Finite Group Databasebeta

← #27484116

Subgroups — #27484116

Order 13568.

828 conjugacy classes of subgroups.

The subgroup lattice has 634 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20283066C211yesmaximal, characteristic
2C4×#13133193C211yesmaximal, characteristic
3#20276280C211yesmaximal, characteristic, complemented
4#20276251C211yesmaximal, characteristic, complemented
5#20276041C211yesmaximal, characteristic, complemented
6#20275854C211yesmaximal, characteristic, complemented
7#20275308C211yesmaximal, characteristic, complemented
8#20275304C211yesmaximal, characteristic, complemented
9#20275135C211yesmaximal, characteristic
10#20275135C211yesmaximal, characteristic
11#20274969C211yesmaximal, characteristic, complemented
12#20274743C211yesmaximal, characteristic
13#20274741C211yesmaximal, characteristic
14#336×D53C211yesmaximal, characteristic
15C53×#1863C211yesmaximal, characteristic, complemented
16C22×C8×D53C2211yescharacteristic
17C22×C8×D53C2211yescharacteristic
18C2×#13133216C2211yescharacteristic
19C2×#13133195—21no—
20C2×#13133193C2211yescharacteristic
21C2×C42×D53C2211yescharacteristic
22C2×#13133151C2211yescharacteristic
23C2×#13133150C2211yescharacteristic
24C2×#13133150C2211yescharacteristic
25C2×#13133141C2211yescharacteristic
26C2×#13133141C2211yescharacteristic
27#16810530C2211yescharacteristic
28#16810527—21no—
29#16810521C2211yescharacteristic
30#16810519C2211yescharacteristic
31#16810518—21no—
32#16810504—21no—
33#16810494—21no—
34#16810492C2211yescharacteristic
35#16810482—21no—
36#16810480C2211yescharacteristic
37#16810480C2211yescharacteristic
38#16810480C2211yescharacteristic
39C4×#11719919—21no—
40C4×C4⋊Dic53C2211yescharacteristic
41C4×#11719921—21no—
42#16810386C2211yescharacteristic
43#16810386C2211yescharacteristic
44#16810386C2211yescharacteristic
45#16810385C2211yescharacteristic
46#16810385C2211yescharacteristic
47#16810385C2211yescharacteristic
48#16810385C2211yescharacteristic
49#16810378C2211yescharacteristic
50#16810377C2211yescharacteristic
51#16810377C2211yescharacteristic
52C2×C4×Q8×C53C2211yescharacteristic
53C2×#107×C53C2211yescharacteristic
54C2×#103×C53C2211yescharacteristic
55C2×#103×C53C2211yescharacteristic
56C53×#389C2211yescharacteristic
57C53×#340C2211yescharacteristic
58C53×#336C2211yescharacteristic
59C22×Dic106—21no—
60C22×C4×D53D411yescharacteristic
61C22×C4×D53D411yescharacteristic
62C22×C4×D53C2311yescharacteristic
63C22×#88322C2311yescharacteristic
64C22×#88322C2311yescharacteristic
65C2×C8×D53—22no—
66C2×C8×D53—22no—
67C2×C8×D53—22no—
68C2×C8×D53—22no—
69C2×C8×D53—41no—
70C2×C8×D53—41no—
71#13133216—41no—
72#13133216—41no—
73#13133195—44no—
74#13133193—22no—
75#13133193Q814yescomplemented
76C2×C4⋊Dic53C2311yescharacteristic
77C2×C4⋊Dic53C2311yescharacteristic
78C2×C4⋊Dic53C2311yescharacteristic
79C2×C4⋊Dic53C2311yescharacteristic
80C2×#11719919—21no—
81C2×#11719919—21no—
82C2×#11719919—21no—
83C2×#11719919—21no—
84C2×#11719919—21no—
85C2×#11719919—21no—
86C2×#11719921—21no—
87C2×#11719921—21no—
88C2×#11719921—21no—
89C2×#11719921—21no—
90C2×C4×Dic53—21no—
91C2×C4×Dic53D411yescharacteristic
92C2×C4×Dic53D411yescharacteristic
93C2×C4×Dic53C2311yescharacteristic
94C42×D53—41no—
95C42×D53—41no—
96#13133151—22no—
97#13133150—22no—
98#13133150—41no—
99#13133141—22no—
100#13133141—41no—
101#13133124—21no—
102#13133124—21no—
103#13133124—21no—
104#13133124—21no—
105#13133124—21no—
106#13133124—21no—
107#13133124—21no—
108#13133124—21no—
109C22×Q8×C53C2311yescharacteristic
110C22×C8×C53C2311yescharacteristic
111C22×C8×C53C2311yescharacteristic
112C4×Q8×C53—41no—
113C4×Q8×C53—41no—
114C2×C4⋊C4×C53—21no—
115C2×C4⋊C4×C53—21no—
116C2×C4⋊C4×C53C2311yescharacteristic
117C2×C4⋊C4×C53C2311yescharacteristic
118C2×C4⋊C4×C53C2311yescharacteristic
119C2×C42×C53—21no—
120C2×C42×C53C2311yescharacteristic
121#107×C53—41no—
122#103×C53—22no—
123#103×C53—41no—
124#95×C53—21no—
125C23×D53C2×D411yescharacteristic
126C2×Q8×C53—21no—
127C2×Q8×C53—21no—
128C2×Q8×C53—21no—
129C2×Q8×C53—21no—
130C2×Q8×C53—21no—
131C2×Q8×C53C2×Q812yes—
132C22×C4×C53—21no—
133C22×C4×C53—21no—
134C22×C4×C53—21no—
135C22×C4×C53—21no—
136C22×C4×C53C2×D411yescharacteristic
137C22×C4×C53C2×D411yescharacteristic
138C22×C4×C53C2411yescharacteristic
139C22×Dic53—21no—
140C22×Dic53—21no—
141C22×Dic53—21no—
142C22×Dic53—21no—
143C22×Dic53C2×D411yescharacteristic
144C22×Dic53C2×D411yescharacteristic
145C22×Dic53C2×D411yescharacteristic
146C2×C4×D53—21no—
147C2×C4×D53—21no—
148C2×C4×D53—21no—
149C2×C4×D53—21no—
150C2×C4×D53—22no—
151C2×C4×D53—22no—
152C2×C4×D53—41no—
153C2×C4×D53—41no—
154C2×C4×D53—41no—
155C2×C4×D53Q1614yes—
156C2×C4×D53C2×Q812yes—
157C2×C4×D53#4112yes—
158C2×Dic106—42no—
159C2×Dic106—42no—
160C2×Dic106—42no—
161C2×C8×C53—21no—
162C2×C8×C53—21no—
163C2×C8×C53—21no—
164C2×C8×C53—21no—
165C2×C8×C53—21no—
166C2×C8×C53#4112yes—
167C4⋊C4×C53—21no—
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×C53—42no—
175C42×C53—21no—
176C42×C53—21no—
177C42×C53—42no—
178#11719921—24no—
179#11719921—42no—
180#11719921—42no—
181#11719921—42no—
182C4⋊Dic53—21no—
183C4⋊Dic53—21no—
184C4⋊Dic53—21no—
185C4⋊Dic53—21no—
186C4⋊Dic53—21no—
187C4⋊Dic53—21no—
188C4⋊Dic53—21no—
189C4⋊Dic53C2×Q812yes—
190#11719919—24no—
191#11719919—42no—
192#11719919—42no—
193#11719919—42no—
194#11719919—42no—
195#11719919—42no—
196C4×Dic53—21no—
197C4×Dic53—21no—
198C4×Dic53—41no—
199C4×Dic53—41no—
200C4×Dic53—42no—
201C2×#88322—21no—
202C2×#88322—21no—
203C2×#88322—21no—
204C2×#88322—21no—
205C2×#88322—21no—
206C2×#88322#4112yes—
207C8×D53—44no—
208C8×D53—44no—
209C23×C53C22×D411yescharacteristic
210C22×D53—21no—
211C22×D53—21no—
212C22×D53—21no—
213C22×D53—21no—
214C22×D53C22⋊Q812yes—
215C22×D53C2×Q1612yes—
216C22×D53AGL(1,ℤ/8)12yes—
217Q8×C53—22no—
218Q8×C53—22no—
219Q8×C53—41no—
220Q8×C53—41no—
221C2×C4×C53—21no—
222C2×C4×C53—21no—
223C2×C4×C53—22no—
224C2×C4×C53—21no—
225C2×C4×C53—21no—
226C2×C4×C53—21no—
227C2×C4×C53—41no—
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×C53C2×Q1612yes—
239C2×C4×C53C22×Q811yescharacteristic
240C2×C4×C53C2×#4111yescharacteristic
241C2×C4×C53C2×#4111yescharacteristic
242C2×C4×C53C2×#4111yescharacteristic
243C2×C4×C53H(5,𝔽2)11yescharacteristic
244C2×C4×C53H(5,𝔽2)11yescharacteristic
245C2×Dic53—21no—
246C2×Dic53—21no—
247C2×Dic53—21no—
248C2×Dic53—21no—
249C2×Dic53—21no—
250C2×Dic53—21no—
251C2×Dic53—22no—
252C2×Dic53—41no—
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×Dic53—41no—
262C2×Dic53—41no—
263C2×Dic53C22⋊Q812yes—
264C2×Dic53C2×Q1612yes—
265C2×Dic53#13512yes—
266C4×D53—22no—
267C4×D53—22no—
268C4×D53—22no—
269C4×D53—22no—
270C4×D53—44no—
271C4×D53—44no—
272Dic106—84no—
273C8×C53—22no—
274C8×C53—41no—
275#88322—22no—
276#88322—41no—
277#29160—531nomaximal, Sylow
278C22×C53C2×C22⋊Q811yescharacteristic
279C22×C53#53211yescharacteristic
280C22×C53#53411yescharacteristic
281C22×C53C22×Q1611yescharacteristic
282C22×C53C2×#13511yescharacteristic
283C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
284C22×C53#57511yescharacteristic
285C2×D53—22no—
286C2×D53—22no—
287C2×D53—22no—
288C2×D53—22no—
289C2×D53—22no—
290C2×D53—22no—
291C2×D53#47714yes—
292C4×C53—22no—
293C4×C53—42no—
294C4×C53—41no—
295C4×C53—81no—
296C4×C53—81no—
297C4×C53—81no—
298C4×C53#55111yescharacteristic
299C4×C53#55111yescharacteristic
300C4×C53#55411yescharacteristic
301C4×C53#55411yescharacteristic
302Dic53—22no—
303Dic53—22no—
304Dic53—22no—
305Dic53—42no—
306Dic53—81no—
307Dic53—81no—
308Dic53—81no—
309#2857—531no—
310C4×C22⋊Q8—531no—
311#2100—531no—
312#2095—531no—
313#2057—531no—
314#2024—531no—
315#1921—531no—
316#1921—531no—
317#1892—531no—
318#1892—531no—
319#1863—531no—
320#1863D5311yescharacteristic, complemented
321#1808—531no—
322#1808—531no—
323C2×#336—531no—
324C2×C53#256411yescharacteristic
325C2×C53C2×#47711yescharacteristic
326C2×C53#307611yescharacteristic
327C2×C53#317011yescharacteristic
328C2×C53#318011yescharacteristic
329C2×C53#318811yescharacteristic
330C2×C53#318911yescharacteristic
331D53—24no—
332C23×C8—531no—
333C23×C8—531no—
334C2×C4⋊Q8—531no—
335C2×C22⋊Q8—531no—
336C2×C22⋊Q8—1061no—
337C2×C4×Q8C2×D5311yescharacteristic
338C22×C42—531no—
339C2×#107—531no—
340C2×#107—531no—
341C2×#107—531no—
342C2×#107C2×D5311yescharacteristic
343C2×#103—531no—
344C2×#103—531no—
345C2×#103C2×D5311yescharacteristic
346C2×#103C2×D5311yescharacteristic
347#391—531no—
348#391—1061no—
349#389—531no—
350#389—531no—
351#389—1061no—
352#389C2×D5311yescharacteristic
353#387—1061no—
354#385—1061no—
355#384—531no—
356#380—531no—
357#380—531no—
358#380—531no—
359#380—1061no—
360C4×C4⋊C4—531no—
361C4×C4⋊C4—1061no—
362C4×C22⋊C4—1061no—
363#340—531no—
364#340—531no—
365#340—531no—
366#340—531no—
367#340—531no—
368#340—531no—
369#340—531no—
370#340C2×D5311yescharacteristic
371#336—531no—
372#336—531no—
373#336—531no—
374#336C2×D5311yescharacteristic
375C53#2916011yescharacteristic, complemented, Sylow
376C22×Q8—1061no—
377C22×Q8C22×D5311yescharacteristic
378C23×C4—531no—
379C23×C4—531no—
380C23×C4—531no—
381C22×C8—531no—
382C22×C8—531no—
383C22×C8—1062no—
384C22×C8—1062no—
385C22×C8—1062no—
386C22×C8—1062no—
387C22×C8—2121no—
388C22×C8—2121no—
389C22×C8C22×D5311yescharacteristic
390C22×C8C22×D5311yescharacteristic
391C4⋊Q8—2121no—
392C4⋊Q8—2121no—
393C22⋊Q8—534no—
394C22⋊Q8—1062no—
395C22⋊Q8—2124no—
396C4×Q8—41no—
397C4×Q8—41no—
398C2×C4⋊C4—21no—
399C2×C4⋊C4—21no—
400C2×C4⋊C4—531no—
401C2×C4⋊C4—531no—
402C2×C4⋊C4—531no—
403C2×C4⋊C4—531no—
404C2×C4⋊C4—1061no—
405C2×C4⋊C4—1061no—
406C2×C4⋊C4—1061no—
407C2×C4⋊C4—1061no—
408C2×C4⋊C4—1061no—
409C2×C4⋊C4—1061no—
410C2×C4⋊C4C22×D5311yescharacteristic
411C2×C4⋊C4C22×D5311yescharacteristic
412C2×C4⋊C4C22×D5311yescharacteristic
413C2×C22⋊C4—1061no—
414C2×C22⋊C4—1061no—
415C2×C22⋊C4—1061no—
416C2×C22⋊C4—1061no—
417C2×C42—21no—
418C2×C42—531no—
419C2×C42—531no—
420C2×C42—531no—
421C2×C42—1061no—
422C2×C42—2121no—
423C2×C42—2121no—
424C2×C42C22×D5311yescharacteristic
425#107—41no—
426#107—1062no—
427#107—1062no—
428#107—2121no—
429#103—22no—
430#103—41no—
431#103—1062no—
432#103—2121no—
433#95—21no—
434#95—1061no—
435#95—1061no—
436#95—1061no—
437#95—1061no—
438#95—1061no—
439#95—1061no—
440#95—1061no—
441#95—1061no—
442C24—531no—
443C2×Q8—21no—
444C2×Q8—21no—
445C2×Q8—21no—
446C2×Q8—21no—
447C2×Q8—21no—
448C2×Q8—2122no—
449C2×Q8—2122no—
450C2×Q8—2122no—
451C2×Q8Q8×D5312yes—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—531no—
457C22×C4—531no—
458C22×C4—531no—
459C22×C4—532no—
460C22×C4—532no—
461C22×C4—534no—
462C22×C4—1062no—
463C22×C4—1062no—
464C22×C4—1061no—
465C22×C4—1061no—
466C22×C4—1061no—
467C22×C4—1061no—
468C22×C4—1061no—
469C22×C4—1061no—
470C22×C4—1061no—
471C22×C4—1061no—
472C22×C4—2121no—
473C22×C4—2121no—
474C22×C4—2121no—
475C22×C4D4×D5311yescharacteristic
476C22×C4D4×D5311yescharacteristic
477C22×C4C23×D5311yescharacteristic
478C2×C8—21no—
479C2×C8—21no—
480C2×C8—21no—
481C2×C8—21no—
482C2×C8—21no—
483C2×C8—532no—
484C2×C8—1061no—
485C2×C8—1061no—
486C2×C8—1061no—
487C2×C8—1061no—
488C2×C8—1061no—
489C2×C8—2124no—
490C2×C8—2124no—
491C2×C8#1171994512yes—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—42no—
499C4⋊C4—42no—
500C4⋊C4—532no—
501C4⋊C4—1064no—
502C4⋊C4—1061no—
503C4⋊C4—1061no—
504C4⋊C4—1061no—
505C4⋊C4—1061no—
506C4⋊C4—1061no—
507C4⋊C4—1061no—
508C4⋊C4—1061no—
509C4⋊C4—2122no—
510C4⋊C4—2122no—
511C4⋊C4—2122no—
512C4⋊C4—2122no—
513C4⋊C4—2122no—
514C22⋊C4—1064no—
515C22⋊C4—2122no—
516C22⋊C4—2122no—
517C22⋊C4—2122no—
518C42—21no—
519C42—21no—
520C42—42no—
521C42—1061no—
522C42—1061no—
523C42—2121no—
524C42—2121no—
525C42—2122no—
526C23—532no—
527C23—532no—
528C23—532no—
529C23—1061no—
530C23—1061no—
531C23—1061no—
532C23—1061no—
533C23C2×D4×D5311yescharacteristic
534Q8—22no—
535Q8—22no—
536Q8—41no—
537Q8—41no—
538Q8—4244no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—22no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
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—532no—
557C2×C4—532no—
558C2×C4—532no—
559C2×C4—1061no—
560C2×C4—1061no—
561C2×C4—1061no—
562C2×C4—1061no—
563C2×C4—1061no—
564C2×C4—1061no—
565C2×C4—1062no—
566C2×C4—1062no—
567C2×C4—1062no—
568C2×C4—1062no—
569C2×C4—1062no—
570C2×C4—2124no—
571C2×C4—2124no—
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—2121no—
582C2×C4—2121no—
583C2×C4Q16×D5312yes—
584C2×C4C2×Q8×D5311yescharacteristic
585C2×C4#41×D5311yescharacteristic
586C2×C4#41×D5311yescharacteristic
587C2×C4C2×#1171994511yescharacteristic
588C2×C4#1313326211yescharacteristic
589C2×C4#1313326211yescharacteristic
590C8—22no—
591C8—41no—
592C8—1062no—
593C8—2121no—
594C22—534no—
595C22—1062no—
596C22—1062no—
597C22—1062no—
598C22—1062no—
599C22—1062no—
600C22—1062no—
601C22C22⋊Q8×D5311yescharacteristic
602C22#1681117011yescharacteristic
603C22#1681118311yescharacteristic
604C22C2×Q16×D5311yescharacteristic
605C22#135×D5311yescharacteristic
606C22AGL(1,ℤ/8)×D5311yescharacteristic
607C22#1681142211yescharacteristic
608C4—22no—
609C4—41no—
610C4—42no—
611C4—81no—
612C4—81no—
613C4—81no—
614C4—1062no—
615C4—1062no—
616C4—1062no—
617C4—4241no—
618C4—4241no—
619C4—4241no—
620C4—2122no—
621C4#1681130111yescharacteristic
622C4#1681130211yescharacteristic
623C4#1681132711yescharacteristic
624C4#1681132811yescharacteristic
625C2—1064no—
626C2#2027990011yescharacteristic
627C2#477×D5311yescharacteristic
628C2#2028450211yescharacteristic
629C2#2028529811yescharacteristic
630C2#2028540911yescharacteristic
631C2#2028547811yescharacteristic
632C2#2028548811yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#1863C2
socleC23×C53C22×D4
prime radicalC23×D53C2×D4
hypercenter#1863D53
hypocenterC53#29160
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1863

Complements

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

NH ≅ G/Ntype
3. #20276280C2semidirect
4. #20276251C2semidirect
5. #20276041C2semidirect
6. #20275854C2semidirect
7. #20275308C2semidirect
8. #20275304C2semidirect
11. #20274969C2semidirect
15. C53×#1863C2semidirect
75. #13133193Q8semidirect
320. #1863D53semidirect
375. C53#29160semidirect

Maximal subgroups

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

subgroupindex
#202830662
C4×#131331932
#202762802
#202762512
#202760412
#202758542
#202753082
#202753042
#202751352
#202749692
#202747432
#202747412
#336×D532
C53×#18632
#2916053