Finite Group Databasebeta

← #27465230

Subgroups — #27465230

Order 13568.

1052 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20277098C211yesmaximal, characteristic
2#20276152C212yesmaximal, complemented
3#20276146C212yesmaximal, complemented
4#20274912C212yesmaximal, complemented
5#20274908C211yesmaximal, characteristic, complemented
6#20274903C211yesmaximal, characteristic, complemented
7#20274807C212yesmaximal, complemented
8#20274801C211yesmaximal, characteristic, complemented
9#20274801C211yesmaximal, characteristic, complemented
10#20274625C211yesmaximal, characteristic
11C2×C53×#380C211yesmaximal, characteristic
12C2×#13133176C2211yescharacteristic
13C2×#13133176C2211yescharacteristic
14C2×#13133150C2212yescomplemented
15C2×#13133148C2212yescomplemented
16C2×C22⋊#88322C2211yescharacteristic
17C2×C22⋊#88322C2211yescharacteristic
18C2×C22⋊#88322C2212yes—
19C2×C22⋊#88322C2212yes—
20#16810486—21no—
21#16810483C2211yescharacteristic
22#16810483C2211yescharacteristic
23#16810483C2211yescharacteristic
24#16810483C2211yescharacteristic
25#16810479—21no—
26C4×C22⋊Dic53—21no—
27C4×C22⋊Dic53—21no—
28#16810385C2212yescomplemented
29#16810385C2212yescomplemented
30#16810385C2212yescomplemented
31#16810385C2212yescomplemented
32#16810385C2212yescomplemented
33#16810385C2212yescomplemented
34C24×C4×C53C2211yescharacteristic
35C22×C4⋊C4×C53C2211yescharacteristic
36C22×C4⋊C4×C53C2211yescharacteristic
37C22×C22⋊C4×C53—21no—
38C53×#380—21no—
39C53×#380—21no—
40C53×#380C2212yes—
41C53×#380C2212yes—
42C2×#95×C53—21no—
43C22×#88322C2312yes—
44C22×#88322C2312yes—
45#13133176—24no—
46#13133176—42no—
47C2×C4⋊Dic53C2311yescharacteristic
48C2×C4⋊Dic53C2311yescharacteristic
49C2×C4⋊Dic53C2311yescharacteristic
50C2×C4⋊Dic53C2311yescharacteristic
51C2×C22⋊Dic53—21no—
52C2×C22⋊Dic53—21no—
53C2×C22⋊Dic53—21no—
54C2×C22⋊Dic53—21no—
55C2×C22⋊Dic53—21no—
56C2×C22⋊Dic53—21no—
57C2×C4×Dic53—21no—
58C2×C4×Dic53—21no—
59#13133150—42no—
60#13133148—42no—
61C22⋊#88322—24no—
62C22⋊#88322—22no—
63C22⋊#88322—22no—
64C22⋊#88322—42no—
65#13133124—21no—
66#13133124—21no—
67#13133124—21no—
68#13133124—21no—
69C25×C53D411yescharacteristic
70C23×C4×C53—24no—
71C23×C4×C53—22no—
72C23×C4×C53—22no—
73C23×C4×C53—21no—
74C23×C4×C53—21no—
75C23×C4×C53—42no—
76C23×C4×C53D411yescharacteristic
77C23×C4×C53D411yescharacteristic
78C23×C4×C53D411yescharacteristic
79C23×C4×C53C2311yescharacteristic
80C23×C4×C53C2311yescharacteristic
81C23×C4×C53C2311yescharacteristic
82C2×C4⋊C4×C53—21no—
83C2×C4⋊C4×C53—21no—
84C2×C4⋊C4×C53—22no—
85C2×C4⋊C4×C53—22no—
86C2×C4⋊C4×C53—42no—
87C2×C4⋊C4×C53C2312yes—
88C2×C4⋊C4×C53C2312yes—
89C2×C22⋊C4×C53—22no—
90C2×C22⋊C4×C53—22no—
91C2×C22⋊C4×C53—44no—
92#95×C53—22no—
93#95×C53—22no—
94C24×C53—22no—
95C24×C53—22no—
96C24×C53—22no—
97C24×C53—22no—
98C24×C53—22no—
99C24×C53—41no—
100C24×C53—41no—
101C24×C53C2×D411yescharacteristic
102C24×C53C2×D411yescharacteristic
103C24×C53C2×D411yescharacteristic
104C22×C4×C53—21no—
105C22×C4×C53—21no—
106C22×C4×C53—22no—
107C22×C4×C53—22no—
108C22×C4×C53—22no—
109C22×C4×C53—22no—
110C22×C4×C53—22no—
111C22×C4×C53—22no—
112C22×C4×C53—22no—
113C22×C4×C53—22no—
114C22×C4×C53—22no—
115C22×C4×C53—21no—
116C22×C4×C53—21no—
117C22×C4×C53—22no—
118C22×C4×C53—22no—
119C22×C4×C53—22no—
120C22×C4×C53—22no—
121C22×C4×C53—41no—
122C22×C4×C53—41no—
123C22×C4×C53—41no—
124C22×C4×C53—48no—
125C22×C4×C53—48no—
126C22×C4×C53—42no—
127C22×C4×C53—42no—
128C22×C4×C53—42no—
129C22×C4×C53—42no—
130C22×C4×C53—42no—
131C22×C4×C53—42no—
132C22×C4×C53C2×D411yescharacteristic
133C22×C4×C53C2×D411yescharacteristic
134C22×C4×C53C2×D411yescharacteristic
135C22×C4×C53#4114yes—
136C22×C4×C53#4112yes—
137C22×C4×C53#4112yes—
138C22×C4×C53C2411yescharacteristic
139C22×Dic53—21no—
140C22×Dic53—21no—
141C22×Dic53—21no—
142C22×Dic53—21no—
143C4⋊C4×C53—22no—
144C4⋊C4×C53—22no—
145C4⋊C4×C53—22no—
146C4⋊C4×C53—22no—
147C4⋊C4×C53—42no—
148C4⋊C4×C53—42no—
149C22⋊C4×C53—44no—
150C22⋊C4×C53—44no—
151C22⋊Dic53—42no—
152C22⋊Dic53—42no—
153C22⋊Dic53—42no—
154C22⋊Dic53—42no—
155C22⋊Dic53—42no—
156C22⋊Dic53—42no—
157C4⋊Dic53—22no—
158C4⋊Dic53—22no—
159C4⋊Dic53—22no—
160C4⋊Dic53—22no—
161C4×Dic53—42no—
162C4×Dic53—42no—
163C2×#88322—22no—
164C2×#88322—22no—
165C2×#88322—22no—
166C2×#88322—22no—
167C2×#88322—22no—
168C2×#88322—22no—
169C23×C53—21no—
170C23×C53—21no—
171C23×C53—21no—
172C23×C53—21no—
173C23×C53—28no—
174C23×C53—21no—
175C23×C53—21no—
176C23×C53—21no—
177C23×C53—21no—
178C23×C53—21no—
179C23×C53—21no—
180C23×C53—21no—
181C23×C53—21no—
182C23×C53—21no—
183C23×C53—21no—
184C23×C53—21no—
185C23×C53—21no—
186C23×C53—44no—
187C23×C53—44no—
188C23×C53—44no—
189C23×C53—44no—
190C23×C53—44no—
191C23×C53—44no—
192C23×C53#12312yes—
193C23×C53#12312yes—
194C23×C53#13512yes—
195C23×C53AGL(1,ℤ/8)12yes—
196C23×C53#13712yes—
197C23×C53C22×D411yescharacteristic
198C2×C4×C53—21no—
199C2×C4×C53—21no—
200C2×C4×C53—21no—
201C2×C4×C53—21no—
202C2×C4×C53—24no—
203C2×C4×C53—24no—
204C2×C4×C53—24no—
205C2×C4×C53—24no—
206C2×C4×C53—21no—
207C2×C4×C53—21no—
208C2×C4×C53—21no—
209C2×C4×C53—44no—
210C2×C4×C53—44no—
211C2×C4×C53—44no—
212C2×C4×C53—42no—
213C2×C4×C53—42no—
214C2×C4×C53—42no—
215C2×C4×C53—42no—
216C2×C4×C53—42no—
217C2×C4×C53—42no—
218C2×C4×C53—42no—
219C2×C4×C53—42no—
220C2×C4×C53—84no—
221C2×C4×C53—84no—
222C2×C4×C53#12312yes—
223C2×C4×C53#12312yes—
224C2×C4×C53C2×#4111yescharacteristic
225C2×C4×C53C2×#4111yescharacteristic
226C2×C4×C53C2×#4112yes—
227C2×C4×C53H(5,𝔽2)12yes—
228C2×Dic53—41no—
229C2×Dic53—41no—
230C2×Dic53—41no—
231C2×Dic53—41no—
232C2×Dic53—41no—
233C2×Dic53—41no—
234C2×Dic53—41no—
235C2×Dic53—41no—
236C2×Dic53—41no—
237C2×Dic53—41no—
238C2×Dic53—41no—
239C2×Dic53—41no—
240#88322—42no—
241#88322—42no—
242#27602—531nomaximal, Sylow
243C22×C53—22no—
244C22×C53—22no—
245C22×C53—22no—
246C22×C53—22no—
247C22×C53—22no—
248C22×C53—22no—
249C22×C53—22no—
250C22×C53—22no—
251C22×C53—22no—
252C22×C53—22no—
253C22×C53—22no—
254C22×C53—22no—
255C22×C53—416no—
256C22×C53—41no—
257C22×C53—41no—
258C22×C53—41no—
259C22×C53—41no—
260C22×C53—41no—
261C22×C53—41no—
262C22×C53—41no—
263C22×C53—41no—
264C22×C53#48214yes—
265C22×C53C2×#12311yescharacteristic
266C22×C53C2×#12311yescharacteristic
267C22×C53#53411yescharacteristic
268C22×C53C2×#13511yescharacteristic
269C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
270C22×C53C2×#13711yescharacteristic
271C22×C53#57511yescharacteristic
272C4×C53—22no—
273C4×C53—22no—
274C4×C53—41no—
275C4×C53—82no—
276C4×C53—82no—
277C4×C53#55112yes—
278C4×C53#55112yes—
279Dic53—81no—
280Dic53—81no—
281Dic53—81no—
282Dic53—81no—
283#2280—531no—
284C2×#380D5311yescharacteristic
285#2077—532no—
286#2076—532no—
287#1854—532no—
288#1853—531no—
289#1852—531no—
290#1825—532no—
291#1823—531no—
292#1823—531no—
293#1779—531no—
294C2×C53—24no—
295C2×C53—42no—
296C2×C53—42no—
297C2×C53#248311yescharacteristic
298C2×C53C2×#48211yescharacteristic
299C2×C53#309111yescharacteristic
300C2×C53#309211yescharacteristic
301C2×C53#309511yescharacteristic
302C2×C53#319011yescharacteristic
303C2×C53#319411yescharacteristic
304C24×C4C2×D5311yescharacteristic
305C2×#117—531no—
306C2×#117—531no—
307C22×C4⋊C4C2×D5311yescharacteristic
308C22×C4⋊C4C2×D5311yescharacteristic
309C22×C22⋊C4—21no—
310C2×#107—532no—
311C2×#106—532no—
312C2×C22⋊C8—531no—
313C2×C22⋊C8—531no—
314C2×C22⋊C8—532no—
315C2×C22⋊C8—532no—
316#381—1061no—
317#380—21no—
318#380—21no—
319#380—531no—
320#380—531no—
321#380—531no—
322#380—531no—
323#380C2×D5312yes—
324#380C2×D5312yes—
325#379—1061no—
326C4×C22⋊C4—1061no—
327C4×C22⋊C4—1061no—
328C2×#95—21no—
329#340—532no—
330#340—532no—
331#340—532no—
332#340—532no—
333#340—532no—
334#340—532no—
335C53#2760211yescharacteristic, complemented, Sylow
336C25C22⋊D5311yescharacteristic
337C23×C4—24no—
338C23×C4—22no—
339C23×C4—22no—
340C23×C4—21no—
341C23×C4—21no—
342C23×C4—42no—
343C23×C4C22⋊D5311yescharacteristic
344C23×C4C22⋊D5311yescharacteristic
345C23×C4C22⋊D5311yescharacteristic
346C23×C4C22×D5311yescharacteristic
347C23×C4C22×D5311yescharacteristic
348C23×C4C22×D5311yescharacteristic
349C22×C8—532no—
350C22×C8—532no—
351#117—1064no—
352#117—2122no—
353C2×C4⋊C4—22no—
354C2×C4⋊C4—22no—
355C2×C4⋊C4—21no—
356C2×C4⋊C4—21no—
357C2×C4⋊C4—42no—
358C2×C4⋊C4—531no—
359C2×C4⋊C4—531no—
360C2×C4⋊C4—531no—
361C2×C4⋊C4—531no—
362C2×C4⋊C4C22×D5312yes—
363C2×C4⋊C4C22×D5312yes—
364C2×C22⋊C4—22no—
365C2×C22⋊C4—22no—
366C2×C22⋊C4—44no—
367C2×C22⋊C4—1061no—
368C2×C22⋊C4—1061no—
369C2×C22⋊C4—1061no—
370C2×C22⋊C4—1061no—
371C2×C22⋊C4—1061no—
372C2×C22⋊C4—1061no—
373C2×C42—1061no—
374C2×C42—1061no—
375#107—2122no—
376#106—2122no—
377C22⋊C8—1064no—
378C22⋊C8—1062no—
379C22⋊C8—1062no—
380C22⋊C8—2122no—
381#95—22no—
382#95—22no—
383#95—1061no—
384#95—1061no—
385#95—1061no—
386#95—1061no—
387C24—22no—
388C24—22no—
389C24—22no—
390C24—22no—
391C24—22no—
392C24—41no—
393C24—41no—
394C24C2×C22⋊D5311yescharacteristic
395C24C2×C22⋊D5311yescharacteristic
396C24C2×C22⋊D5311yescharacteristic
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—41no—
415C22×C4—41no—
416C22×C4—41no—
417C22×C4—48no—
418C22×C4—48no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—42no—
425C22×C4—1061no—
426C22×C4—1061no—
427C22×C4—1061no—
428C22×C4—1061no—
429C22×C4#1171994514yes—
430C22×C4#1171994712yes—
431C22×C4#1171994712yes—
432C22×C4C2×C22⋊D5311yescharacteristic
433C22×C4C2×C22⋊D5311yescharacteristic
434C22×C4C2×C22⋊D5311yescharacteristic
435C22×C4C23×D5311yescharacteristic
436C2×C8—1062no—
437C2×C8—1062no—
438C2×C8—1062no—
439C2×C8—1062no—
440C2×C8—1062no—
441C2×C8—1062no—
442C4⋊C4—22no—
443C4⋊C4—22no—
444C4⋊C4—22no—
445C4⋊C4—22no—
446C4⋊C4—42no—
447C4⋊C4—42no—
448C4⋊C4—1062no—
449C4⋊C4—1062no—
450C4⋊C4—1062no—
451C4⋊C4—1062no—
452C22⋊C4—44no—
453C22⋊C4—44no—
454C22⋊C4—2122no—
455C22⋊C4—2122no—
456C22⋊C4—2122no—
457C22⋊C4—2122no—
458C22⋊C4—2122no—
459C22⋊C4—2122no—
460C42—2122no—
461C42—2122no—
462C23—21no—
463C23—21no—
464C23—21no—
465C23—21no—
466C23—28no—
467C23—21no—
468C23—21no—
469C23—21no—
470C23—21no—
471C23—21no—
472C23—21no—
473C23—21no—
474C23—21no—
475C23—21no—
476C23—21no—
477C23—21no—
478C23—21no—
479C23—44no—
480C23—44no—
481C23—44no—
482C23—44no—
483C23—44no—
484C23—44no—
485C23#1313319812yes—
486C23#1313320012yes—
487C23#1313323712yes—
488C23#1313324312yes—
489C23#1313324812yes—
490C23C22×C22⋊D5311yescharacteristic
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—24no—
496C2×C4—24no—
497C2×C4—24no—
498C2×C4—24no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—44no—
503C2×C4—44no—
504C2×C4—44no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—84no—
514C2×C4—84no—
515C2×C4—2121no—
516C2×C4—2121no—
517C2×C4—2121no—
518C2×C4—2121no—
519C2×C4—2121no—
520C2×C4—2121no—
521C2×C4—2121no—
522C2×C4—2121no—
523C2×C4—2121no—
524C2×C4—2121no—
525C2×C4—2121no—
526C2×C4—2121no—
527C2×C4#1313319812yes—
528C2×C4#1313320012yes—
529C2×C4C2×#1171994711yescharacteristic
530C2×C4C2×#1171994711yescharacteristic
531C2×C4C2×#1171994512yes—
532C2×C4#1313326212yes—
533C8—2122no—
534C8—2122no—
535C22—22no—
536C22—22no—
537C22—22no—
538C22—22no—
539C22—22no—
540C22—22no—
541C22—22no—
542C22—22no—
543C22—22no—
544C22—22no—
545C22—22no—
546C22—22no—
547C22—416no—
548C22—41no—
549C22—41no—
550C22—41no—
551C22—41no—
552C22—41no—
553C22—41no—
554C22—41no—
555C22—41no—
556C22#1681094414yes—
557C22C2×#1313319811yescharacteristic
558C22C2×#1313320011yescharacteristic
559C22#1681118211yescharacteristic
560C22C2×#1313323711yescharacteristic
561C22C2×#1313324311yescharacteristic
562C22C2×#1313324811yescharacteristic
563C22#1681142111yescharacteristic
564C4—22no—
565C4—22no—
566C4—41no—
567C4—82no—
568C4—82no—
569C4—4241no—
570C4—4241no—
571C4—4241no—
572C4—4241no—
573C4#1681129912yes—
574C4#1681129912yes—
575C2—24no—
576C2—42no—
577C2—42no—
578C2#2027899311yescharacteristic
579C2C2×#1681094411yescharacteristic
580C2#2028462811yescharacteristic
581C2#2028463811yescharacteristic
582C2#2028465911yescharacteristic
583C2#2028549611yescharacteristic
584C2#2028552911yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC2×C53×#380C2
socleC23×C53C22×D4
prime radicalC25×C53D4
hypercenterC2×#380D53
hypocenterC53#27602
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C24×C4 ◁ C2×#380

Complements

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

NH ≅ G/Ntype
2. #20276152C2semidirect
3. #20276146C2semidirect
4. #20274912C2semidirect
5. #20274908C2semidirect
6. #20274903C2semidirect
7. #20274807C2semidirect
8. #20274801C2semidirect
9. #20274801C2semidirect
14. C2×#13133150C22semidirect
15. C2×#13133148C22semidirect
28. #16810385C22semidirect
29. #16810385C22semidirect
30. #16810385C22semidirect
31. #16810385C22semidirect
32. #16810385C22semidirect
33. #16810385C22semidirect
335. C53#27602semidirect

Maximal subgroups

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

subgroupindex
#202770982
#202761522
#202761462
#202749122
#202749082
#202749032
#202748072
#202748012
#202746252
C2×C53×#3802
#2760253