Finite Group Databasebeta

← #27452059

Subgroups — #27452059

Order 13568.

772 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20277809C211yesmaximal, characteristic, complemented
2#20276216C211yesmaximal, characteristic, complemented
3#20275946C211yesmaximal, characteristic
4#20275861C211yesmaximal, characteristic, complemented
5#20275861C211yesmaximal, characteristic, complemented
6#20275643C211yesmaximal, characteristic
7#20275415C211yesmaximal, characteristic, complemented
8#20275339C211yesmaximal, characteristic, complemented
9#20275332C211yesmaximal, characteristic, complemented
10#20275332C211yesmaximal, characteristic, complemented
11#20275283C211yesmaximal, characteristic, complemented
12#20275281C211yesmaximal, characteristic, complemented
13#20274976C211yesmaximal, characteristic, complemented
14#20274540C211yesmaximal, characteristic, complemented
15C53×#2445C211yesmaximal, characteristic
16C2×#13133209C2211yescharacteristic
17C2×#13133150C2211yescharacteristic
18C2×#13133148C2211yescharacteristic
19C2×C4⋊#88322C2211yescharacteristic
20C2×#13133138C2211yescharacteristic
21C2×#13133138C2211yescharacteristic
22C2×#13133138C2211yescharacteristic
23C2×#13133138C2211yescharacteristic
24C2×#13133138C2211yescharacteristic
25C2×#13133138C2211yescharacteristic
26C2×#13133138C2211yescharacteristic
27C2×#13133138C2211yescharacteristic
28C2×#13133128C2211yescharacteristic
29#16810521C2211yescharacteristic
30#16810521C2211yescharacteristic
31#16810518—21no—
32#16810505—21no—
33#16810504—21no—
34C4×C4⋊Dic53C2211yescharacteristic
35C4×C4⋊Dic53C2211yescharacteristic
36C4×C4⋊Dic53C2211yescharacteristic
37#16810474—21no—
38#16810385C2211yescharacteristic
39#16810385C2211yescharacteristic
40#16810385C2211yescharacteristic
41#16810385C2211yescharacteristic
42#16810385C2211yescharacteristic
43#16810385C2211yescharacteristic
44#16810378C2211yescharacteristic
45#16810378C2211yescharacteristic
46#16810378C2211yescharacteristic
47#16810378C2211yescharacteristic
48C2×#127×C53C2211yescharacteristic
49C2×#123×C53—21no—
50C2×C4×D4×C53C2211yescharacteristic
51C53×#394—21no—
52C53×#394—21no—
53C53×#390C2211yescharacteristic
54C53×#390C2211yescharacteristic
55C53×#390C2211yescharacteristic
56C53×#389C2211yescharacteristic
57C53×#386—21no—
58C4×C4⋊C4×C53C2211yescharacteristic
59C22×#88322C2311yescharacteristic
60C22×#88322C2311yescharacteristic
61C22×#88322C2311yescharacteristic
62C22×#88322C2311yescharacteristic
63#13133209—22no—
64#13133209—22no—
65C2×C4⋊Dic53C2311yescharacteristic
66C2×C4⋊Dic53C2311yescharacteristic
67C2×C4⋊Dic53C2311yescharacteristic
68C2×C4⋊Dic53C2311yescharacteristic
69C2×#11719919—21no—
70C2×#11719919—21no—
71C2×#11719919—21no—
72C2×#11719919—21no—
73C2×C4×Dic53—21no—
74C2×C4×Dic53—21no—
75C2×C4×Dic53—21no—
76#13133150—22no—
77#13133148—41no—
78C4⋊#88322—41no—
79#13133138—22no—
80#13133138—22no—
81#13133138—22no—
82#13133138—22no—
83#13133138—22no—
84#13133138—22no—
85#13133138—22no—
86#13133138—41no—
87#13133128—41no—
88#13133124—21no—
89#13133124—21no—
90#13133124—21no—
91#13133124—21no—
92#13133124—21no—
93C22×D4×C53—21no—
94C22×D4×C53—21no—
95C22×D4×C53C2311yescharacteristic
96C22×D4×C53C2311yescharacteristic
97C23×C4×C53—21no—
98#127×C53—22no—
99#127×C53—41no—
100#123×C53—44no—
101C4×D4×C53—22no—
102C4×D4×C53—41no—
103C2×C4⋊C4×C53—21no—
104C2×C4⋊C4×C53D411yescharacteristic
105C2×C4⋊C4×C53D411yescharacteristic
106C2×C4⋊C4×C53C2311yescharacteristic
107C2×C4⋊C4×C53C2311yescharacteristic
108C2×C22⋊C4×C53—21no—
109C2×C22⋊C4×C53—21no—
110C2×C22⋊C4×C53—21no—
111C2×C22⋊C4×C53—21no—
112C2×C22⋊C4×C53—21no—
113C2×C22⋊C4×C53—21no—
114C2×C22⋊C4×C53—21no—
115C2×C42×C53C2311yescharacteristic
116C2×C42×C53C2311yescharacteristic
117C2×C42×C53C2311yescharacteristic
118#95×C53—21no—
119#95×C53D411yescharacteristic
120#95×C53D411yescharacteristic
121C24×C53—21no—
122C24×C53—21no—
123C2×D4×C53—21no—
124C2×D4×C53—21no—
125C2×D4×C53—21no—
126C2×D4×C53—21no—
127C2×D4×C53—21no—
128C2×D4×C53—21no—
129C2×D4×C53—21no—
130C2×D4×C53—21no—
131C2×D4×C53—21no—
132C2×D4×C53—21no—
133C2×D4×C53—21no—
134C2×D4×C53—42no—
135C2×D4×C53—42no—
136C2×D4×C53—42no—
137C2×D4×C53—42no—
138C2×D4×C53—81no—
139C2×D4×C53—81no—
140C2×D4×C53#4112yes—
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×C53C2×D411yescharacteristic
152C22×C4×C53C2411yescharacteristic
153C22×Dic53—21no—
154C22×Dic53—21no—
155C22×Dic53—21no—
156C22×Dic53—21no—
157C4⋊C4×C53—22no—
158C4⋊C4×C53—21no—
159C4⋊C4×C53—21no—
160C4⋊C4×C53—21no—
161C4⋊C4×C53—21no—
162C4⋊C4×C53—41no—
163C4⋊C4×C53—42no—
164C22⋊C4×C53—42no—
165C22⋊C4×C53—42no—
166C22⋊C4×C53—42no—
167C22⋊C4×C53—42no—
168C22⋊C4×C53—81no—
169C22⋊C4×C53—81no—
170C22⋊C4×C53—81no—
171C42×C53—21no—
172C42×C53—21no—
173C42×C53—21no—
174C42×C53—21no—
175C42×C53—21no—
176C42×C53—21no—
177C4⋊Dic53—21no—
178C4⋊Dic53—21no—
179C4⋊Dic53—21no—
180C4⋊Dic53—21no—
181C4⋊Dic53—21no—
182C4⋊Dic53#4112yes—
183C4⋊Dic53#4112yes—
184C4⋊Dic53#4112yes—
185#11719919—24no—
186#11719919—24no—
187#11719919—42no—
188#11719919—81no—
189C4×Dic53—42no—
190C4×Dic53—81no—
191C4×Dic53—81no—
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×#88322—21no—
203C2×#88322—21no—
204C23×C53—22no—
205C23×C53—41no—
206C23×C53—41no—
207C23×C53—41no—
208C23×C53—41no—
209C23×C53—41no—
210C23×C53—41no—
211C23×C53—41no—
212C23×C53—41no—
213C23×C53—41no—
214C23×C53—41no—
215C23×C53—41no—
216C23×C53—41no—
217C23×C53—41no—
218C23×C53C22×D411yescharacteristic
219D4×C53—22no—
220D4×C53—22no—
221D4×C53—41no—
222D4×C53—41no—
223D4×C53—41no—
224D4×C53—41no—
225D4×C53—41no—
226D4×C53—41no—
227D4×C53—84no—
228D4×C53—84no—
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—21no—
236C2×C4×C53—21no—
237C2×C4×C53—21no—
238C2×C4×C53—21no—
239C2×C4×C53—21no—
240C2×C4×C53—21no—
241C2×C4×C53—21no—
242C2×C4×C53—21no—
243C2×C4×C53—21no—
244C2×C4×C53—21no—
245C2×C4×C53—21no—
246C2×C4×C53—41no—
247C2×C4×C53—41no—
248C2×C4×C53—41no—
249C2×C4×C53—41no—
250C2×C4×C53—41no—
251C2×C4×C53—41no—
252C2×C4×C53—84no—
253C2×C4×C53AGL(1,ℤ/8)12yes—
254C2×C4×C53C2×#4111yescharacteristic
255C2×C4×C53C2×#4111yescharacteristic
256C2×C4×C53C2×#4111yescharacteristic
257C2×C4×C53C2×#4111yescharacteristic
258C2×C4×C53H(5,𝔽2)11yescharacteristic
259C2×C4×C532−1+411yescharacteristic
260C2×Dic53—22no—
261C2×Dic53—22no—
262C2×Dic53—22no—
263C2×Dic53—41no—
264C2×Dic53—41no—
265C2×Dic53—41no—
266C2×Dic53—41no—
267C2×Dic53—41no—
268C2×Dic53—41no—
269C2×Dic53—41no—
270C2×Dic53—41no—
271C2×Dic53—41no—
272#88322—41no—
273#88322—41no—
274#88322—41no—
275#88322—41no—
276#26716—531nomaximal, Sylow
277C22×C53—42no—
278C22×C53—42no—
279C22×C53—42no—
280C22×C53—81no—
281C22×C53—81no—
282C22×C53—81no—
283C22×C53—81no—
284C22×C53—81no—
285C22×C53—81no—
286C22×C53—81no—
287C22×C53—81no—
288C22×C53—81no—
289C22×C53—81no—
290C22×C53—81no—
291C22×C53#52511yescharacteristic
292C22×C53#54611yescharacteristic
293C22×C53#54811yescharacteristic
294C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
295C22×C53#57511yescharacteristic
296C22×C53#57611yescharacteristic
297C22×C53#57711yescharacteristic
298C4×C53—22no—
299C4×C53—41no—
300C4×C53—41no—
301C4×C53—41no—
302C4×C53—41no—
303C4×C53—41no—
304C4×C53—81no—
305C4×C53—81no—
306C4×C53#52911yescharacteristic
307C4×C53#54211yescharacteristic
308C4×C53#55211yescharacteristic
309C4×C53#55911yescharacteristic
310Dic53—42no—
311Dic53—42no—
312Dic53—42no—
313Dic53—81no—
314#2445D5311yescharacteristic
315#2370—531no—
316#2088—531no—
317#2041—531no—
318#2025—531no—
319#2025—531no—
320#1982—531no—
321#1936—531no—
322#1926—531no—
323#1925—531no—
324#1925—531no—
325#1918—531no—
326#1917—531no—
327#1864—531no—
328#1762—531no—
329C2×C53—82no—
330C2×C53—161no—
331C2×C53#245811yescharacteristic
332C2×C53#311511yescharacteristic
333C2×C53#311911yescharacteristic
334C2×C53#331011yescharacteristic
335C2×C53#332211yescharacteristic
336C2×C53#336611yescharacteristic
337C2×C53#337111yescharacteristic
338C2×#127C2×D5311yescharacteristic
339C2×#125—531no—
340C2×#123—21no—
341C2×C4×D4C2×D5311yescharacteristic
342C2×#107—531no—
343C2×#106—531no—
344C2×C4⋊C8—531no—
345C2×#102—531no—
346C2×#102—531no—
347C2×#102—531no—
348C2×#102—531no—
349C2×#102—531no—
350C2×#102—531no—
351C2×#102—531no—
352C2×#102—531no—
353C2×#97—531no—
354#394—21no—
355#394—21no—
356#390C2×D5311yescharacteristic
357#390C2×D5311yescharacteristic
358#390C2×D5311yescharacteristic
359#389—531no—
360#389—531no—
361#389—1061no—
362#389C2×D5311yescharacteristic
363#387—1061no—
364#387—1061no—
365#386—21no—
366C4×C4⋊C4—531no—
367C4×C4⋊C4—531no—
368C4×C4⋊C4—531no—
369C4×C4⋊C4—1061no—
370C4×C4⋊C4C2×D5311yescharacteristic
371#340—531no—
372#340—531no—
373#340—531no—
374#340—531no—
375#340—531no—
376#340—531no—
377#336—531no—
378#336—531no—
379#336—531no—
380#336—531no—
381C53#2671611yescharacteristic, complemented, Sylow
382C22×D4—21no—
383C22×D4—21no—
384C22×D4C22×D5311yescharacteristic
385C22×D4C22×D5311yescharacteristic
386C23×C4—21no—
387C22×C8—531no—
388C22×C8—531no—
389C22×C8—531no—
390C22×C8—531no—
391#127—22no—
392#127—41no—
393#125—1062no—
394#125—1062no—
395#123—44no—
396C4×D4—22no—
397C4×D4—41no—
398C2×C4⋊C4—21no—
399C2×C4⋊C4—531no—
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⋊C4—1061no—
407C2×C4⋊C4C22⋊D5311yescharacteristic
408C2×C4⋊C4C22⋊D5311yescharacteristic
409C2×C4⋊C4C22×D5311yescharacteristic
410C2×C4⋊C4C22×D5311yescharacteristic
411C2×C22⋊C4—21no—
412C2×C22⋊C4—21no—
413C2×C22⋊C4—21no—
414C2×C22⋊C4—21no—
415C2×C22⋊C4—21no—
416C2×C22⋊C4—21no—
417C2×C22⋊C4—21no—
418C2×C42—1061no—
419C2×C42—1061no—
420C2×C42—1061no—
421C2×C42C22×D5311yescharacteristic
422C2×C42C22×D5311yescharacteristic
423C2×C42C22×D5311yescharacteristic
424#107—1062no—
425#106—2121no—
426C4⋊C8—2121no—
427#102—1062no—
428#102—1062no—
429#102—1062no—
430#102—1062no—
431#102—1062no—
432#102—1062no—
433#102—1062no—
434#102—2121no—
435#97—2121no—
436#95—21no—
437#95—1061no—
438#95—1061no—
439#95—1061no—
440#95—1061no—
441#95—1061no—
442#95C22⋊D5311yescharacteristic
443#95C22⋊D5311yescharacteristic
444C24—21no—
445C24—21no—
446C2×D4—21no—
447C2×D4—21no—
448C2×D4—21no—
449C2×D4—21no—
450C2×D4—21no—
451C2×D4—21no—
452C2×D4—21no—
453C2×D4—21no—
454C2×D4—21no—
455C2×D4—21no—
456C2×D4—21no—
457C2×D4—42no—
458C2×D4—42no—
459C2×D4—42no—
460C2×D4—42no—
461C2×D4—81no—
462C2×D4—81no—
463C2×D4#1171994712yes—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—42no—
467C22×C4—42no—
468C22×C4—81no—
469C22×C4—1061no—
470C22×C4—1061no—
471C22×C4—1061no—
472C22×C4—1061no—
473C22×C4C2×C22⋊D5311yescharacteristic
474C22×C4C2×C22⋊D5311yescharacteristic
475C22×C4C2×C22⋊D5311yescharacteristic
476C22×C4C2×C22⋊D5311yescharacteristic
477C22×C4C2×C22⋊D5311yescharacteristic
478C22×C4C2×C22⋊D5311yescharacteristic
479C22×C4C23×D5311yescharacteristic
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—
491C2×C8—1061no—
492C4⋊C4—22no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—41no—
498C4⋊C4—42no—
499C4⋊C4—532no—
500C4⋊C4—532no—
501C4⋊C4—532no—
502C4⋊C4—1064no—
503C4⋊C4—1064no—
504C4⋊C4—1061no—
505C4⋊C4—1061no—
506C4⋊C4—1061no—
507C4⋊C4—1061no—
508C4⋊C4—1061no—
509C4⋊C4—4241no—
510C4⋊C4—2122no—
511C22⋊C4—42no—
512C22⋊C4—42no—
513C22⋊C4—42no—
514C22⋊C4—42no—
515C22⋊C4—81no—
516C22⋊C4—81no—
517C22⋊C4—81no—
518C42—21no—
519C42—21no—
520C42—21no—
521C42—21no—
522C42—21no—
523C42—21no—
524C42—4241no—
525C42—4241no—
526C42—2122no—
527C23—22no—
528C23—41no—
529C23—41no—
530C23—41no—
531C23—41no—
532C23—41no—
533C23—41no—
534C23—41no—
535C23—41no—
536C23—41no—
537C23—41no—
538C23—41no—
539C23—41no—
540C23—41no—
541C23C22×C22⋊D5311yescharacteristic
542D4—22no—
543D4—22no—
544D4—41no—
545D4—41no—
546D4—41no—
547D4—41no—
548D4—41no—
549D4—41no—
550D4—84no—
551D4—84no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—84no—
576C2×C4—1062no—
577C2×C4—1062no—
578C2×C4—1062no—
579C2×C4—2121no—
580C2×C4—2121no—
581C2×C4—2121no—
582C2×C4—2121no—
583C2×C4—2121no—
584C2×C4—2121no—
585C2×C4—2121no—
586C2×C4—2121no—
587C2×C4—2121no—
588C2×C4#1313324312yes—
589C2×C4#41×D5311yescharacteristic
590C2×C4#41×D5311yescharacteristic
591C2×C4#41×D5311yescharacteristic
592C2×C4C2×#1171994711yescharacteristic
593C2×C4#1313326211yescharacteristic
594C2×C4#1313326411yescharacteristic
595C8—2121no—
596C8—2121no—
597C8—2121no—
598C8—2121no—
599C22—42no—
600C22—42no—
601C22—42no—
602C22—81no—
603C22—81no—
604C22—81no—
605C22—81no—
606C22—81no—
607C22—81no—
608C22—81no—
609C22—81no—
610C22—81no—
611C22—81no—
612C22—81no—
613C22#1681113011yescharacteristic
614C22#1681126711yescharacteristic
615C22#1681128011yescharacteristic
616C22C2×#1313324311yescharacteristic
617C22#1681142111yescharacteristic
618C22#1681142611yescharacteristic
619C22#1681143411yescharacteristic
620C4—22no—
621C4—41no—
622C4—41no—
623C4—41no—
624C4—41no—
625C4—41no—
626C4—81no—
627C4—81no—
628C4—4241no—
629C4—2122no—
630C4—2122no—
631C4—2122no—
632C4#1681115011yescharacteristic
633C4#1681124111yescharacteristic
634C4#1681131011yescharacteristic
635C4#1681135711yescharacteristic
636C2—82no—
637C2—161no—
638C2#2027863311yescharacteristic
639C2#2028486011yescharacteristic
640C2#2028490811yescharacteristic
641C2#2028673611yescharacteristic
642C2#2028691611yescharacteristic
643C2#2028741811yescharacteristic
644C2#2028747711yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2445C2
socleC23×C53C22×D4
prime radicalC2×#127×C53C22
hypercenter#2445D53
hypocenterC53#26716
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2445

Complements

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

NH ≅ G/Ntype
1. #20277809C2semidirect
2. #20276216C2semidirect
4. #20275861C2semidirect
5. #20275861C2semidirect
7. #20275415C2semidirect
8. #20275339C2semidirect
9. #20275332C2semidirect
10. #20275332C2semidirect
11. #20275283C2semidirect
12. #20275281C2semidirect
13. #20274976C2semidirect
14. #20274540C2semidirect
381. C53#26716semidirect

Maximal subgroups

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

subgroupindex
#202778092
#202762162
#202759462
#202758612
#202756432
#202754152
#202753392
#202753322
#202752832
#202752812
#202749762
#202745402
C53×#24452
#2671653