Finite Group Databasebeta

← #27456773

Subgroups — #27456773

Order 13568.

792 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20279148C211yesmaximal, characteristic, complemented
2#20276118C211yesmaximal, characteristic, complemented
3#20276089C211yesmaximal, characteristic, complemented
4#20275901C211yesmaximal, characteristic
5#20275901C211yesmaximal, characteristic
6#20275478C211yesmaximal, characteristic, complemented
7#20275440C211yesmaximal, characteristic, complemented
8#20275323C211yesmaximal, characteristic, complemented
9#20275300C211yesmaximal, characteristic, complemented
10#20275290C211yesmaximal, characteristic, complemented
11#20274976C211yesmaximal, characteristic, complemented
12#20274908C211yesmaximal, characteristic, complemented
13#20274801C211yesmaximal, characteristic, complemented
14C4×#13133138C211yesmaximal, characteristic, complemented
15C53×#2361C211yesmaximal, characteristic
16C2×#13133148C2211yescharacteristic
17C2×#13133148C2211yescharacteristic
18C2×#13133148C2211yescharacteristic
19C2×#13133148C2211yescharacteristic
20C2×C4⋊#88322C2211yescharacteristic
21C2×#13133138C2211yescharacteristic
22C2×#13133138C2211yescharacteristic
23C2×#13133138C2211yescharacteristic
24C2×#13133138C2211yescharacteristic
25C2×C22⋊#88322C2211yescharacteristic
26C2×C22⋊#88322C2211yescharacteristic
27C2×C4×#88322C2211yescharacteristic
28#16810565—21no—
29#16810546—21no—
30#16810521C2211yescharacteristic
31#16810521C2211yescharacteristic
32#16810516—21no—
33#16810510—21no—
34#16810493C2211yescharacteristic
35#16810483C2211yescharacteristic
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×#121×C53C2211yescharacteristic
49C2×C4×D4×C53—21no—
50C2×C4×D4×C53C2211yescharacteristic
51C22×C4⋊C4×C53C2211yescharacteristic
52C53×#390C2211yescharacteristic
53C53×#389C2211yescharacteristic
54C53×#388—21no—
55C53×#386—21no—
56C53×#385—21no—
57C4×C4⋊C4×C53C2211yescharacteristic
58C4×C22⋊C4×C53C2211yescharacteristic
59C22×#88322C2311yescharacteristic
60C22×#88322C2311yescharacteristic
61C22×#88322C2311yescharacteristic
62C22×#88322C2311yescharacteristic
63C2×C4⋊Dic53—21no—
64C2×C4⋊Dic53—21no—
65C2×C4⋊Dic53C2311yescharacteristic
66C2×C4⋊Dic53C2311yescharacteristic
67C2×C4⋊Dic53C2311yescharacteristic
68C2×C4⋊Dic53C2311yescharacteristic
69C2×#11719919—21no—
70C2×#11719919—21no—
71C2×C22⋊Dic53—21no—
72C2×C22⋊Dic53—21no—
73C2×C4×Dic53—21no—
74#13133148—22no—
75#13133148—22no—
76#13133148—22no—
77#13133148—41no—
78C4⋊#88322—22no—
79#13133138—22no—
80#13133138—22no—
81#13133138—22no—
82#13133138—41no—
83C22⋊#88322—22no—
84C22⋊#88322—41no—
85C4×#88322—41no—
86#13133124—21no—
87#13133124—21no—
88#13133124—21no—
89#13133124—21no—
90#13133124—21no—
91C22×D4×C53—21no—
92C22×D4×C53C2311yescharacteristic
93C23×C4×C53—21no—
94C23×C4×C53—21no—
95C23×C4×C53C2311yescharacteristic
96#121×C53—22no—
97#121×C53—22no—
98C4×D4×C53—22no—
99C4×D4×C53—41no—
100C4×D4×C53—44no—
101C2×C4⋊C4×C53—21no—
102C2×C4⋊C4×C53—21no—
103C2×C4⋊C4×C53—22no—
104C2×C4⋊C4×C53—41no—
105C2×C4⋊C4×C53C2311yescharacteristic
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×C53D411yescharacteristic
113C2×C22⋊C4×C53D411yescharacteristic
114C2×C42×C53—21no—
115C2×C42×C53C2311yescharacteristic
116C2×C42×C53C2311yescharacteristic
117#95×C53—21no—
118#95×C53—21no—
119#95×C53D411yescharacteristic
120#95×C53D411yescharacteristic
121C24×C53—21no—
122C24×C53C2×D411yescharacteristic
123C2×D4×C53—24no—
124C2×D4×C53—21no—
125C2×D4×C53—21no—
126C2×D4×C53—21no—
127C2×D4×C53—21no—
128C2×D4×C53—21no—
129C2×D4×C53—42no—
130C2×D4×C53—42no—
131C2×D4×C53#4112yes—
132C22×C4×C53—21no—
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—42no—
142C22×C4×C53—42no—
143C22×C4×C53—42no—
144C22×C4×C53—42no—
145C22×C4×C53—81no—
146C22×C4×C53—81no—
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—21no—
158C4⋊C4×C53—21no—
159C4⋊C4×C53—21no—
160C4⋊C4×C53—21no—
161C4⋊C4×C53—42no—
162C4⋊C4×C53—42no—
163C4⋊C4×C53#4112yes—
164C4⋊C4×C53#4112yes—
165C22⋊C4×C53—24no—
166C22⋊C4×C53—41no—
167C22⋊C4×C53—42no—
168C22⋊C4×C53—42no—
169C22⋊C4×C53—81no—
170C22⋊C4×C53SD1614yes—
171C42×C53—21no—
172C42×C53—21no—
173C42×C53—21no—
174C42×C53—21no—
175C42×C53—42no—
176C22⋊Dic53—81no—
177C22⋊Dic53—81no—
178C4⋊Dic53—21no—
179C4⋊Dic53—21no—
180C4⋊Dic53—21no—
181C4⋊Dic53—21no—
182C4⋊Dic53—21no—
183C4⋊Dic53—21no—
184C4⋊Dic53—21no—
185C4⋊Dic53—21no—
186C4⋊Dic53—81no—
187C4⋊Dic53—81no—
188#11719919—42no—
189#11719919—81no—
190C4×Dic53—81no—
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#4112yes—
203C23×C53—21no—
204C23×C53—21no—
205C23×C53—21no—
206C23×C53—21no—
207C23×C53—21no—
208C23×C53—21no—
209C23×C53—22no—
210C23×C53—41no—
211C23×C53—41no—
212C23×C53—41no—
213C23×C53—41no—
214C23×C53—41no—
215C23×C53—41no—
216C23×C53C2×SD1612yes—
217C23×C53C22×D411yescharacteristic
218D4×C53—22no—
219D4×C53—22no—
220D4×C53—48no—
221D4×C53—41no—
222D4×C53—41no—
223C2×C4×C53—21no—
224C2×C4×C53—21no—
225C2×C4×C53—21no—
226C2×C4×C53—21no—
227C2×C4×C53—21no—
228C2×C4×C53—22no—
229C2×C4×C53—21no—
230C2×C4×C53—22no—
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—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—41no—
249C2×C4×C53—84no—
250C2×C4×C53—84no—
251C2×C4×C53C2×SD1612yes—
252C2×C4×C53C2×SD1612yes—
253C2×C4×C53C2×#4111yescharacteristic
254C2×C4×C53C2×#4111yescharacteristic
255C2×C4×C53C2×#4111yescharacteristic
256C2×C4×C53C2×#4111yescharacteristic
257C2×C4×C53H(5,𝔽2)11yescharacteristic
258C2×C4×C532−1+411yescharacteristic
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—
268C2×Dic53—41no—
269C2×Dic53—41no—
270C2×Dic53—41no—
271#88322—22no—
272#88322—41no—
273#88322—41no—
274#88322—41no—
275#27032—531nomaximal, Sylow
276C22×C53—22no—
277C22×C53—22no—
278C22×C53—22no—
279C22×C53—41no—
280C22×C53—41no—
281C22×C53—41no—
282C22×C53—41no—
283C22×C53—42no—
284C22×C53—42no—
285C22×C53—42no—
286C22×C53—81no—
287C22×C53—81no—
288C22×C53—81no—
289C22×C53—81no—
290C22×C53#52511yescharacteristic
291C22×C53#54611yescharacteristic
292C22×C53#54711yescharacteristic
293C22×C53C22×SD1611yescharacteristic
294C22×C53#57511yescharacteristic
295C22×C53#57611yescharacteristic
296C22×C53#57711yescharacteristic
297C4×C53—22no—
298C4×C53—22no—
299C4×C53—41no—
300C4×C53—42no—
301C4×C53—41no—
302C4×C53—42no—
303C4×C53—41no—
304C4×C53—81no—
305C4×C53#52911yescharacteristic
306C4×C53#53911yescharacteristic
307C4×C53#55311yescharacteristic
308C4×C53#55611yescharacteristic
309Dic53—81no—
310Dic53—81no—
311Dic53—81no—
312Dic53—81no—
313#2497—531no—
314#2361D5311yescharacteristic
315#2071—531no—
316#2065—531no—
317#2034—531no—
318#2034—531no—
319#1948—531no—
320#1941—531no—
321#1923—531no—
322#1920—531no—
323#1919—531no—
324#1864—531no—
325#1853—531no—
326#1823—531no—
327C4×#102—531no—
328C2×C53—42no—
329C2×C53—82no—
330C2×C53#246611yescharacteristic
331C2×C53#310111yescharacteristic
332C2×C53#311911yescharacteristic
333C2×C53#329511yescharacteristic
334C2×C53#332111yescharacteristic
335C2×C53#333511yescharacteristic
336C2×C53#335511yescharacteristic
337C2×#121C2×D5311yescharacteristic
338C2×C4×D4—21no—
339C2×C4×D4C2×D5311yescharacteristic
340C22×C4⋊C4C2×D5311yescharacteristic
341C2×#106—531no—
342C2×#106—531no—
343C2×#106—531no—
344C2×#106—531no—
345C2×C4⋊C8—531no—
346C2×#102—531no—
347C2×#102—531no—
348C2×#102—531no—
349C2×#102—531no—
350C2×C22⋊C8—531no—
351C2×C22⋊C8—531no—
352C2×C4×C8—531no—
353#400—1061no—
354#396—1061no—
355#390C2×D5311yescharacteristic
356#389—531no—
357#389—531no—
358#389C2×D5311yescharacteristic
359#388—21no—
360#388—1061no—
361#387—1061no—
362#386—21no—
363#385—21no—
364#384—531no—
365#380—531no—
366#380—531no—
367C4×C4⋊C4—531no—
368C4×C4⋊C4C2×D5311yescharacteristic
369C4×C22⋊C4C2×D5311yescharacteristic
370#340—531no—
371#340—531no—
372#340—531no—
373#340—531no—
374#340—531no—
375#340—531no—
376#340—531no—
377#340—531no—
378#336—531no—
379#336—531no—
380C53#2703211yescharacteristic, complemented, Sylow
381C22×D4—21no—
382C22×D4C22×D5311yescharacteristic
383C23×C4—21no—
384C23×C4—21no—
385C23×C4C22×D5311yescharacteristic
386C22×C8—531no—
387C22×C8—531no—
388C22×C8—531no—
389C22×C8—531no—
390#121—22no—
391#121—22no—
392C4×D4—22no—
393C4×D4—41no—
394C4×D4—44no—
395C2×C4⋊C4—22no—
396C2×C4⋊C4—21no—
397C2×C4⋊C4—21no—
398C2×C4⋊C4—41no—
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×C22⋊C4—21no—
411C2×C22⋊C4—21no—
412C2×C22⋊C4—21no—
413C2×C22⋊C4—21no—
414C2×C22⋊C4—1061no—
415C2×C22⋊C4—1061no—
416C2×C22⋊C4C22⋊D5311yescharacteristic
417C2×C22⋊C4C22⋊D5311yescharacteristic
418C2×C42—21no—
419C2×C42—1061no—
420C2×C42C22×D5311yescharacteristic
421C2×C42C22×D5311yescharacteristic
422#106—1062no—
423#106—1062no—
424#106—1062no—
425#106—2121no—
426C4⋊C8—1062no—
427#102—1062no—
428#102—1062no—
429#102—1062no—
430#102—2121no—
431C22⋊C8—1062no—
432C22⋊C8—2121no—
433C4×C8—2121no—
434#95—21no—
435#95—21no—
436#95—1061no—
437#95—1061no—
438#95—1061no—
439#95—1061no—
440#95—1061no—
441#95C22⋊D5311yescharacteristic
442#95C22⋊D5311yescharacteristic
443C24—21no—
444C24C2×C22⋊D5311yescharacteristic
445C2×D4—24no—
446C2×D4—21no—
447C2×D4—21no—
448C2×D4—21no—
449C2×D4—21no—
450C2×D4—21no—
451C2×D4—42no—
452C2×D4—42no—
453C2×D4#1171994512yes—
454C22×C4—21no—
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—42no—
464C22×C4—42no—
465C22×C4—42no—
466C22×C4—42no—
467C22×C4—81no—
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×C4C23×D5311yescharacteristic
479C2×C8—532no—
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—21no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—42no—
496C4⋊C4—42no—
497C4⋊C4—1061no—
498C4⋊C4—1061no—
499C4⋊C4—1061no—
500C4⋊C4—1061no—
501C4⋊C4—1061no—
502C4⋊C4—1061no—
503C4⋊C4—1061no—
504C4⋊C4—1061no—
505C4⋊C4—4241no—
506C4⋊C4—4241no—
507C4⋊C4—4241no—
508C4⋊C4—2122no—
509C4⋊C4#1171994512yes—
510C4⋊C4#1171994712yes—
511C22⋊C4—24no—
512C22⋊C4—41no—
513C22⋊C4—42no—
514C22⋊C4—42no—
515C22⋊C4—81no—
516C22⋊C4—4241no—
517C22⋊C4—4241no—
518C22⋊C4#1171992314yes—
519C42—21no—
520C42—21no—
521C42—21no—
522C42—21no—
523C42—42no—
524C42—4241no—
525C23—21no—
526C23—21no—
527C23—21no—
528C23—21no—
529C23—21no—
530C23—21no—
531C23—22no—
532C23—41no—
533C23—41no—
534C23—41no—
535C23—41no—
536C23—41no—
537C23—41no—
538C23C2×#1171992312yes—
539C23C22×C22⋊D5311yescharacteristic
540D4—22no—
541D4—22no—
542D4—48no—
543D4—41no—
544D4—41no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—22no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—22no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
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—84no—
572C2×C4—84no—
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×C4—2121no—
584C2×C4—2121no—
585C2×C4C2×#1171992312yes—
586C2×C4C2×#1171992312yes—
587C2×C4#41×D5311yescharacteristic
588C2×C4C2×#1171994711yescharacteristic
589C2×C4C2×#1171994511yescharacteristic
590C2×C4C2×#1171994511yescharacteristic
591C2×C4#1313326211yescharacteristic
592C2×C4#1313326411yescharacteristic
593C8—1062no—
594C8—2121no—
595C8—2121no—
596C8—2121no—
597C22—22no—
598C22—22no—
599C22—22no—
600C22—41no—
601C22—41no—
602C22—41no—
603C22—41no—
604C22—42no—
605C22—42no—
606C22—42no—
607C22—81no—
608C22—81no—
609C22—81no—
610C22—81no—
611C22#1681113411yescharacteristic
612C22#1681126711yescharacteristic
613C22#1681127211yescharacteristic
614C22C22×#1171992311yescharacteristic
615C22#1681142111yescharacteristic
616C22#1681142611yescharacteristic
617C22#1681143111yescharacteristic
618C4—22no—
619C4—22no—
620C4—42no—
621C4—42no—
622C4—41no—
623C4—41no—
624C4—41no—
625C4—81no—
626C4—4241no—
627C4—4241no—
628C4—4241no—
629C4—4241no—
630C4#1681115311yescharacteristic
631C4#1681121511yescharacteristic
632C4#1681132311yescharacteristic
633C4#1681134311yescharacteristic
634C2—42no—
635C2—82no—
636C2#2027874511yescharacteristic
637C2#2028472311yescharacteristic
638C2#2028491411yescharacteristic
639C2#2028653911yescharacteristic
640C2#2028690111yescharacteristic
641C2#2028707211yescharacteristic
642C2#2028731511yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2361C2
socleC23×C53C22×D4
prime radicalC2×#121×C53C22
hypercenter#2361D53
hypocenterC53#27032
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2361

Complements

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

NH ≅ G/Ntype
1. #20279148C2semidirect
2. #20276118C2semidirect
3. #20276089C2semidirect
6. #20275478C2semidirect
7. #20275440C2semidirect
8. #20275323C2semidirect
9. #20275300C2semidirect
10. #20275290C2semidirect
11. #20274976C2semidirect
12. #20274908C2semidirect
13. #20274801C2semidirect
14. C4×#13133138C2semidirect
380. C53#27032semidirect

Maximal subgroups

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

subgroupindex
#202791482
#202761182
#202760892
#202759012
#202754782
#202754402
#202753232
#202753002
#202752902
#202749762
#202749082
#202748012
C4×#131331382
C53×#23612
#2703253