Finite Group Databasebeta

← #27452192

Subgroups — #27452192

Order 13568.

748 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
1#20279056C211yesmaximal, characteristic
2#20276037C211yesmaximal, characteristic, complemented
3#20275908C211yesmaximal, characteristic
4#20275866C211yesmaximal, characteristic, complemented
5#20275723C211yesmaximal, characteristic
6#20275672C211yesmaximal, characteristic, complemented
7#20275390C211yesmaximal, characteristic, complemented
8#20275349C211yesmaximal, characteristic, complemented
9#20275323C211yesmaximal, characteristic
10#20275297C211yesmaximal, characteristic, complemented
11#20275056C211yesmaximal, characteristic, complemented
12#20275022C211yesmaximal, characteristic
13#20274801C211yesmaximal, characteristic
14C4×#13133141C211yesmaximal, characteristic, complemented
15C53×#2360C211yesmaximal, characteristic
16C22×#11719925C2211yescharacteristic
17C2×#13133209C2211yescharacteristic
18C2×#13133196C2211yescharacteristic
19C2×#13133148C2211yescharacteristic
20C2×C4⋊#88322C2211yescharacteristic
21C2×#13133141C2211yescharacteristic
22C2×#13133141C2211yescharacteristic
23C2×#13133141C2211yescharacteristic
24C2×#13133140C2211yescharacteristic
25C2×#13133140C2211yescharacteristic
26C2×#13133140C2211yescharacteristic
27C2×C22⋊#88322C2211yescharacteristic
28C2×C22⋊#88322C2211yescharacteristic
29C2×C4×#88322C2211yescharacteristic
30#16810564—21no—
31#16810551—21no—
32#16810529C2211yescharacteristic
33#16810529C2211yescharacteristic
34#16810511—21no—
35#16810487—21no—
36#16810483C2211yescharacteristic
37C4×C4⋊Dic53C2211yescharacteristic
38#16810385C2211yescharacteristic
39#16810385C2211yescharacteristic
40#16810385C2211yescharacteristic
41#16810385C2211yescharacteristic
42#16810385C2211yescharacteristic
43#16810385C2211yescharacteristic
44#16810385C2211yescharacteristic
45#16810385C2211yescharacteristic
46#16810378C2211yescharacteristic
47#16810378C2211yescharacteristic
48C2×C22⋊Q8×C53C2211yescharacteristic
49C2×C4×Q8×C53C2211yescharacteristic
50C22×C4⋊C4×C53C2211yescharacteristic
51C53×#391C2211yescharacteristic
52C53×#389—21no—
53C53×#389C2211yescharacteristic
54C53×#387—21no—
55C53×#385—21no—
56C4×C4⋊C4×C53C2211yescharacteristic
57C4×C22⋊C4×C53—21no—
58C4×C22⋊C4×C53C2211yescharacteristic
59C22×Dic106C2311yescharacteristic
60C2×#11719925—22no—
61C2×#11719925—41no—
62C2×#11719925—41no—
63C22×#88322C2311yescharacteristic
64C22×#88322C2311yescharacteristic
65C22×#88322C2311yescharacteristic
66C22×#88322C2311yescharacteristic
67#13133209—22no—
68#13133209—41no—
69#13133196—22no—
70#13133196—41no—
71C2×C4⋊Dic53C2311yescharacteristic
72C2×C4⋊Dic53C2311yescharacteristic
73C2×C4⋊Dic53C2311yescharacteristic
74C2×#11719919—21no—
75C2×#11719919—21no—
76C2×#11719919—21no—
77C2×C22⋊Dic53—21no—
78C2×C22⋊Dic53—21no—
79C2×C4×Dic53—21no—
80#13133148—22no—
81C4⋊#88322—41no—
82#13133141—22no—
83#13133141—22no—
84#13133141—41no—
85#13133140—22no—
86#13133140—22no—
87#13133140—41no—
88C22⋊#88322—41no—
89C22⋊#88322—41no—
90C4×#88322—41no—
91#13133124—21no—
92#13133124—21no—
93#13133124—21no—
94#13133124—21no—
95#13133124—21no—
96#13133124—21no—
97C22×Q8×C53C2311yescharacteristic
98C23×C4×C53—21no—
99C23×C4×C53C2311yescharacteristic
100C22⋊Q8×C53—22no—
101C22⋊Q8×C53—22no—
102C4×Q8×C53—22no—
103C4×Q8×C53—41no—
104C2×C4⋊C4×C53—21no—
105C2×C4⋊C4×C53—21no—
106C2×C4⋊C4×C53—21no—
107C2×C4⋊C4×C53—21no—
108C2×C4⋊C4×C53—22no—
109C2×C4⋊C4×C53—41no—
110C2×C4⋊C4×C53C2311yescharacteristic
111C2×C4⋊C4×C53C2311yescharacteristic
112C2×C4⋊C4×C53C2311yescharacteristic
113C2×C22⋊C4×C53—21no—
114C2×C22⋊C4×C53D411yescharacteristic
115C2×C22⋊C4×C53D411yescharacteristic
116C2×C42×C53—21no—
117C2×C42×C53—21no—
118C2×C42×C53C2311yescharacteristic
119C2×C42×C53C2311yescharacteristic
120#95×C53—21no—
121#95×C53—21no—
122#95×C53—21no—
123#95×C53—21no—
124#95×C53D411yescharacteristic
125#95×C53D411yescharacteristic
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—21no—
134C22×C4×C53—21no—
135C22×C4×C53—21no—
136C22×C4×C53—21no—
137C22×C4×C53—21no—
138C22×C4×C53—21no—
139C22×C4×C53—21no—
140C22×C4×C53—21no—
141C22×C4×C53—21no—
142C22×C4×C53—21no—
143C22×C4×C53—42no—
144C22×C4×C53—42no—
145C22×C4×C53—81no—
146C22×C4×C53C2×D411yescharacteristic
147C22×C4×C53C2×D411yescharacteristic
148C22×C4×C53C2×D411yescharacteristic
149C22×C4×C53C2×D411yescharacteristic
150C22×C4×C53C2×D411yescharacteristic
151C22×C4×C53C2411yescharacteristic
152C22×Dic53—21no—
153C22×Dic53—21no—
154C22×Dic53—21no—
155C22×Dic53—21no—
156C2×Dic106—21no—
157C2×Dic106—21no—
158C2×Dic106—21no—
159C2×Dic106—21no—
160C2×Dic106—21no—
161C2×Dic106—21no—
162C4⋊C4×C53—24no—
163C4⋊C4×C53—21no—
164C4⋊C4×C53—21no—
165C4⋊C4×C53—21no—
166C4⋊C4×C53—21no—
167C4⋊C4×C53—42no—
168C4⋊C4×C53—42no—
169C4⋊C4×C53—81no—
170C4⋊C4×C53#4112yes—
171C4⋊C4×C53#4112yes—
172C22⋊C4×C53—24no—
173C22⋊C4×C53—22no—
174C22⋊C4×C53—22no—
175C42×C53—21no—
176C42×C53—21no—
177C42×C53—21no—
178C42×C53—21no—
179C42×C53—42no—
180C42×C53—81no—
181C22⋊Dic53—42no—
182C22⋊Dic53—42no—
183#11719925—44no—
184C4⋊Dic53—21no—
185C4⋊Dic53—21no—
186C4⋊Dic53—21no—
187C4⋊Dic53—21no—
188C4⋊Dic53—21no—
189C4⋊Dic53#4112yes—
190#11719919—42no—
191#11719919—42no—
192#11719919—42no—
193C4×Dic53—42no—
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×#88322—21no—
204C2×#88322—21no—
205C2×#88322—21no—
206C23×C53—21no—
207C23×C53—21no—
208C23×C53—21no—
209C23×C53—21no—
210C23×C53—21no—
211C23×C53—21no—
212C23×C53#13512yes—
213C23×C53C22×D411yescharacteristic
214Q8×C53—22no—
215Q8×C53—22no—
216Q8×C53—41no—
217Q8×C53—41no—
218C2×C4×C53—21no—
219C2×C4×C53—21no—
220C2×C4×C53—21no—
221C2×C4×C53—22no—
222C2×C4×C53—22no—
223C2×C4×C53—21no—
224C2×C4×C53—21no—
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—21no—
232C2×C4×C53—21no—
233C2×C4×C53—21no—
234C2×C4×C53—21no—
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—41no—
248C2×C4×C53—84no—
249C2×C4×C53#13712yes—
250C2×C4×C53C2×#4111yescharacteristic
251C2×C4×C53C2×#4111yescharacteristic
252C2×C4×C53C2×#4111yescharacteristic
253C2×C4×C53C2×#4111yescharacteristic
254C2×C4×C53H(5,𝔽2)11yescharacteristic
255C2×C4×C53H(5,𝔽2)11yescharacteristic
256C2×Dic53—22no—
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—
267C2×Dic53—41no—
268Dic106—41no—
269Dic106—41no—
270Dic106—41no—
271Dic106—41no—
272#88322—41no—
273#88322—41no—
274#88322—41no—
275#88322—41no—
276#26725—531nomaximal, Sylow
277C22×C53—22no—
278C22×C53—22no—
279C22×C53—22no—
280C22×C53—41no—
281C22×C53—41no—
282C22×C53—41no—
283C22×C53—41no—
284C22×C53#52511yescharacteristic
285C22×C53#54611yescharacteristic
286C22×C53#54611yescharacteristic
287C22×C53C2×#13511yescharacteristic
288C22×C53C2×#13711yescharacteristic
289C22×C53#57711yescharacteristic
290C22×C53#57811yescharacteristic
291C4×C53—22no—
292C4×C53—41no—
293C4×C53—42no—
294C4×C53—41no—
295C4×C53—42no—
296C4×C53—41no—
297C4×C53—42no—
298C4×C53—41no—
299C4×C53—81no—
300C4×C53#52911yescharacteristic
301C4×C53#53811yescharacteristic
302C4×C53#55511yescharacteristic
303C4×C53#55911yescharacteristic
304Dic53—42no—
305Dic53—81no—
306Dic53—81no—
307Dic53—81no—
308#2489—531no—
309#2360D5311yescharacteristic
310#2056—531no—
311#2035—531no—
312#2026—531no—
313#2001—531no—
314#1989—531no—
315#1933—531no—
316#1927—531no—
317#1923—531no—
318#1920—531no—
319#1878—531no—
320#1872—531no—
321#1823—531no—
322C4×#103—531no—
323C2×C53—42no—
324C2×C53#245911yescharacteristic
325C2×C53#310811yescharacteristic
326C2×C53#312011yescharacteristic
327C2×C53#329611yescharacteristic
328C2×C53#330011yescharacteristic
329C2×C53#331211yescharacteristic
330C2×C53#333111yescharacteristic
331C22×Q16—531no—
332C2×#125—531no—
333C2×C22⋊Q8—531no—
334C2×C22⋊Q8C2×D5311yescharacteristic
335C2×C4×Q8C2×D5311yescharacteristic
336C22×C4⋊C4C2×D5311yescharacteristic
337C2×#106—531no—
338C2×C4⋊C8—531no—
339C2×#103—531no—
340C2×#103—531no—
341C2×#103—531no—
342C2×#103—531no—
343C2×#103—531no—
344C2×#103—531no—
345C2×C22⋊C8—531no—
346C2×C22⋊C8—531no—
347C2×C4×C8—531no—
348#400—1061no—
349#397—1061no—
350#391—531no—
351#391—531no—
352#391C2×D5311yescharacteristic
353#389—21no—
354#389C2×D5311yescharacteristic
355#388—1061no—
356#387—21no—
357#385—21no—
358#382—1061no—
359#380—531no—
360C4×C4⋊C4—531no—
361C4×C4⋊C4C2×D5311yescharacteristic
362C4×C22⋊C4—21no—
363C4×C22⋊C4C2×D5311yescharacteristic
364#340—531no—
365#340—531no—
366#340—531no—
367#340—531no—
368#340—531no—
369#340—531no—
370#340—531no—
371#340—531no—
372#336—531no—
373#336—531no—
374C53#2672511yescharacteristic, complemented, Sylow
375C22×Q8—531no—
376C22×Q8C22×D5311yescharacteristic
377C23×C4—21no—
378C23×C4C22×D5311yescharacteristic
379C2×Q16—1062no—
380C2×Q16—2121no—
381C2×Q16—2121no—
382C22×C8—531no—
383C22×C8—531no—
384C22×C8—531no—
385C22×C8—531no—
386#125—1062no—
387#125—2121no—
388C22⋊Q8—22no—
389C22⋊Q8—22no—
390C22⋊Q8—1062no—
391C22⋊Q8—2121no—
392C4×Q8—22no—
393C4×Q8—41no—
394C2×C4⋊C4—21no—
395C2×C4⋊C4—22no—
396C2×C4⋊C4—21no—
397C2×C4⋊C4—21no—
398C2×C4⋊C4—21no—
399C2×C4⋊C4—41no—
400C2×C4⋊C4—531no—
401C2×C4⋊C4—531no—
402C2×C4⋊C4—531no—
403C2×C4⋊C4—1061no—
404C2×C4⋊C4—1061no—
405C2×C4⋊C4—1061no—
406C2×C4⋊C4C22×D5311yescharacteristic
407C2×C4⋊C4C22×D5311yescharacteristic
408C2×C4⋊C4C22×D5311yescharacteristic
409C2×C22⋊C4—21no—
410C2×C22⋊C4—1061no—
411C2×C22⋊C4—1061no—
412C2×C22⋊C4C22⋊D5311yescharacteristic
413C2×C22⋊C4C22⋊D5311yescharacteristic
414C2×C42—21no—
415C2×C42—21no—
416C2×C42—1061no—
417C2×C42C22×D5311yescharacteristic
418C2×C42C22×D5311yescharacteristic
419#106—1062no—
420C4⋊C8—2121no—
421#103—1062no—
422#103—1062no—
423#103—1062no—
424#103—1062no—
425#103—2121no—
426#103—2121no—
427C22⋊C8—2121no—
428C22⋊C8—2121no—
429C4×C8—2121no—
430#95—21no—
431#95—21no—
432#95—21no—
433#95—21no—
434#95—1061no—
435#95—1061no—
436#95—1061no—
437#95—1061no—
438#95—1061no—
439#95—1061no—
440#95C22⋊D5311yescharacteristic
441#95C22⋊D5311yescharacteristic
442C24C2×C22⋊D5311yescharacteristic
443C2×Q8—21no—
444C2×Q8—21no—
445C2×Q8—21no—
446C2×Q8—21no—
447C2×Q8—21no—
448C2×Q8—1061no—
449C2×Q8—1061no—
450C2×Q8—1061no—
451C2×Q8—1061no—
452C2×Q8—1061no—
453C2×Q8—1061no—
454C2×Q8#1171994512yes—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—42no—
466C22×C4—42no—
467C22×C4—81no—
468C22×C4—1061no—
469C22×C4—1061no—
470C22×C4—1061no—
471C22×C4—1061no—
472C22×C4C2×C22⋊D5311yescharacteristic
473C22×C4C2×C22⋊D5311yescharacteristic
474C22×C4C2×C22⋊D5311yescharacteristic
475C22×C4C2×C22⋊D5311yescharacteristic
476C22×C4C2×C22⋊D5311yescharacteristic
477C22×C4C23×D5311yescharacteristic
478Q16—2124no—
479C2×C8—1061no—
480C2×C8—1061no—
481C2×C8—1061no—
482C2×C8—1061no—
483C2×C8—1061no—
484C2×C8—1061no—
485C2×C8—1061no—
486C2×C8—1061no—
487C2×C8—1061no—
488C2×C8—1061no—
489C2×C8—1061no—
490C2×C8—1061no—
491C4⋊C4—24no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—42no—
497C4⋊C4—42no—
498C4⋊C4—81no—
499C4⋊C4—532no—
500C4⋊C4—1061no—
501C4⋊C4—1061no—
502C4⋊C4—1061no—
503C4⋊C4—1061no—
504C4⋊C4—1061no—
505C4⋊C4—2122no—
506C4⋊C4—2122no—
507C4⋊C4—2122no—
508C4⋊C4#1171994512yes—
509C4⋊C4#1171994712yes—
510C22⋊C4—24no—
511C22⋊C4—22no—
512C22⋊C4—22no—
513C22⋊C4—2122no—
514C22⋊C4—2122no—
515C42—21no—
516C42—21no—
517C42—21no—
518C42—21no—
519C42—42no—
520C42—81no—
521C42—2122no—
522C23—21no—
523C23—21no—
524C23—21no—
525C23—21no—
526C23—21no—
527C23—21no—
528C23#1313323712yes—
529C23C22×C22⋊D5311yescharacteristic
530Q8—22no—
531Q8—22no—
532Q8—41no—
533Q8—41no—
534Q8—2121no—
535Q8—2121no—
536Q8—2121no—
537Q8—2121no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—22no—
542C2×C4—21no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—84no—
569C2×C4—1062no—
570C2×C4—2121no—
571C2×C4—2121no—
572C2×C4—2121no—
573C2×C4—2121no—
574C2×C4—2121no—
575C2×C4—2121no—
576C2×C4—2121no—
577C2×C4—2121no—
578C2×C4—2121no—
579C2×C4—2121no—
580C2×C4—2121no—
581C2×C4#1313324812yes—
582C2×C4#41×D5311yescharacteristic
583C2×C4C2×#1171994711yescharacteristic
584C2×C4C2×#1171994511yescharacteristic
585C2×C4C2×#1171994511yescharacteristic
586C2×C4#1313326211yescharacteristic
587C2×C4#1313326311yescharacteristic
588C8—2121no—
589C8—2121no—
590C8—2121no—
591C8—2121no—
592C22—22no—
593C22—22no—
594C22—22no—
595C22—41no—
596C22—41no—
597C22—41no—
598C22—41no—
599C22#1681113411yescharacteristic
600C22#1681126111yescharacteristic
601C22#1681126711yescharacteristic
602C22C2×#1313323711yescharacteristic
603C22C2×#1313324811yescharacteristic
604C22#1681143411yescharacteristic
605C22#1681143811yescharacteristic
606C4—22no—
607C4—42no—
608C4—41no—
609C4—41no—
610C4—42no—
611C4—41no—
612C4—42no—
613C4—41no—
614C4—81no—
615C4—4241no—
616C4—4241no—
617C4—4241no—
618C4—2122no—
619C4#1681115311yescharacteristic
620C4#1681120611yescharacteristic
621C4#1681133811yescharacteristic
622C4#1681135611yescharacteristic
623C2—42no—
624C2#2027864611yescharacteristic
625C2#2028479511yescharacteristic
626C2#2028492911yescharacteristic
627C2#2028655011yescharacteristic
628C2#2028659611yescharacteristic
629C2#2028676611yescharacteristic
630C2#2028702511yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2360C2
socleC23×C53C22×D4
prime radicalC24×C53C2×D4
hypercenter#2360D53
hypocenterC53#26725
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2360

Complements

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

NH ≅ G/Ntype
2. #20276037C2semidirect
4. #20275866C2semidirect
6. #20275672C2semidirect
7. #20275390C2semidirect
8. #20275349C2semidirect
10. #20275297C2semidirect
11. #20275056C2semidirect
14. C4×#13133141C2semidirect
374. C53#26725semidirect

Maximal subgroups

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

subgroupindex
#202790562
#202760372
#202759082
#202758662
#202757232
#202756722
#202753902
#202753492
#202753232
#202752972
#202750562
#202750222
#202748012
C4×#131331412
C53×#23602
#2672553