—71no—
139C4×#103—71no—
140C4×C22⋊C8—71no—
141C24×C7C2×D411yescharacteristic
142C2×C7×Q8—21no—
143C2×C7×Q8—21no—
144C2×C7×Q8—21no—
145C2×C7×Q8—21no—
146C2×C7×Q8—21no—
147C2×C7×Q8C22×C412yes—
148C22×C4×C7—21no—
149C22×C4×C7—21no—
150C22×C4×C7—21no—
151C22×C4×C7—21no—
152C22×C4×C7—21no—
153C22×C4×C7—21no—
154C22×C4×C7—21no—
155C22×C4×C7—21no—
156C22×C4×C7C22×C412yes—
157C22×C4×C7C2×D411yescharacteristic
158C22×C4×C7#4112yes—
159C22×C4×C7C2411yescharacteristic
160C22×Dic7—21no—
161C22×Dic7—21no—
162C22×Dic7—22no—
163C22×Dic7—22no—
164C22×Dic7—22no—
165C22×Dic7—22no—
166C22×Dic7—22no—
167C22×Dic7—21no—
168C22×Dic7—21no—
169C22×Dic7—41no—
170C22×Dic7C2×D411yescharacteristic
171C22×Dic7C2×D411yescharacteristic
172C22×Dic7C2×D411yescharacteristic
173C22×Dic7C2×D411yescharacteristic
174C2×C7×C8—21no—
175C2×C7×C8—21no—
176C2×C7×C8—21no—
177C2×C7×C8—21no—
178C2×C7×C8C22×C412yes—
179C2×C7×C8C22×C412yes—
180C7×C4⋊C4—24no—
181C7×C4⋊C4—21no—
182C7×C4⋊C4—21no—
183C7×C4⋊C4—21no—
184C7×C4⋊C4C22×C412yes—
185C7×C4⋊C4C22×C412yes—
186C7×C4⋊C4C22×C412yes—
187C7×C22⋊C4—24no—
188C7×C22⋊C4—24no—
189C42×C7—24no—
190C22⋊Dic7—24no—
191C22⋊Dic7—42no—
192C4⋊Dic7—21no—
193C4⋊Dic7—21no—
194C4⋊Dic7—21no—
195C4⋊Dic7—21no—
196C4⋊Dic7—21no—
197C4⋊Dic7—21no—
198C4⋊Dic7—42no—
199C4⋊Dic7—42no—
200C4⋊Dic7#4112yes—
201C4⋊Dic7#4112yes—
202C4×Dic7—24no—
203C4×Dic7—21no—
204C4×Dic7—21no—
205C4×Dic7—22no—
206C4×Dic7—22no—
207C4×Dic7—42no—
208C4×Dic7—42no—
209C4×Dic7—42no—
210C4×Dic7—42no—
211C4×Dic7—42no—
212C4×Dic7—42no—
213C4×Dic7#4112yes—
214C4×Dic7#4112yes—
215C2×C7⋊C8—21no—
216C2×C7⋊C8—21no—
217C2×C7⋊C8—21no—
218C2×C7⋊C8—21no—
219C2×C7⋊C8—21no—
220C2×C7⋊C8—21no—
221C2×C22⋊Q8C2×D711yescharacteristic
222C2×C4×Q8—71no—
223C2×#117—71no—
224C2×#117C2×D711yescharacteristic
225C22×C42—71no—
226#485Dic718yesdirect product factor
227C2×#107—71no—
228C2×#107C2×D711yescharacteristic
229C2×#106—71no—
230C2×#106C2×D711yescharacteristic
231C2×#103—71no—
232C2×#103—71no—
233C2×#103C2×D711yescharacteristic
234C2×#103C2×D711yescharacteristic
235C2×C22⋊C8—71no—
236C2×C22⋊C8C2×D711yescharacteristic
237C2×C4×C8—71no—
238C2×C4×C8—71no—
239#391—71no—
240#389—71no—
241#387—141no—
242#385—141no—
243#380—71no—
244C4×C4⋊C4—71no—
245C4×C4⋊C4—71no—
246C4×C4⋊C4—71no—
247C4×C4⋊C4—71no—
248C4×C4⋊C4—71no—
249C4×C4⋊C4—141no—
250C4×C22⋊C4—141no—
251C4×C22⋊C4—141no—
252C4×C22⋊C4—141no—
253#376—141no—
254C43—141no—
255#340—71no—
256#340—71no—
257#340—71no—
258#340—71no—
259#340—71no—
260#340—71no—
261#340—71no—
262#340—71no—
263#336—71no—
264#336—71no—
265C23×C7—21no—
266C23×C7—21no—
267C23×C7—21no—
268C23×C7—21no—
269C23×C7—21no—
270C23×C7—21no—
271C23×C7C4×D412yes—
272C23×C7C22×D411yescharacteristic
273C7×Q8—22no—
274C7×Q8—22no—
275C7×Q8—41no—
276C7×Q8—41no—
277C2×C4×C7—21no—
278C2×C4×C7—21no—
279C2×C4×C7—22no—
280C2×C4×C7—22no—
281C2×C4×C7—22no—
282C2×C4×C7—22no—
283C2×C4×C7—22no—
284C2×C4×C7—22no—
285C2×C4×C7—22no—
286C2×C4×C7—22no—
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—41no—
295C2×C4×C7C4×D412yes—
296C2×C4×C7C23×C411yescharacteristic
297C2×C4×C7C2×#4111yescharacteristic
298C2×C4×C7C2×#4111yescharacteristic
299C2×C4×C7C2×#4111yescharacteristic
300C2×C4×C7C2×#4111yescharacteristic
301C2×C4×C7C2×#4111yescharacteristic
302C2×Dic7—21no—
303C2×Dic7—21no—
304C2×Dic7—21no—
305C2×Dic7—21no—
306C2×Dic7—21no—
307C2×Dic7—22no—
308C2×Dic7—22no—
309C2×Dic7—21no—
310C2×Dic7—21no—
311C2×Dic7—44no—
312C2×Dic7—41no—
313C2×Dic7—41no—
314C2×Dic7—41no—
315C2×Dic7—41no—
316C2×Dic7—44no—
317C2×Dic7—41no—
318C2×Dic7—41no—
319C2×Dic7—41no—
320C2×Dic7—41no—
321C2×Dic7—41no—
322C2×Dic7—41no—
323C2×Dic7#12312yes—
324C2×Dic7#12312yes—
325C2×Dic7#13512yes—
326C2×Dic7#13512yes—
327C2×Dic7#13712yes—
328C7×C8—22no—
329C7×C8—22no—
330C7⋊C8—41no—
331C7⋊C8—41no—
332C22×Q8C22×D711yescharacteristic
333C23×C4—71no—
334C23×C4—71no—
335C23×C4C22×D711yescharacteristic
336C22×C8—71no—
337C22×C8—71no—
338C22×C8C22×D711yescharacteristic
339C22×C8C22×D711yescharacteristic
340C22⋊Q8—22no—
341C22⋊Q8C2×Dic714yes—
342C4×Q8—142no—
343C4×Q8—142no—
344#117—22no—
345#117—142no—
346#117—142no—
347#117C2×Dic714yes—
348C2×C4⋊C4—21no—
349C2×C4⋊C4—71no—
350C2×C4⋊C4—71no—
351C2×C4⋊C4—71no—
352C2×C4⋊C4—71no—
353C2×C4⋊C4—141no—
354C2×C4⋊C4—141no—
355C2×C4⋊C4C22×D711yescharacteristic
356C2×C4⋊C4C22×D711yescharacteristic
357C2×C4⋊C4C22×D711yescharacteristic
358C2×C22⋊C4—21no—
359C2×C22⋊C4—21no—
360C2×C22⋊C4—141no—
361C2×C22⋊C4—141no—
362C2×C42—21no—
363C2×C42—71no—
364C2×C42—71no—
365C2×C42—71no—
366C2×C42—71no—
367C2×C42—142no—
368C2×C42—142no—
369C2×C42—141no—
370C2×C42—141no—
371C2×C42—141no—
372C2×C42—141no—
373C2×C42—141no—
374C2×C42—141no—
375C2×C42—141no—
376#107—142no—
377#107C2×Dic714yescomplemented
378#106—142no—
379#106C2×Dic714yescomplemented
380#103—142no—
381#103—142no—
382#103C2×Dic714yescomplemented
383#103C2×Dic714yescomplemented
384C22⋊C8—142no—
385C22⋊C8C2×Dic714yes—
386C4×C8—142no—
387C4×C8—142no—
388#95—141no—
389#95—141no—
390#95—141no—
391#95—141no—
392#95—141no—
393#95—141no—
394#95—141no—
395#95—141no—
396#95—141no—
397C22×C7—22no—
398C22×C7—22no—
399C22×C7—22no—
400C22×C7—41no—
401C22×C7—41no—
402C22×C7—41no—
403C22×C7—41no—
404C22×C7C2×C4×D411yescharacteristic
405C22×C7C2×#12311yescharacteristic
406C22×C7#52511yescharacteristic
407C22×C7C2×#13511yescharacteristic
408C22×C7C2×#13511yescharacteristic
409C22×C7C2×#13711yescharacteristic
410C22×C7#57511yescharacteristic
411C4×C7—22no—
412C4×C7—42no—
413C4×C7—42no—
414C4×C7—42no—
415C4×C7—42no—
416C4×C7C4×#4111yescharacteristic
417C4×C7C4×#4111yescharacteristic
418C4×C7#52911yescharacteristic
419C4×C7#52911yescharacteristic
420Dic7—22no—
421Dic7—22no—
422Dic7—41no—
423Dic7—42no—
424Dic7—42no—
425Dic7—81no—
426Dic7—81no—
427Dic7#48514yesdirect product factor
428C24D4×D711yescharacteristic
429C2×Q8—21no—
430C2×Q8—21no—
431C2×Q8—21no—
432C2×Q8—21no—
433C2×Q8—21no—
434C2×Q8C22×Dic712yes—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—71no—
444C22×C4—71no—
445C22×C4—71no—
446C22×C4—71no—
447C22×C4—142no—
448C22×C4—142no—
449C22×C4—142no—
450C22×C4—142no—
451C22×C4—142no—
452C22×C4—141no—
453C22×C4—141no—
454C22×C4—141no—
455C22×C4—141no—
456C22×C4—281no—
457C22×C4D4×D711yescharacteristic
458C22×C4#115912yes—
459C22×C4C22×Dic712yes—
460C22×C4C23×D711yescharacteristic
461C2×C8—21no—
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—21no—
465C2×C8—141no—
466C2×C8—141no—
467C2×C8—141no—
468C2×C8—141no—
469C2×C8—141no—
470C2×C8—141no—
471C2×C8C22×Dic712yes—
472C2×C8C22×Dic712yes—
473C4⋊C4—24no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—72no—
478C4⋊C4—72no—
479C4⋊C4—141no—
480C4⋊C4—141no—
481C4⋊C4—141no—
482C4⋊C4—141no—
483C4⋊C4—141no—
484C4⋊C4—141no—
485C4⋊C4—282no—
486C4⋊C4—282no—
487C4⋊C4C22×Dic712yes—
488C4⋊C4C22×Dic712yes—
489C4⋊C4C22×Dic712yes—
490C22⋊C4—24no—
491C22⋊C4—24no—
492C22⋊C4—144no—
493C22⋊C4—282no—
494C42—24no—
495C42—72no—
496C42—72no—
497C42—142no—
498C42—142no—
499C42—144no—
500C42—141no—
501C42—141no—
502C42—282no—
503C42—282no—
504C42—282no—
505C42—282no—
506C42—282no—
507C42—282no—
508C2×C7—42no—
509C2×C7C4×#12311yescharacteristic
510C2×C7C4×#13511yescharacteristic
511C2×C7C4×#13711yescharacteristic
512C2×C7C2×#48511yescharacteristic
513C2×C7#309111yescharacteristic
514C2×C7#310211yescharacteristic
515C2×C7#310711yescharacteristic
516C23—21no—
517C23—21no—
518C23—21no—
519C23—21no—
520C23—21no—
521C23—21no—
522C23D4×Dic712yes—
523C23C2×D4×D711yescharacteristic
524Q8—22no—
525Q8—22no—
526Q8—41no—
527Q8—41no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—72no—
547C2×C4—72no—
548C2×C4—72no—
549C2×C4—72no—
550C2×C4—72no—
551C2×C4—141no—
552C2×C4—141no—
553C2×C4—141no—
554C2×C4—141no—
555C2×C4—141no—
556C2×C4—141no—
557C2×C4—141no—
558C2×C4—142no—
559C2×C4—142no—
560C2×C4—284no—
561C2×C4—284no—
562C2×C4—281no—
563C2×C4—281no—
564C2×C4—281no—
565C2×C4—281no—
566C2×C4—281no—
567C2×C4—281no—
568C2×C4—281no—
569C2×C4—281no—
570C2×C4—281no—
571C2×C4—281no—
572C2×C4D4×Dic712yes—
573C2×C4C2×#115911yescharacteristic
574C2×C4D7×#4111yescharacteristic
575C2×C4D7×#4111yescharacteristic
576C2×C4D7×#4111yescharacteristic
577C2×C4D7×#4111yescharacteristic
578C2×C4C23×Dic711yescharacteristic
579C8—22no—
580C8—22no—
581C8—281no—
582C8—281no—
583C7C4×#48511yescharacteristic, complemented, Sylow
584C22—22no—
585C22—22no—
586C22—22no—
587C22—41no—
588C22—41no—
589C22—41no—
590C22—41no—
591C22D7×#12311yescharacteristic
592C22#9034711yescharacteristic
593C22D7×#13511yescharacteristic
594C22D7×#13511yescharacteristic
595C22D7×#13711yescharacteristic
596C22#9046811yescharacteristic
597C22C2×D4×Dic711yescharacteristic
598C4—22no—
599C4—42no—
600C4—42no—
601C4—42no—
602C4—42no—
603C4—74no—
604C4—142no—
605C4—142no—
606C4—282no—
607C4—282no—
608C4—281no—
609C4—561no—
610C4—561no—
611C4#9034011yescharacteristic
612C4#9034011yescharacteristic
613C4#41×Dic711yescharacteristic
614C4#41×Dic711yescharacteristic
615C2—42no—
616C2#1173893711yescharacteristic
617C2#1173893811yescharacteristic
618C2D7×#48511yescharacteristic
619C2#1173896011yescharacteristic
620C2Dic7×#12311yescharacteristic
621C2Dic7×#13511yescharacteristic
622C2Dic7×#13711yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC2×C7×#485C2
socleC23×C7C22×D4
prime radicalC24×C7C2×D4
cosocleC22×C4×C7C24
hypercenterC2×#485D7
hypocenterC7C4×#485
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#485

Complements

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

NH ≅ G/Ntype
6. #11728363C2semidirect
7. Dic7×#107C2semidirect
8. #11728311C2semidirect
9. Dic7×#106C2semidirect
10. #11727978C2semidirect
11. #11727977C2semidirect
12. Dic7×#103C2semidirect
13. Dic7×#103C2semidirect
21. C7×#485C4semidirect
105. C7×#107C2×C4semidirect
106. C7×#106C2×C4semidirect
107. C7×#103C2×C4semidirect
108. C7×#103C2×C4semidirect
226. #485Dic7direct (N × H)
377. #107C2×Dic7semidirect
379. #106C2×Dic7semidirect
382. #103C2×Dic7semidirect
383. #103C2×Dic7semidirect
427. Dic7#485direct (N × H)
583. C7C4×#485semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#4852
Dic7×C22⋊Q82
Dic7×#1172
#117294472
#117293292
#117283632
Dic7×#1072
#117283112
Dic7×#1062
#117279782
#117279772
Dic7×#1032
#117276832
Dic7×C22⋊C82
C4×#4857