Finite Group Databasebeta

← #27384814

Subgroups — #27384814

Order 13568.

1590 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×#13133175C211yesmaximal, characteristic
2C2×#16810660C212yesmaximal, complemented
3C2×C22⋊C8×D53C211yesmaximal, characteristic
4#20277097C211yesmaximal, characteristic
5#20274917C212yesmaximal, complemented
6#20274911C212yesmaximal, complemented
7#20274908C211yesmaximal, characteristic, complemented
8#20274907C211yesmaximal, characteristic, complemented
9#20274905C211yesmaximal, characteristic, complemented
10#20274430C212yesmaximal, complemented
11C53×#1852C211yesmaximal, characteristic, complemented
12C23×C4×D53C2211yescharacteristic
13C22×C8×D53C2212yes—
14C2×#13133175—21no—
15C2×#13133175—21no—
16C2×#13133175—28no—
17C2×#13133175C2211yescharacteristic
18C2×#13133175C2211yescharacteristic
19C2×#13133175C2212yes—
20C2×#13133175C2212yes—
21C22×C4⋊Dic53C2211yescharacteristic
22C22×#11719921—21no—
23C22×C4×Dic53—21no—
24#16810660—28no—
25C2×#13133151C2212yescomplemented
26C2×#13133148C2212yescomplemented
27C2×#13133148C2212yescomplemented
28C2×C22⋊#88322C2211yescharacteristic
29C2×#13133129C2212yes—
30C22⋊C8×D53C418yescomplemented
31#16810485—21no—
32#16810483C2211yescharacteristic
33#16810480C2211yescharacteristic
34#16810480C2211yescharacteristic
35#16810478—21no—
36C22⋊C4×Dic53—21no—
37#16810470—21no—
38#16810386C2212yescomplemented
39#16810385C2212yescomplemented
40#16810385C2212yescomplemented
41C22×C4⋊C4×C53C2211yescharacteristic
42C2×#107×C53C2212yescomplemented
43C2×C22⋊C8×C53C2211yescharacteristic
44C53×#380C2211yescharacteristic
45C53×#340C2212yescomplemented
46C24×D53D411yescharacteristic
47C23×Dic53—21no—
48C23×Dic53D411yescharacteristic
49C22×C4×D53—24no—
50C22×C4×D53—24no—
51C22×C4×D53C2×C414yes—
52C22×C4×D53D411yescharacteristic
53C22×C4×D53D411yescharacteristic
54C22×C4×D53Q814yes—
55C22×C4×D53C2311yescharacteristic
56C22×C4×D53C2311yescharacteristic
57C22×#88322C2312yes—
58C2×C8×D53—24no—
59C2×C8×D53—24no—
60C2×C8×D53C2×C418yescomplemented
61#13133175—28no—
62#13133175—28no—
63#13133175—24no—
64#13133175—24no—
65#13133175—24no—
66#13133175—24no—
67#13133175—24no—
68#13133175D418yes—
69C2×C4⋊Dic53—24no—
70C2×C4⋊Dic53—21no—
71C2×C4⋊Dic53C2311yescharacteristic
72C2×C4⋊Dic53C2311yescharacteristic
73C2×C4⋊Dic53C2312yes—
74C2×C22⋊Dic53—21no—
75C2×#11719921—28no—
76C2×#11719921—22no—
77C2×#11719921—22no—
78C2×#11719921—21no—
79C2×#11719921—21no—
80C2×#11719921—21no—
81C2×#11719921—21no—
82C2×C4×Dic53—28no—
83C2×C4×Dic53—22no—
84C2×C4×Dic53—22no—
85C2×C4×Dic53—21no—
86C2×C4×Dic53—21no—
87#13133151—24no—
88#13133148—24no—
89#13133148—24no—
90C22⋊#88322C2×C414yes—
91#13133129C2×C418yescomplemented
92#13133124—21no—
93#13133124—21no—
94#13133124—21no—
95C23×C4×C53—21no—
96C23×C4×C53C2311yescharacteristic
97C22×C8×C53C2312yes—
98C2×C4⋊C4×C53—24no—
99C2×C4⋊C4×C53—21no—
100C2×C4⋊C4×C53C2311yescharacteristic
101C2×C4⋊C4×C53C2311yescharacteristic
102C2×C4⋊C4×C53C2312yes—
103C2×C22⋊C4×C53—21no—
104#107×C53—24no—
105C22⋊C8×C53C2×C414yes—
106#95×C53—21no—
107C24×C53C2×D411yescharacteristic
108C23×D53—22no—
109C23×D53—22no—
110C23×D53—22no—
111C23×D53—22no—
112C23×D53—41no—
113C23×D53—41no—
114C23×D53C4⋊C414yes—
115C23×D53C2×D411yescharacteristic
116C23×D53C2×D411yescharacteristic
117C22×C4×C53—22no—
118C22×C4×C53—22no—
119C22×C4×C53—24no—
120C22×C4×C53—22no—
121C22×C4×C53—21no—
122C22×C4×C53—21no—
123C22×C4×C53—42no—
124C22×C4×C53—42no—
125C22×C4×C53C22×C412yes—
126C22×C4×C53C2×D411yescharacteristic
127C22×C4×C53C2×Q812yes—
128C22×C4×C53C2411yescharacteristic
129C22×Dic53—22no—
130C22×Dic53—22no—
131C22×Dic53—24no—
132C22×Dic53—22no—
133C22×Dic53—21no—
134C22×Dic53—21no—
135C22×Dic53—42no—
136C22×Dic53—42no—
137C22×Dic53C4⋊C414yes—
138C22×Dic53C2×D411yescharacteristic
139C22×Dic53C2×D411yescharacteristic
140C2×C4×D53—22no—
141C2×C4×D53—22no—
142C2×C4×D53—22no—
143C2×C4×D53—22no—
144C2×C4×D53—22no—
145C2×C4×D53—22no—
146C2×C4×D53—216no—
147C2×C4×D53—216no—
148C2×C4×D53C4⋊C414yes—
149C2×C4×D53C4⋊C414yes—
150C2×C4×D53C22×C412yes—
151C2×C4×D53C22×C412yes—
152C2×C4×D53C2×D414yes—
153C2×C4×D53C2×Q812yes—
154C2×C4×D53C2×Q812yes—
155C2×C4×D53#4114yes—
156C2×C8×C53—22no—
157C2×C8×C53—22no—
158C2×C8×C53C22×C414yes—
159C4⋊C4×C53—24no—
160C4⋊C4×C53—22no—
161C4⋊C4×C53—22no—
162C4⋊C4×C53—22no—
163C4⋊C4×C53C2×D414yes—
164C22⋊C4×C53—42no—
165C22⋊Dic53—42no—
166#11719921—28no—
167#11719921—28no—
168#11719921—42no—
169#11719921—42no—
170#11719921—42no—
171#11719921—42no—
172C4⋊Dic53—24no—
173C4⋊Dic53—22no—
174C4⋊Dic53—22no—
175C4⋊Dic53—22no—
176C4⋊Dic53C2×D414yes—
177C4×Dic53—28no—
178C4×Dic53—28no—
179C4×Dic53—42no—
180C4×Dic53—42no—
181C2×#88322—22no—
182C2×#88322—22no—
183C2×#88322C22×C414yes—
184C8×D53—216no—
185C23×C53—21no—
186C23×C53—21no—
187C23×C53—21no—
188C23×C53—21no—
189C23×C53C2×C4⋊C412yes—
190C23×C53#13512yes—
191C23×C53#13512yes—
192C23×C53C22×D411yescharacteristic
193C22×D53—21no—
194C22×D53—21no—
195C22×D53—21no—
196C22×D53—21no—
197C22×D53—21no—
198C22×D53—21no—
199C22×D53—21no—
200C22×D53—21no—
201C22×D53—21no—
202C22×D53—21no—
203C22×D53—28no—
204C22×D53—28no—
205C22×D53—44no—
206C22×D53—44no—
207C22×D53—44no—
208C22×D53—44no—
209C22×D53—44no—
210C22×D53C2×C4⋊C412yes—
211C22×D53C2×C4⋊C412yes—
212C22×D53#12012yes—
213C22×D53#12312yes—
214C2×C4×C53—21no—
215C2×C4×C53—21no—
216C2×C4×C53—24no—
217C2×C4×C53—24no—
218C2×C4×C53—24no—
219C2×C4×C53—48no—
220C2×C4×C53—42no—
221C2×C4×C53—42no—
222C2×C4×C53—41no—
223C2×C4×C53—41no—
224C2×C4×C53—41no—
225C2×C4×C53—41no—
226C2×C4×C53—41no—
227C2×C4×C53—41no—
228C2×C4×C53C2×C4⋊C412yes—
229C2×C4×C53C23×C411yescharacteristic
230C2×C4×C53C22×D412yes—
231C2×C4×C53C22×Q811yescharacteristic
232C2×C4×C53C2×#4112yes—
233C2×Dic53—21no—
234C2×Dic53—21no—
235C2×Dic53—28no—
236C2×Dic53—24no—
237C2×Dic53—48no—
238C2×Dic53—42no—
239C2×Dic53—42no—
240C2×Dic53—41no—
241C2×Dic53—41no—
242C2×Dic53—41no—
243C2×Dic53—41no—
244C2×Dic53—41no—
245C2×Dic53—41no—
246C2×Dic53C2×C4⋊C412yes—
247C2×Dic53C2×C4⋊C412yes—
248C2×Dic53#12012yes—
249C2×Dic53#12312yes—
250C4×D53—28no—
251C4×D53—24no—
252C4×D53—24no—
253C4×D53—28no—
254C4×D53C4×D418yes—
255C4×D53C22⋊Q818yes—
256C8×C53—24no—
257#88322—24no—
258#20843—531nomaximal, Sylow
259C22×C53—22no—
260C22×C53—22no—
261C22×C53—22no—
262C22×C53—22no—
263C22×C53—22no—
264C22×C53—22no—
265C22×C53#42714yescomplemented
266C22×C53C22×C4⋊C411yescharacteristic
267C22×C53C2×#12011yescharacteristic
268C22×C53C2×#12311yescharacteristic
269C22×C53C2×#13511yescharacteristic
270C22×C53C2×#13511yescharacteristic
271C22×C53#57511yescharacteristic
272C22×C53#57511yescharacteristic
273C2×D53—22no—
274C2×D53—22no—
275C2×D53—22no—
276C2×D53—22no—
277C2×D53—22no—
278C2×D53—22no—
279C2×D53—22no—
280C2×D53—22no—
281C2×D53—22no—
282C2×D53—416no—
283C2×D53—41no—
284C2×D53—41no—
285C2×D53—41no—
286C2×D53—41no—
287C2×D53#38514yes—
288C4×C53—22no—
289C4×C53—44no—
290C4×C53—81no—
291C4×C53—81no—
292C4×C53C2×C4×D412yes—
293C4×C53C2×C22⋊Q812yes—
294Dic53—22no—
295Dic53—44no—
296Dic53—81no—
297Dic53—81no—
298Dic53#38514yes—
299C22×#117—531no—
300C2×#427—532no—
301C22×C22⋊C8—531no—
302#2280—531no—
303#1854—532no—
304#1854—532no—
305#1853—531no—
306#1853—531no—
307#1852—531no—
308#1852D5311yescharacteristic, complemented
309#1735—532no—
310C2×C53—24no—
311C2×C53C2×#38511yescharacteristic
312C2×C53C2×#42711yescharacteristic
313C2×C53#291111yescharacteristic
314C2×C53#300211yescharacteristic
315C2×C53#300211yescharacteristic
316C2×C53#309111yescharacteristic
317C2×C53#309111yescharacteristic
318D53—24no—
319D53—42no—
320C24×C4—531no—
321C23×C8—532no—
322C2×#117—532no—
323C2×#117—532no—
324C2×#117—531no—
325C2×#117—531no—
326C2×#117—1068no—
327C2×#117—1061no—
328C2×#117—1061no—
329C22×C4⋊C4—531no—
330C22×C4⋊C4C2×D5311yescharacteristic
331C22×C22⋊C4—1061no—
332C22×C42—1061no—
333#427—1068no—
334C2×#107—532no—
335C2×#107C2×D5312yescomplemented
336C2×#106—532no—
337C2×#106—532no—
338C2×C22⋊C8—531no—
339C2×C22⋊C8—532no—
340C2×C22⋊C8—538no—
341C2×C22⋊C8C2×D5311yescharacteristic
342#381—1061no—
343#380—531no—
344#380—531no—
345#380—531no—
346#380C2×D5311yescharacteristic
347#379—1061no—
348C4×C22⋊C4—1061no—
349C4×C22⋊C4—1061no—
350#340—532no—
351#340—532no—
352#340—532no—
353#340C2×D5312yescomplemented
354C53#2084311yescharacteristic, complemented, Sylow
355C25—531no—
356C23×C4—21no—
357C23×C4—531no—
358C23×C4—531no—
359C23×C4—534no—
360C23×C4—531no—
361C23×C4—531no—
362C23×C4—534no—
363C23×C4—531no—
364C23×C4—1064no—
365C23×C4—1064no—
366C23×C4—1061no—
367C23×C4C22×D5311yescharacteristic
368C22×C8—532no—
369C22×C8—538no—
370C22×C8—1064no—
371C22×C8—1064no—
372C22×C8C22×D5312yes—
373#117—538no—
374#117—1064no—
375#117—1064no—
376#117—1068no—
377#117—1068no—
378#117—1064no—
379#117—1064no—
380#117—1064no—
381C2×C4⋊C4—21no—
382C2×C4⋊C4—24no—
383C2×C4⋊C4—532no—
384C2×C4⋊C4—531no—
385C2×C4⋊C4—531no—
386C2×C4⋊C4—1064no—
387C2×C4⋊C4—1061no—
388C2×C4⋊C4C22×D5311yescharacteristic
389C2×C4⋊C4C22×D5311yescharacteristic
390C2×C4⋊C4C22×D5312yes—
391C2×C22⋊C4—21no—
392C2×C22⋊C4—1062no—
393C2×C22⋊C4—1062no—
394C2×C22⋊C4—1068no—
395C2×C22⋊C4—1061no—
396C2×C22⋊C4—1061no—
397C2×C22⋊C4—1061no—
398C2×C22⋊C4—1061no—
399C2×C22⋊C4—1061no—
400C2×C42—1062no—
401C2×C42—1062no—
402C2×C42—1068no—
403C2×C42—1061no—
404C2×C42—1061no—
405#107—24no—
406#107—1064no—
407#106—1064no—
408#106—1064no—
409C22⋊C8—534no—
410C22⋊C8—538no—
411C22⋊C8C4×D5314yes—
412#95—21no—
413#95—1061no—
414#95—1061no—
415#95—1061no—
416C24—531no—
417C24—531no—
418C24—534no—
419C24—1062no—
420C24—1062no—
421C24—1062no—
422C24—1062no—
423C24—2121no—
424C24—2121no—
425C24D4×D5311yescharacteristic
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—24no—
429C22×C4—22no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—531no—
435C22×C4—532no—
436C22×C4—534no—
437C22×C4—532no—
438C22×C4—534no—
439C22×C4—534no—
440C22×C4—531no—
441C22×C4—534no—
442C22×C4—534no—
443C22×C4—532no—
444C22×C4—532no—
445C22×C4—1062no—
446C22×C4—1062no—
447C22×C4—10616no—
448C22×C4—10616no—
449C22×C4—1062no—
450C22×C4—1062no—
451C22×C4—1062no—
452C22×C4—1062no—
453C22×C4—1062no—
454C22×C4—1062no—
455C22×C4—1064no—
456C22×C4—1062no—
457C22×C4—1061no—
458C22×C4—1061no—
459C22×C4—2122no—
460C22×C4—2122no—
461C22×C4C2×C4×D5312yes—
462C22×C4D4×D5311yescharacteristic
463C22×C4Q8×D5312yes—
464C22×C4C23×D5311yescharacteristic
465C2×C8—22no—
466C2×C8—22no—
467C2×C8—534no—
468C2×C8—10616no—
469C2×C8—1062no—
470C2×C8—1062no—
471C2×C8C2×C4×D5314yes—
472C4⋊C4—24no—
473C4⋊C4—22no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—534no—
477C4⋊C4—1064no—
478C4⋊C4—1062no—
479C4⋊C4—1062no—
480C4⋊C4—1062no—
481C4⋊C4D4×D5314yes—
482C22⋊C4—42no—
483C22⋊C4—1068no—
484C22⋊C4—1068no—
485C22⋊C4—2122no—
486C22⋊C4—2122no—
487C22⋊C4—2122no—
488C22⋊C4—2122no—
489C22⋊C4—2122no—
490C42—1068no—
491C42—1068no—
492C42—2122no—
493C42—2122no—
494C23—21no—
495C23—21no—
496C23—21no—
497C23—21no—
498C23—532no—
499C23—532no—
500C23—532no—
501C23—532no—
502C23—1068no—
503C23—1061no—
504C23—1061no—
505C23—1061no—
506C23—1061no—
507C23—1061no—
508C23—1061no—
509C23—1061no—
510C23—1061no—
511C23—1061no—
512C23—1061no—
513C23—1068no—
514C23—2124no—
515C23—2124no—
516C23—2124no—
517C23—2124no—
518C23—2124no—
519C23C4⋊C4×D5312yes—
520C23#1313323612yes—
521C23#1313324012yes—
522C23C2×D4×D5311yescharacteristic
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—48no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—532no—
538C2×C4—532no—
539C2×C4—532no—
540C2×C4—538no—
541C2×C4—538no—
542C2×C4—532no—
543C2×C4—1068no—
544C2×C4—1068no—
545C2×C4—1068no—
546C2×C4—1061no—
547C2×C4—1061no—
548C2×C4—1064no—
549C2×C4—1064no—
550C2×C4—1064no—
551C2×C4—2121no—
552C2×C4—2121no—
553C2×C4—2121no—
554C2×C4—2121no—
555C2×C4—2121no—
556C2×C4—2121no—
557C2×C4—2128no—
558C2×C4—2122no—
559C2×C4—2122no—
560C2×C4C4⋊C4×D5312yes—
561C2×C4C22×C4×D5311yescharacteristic
562C2×C4C2×D4×D5312yes—
563C2×C4C2×Q8×D5311yescharacteristic
564C2×C4#41×D5312yes—
565C8—24no—
566C8—1064no—
567C22—22no—
568C22—22no—
569C22—22no—
570C22—22no—
571C22—22no—
572C22—22no—
573C22—534no—
574C22—1062no—
575C22—1062no—
576C22—1062no—
577C22—1062no—
578C22—1062no—
579C22—1062no—
580C22—1062no—
581C22—1062no—
582C22—1062no—
583C22—21216no—
584C22—2121no—
585C22—2121no—
586C22—2121no—
587C22—2121no—
588C22#1681066014yescomplemented
589C22C2×C4⋊C4×D5311yescharacteristic
590C22#120×D5311yescharacteristic
591C22#123×D5311yescharacteristic
592C22C2×#1313323611yescharacteristic
593C22C2×#1313324011yescharacteristic
594C22#1681141911yescharacteristic
595C22#1681142211yescharacteristic
596C4—22no—
597C4—44no—
598C4—81no—
599C4—81no—
600C4—534no—
601C4—1062no—
602C4—4241no—
603C4—4241no—
604C4—2124no—
605C4C4×D4×D5312yes—
606C4C22⋊Q8×D5312yes—
607C2—24no—
608C2—1064no—
609C2—2122no—
610C2#385×D5311yescharacteristic
611C2C2×#1681066011yescharacteristic
612C2#2028330211yescharacteristic
613C2#2028392811yescharacteristic
614C2#2028393811yescharacteristic
615C2#2028462311yescharacteristic
616C2#2028463111yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#1852C2
socleC23×C53C22×D4
prime radicalC24×D53D4
cosocleC22×C4×C53C24
hypercenter#1852D53
hypocenterC53#20843
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1852

Complements

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

NH ≅ G/Ntype
2. C2×#16810660C2semidirect
5. #20274917C2semidirect
6. #20274911C2semidirect
7. #20274908C2semidirect
8. #20274907C2semidirect
9. #20274905C2semidirect
10. #20274430C2semidirect
11. C53×#1852C2semidirect
25. C2×#13133151C22semidirect
26. C2×#13133148C22semidirect
27. C2×#13133148C22semidirect
30. C22⋊C8×D53C4semidirect
38. #16810386C22semidirect
39. #16810385C22semidirect
40. #16810385C22semidirect
42. C2×#107×C53C22semidirect
45. C53×#340C22semidirect
60. C2×C8×D53C2×C4semidirect
91. #13133129C2×C4semidirect
265. C22×C53#427semidirect
308. #1852D53semidirect
335. C2×#107C2×D53semidirect
353. #340C2×D53semidirect
354. C53#20843semidirect
588. C22#16810660semidirect

Maximal subgroups

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

subgroupindex
C22×#131331752
C2×#168106602
C2×C22⋊C8×D532
#202770972
#202749172
#202749112
#202749082
#202749072
#202749052
#202744302
C53×#18522
#2084353