Finite Group Databasebeta

← #27465554

Subgroups — #27465554

Order 13568.

810 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
1C4×#13133191C211yesmaximal, characteristic
2#20276152C211yesmaximal, characteristic, complemented
3#20276141C211yesmaximal, characteristic, complemented
4#20275904C211yesmaximal, characteristic
5#20275897C211yesmaximal, characteristic
6#20275481C211yesmaximal, characteristic, complemented
7#20275481C211yesmaximal, characteristic, complemented
8#20275342C211yesmaximal, characteristic, complemented
9#20275335C211yesmaximal, characteristic, complemented
10#20275307C211yesmaximal, characteristic, complemented
11#20275300C211yesmaximal, characteristic, complemented
12#20274722C211yesmaximal, characteristic, complemented
13#20274722C211yesmaximal, characteristic, complemented
14#20274695C211yesmaximal, characteristic, complemented
15C53×#2283C211yesmaximal, characteristic, complemented
16C2×#13133191C2211yescharacteristic
17C2×C4×D212C2211yescharacteristic
18C2×C4×C22⋊D53—21no—
19C2×#13133150C2211yescharacteristic
20C2×#13133150C2211yescharacteristic
21C2×#13133148C2211yescharacteristic
22C2×#13133148C2211yescharacteristic
23C2×C4⋊#88322C2211yescharacteristic
24C2×C4⋊#88322C2211yescharacteristic
25C2×#13133137C2211yescharacteristic
26C2×#13133137C2211yescharacteristic
27C2×#13133137C2211yescharacteristic
28C2×#13133137C2211yescharacteristic
29C2×C22⋊#88322C2211yescharacteristic
30C2×C22⋊#88322C2211yescharacteristic
31#16810525C2211yescharacteristic
32#16810521C2211yescharacteristic
33#16810514—21no—
34#16810502—21no—
35#16810483C2211yescharacteristic
36C4×C4⋊Dic53C2211yescharacteristic
37C4×#11719921—21no—
38#16810385C2211yescharacteristic
39#16810385C2211yescharacteristic
40#16810385C2211yescharacteristic
41#16810385C2211yescharacteristic
42#16810385C2211yescharacteristic
43#16810385C2211yescharacteristic
44#16810385C2211yescharacteristic
45#16810385C2211yescharacteristic
46#16810378C2211yescharacteristic
47#16810378C2211yescharacteristic
48C22×C42×C53C2211yescharacteristic
49C53×#383—21no—
50C53×#382C2211yescharacteristic
51C53×#382C2211yescharacteristic
52C53×#381—21no—
53C53×#380C2211yescharacteristic
54C53×#380C2211yescharacteristic
55C4×C4⋊C4×C53C2211yescharacteristic
56C4×C4⋊C4×C53C2211yescharacteristic
57C4×C22⋊C4×C53—21no—
58C53×#376—21no—
59C22×D212C2311yescharacteristic
60C22×C22⋊D53—21no—
61C22×C4×D53—21no—
62C22×#88322C2311yescharacteristic
63C22×#88322C2311yescharacteristic
64C22×#88322C2311yescharacteristic
65C22×#88322C2311yescharacteristic
66#13133191—22no—
67#13133191—22no—
68C4×D212—22no—
69C4×D212—22no—
70C4×C22⋊D53—44no—
71C2×C4⋊Dic53C2311yescharacteristic
72C2×C4⋊Dic53C2311yescharacteristic
73C2×C4⋊Dic53C2311yescharacteristic
74C2×#11719919—21no—
75C2×C22⋊Dic53—21no—
76C2×#11719921—21no—
77C2×#11719921—21no—
78C2×#11719921—21no—
79C2×#11719921—21no—
80C2×C4×Dic53—21no—
81#13133150—22no—
82#13133150—41no—
83#13133148—22no—
84#13133148—41no—
85C4⋊#88322—22no—
86C4⋊#88322—22no—
87#13133137—22no—
88#13133137—22no—
89#13133137—41no—
90#13133137—41no—
91C22⋊#88322—41no—
92C22⋊#88322—41no—
93#13133124—21no—
94#13133124—21no—
95#13133124—21no—
96C23×C4×C53D411yescharacteristic
97C23×C4×C53D411yescharacteristic
98C23×C4×C53C2311yescharacteristic
99C2×C4⋊C4×C53C2311yescharacteristic
100C2×C4⋊C4×C53C2311yescharacteristic
101C2×C4⋊C4×C53C2311yescharacteristic
102C2×C4⋊C4×C53C2311yescharacteristic
103C2×C22⋊C4×C53—21no—
104C2×C22⋊C4×C53—21no—
105C2×C42×C53—22no—
106C2×C42×C53—22no—
107C2×C42×C53—21no—
108C2×C42×C53—21no—
109C2×C42×C53D411yescharacteristic
110C2×C42×C53D411yescharacteristic
111C2×C42×C53C2311yescharacteristic
112C2×C42×C53C2311yescharacteristic
113#95×C53—21no—
114#95×C53—21no—
115#95×C53—21no—
116#95×C53—21no—
117#95×C53—21no—
118#95×C53—21no—
119#95×C53—21no—
120#95×C53—21no—
121C24×C53C2×D411yescharacteristic
122C23×D53—21no—
123C22×C4×C53—22no—
124C22×C4×C53—22no—
125C22×C4×C53—22no—
126C22×C4×C53—22no—
127C22×C4×C53—21no—
128C22×C4×C53—21no—
129C22×C4×C53—21no—
130C22×C4×C53—21no—
131C22×C4×C53—21no—
132C22×C4×C53—21no—
133C22×C4×C53—21no—
134C22×C4×C53—21no—
135C22×C4×C53—21no—
136C22×C4×C53—21no—
137C22×C4×C53—41no—
138C22×C4×C53—41no—
139C22×C4×C53C2×D411yescharacteristic
140C22×C4×C53C2×D411yescharacteristic
141C22×C4×C53C2×D411yescharacteristic
142C22×C4×C53C2×D411yescharacteristic
143C22×C4×C53C2×D411yescharacteristic
144C22×C4×C53C2411yescharacteristic
145C2×C22⋊D53—24no—
146C2×C22⋊D53—42no—
147C2×C22⋊D53—42no—
148C22×Dic53—21no—
149C22×Dic53—21no—
150C22×Dic53—21no—
151C2×D212—21no—
152C2×D212—21no—
153C2×D212—21no—
154C2×D212—21no—
155C2×D212—21no—
156C2×D212#4112yes—
157C2×C4×D53—42no—
158C2×C4×D53—42no—
159C2×C4×D53—42no—
160C4⋊C4×C53—21no—
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—21no—
168C22⋊C4×C53—42no—
169C22⋊C4×C53—42no—
170C42×C53—22no—
171C42×C53—22no—
172C42×C53—21no—
173C42×C53—21no—
174C42×C53—42no—
175C42×C53—42no—
176C42×C53#4112yes—
177C42×C53#4112yes—
178C22⋊Dic53—42no—
179#11719921—24no—
180#11719921—42no—
181#11719921—42no—
182#11719921—42no—
183C4⋊Dic53—21no—
184C4⋊Dic53—21no—
185C4⋊Dic53—21no—
186C4⋊Dic53—21no—
187C4⋊Dic53—21no—
188C4⋊Dic53#4112yes—
189#11719919—42no—
190C4×Dic53—42no—
191C2×#88322—21no—
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—
203C23×C53—21no—
204C23×C53—21no—
205C23×C53—21no—
206C23×C53—21no—
207C23×C53—21no—
208C23×C53—21no—
209C23×C53AGL(1,ℤ/8)12yes—
210C23×C53C22×D411yescharacteristic
211C22×D53—22no—
212C22×D53—41no—
213C22×D53—41no—
214C22×D53—41no—
215C22×D53—41no—
216C22×D53—41no—
217C22×D53—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—44no—
231C2×C4×C53—44no—
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×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#12312yes—
249C2×C4×C53#12312yes—
250C2×C4×C53#13512yes—
251C2×C4×C53C2×#4111yescharacteristic
252C2×C4×C53C2×#4111yescharacteristic
253C2×C4×C53C2×#4111yescharacteristic
254C2×C4×C53C2×#4111yescharacteristic
255C2×C4×C53H(5,𝔽2)11yescharacteristic
256C2×C4×C532−1+411yescharacteristic
257C22⋊D53—48no—
258C2×Dic53—22no—
259C2×Dic53—41no—
260C2×Dic53—41no—
261C2×Dic53—41no—
262C2×Dic53—41no—
263C2×Dic53—41no—
264C2×Dic53—41no—
265C2×Dic53—41no—
266C2×Dic53—41no—
267D212—22no—
268D212—22no—
269D212—41no—
270D212—41no—
271C4×D53—84no—
272#88322—41no—
273#88322—41no—
274#88322—41no—
275#88322—41no—
276#27633—531nomaximal, Sylow
277C22×C53—22no—
278C22×C53—22no—
279C22×C53—22no—
280C22×C53—41no—
281C22×C53—41no—
282C22×C53—41no—
283C22×C53—41no—
284C22×C53C2×#12311yescharacteristic
285C22×C53#52511yescharacteristic
286C22×C53#53711yescharacteristic
287C22×C53C2×#13511yescharacteristic
288C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
289C22×C53#57511yescharacteristic
290C22×C53#57511yescharacteristic
291C2×D53—42no—
292C2×D53—42no—
293C2×D53—42no—
294C2×D53—81no—
295C2×D53—81no—
296C2×D53—81no—
297C2×D53—81no—
298C4×C53—22no—
299C4×C53—22no—
300C4×C53—22no—
301C4×C53—41no—
302C4×C53—41no—
303C4×C53—81no—
304C4×C53—81no—
305C4×C53—81no—
306C4×C53—81no—
307C4×C53#55211yescharacteristic
308C4×C53#55311yescharacteristic
309C4×C53#55511yescharacteristic
310C4×C53#55611yescharacteristic
311Dic53—42no—
312Dic53—81no—
313Dic53—81no—
314C4×#121—531no—
315#2283D5311yescharacteristic, complemented
316#2077—531no—
317#2075—531no—
318#2034—531no—
319#2033—531no—
320#1949—531no—
321#1949—531no—
322#1926—531no—
323#1925—531no—
324#1921—531no—
325#1920—531no—
326#1804—531no—
327#1804—531no—
328#1797—531no—
329C2×C53—42no—
330C2×C53#248511yescharacteristic
331C2×C53#309111yescharacteristic
332C2×C53#309411yescharacteristic
333C2×C53#310211yescharacteristic
334C2×C53#311411yescharacteristic
335C2×C53#324711yescharacteristic
336C2×C53#325611yescharacteristic
337D53—82no—
338C2×#121—531no—
339C2×C4×D4—531no—
340C2×C4×D4—1061no—
341C22×C42C2×D5311yescharacteristic
342C2×#107—531no—
343C2×#107—531no—
344C2×#106—531no—
345C2×#106—531no—
346C2×C4⋊C8—531no—
347C2×C4⋊C8—531no—
348C2×#102—531no—
349C2×#102—531no—
350C2×#102—531no—
351C2×#102—531no—
352C2×C22⋊C8—531no—
353C2×C22⋊C8—531no—
354#390—531no—
355#389—531no—
356#388—1061no—
357#386—1061no—
358#383—21no—
359#382C2×D5311yescharacteristic
360#382C2×D5311yescharacteristic
361#381—21no—
362#380—531no—
363#380C2×D5311yescharacteristic
364#380C2×D5311yescharacteristic
365C4×C4⋊C4—531no—
366C4×C4⋊C4C2×D5311yescharacteristic
367C4×C4⋊C4C2×D5311yescharacteristic
368C4×C22⋊C4—21no—
369C4×C22⋊C4—1061no—
370#376—21no—
371#340—531no—
372#340—531no—
373#340—531no—
374#340—531no—
375#340—531no—
376#340—531no—
377#340—531no—
378#340—531no—
379#336—531no—
380#336—531no—
381C53#2763311yescharacteristic, complemented, Sylow
382C22×D4—531no—
383C22×D4—1061no—
384C23×C4—1061no—
385C23×C4C22⋊D5311yescharacteristic
386C23×C4C22⋊D5311yescharacteristic
387C23×C4C22×D5311yescharacteristic
388C22×C8—531no—
389C22×C8—531no—
390C22×C8—531no—
391C22×C8—531no—
392#121—1062no—
393#121—1062no—
394C4×D4—1062no—
395C4×D4—1062no—
396C4×D4—2124no—
397C2×C4⋊C4—531no—
398C2×C4⋊C4—531no—
399C2×C4⋊C4—531no—
400C2×C4⋊C4—1061no—
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—1061no—
408C2×C22⋊C4—1061no—
409C2×C22⋊C4—1061no—
410C2×C22⋊C4—1061no—
411C2×C22⋊C4—1061no—
412C2×C42—22no—
413C2×C42—22no—
414C2×C42—21no—
415C2×C42—21no—
416C2×C42—1061no—
417C2×C42C22⋊D5311yescharacteristic
418C2×C42C22⋊D5311yescharacteristic
419C2×C42C22×D5311yescharacteristic
420C2×C42C22×D5311yescharacteristic
421#107—1062no—
422#107—2121no—
423#106—1062no—
424#106—2121no—
425C4⋊C8—1062no—
426C4⋊C8—1062no—
427#102—1062no—
428#102—1062no—
429#102—2121no—
430#102—2121no—
431C22⋊C8—2121no—
432C22⋊C8—2121no—
433#95—21no—
434#95—21no—
435#95—21no—
436#95—21no—
437#95—21no—
438#95—21no—
439#95—21no—
440#95—21no—
441#95—1061no—
442#95—1061no—
443#95—1061no—
444C24—1061no—
445C24C2×C22⋊D5311yescharacteristic
446C2×D4—532no—
447C2×D4—1064no—
448C2×D4—1061no—
449C2×D4—1061no—
450C2×D4—1061no—
451C2×D4—1061no—
452C2×D4—1061no—
453C2×D4—2122no—
454C2×D4—2122no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—41no—
470C22×C4—41no—
471C22×C4—1061no—
472C22×C4—1061no—
473C22×C4—1061no—
474C22×C4—2122no—
475C22×C4—2122no—
476C22×C4—2122no—
477C22×C4C2×C22⋊D5311yescharacteristic
478C22×C4C2×C22⋊D5311yescharacteristic
479C22×C4C2×C22⋊D5311yescharacteristic
480C22×C4C2×C22⋊D5311yescharacteristic
481C22×C4C2×C22⋊D5311yescharacteristic
482C22×C4C23×D5311yescharacteristic
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—
492C2×C8—1061no—
493C2×C8—1061no—
494C2×C8—1061no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—21no—
503C4⋊C4—532no—
504C4⋊C4—1061no—
505C4⋊C4—1061no—
506C4⋊C4—1061no—
507C4⋊C4—1061no—
508C4⋊C4—1061no—
509C4⋊C4—2122no—
510C22⋊C4—42no—
511C22⋊C4—42no—
512C22⋊C4—1064no—
513C22⋊C4—2122no—
514C22⋊C4—2122no—
515C22⋊C4—2122no—
516C22⋊C4—2122no—
517C42—22no—
518C42—22no—
519C42—21no—
520C42—21no—
521C42—42no—
522C42—42no—
523C42—2122no—
524C42#1171994512yes—
525C42#1171994512yes—
526C23—21no—
527C23—21no—
528C23—21no—
529C23—21no—
530C23—21no—
531C23—21no—
532C23—1062no—
533C23—2121no—
534C23—2121no—
535C23—2121no—
536C23—2121no—
537C23—2121no—
538C23—2121no—
539C23#1313324212yes—
540C23C22×C22⋊D5311yescharacteristic
541D4—1062no—
542D4—1062no—
543D4—2128no—
544D4—2121no—
545D4—2121no—
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—21no—
558C2×C4—44no—
559C2×C4—44no—
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—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—1062no—
577C2×C4—4244no—
578C2×C4—2121no—
579C2×C4—2121no—
580C2×C4—2121no—
581C2×C4—2121no—
582C2×C4—2121no—
583C2×C4—2121no—
584C2×C4—2121no—
585C2×C4—2121no—
586C2×C4#1313320012yes—
587C2×C4#1313320012yes—
588C2×C4#1313323712yes—
589C2×C4#41×D5311yescharacteristic
590C2×C4#41×D5311yescharacteristic
591C2×C4C2×#1171994511yescharacteristic
592C2×C4C2×#1171994511yescharacteristic
593C2×C4#1313326311yescharacteristic
594C2×C4#1313326411yescharacteristic
595C8—2121no—
596C8—2121no—
597C8—2121no—
598C8—2121no—
599C22—22no—
600C22—22no—
601C22—22no—
602C22—41no—
603C22—41no—
604C22—41no—
605C22—41no—
606C22—4241no—
607C22—4241no—
608C22—4241no—
609C22—4241no—
610C22—2122no—
611C22—2122no—
612C22—2122no—
613C22C2×#1313320011yescharacteristic
614C22#1681113011yescharacteristic
615C22#1681120111yescharacteristic
616C22C2×#1313323711yescharacteristic
617C22C2×#1313324211yescharacteristic
618C22#1681142111yescharacteristic
619C22#1681142111yescharacteristic
620C4—22no—
621C4—22no—
622C4—22no—
623C4—41no—
624C4—41no—
625C4—81no—
626C4—81no—
627C4—81no—
628C4—81no—
629C4—4241no—
630C4—4241no—
631C4—2122no—
632C4#1681131211yescharacteristic
633C4#1681131711yescharacteristic
634C4#1681133611yescharacteristic
635C4#1681134111yescharacteristic
636C2—42no—
637C2—4242no—
638C2#2027900611yescharacteristic
639C2#2028462811yescharacteristic
640C2#2028465511yescharacteristic
641C2#2028472711yescharacteristic
642C2#2028484911yescharacteristic
643C2#2028604111yescharacteristic
644C2#2028611111yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2283C2
socleC23×C53C22×D4
prime radicalC2×#13133191C22
hypercenter#2283D53
hypocenterC53#27633
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C22×C42 ◁ #2283

Complements

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

NH ≅ G/Ntype
2. #20276152C2semidirect
3. #20276141C2semidirect
6. #20275481C2semidirect
7. #20275481C2semidirect
8. #20275342C2semidirect
9. #20275335C2semidirect
10. #20275307C2semidirect
11. #20275300C2semidirect
12. #20274722C2semidirect
13. #20274722C2semidirect
14. #20274695C2semidirect
15. C53×#2283C2semidirect
315. #2283D53semidirect
381. C53#27633semidirect

Maximal subgroups

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

subgroupindex
C4×#131331912
#202761522
#202761412
#202759042
#202758972
#202754812
#202753422
#202753352
#202753072
#202753002
#202747222
#202746952
C53×#22832
#2763353