Finite Group Databasebeta

← #27425370

Subgroups — #27425370

Order 13568.

784 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810930C211yesmaximal, characteristic, complemented
2C2×#16810680C211yesmaximal, characteristic
3#20283025C211yesmaximal, characteristic
4#20276152C211yesmaximal, characteristic
5#20276146C211yesmaximal, characteristic
6#20275915C211yesmaximal, characteristic
7#20275908C211yesmaximal, characteristic
8#20275672C211yesmaximal, characteristic, complemented
9#20275483C211yesmaximal, characteristic, complemented
10#20275455C211yesmaximal, characteristic, complemented
11#20275450C211yesmaximal, characteristic, complemented
12#20275408C211yesmaximal, characteristic, complemented
13#20275408C211yesmaximal, characteristic, complemented
14#20274730C211yesmaximal, characteristic, complemented
15C4×C22⋊Q8×C53C211yesmaximal, characteristic
16C2×#13133209C2211yescharacteristic
17#16810930—24no—
18#16810680—24no—
19C2×#13133150C2211yescharacteristic
20C2×#13133150C2211yescharacteristic
21C2×#13133148C2211yescharacteristic
22C2×#13133148C2211yescharacteristic
23C2×C4⋊#88322C2211yescharacteristic
24C2×C4⋊#88322C2211yescharacteristic
25C2×#13133141C2211yescharacteristic
26C2×#13133141C2211yescharacteristic
27C2×#13133141C2211yescharacteristic
28C2×#13133141C2211yescharacteristic
29C2×C22⋊#88322C2211yescharacteristic
30C2×C22⋊#88322C2211yescharacteristic
31C2×#13133128C2211yescharacteristic
32C2×#13133128C2211yescharacteristic
33#16810521C2211yescharacteristic
34#16810521C2211yescharacteristic
35#16810507—21no—
36#16810507—21no—
37#16810496—21no—
38#16810496—21no—
39#16810489C2211yescharacteristic
40#16810483C2211yescharacteristic
41#16810483C2211yescharacteristic
42#16810385C2211yescharacteristic
43#16810385C2211yescharacteristic
44#16810385C2211yescharacteristic
45#16810385C2211yescharacteristic
46#16810385C2211yescharacteristic
47#16810385C2211yescharacteristic
48#16810385C2211yescharacteristic
49#16810385C2211yescharacteristic
50C2×C22⋊Q8×C53C2211yescharacteristic
51C2×C4×Q8×C53C2211yescharacteristic
52C22×C42×C53C2211yescharacteristic
53C53×#391C2211yescharacteristic
54C53×#389C2211yescharacteristic
55C53×#387—21no—
56C53×#385—21no—
57C53×#380C2211yescharacteristic
58C4×C4⋊C4×C53—21no—
59C4×C4⋊C4×C53C2211yescharacteristic
60C4×C22⋊C4×C53—21no—
61C22×#88322C2311yescharacteristic
62C22×#88322C2311yescharacteristic
63C22×#88322C2311yescharacteristic
64C22×#88322C2311yescharacteristic
65#13133209—22no—
66#13133209—22no—
67C2×C4⋊Dic53C2311yescharacteristic
68C2×C4⋊Dic53C2311yescharacteristic
69C2×C4⋊Dic53C2311yescharacteristic
70C2×C4⋊Dic53C2311yescharacteristic
71C2×#11719919—21no—
72C2×#11719919—21no—
73C2×#11719919—21no—
74C2×#11719919—21no—
75C2×C22⋊Dic53—21no—
76C2×C22⋊Dic53—21no—
77#13133150—22no—
78#13133150—22no—
79#13133148—22no—
80#13133148—22no—
81C4⋊#88322—22no—
82C4⋊#88322D414yes—
83#13133141—22no—
84#13133141—22no—
85#13133141—22no—
86#13133141—22no—
87C22⋊#88322—22no—
88C22⋊#88322—22no—
89#13133128—22no—
90#13133128—22no—
91#13133124—21no—
92#13133124—21no—
93#13133124—21no—
94#13133124—21no—
95#13133124—21no—
96#13133124—21no—
97C22×Q8×C53C2311yescharacteristic
98C23×C4×C53D411yescharacteristic
99C23×C4×C53D411yescharacteristic
100C23×C4×C53C2311yescharacteristic
101C22⋊Q8×C53—22no—
102C22⋊Q8×C53—22no—
103C4×Q8×C53—22no—
104C4×Q8×C53—22no—
105C2×C4⋊C4×C53—21no—
106C2×C4⋊C4×C53—21no—
107C2×C4⋊C4×C53—21no—
108C2×C4⋊C4×C53C2311yescharacteristic
109C2×C4⋊C4×C53C2311yescharacteristic
110C2×C4⋊C4×C53C2311yescharacteristic
111C2×C22⋊C4×C53—21no—
112C2×C22⋊C4×C53—21no—
113C2×C42×C53—22no—
114C2×C42×C53—22no—
115C2×C42×C53—21no—
116C2×C42×C53—21no—
117C2×C42×C53D411yescharacteristic
118C2×C42×C53D411yescharacteristic
119C2×C42×C53C2311yescharacteristic
120C2×C42×C53C2311yescharacteristic
121#95×C53—21no—
122#95×C53—21no—
123#95×C53—21no—
124#95×C53—21no—
125#95×C53—21no—
126C24×C53C2×D411yescharacteristic
127C2×Q8×C53—21no—
128C2×Q8×C53—21no—
129C2×Q8×C53—21no—
130C2×Q8×C53—21no—
131C2×Q8×C53—21no—
132C2×Q8×C53#4112yes—
133C22×C4×C53—22no—
134C22×C4×C53—21no—
135C22×C4×C53—21no—
136C22×C4×C53—22no—
137C22×C4×C53—22no—
138C22×C4×C53—21no—
139C22×C4×C53—21no—
140C22×C4×C53—21no—
141C22×C4×C53—21no—
142C22×C4×C53—21no—
143C22×C4×C53—21no—
144C22×C4×C53—21no—
145C22×C4×C53—21no—
146C22×C4×C53—41no—
147C22×C4×C53—41no—
148C22×C4×C53—41no—
149C22×C4×C53C2×D411yescharacteristic
150C22×C4×C53C2×D411yescharacteristic
151C22×C4×C53C2×D411yescharacteristic
152C22×C4×C53C2×D411yescharacteristic
153C22×C4×C53C2×D411yescharacteristic
154C22×C4×C53C2411yescharacteristic
155C22×Dic53—21no—
156C22×Dic53—21no—
157C22×Dic53—21no—
158C22×Dic53—21no—
159C4⋊C4×C53—24no—
160C4⋊C4×C53—21no—
161C4⋊C4×C53—21no—
162C4⋊C4×C53—21no—
163C4⋊C4×C53—21no—
164C4⋊C4×C53—21no—
165C4⋊C4×C53—42no—
166C4⋊C4×C53—42no—
167C4⋊C4×C53#4112yes—
168C22⋊C4×C53—24no—
169C22⋊C4×C53—42no—
170C42×C53—22no—
171C42×C53—22no—
172C42×C53—21no—
173C42×C53—21no—
174C42×C53—42no—
175C42×C53—42no—
176C42×C53C2×D412yes—
177C42×C53#4112yes—
178C22⋊Dic53—42no—
179C22⋊Dic53—42no—
180C4⋊Dic53—21no—
181C4⋊Dic53—21no—
182C4⋊Dic53—21no—
183C4⋊Dic53—21no—
184C4⋊Dic53—21no—
185C4⋊Dic53—21no—
186C4⋊Dic53—21no—
187C4⋊Dic53—21no—
188#11719919—42no—
189#11719919—42no—
190#11719919—42no—
191#11719919—42no—
192C2×#88322—21no—
193C2×#88322—21no—
194C2×#88322—21no—
195C2×#88322—21no—
196C2×#88322—21no—
197C2×#88322—21no—
198C2×#88322—21no—
199C2×#88322—21no—
200C2×#88322—21no—
201C2×#88322—21no—
202C2×#88322C2×D412yes—
203C2×#88322C2×D412yes—
204C23×C53—21no—
205C23×C53—21no—
206C23×C53—21no—
207C23×C53—21no—
208C23×C53—21no—
209C23×C53—21no—
210C23×C53#13712yes—
211C23×C53C22×D411yescharacteristic
212Q8×C53—22no—
213Q8×C53—22no—
214Q8×C53—41no—
215Q8×C53—41no—
216C2×C4×C53—21no—
217C2×C4×C53—21no—
218C2×C4×C53—21no—
219C2×C4×C53—21no—
220C2×C4×C53—21no—
221C2×C4×C53—21no—
222C2×C4×C53—21no—
223C2×C4×C53—21no—
224C2×C4×C53—22no—
225C2×C4×C53—22no—
226C2×C4×C53—21no—
227C2×C4×C53—21no—
228C2×C4×C53—21no—
229C2×C4×C53—21no—
230C2×C4×C53—21no—
231C2×C4×C53—44no—
232C2×C4×C53—44no—
233C2×C4×C53—41no—
234C2×C4×C53—41no—
235C2×C4×C53—41no—
236C2×C4×C53—41no—
237C2×C4×C53—41no—
238C2×C4×C53—41no—
239C2×C4×C53—41no—
240C2×C4×C53—41no—
241C2×C4×C53—41no—
242C2×C4×C53—41no—
243C2×C4×C53—41no—
244C2×C4×C53—41no—
245C2×C4×C53—41no—
246C2×C4×C53—41no—
247C2×C4×C53#12112yes—
248C2×C4×C53#12112yes—
249C2×C4×C53C22×D411yescharacteristic
250C2×C4×C53C2×#4111yescharacteristic
251C2×C4×C53C2×#4111yescharacteristic
252C2×C4×C53C2×#4111yescharacteristic
253C2×C4×C532−1+411yescharacteristic
254C2×C4×C532−1+411yescharacteristic
255C2×Dic53—41no—
256C2×Dic53—41no—
257C2×Dic53—41no—
258C2×Dic53—41no—
259C2×Dic53—41no—
260C2×Dic53—41no—
261C2×Dic53—41no—
262C2×Dic53—41no—
263C2×Dic53—41no—
264C2×Dic53—41no—
265C2×Dic53—41no—
266C2×Dic53—41no—
267#88322—22no—
268#88322—22no—
269#88322—41no—
270#88322—41no—
271#24707—531nomaximal, Sylow
272C22×C53—22no—
273C22×C53—22no—
274C22×C53—22no—
275C22×C53—41no—
276C22×C53—41no—
277C22×C53—41no—
278C22×C53—41no—
279C22×C53C2×#12111yescharacteristic
280C22×C53#52511yescharacteristic
281C22×C53#53611yescharacteristic
282C22×C53C2×#13711yescharacteristic
283C22×C53#57511yescharacteristic
284C22×C53#57511yescharacteristic
285C22×C53#57511yescharacteristic
286C4×C53—22no—
287C4×C53—22no—
288C4×C53—42no—
289C4×C53—41no—
290C4×C53—42no—
291C4×C53—41no—
292C4×C53—41no—
293C4×C53—81no—
294C4×C53—81no—
295C4×C53#54711yescharacteristic
296C4×C53#54811yescharacteristic
297C4×C53#55211yescharacteristic
298C4×C53#55611yescharacteristic
299Dic53—81no—
300Dic53—81no—
301Dic53—81no—
302Dic53—81no—
303C2×#479—531no—
304C2×#432—531no—
305#2851—531no—
306C4×C22⋊Q8D5311yescharacteristic
307#2077—531no—
308#2076—531no—
309#2036—531no—
310#2035—531no—
311#1989—531no—
312#1949—531no—
313#1944—531no—
314#1943—531no—
315#1935—531no—
316#1935—531no—
317#1805—531no—
318C2×C53—42no—
319C2×C53#241511yescharacteristic
320C2×C53#305411yescharacteristic
321C2×C53#305611yescharacteristic
322C2×C53#311111yescharacteristic
323C2×C53#311711yescharacteristic
324C2×C53#323611yescharacteristic
325C2×C53#324011yescharacteristic
326C2×#125—531no—
327C2×C22⋊Q8C2×D5311yescharacteristic
328C2×C4×Q8C2×D5311yescharacteristic
329C22×C42C2×D5311yescharacteristic
330#479—1064no—
331#432—1064no—
332C2×#107—531no—
333C2×#107—531no—
334C2×#106—531no—
335C2×#106—531no—
336C2×C4⋊C8—531no—
337C2×C4⋊C8—531no—
338C2×#103—531no—
339C2×#103—531no—
340C2×#103—531no—
341C2×#103—531no—
342C2×C22⋊C8—531no—
343C2×C22⋊C8—531no—
344C2×#97—531no—
345C2×#97—531no—
346#391C2×D5311yescharacteristic
347#389—531no—
348#389—531no—
349#389C2×D5311yescharacteristic
350#387—21no—
351#387—1061no—
352#387—1061no—
353#385—21no—
354#385—1061no—
355#385—1061no—
356#382—531no—
357#380—531no—
358#380—531no—
359#380C2×D5311yescharacteristic
360C4×C4⋊C4—21no—
361C4×C4⋊C4C2×D5311yescharacteristic
362C4×C22⋊C4—21no—
363#340—531no—
364#340—531no—
365#340—531no—
366#340—531no—
367#340—531no—
368#340—531no—
369#340—531no—
370#340—531no—
371C53#2470711yescharacteristic, complemented, Sylow
372C22×Q8C22×D5311yescharacteristic
373C23×C4C22⋊D5311yescharacteristic
374C23×C4C22⋊D5311yescharacteristic
375C23×C4C22×D5311yescharacteristic
376C22×C8—531no—
377C22×C8—531no—
378C22×C8—531no—
379C22×C8—531no—
380#125—1062no—
381#125—1062no—
382C22⋊Q8—22no—
383C22⋊Q8—22no—
384C4×Q8—22no—
385C4×Q8—22no—
386C2×C4⋊C4—21no—
387C2×C4⋊C4—21no—
388C2×C4⋊C4—21no—
389C2×C4⋊C4—531no—
390C2×C4⋊C4—531no—
391C2×C4⋊C4—531no—
392C2×C4⋊C4—531no—
393C2×C4⋊C4—1061no—
394C2×C4⋊C4—1061no—
395C2×C4⋊C4—1061no—
396C2×C4⋊C4—1061no—
397C2×C4⋊C4C22×D5311yescharacteristic
398C2×C4⋊C4C22×D5311yescharacteristic
399C2×C4⋊C4C22×D5311yescharacteristic
400C2×C22⋊C4—21no—
401C2×C22⋊C4—21no—
402C2×C22⋊C4—1061no—
403C2×C22⋊C4—1061no—
404C2×C42—22no—
405C2×C42—22no—
406C2×C42—21no—
407C2×C42—21no—
408C2×C42C22⋊D5311yescharacteristic
409C2×C42C22⋊D5311yescharacteristic
410C2×C42C22×D5311yescharacteristic
411C2×C42C22×D5311yescharacteristic
412#107—1062no—
413#107—1062no—
414#106—1062no—
415#106—1062no—
416C4⋊C8—534no—
417C4⋊C8—1062no—
418#103—1062no—
419#103—1062no—
420#103—1062no—
421#103—1062no—
422C22⋊C8—1062no—
423C22⋊C8—1062no—
424#97—1062no—
425#97—1062no—
426#95—21no—
427#95—21no—
428#95—21no—
429#95—21no—
430#95—21no—
431#95—1061no—
432#95—1061no—
433#95—1061no—
434#95—1061no—
435#95—1061no—
436#95—1061no—
437C24C2×C22⋊D5311yescharacteristic
438C2×Q8—21no—
439C2×Q8—21no—
440C2×Q8—21no—
441C2×Q8—21no—
442C2×Q8—21no—
443C2×Q8#1171994512yes—
444C22×C4—22no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—41no—
458C22×C4—41no—
459C22×C4—41no—
460C22×C4—1061no—
461C22×C4—1061no—
462C22×C4—1061no—
463C22×C4—1061no—
464C22×C4C2×C22⋊D5311yescharacteristic
465C22×C4C2×C22⋊D5311yescharacteristic
466C22×C4C2×C22⋊D5311yescharacteristic
467C22×C4C2×C22⋊D5311yescharacteristic
468C22×C4C2×C22⋊D5311yescharacteristic
469C22×C4C23×D5311yescharacteristic
470C2×C8—532no—
471C2×C8—532no—
472C2×C8—1061no—
473C2×C8—1061no—
474C2×C8—1061no—
475C2×C8—1061no—
476C2×C8—1061no—
477C2×C8—1061no—
478C2×C8—1061no—
479C2×C8—1061no—
480C2×C8—1061no—
481C2×C8—1061no—
482C4⋊C4—24no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—42no—
489C4⋊C4—42no—
490C4⋊C4—1061no—
491C4⋊C4—1061no—
492C4⋊C4—1061no—
493C4⋊C4—1061no—
494C4⋊C4—1061no—
495C4⋊C4—1061no—
496C4⋊C4—1061no—
497C4⋊C4—1061no—
498C4⋊C4—2122no—
499C4⋊C4—2122no—
500C4⋊C4—2122no—
501C4⋊C4—2122no—
502C4⋊C4#1171994512yes—
503C22⋊C4—24no—
504C22⋊C4—42no—
505C22⋊C4—2122no—
506C22⋊C4—2122no—
507C42—22no—
508C42—22no—
509C42—21no—
510C42—21no—
511C42—42no—
512C42—42no—
513C42D4×D5312yes—
514C42#1171994712yes—
515C23—21no—
516C23—21no—
517C23—21no—
518C23—21no—
519C23—21no—
520C23—21no—
521C23#1313324812yes—
522C23C22×C22⋊D5311yescharacteristic
523Q8—22no—
524Q8—22no—
525Q8—41no—
526Q8—41no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—2121no—
559C2×C4—2121no—
560C2×C4—2121no—
561C2×C4—2121no—
562C2×C4—2121no—
563C2×C4—2121no—
564C2×C4—2121no—
565C2×C4—2121no—
566C2×C4—2121no—
567C2×C4—2121no—
568C2×C4—2121no—
569C2×C4—2121no—
570C2×C4#1313318912yes—
571C2×C4#1313318912yes—
572C2×C4C2×D4×D5311yescharacteristic
573C2×C4C2×#1171994711yescharacteristic
574C2×C4C2×#1171994511yescharacteristic
575C2×C4C2×#1171994511yescharacteristic
576C2×C4#1313326411yescharacteristic
577C2×C4#1313326411yescharacteristic
578C8—1062no—
579C8—1062no—
580C8—2121no—
581C8—2121no—
582C22—22no—
583C22—22no—
584C22—22no—
585C22—41no—
586C22—41no—
587C22—41no—
588C22—41no—
589C22C2×#1313318911yescharacteristic
590C22#1681113411yescharacteristic
591C22#1681119411yescharacteristic
592C22C2×#1313324811yescharacteristic
593C22#1681142111yescharacteristic
594C22#1681142111yescharacteristic
595C22#1681142111yescharacteristic
596C4—22no—
597C4—22no—
598C4—41no—
599C4—42no—
600C4—42no—
601C4—41no—
602C4—41no—
603C4—81no—
604C4—81no—
605C4—4241no—
606C4—4241no—
607C4—4241no—
608C4—4241no—
609C4#1681127811yescharacteristic
610C4#1681128211yescharacteristic
611C4#1681131311yescharacteristic
612C4#1681134311yescharacteristic
613C2—42no—
614C2#2027814411yescharacteristic
615C2#2028432111yescharacteristic
616C2#2028434111yescharacteristic
617C2#2028482411yescharacteristic
618C2#2028488611yescharacteristic
619C2#2028591711yescharacteristic
620C2#2028595111yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC4×C22⋊Q8×C53C2
socleC23×C53C22×D4
prime radicalC24×C53C2×D4
hypercenterC4×C22⋊Q8D53
hypocenterC53#24707
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C22×C42 ◁ C4×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C2×#16810930C2semidirect
8. #20275672C2semidirect
9. #20275483C2semidirect
10. #20275455C2semidirect
11. #20275450C2semidirect
12. #20275408C2semidirect
13. #20275408C2semidirect
14. #20274730C2semidirect
371. C53#24707semidirect

Maximal subgroups

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

subgroupindex
C2×#168109302
C2×#168106802
#202830252
#202761522
#202761462
#202759152
#202759082
#202756722
#202754832
#202754552
#202754502
#202754082
#202747302
C4×C22⋊Q8×C532
#2470753