Finite Group Databasebeta

← #27457330

Subgroups — #27457330

Order 13568.

734 conjugacy classes of subgroups.

The subgroup lattice has 630 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20277771C211yesmaximal, characteristic, complemented
2#20276266C211yesmaximal, characteristic, complemented
3#20276032C211yesmaximal, characteristic
4#20275961C211yesmaximal, characteristic, complemented
5#20275797C211yesmaximal, characteristic
6#20275568C211yesmaximal, characteristic
7#20275356C211yesmaximal, characteristic, complemented
8#20275332C211yesmaximal, characteristic
9#20275297C211yesmaximal, characteristic, complemented
10#20275274C211yesmaximal, characteristic, complemented
11#20275073C211yesmaximal, characteristic
12#20275073C211yesmaximal, characteristic
13#20274727C211yesmaximal, characteristic, complemented
14C4×#13133140C211yesmaximal, characteristic, complemented
15C53×#2513C211yesmaximal, characteristic
16C22×#11719923C2211yescharacteristic
17C22×#11719923C2211yescharacteristic
18C2×#13133217C2211yescharacteristic
19C2×C4×Dic106C2211yescharacteristic
20C2×C4×Dic106C2211yescharacteristic
21C2×#13133150C2211yescharacteristic
22C2×#13133148C2211yescharacteristic
23C2×C4⋊#88322C2211yescharacteristic
24C2×#13133140C2211yescharacteristic
25C2×#13133140C2211yescharacteristic
26C2×#13133140C2211yescharacteristic
27C2×#13133140C2211yescharacteristic
28C2×#13133140C2211yescharacteristic
29C2×#13133140C2211yescharacteristic
30C2×#13133138C2211yescharacteristic
31C2×#13133138C2211yescharacteristic
32C2×C4×#88322C2211yescharacteristic
33#16810529C2211yescharacteristic
34#16810529C2211yescharacteristic
35#16810519—21no—
36#16810506—21no—
37#16810506—21no—
38C4⋊C4×Dic53—21no—
39C4×C4⋊Dic53C2211yescharacteristic
40#16810385C2211yescharacteristic
41#16810385C2211yescharacteristic
42#16810385C2211yescharacteristic
43#16810385C2211yescharacteristic
44#16810378C2211yescharacteristic
45#16810378C2211yescharacteristic
46#16810378C2211yescharacteristic
47#16810378C2211yescharacteristic
48C53×#397—21no—
49C53×#397—21no—
50C53×#390C2211yescharacteristic
51C53×#390C2211yescharacteristic
52C53×#390C2211yescharacteristic
53C53×#389C2211yescharacteristic
54C53×#388—21no—
55C53×#388—21no—
56C53×#382C2211yescharacteristic
57C4×C4⋊C4×C53C2211yescharacteristic
58C4×C4⋊C4×C53C2211yescharacteristic
59C22×Dic106C2311yescharacteristic
60C22×Dic106C2311yescharacteristic
61C2×#11719923—22no—
62C2×#11719923—22no—
63C2×#11719923—22no—
64C2×#11719923—41no—
65C2×#11719923—41no—
66C2×#11719923—41no—
67C22×#88322C2311yescharacteristic
68C22×#88322C2311yescharacteristic
69C22×#88322C2311yescharacteristic
70C22×#88322C2311yescharacteristic
71#13133217—22no—
72#13133217—22no—
73C4×Dic106—22no—
74C4×Dic106—22no—
75C4×Dic106—41no—
76C4×Dic106—41no—
77C2×C4⋊Dic53—21no—
78C2×C4⋊Dic53—21no—
79C2×C4⋊Dic53C2311yescharacteristic
80C2×C4⋊Dic53C2311yescharacteristic
81C2×#11719919—21no—
82C2×#11719919—21no—
83C2×C4×Dic53—21no—
84C2×C4×Dic53—21no—
85C2×C4×Dic53—21no—
86#13133150—41no—
87#13133148—22no—
88C4⋊#88322—41no—
89#13133140—22no—
90#13133140—22no—
91#13133140—22no—
92#13133140—41no—
93#13133140—41no—
94#13133140—41no—
95#13133138—22no—
96#13133138—41no—
97C4×#88322—41no—
98#13133124—21no—
99#13133124—21no—
100#13133124—21no—
101#13133124—21no—
102#13133124—21no—
103C22×D4×C53C2311yescharacteristic
104C2×C4⋊C4×C53—21no—
105C2×C4⋊C4×C53D411yescharacteristic
106C2×C4⋊C4×C53D411yescharacteristic
107C2×C4⋊C4×C53C2311yescharacteristic
108C2×C4⋊C4×C53C2311yescharacteristic
109C2×C4⋊C4×C53C2311yescharacteristic
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×C22⋊C4×C53—21no—
116C2×C42×C53—21no—
117C2×C42×C53C2311yescharacteristic
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×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—
133C22×C4×C53—21no—
134C22×C4×C53—21no—
135C22×C4×C53—21no—
136C22×C4×C53C2×D411yescharacteristic
137C22×C4×C53C2×D411yescharacteristic
138C22×C4×C53C2×D411yescharacteristic
139C22×C4×C53C2×D411yescharacteristic
140C22×C4×C53C2×D411yescharacteristic
141C22×C4×C53C2×D411yescharacteristic
142C22×C4×C53C2411yescharacteristic
143C22×Dic53—21no—
144C22×Dic53—21no—
145C22×Dic53—21no—
146C22×Dic53—21no—
147C2×Dic106—21no—
148C2×Dic106—21no—
149C2×Dic106—21no—
150C2×Dic106—21no—
151C2×Dic106—21no—
152C2×Dic106—21no—
153C2×Dic106—21no—
154C2×Dic106—21no—
155C2×Dic106—21no—
156C2×Dic106—21no—
157C2×Dic106#4112yes—
158C2×Dic106#4112yes—
159C4⋊C4×C53—22no—
160C4⋊C4×C53—22no—
161C4⋊C4×C53—21no—
162C4⋊C4×C53—21no—
163C4⋊C4×C53—21no—
164C4⋊C4×C53—21no—
165C4⋊C4×C53—21no—
166C4⋊C4×C53—21no—
167C4⋊C4×C53—81no—
168C22⋊C4×C53—42no—
169C22⋊C4×C53—42no—
170C22⋊C4×C53—81no—
171C22⋊C4×C53—81no—
172C22⋊C4×C53—81no—
173C22⋊C4×C53—81no—
174C42×C53—21no—
175C42×C53—21no—
176C42×C53—21no—
177C42×C53—21no—
178C42×C53—21no—
179C42×C53—21no—
180C42×C53—81no—
181#11719923—44no—
182#11719923—44no—
183C4⋊Dic53—24no—
184C4⋊Dic53—24no—
185C4⋊Dic53—21no—
186C4⋊Dic53—21no—
187C4⋊Dic53—21no—
188C4⋊Dic53#4112yes—
189#11719919—42no—
190#11719919—42no—
191C4×Dic53—42no—
192C4×Dic53—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#4112yes—
206C23×C53—41no—
207C23×C53—41no—
208C23×C53—41no—
209C23×C53—41no—
210C23×C53—41no—
211C23×C53—41no—
212C23×C53—41no—
213C23×C53C22×D411yescharacteristic
214D4×C53—41no—
215D4×C53—41no—
216D4×C53—41no—
217D4×C53—41no—
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—21no—
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—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#13512yes—
244C2×C4×C53#13712yes—
245C2×C4×C53C2×#4111yescharacteristic
246C2×C4×C53C2×#4111yescharacteristic
247C2×C4×C53C2×#4111yescharacteristic
248C2×C4×C53C2×#4111yescharacteristic
249C2×C4×C53H(5,𝔽2)11yescharacteristic
250C2×C4×C53H(5,𝔽2)11yescharacteristic
251C2×Dic53—22no—
252C2×Dic53—22no—
253C2×Dic53—22no—
254C2×Dic53—41no—
255C2×Dic53—41no—
256C2×Dic53—41no—
257C2×Dic53—41no—
258C2×Dic53—41no—
259C2×Dic53—41no—
260C2×Dic53—41no—
261C2×Dic53—41no—
262C2×Dic53—41no—
263Dic106—22no—
264Dic106—22no—
265Dic106—22no—
266Dic106—22no—
267Dic106—41no—
268Dic106—41no—
269Dic106—41no—
270Dic106—41no—
271#88322—22no—
272#88322—41no—
273#88322—41no—
274#88322—41no—
275#27069—531nomaximal, Sylow
276C22×C53—81no—
277C22×C53—81no—
278C22×C53—81no—
279C22×C53—81no—
280C22×C53—81no—
281C22×C53—81no—
282C22×C53—81no—
283C22×C53#52511yescharacteristic
284C22×C53#54611yescharacteristic
285C22×C53#55011yescharacteristic
286C22×C53C2×#13511yescharacteristic
287C22×C53C2×#13711yescharacteristic
288C22×C53#57711yescharacteristic
289C22×C53#57711yescharacteristic
290C4×C53—22no—
291C4×C53—22no—
292C4×C53—41no—
293C4×C53—41no—
294C4×C53—41no—
295C4×C53—41no—
296C4×C53—81no—
297C4×C53—81no—
298C4×C53—81no—
299C4×C53#52911yescharacteristic
300C4×C53#53811yescharacteristic
301C4×C53#55111yescharacteristic
302C4×C53#55311yescharacteristic
303Dic53—42no—
304Dic53—42no—
305Dic53—42no—
306Dic53—81no—
307#2513D5311yescharacteristic
308#2365—531no—
309#2097—531no—
310#2055—531no—
311#2043—531no—
312#2013—531no—
313#1968—531no—
314#1928—531no—
315#1925—531no—
316#1920—531no—
317#1916—531no—
318#1880—531no—
319#1880—531no—
320#1805—531no—
321C4×#103—531no—
322C2×C53—161no—
323C2×C53#246711yescharacteristic
324C2×C53#310811yescharacteristic
325C2×C53#312611yescharacteristic
326C2×C53#330311yescharacteristic
327C2×C53#332011yescharacteristic
328C2×C53#339411yescharacteristic
329C2×C53#340011yescharacteristic
330C22×SD16—531no—
331C22×SD16—531no—
332C2×C4⋊Q8—531no—
333C2×C4×Q8—531no—
334C2×C4×Q8—531no—
335C2×#107—531no—
336C2×#106—531no—
337C2×C4⋊C8—531no—
338C2×#103—531no—
339C2×#103—531no—
340C2×#103—531no—
341C2×#103—531no—
342C2×#103—531no—
343C2×#103—531no—
344C2×#102—531no—
345C2×#102—531no—
346C2×C4×C8—531no—
347#397—21no—
348#397—21no—
349#391—531no—
350#391—531no—
351#390C2×D5311yescharacteristic
352#390C2×D5311yescharacteristic
353#390C2×D5311yescharacteristic
354#389—1061no—
355#389C2×D5311yescharacteristic
356#388—21no—
357#388—21no—
358#387—1061no—
359#387—1061no—
360#382C2×D5311yescharacteristic
361C4×C4⋊C4—531no—
362C4×C4⋊C4—1061no—
363C4×C4⋊C4C2×D5311yescharacteristic
364C4×C4⋊C4C2×D5311yescharacteristic
365#340—531no—
366#340—531no—
367#340—531no—
368#340—531no—
369#336—531no—
370#336—531no—
371#336—531no—
372#336—531no—
373C53#2706911yescharacteristic, complemented, Sylow
374C22×Q8—531no—
375C22×Q8—531no—
376C22×D4C22×D5311yescharacteristic
377C2×SD16—1062no—
378C2×SD16—1062no—
379C2×SD16—1062no—
380C2×SD16—2121no—
381C2×SD16—2121no—
382C2×SD16—2121no—
383C22×C8—531no—
384C22×C8—531no—
385C22×C8—531no—
386C22×C8—531no—
387C4⋊Q8—1062no—
388C4⋊Q8—1062no—
389C4×Q8—1062no—
390C4×Q8—1062no—
391C4×Q8—2121no—
392C4×Q8—2121no—
393C2×C4⋊C4—21no—
394C2×C4⋊C4—531no—
395C2×C4⋊C4—531no—
396C2×C4⋊C4—1061no—
397C2×C4⋊C4—1061no—
398C2×C4⋊C4—1061no—
399C2×C4⋊C4—1061no—
400C2×C4⋊C4C22⋊D5311yescharacteristic
401C2×C4⋊C4C22⋊D5311yescharacteristic
402C2×C4⋊C4C22×D5311yescharacteristic
403C2×C4⋊C4C22×D5311yescharacteristic
404C2×C4⋊C4C22×D5311yescharacteristic
405C2×C22⋊C4—21no—
406C2×C22⋊C4—21no—
407C2×C22⋊C4—21no—
408C2×C22⋊C4—21no—
409C2×C22⋊C4—21no—
410C2×C22⋊C4—21no—
411C2×C42—21no—
412C2×C42—1061no—
413C2×C42—1061no—
414C2×C42—1061no—
415C2×C42C22×D5311yescharacteristic
416C2×C42C22×D5311yescharacteristic
417C2×C42C22×D5311yescharacteristic
418#107—2121no—
419#106—1062no—
420C4⋊C8—2121no—
421#103—1062no—
422#103—1062no—
423#103—1062no—
424#103—2121no—
425#103—2121no—
426#103—2121no—
427#102—1062no—
428#102—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#95C22⋊D5311yescharacteristic
440#95C22⋊D5311yescharacteristic
441C24—21no—
442C2×Q8—532no—
443C2×Q8—532no—
444C2×Q8—1061no—
445C2×Q8—1061no—
446C2×Q8—1061no—
447C2×Q8—1061no—
448C2×Q8—1061no—
449C2×Q8—1061no—
450C2×Q8—1061no—
451C2×Q8—1061no—
452C2×Q8—1061no—
453C2×Q8—1061no—
454C2×D4—21no—
455C2×D4—21no—
456C2×D4—21no—
457C2×D4—21no—
458C2×D4—21no—
459C2×D4—21no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—1061no—
464C22×C4—1061no—
465C22×C4—1061no—
466C22×C4—1061no—
467C22×C4C2×C22⋊D5311yescharacteristic
468C22×C4C2×C22⋊D5311yescharacteristic
469C22×C4C2×C22⋊D5311yescharacteristic
470C22×C4C2×C22⋊D5311yescharacteristic
471C22×C4C2×C22⋊D5311yescharacteristic
472C22×C4C2×C22⋊D5311yescharacteristic
473C22×C4C23×D5311yescharacteristic
474SD16—2124no—
475SD16—2124no—
476C2×C8—532no—
477C2×C8—1061no—
478C2×C8—1061no—
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—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—81no—
497C4⋊C4—532no—
498C4⋊C4—1064no—
499C4⋊C4—1064no—
500C4⋊C4—1061no—
501C4⋊C4—1061no—
502C4⋊C4—1061no—
503C4⋊C4—2122no—
504C4⋊C4—2122no—
505C22⋊C4—42no—
506C22⋊C4—42no—
507C22⋊C4—81no—
508C22⋊C4—81no—
509C22⋊C4—81no—
510C22⋊C4—81no—
511C42—21no—
512C42—21no—
513C42—21no—
514C42—21no—
515C42—21no—
516C42—21no—
517C42—81no—
518C42—2122no—
519C42—2122no—
520C42—2122no—
521C23—41no—
522C23—41no—
523C23—41no—
524C23—41no—
525C23—41no—
526C23—41no—
527C23—41no—
528C23C22×C22⋊D5311yescharacteristic
529Q8—1062no—
530Q8—1062no—
531Q8—1062no—
532Q8—1062no—
533Q8—2121no—
534Q8—2121no—
535Q8—2121no—
536Q8—2121no—
537D4—41no—
538D4—41no—
539D4—41no—
540D4—41no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
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—21no—
556C2×C4—21no—
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—1062no—
567C2×C4—1062no—
568C2×C4—1062no—
569C2×C4—2121no—
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#1313323712yes—
579C2×C4#1313324712yes—
580C2×C4#41×D5311yescharacteristic
581C2×C4#41×D5311yescharacteristic
582C2×C4#41×D5311yescharacteristic
583C2×C4#41×D5311yescharacteristic
584C2×C4#1313326311yescharacteristic
585C2×C4#1313326311yescharacteristic
586C8—1062no—
587C8—2121no—
588C8—2121no—
589C8—2121no—
590C22—81no—
591C22—81no—
592C22—81no—
593C22—81no—
594C22—81no—
595C22—81no—
596C22—81no—
597C22#1681113011yescharacteristic
598C22#1681126111yescharacteristic
599C22#1681129211yescharacteristic
600C22C2×#1313323711yescharacteristic
601C22C2×#1313324711yescharacteristic
602C22#1681143411yescharacteristic
603C22#1681143411yescharacteristic
604C4—22no—
605C4—22no—
606C4—41no—
607C4—41no—
608C4—41no—
609C4—41no—
610C4—81no—
611C4—81no—
612C4—81no—
613C4—4241no—
614C4—2122no—
615C4—2122no—
616C4—2122no—
617C4#1681115611yescharacteristic
618C4#1681121211yescharacteristic
619C4#1681129811yescharacteristic
620C4#1681132411yescharacteristic
621C2—161no—
622C2#2027876211yescharacteristic
623C2#2028479111yescharacteristic
624C2#2028500111yescharacteristic
625C2#2028663711yescharacteristic
626C2#2028688011yescharacteristic
627C2#2028770711yescharacteristic
628C2#2028776811yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2513C2
socleC23×C53C22×D4
prime radicalC22×D4×C53C23
hypercenter#2513D53
hypocenterC53#27069
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2513

Complements

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

NH ≅ G/Ntype
1. #20277771C2semidirect
2. #20276266C2semidirect
4. #20275961C2semidirect
7. #20275356C2semidirect
9. #20275297C2semidirect
10. #20275274C2semidirect
13. #20274727C2semidirect
14. C4×#13133140C2semidirect
373. C53#27069semidirect

Maximal subgroups

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

subgroupindex
#202777712
#202762662
#202760322
#202759612
#202757972
#202755682
#202753562
#202753322
#202752972
#202752742
#202750732
#202747272
C4×#131331402
C53×#25132
#2706953