Finite Group Databasebeta

← #450×Dic53 #27389673

Subgroups — #450×Dic53 #27389673

Order 13568.

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×Dic53C211yesmaximal, characteristic
2C2×SD16×Dic53C212yesmaximal, complemented
3C22⋊Q8×Dic53C211yesmaximal, characteristic, complemented
4#20275051C212yesmaximal, complemented
5#20275023C211yesmaximal, characteristic, complemented
6#20274978C212yesmaximal, complemented
7#20274910C211yesmaximal, characteristic, complemented
8#20274646C211yesmaximal, characteristic
9#102×Dic53C212yesmaximal, complemented
10C22⋊C8×Dic53C211yesmaximal, characteristic, complemented
11C2×C53×#450C211yesmaximal, characteristic
12C24×Dic53—21no—
13C2×Q8×Dic53C2211yescharacteristic, complemented
14C2×D4×Dic53—28no—
15C2×D4×Dic53—21no—
16C2×D4×Dic53—21no—
17C2×D4×Dic53C2212yes—
18C2×D4×Dic53C2212yes—
19C22×C4⋊Dic53C2211yescharacteristic
20C22×C22⋊Dic53—21no—
21C22×C4×Dic53C2211yescharacteristic
22SD16×Dic53—28no—
23C2×#13133149C2212yescomplemented
24C2×#13133141C2212yescomplemented
25C2×#13133138C2212yes—
26C2×#13133138C2212yes—
27C2×C22⋊#88322C2211yescharacteristic
28C2×C8×Dic53C2212yescomplemented
29#16810530C2211yescharacteristic, complemented
30#16810522C2211yescharacteristic, complemented
31#16810509—21no—
32#16810498—21no—
33#16810483C2211yescharacteristic
34C4⋊C4×Dic53—21no—
35C4⋊C4×Dic53C2211yescharacteristic, complemented
36C22⋊C4×Dic53—21no—
37#16810386C2212yescomplemented
38#16810385C2212yescomplemented
39#16810377C2212yescomplemented
40C23×D4×C53C2211yescharacteristic
41C22×SD16×C53C2212yes—
42C2×C22⋊Q8×C53C2211yescharacteristic
43C53×#450C418yescomplemented
44C2×#102×C53C2212yes—
45C2×C22⋊C8×C53C2211yescharacteristic
46C23×Dic53—28no—
47C23×Dic53—21no—
48C23×Dic53—22no—
49C23×Dic53—22no—
50C23×Dic53—44no—
51C23×Dic53—44no—
52C23×Dic53D411yescharacteristic
53C23×Dic53D411yescharacteristic
54C22×#88322C2312yes—
55Q8×Dic53—24no—
56D4×Dic53—28no—
57D4×Dic53—28no—
58D4×Dic53—24no—
59D4×Dic53—24no—
60D4×Dic53—24no—
61D4×Dic53D418yes—
62C2×C4⋊Dic53—24no—
63C2×C4⋊Dic53—21no—
64C2×C4⋊Dic53—21no—
65C2×C4⋊Dic53—21no—
66C2×C4⋊Dic53C2311yescharacteristic
67C2×C4⋊Dic53C2311yescharacteristic
68C2×C4⋊Dic53C2312yes—
69C2×C22⋊Dic53—28no—
70C2×C22⋊Dic53—21no—
71C2×C22⋊Dic53—22no—
72C2×C22⋊Dic53—22no—
73C2×C4×Dic53—24no—
74C2×C4×Dic53—21no—
75C2×C4×Dic53—21no—
76C2×C4×Dic53D411yescharacteristic
77C2×C4×Dic53D411yescharacteristic
78C2×C4×Dic53C2311yescharacteristic
79C2×C4×Dic53C2311yescharacteristic
80#13133149—24no—
81#13133141—24no—
82#13133138—24no—
83#13133138—24no—
84C22⋊#88322—22no—
85C8×Dic53—24no—
86#13133124—21no—
87#13133124—21no—
88#13133124—21no—
89#13133124—21no—
90#13133124—21no—
91C25×C53—21no—
92C22×Q8×C53C2311yescharacteristic
93C22×D4×C53—24no—
94C22×D4×C53—24no—
95C22×D4×C53—21no—
96C22×D4×C53—22no—
97C22×D4×C53C2×C414yes—
98C22×D4×C53C2312yes—
99C23×C4×C53C2311yescharacteristic
100C2×SD16×C53—24no—
101C2×SD16×C53—24no—
102C2×SD16×C53C2×C418yescomplemented
103C22×C8×C53C2312yes—
104C22⋊Q8×C53—22no—
105C22⋊Q8×C53C2×C414yescomplemented
106C2×C4⋊C4×C53—21no—
107C2×C4⋊C4×C53C2311yescharacteristic
108C2×C22⋊C4×C53—21no—
109#102×C53C2×C418yescomplemented
110C22⋊C8×C53C2×C414yescomplemented
111C24×C53—24no—
112C24×C53—24no—
113C24×C53—24no—
114C24×C53—22no—
115C24×C53—42no—
116C24×C53—42no—
117C24×C53—42no—
118C24×C53—42no—
119C24×C53C2×D411yescharacteristic
120C2×Q8×C53—21no—
121C2×Q8×C53—22no—
122C2×Q8×C53—22no—
123C2×Q8×C53C22×C412yescomplemented
124C2×D4×C53—24no—
125C2×D4×C53—216no—
126C2×D4×C53—24no—
127C2×D4×C53—216no—
128C2×D4×C53—24no—
129C2×D4×C53—22no—
130C2×D4×C53—22no—
131C2×D4×C53—22no—
132C2×D4×C53C22×C414yes—
133C2×D4×C53C2×D414yes—
134C2×D4×C53#4114yes—
135C22×C4×C53—22no—
136C22×C4×C53—22no—
137C22×C4×C53—21no—
138C22×C4×C53—21no—
139C22×C4×C53C22×C412yes—
140C22×C4×C53C2×D411yescharacteristic
141C22×C4×C53#4112yes—
142C22×C4×C53C2411yescharacteristic
143C22×Dic53—22no—
144C22×Dic53—22no—
145C22×Dic53—24no—
146C22×Dic53—28no—
147C22×Dic53—22no—
148C22×Dic53—22no—
149C22×Dic53—22no—
150C22×Dic53—28no—
151C22×Dic53—21no—
152C22×Dic53—21no—
153C22×Dic53—41no—
154C22×Dic53—44no—
155C22×Dic53—432no—
156C22×Dic53—42no—
157C22×Dic53—42no—
158C22×Dic53—44no—
159C22×Dic53—44no—
160C22×Dic53—44no—
161C22×Dic53SD1614yescomplemented
162C22×Dic53C2×D411yescharacteristic
163C22×Dic53C2×D411yescharacteristic
164C22×Dic53C2×D411yescharacteristic
165C22×Dic53C2×D411yescharacteristic
166SD16×C53—216no—
167C2×C8×C53—22no—
168C2×C8×C53—22no—
169C2×C8×C53C22×C414yescomplemented
170C4⋊C4×C53—24no—
171C4⋊C4×C53—21no—
172C4⋊C4×C53C22×C412yescomplemented
173C22⋊C4×C53—24no—
174C22⋊Dic53—28no—
175C22⋊Dic53—28no—
176C22⋊Dic53—42no—
177C4⋊Dic53—24no—
178C4⋊Dic53—22no—
179C4⋊Dic53—22no—
180C4⋊Dic53—22no—
181C4⋊Dic53—42no—
182C4⋊Dic53—42no—
183C4⋊Dic53C2×D414yes—
184C4×Dic53—22no—
185C4×Dic53—22no—
186C4×Dic53—22no—
187C4×Dic53—42no—
188C4×Dic53—42no—
189C4×Dic53C2×D414yes—
190C2×#88322—22no—
191C2×#88322—22no—
192C2×#88322—22no—
193C23×C53—21no—
194C23×C53—21no—
195C23×C53—21no—
196C23×C53—21no—
197C23×C53—216no—
198C23×C53—24no—
199C23×C53—24no—
200C23×C53—24no—
201C23×C53—48no—
202C23×C53—48no—
203C23×C53—48no—
204C23×C53—48no—
205C23×C53—48no—
206C23×C53—42no—
207C23×C53—42no—
208C23×C53—42no—
209C23×C53—42no—
210C23×C53—48no—
211C23×C53C4×D412yes—
212C23×C53C2×SD1612yes—
213C23×C53#13512yes—
214C23×C53C22×D411yescharacteristic
215Q8×C53—24no—
216Q8×C53—42no—
217D4×C53—216no—
218D4×C53—24no—
219D4×C53—24no—
220D4×C53—24no—
221D4×C53C4×D418yes—
222C2×C4×C53—21no—
223C2×C4×C53—21no—
224C2×C4×C53—24no—
225C2×C4×C53—24no—
226C2×C4×C53—22no—
227C2×C4×C53—22no—
228C2×C4×C53—41no—
229C2×C4×C53—41no—
230C2×C4×C53—41no—
231C2×C4×C53—41no—
232C2×C4×C53C4×D412yes—
233C2×C4×C53C23×C411yescharacteristic
234C2×C4×C53C22×D412yes—
235C2×C4×C53C2×#4112yes—
236C2×C4×C53C2×#4111yescharacteristic
237C2×Dic53—21no—
238C2×Dic53—21no—
239C2×Dic53—21no—
240C2×Dic53—21no—
241C2×Dic53—28no—
242C2×Dic53—24no—
243C2×Dic53—21no—
244C2×Dic53—21no—
245C2×Dic53—21no—
246C2×Dic53—44no—
247C2×Dic53—416no—
248C2×Dic53—48no—
249C2×Dic53—42no—
250C2×Dic53—42no—
251C2×Dic53—416no—
252C2×Dic53—41no—
253C2×Dic53—41no—
254C2×Dic53—41no—
255C2×Dic53—41no—
256C2×Dic53—41no—
257C2×Dic53—41no—
258C2×Dic53#12012yes—
259C2×Dic53#12012yes—
260C2×Dic53C2×SD1612yes—
261C2×Dic53C2×SD1612yes—
262C2×Dic53AGL(1,ℤ/8)12yes—
263C8×C53—24no—
264#88322—42no—
265C4×#450—531nomaximal, Sylow
266C22×C53—22no—
267C22×C53—22no—
268C22×C53—22no—
269C22×C53—22no—
270C22×C53—22no—
271C22×C53—22no—
272C22×C53—28no—
273C22×C53—432no—
274C22×C53—44no—
275C22×C53—44no—
276C22×C53—44no—
277C22×C53—44no—
278C22×C53—44no—
279C22×C53—44no—
280C22×C53C4×SD1614yescomplemented
281C22×C53C2×C4×D411yescharacteristic
282C22×C53C2×#12011yescharacteristic
283C22×C53#52511yescharacteristic
284C22×C53C22×SD1611yescharacteristic
285C22×C53C2×#13511yescharacteristic
286C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
287C22×C53#57511yescharacteristic
288C4×C53—22no—
289C4×C53—42no—
290C4×C53—42no—
291C4×C53C2×C4×D412yes—
292C4×C53#52512yes—
293Dic53—22no—
294Dic53—22no—
295Dic53—44no—
296Dic53—41no—
297Dic53—81no—
298Dic53—81no—
299Dic53#45014yesdirect product factor
300C22×C4×D4—531no—
301C2×#450D5311yescharacteristic
302C2×C4×SD16—532no—
303C4×C22⋊Q8—531no—
304#1877—532no—
305#1872—531no—
306#1864—532no—
307#1853—531no—
308#1786—531no—
309C4×#102—532no—
310C4×C22⋊C8—531no—
311C2×C53—24no—
312C2×C53—48no—
313C2×C53C4×#12011yescharacteristic
314C2×C53C2×C4×SD1611yescharacteristic
315C2×C53C4×AGL(1,ℤ/8)11yescharacteristic
316C2×C53C2×#45011yescharacteristic
317C2×C53#300211yescharacteristic
318C2×C53#310111yescharacteristic
319C2×C53#310511yescharacteristic
320C23×D4C2×D5311yescharacteristic
321C24×C4—1061no—
322C22×SD16C2×D5312yes—
323C2×C22⋊Q8C2×D5311yescharacteristic
324C2×C4×Q8—531no—
325C2×C4×D4—532no—
326C2×C4×D4—532no—
327C2×C4×D4—1061no—
328C2×C4×D4—1061no—
329C2×C4×D4—1068no—
330C22×C4⋊C4—531no—
331C22×C22⋊C4—1061no—
332C22×C42—531no—
333#450Dic5318yesdirect product factor
334C4×SD16—1068no—
335C2×#106—532no—
336C2×#103—532no—
337C2×#102—532no—
338C2×#102—532no—
339C2×#102C2×D5312yes—
340C2×C22⋊C8—531no—
341C2×C22⋊C8C2×D5311yescharacteristic
342C2×C4×C8—532no—
343#391—531no—
344#389—531no—
345#387—1061no—
346#385—1061no—
347#380—531no—
348C4×C4⋊C4—531no—
349C4×C4⋊C4—1061no—
350C4×C22⋊C4—1061no—
351#340—532no—
352#340—532no—
353#336—532no—
354C53C4×#45011yescharacteristic, complemented, Sylow
355C25—21no—
356C22×Q8C22×D5311yescharacteristic
357C22×D4—24no—
358C22×D4—24no—
359C22×D4—21no—
360C22×D4—22no—
361C22×D4C2×Dic5314yes—
362C22×D4C22×D5312yes—
363C23×C4—531no—
364C23×C4—531no—
365C23×C4—1062no—
366C23×C4—1062no—
367C23×C4—1068no—
368C23×C4—1061no—
369C23×C4—2124no—
370C23×C4—2124no—
371C23×C4C22×D5311yescharacteristic
372C2×SD16—24no—
373C2×SD16—24no—
374C2×SD16C2×Dic5318yescomplemented
375C22×C8—532no—
376C22×C8C22×D5312yes—
377C22⋊Q8—22no—
378C22⋊Q8C2×Dic5314yescomplemented
379C4×Q8—1064no—
380C4×D4—538no—
381C4×D4—1064no—
382C4×D4—1064no—
383C4×D4—1068no—
384C4×D4—1068no—
385C4×D4—1064no—
386C2×C4⋊C4—21no—
387C2×C4⋊C4—532no—
388C2×C4⋊C4—531no—
389C2×C4⋊C4—531no—
390C2×C4⋊C4—1064no—
391C2×C4⋊C4—1061no—
392C2×C4⋊C4—1061no—
393C2×C4⋊C4—1061no—
394C2×C4⋊C4C22×D5311yescharacteristic
395C2×C22⋊C4—21no—
396C2×C22⋊C4—1062no—
397C2×C22⋊C4—1062no—
398C2×C22⋊C4—1068no—
399C2×C22⋊C4—1061no—
400C2×C42—531no—
401C2×C42—531no—
402C2×C42—531no—
403C2×C42—531no—
404C2×C42—1064no—
405C2×C42—1061no—
406C2×C42—1061no—
407#106—1064no—
408#103—1064no—
409#102—1064no—
410#102—1064no—
411#102C2×Dic5318yescomplemented
412C22⋊C8—1062no—
413C22⋊C8C2×Dic5314yescomplemented
414C4×C8—1064no—
415#95—1061no—
416#95—1061no—
417#95—1061no—
418#95—1061no—
419#95—1061no—
420C24—24no—
421C24—24no—
422C24—24no—
423C24—22no—
424C24—42no—
425C24—42no—
426C24—42no—
427C24—42no—
428C24D4×D5311yescharacteristic
429C2×Q8—21no—
430C2×Q8—22no—
431C2×Q8—22no—
432C2×Q8C22×Dic5312yescomplemented
433C2×D4—24no—
434C2×D4—24no—
435C2×D4—216no—
436C2×D4—216no—
437C2×D4—24no—
438C2×D4—22no—
439C2×D4—22no—
440C2×D4—22no—
441C2×D4D4×D5314yes—
442C2×D4#1171994714yes—
443C2×D4C22×Dic5314yes—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—531no—
449C22×C4—534no—
450C22×C4—531no—
451C22×C4—531no—
452C22×C4—531no—
453C22×C4—1062no—
454C22×C4—1062no—
455C22×C4—1062no—
456C22×C4—1068no—
457C22×C4—1064no—
458C22×C4—1068no—
459C22×C4—1062no—
460C22×C4—1062no—
461C22×C4—1061no—
462C22×C4—1061no—
463C22×C4—2124no—
464C22×C4—2124no—
465C22×C4—2124no—
466C22×C4—2121no—
467C22×C4—21232no—
468C22×C4—2124no—
469C22×C4—2122no—
470C22×C4—2122no—
471C22×C4D4×D5311yescharacteristic
472C22×C4#1171994712yes—
473C22×C4C22×Dic5312yes—
474C22×C4C23×D5311yescharacteristic
475SD16—216no—
476C2×C8—22no—
477C2×C8—22no—
478C2×C8—1062no—
479C2×C8—1062no—
480C2×C8—1062no—
481C2×C8C22×Dic5314yescomplemented
482C4⋊C4—24no—
483C4⋊C4—21no—
484C4⋊C4—534no—
485C4⋊C4—1064no—
486C4⋊C4—1062no—
487C4⋊C4—1062no—
488C4⋊C4—1062no—
489C4⋊C4—2122no—
490C4⋊C4—2122no—
491C4⋊C4C22×Dic5312yescomplemented
492C22⋊C4—24no—
493C22⋊C4—1068no—
494C22⋊C4—1068no—
495C22⋊C4—2122no—
496C42—534no—
497C42—1062no—
498C42—1062no—
499C42—1062no—
500C42—2122no—
501C42—2122no—
502C23—21no—
503C23—21no—
504C23—21no—
505C23—21no—
506C23—216no—
507C23—24no—
508C23—24no—
509C23—24no—
510C23—48no—
511C23—48no—
512C23—48no—
513C23—48no—
514C23—48no—
515C23—42no—
516C23—42no—
517C23—42no—
518C23—42no—
519C23—48no—
520C23D4×Dic5312yes—
521C23SD16×D5312yes—
522C23#1313323812yes—
523C23C2×D4×D5311yescharacteristic
524Q8—24no—
525Q8—42no—
526D4—216no—
527D4—24no—
528D4—24no—
529D4—24no—
530D4D4×Dic5318yes—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—532no—
542C2×C4—532no—
543C2×C4—532no—
544C2×C4—532no—
545C2×C4—532no—
546C2×C4—1068no—
547C2×C4—1061no—
548C2×C4—1061no—
549C2×C4—1061no—
550C2×C4—1061no—
551C2×C4—1061no—
552C2×C4—1064no—
553C2×C4—1061no—
554C2×C4—1061no—
555C2×C4—21216no—
556C2×C4—2124no—
557C2×C4—21216no—
558C2×C4—2121no—
559C2×C4—2121no—
560C2×C4—2121no—
561C2×C4—2121no—
562C2×C4—2121no—
563C2×C4—2121no—
564C2×C4—2128no—
565C2×C4—2122no—
566C2×C4—2122no—
567C2×C4D4×Dic5312yes—
568C2×C4C23×Dic5311yescharacteristic
569C2×C4C2×D4×D5312yes—
570C2×C4C2×#1171994712yes—
571C2×C4C2×#1171994711yescharacteristic
572C8—24no—
573C8—2122no—
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×Dic5314yescomplemented
589C22C2×D4×Dic5311yescharacteristic
590C22#120×D5311yescharacteristic
591C22#1681112911yescharacteristic
592C22C2×SD16×D5311yescharacteristic
593C22C2×#1313323811yescharacteristic
594C22AGL(1,ℤ/8)×D5311yescharacteristic
595C22#1681141911yescharacteristic
596C4—22no—
597C4—42no—
598C4—42no—
599C4—534no—
600C4—1062no—
601C4—1062no—
602C4—4241no—
603C4—4241no—
604C4—2121no—
605C4—2124no—
606C4C2×D4×Dic5312yes—
607C4#1681112912yes—
608C2—24no—
609C2—48no—
610C2#120×Dic5311yescharacteristic
611C2C2×SD16×Dic5311yescharacteristic
612C2AGL(1,ℤ/8)×Dic5311yescharacteristic
613C2#450×D5311yescharacteristic
614C2#2028393411yescharacteristic
615C2#2028471811yescharacteristic
616C2#2028476311yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC2×C53×#450C2
socleC23×C53C22×D4
prime radicalC23×D4×C53C22
cosocleC22×C4×C53C24
hypercenterC2×#450D53
hypocenterC53C4×#450
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#450

Complements

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

NH ≅ G/Ntype
2. C2×SD16×Dic53C2semidirect
3. C22⋊Q8×Dic53C2semidirect
4. #20275051C2semidirect
5. #20275023C2semidirect
6. #20274978C2semidirect
7. #20274910C2semidirect
9. #102×Dic53C2semidirect
10. C22⋊C8×Dic53C2semidirect
13. C2×Q8×Dic53C22semidirect
23. C2×#13133149C22semidirect
24. C2×#13133141C22semidirect
28. C2×C8×Dic53C22semidirect
29. #16810530C22semidirect
30. #16810522C22semidirect
35. C4⋊C4×Dic53C22semidirect
37. #16810386C22semidirect
38. #16810385C22semidirect
39. #16810377C22semidirect
43. C53×#450C4semidirect
102. C2×SD16×C53C2×C4semidirect
105. C22⋊Q8×C53C2×C4semidirect
109. #102×C53C2×C4semidirect
110. C22⋊C8×C53C2×C4semidirect
123. C2×Q8×C53C22×C4semidirect
161. C22×Dic53SD16semidirect
169. C2×C8×C53C22×C4semidirect
172. C4⋊C4×C53C22×C4semidirect
280. C22×C53C4×SD16semidirect
299. Dic53#450direct (N × H)
333. #450Dic53direct (N × H)
354. C53C4×#450semidirect
374. C2×SD16C2×Dic53semidirect
378. C22⋊Q8C2×Dic53semidirect
411. #102C2×Dic53semidirect
413. C22⋊C8C2×Dic53semidirect
432. C2×Q8C22×Dic53semidirect
481. C2×C8C22×Dic53semidirect
491. C4⋊C4C22×Dic53semidirect
588. C22SD16×Dic53semidirect

Maximal subgroups

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

subgroupindex
C22×D4×Dic532
C2×SD16×Dic532
C22⋊Q8×Dic532
#202750512
#202750232
#202749782
#202749102
#202746462
#102×Dic532
C22⋊C8×Dic532
C2×C53×#4502
C4×#45053