Finite Group Databasebeta

← #27391328

Subgroups — #27391328

Order 13568.

876 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810961C211yesmaximal, characteristic
2C4×#13133176C211yesmaximal, characteristic
3#20275353C211yesmaximal, characteristic, complemented
4#20275346C211yesmaximal, characteristic, complemented
5#20275304C211yesmaximal, characteristic, complemented
6#20275297C211yesmaximal, characteristic, complemented
7#20275028C211yesmaximal, characteristic
8#20274912C211yesmaximal, characteristic
9#20274807C211yesmaximal, characteristic
10#20274613C211yesmaximal, characteristic, complemented
11#20274613C211yesmaximal, characteristic, complemented
12#20274545C211yesmaximal, characteristic, complemented
13#20274545C211yesmaximal, characteristic, complemented
14#20274477C211yesmaximal, characteristic
15C4×C22⋊Q8×C53C211yesmaximal, characteristic
16C2×#13133176C2211yescharacteristic
17C2×#13133176C2211yescharacteristic
18#16810961C418yescomplemented
19C2×#13133150C2211yescharacteristic
20C2×#13133150C2211yescharacteristic
21C2×#13133148C2211yescharacteristic
22C2×#13133148C2211yescharacteristic
23C2×#13133141C2211yescharacteristic
24C2×#13133141C2211yescharacteristic
25C2×#13133141C2211yescharacteristic
26C2×#13133141C2211yescharacteristic
27C2×C22⋊#88322C2211yescharacteristic
28C2×C22⋊#88322C2211yescharacteristic
29C2×#13133128C2211yescharacteristic
30C2×#13133128C2211yescharacteristic
31C4×C4⋊Dic53C2211yescharacteristic
32C4×C4⋊Dic53C2211yescharacteristic
33C4×C4⋊Dic53C2211yescharacteristic
34C4×C4⋊Dic53C2211yescharacteristic
35C4×C22⋊Dic53—21no—
36C4×C22⋊Dic53—21no—
37#16810469—21no—
38C42×Dic53—21no—
39#16810385C2211yescharacteristic
40#16810385C2211yescharacteristic
41#16810385C2211yescharacteristic
42#16810385C2211yescharacteristic
43#16810385C2211yescharacteristic
44#16810385C2211yescharacteristic
45#16810385C2211yescharacteristic
46#16810385C2211yescharacteristic
47#16810378C2211yescharacteristic
48#16810378C2211yescharacteristic
49C2×C22⋊Q8×C53C2211yescharacteristic
50C2×C4×Q8×C53C2211yescharacteristic
51C22×C42×C53C2211yescharacteristic
52C53×#391C2211yescharacteristic
53C53×#389C2211yescharacteristic
54C53×#387—21no—
55C53×#385—21no—
56C53×#380C2211yescharacteristic
57C4×C4⋊C4×C53—21no—
58C4×C4⋊C4×C53C2211yescharacteristic
59C4×C22⋊C4×C53—21no—
60C22×#88322C2311yescharacteristic
61C22×#88322C2311yescharacteristic
62C22×#88322C2311yescharacteristic
63C22×#88322C2311yescharacteristic
64#13133176—22no—
65#13133176—22no—
66#13133176—22no—
67#13133176C2×C414yes—
68C2×C4⋊Dic53C2311yescharacteristic
69C2×C4⋊Dic53C2311yescharacteristic
70C2×C4⋊Dic53C2311yescharacteristic
71C2×C4⋊Dic53C2311yescharacteristic
72C2×C22⋊Dic53—21no—
73C2×C22⋊Dic53—21no—
74C2×C4×Dic53—21no—
75C2×C4×Dic53—21no—
76C2×C4×Dic53—21no—
77C2×C4×Dic53—21no—
78C2×C4×Dic53—21no—
79C2×C4×Dic53—21no—
80#13133150—22no—
81#13133150C2×C414yescomplemented
82#13133148—22no—
83#13133148C2×C414yescomplemented
84#13133141—22no—
85#13133141—22no—
86#13133141C2×C414yescomplemented
87#13133141C2×C414yescomplemented
88C22⋊#88322—22no—
89C22⋊#88322C2×C414yes—
90#13133128—22no—
91#13133128—22no—
92#13133124—21no—
93#13133124—21no—
94#13133124—21no—
95#13133124—21no—
96C22×Q8×C53C2311yescharacteristic
97C23×C4×C53D411yescharacteristic
98C23×C4×C53D411yescharacteristic
99C23×C4×C53C2311yescharacteristic
100C22⋊Q8×C53—22no—
101C22⋊Q8×C53C2×C414yes—
102C4×Q8×C53—22no—
103C4×Q8×C53—22no—
104C2×C4⋊C4×C53—21no—
105C2×C4⋊C4×C53—21no—
106C2×C4⋊C4×C53—21no—
107C2×C4⋊C4×C53C2311yescharacteristic
108C2×C4⋊C4×C53C2311yescharacteristic
109C2×C4⋊C4×C53C2311yescharacteristic
110C2×C22⋊C4×C53—21no—
111C2×C22⋊C4×C53—21no—
112C2×C42×C53—22no—
113C2×C42×C53—22no—
114C2×C42×C53—21no—
115C2×C42×C53—21no—
116C2×C42×C53D411yescharacteristic
117C2×C42×C53D411yescharacteristic
118C2×C42×C53C2311yescharacteristic
119C2×C42×C53C2311yescharacteristic
120#95×C53—21no—
121#95×C53—21no—
122#95×C53—21no—
123#95×C53—21no—
124#95×C53—21no—
125C24×C53C2×D411yescharacteristic
126C2×Q8×C53—21no—
127C2×Q8×C53—21no—
128C2×Q8×C53—21no—
129C2×Q8×C53—21no—
130C2×Q8×C53—21no—
131C2×Q8×C53C22×C412yes—
132C22×C4×C53—21no—
133C22×C4×C53—21no—
134C22×C4×C53—21no—
135C22×C4×C53—21no—
136C22×C4×C53—21no—
137C22×C4×C53—21no—
138C22×C4×C53—22no—
139C22×C4×C53—22no—
140C22×C4×C53—22no—
141C22×C4×C53—21no—
142C22×C4×C53—21no—
143C22×C4×C53—41no—
144C22×C4×C53—41no—
145C22×C4×C53—41no—
146C22×C4×C53C22×C412yes—
147C22×C4×C53C2×D411yescharacteristic
148C22×C4×C53C2×D411yescharacteristic
149C22×C4×C53C2×D411yescharacteristic
150C22×C4×C53C2×D411yescharacteristic
151C22×C4×C53C2×D411yescharacteristic
152C22×C4×C53#4112yes—
153C22×C4×C53C2411yescharacteristic
154C22×Dic53—21no—
155C22×Dic53—21no—
156C22×Dic53—21no—
157C22×Dic53—21no—
158C4⋊C4×C53—24no—
159C4⋊C4×C53—21no—
160C4⋊C4×C53—21no—
161C4⋊C4×C53—21no—
162C4⋊C4×C53—21no—
163C4⋊C4×C53—21no—
164C4⋊C4×C53—42no—
165C4⋊C4×C53—42no—
166C4⋊C4×C53C22×C412yes—
167C22⋊C4×C53—24no—
168C22⋊C4×C53—42no—
169C42×C53—22no—
170C42×C53—22no—
171C42×C53—21no—
172C42×C53—21no—
173C42×C53—42no—
174C42×C53—42no—
175C42×C53#4112yes—
176C42×C53#4112yes—
177C22⋊Dic53—24no—
178C22⋊Dic53—24no—
179C4⋊Dic53—21no—
180C4⋊Dic53—21no—
181C4⋊Dic53—21no—
182C4⋊Dic53—21no—
183C4⋊Dic53C22×C412yes—
184C4⋊Dic53C22×C412yes—
185C4⋊Dic53#4112yes—
186C4⋊Dic53#4112yes—
187C4×Dic53—24no—
188C4×Dic53—24no—
189C4×Dic53—42no—
190C4×Dic53—42no—
191C4×Dic53—42no—
192C4×Dic53—42no—
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×#88322—21no—
203C2×#88322C22×C412yes—
204C2×#88322C22×C412yes—
205C23×C53—21no—
206C23×C53—21no—
207C23×C53—21no—
208C23×C53—21no—
209C23×C53—21no—
210C23×C53—21no—
211C23×C53C4×D412yes—
212C23×C53C22×D411yescharacteristic
213Q8×C53—22no—
214Q8×C53—22no—
215Q8×C53—41no—
216Q8×C53—41no—
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—21no—
225C2×C4×C53—22no—
226C2×C4×C53—22no—
227C2×C4×C53—22no—
228C2×C4×C53—22no—
229C2×C4×C53—22no—
230C2×C4×C53—22no—
231C2×C4×C53—21no—
232C2×C4×C53—21no—
233C2×C4×C53—21no—
234C2×C4×C53—44no—
235C2×C4×C53—44no—
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×C53C4×D412yes—
247C2×C4×C53#12312yes—
248C2×C4×C53#12312yes—
249C2×C4×C53#13712yes—
250C2×C4×C53C23×C411yescharacteristic
251C2×C4×C53C2×#4111yescharacteristic
252C2×C4×C53C2×#4111yescharacteristic
253C2×C4×C53C2×#4111yescharacteristic
254C2×C4×C53C2×#4111yescharacteristic
255C2×C4×C53C2×#4111yescharacteristic
256C2×Dic53—22no—
257C2×Dic53—22no—
258C2×Dic53—22no—
259C2×Dic53—22no—
260C2×Dic53—41no—
261C2×Dic53—41no—
262C2×Dic53—41no—
263C2×Dic53—41no—
264C2×Dic53—41no—
265C2×Dic53—41no—
266C2×Dic53—41no—
267C2×Dic53—41no—
268#88322—22no—
269#88322—22no—
270#88322—41no—
271#88322—41no—
272#21583—531nomaximal, Sylow
273C22×C53—22no—
274C22×C53—22no—
275C22×C53—22no—
276C22×C53—41no—
277C22×C53—41no—
278C22×C53—41no—
279C22×C53—41no—
280C22×C53C2×C4×D411yescharacteristic
281C22×C53C2×#12311yescharacteristic
282C22×C53#52511yescharacteristic
283C22×C53C2×#13711yescharacteristic
284C22×C53#57511yescharacteristic
285C22×C53#57511yescharacteristic
286C22×C53#57511yescharacteristic
287C4×C53—22no—
288C4×C53—22no—
289C4×C53—22no—
290C4×C53—22no—
291C4×C53—41no—
292C4×C53—42no—
293C4×C53—42no—
294C4×C53—81no—
295C4×C53—81no—
296C4×C53C4×#4111yescharacteristic
297C4×C53C4×#4111yescharacteristic
298C4×C53#52911yescharacteristic
299C4×C53#52911yescharacteristic
300Dic53—42no—
301Dic53—42no—
302Dic53—42no—
303Dic53—42no—
304C2×#485—531no—
305C4×C22⋊Q8D5311yescharacteristic
306C4×#117—531no—
307#1928—531no—
308#1927—531no—
309#1921—531no—
310#1920—531no—
311#1873—531no—
312#1854—531no—
313#1825—531no—
314#1776—531no—
315#1776—531no—
316#1763—531no—
317#1763—531no—
318#1749—531no—
319C2×C53—42no—
320C2×C53C4×#12311yescharacteristic
321C2×C53#294311yescharacteristic
322C2×C53C4×#13711yescharacteristic
323C2×C53#309311yescharacteristic
324C2×C53#309511yescharacteristic
325C2×C53#311111yescharacteristic
326C2×C53#311611yescharacteristic
327C2×C22⋊Q8C2×D5311yescharacteristic
328C2×C4×Q8C2×D5311yescharacteristic
329C2×#117—531no—
330C2×#117—531no—
331C22×C42C2×D5311yescharacteristic
332#485—538no—
333C2×#107—531no—
334C2×#107—531no—
335C2×#106—531no—
336C2×#106—531no—
337C2×#103—531no—
338C2×#103—531no—
339C2×#103—531no—
340C2×#103—531no—
341C2×C22⋊C8—531no—
342C2×C22⋊C8—531no—
343C2×#97—531no—
344C2×#97—531no—
345#391C2×D5311yescharacteristic
346#389C2×D5311yescharacteristic
347#387—21no—
348#385—21no—
349#380C2×D5311yescharacteristic
350C4×C4⋊C4—21no—
351C4×C4⋊C4—531no—
352C4×C4⋊C4—531no—
353C4×C4⋊C4—531no—
354C4×C4⋊C4—531no—
355C4×C4⋊C4C2×D5311yescharacteristic
356C4×C22⋊C4—21no—
357C4×C22⋊C4—1061no—
358C4×C22⋊C4—1061no—
359#376—1061no—
360C43—1061no—
361#340—531no—
362#340—531no—
363#340—531no—
364#340—531no—
365#340—531no—
366#340—531no—
367#340—531no—
368#340—531no—
369#336—531no—
370#336—531no—
371C53#2158311yescharacteristic, 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—
380C22⋊Q8—22no—
381C22⋊Q8C4×D5314yes—
382C4×Q8—22no—
383C4×Q8—22no—
384#117—534no—
385#117—1062no—
386#117—1062no—
387#117—1062no—
388C2×C4⋊C4—21no—
389C2×C4⋊C4—21no—
390C2×C4⋊C4—21no—
391C2×C4⋊C4—531no—
392C2×C4⋊C4—531no—
393C2×C4⋊C4—531no—
394C2×C4⋊C4—531no—
395C2×C4⋊C4C22×D5311yescharacteristic
396C2×C4⋊C4C22×D5311yescharacteristic
397C2×C4⋊C4C22×D5311yescharacteristic
398C2×C22⋊C4—21no—
399C2×C22⋊C4—21no—
400C2×C22⋊C4—1061no—
401C2×C22⋊C4—1061no—
402C2×C42—22no—
403C2×C42—22no—
404C2×C42—21no—
405C2×C42—21no—
406C2×C42—1061no—
407C2×C42—1061no—
408C2×C42—1061no—
409C2×C42—1061no—
410C2×C42—1061no—
411C2×C42—1061no—
412C2×C42C22⋊D5311yescharacteristic
413C2×C42C22⋊D5311yescharacteristic
414C2×C42C22×D5311yescharacteristic
415C2×C42C22×D5311yescharacteristic
416#107—534no—
417#107—1062no—
418#106—534no—
419#106—1062no—
420#103—534no—
421#103—534no—
422#103—1062no—
423#103—1062no—
424C22⋊C8—534no—
425C22⋊C8—1062no—
426#97—1062no—
427#97—1062no—
428#95—21no—
429#95—21no—
430#95—21no—
431#95—21no—
432#95—21no—
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×Q8C2×C4×D5312yes—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—41no—
456C22×C4—41no—
457C22×C4—41no—
458C22×C4—1061no—
459C22×C4—1061no—
460C22×C4—1061no—
461C22×C4—1061no—
462C22×C4C2×C4×D5312yes—
463C22×C4#1171994512yes—
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—532no—
491C4⋊C4—532no—
492C4⋊C4—532no—
493C4⋊C4—532no—
494C4⋊C4—1061no—
495C4⋊C4—1061no—
496C4⋊C4—1061no—
497C4⋊C4—1061no—
498C4⋊C4C2×C4×D5312yes—
499C22⋊C4—24no—
500C22⋊C4—42no—
501C22⋊C4—1064no—
502C22⋊C4—1064no—
503C42—22no—
504C42—22no—
505C42—21no—
506C42—21no—
507C42—42no—
508C42—42no—
509C42—1064no—
510C42—1064no—
511C42—2122no—
512C42—2122no—
513C42—2122no—
514C42—2122no—
515C42#1171994712yes—
516C42#1171994712yes—
517C23—21no—
518C23—21no—
519C23—21no—
520C23—21no—
521C23—21no—
522C23—21no—
523C23C4×C22⋊D5312yes—
524C23C22×C22⋊D5311yescharacteristic
525Q8—22no—
526Q8—22no—
527Q8—41no—
528Q8—41no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—44no—
547C2×C4—44no—
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—1062no—
559C2×C4—1062no—
560C2×C4—1062no—
561C2×C4—1062no—
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×C4C4×C22⋊D5312yes—
571C2×C4#1313319812yes—
572C2×C4#1313319812yes—
573C2×C4#1313324812yes—
574C2×C4C22×C4×D5311yescharacteristic
575C2×C4#41×D5311yescharacteristic
576C2×C4#41×D5311yescharacteristic
577C2×C4C2×#1171994711yescharacteristic
578C2×C4C2×#1171994711yescharacteristic
579C2×C4C2×#1171994511yescharacteristic
580C8—1062no—
581C8—1062no—
582C8—2121no—
583C8—2121no—
584C22—22no—
585C22—22no—
586C22—22no—
587C22—41no—
588C22—41no—
589C22—41no—
590C22—41no—
591C22C2×C4×C22⋊D5311yescharacteristic
592C22C2×#1313319811yescharacteristic
593C22#1681113011yescharacteristic
594C22C2×#1313324811yescharacteristic
595C22#1681142111yescharacteristic
596C22#1681142111yescharacteristic
597C22#1681142111yescharacteristic
598C4—22no—
599C4—22no—
600C4—22no—
601C4—22no—
602C4—42no—
603C4—41no—
604C4—42no—
605C4—81no—
606C4—81no—
607C4—2122no—
608C4—2122no—
609C4—2122no—
610C4—2122no—
611C4C4×#1171994711yescharacteristic
612C4C4×#1171994711yescharacteristic
613C4#1681115311yescharacteristic
614C4#1681115311yescharacteristic
615C2—42no—
616C2C4×#1313319811yescharacteristic
617C2#2028351311yescharacteristic
618C2C4×#1313324811yescharacteristic
619C2#2028464411yescharacteristic
620C2#2028465911yescharacteristic
621C2#2028481811yescharacteristic
622C2#2028487111yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC4×C22⋊Q8×C53C2
socleC23×C53C22×D4
prime radicalC24×C53C2×D4
cosocleC22×C4×C53C24
hypercenterC4×C22⋊Q8D53
hypocenterC53#21583
LCSG ▷ C2×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
3. #20275353C2semidirect
4. #20275346C2semidirect
5. #20275304C2semidirect
6. #20275297C2semidirect
10. #20274613C2semidirect
11. #20274613C2semidirect
12. #20274545C2semidirect
13. #20274545C2semidirect
18. #16810961C4semidirect
81. #13133150C2×C4semidirect
83. #13133148C2×C4semidirect
86. #13133141C2×C4semidirect
87. #13133141C2×C4semidirect
371. C53#21583semidirect

Maximal subgroups

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

subgroupindex
C2×#168109612
C4×#131331762
#202753532
#202753462
#202753042
#202752972
#202750282
#202749122
#202748072
#202746132
#202745452
#202744772
C4×C22⋊Q8×C532
#2158353