Finite Group Databasebeta

← #27411572

Subgroups — #27411572

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×#16811130C211yesmaximal, characteristic
2C2×#16810661C211yesmaximal, characteristic, complemented
3C2×#16810619C211yesmaximal, characteristic, complemented
4C22×#13133129C211yesmaximal, characteristic, complemented
5#20277094C211yesmaximal, characteristic, complemented
6#20275917C212yesmaximal, complemented
7#20275910C212yesmaximal, complemented
8#20275903C212yesmaximal, complemented
9#20275896C212yesmaximal, complemented
10#20274431C211yesmaximal, characteristic, complemented
11C2×C53×#416C211yesmaximal, characteristic, complemented
12C23×C4×D53C2211yescharacteristic
13C23×#88322C2211yescharacteristic
14#16811130—24no—
15#16811130—24no—
16C2×#13133198—21no—
17C2×#13133193C2212yes—
18C2×#13133188C2212yes—
19C2×#13133185—21no—
20C2×C4×C22⋊D53—22no—
21C2×#13133176C2211yescharacteristic
22C2×#13133176C2211yescharacteristic
23#16810661—24no—
24C2×#13133151C2212yescomplemented
25C2×#13133149C2212yescomplemented
26#16810619—24no—
27C2×#13133141C2212yescomplemented
28C2×#13133138C2212yescomplemented
29C2×#13133129—24no—
30C2×#13133129C2212yes—
31C2×#13133129C2212yes—
32#16810484—21no—
33#16810480C2212yes—
34#16810480C2212yes—
35#16810478—21no—
36C4×#11719921—22no—
37#16810385C2212yescomplemented
38#16810385C2212yescomplemented
39C22×#41×C53C2211yescharacteristic
40C23×C8×C53C2211yescharacteristic
41C2×#117×C53C2211yescharacteristic
42C53×#416—24no—
43C2×#103×C53C2211yescharacteristic, complemented
44C2×#103×C53C2211yescharacteristic, complemented
45C2×#102×C53C2211yescharacteristic, complemented
46C2×#102×C53C2211yescharacteristic, complemented
47C24×D53D411yescharacteristic
48C22×C22⋊D53—22no—
49C23×Dic53D411yescharacteristic
50C22×C4×D53—22no—
51C22×C4×D53—22no—
52C22×C4×D53—22no—
53C22×C4×D53—22no—
54C22×C4×D53—22no—
55C22×C4×D53—22no—
56C22×C4×D53D412yes—
57C22×C4×D53C2312yes—
58C22×#88322—22no—
59C22×#88322—22no—
60C22×#88322—22no—
61C22×#88322C2312yes—
62#13133198—28no—
63#13133193—24no—
64#13133193—24no—
65#13133188—24no—
66#13133188—24no—
67#13133185—28no—
68C4×C22⋊D53—216no—
69#13133176—22no—
70#13133176—22no—
71#13133176—22no—
72#13133176D414yes—
73C2×C4⋊Dic53C2312yes—
74C2×C4⋊Dic53C2312yes—
75C2×#11719919—22no—
76C2×C22⋊Dic53—21no—
77C2×C22⋊Dic53—21no—
78C2×C22⋊Dic53—22no—
79C2×#11719921—22no—
80C2×#11719921—22no—
81C2×#11719921—22no—
82C2×C4×Dic53—21no—
83C2×C4×Dic53—21no—
84#13133151—24no—
85#13133149—24no—
86#13133141—24no—
87#13133138—24no—
88#13133129—24no—
89#13133129—24no—
90#13133124—22no—
91#13133124—22no—
92C2×#41×C53—22no—
93C2×#41×C53—22no—
94C2×#41×C53—22no—
95C2×#41×C53—22no—
96C2×#41×C53—22no—
97C2×#41×C53D414yes—
98C22×Q8×C53C2311yescharacteristic
99C22×D4×C53—21no—
100C22×D4×C53C2311yescharacteristic
101C23×C4×C53—21no—
102C23×C4×C53C2311yescharacteristic
103C22×C8×C53—22no—
104C22×C8×C53—22no—
105C22×C8×C53D414yescomplemented
106C22×C8×C53C2311yescharacteristic
107C22×C8×C53C2311yescharacteristic
108#117×C53—22no—
109#117×C53—22no—
110C2×C4⋊C4×C53C2311yescharacteristic
111C2×C4⋊C4×C53C2311yescharacteristic
112C2×C22⋊C4×C53—21no—
113C2×C42×C53—21no—
114#103×C53—22no—
115#103×C53—22no—
116#102×C53—22no—
117#102×C53—22no—
118C24×C53—21no—
119C24×C53C2×D411yescharacteristic
120C23×D53—22no—
121C23×D53—24no—
122C23×D53—22no—
123C23×D53—22no—
124C23×D53—42no—
125C23×D53C2×D412yes—
126#41×C53—28no—
127#41×C53—28no—
128#41×C53—44no—
129#41×C53—44no—
130C2×Q8×C53—21no—
131C2×Q8×C53—21no—
132C2×Q8×C53—21no—
133C2×Q8×C53—21no—
134C2×Q8×C53—21no—
135C2×Q8×C53C2×D412yes—
136C2×D4×C53—24no—
137C2×D4×C53—21no—
138C2×D4×C53—21no—
139C2×D4×C53—21no—
140C2×D4×C53—21no—
141C2×D4×C53—21no—
142C2×D4×C53—42no—
143C2×D4×C53—42no—
144C2×D4×C53C2×D412yes—
145C22×C4×C53—21no—
146C22×C4×C53—24no—
147C22×C4×C53—21no—
148C22×C4×C53—21no—
149C22×C4×C53—42no—
150C22×C4×C53—42no—
151C22×C4×C53C2×D411yescharacteristic
152C22×C4×C53C2×D412yes—
153C22×C4×C53C2×D412yes—
154C22×C4×C53C2×D412yes—
155C22×C4×C53#4112yes—
156C22×C4×C53#4112yes—
157C22×C4×C53#4112yes—
158C22×C4×C53C2411yescharacteristic
159C2×C22⋊D53—28no—
160C2×C22⋊D53—44no—
161C2×C22⋊D53—44no—
162C22×Dic53—22no—
163C22×Dic53—22no—
164C22×Dic53—22no—
165C22×Dic53—22no—
166C22×Dic53—22no—
167C22×Dic53C2×D412yes—
168C2×C4×D53—24no—
169C2×C4×D53—24no—
170C2×C4×D53—24no—
171C2×C4×D53—22no—
172C2×C4×D53—22no—
173C2×C4×D53—22no—
174C2×C4×D53—22no—
175C2×C4×D53—22no—
176C2×C4×D53—22no—
177C2×C4×D53—44no—
178C2×C4×D53—44no—
179C2×C4×D53—44no—
180C2×C4×D53—44no—
181C2×C8×C53—28no—
182C2×C8×C53—21no—
183C2×C8×C53—21no—
184C2×C8×C53—21no—
185C2×C8×C53—21no—
186C2×C8×C53C2×D412yes—
187C2×C8×C53C2×D412yes—
188C4⋊C4×C53—21no—
189C4⋊C4×C53—21no—
190C4⋊C4×C53—21no—
191C4⋊C4×C53—21no—
192C22⋊C4×C53—42no—
193C42×C53—42no—
194C22⋊Dic53—28no—
195C22⋊Dic53—24no—
196C22⋊Dic53—42no—
197#11719921—28no—
198#11719921—44no—
199#11719921—44no—
200C4⋊Dic53—22no—
201C4⋊Dic53—22no—
202C4⋊Dic53—22no—
203C4⋊Dic53C2×D414yes—
204#11719919—28no—
205C4×Dic53—24no—
206C4×Dic53—42no—
207C2×#88322—22no—
208C2×#88322—22no—
209C2×#88322—22no—
210C2×#88322—44no—
211C23×C53—22no—
212C23×C53—22no—
213C23×C53—22no—
214C23×C53—41no—
215C23×C53—41no—
216C23×C53—42no—
217C23×C53—42no—
218C23×C53#12012yes—
219C23×C53#12112yes—
220C23×C53#12312yes—
221C23×C53C22×D411yescharacteristic
222C22×D53—28no—
223C22×D53—22no—
224C22×D53—22no—
225C22×D53—22no—
226C22×D53—22no—
227C22×D53—22no—
228C22×D53—22no—
229C22×D53—48no—
230C22×D53—48no—
231C22×D53—44no—
232C22×D53—44no—
233C22×D53#13514yes—
234Q8×C53—22no—
235Q8×C53—22no—
236Q8×C53—41no—
237Q8×C53—41no—
238D4×C53—22no—
239D4×C53—22no—
240D4×C53—48no—
241D4×C53—41no—
242D4×C53—41no—
243C2×C4×C53—22no—
244C2×C4×C53—41no—
245C2×C4×C53—41no—
246C2×C4×C53—48no—
247C2×C4×C53—41no—
248C2×C4×C53—41no—
249C2×C4×C53—42no—
250C2×C4×C53—42no—
251C2×C4×C53#12012yes—
252C2×C4×C53#12112yes—
253C2×C4×C53#12114yes—
254C2×C4×C53#12114yes—
255C2×C4×C53#12312yes—
256C2×C4×C53#12314yes—
257C2×C4×C53#12414yes—
258C2×C4×C53C22×D411yescharacteristic
259C2×C4×C53C22×D411yescharacteristic
260C2×C4×C53C22×D411yescharacteristic
261C2×C4×C53C2×#4111yescharacteristic
262C2×C4×C53C2×#4111yescharacteristic
263C2×C4×C53C2×#4111yescharacteristic
264C22⋊D53—416no—
265C2×Dic53—22no—
266C2×Dic53—24no—
267C2×Dic53—22no—
268C2×Dic53—22no—
269C2×Dic53—42no—
270C2×Dic53—42no—
271C2×Dic53—44no—
272C2×Dic53—42no—
273C2×Dic53—42no—
274C2×Dic53—42no—
275C4×D53—42no—
276C4×D53—42no—
277C4×D53—42no—
278C4×D53—42no—
279C4×D53—48no—
280C8×C53—22no—
281C8×C53—22no—
282#88322—42no—
283#23479—531nomaximal, Sylow
284C22×C53—24no—
285C22×C53—24no—
286C22×C53—22no—
287C22×C53—22no—
288C22×C53—42no—
289C22×C53—44no—
290C22×C53—82no—
291C22×C53—81no—
292C22×C53—81no—
293C22×C53#39414yes—
294C22×C53C2×#12011yescharacteristic
295C22×C53C2×#12111yescharacteristic
296C22×C53C2×#12311yescharacteristic
297C22×C53C2×#13512yes—
298C22×C53#57512yes—
299C2×D53—24no—
300C2×D53—24no—
301C2×D53—24no—
302C2×D53—416no—
303C2×D53—42no—
304C2×D53—42no—
305C2×D53—42no—
306C2×D53—42no—
307C4×C53—42no—
308C4×C53—81no—
309C4×C53—81no—
310C4×C53#39414yes—
311C4×C53C2×#12111yescharacteristic
312C4×C53C2×#12111yescharacteristic
313C4×C53C2×#12311yescharacteristic
314C4×C53C2×#12411yescharacteristic
315Dic53—42no—
316Dic53—44no—
317Dic53—82no—
318C2×#525—531no—
319C2×#427—531no—
320C2×#416—531no—
321C2×#416D5311yescharacteristic, complemented
322C22×C22⋊C8—531no—
323#2280—531no—
324#2036—532no—
325#2035—532no—
326#2034—532no—
327#2033—532no—
328#1735—531no—
329C2×C53—24no—
330C2×C53—82no—
331C2×C53C2×#39411yescharacteristic
332C2×C53#300212yes—
333C2×C53#305011yescharacteristic
334C2×C53#305411yescharacteristic
335C2×C53#309112yes—
336D53—44no—
337C22×#41C2×D5311yescharacteristic
338C24×C4—531no—
339C23×C8—531no—
340C23×C8C2×D5311yescharacteristic
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×#117—531no—
350C2×#117C2×D5311yescharacteristic
351#427—1064no—
352C2×#107—532no—
353C2×#106—532no—
354#416—24no—
355#416—1064no—
356C2×#103—532no—
357C2×#103C2×D5311yescharacteristic, complemented
358C2×#103C2×D5311yescharacteristic, complemented
359C2×#102—532no—
360C2×#102C2×D5311yescharacteristic, complemented
361C2×#102C2×D5311yescharacteristic, complemented
362C2×C22⋊C8—532no—
363C2×C22⋊C8—532no—
364C2×C22⋊C8—1064no—
365#381—1061no—
366#380—532no—
367#380—532no—
368#379—1061no—
369C4×C22⋊C4—1062no—
370#340—532no—
371#340—532no—
372C53#2347911yescharacteristic, complemented, Sylow
373C25—531no—
374C2×#41—22no—
375C2×#41—22no—
376C2×#41—22no—
377C2×#41—22no—
378C2×#41—22no—
379C2×#41D21214yes—
380C22×Q8C22×D5311yescharacteristic
381C22×D4—21no—
382C22×D4—1062no—
383C22×D4C22×D5311yescharacteristic
384C23×C4—21no—
385C23×C4—531no—
386C23×C4—532no—
387C23×C4—532no—
388C23×C4—1062no—
389C23×C4—1062no—
390C23×C4—1062no—
391C23×C4—1062no—
392C23×C4—1062no—
393C23×C4—1062no—
394C23×C4C22×D5311yescharacteristic
395C22×C8—22no—
396C22×C8—22no—
397C22×C8—532no—
398C22×C8—1062no—
399C22×C8—1062no—
400C22×C8—1062no—
401C22×C8C22⋊D5314yescomplemented
402C22×C8C22×D5311yescharacteristic
403C22×C8C22×D5311yescharacteristic
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—534no—
414#117—1062no—
415#117—1062no—
416#117—1062no—
417C2×C4⋊C4—532no—
418C2×C4⋊C4—532no—
419C2×C4⋊C4—1062no—
420C2×C4⋊C4C22×D5311yescharacteristic
421C2×C4⋊C4C22×D5311yescharacteristic
422C2×C22⋊C4—21no—
423C2×C22⋊C4—1062no—
424C2×C22⋊C4—1062no—
425C2×C22⋊C4—1061no—
426C2×C22⋊C4—1062no—
427C2×C22⋊C4—1062no—
428C2×C22⋊C4—1061no—
429C2×C42—21no—
430C2×C42—1061no—
431C2×C42—1061no—
432#107—1064no—
433#106—1064no—
434#103—22no—
435#103—22no—
436#103—1064no—
437#102—22no—
438#102—22no—
439#102—1064no—
440C22⋊C8—1064no—
441C22⋊C8—1064no—
442#95—1062no—
443#95—1062no—
444C24—21no—
445C24—532no—
446C24—1064no—
447C24—1062no—
448C24—1062no—
449C24—1062no—
450C24—2122no—
451C24D4×D5311yescharacteristic
452#41—28no—
453#41—28no—
454#41—44no—
455#41—44no—
456C2×Q8—21no—
457C2×Q8—21no—
458C2×Q8—21no—
459C2×Q8—21no—
460C2×Q8—21no—
461C2×Q8C2×D21212yes—
462C2×D4—24no—
463C2×D4—21no—
464C2×D4—21no—
465C2×D4—21no—
466C2×D4—21no—
467C2×D4—21no—
468C2×D4—42no—
469C2×D4—42no—
470C2×D4—1068no—
471C2×D4—2124no—
472C2×D4—2124no—
473C2×D4C2×D21212yes—
474C22×C4—21no—
475C22×C4—24no—
476C22×C4—21no—
477C22×C4—21no—
478C22×C4—42no—
479C22×C4—42no—
480C22×C4—532no—
481C22×C4—1062no—
482C22×C4—1062no—
483C22×C4—1062no—
484C22×C4—1062no—
485C22×C4—1064no—
486C22×C4—1064no—
487C22×C4—1064no—
488C22×C4—1062no—
489C22×C4—1062no—
490C22×C4—1062no—
491C22×C4—1062no—
492C22×C4—1062no—
493C22×C4—1062no—
494C22×C4—1062no—
495C22×C4—2124no—
496C22×C4—2124no—
497C22×C4—2124no—
498C22×C4—2124no—
499C22×C4C2×D21212yes—
500C22×C4#1171994512yes—
501C22×C4D4×D5312yes—
502C22×C4D4×D5311yescharacteristic
503C22×C4#1171994712yes—
504C22×C4#1171994712yes—
505C22×C4C2×C22⋊D5312yes—
506C22×C4C23×D5311yescharacteristic
507C2×C8—28no—
508C2×C8—21no—
509C2×C8—21no—
510C2×C8—21no—
511C2×C8—21no—
512C2×C8—1062no—
513C2×C8—1062no—
514C2×C8—1062no—
515C2×C8—2124no—
516C2×C8C2×C22⋊D5312yes—
517C2×C8C2×C22⋊D5312yes—
518C4⋊C4—21no—
519C4⋊C4—21no—
520C4⋊C4—21no—
521C4⋊C4—21no—
522C4⋊C4—534no—
523C4⋊C4—1068no—
524C4⋊C4—1062no—
525C4⋊C4—1062no—
526C4⋊C4—1062no—
527C22⋊C4—42no—
528C22⋊C4—1068no—
529C22⋊C4—1068no—
530C22⋊C4—1064no—
531C22⋊C4—2124no—
532C22⋊C4—2124no—
533C22⋊C4—2122no—
534C42—42no—
535C42—1064no—
536C42—2122no—
537C23—22no—
538C23—22no—
539C23—22no—
540C23—41no—
541C23—41no—
542C23—42no—
543C23—42no—
544C23—534no—
545C23—1068no—
546C23—1062no—
547C23—1062no—
548C23—1062no—
549C23—1062no—
550C23—1062no—
551C23—1062no—
552C23—2128no—
553C23—2128no—
554C23—2124no—
555C23—2124no—
556C23#1313318712yes—
557C23#1313318812yes—
558C23#1313319912yes—
559C23C2×D4×D5311yescharacteristic
560Q8—22no—
561Q8—22no—
562Q8—41no—
563Q8—41no—
564D4—22no—
565D4—22no—
566D4—48no—
567D4—41no—
568D4—41no—
569D4—21216no—
570C2×C4—22no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—48no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—1064no—
579C2×C4—1062no—
580C2×C4—1062no—
581C2×C4—1062no—
582C2×C4—2122no—
583C2×C4—2122no—
584C2×C4—2122no—
585C2×C4—2124no—
586C2×C4—2122no—
587C2×C4—2122no—
588C2×C4—2128no—
589C2×C4—2122no—
590C2×C4—2122no—
591C2×C4—2122no—
592C2×C4—2122no—
593C2×C4#1313318712yes—
594C2×C4#1313318812yes—
595C2×C4#1313318914yes—
596C2×C4#1313319114yes—
597C2×C4#1313319912yes—
598C2×C4#1313320114yes—
599C2×C4#1313320314yes—
600C2×C4C2×D4×D5311yescharacteristic
601C2×C4C22×C22⋊D5311yescharacteristic
602C2×C4C22×D21211yescharacteristic
603C2×C4C2×#1171994711yescharacteristic
604C2×C4C2×#1171994711yescharacteristic
605C2×C4C2×#1171994511yescharacteristic
606C8—22no—
607C8—22no—
608C8—2122no—
609C22—24no—
610C22—22no—
611C22—22no—
612C22—24no—
613C22—44no—
614C22—42no—
615C22—81no—
616C22—81no—
617C22—82no—
618C22—1064no—
619C22—1064no—
620C22—1064no—
621C22—21216no—
622C22—2122no—
623C22—2122no—
624C22—2122no—
625C22—2122no—
626C22#1681053914yes—
627C22C2×#1313318711yescharacteristic
628C22C2×#1313318811yescharacteristic
629C22C2×#1313319911yescharacteristic
630C22#135×D5312yes—
631C22#1681142012yes—
632C4—42no—
633C4—81no—
634C4—81no—
635C4—4242no—
636C4—2124no—
637C4—2122no—
638C4#1681053914yes—
639C4C2×#1313318911yescharacteristic
640C4C2×#1313319111yescharacteristic
641C4C2×#1313320111yescharacteristic
642C4C2×#1313320311yescharacteristic
643C2—24no—
644C2—82no—
645C2—2124no—
646C2C2×#1681053911yescharacteristic
647C2#2028393512yes—
648C2#2028429211yescharacteristic
649C2#2028432011yescharacteristic
650C2#2028462612yes—
subgroupG/H
centerC22×C4D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC2×C53×#416C2
socleC23×C53C22×D4
prime radicalC2×#16811130C2
hypercenterC2×#416D53
hypocenterC53#23479
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22×C4 ◁ C23×C4 ◁ C2×#416

Complements

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

NH ≅ G/Ntype
2. C2×#16810661C2semidirect
3. C2×#16810619C2semidirect
4. C22×#13133129C2semidirect
5. #20277094C2semidirect
6. #20275917C2semidirect
7. #20275910C2semidirect
8. #20275903C2semidirect
9. #20275896C2semidirect
10. #20274431C2semidirect
11. C2×C53×#416C2semidirect
24. C2×#13133151C22semidirect
25. C2×#13133149C22semidirect
27. C2×#13133141C22semidirect
28. C2×#13133138C22semidirect
37. #16810385C22semidirect
38. #16810385C22semidirect
43. C2×#103×C53C22semidirect
44. C2×#103×C53C22semidirect
45. C2×#102×C53C22semidirect
46. C2×#102×C53C22semidirect
105. C22×C8×C53D4semidirect
321. C2×#416D53semidirect
357. C2×#103C2×D53semidirect
358. C2×#103C2×D53semidirect
360. C2×#102C2×D53semidirect
361. C2×#102C2×D53semidirect
372. C53#23479semidirect
401. C22×C8C22⋊D53semidirect

Maximal subgroups

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

subgroupindex
C2×#168111302
C2×#168106612
C2×#168106192
C22×#131331292
#202770942
#202759172
#202759102
#202759032
#202758962
#202744312
C2×C53×#4162
#2347953