Finite Group Databasebeta

← #27403290

Subgroups — #27403290

Order 13568.

902 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810708C211yesmaximal, characteristic
2#20277671C211yesmaximal, characteristic, complemented
3#20277616C211yesmaximal, characteristic
4#20275407C211yesmaximal, characteristic, complemented
5#20275303C211yesmaximal, characteristic, complemented
6#20275297C211yesmaximal, characteristic, complemented
7#20275052C211yesmaximal, characteristic
8#20275046C211yesmaximal, characteristic
9#20274841C211yesmaximal, characteristic, complemented
10#20274718C211yesmaximal, characteristic, complemented
11#20274714C211yesmaximal, characteristic
12#20274583C211yesmaximal, characteristic, complemented
13#20274541C211yesmaximal, characteristic
14#20274539C211yesmaximal, characteristic
15C53×#1935C211yesmaximal, characteristic, complemented
16C22×#11719924C2211yescharacteristic
17C2×Q8×Dic53C2211yescharacteristic
18C2×C4×D212C2211yescharacteristic
19C2×#13133179C2211yescharacteristic
20C2×#13133179C2211yescharacteristic
21C2×C4⋊C4×D53—21no—
22#16810708C418yescomplemented
23C2×#13133148C2211yescharacteristic
24C2×#13133146C2211yescharacteristic
25C2×#13133141C2211yescharacteristic
26C2×#13133139C2211yescharacteristic
27C2×#13133139C2211yescharacteristic
28C2×#13133137C2211yescharacteristic
29C2×#13133127C2211yescharacteristic
30C2×C8×Dic53C2211yescharacteristic
31#16810530C2211yescharacteristic
32#16810522C2211yescharacteristic
33#16810508—21no—
34#16810506—21no—
35#16810497—21no—
36#16810494—21no—
37#16810492C2211yescharacteristic
38#16810492C2211yescharacteristic
39#16810487—21no—
40#16810482—21no—
41C4⋊C4×Dic53C2211yescharacteristic
42C4×#11719919C2211yescharacteristic
43C4×C4⋊Dic53C2211yescharacteristic
44#16810474—21no—
45#16810386C2211yescharacteristic
46#16810386C2211yescharacteristic
47#16810385C2211yescharacteristic
48#16810385C2211yescharacteristic
49#16810385C2211yescharacteristic
50#16810385C2211yescharacteristic
51#16810378C2211yescharacteristic
52#16810377C2211yescharacteristic
53C2×#103×C53C2211yescharacteristic
54C2×#103×C53C2211yescharacteristic
55C2×#97×C53C2211yescharacteristic
56C53×#391C2211yescharacteristic
57C53×#389C2211yescharacteristic
58C53×#340C2211yescharacteristic
59C53×#340C2211yescharacteristic
60C22×D212C2311yescharacteristic
61C22×C4×D53—21no—
62C22×C4×D53—21no—
63C22×C4×D53—21no—
64C2×#11719924—22no—
65C2×#11719924—22no—
66C2×#11719924C2×C414yes—
67C22×#88322C2311yescharacteristic
68C22×#88322C2311yescharacteristic
69Q8×Dic53—22no—
70Q8×Dic53C2×C414yescomplemented
71C4×D212—22no—
72C4×D212—41no—
73#13133179—22no—
74#13133179—22no—
75#13133179—41no—
76#13133179C2×C414yes—
77C2×C4⋊Dic53—21no—
78C2×C4⋊Dic53—21no—
79C2×C4⋊Dic53C2311yescharacteristic
80C2×C4⋊Dic53C2311yescharacteristic
81C2×C4⋊Dic53C2311yescharacteristic
82C2×#11719919—21no—
83C2×#11719919—21no—
84C2×#11719919—21no—
85C2×#11719919—21no—
86C2×#11719919D411yescharacteristic
87C2×#11719919D411yescharacteristic
88C4⋊C4×D53—44no—
89C2×#11719921—21no—
90C2×#11719921—21no—
91C2×#11719921—21no—
92C2×C4×Dic53—21no—
93C2×C4×Dic53—21no—
94C2×C4×Dic53—21no—
95C2×C4×Dic53C2311yescharacteristic
96C2×C4×Dic53C2311yescharacteristic
97#13133148C2×C414yescomplemented
98#13133146—22no—
99#13133141—41no—
100#13133139—22no—
101#13133139C2×C414yes—
102#13133137—22no—
103#13133127—41no—
104C8×Dic53C2×C414yescomplemented
105#13133124—21no—
106#13133124—21no—
107#13133124—21no—
108#13133124—21no—
109#13133124—21no—
110#13133124D411yescharacteristic
111#13133124D411yescharacteristic
112C22×Q8×C53C2311yescharacteristic
113C22×C8×C53C2311yescharacteristic
114C22×C8×C53C2311yescharacteristic
115C2×C4⋊C4×C53—21no—
116C2×C4⋊C4×C53C2311yescharacteristic
117C2×C4⋊C4×C53C2311yescharacteristic
118C2×C4⋊C4×C53C2311yescharacteristic
119C2×C42×C53C2311yescharacteristic
120#103×C53—41no—
121#103×C53C2×C414yescomplemented
122#97×C53—41no—
123#95×C53—21no—
124#95×C53—21no—
125#95×C53—21no—
126C23×D53—21no—
127C2×Q8×C53—21no—
128C2×Q8×C53—21no—
129C2×Q8×C53—21no—
130C2×Q8×C53—21no—
131C2×Q8×C53—21no—
132C2×Q8×C53C22×C412yes—
133C22×C4×C53—21no—
134C22×C4×C53—21no—
135C22×C4×C53—21no—
136C22×C4×C53—21no—
137C22×C4×C53C2×D411yescharacteristic
138C22×C4×C53C2×D411yescharacteristic
139C22×C4×C53C2411yescharacteristic
140C22×Dic53—21no—
141C22×Dic53—21no—
142C22×Dic53—21no—
143C22×Dic53C2×D411yescharacteristic
144C22×Dic53C2×D411yescharacteristic
145C22×Dic53C2×D411yescharacteristic
146C22×Dic53C2×D411yescharacteristic
147C2×D212—21no—
148C2×D212—21no—
149C2×D212—21no—
150C2×D212C22×C412yes—
151C2×D212#4112yes—
152C2×D212#4112yes—
153C2×C4×D53—24no—
154C2×C4×D53—42no—
155C2×C4×D53—42no—
156C2×C4×D53—42no—
157C2×C4×D53—42no—
158C2×C4×D53—42no—
159C2×C4×D53—42no—
160C2×C4×D53—81no—
161C2×C4×D53—81no—
162C2×C8×C53—21no—
163C2×C8×C53—21no—
164C2×C8×C53—21no—
165C2×C8×C53—21no—
166C2×C8×C53—21no—
167C2×C8×C53C22×C412yes—
168C4⋊C4×C53—21no—
169C4⋊C4×C53—21no—
170C4⋊C4×C53—21no—
171C4⋊C4×C53—21no—
172C4⋊C4×C53—21no—
173C4⋊C4×C53—42no—
174C4⋊C4×C53C22×C412yes—
175C42×C53—21no—
176C42×C53—21no—
177#11719924—28no—
178#11719921—24no—
179#11719921—42no—
180#11719921—42no—
181C4⋊Dic53—21no—
182C4⋊Dic53—24no—
183C4⋊Dic53—21no—
184C4⋊Dic53—21no—
185C4⋊Dic53—21no—
186C4⋊Dic53—21no—
187C4⋊Dic53—42no—
188C4⋊Dic53C22×C412yes—
189#11719919—22no—
190#11719919—22no—
191#11719919—42no—
192#11719919—42no—
193#11719919—81no—
194#11719919—81no—
195C4×Dic53—21no—
196C4×Dic53—21no—
197C4×Dic53—24no—
198C4×Dic53—81no—
199C4×Dic53—81no—
200C4×Dic53C22×C412yes—
201C4×Dic53#4112yes—
202C2×#88322—21no—
203C2×#88322—21no—
204C2×#88322—21no—
205C2×#88322—21no—
206C2×#88322—21no—
207C2×#88322C22×C412yes—
208C23×C53C22×D411yescharacteristic
209C22×D53—22no—
210C22×D53—22no—
211C22×D53—22no—
212C22×D53—41no—
213C22×D53—41no—
214C22×D53—41no—
215C22×D53—41no—
216Q8×C53—22no—
217Q8×C53—22no—
218Q8×C53—41no—
219Q8×C53—41no—
220C2×C4×C53—22no—
221C2×C4×C53—22no—
222C2×C4×C53—21no—
223C2×C4×C53—21no—
224C2×C4×C53—21no—
225C2×C4×C53—21no—
226C2×C4×C53—21no—
227C2×C4×C53—21no—
228C2×C4×C53—41no—
229C2×C4×C53—41no—
230C2×C4×C53—41no—
231C2×C4×C53—41no—
232C2×C4×C53—41no—
233C2×C4×C53—41no—
234C2×C4×C53—41no—
235C2×C4×C53—41no—
236C2×C4×C53—41no—
237C2×C4×C53—41no—
238C2×C4×C53C23×C411yescharacteristic
239C2×C4×C53C2×#4111yescharacteristic
240C2×C4×C53C2×#4111yescharacteristic
241C2×C4×C53C2×#4111yescharacteristic
242C2×C4×C53H(5,𝔽2)11yescharacteristic
243C2×C4×C532−1+411yescharacteristic
244C2×Dic53—21no—
245C2×Dic53—21no—
246C2×Dic53—22no—
247C2×Dic53—21no—
248C2×Dic53—21no—
249C2×Dic53—21no—
250C2×Dic53—21no—
251C2×Dic53—21no—
252C2×Dic53—21no—
253C2×Dic53—41no—
254C2×Dic53—41no—
255C2×Dic53—41no—
256C2×Dic53—41no—
257C2×Dic53—41no—
258C2×Dic53—41no—
259C2×Dic53—41no—
260C2×Dic53—41no—
261C2×Dic53C4×D412yes—
262C2×Dic53C4×D412yes—
263C2×Dic53AGL(1,ℤ/8)12yes—
264C2×Dic53#13712yes—
265D212—22no—
266D212—22no—
267D212—22no—
268D212#11714yes—
269C4×D53—48no—
270C4×D53—84no—
271C4×D53—84no—
272C8×C53—22no—
273C8×C53—41no—
274#88322—22no—
275#88322—41no—
276#22762—531nomaximal, Sylow
277C22×C53C2×C4×D411yescharacteristic
278C22×C53#54611yescharacteristic
279C22×C53#54811yescharacteristic
280C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
281C22×C53C2×#13711yescharacteristic
282C22×C53#57711yescharacteristic
283C22×C53#57711yescharacteristic
284C2×D53—24no—
285C2×D53—42no—
286C2×D53—42no—
287C2×D53—42no—
288C2×D53—42no—
289C2×D53—42no—
290C2×D53—42no—
291C4×C53—41no—
292C4×C53—42no—
293C4×C53—42no—
294C4×C53—41no—
295C4×C53—81no—
296C4×C53—81no—
297C4×C53C2×#11711yescharacteristic
298C4×C53#52011yescharacteristic
299C4×C53#55211yescharacteristic
300C4×C53#55311yescharacteristic
301Dic53—22no—
302Dic53—41no—
303Dic53—42no—
304Dic53—41no—
305Dic53—81no—
306Dic53—81no—
307Dic53#44014yes—
308C2×C4×SD16—531no—
309#2355—531no—
310D4×C4⋊C4—531no—
311#1935—531no—
312#1935D5311yescharacteristic, complemented
313#1920—531no—
314#1920—531no—
315#1877—531no—
316#1877—531no—
317#1834—531no—
318#1803—531no—
319#1802—531no—
320#1770—531no—
321#1762—531no—
322#1762—531no—
323C2×C53#235211yescharacteristic
324C2×C53C2×#44011yescharacteristic
325C2×C53#295211yescharacteristic
326C2×C53#332411yescharacteristic
327C2×C53#333111yescharacteristic
328C2×C53#337211yescharacteristic
329C2×C53#337311yescharacteristic
330D53—44no—
331C22×SD16—531no—
332C2×C4×Q8—531no—
333C2×C4×D4—531no—
334C2×C4×D4—531no—
335C2×C4×D4—531no—
336C22×C4⋊C4—1061no—
337C4×SD16—538no—
338C2×#106—531no—
339C2×C4⋊C8—531no—
340C2×#103—531no—
341C2×#103C2×D5311yescharacteristic
342C2×#103C2×D5311yescharacteristic
343C2×#102—531no—
344C2×#102—531no—
345C2×#102—531no—
346C2×#97—531no—
347C2×#97C2×D5311yescharacteristic
348C2×C4×C8—531no—
349#391—531no—
350#391C2×D5311yescharacteristic
351#389—531no—
352#389C2×D5311yescharacteristic
353#387—1061no—
354#387—1061no—
355#385—1061no—
356#385—1061no—
357#384—531no—
358#384—531no—
359#382—1061no—
360#380—1061no—
361C4×C4⋊C4—531no—
362C4×C4⋊C4—531no—
363C4×C4⋊C4—531no—
364C4×C4⋊C4—1061no—
365#340—531no—
366#340—531no—
367#340—531no—
368#340—531no—
369#340—531no—
370#340—531no—
371#340C2×D5311yescharacteristic
372#340C2×D5311yescharacteristic
373#336—531no—
374#336—531no—
375C53#2276211yescharacteristic, complemented, Sylow
376C22×Q8C22×D5311yescharacteristic
377C22×D4—531no—
378C23×C4—1061no—
379C23×C4—1061no—
380C23×C4—1061no—
381C2×SD16—534no—
382C2×SD16—1062no—
383C2×SD16—1062no—
384C22×C8—531no—
385C22×C8—531no—
386C22×C8C22×D5311yescharacteristic
387C22×C8C22×D5311yescharacteristic
388C4×Q8—534no—
389C4×Q8—1062no—
390C4×D4—534no—
391C4×D4—1062no—
392C4×D4—1062no—
393C4×D4—1062no—
394C4×D4—2121no—
395C4×D4—2121no—
396C2×C4⋊C4—21no—
397C2×C4⋊C4—531no—
398C2×C4⋊C4—531no—
399C2×C4⋊C4—531no—
400C2×C4⋊C4—531no—
401C2×C4⋊C4—531no—
402C2×C4⋊C4—1061no—
403C2×C4⋊C4—1061no—
404C2×C4⋊C4—1061no—
405C2×C4⋊C4—1061no—
406C2×C4⋊C4—1061no—
407C2×C4⋊C4—1061no—
408C2×C4⋊C4—2124no—
409C2×C4⋊C4C22×D5311yescharacteristic
410C2×C4⋊C4C22×D5311yescharacteristic
411C2×C4⋊C4C22×D5311yescharacteristic
412C2×C22⋊C4—1061no—
413C2×C22⋊C4—1061no—
414C2×C22⋊C4—1061no—
415C2×C42—531no—
416C2×C42—531no—
417C2×C42—1061no—
418C2×C42—1061no—
419C2×C42—1061no—
420C2×C42C22×D5311yescharacteristic
421#106—534no—
422C4⋊C8—1062no—
423#103—41no—
424#103—2121no—
425#103C4×D5314yescomplemented
426#102—534no—
427#102—1062no—
428#102—1062no—
429#97—41no—
430#97—2121no—
431C4×C8—534no—
432#95—21no—
433#95—21no—
434#95—21no—
435#95—531no—
436#95—531no—
437#95—1061no—
438#95—1061no—
439#95—1061no—
440#95—1061no—
441#95—1061no—
442C24—1061no—
443C2×Q8—21no—
444C2×Q8—21no—
445C2×Q8—21no—
446C2×Q8—21no—
447C2×Q8—21no—
448C2×Q8C2×C4×D5312yes—
449C2×D4—532no—
450C2×D4—532no—
451C2×D4—532no—
452C2×D4—1061no—
453C2×D4—1061no—
454C2×D4—1061no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—531no—
460C22×C4—531no—
461C22×C4—531no—
462C22×C4—531no—
463C22×C4—1064no—
464C22×C4—1061no—
465C22×C4—1061no—
466C22×C4—1061no—
467C22×C4—4241no—
468C22×C4—4241no—
469C22×C4—2122no—
470C22×C4—2122no—
471C22×C4—2122no—
472C22×C4—2122no—
473C22×C4—2122no—
474C22×C4—2122no—
475C22×C4D4×D5311yescharacteristic
476C22×C4D4×D5311yescharacteristic
477C22×C4C23×D5311yescharacteristic
478SD16—1068no—
479C2×C8—21no—
480C2×C8—21no—
481C2×C8—21no—
482C2×C8—21no—
483C2×C8—21no—
484C2×C8—532no—
485C2×C8—1061no—
486C2×C8—1061no—
487C2×C8—1061no—
488C2×C8—1061no—
489C2×C8—1061no—
490C2×C8C2×C4×D5312yes—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—42no—
497C4⋊C4—532no—
498C4⋊C4—1062no—
499C4⋊C4—1062no—
500C4⋊C4—1064no—
501C4⋊C4—1061no—
502C4⋊C4—1061no—
503C4⋊C4—1061no—
504C4⋊C4—1061no—
505C4⋊C4—1061no—
506C4⋊C4—4241no—
507C4⋊C4—4241no—
508C4⋊C4—2122no—
509C4⋊C4—2122no—
510C4⋊C4—2122no—
511C4⋊C4C2×C4×D5312yes—
512C22⋊C4—1064no—
513C22⋊C4—2122no—
514C22⋊C4—2122no—
515C42—21no—
516C42—21no—
517C42—532no—
518C42—532no—
519C42—1064no—
520C42—1061no—
521C42—1061no—
522C42—4241no—
523C42—4241no—
524C23—1062no—
525C23—1062no—
526C23—1062no—
527C23—2121no—
528C23—2121no—
529C23—2121no—
530C23—2121no—
531C23C2×D4×D5311yescharacteristic
532Q8—22no—
533Q8—22no—
534Q8—41no—
535Q8—41no—
536D4—534no—
537D4—1062no—
538D4—1062no—
539D4—1062no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
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—532no—
559C2×C4—532no—
560C2×C4—532no—
561C2×C4—532no—
562C2×C4—1061no—
563C2×C4—1061no—
564C2×C4—1062no—
565C2×C4—1061no—
566C2×C4—1061no—
567C2×C4—1061no—
568C2×C4—1061no—
569C2×C4—1061no—
570C2×C4—1061no—
571C2×C4—4244no—
572C2×C4—4244no—
573C2×C4—2128no—
574C2×C4—2121no—
575C2×C4—2121no—
576C2×C4—2121no—
577C2×C4—2121no—
578C2×C4—2121no—
579C2×C4—2121no—
580C2×C4—2121no—
581C2×C4—2121no—
582C2×C4C22×C4×D5311yescharacteristic
583C2×C4#41×D5311yescharacteristic
584C2×C4#41×D5311yescharacteristic
585C2×C4#41×D5311yescharacteristic
586C2×C4#1313326211yescharacteristic
587C2×C4#1313326411yescharacteristic
588C8—22no—
589C8—41no—
590C8—1062no—
591C8—2121no—
592C22—1064no—
593C22—2122no—
594C22—2122no—
595C22—2122no—
596C22—2122no—
597C22—2122no—
598C22—2122no—
599C22C4×D4×D5311yescharacteristic
600C22#1681126211yescharacteristic
601C22#1681128111yescharacteristic
602C22AGL(1,ℤ/8)×D5311yescharacteristic
603C22#137×D5311yescharacteristic
604C22#1681143211yescharacteristic
605C22#1681143211yescharacteristic
606C4—41no—
607C4—42no—
608C4—42no—
609C4—41no—
610C4—81no—
611C4—81no—
612C4—534no—
613C4—1062no—
614C4—4241no—
615C4—4241no—
616C4—2121no—
617C4—2122no—
618C4—2121no—
619C4#117×D5311yescharacteristic
620C4#1681110211yescharacteristic
621C4#1681130911yescharacteristic
622C4#1681131811yescharacteristic
623C2—2124no—
624C2#2027764311yescharacteristic
625C2#440×D5311yescharacteristic
626C2#2028359511yescharacteristic
627C2#2028694111yescharacteristic
628C2#2028702611yescharacteristic
629C2#2028749311yescharacteristic
630C2#2028750411yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#1935C2
socleC23×C53C22×D4
prime radicalC22×D212C23
cosocleC22×C4×C53C24
hypercenter#1935D53
hypocenterC53#22762
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1935

Complements

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

NH ≅ G/Ntype
2. #20277671C2semidirect
4. #20275407C2semidirect
5. #20275303C2semidirect
6. #20275297C2semidirect
9. #20274841C2semidirect
10. #20274718C2semidirect
12. #20274583C2semidirect
15. C53×#1935C2semidirect
22. #16810708C4semidirect
70. Q8×Dic53C2×C4semidirect
97. #13133148C2×C4semidirect
104. C8×Dic53C2×C4semidirect
121. #103×C53C2×C4semidirect
312. #1935D53semidirect
375. C53#22762semidirect
425. #103C4×D53semidirect

Maximal subgroups

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

subgroupindex
C2×#168107082
#202776712
#202776162
#202754072
#202753032
#202752972
#202750522
#202750462
#202748412
#202747182
#202747142
#202745832
#202745412
#202745392
C53×#19352
#2276253