Finite Group Databasebeta

← #27411573

Subgroups — #27411573

Order 13568.

1610 conjugacy classes of subgroups.

The subgroup lattice has 652 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811134C211yesmaximal, characteristic
2C2×#16810659C211yesmaximal, characteristic, complemented
3C2×#16810617C211yesmaximal, characteristic, complemented
4C2×#16810617C211yesmaximal, characteristic, complemented
5C22×C22⋊#88322C211yesmaximal, characteristic, complemented
6#20275918C212yesmaximal, complemented
7#20275911C212yesmaximal, complemented
8#20275904C212yesmaximal, complemented
9#20275897C212yesmaximal, complemented
10#20274431C211yesmaximal, characteristic, complemented
11C53×#2280C211yesmaximal, characteristic, complemented
12C22×#11719945C2211yescharacteristic
13C23×#88322C2211yescharacteristic
14C23×#88322C2211yescharacteristic
15#16811134—24no—
16#16811134—24no—
17C2×#13133200—21no—
18C2×#13133196C2212yes—
19C2×#13133191C2212yes—
20C2×#13133186—21no—
21C2×C4×C22⋊D53—22no—
22C2×#13133176C2211yescharacteristic
23#16810659—24no—
24C2×#13133150C2212yescomplemented
25C2×#13133148C2212yescomplemented
26#16810617—24no—
27#16810617—24no—
28C2×#13133140C2211yescharacteristic, complemented
29C2×#13133140C2212yescomplemented
30C2×#13133140C2211yescharacteristic, complemented
31C2×#13133137C2211yescharacteristic, complemented
32C2×#13133137C2212yescomplemented
33C2×#13133137C2211yescharacteristic, complemented
34C2×C22⋊#88322—24no—
35C2×C22⋊#88322C2212yes—
36C2×C22⋊#88322C2212yes—
37#16810385C2212yescomplemented
38#16810385C2212yescomplemented
39C24×C4×C53C2211yescharacteristic
40C2×#117×C53C2211yescharacteristic
41C2×#117×C53C2211yescharacteristic
42C53×#381—21no—
43C53×#380C2212yes—
44C53×#380C2212yes—
45C53×#379—21no—
46C4×C22⋊C4×C53—22no—
47C2×#11719945—22no—
48C2×#11719945—22no—
49C2×#11719945—22no—
50C2×#11719945—22no—
51C2×#11719945—22no—
52C2×#11719945D414yes—
53C22×Dic106C2311yescharacteristic
54C22×D212C2311yescharacteristic
55C22×C22⋊D53—22no—
56C22×C22⋊D53—21no—
57C22×C4×D53—21no—
58C22×#88322—22no—
59C22×#88322—22no—
60C22×#88322—22no—
61C22×#88322—22no—
62C22×#88322—22no—
63C22×#88322D414yescomplemented
64C22×#88322C2312yes—
65C22×#88322C2311yescharacteristic
66C22×#88322C2311yescharacteristic
67#13133200—28no—
68#13133196—24no—
69#13133196—24no—
70#13133191—24no—
71#13133191—24no—
72#13133186—28no—
73C4×C22⋊D53—216no—
74#13133176—22no—
75#13133176D414yes—
76C2×C4⋊Dic53C2312yes—
77C2×#11719919—22no—
78C2×C22⋊Dic53—21no—
79C2×C22⋊Dic53—22no—
80C2×#11719921—22no—
81C2×C4×Dic53—21no—
82#13133150—24no—
83#13133148—24no—
84#13133140—22no—
85#13133140—22no—
86#13133140—24no—
87#13133137—22no—
88#13133137—22no—
89#13133137—24no—
90C22⋊#88322—24no—
91C22⋊#88322—24no—
92C25×C53D411yescharacteristic
93C23×C4×C53—22no—
94C23×C4×C53—22no—
95C23×C4×C53—22no—
96C23×C4×C53—22no—
97C23×C4×C53—22no—
98C23×C4×C53—22no—
99C23×C4×C53D412yes—
100C23×C4×C53D411yescharacteristic
101C23×C4×C53C2312yes—
102C23×C4×C53C2311yescharacteristic
103#117×C53—22no—
104#117×C53—22no—
105#117×C53—22no—
106#117×C53—22no—
107C2×C4⋊C4×C53C2311yescharacteristic
108C2×C4⋊C4×C53C2311yescharacteristic
109C2×C4⋊C4×C53C2312yes—
110C2×C22⋊C4×C53—21no—
111C2×C22⋊C4×C53—21no—
112C2×C22⋊C4×C53—22no—
113C2×C22⋊C4×C53—22no—
114C2×C42×C53—21no—
115C2×C42×C53—21no—
116#95×C53—22no—
117#95×C53—22no—
118C24×C53—22no—
119C24×C53—24no—
120C24×C53—22no—
121C24×C53—22no—
122C24×C53—42no—
123C24×C53C2×D412yes—
124C24×C53C2×D411yescharacteristic
125C23×D53—21no—
126C22×C4×C53—22no—
127C22×C4×C53—22no—
128C22×C4×C53—22no—
129C22×C4×C53—24no—
130C22×C4×C53—24no—
131C22×C4×C53—24no—
132C22×C4×C53—22no—
133C22×C4×C53—22no—
134C22×C4×C53—22no—
135C22×C4×C53—22no—
136C22×C4×C53—22no—
137C22×C4×C53—22no—
138C22×C4×C53—22no—
139C22×C4×C53—21no—
140C22×C4×C53—21no—
141C22×C4×C53—44no—
142C22×C4×C53—44no—
143C22×C4×C53—44no—
144C22×C4×C53—44no—
145C22×C4×C53C2×D412yes—
146C22×C4×C53C2×D411yescharacteristic
147C22×C4×C53C2×D412yes—
148C22×C4×C53C2×D412yes—
149C22×C4×C53C2×D412yes—
150C22×C4×C53#4112yes—
151C22×C4×C53#4112yes—
152C22×C4×C53#4112yes—
153C22×C4×C53C2411yescharacteristic
154C2×C22⋊D53—28no—
155C2×C22⋊D53—24no—
156C2×C22⋊D53—44no—
157C2×C22⋊D53—42no—
158C2×C22⋊D53—42no—
159C2×C22⋊D53—44no—
160C22×Dic53—22no—
161C22×Dic53—21no—
162#11719945—28no—
163#11719945—28no—
164#11719945—44no—
165#11719945—44no—
166C2×D212—21no—
167C2×D212—21no—
168C2×D212—21no—
169C2×D212—21no—
170C2×D212—21no—
171C2×D212C2×D412yes—
172C2×C4×D53—24no—
173C2×C4×D53—42no—
174C2×C4×D53—42no—
175C2×Dic106—21no—
176C2×Dic106—21no—
177C2×Dic106—21no—
178C2×Dic106—21no—
179C2×Dic106—21no—
180C2×Dic106C2×D412yes—
181C4⋊C4×C53—22no—
182C4⋊C4×C53—21no—
183C4⋊C4×C53—21no—
184C4⋊C4×C53—22no—
185C4⋊C4×C53—21no—
186C4⋊C4×C53—21no—
187C22⋊C4×C53—44no—
188C22⋊C4×C53—42no—
189C22⋊C4×C53—42no—
190C22⋊C4×C53—44no—
191C42×C53—42no—
192C42×C53—42no—
193C22⋊Dic53—28no—
194C22⋊Dic53—24no—
195#11719921—28no—
196C4⋊Dic53—22no—
197C4⋊Dic53C2×D414yes—
198#11719919—28no—
199C4×Dic53—24no—
200C2×#88322—28no—
201C2×#88322—21no—
202C2×#88322—21no—
203C2×#88322—21no—
204C2×#88322—21no—
205C2×#88322—22no—
206C2×#88322—22no—
207C2×#88322—22no—
208C2×#88322—44no—
209C2×#88322C2×D412yes—
210C2×#88322C2×D412yes—
211C23×C53—22no—
212C23×C53—22no—
213C23×C53—28no—
214C23×C53—22no—
215C23×C53—22no—
216C23×C53—22no—
217C23×C53—22no—
218C23×C53—22no—
219C23×C53—22no—
220C23×C53—48no—
221C23×C53—48no—
222C23×C53—44no—
223C23×C53—44no—
224C23×C53#12012yes—
225C23×C53#12112yes—
226C23×C53#12312yes—
227C23×C53#13514yes—
228C23×C53C22×D411yescharacteristic
229C22×D53—22no—
230C22×D53—41no—
231C22×D53—41no—
232C22×D53—42no—
233C22×D53—42no—
234C2×C4×C53—22no—
235C2×C4×C53—22no—
236C2×C4×C53—22no—
237C2×C4×C53—44no—
238C2×C4×C53—42no—
239C2×C4×C53—42no—
240C2×C4×C53—42no—
241C2×C4×C53—42no—
242C2×C4×C53—48no—
243C2×C4×C53—42no—
244C2×C4×C53—41no—
245C2×C4×C53—41no—
246C2×C4×C53—42no—
247C2×C4×C53—42no—
248C2×C4×C53—42no—
249C2×C4×C53—42no—
250C2×C4×C53#12012yes—
251C2×C4×C53#12112yes—
252C2×C4×C53#12114yes—
253C2×C4×C53#12114yes—
254C2×C4×C53#12314yes—
255C2×C4×C53#12312yes—
256C2×C4×C53#12414yes—
257C2×C4×C53C22×D411yescharacteristic
258C2×C4×C53C22×D411yescharacteristic
259C2×C4×C53C22×D411yescharacteristic
260C2×C4×C53C2×#4111yescharacteristic
261C2×C4×C53C2×#4111yescharacteristic
262C2×C4×C53C2×#4111yescharacteristic
263C22⋊D53—416no—
264C22⋊D53—48no—
265C2×Dic53—22no—
266C2×Dic53—24no—
267C2×Dic53—41no—
268C2×Dic53—41no—
269C2×Dic53—42no—
270C2×Dic53—42no—
271D212—22no—
272D212—22no—
273D212—41no—
274D212—41no—
275C4×D53—48no—
276Dic106—22no—
277Dic106—22no—
278Dic106—41no—
279Dic106—41no—
280#88322—22no—
281#88322—22no—
282#88322—42no—
283#23479—531nomaximal, Sylow
284C22×C53—24no—
285C22×C53—24no—
286C22×C53—22no—
287C22×C53—22no—
288C22×C53—24no—
289C22×C53—24no—
290C22×C53—24no—
291C22×C53—416no—
292C22×C53—42no—
293C22×C53—42no—
294C22×C53—42no—
295C22×C53—42no—
296C22×C53#39414yes—
297C22×C53C2×#12011yescharacteristic
298C22×C53C2×#12111yescharacteristic
299C22×C53C2×#12311yescharacteristic
300C22×C53C2×#13512yes—
301C22×C53#57512yes—
302C2×D53—42no—
303C2×D53—44no—
304C2×D53—82no—
305C2×D53—81no—
306C2×D53—81no—
307C4×C53—42no—
308C4×C53—82no—
309C4×C53—81no—
310C4×C53—81no—
311C4×C53#39414yes—
312C4×C53C2×#12111yescharacteristic
313C4×C53C2×#12111yescharacteristic
314C4×C53C2×#12311yescharacteristic
315C4×C53C2×#12411yescharacteristic
316Dic53—42no—
317Dic53—44no—
318C2×#525—531no—
319C2×#427—531no—
320C2×#416—531no—
321C2×#416—531no—
322C22×C22⋊C8—531no—
323#2280D5311yescharacteristic, complemented
324#2036—532no—
325#2035—532no—
326#2034—532no—
327#2033—532no—
328#1735—531no—
329C2×C53—24no—
330C2×C53—44no—
331C2×C53C2×#39411yescharacteristic
332C2×C53#300212yes—
333C2×C53#305011yescharacteristic
334C2×C53#305411yescharacteristic
335C2×C53#309112yes—
336D53—82no—
337C22×#41—531no—
338C24×C4C2×D5311yescharacteristic
339C23×C8—531no—
340C23×C8—531no—
341#525—1064no—
342#525—1064no—
343C2×#123—1061no—
344C2×C22⋊Q8—532no—
345C2×#121—532no—
346C2×#120—1061no—
347C2×C4×D4—1062no—
348C2×#117—531no—
349C2×#117C2×D5311yescharacteristic
350C2×#117C2×D5311yescharacteristic
351#427—1064no—
352C2×#107—532no—
353C2×#106—532no—
354#416—1064no—
355#416—1064no—
356C2×#103—531no—
357C2×#103—532no—
358C2×#103—531no—
359C2×#102—531no—
360C2×#102—532no—
361C2×#102—531no—
362C2×C22⋊C8—532no—
363C2×C22⋊C8—532no—
364C2×C22⋊C8—1064no—
365#381—21no—
366#380C2×D5312yes—
367#380C2×D5312yes—
368#379—21no—
369C4×C22⋊C4—22no—
370#340—532no—
371#340—532no—
372C53#2347911yescharacteristic, complemented, Sylow
373C25C22⋊D5311yescharacteristic
374C2×#41—534no—
375C2×#41—1062no—
376C2×#41—1062no—
377C2×#41—1062no—
378C2×#41—1062no—
379C2×#41—1062no—
380C22×Q8—531no—
381C22×D4—531no—
382C22×D4—1062no—
383C22×D4—1061no—
384C23×C4—22no—
385C23×C4—22no—
386C23×C4—22no—
387C23×C4—22no—
388C23×C4—22no—
389C23×C4—22no—
390C23×C4—1061no—
391C23×C4C22⋊D5311yescharacteristic
392C23×C4C22⋊D5312yes—
393C23×C4C22×D5312yes—
394C23×C4C22×D5311yescharacteristic
395C22×C8—534no—
396C22×C8—531no—
397C22×C8—531no—
398C22×C8—532no—
399C22×C8—1062no—
400C22×C8—1062no—
401C22×C8—1062no—
402C22×C8—1062no—
403C22×C8—1062no—
404#123—1068no—
405C22⋊Q8—1064no—
406C22⋊Q8—1064no—
407#121—1064no—
408#121—1064no—
409#120—1068no—
410C4×D4—10616no—
411#117—22no—
412#117—22no—
413#117—22no—
414#117—22no—
415#117—534no—
416#117—1062no—
417C2×C4⋊C4—532no—
418C2×C4⋊C4—1062no—
419C2×C4⋊C4C22×D5311yescharacteristic
420C2×C4⋊C4C22×D5311yescharacteristic
421C2×C4⋊C4C22×D5312yes—
422C2×C22⋊C4—21no—
423C2×C22⋊C4—21no—
424C2×C22⋊C4—22no—
425C2×C22⋊C4—22no—
426C2×C22⋊C4—1061no—
427C2×C22⋊C4—1062no—
428C2×C22⋊C4—1062no—
429C2×C42—21no—
430C2×C42—21no—
431C2×C42—1061no—
432#107—1064no—
433#106—1064no—
434#103—1062no—
435#103—1062no—
436#103—1064no—
437#102—1062no—
438#102—1062no—
439#102—1064no—
440C22⋊C8—1064no—
441C22⋊C8—1064no—
442#95—22no—
443#95—22no—
444C24—22no—
445C24—24no—
446C24—22no—
447C24—22no—
448C24—42no—
449C24—1061no—
450C24C2×C22⋊D5312yes—
451C24C2×C22⋊D5311yescharacteristic
452#41—1068no—
453#41—1068no—
454#41—2124no—
455#41—2124no—
456C2×Q8—532no—
457C2×Q8—1061no—
458C2×Q8—1061no—
459C2×Q8—1061no—
460C2×Q8—1061no—
461C2×Q8—1061no—
462C2×D4—532no—
463C2×D4—1068no—
464C2×D4—1064no—
465C2×D4—1061no—
466C2×D4—1061no—
467C2×D4—1061no—
468C2×D4—1061no—
469C2×D4—1061no—
470C2×D4—2124no—
471C2×D4—2122no—
472C2×D4—2122no—
473C2×D4—2124no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—24no—
478C22×C4—24no—
479C22×C4—24no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—22no—
485C22×C4—22no—
486C22×C4—22no—
487C22×C4—21no—
488C22×C4—21no—
489C22×C4—44no—
490C22×C4—44no—
491C22×C4—44no—
492C22×C4—44no—
493C22×C4—1062no—
494C22×C4—1061no—
495C22×C4—1064no—
496C22×C4—2122no—
497C22×C4—2122no—
498C22×C4#1171994512yes—
499C22×C4#1171994512yes—
500C22×C4D4×D5312yes—
501C22×C4D4×D5312yes—
502C22×C4D4×D5312yes—
503C22×C4#1171994712yes—
504C22×C4C2×C22⋊D5312yes—
505C22×C4C2×C22⋊D5311yescharacteristic
506C22×C4C23×D5311yescharacteristic
507C2×C8—532no—
508C2×C8—532no—
509C2×C8—1068no—
510C2×C8—1061no—
511C2×C8—1061no—
512C2×C8—1061no—
513C2×C8—1061no—
514C2×C8—1062no—
515C2×C8—1062no—
516C2×C8—1062no—
517C2×C8—2124no—
518C4⋊C4—22no—
519C4⋊C4—21no—
520C4⋊C4—21no—
521C4⋊C4—22no—
522C4⋊C4—21no—
523C4⋊C4—21no—
524C4⋊C4—534no—
525C4⋊C4—1068no—
526C4⋊C4—1062no—
527C22⋊C4—44no—
528C22⋊C4—42no—
529C22⋊C4—42no—
530C22⋊C4—44no—
531C22⋊C4—1068no—
532C22⋊C4—1068no—
533C22⋊C4—1064no—
534C42—42no—
535C42—42no—
536C42—1064no—
537C23—22no—
538C23—22no—
539C23—28no—
540C23—22no—
541C23—22no—
542C23—22no—
543C23—22no—
544C23—22no—
545C23—22no—
546C23—48no—
547C23—48no—
548C23—44no—
549C23—44no—
550C23—1062no—
551C23—2121no—
552C23—2121no—
553C23—2122no—
554C23—2122no—
555C23#1313318512yes—
556C23#1313318912yes—
557C23#1313320012yes—
558C23#1313323714yes—
559C23C22×C22⋊D5311yescharacteristic
560Q8—1062no—
561Q8—1062no—
562Q8—2121no—
563Q8—2121no—
564D4—1062no—
565D4—1062no—
566D4—21216no—
567D4—2128no—
568D4—2121no—
569D4—2121no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—44no—
574C2×C4—48no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—1064no—
587C2×C4—1062no—
588C2×C4—2121no—
589C2×C4—2121no—
590C2×C4—2122no—
591C2×C4—2122no—
592C2×C4—2128no—
593C2×C4#1313318512yes—
594C2×C4#1313318912yes—
595C2×C4#1313319014yes—
596C2×C4#1313319014yes—
597C2×C4#1313319914yes—
598C2×C4#1313320012yes—
599C2×C4#1313320314yes—
600C2×C4C2×D4×D5311yescharacteristic
601C2×C4C2×D4×D5311yescharacteristic
602C2×C4C2×D4×D5311yescharacteristic
603C2×C4C2×#1171994711yescharacteristic
604C2×C4C2×#1171994511yescharacteristic
605C2×C4C2×#1171994511yescharacteristic
606C8—1062no—
607C8—1062no—
608C8—2122no—
609C22—24no—
610C22—22no—
611C22—22no—
612C22—24no—
613C22—24no—
614C22—24no—
615C22—24no—
616C22—416no—
617C22—42no—
618C22—42no—
619C22—42no—
620C22—42no—
621C22—4241no—
622C22—4241no—
623C22—4242no—
624C22—2124no—
625C22—2122no—
626C22#1681053814yes—
627C22C2×#1313318511yescharacteristic
628C22C2×#1313318911yescharacteristic
629C22C2×#1313320011yescharacteristic
630C22C2×#1313323712yes—
631C22#1681142112yes—
632C4—42no—
633C4—82no—
634C4—81no—
635C4—81no—
636C4—2124no—
637C4—2122no—
638C4#1681053814yes—
639C4C2×#1313319011yescharacteristic
640C4C2×#1313319011yescharacteristic
641C4C2×#1313319911yescharacteristic
642C4C2×#1313320311yescharacteristic
643C2—24no—
644C2—44no—
645C2—4242no—
646C2C2×#1681053811yescharacteristic
647C2#2028393112yes—
648C2#2028429311yescharacteristic
649C2#2028432111yescharacteristic
650C2#2028462812yes—
subgroupG/H
centerC22×C4C2×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2280C2
socleC23×C53C22×D4
prime radicalC2×#16811134C2
hypercenter#2280D53
hypocenterC53#23479
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22×C4 ◁ C24×C4 ◁ #2280

Complements

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

NH ≅ G/Ntype
2. C2×#16810659C2semidirect
3. C2×#16810617C2semidirect
4. C2×#16810617C2semidirect
5. C22×C22⋊#88322C2semidirect
6. #20275918C2semidirect
7. #20275911C2semidirect
8. #20275904C2semidirect
9. #20275897C2semidirect
10. #20274431C2semidirect
11. C53×#2280C2semidirect
24. C2×#13133150C22semidirect
25. C2×#13133148C22semidirect
28. C2×#13133140C22semidirect
29. C2×#13133140C22semidirect
30. C2×#13133140C22semidirect
31. C2×#13133137C22semidirect
32. C2×#13133137C22semidirect
33. C2×#13133137C22semidirect
37. #16810385C22semidirect
38. #16810385C22semidirect
63. C22×#88322D4semidirect
323. #2280D53semidirect
372. C53#23479semidirect

Maximal subgroups

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

subgroupindex
C2×#168111342
C2×#168106592
C2×#168106172
C22×C22⋊#883222
#202759182
#202759112
#202759042
#202758972
#202744312
C53×#22802
#2347953