Finite Group Databasebeta

← #27487113

Subgroups — #27487113

Order 13568.

800 conjugacy classes of subgroups.

The subgroup lattice has 642 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20277179C211yesmaximal, characteristic, complemented
2#20276206C211yesmaximal, characteristic, complemented
3#20276103C211yesmaximal, characteristic, complemented
4#20276100C211yesmaximal, characteristic, complemented
5#20276025C211yesmaximal, characteristic
6#20275580C211yesmaximal, characteristic
7#20275445C211yesmaximal, characteristic, complemented
8#20275401C211yesmaximal, characteristic, complemented
9#20275397C211yesmaximal, characteristic, complemented
10#20275332C211yesmaximal, characteristic, complemented
11#20275283C211yesmaximal, characteristic, complemented
12#20275066C211yesmaximal, characteristic
13#20274961C211yesmaximal, characteristic, complemented
14C4×C4⋊#88322C211yesmaximal, characteristic, complemented
15C53×#2860C211yesmaximal, characteristic, complemented
16C22×#11719922C2211yescharacteristic
17C2×C4×D212C2211yescharacteristic
18C2×#13133174—21no—
19C2×#13133150C2211yescharacteristic
20C2×#13133150C2211yescharacteristic
21C2×#13133150C2211yescharacteristic
22C2×C4⋊#88322C2211yescharacteristic
23C2×C4⋊#88322C2211yescharacteristic
24C2×#13133138C2211yescharacteristic
25C2×#13133138C2211yescharacteristic
26C2×#13133138C2211yescharacteristic
27C2×#13133137C2211yescharacteristic
28C2×#13133137C2211yescharacteristic
29C2×#13133137C2211yescharacteristic
30C2×C4×#88322C2211yescharacteristic
31C2×C4×#88322C2211yescharacteristic
32#16810525C2211yescharacteristic
33#16810525C2211yescharacteristic
34#16810521C2211yescharacteristic
35#16810521C2211yescharacteristic
36#16810514—21no—
37#16810514—21no—
38C4×C4⋊Dic53C2211yescharacteristic
39C4×#11719921—21no—
40#16810385C2211yescharacteristic
41#16810385C2211yescharacteristic
42#16810385C2211yescharacteristic
43#16810385C2211yescharacteristic
44#16810385C2211yescharacteristic
45#16810385C2211yescharacteristic
46#16810378C2211yescharacteristic
47#16810378C2211yescharacteristic
48C2×#126×C53—21no—
49C53×#390C2211yescharacteristic
50C53×#390C2211yescharacteristic
51C53×#390C2211yescharacteristic
52C53×#389C2211yescharacteristic
53C53×#389C2211yescharacteristic
54C53×#389C2211yescharacteristic
55C53×#388—21no—
56C53×#388—21no—
57C53×#388—21no—
58C43×C53C2211yescharacteristic
59C22×D212C2311yescharacteristic
60C22×C4×D53—21no—
61C2×#11719922—22no—
62C2×#11719922—22no—
63C2×#11719922—41no—
64C22×#88322C2311yescharacteristic
65C22×#88322C2311yescharacteristic
66C22×#88322C2311yescharacteristic
67C22×#88322C2311yescharacteristic
68C4×D212—22no—
69C4×D212—22no—
70#13133174—44no—
71C2×C4⋊Dic53C2311yescharacteristic
72C2×C4⋊Dic53C2311yescharacteristic
73C2×C4⋊Dic53C2311yescharacteristic
74C2×#11719919—21no—
75C2×#11719919—21no—
76C2×#11719921—21no—
77C2×#11719921—21no—
78C2×#11719921—21no—
79C2×#11719921—21no—
80C2×#11719921—21no—
81C2×C4×Dic53—21no—
82#13133150—22no—
83#13133150—22no—
84#13133150—41no—
85C4⋊#88322—22no—
86C4⋊#88322—22no—
87#13133138—22no—
88#13133138—22no—
89#13133138—41no—
90#13133137—22no—
91#13133137—22no—
92#13133137—41no—
93C4×#88322—41no—
94C4×#88322—41no—
95#13133124—21no—
96#13133124—21no—
97#13133124—21no—
98C22×D4×C53C2311yescharacteristic
99#126×C53—44no—
100C2×C4⋊C4×C53—21no—
101C2×C4⋊C4×C53—21no—
102C2×C4⋊C4×C53—21no—
103C2×C4⋊C4×C53C2311yescharacteristic
104C2×C4⋊C4×C53C2311yescharacteristic
105C2×C4⋊C4×C53C2311yescharacteristic
106C2×C22⋊C4×C53—21no—
107C2×C22⋊C4×C53—21no—
108C2×C22⋊C4×C53—21no—
109C2×C22⋊C4×C53—21no—
110C2×C22⋊C4×C53—21no—
111C2×C22⋊C4×C53—21no—
112C2×C42×C53D411yescharacteristic
113C2×C42×C53D411yescharacteristic
114C2×C42×C53D411yescharacteristic
115C2×C42×C53D411yescharacteristic
116C2×C42×C53C2311yescharacteristic
117C2×C42×C53C2311yescharacteristic
118C2×C42×C53C2311yescharacteristic
119#95×C53—21no—
120#95×C53—21no—
121#95×C53—21no—
122C24×C53—21no—
123C23×D53—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—
130C22×C4×C53—21no—
131C22×C4×C53—21no—
132C22×C4×C53—21no—
133C22×C4×C53C2×D411yescharacteristic
134C22×C4×C53C2×D411yescharacteristic
135C22×C4×C53C2×D411yescharacteristic
136C22×C4×C53C2×D411yescharacteristic
137C22×C4×C53C2×D411yescharacteristic
138C22×C4×C53C2×D411yescharacteristic
139C22×C4×C53C2411yescharacteristic
140C22×Dic53—21no—
141C22×Dic53—21no—
142C22×Dic53—21no—
143C2×D212—21no—
144C2×D212—21no—
145C2×D212—21no—
146C2×D212—21no—
147C2×D212—21no—
148C2×D212#4112yes—
149C2×C4×D53—24no—
150C2×C4×D53—42no—
151C2×C4×D53—42no—
152C4⋊C4×C53—21no—
153C4⋊C4×C53—21no—
154C4⋊C4×C53—21no—
155C4⋊C4×C53—21no—
156C4⋊C4×C53—21no—
157C4⋊C4×C53—21no—
158C4⋊C4×C53—42no—
159C4⋊C4×C53—42no—
160C4⋊C4×C53—42no—
161C22⋊C4×C53—42no—
162C22⋊C4×C53—42no—
163C22⋊C4×C53—42no—
164C22⋊C4×C53—42no—
165C22⋊C4×C53—42no—
166C22⋊C4×C53—42no—
167C42×C53—22no—
168C42×C53—21no—
169C42×C53—21no—
170C42×C53—21no—
171C42×C53—21no—
172C42×C53—21no—
173C42×C53—21no—
174C42×C53—41no—
175C42×C53—41no—
176C42×C53D814yescomplemented
177#11719922—44no—
178#11719921—24no—
179#11719921—42no—
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—
190#11719919—42no—
191C4×Dic53—42no—
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—
203C2×#88322#4112yes—
204C23×C53—41no—
205C23×C53—41no—
206C23×C53—41no—
207C23×C53—41no—
208C23×C53—41no—
209C23×C53—41no—
210C23×C53—41no—
211C23×C53C22×D411yescharacteristic
212C22×D53—22no—
213C22×D53—41no—
214C22×D53—41no—
215C22×D53—41no—
216C22×D53—41no—
217C22×D53—41no—
218C22×D53—41no—
219D4×C53—41no—
220D4×C53—41no—
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—21no—
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—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×C53C2×D812yes—
247C2×C4×C53C2×D812yes—
248C2×C4×C53C2×D812yes—
249C2×C4×C53#13512yes—
250C2×C4×C53C2×#4111yescharacteristic
251C2×C4×C53C2×#4111yescharacteristic
252C2×C4×C53C2×#4111yescharacteristic
253C2×C4×C53C2×#4111yescharacteristic
254C2×C4×C53H(5,𝔽2)11yescharacteristic
255C2×C4×C532−1+411yescharacteristic
256C2×Dic53—22no—
257C2×Dic53—41no—
258C2×Dic53—41no—
259C2×Dic53—41no—
260C2×Dic53—41no—
261C2×Dic53—41no—
262C2×Dic53—41no—
263C2×Dic53—41no—
264C2×Dic53—41no—
265D212—22no—
266D212—22no—
267D212—41no—
268D212—41no—
269C4×D53—48no—
270#88322—22no—
271#88322—22no—
272#88322—41no—
273#88322—41no—
274#29428—531nomaximal, Sylow
275C22×C53—81no—
276C22×C53—81no—
277C22×C53—81no—
278C22×C53—81no—
279C22×C53—81no—
280C22×C53—81no—
281C22×C53—81no—
282C22×C53#52511yescharacteristic
283C22×C53#53211yescharacteristic
284C22×C53#53711yescharacteristic
285C22×C53C22×D811yescharacteristic
286C22×C53C2×#13511yescharacteristic
287C22×C53#57511yescharacteristic
288C22×C53#57511yescharacteristic
289C2×D53—42no—
290C2×D53—42no—
291C2×D53—42no—
292C2×D53—81no—
293C2×D53—81no—
294C2×D53—81no—
295C2×D53—81no—
296C4×C53—22no—
297C4×C53—22no—
298C4×C53—22no—
299C4×C53—22no—
300C4×C53—41no—
301C4×C53—41no—
302C4×C53—81no—
303C4×C53—81no—
304C4×C53—81no—
305C4×C53#55111yescharacteristic
306C4×C53#55211yescharacteristic
307C4×C53#55311yescharacteristic
308C4×C53#55611yescharacteristic
309Dic53—42no—
310Dic53—81no—
311Dic53—81no—
312#2860D5311yescharacteristic, complemented
313#2293—531no—
314#2086—531no—
315#2067—531no—
316#2067—531no—
317#2054—531no—
318#1970—531no—
319#1942—531no—
320#1934—531no—
321#1934—531no—
322#1925—531no—
323#1918—531no—
324#1879—531no—
325#1862—531no—
326C4×C4⋊C8—531no—
327C2×C53—161no—
328C2×C53#260111yescharacteristic
329C2×C53#310011yescharacteristic
330C2×C53#310211yescharacteristic
331C2×C53#316911yescharacteristic
332C2×C53#318311yescharacteristic
333C2×C53#324511yescharacteristic
334C2×C53#325011yescharacteristic
335D53—82no—
336C22×D8—531no—
337C2×#126—21no—
338C2×C4×D4—531no—
339C2×#117—1061no—
340C2×#107—531no—
341C2×#107—531no—
342C2×#107—531no—
343C2×C4⋊C8—531no—
344C2×C4⋊C8—531no—
345C2×#102—531no—
346C2×#102—531no—
347C2×#102—531no—
348C2×#102—531no—
349C2×#102—531no—
350C2×#102—531no—
351C2×C4×C8—531no—
352C2×C4×C8—531no—
353#390—531no—
354#390—531no—
355#390C2×D5311yescharacteristic
356#390C2×D5311yescharacteristic
357#390C2×D5311yescharacteristic
358#389—531no—
359#389—531no—
360#389C2×D5311yescharacteristic
361#389C2×D5311yescharacteristic
362#389C2×D5311yescharacteristic
363#388—21no—
364#388—21no—
365#388—21no—
366#388—1061no—
367#388—1061no—
368C4×C4⋊C4—531no—
369C4×C22⋊C4—1061no—
370C43C2×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—
379C53#2942811yescharacteristic, complemented, Sylow
380C22×D4—531no—
381C22×D4C22×D5311yescharacteristic
382C23×C4—1061no—
383C2×D8—1062no—
384C2×D8—1062no—
385C2×D8—2121no—
386C22×C8—531no—
387C22×C8—531no—
388C22×C8—531no—
389C22×C8—531no—
390#126—44no—
391C4×D4—1062no—
392C4×D4—1062no—
393#117—2124no—
394C2×C4⋊C4—21no—
395C2×C4⋊C4—21no—
396C2×C4⋊C4—21no—
397C2×C4⋊C4—531no—
398C2×C4⋊C4—531no—
399C2×C4⋊C4—531no—
400C2×C4⋊C4—1061no—
401C2×C4⋊C4—1061no—
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×C22⋊C4—1061no—
412C2×C22⋊C4—1061no—
413C2×C22⋊C4—1061no—
414C2×C22⋊C4—1061no—
415C2×C22⋊C4—1061no—
416C2×C42—1061no—
417C2×C42C22⋊D5311yescharacteristic
418C2×C42C22⋊D5311yescharacteristic
419C2×C42C22⋊D5311yescharacteristic
420C2×C42C22⋊D5311yescharacteristic
421C2×C42C22×D5311yescharacteristic
422C2×C42C22×D5311yescharacteristic
423C2×C42C22×D5311yescharacteristic
424#107—1062no—
425#107—1062no—
426#107—2121no—
427C4⋊C8—1062no—
428C4⋊C8—1062no—
429#102—1062no—
430#102—1062no—
431#102—1062no—
432#102—1062no—
433#102—2121no—
434#102—2121no—
435C4×C8—2121no—
436C4×C8—2121no—
437#95—21no—
438#95—21no—
439#95—21no—
440#95—1061no—
441#95—1061no—
442#95—1061no—
443C24—21no—
444C24—1061no—
445C2×D4—21no—
446C2×D4—21no—
447C2×D4—21no—
448C2×D4—21no—
449C2×D4—21no—
450C2×D4—21no—
451C2×D4—532no—
452C2×D4—1061no—
453C2×D4—1061no—
454C2×D4—1061no—
455C2×D4—1061no—
456C2×D4—1061no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—1064no—
461C22×C4—1061no—
462C22×C4—1061no—
463C22×C4—1061no—
464C22×C4—2122no—
465C22×C4—2122no—
466C22×C4C2×C22⋊D5311yescharacteristic
467C22×C4C2×C22⋊D5311yescharacteristic
468C22×C4C2×C22⋊D5311yescharacteristic
469C22×C4C2×C22⋊D5311yescharacteristic
470C22×C4C2×C22⋊D5311yescharacteristic
471C22×C4C2×C22⋊D5311yescharacteristic
472C22×C4C23×D5311yescharacteristic
473D8—2124no—
474C2×C8—532no—
475C2×C8—532no—
476C2×C8—1061no—
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—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—42no—
493C4⋊C4—42no—
494C4⋊C4—42no—
495C4⋊C4—532no—
496C4⋊C4—1061no—
497C4⋊C4—1061no—
498C4⋊C4—1061no—
499C4⋊C4—1061no—
500C4⋊C4—1061no—
501C4⋊C4—2122no—
502C4⋊C4—2122no—
503C22⋊C4—42no—
504C22⋊C4—42no—
505C22⋊C4—42no—
506C22⋊C4—42no—
507C22⋊C4—42no—
508C22⋊C4—42no—
509C22⋊C4—1064no—
510C22⋊C4—2122no—
511C22⋊C4—2122no—
512C22⋊C4—2122no—
513C22⋊C4—2122no—
514C42—22no—
515C42—21no—
516C42—21no—
517C42—21no—
518C42—21no—
519C42—21no—
520C42—21no—
521C42—41no—
522C42—41no—
523C42—2122no—
524C42#1171992214yescomplemented
525C23—41no—
526C23—41no—
527C23—41no—
528C23—41no—
529C23—41no—
530C23—41no—
531C23—41no—
532C23—1062no—
533C23—2121no—
534C23—2121no—
535C23—2121no—
536C23—2121no—
537C23—2121no—
538C23—2121no—
539C23C22×C22⋊D5311yescharacteristic
540D4—41no—
541D4—41no—
542D4—41no—
543D4—41no—
544D4—1062no—
545D4—1062no—
546D4—2121no—
547D4—2121no—
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—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
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—1062no—
572C2×C4—2128no—
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×C4C2×#1171992212yes—
582C2×C4C2×#1171992212yes—
583C2×C4C2×#1171992212yes—
584C2×C4#1313323712yes—
585C2×C4#41×D5311yescharacteristic
586C2×C4#41×D5311yescharacteristic
587C2×C4#41×D5311yescharacteristic
588C2×C4#41×D5311yescharacteristic
589C2×C4#1313326311yescharacteristic
590C2×C4#1313326411yescharacteristic
591C8—1062no—
592C8—1062no—
593C8—2121no—
594C8—2121no—
595C22—81no—
596C22—81no—
597C22—81no—
598C22—81no—
599C22—81no—
600C22—81no—
601C22—81no—
602C22—4241no—
603C22—4241no—
604C22—4241no—
605C22—4241no—
606C22—2122no—
607C22—2122no—
608C22—2122no—
609C22#1681113011yescharacteristic
610C22#1681116911yescharacteristic
611C22#1681120111yescharacteristic
612C22C22×#1171992211yescharacteristic
613C22C2×#1313323711yescharacteristic
614C22#1681142111yescharacteristic
615C22#1681142111yescharacteristic
616C4—22no—
617C4—22no—
618C4—22no—
619C4—22no—
620C4—41no—
621C4—41no—
622C4—81no—
623C4—81no—
624C4—81no—
625C4—4241no—
626C4—4241no—
627C4—2122no—
628C4#1681129811yescharacteristic
629C4#1681130911yescharacteristic
630C4#1681132411yescharacteristic
631C4#1681134011yescharacteristic
632C2—161no—
633C2—4242no—
634C2#2028031411yescharacteristic
635C2#2028471111yescharacteristic
636C2#2028472711yescharacteristic
637C2#2028529011yescharacteristic
638C2#2028543811yescharacteristic
639C2#2028602211yescharacteristic
640C2#2028606211yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2860C2
socleC23×C53C22×D4
prime radicalC22×#11719922C22
hypercenter#2860D53
hypocenterC53#29428
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C43 ◁ #2860

Complements

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

NH ≅ G/Ntype
1. #20277179C2semidirect
2. #20276206C2semidirect
3. #20276103C2semidirect
4. #20276100C2semidirect
7. #20275445C2semidirect
8. #20275401C2semidirect
9. #20275397C2semidirect
10. #20275332C2semidirect
11. #20275283C2semidirect
13. #20274961C2semidirect
14. C4×C4⋊#88322C2semidirect
15. C53×#2860C2semidirect
176. C42×C53D8semidirect
312. #2860D53semidirect
379. C53#29428semidirect
524. C42#11719922semidirect

Maximal subgroups

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

subgroupindex
#202771792
#202762062
#202761032
#202761002
#202760252
#202755802
#202754452
#202754012
#202753972
#202753322
#202752832
#202750662
#202749612
C4×C4⋊#883222
C53×#28602
#2942853