Finite Group Databasebeta

← Dic7×#450 #13359377

Subgroups — Dic7×#450 #13359377

Order 1792.

1880 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
1C22×D4×Dic7C211yesmaximal, characteristic
2C2×C7×#450C211yesmaximal, characteristic
3C2×SD16×Dic7C212yesmaximal, complemented
4Dic7×C22⋊Q8C211yesmaximal, characteristic, complemented
5#11729882C211yesmaximal, characteristic
6#11729801C212yesmaximal, complemented
7#11729446C211yesmaximal, characteristic, complemented
8#11727924C212yesmaximal, complemented
9Dic7×#102C212yesmaximal, complemented
10#11727682C211yesmaximal, characteristic, complemented
11Dic7×C22⋊C8C211yesmaximal, characteristic, complemented
12C23×C7×D4C2211yescharacteristic
13C24×Dic7—21no—
14C22×C7×SD16C2212yes—
15C2×C7×C22⋊Q8C2211yescharacteristic
16C22×C22⋊Dic7—21no—
17C2×Q8×Dic7C2211yescharacteristic, complemented
18C2×D4×Dic7—28no—
19C2×D4×Dic7—21no—
20C2×D4×Dic7—21no—
21C2×D4×Dic7C2212yes—
22C2×D4×Dic7C2212yes—
23C22×C4⋊Dic7C2211yescharacteristic
24C22×C4×Dic7C2211yescharacteristic
25C7×#450C418yescomplemented
26C2×C7×#102C2212yes—
27C2×C7×C22⋊C8C2211yescharacteristic
28#90002C2211yescharacteristic, complemented
29C2×#6409C2212yescomplemented
30C2×#6406C2212yes—
31C2×#6406C2212yes—
32#89984C2211yescharacteristic
33C2×C22⋊(C7⋊C8)C2211yescharacteristic
34SD16×Dic7—28no—
35C2×#6391C2212yescomplemented
36C2×C8×Dic7C2212yescomplemented
37#89757C2211yescharacteristic, complemented
38#89753—21no—
39C4⋊C4×Dic7—21no—
40C4⋊C4×Dic7C2211yescharacteristic, complemented
41#89722—21no—
42C22⋊C4×Dic7—21no—
43#89346C2212yescomplemented
44#89345C2212yescomplemented
45#89324C2212yescomplemented
46C4×#450—71nomaximal, Sylow
47C25×C7—21no—
48C22×C7×Q8C2311yescharacteristic
49C22×C7×D4—24no—
50C22×C7×D4—24no—
51C22×C7×D4—21no—
52C22×C7×D4—22no—
53C22×C7×D4C2×C414yes—
54C22×C7×D4C2312yes—
55C23×C4×C7C2311yescharacteristic
56C23×Dic7—28no—
57C23×Dic7—21no—
58C23×Dic7—22no—
59C23×Dic7—22no—
60C23×Dic7—44no—
61C23×Dic7—44no—
62C23×Dic7D411yescharacteristic
63C23×Dic7D411yescharacteristic
64C2×C7×SD16—24no—
65C2×C7×SD16—24no—
66C2×C7×SD16C2×C418yescomplemented
67C22×C7×C8C2312yes—
68C7×C22⋊Q8—22no—
69C7×C22⋊Q8C2×C414yescomplemented
70C2×C7×C4⋊C4—21no—
71C2×C7×C4⋊C4C2311yescharacteristic
72C2×C7×C22⋊C4—21no—
73C2×C22⋊Dic7—28no—
74C2×C22⋊Dic7—22no—
75C2×C22⋊Dic7—22no—
76C2×C22⋊Dic7—21no—
77Q8×Dic7—24no—
78D4×Dic7—28no—
79D4×Dic7—28no—
80D4×Dic7—24no—
81D4×Dic7—24no—
82D4×Dic7—24no—
83D4×Dic7D418yes—
84C2×C4⋊Dic7—21no—
85C2×C4⋊Dic7—24no—
86C2×C4⋊Dic7—21no—
87C2×C4⋊Dic7—21no—
88C2×C4⋊Dic7C2311yescharacteristic
89C2×C4⋊Dic7C2311yescharacteristic
90C2×C4⋊Dic7C2312yes—
91C2×C4×Dic7—24no—
92C2×C4×Dic7—21no—
93C2×C4×Dic7—21no—
94C2×C4×Dic7D411yescharacteristic
95C2×C4×Dic7D411yescharacteristic
96C2×C4×Dic7C2311yescharacteristic
97C2×C4×Dic7C2311yescharacteristic
98C22×C7⋊C8C2312yes—
99C7×#102C2×C418yescomplemented
100C7×C22⋊C8C2×C414yescomplemented
101#6409—24no—
102#6406—24no—
103#6406—24no—
104C7⋊#95—21no—
105C7⋊#95—21no—
106C7⋊#95—21no—
107C7⋊#95—21no—
108C7⋊#95—21no—
109C22⋊(C7⋊C8)—22no—
110#6391—24no—
111C8×Dic7—24no—
112C22×C4×D4—71no—
113C2×#450D711yescharacteristic
114C2×C4×SD16—72no—
115C4×C22⋊Q8—71no—
116#1877—72no—
117#1872—71no—
118#1864—72no—
119#1853—71no—
120#1786—71no—
121C4×#102—72no—
122C4×C22⋊C8—71no—
123C24×C7—24no—
124C24×C7—24no—
125C24×C7—24no—
126C24×C7—22no—
127C24×C7—42no—
128C24×C7—42no—
129C24×C7—42no—
130C24×C7—42no—
131C24×C7C2×D411yescharacteristic
132C2×C7×Q8—21no—
133C2×C7×Q8—22no—
134C2×C7×Q8—22no—
135C2×C7×Q8C22×C412yescomplemented
136C2×C7×D4—24no—
137C2×C7×D4—24no—
138C2×C7×D4—216no—
139C2×C7×D4—216no—
140C2×C7×D4—24no—
141C2×C7×D4—22no—
142C2×C7×D4—22no—
143C2×C7×D4—22no—
144C2×C7×D4C22×C414yes—
145C2×C7×D4C2×D414yes—
146C2×C7×D4#4114yes—
147C22×C4×C7—22no—
148C22×C4×C7—22no—
149C22×C4×C7—21no—
150C22×C4×C7—21no—
151C22×C4×C7C22×C412yes—
152C22×C4×C7C2×D411yescharacteristic
153C22×C4×C7#4112yes—
154C22×C4×C7C2411yescharacteristic
155C22×Dic7—22no—
156C22×Dic7—22no—
157C22×Dic7—24no—
158C22×Dic7—28no—
159C22×Dic7—22no—
160C22×Dic7—22no—
161C22×Dic7—22no—
162C22×Dic7—28no—
163C22×Dic7—21no—
164C22×Dic7—21no—
165C22×Dic7—41no—
166C22×Dic7—44no—
167C22×Dic7—432no—
168C22×Dic7—42no—
169C22×Dic7—42no—
170C22×Dic7—44no—
171C22×Dic7—44no—
172C22×Dic7—44no—
173C22×Dic7SD1614yescomplemented
174C22×Dic7C2×D411yescharacteristic
175C22×Dic7C2×D411yescharacteristic
176C22×Dic7C2×D411yescharacteristic
177C22×Dic7C2×D411yescharacteristic
178C7×SD16—216no—
179C2×C7×C8—22no—
180C2×C7×C8—22no—
181C2×C7×C8C22×C414yescomplemented
182C7×C4⋊C4—24no—
183C7×C4⋊C4—21no—
184C7×C4⋊C4C22×C412yescomplemented
185C7×C22⋊C4—24no—
186C22⋊Dic7—28no—
187C22⋊Dic7—28no—
188C22⋊Dic7—42no—
189C4⋊Dic7—24no—
190C4⋊Dic7—22no—
191C4⋊Dic7—22no—
192C4⋊Dic7—22no—
193C4⋊Dic7—42no—
194C4⋊Dic7—42no—
195C4⋊Dic7C2×D414yes—
196C4×Dic7—22no—
197C4×Dic7—22no—
198C4×Dic7—22no—
199C4×Dic7—42no—
200C4×Dic7—42no—
201C4×Dic7C2×D414yes—
202C2×C7⋊C8—22no—
203C2×C7⋊C8—22no—
204C2×C7⋊C8—22no—
205C23×D4C2×D711yescharacteristic
206C24×C4—141no—
207C22×SD16C2×D712yes—
208C2×C22⋊Q8C2×D711yescharacteristic
209C2×C4×Q8—71no—
210C2×C4×D4—72no—
211C2×C4×D4—72no—
212C2×C4×D4—141no—
213C2×C4×D4—141no—
214C2×C4×D4—148no—
215C22×C4⋊C4—71no—
216C22×C22⋊C4—141no—
217C22×C42—71no—
218#450Dic718yesdirect product factor
219C4×SD16—148no—
220C2×#106—72no—
221C2×#103—72no—
222C2×#102—72no—
223C2×#102—72no—
224C2×#102C2×D712yes—
225C2×C22⋊C8—71no—
226C2×C22⋊C8C2×D711yescharacteristic
227C2×C4×C8—72no—
228#391—71no—
229#389—71no—
230#387—141no—
231#385—141no—
232#380—71no—
233C4×C4⋊C4—71no—
234C4×C4⋊C4—141no—
235C4×C22⋊C4—141no—
236#340—72no—
237#340—72no—
238#336—72no—
239C23×C7—21no—
240C23×C7—21no—
241C23×C7—21no—
242C23×C7—21no—
243C23×C7—216no—
244C23×C7—24no—
245C23×C7—24no—
246C23×C7—24no—
247C23×C7—48no—
248C23×C7—48no—
249C23×C7—48no—
250C23×C7—48no—
251C23×C7—48no—
252C23×C7—42no—
253C23×C7—42no—
254C23×C7—42no—
255C23×C7—42no—
256C23×C7—48no—
257C23×C7C4×D412yes—
258C23×C7C2×SD1612yes—
259C23×C7#13512yes—
260C23×C7C22×D411yescharacteristic
261C7×Q8—24no—
262C7×Q8—42no—
263C7×D4—216no—
264C7×D4—24no—
265C7×D4—24no—
266C7×D4—24no—
267C7×D4C4×D418yes—
268C2×C4×C7—21no—
269C2×C4×C7—21no—
270C2×C4×C7—24no—
271C2×C4×C7—24no—
272C2×C4×C7—22no—
273C2×C4×C7—22no—
274C2×C4×C7—41no—
275C2×C4×C7—41no—
276C2×C4×C7—41no—
277C2×C4×C7—41no—
278C2×C4×C7C4×D412yes—
279C2×C4×C7C23×C411yescharacteristic
280C2×C4×C7C22×D412yes—
281C2×C4×C7C2×#4112yes—
282C2×C4×C7C2×#4111yescharacteristic
283C2×Dic7—21no—
284C2×Dic7—21no—
285C2×Dic7—21no—
286C2×Dic7—21no—
287C2×Dic7—28no—
288C2×Dic7—21no—
289C2×Dic7—24no—
290C2×Dic7—21no—
291C2×Dic7—21no—
292C2×Dic7—44no—
293C2×Dic7—416no—
294C2×Dic7—48no—
295C2×Dic7—42no—
296C2×Dic7—42no—
297C2×Dic7—416no—
298C2×Dic7—41no—
299C2×Dic7—41no—
300C2×Dic7—41no—
301C2×Dic7—41no—
302C2×Dic7—41no—
303C2×Dic7—41no—
304C2×Dic7#12012yes—
305C2×Dic7#12012yes—
306C2×Dic7C2×SD1612yes—
307C2×Dic7C2×SD1612yes—
308C2×Dic7AGL(1,ℤ/8)12yes—
309C7×C8—24no—
310C7⋊C8—42no—
311C25—21no—
312C22×Q8C22×D711yescharacteristic
313C22×D4—24no—
314C22×D4—24no—
315C22×D4—21no—
316C22×D4—22no—
317C22×D4C2×Dic714yes—
318C22×D4C22×D712yes—
319C23×C4—71no—
320C23×C4—71no—
321C23×C4—142no—
322C23×C4—142no—
323C23×C4—148no—
324C23×C4—141no—
325C23×C4—284no—
326C23×C4—284no—
327C23×C4C22×D711yescharacteristic
328C2×SD16—24no—
329C2×SD16—24no—
330C2×SD16C2×Dic718yescomplemented
331C22×C8—72no—
332C22×C8C22×D712yes—
333C22⋊Q8—22no—
334C22⋊Q8C2×Dic714yescomplemented
335C4×Q8—144no—
336C4×D4—78no—
337C4×D4—144no—
338C4×D4—144no—
339C4×D4—148no—
340C4×D4—148no—
341C4×D4—144no—
342C2×C4⋊C4—21no—
343C2×C4⋊C4—72no—
344C2×C4⋊C4—71no—
345C2×C4⋊C4—71no—
346C2×C4⋊C4—144no—
347C2×C4⋊C4—141no—
348C2×C4⋊C4—141no—
349C2×C4⋊C4—141no—
350C2×C4⋊C4C22×D711yescharacteristic
351C2×C22⋊C4—21no—
352C2×C22⋊C4—142no—
353C2×C22⋊C4—142no—
354C2×C22⋊C4—148no—
355C2×C22⋊C4—141no—
356C2×C42—71no—
357C2×C42—71no—
358C2×C42—71no—
359C2×C42—71no—
360C2×C42—144no—
361C2×C42—141no—
362C2×C42—141no—
363#106—144no—
364#103—144no—
365#102—144no—
366#102—144no—
367#102C2×Dic718yescomplemented
368C22⋊C8—142no—
369C22⋊C8C2×Dic714yescomplemented
370C4×C8—144no—
371#95—141no—
372#95—141no—
373#95—141no—
374#95—141no—
375#95—141no—
376C22×C7—22no—
377C22×C7—22no—
378C22×C7—22no—
379C22×C7—22no—
380C22×C7—22no—
381C22×C7—22no—
382C22×C7—28no—
383C22×C7—432no—
384C22×C7—44no—
385C22×C7—44no—
386C22×C7—44no—
387C22×C7—44no—
388C22×C7—44no—
389C22×C7—44no—
390C22×C7C4×SD1614yescomplemented
391C22×C7C2×C4×D411yescharacteristic
392C22×C7C2×#12011yescharacteristic
393C22×C7#52511yescharacteristic
394C22×C7C22×SD1611yescharacteristic
395C22×C7C2×#13511yescharacteristic
396C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
397C22×C7#57511yescharacteristic
398C4×C7—22no—
399C4×C7—42no—
400C4×C7—42no—
401C4×C7C2×C4×D412yes—
402C4×C7#52512yes—
403Dic7—22no—
404Dic7—22no—
405Dic7—44no—
406Dic7—41no—
407Dic7—81no—
408Dic7—81no—
409Dic7#45014yesdirect product factor
410C24—24no—
411C24—24no—
412C24—24no—
413C24—22no—
414C24—42no—
415C24—42no—
416C24—42no—
417C24—42no—
418C24D4×D711yescharacteristic
419C2×Q8—21no—
420C2×Q8—22no—
421C2×Q8—22no—
422C2×Q8C22×Dic712yescomplemented
423C2×D4—24no—
424C2×D4—24no—
425C2×D4—216no—
426C2×D4—216no—
427C2×D4—24no—
428C2×D4—22no—
429C2×D4—22no—
430C2×D4—22no—
431C2×D4D4×D714yes—
432C2×D4#115914yes—
433C2×D4C22×Dic714yes—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—71no—
439C22×C4—74no—
440C22×C4—71no—
441C22×C4—71no—
442C22×C4—71no—
443C22×C4—142no—
444C22×C4—142no—
445C22×C4—142no—
446C22×C4—148no—
447C22×C4—144no—
448C22×C4—148no—
449C22×C4—142no—
450C22×C4—142no—
451C22×C4—141no—
452C22×C4—141no—
453C22×C4—284no—
454C22×C4—284no—
455C22×C4—284no—
456C22×C4—281no—
457C22×C4—2832no—
458C22×C4—284no—
459C22×C4—282no—
460C22×C4—282no—
461C22×C4D4×D711yescharacteristic
462C22×C4#115912yes—
463C22×C4C22×Dic712yes—
464C22×C4C23×D711yescharacteristic
465SD16—216no—
466C2×C8—22no—
467C2×C8—22no—
468C2×C8—142no—
469C2×C8—142no—
470C2×C8—142no—
471C2×C8C22×Dic714yescomplemented
472C4⋊C4—24no—
473C4⋊C4—21no—
474C4⋊C4—74no—
475C4⋊C4—144no—
476C4⋊C4—142no—
477C4⋊C4—142no—
478C4⋊C4—142no—
479C4⋊C4—282no—
480C4⋊C4—282no—
481C4⋊C4C22×Dic712yescomplemented
482C22⋊C4—24no—
483C22⋊C4—148no—
484C22⋊C4—148no—
485C22⋊C4—282no—
486C42—74no—
487C42—142no—
488C42—142no—
489C42—142no—
490C42—282no—
491C42—282no—
492C2×C7—24no—
493C2×C7—48no—
494C2×C7C4×#12011yescharacteristic
495C2×C7C2×C4×SD1611yescharacteristic
496C2×C7C4×AGL(1,ℤ/8)11yescharacteristic
497C2×C7C2×#45011yescharacteristic
498C2×C7#300211yescharacteristic
499C2×C7#310111yescharacteristic
500C2×C7#310511yescharacteristic
501C23—21no—
502C23—21no—
503C23—21no—
504C23—21no—
505C23—216no—
506C23—24no—
507C23—24no—
508C23—24no—
509C23—48no—
510C23—48no—
511C23—48no—
512C23—48no—
513C23—48no—
514C23—48no—
515C23—42no—
516C23—42no—
517C23—42no—
518C23—42no—
519C23D7×SD1612yes—
520C23#647912yes—
521C23D4×Dic712yes—
522C23C2×D4×D711yescharacteristic
523Q8—24no—
524Q8—42no—
525D4—216no—
526D4—24no—
527D4—24no—
528D4—24no—
529D4D4×Dic718yes—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—72no—
541C2×C4—72no—
542C2×C4—72no—
543C2×C4—72no—
544C2×C4—72no—
545C2×C4—148no—
546C2×C4—141no—
547C2×C4—141no—
548C2×C4—141no—
549C2×C4—141no—
550C2×C4—141no—
551C2×C4—141no—
552C2×C4—141no—
553C2×C4—144no—
554C2×C4—2816no—
555C2×C4—284no—
556C2×C4—2816no—
557C2×C4—288no—
558C2×C4—282no—
559C2×C4—282no—
560C2×C4—281no—
561C2×C4—281no—
562C2×C4—281no—
563C2×C4—281no—
564C2×C4—281no—
565C2×C4—281no—
566C2×C4D4×Dic712yes—
567C2×C4C2×D4×D712yes—
568C2×C4C2×#115912yes—
569C2×C4C2×#115911yescharacteristic
570C2×C4C23×Dic711yescharacteristic
571C8—24no—
572C8—282no—
573C7C4×#45011yescharacteristic, complemented, Sylow
574C22—22no—
575C22—22no—
576C22—22no—
577C22—22no—
578C22—22no—
579C22—22no—
580C22—28no—
581C22—432no—
582C22—44no—
583C22—44no—
584C22—44no—
585C22—44no—
586C22—44no—
587C22—44no—
588C22SD16×Dic714yescomplemented
589C22#9027711yescharacteristic
590C22D7×#12011yescharacteristic
591C22C2×D7×SD1611yescharacteristic
592C22C2×#647911yescharacteristic
593C22D7×AGL(1,ℤ/8)11yescharacteristic
594C22#9046411yescharacteristic
595C22C2×D4×Dic711yescharacteristic
596C4—22no—
597C4—42no—
598C4—42no—
599C4—74no—
600C4—142no—
601C4—142no—
602C4—284no—
603C4—281no—
604C4—561no—
605C4—561no—
606C4#9027712yes—
607C4C2×D4×Dic712yes—
608C2—24no—
609C2—48no—
610C2#1173757711yescharacteristic
611C2#1173757811yescharacteristic
612C2D7×#45011yescharacteristic
613C2#1173759011yescharacteristic
614C2Dic7×#12011yescharacteristic
615C2C2×SD16×Dic711yescharacteristic
616C2Dic7×AGL(1,ℤ/8)11yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC2×C7×#450C2
socleC23×C7C22×D4
prime radicalC23×C7×D4C22
cosocleC22×C4×C7C24
hypercenterC2×#450D7
hypocenterC7C4×#450
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#450

Complements

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

NH ≅ G/Ntype
3. C2×SD16×Dic7C2semidirect
4. Dic7×C22⋊Q8C2semidirect
6. #11729801C2semidirect
7. #11729446C2semidirect
8. #11727924C2semidirect
9. Dic7×#102C2semidirect
10. #11727682C2semidirect
11. Dic7×C22⋊C8C2semidirect
17. C2×Q8×Dic7C22semidirect
25. C7×#450C4semidirect
28. #90002C22semidirect
29. C2×#6409C22semidirect
35. C2×#6391C22semidirect
36. C2×C8×Dic7C22semidirect
37. #89757C22semidirect
40. C4⋊C4×Dic7C22semidirect
43. #89346C22semidirect
44. #89345C22semidirect
45. #89324C22semidirect
66. C2×C7×SD16C2×C4semidirect
69. C7×C22⋊Q8C2×C4semidirect
99. C7×#102C2×C4semidirect
100. C7×C22⋊C8C2×C4semidirect
135. C2×C7×Q8C22×C4semidirect
173. C22×Dic7SD16semidirect
181. C2×C7×C8C22×C4semidirect
184. C7×C4⋊C4C22×C4semidirect
218. #450Dic7direct (N × H)
330. C2×SD16C2×Dic7semidirect
334. C22⋊Q8C2×Dic7semidirect
367. #102C2×Dic7semidirect
369. C22⋊C8C2×Dic7semidirect
390. C22×C7C4×SD16semidirect
409. Dic7#450direct (N × H)
422. C2×Q8C22×Dic7semidirect
471. C2×C8C22×Dic7semidirect
481. C4⋊C4C22×Dic7semidirect
573. C7C4×#450semidirect
588. C22SD16×Dic7semidirect

Maximal subgroups

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

subgroupindex
C22×D4×Dic72
C2×C7×#4502
C2×SD16×Dic72
Dic7×C22⋊Q82
#117298822
#117298012
#117294462
#117279242
Dic7×#1022
#117276822
Dic7×C22⋊C82
C4×#4507