Finite Group Databasebeta

← #27416856

Subgroups — #27416856

Order 13568.

1468 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810944C212yesmaximal, complemented
2#20277307C211yesmaximal, characteristic, complemented
3#20275901C212yesmaximal, complemented
4#20275894C212yesmaximal, complemented
5#20275033C212yesmaximal, complemented
6#20274914C212yesmaximal, complemented
7#20274646C212yesmaximal, complemented
8#20274627C211yesmaximal, characteristic, complemented
9C2×C53×#535C211yesmaximal, characteristic
10C2×#13133176C2211yescharacteristic, complemented
11C22×C4⋊Dic53C2211yescharacteristic, complemented
12#16810944—28no—
13C2×#13133150C2212yescomplemented
14C2×#13133148C2212yescomplemented
15C2×#13133138C2212yes—
16C2×#13133138C2212yes—
17C2×#13133138C2212yes—
18C2×#13133138C2212yes—
19C2×C22⋊#88322C2212yes—
20C2×C22⋊#88322C2212yescomplemented
21C2×C22⋊#88322C2211yescharacteristic, complemented
22C2×C22⋊#88322C2211yescharacteristic, complemented
23#16810530—21no—
24#16810526—21no—
25#16810503—21no—
26#16810503—21no—
27#16810483C2212yes—
28#16810483C2212yescomplemented
29#16810385C2212yescomplemented
30#16810385C2212yescomplemented
31C22×#41×C53C2211yescharacteristic
32C23×D4×C53C2211yescharacteristic
33C53×#535—28no—
34C2×#127×C53—21no—
35C2×#124×C53—21no—
36C2×#121×C53C2212yes—
37C2×#121×C53C2212yes—
38C2×#120×C53—21no—
39C2×#120×C53—21no—
40C2×#117×C53C2211yescharacteristic
41C23×Dic53—21no—
42C22×#88322C2312yes—
43C22×#88322C2312yes—
44#13133176—24no—
45C2×C4⋊Dic53—21no—
46C2×C4⋊Dic53—41no—
47C2×C4⋊Dic53—41no—
48C2×C4⋊Dic53C2312yes—
49C2×C4⋊Dic53C2312yes—
50C2×C22⋊Dic53—21no—
51C2×C22⋊Dic53—22no—
52C2×C22⋊Dic53—22no—
53C2×C4×Dic53—21no—
54#13133150—24no—
55#13133148—24no—
56#13133138—24no—
57#13133138—24no—
58#13133138—42no—
59#13133138—42no—
60C22⋊#88322—24no—
61C22⋊#88322—24no—
62C22⋊#88322—41no—
63C22⋊#88322—41no—
64#13133124—22no—
65#13133124—22no—
66C25×C53—21no—
67C2×#41×C53—24no—
68C2×#41×C53—24no—
69C2×#41×C53—41no—
70C2×#41×C53—41no—
71C22×Q8×C53D411yescharacteristic
72C22×D4×C53—22no—
73C22×D4×C53—22no—
74C22×D4×C53—24no—
75C22×D4×C53—21no—
76C22×D4×C53—24no—
77C22×D4×C53—41no—
78C22×D4×C53—41no—
79C22×D4×C53D411yescharacteristic
80C22×D4×C53D411yescharacteristic
81C22×D4×C53D411yescharacteristic
82C22×D4×C53C2312yes—
83C23×C4×C53C2311yescharacteristic
84C23×C4×C53C2311yescharacteristic
85C23×C4×C53C2311yescharacteristic
86#127×C53—28no—
87#124×C53—28no—
88#121×C53—24no—
89#121×C53—24no—
90#121×C53—24no—
91#121×C53D418yes—
92#120×C53—28no—
93#120×C53—28no—
94#117×C53—24no—
95C2×C4⋊C4×C53C2312yes—
96C2×C22⋊C4×C53—21no—
97C2×C22⋊C4×C53—22no—
98C2×C22⋊C4×C53—22no—
99C2×C42×C53—21no—
100C24×C53—24no—
101C24×C53—22no—
102C24×C53—42no—
103C24×C53—42no—
104C24×C53—82no—
105C24×C53—82no—
106C24×C53C2×D411yescharacteristic
107C24×C53C2×D411yescharacteristic
108C24×C53C2×D411yescharacteristic
109#41×C53—48no—
110#41×C53—48no—
111C2×Q8×C53—22no—
112C2×Q8×C53—22no—
113C2×Q8×C53—41no—
114C2×D4×C53—24no—
115C2×D4×C53—22no—
116C2×D4×C53—22no—
117C2×D4×C53—22no—
118C2×D4×C53—22no—
119C2×D4×C53—28no—
120C2×D4×C53—28no—
121C2×D4×C53—22no—
122C2×D4×C53—22no—
123C2×D4×C53—22no—
124C2×D4×C53—22no—
125C2×D4×C53—22no—
126C2×D4×C53—22no—
127C2×D4×C53—22no—
128C2×D4×C53—48no—
129C2×D4×C53—48no—
130C2×D4×C53—41no—
131C2×D4×C53—41no—
132C2×D4×C53—41no—
133C2×D4×C53—42no—
134C2×D4×C53—42no—
135C2×D4×C53—82no—
136C2×D4×C53—82no—
137C2×D4×C53—82no—
138C2×D4×C53—82no—
139C2×D4×C53C2×D414yes—
140C22×C4×C53—22no—
141C22×C4×C53—22no—
142C22×C4×C53—21no—
143C22×C4×C53—21no—
144C22×C4×C53—21no—
145C22×C4×C53—21no—
146C22×C4×C53—22no—
147C22×C4×C53—22no—
148C22×C4×C53—22no—
149C22×C4×C53—21no—
150C22×C4×C53—21no—
151C22×C4×C53C2×D411yescharacteristic
152C22×C4×C53C2×D411yescharacteristic
153C22×C4×C53C2×D414yes—
154C22×C4×C53C2×D411yescharacteristic
155C22×C4×C53#4114yes—
156C22×C4×C53C2411yescharacteristic
157C22×Dic53—22no—
158C22×Dic53—22no—
159C22×Dic53—42no—
160C22×Dic53—82no—
161C4⋊C4×C53—22no—
162C4⋊C4×C53C2×D414yes—
163C22⋊C4×C53—24no—
164C22⋊C4×C53—28no—
165C22⋊C4×C53—28no—
166C42×C53—24no—
167C22⋊Dic53—42no—
168C22⋊Dic53—82no—
169C22⋊Dic53—82no—
170C4⋊Dic53—22no—
171C4⋊Dic53—22no—
172C4⋊Dic53—22no—
173C4⋊Dic53—22no—
174C4⋊Dic53—42no—
175C4×Dic53—42no—
176C2×#88322—22no—
177C2×#88322—22no—
178C2×#88322—22no—
179C2×#88322—22no—
180C2×#88322—22no—
181C2×#88322—22no—
182C23×C53—21no—
183C23×C53—21no—
184C23×C53—21no—
185C23×C53—21no—
186C23×C53—22no—
187C23×C53—22no—
188C23×C53—22no—
189C23×C53—22no—
190C23×C53—21no—
191C23×C53—24no—
192C23×C53—21no—
193C23×C53—21no—
194C23×C53—22no—
195C23×C53—22no—
196C23×C53—48no—
197C23×C53—42no—
198C23×C53—42no—
199C23×C53—42no—
200C23×C53—42no—
201C23×C53—42no—
202C23×C53—42no—
203C23×C53—88no—
204C23×C53—88no—
205C23×C53—84no—
206C23×C53—84no—
207C23×C53#12012yes—
208C23×C53#12312yes—
209C23×C53C22×D411yescharacteristic
210Q8×C53—44no—
211D4×C53—24no—
212D4×C53—24no—
213D4×C53—44no—
214D4×C53—44no—
215D4×C53—48no—
216D4×C53—44no—
217D4×C53—42no—
218D4×C53—42no—
219D4×C53—88no—
220D4×C53—88no—
221C2×C4×C53—21no—
222C2×C4×C53—21no—
223C2×C4×C53—21no—
224C2×C4×C53—21no—
225C2×C4×C53—24no—
226C2×C4×C53—24no—
227C2×C4×C53—24no—
228C2×C4×C53—24no—
229C2×C4×C53—21no—
230C2×C4×C53—24no—
231C2×C4×C53—21no—
232C2×C4×C53—21no—
233C2×C4×C53—42no—
234C2×C4×C53—42no—
235C2×C4×C53—42no—
236C2×C4×C53—42no—
237C2×C4×C53#12012yes—
238C2×C4×C53#12312yes—
239C2×C4×C53C22×D412yes—
240C2×C4×C53C2×#4112yes—
241C2×C4×C53H(5,𝔽2)11yescharacteristic
242C2×C4×C53H(5,𝔽2)11yescharacteristic
243C2×Dic53—42no—
244C2×Dic53—42no—
245C2×Dic53—42no—
246C2×Dic53—42no—
247C2×Dic53—42no—
248C2×Dic53—42no—
249C2×Dic53—84no—
250#88322—42no—
251#88322—42no—
252#24015—531nomaximal, Sylow
253C22×C53—24no—
254C22×C53—22no—
255C22×C53—22no—
256C22×C53—24no—
257C22×C53—42no—
258C22×C53—41no—
259C22×C53—41no—
260C22×C53—41no—
261C22×C53—41no—
262C22×C53—42no—
263C22×C53—42no—
264C22×C53—41no—
265C22×C53—44no—
266C22×C53—41no—
267C22×C53—41no—
268C22×C53—42no—
269C22×C53—44no—
270C22×C53—44no—
271C22×C53—816no—
272C22×C53—82no—
273C22×C53—82no—
274C22×C53—82no—
275C22×C53—82no—
276C22×C53C2×#12011yescharacteristic
277C22×C53C2×#12311yescharacteristic
278C22×C53#53411yescharacteristic
279C22×C53#57612yes—
280C22×C53#57712yes—
281C4×C53—22no—
282C4×C53—22no—
283C4×C53—41no—
284C4×C53—44no—
285C4×C53#54612yes—
286C4×C53#54612yes—
287Dic53—82no—
288Dic53—82no—
289C2×#535D5311yescharacteristic
290C2×#482—532no—
291#2311—531no—
292#2034—532no—
293#2033—532no—
294#1874—532no—
295#1854—532no—
296#1786—532no—
297#1780—531no—
298C2×C53—42no—
299C2×C53—42no—
300C2×C53—81no—
301C2×C53—84no—
302C2×C53#240511yescharacteristic
303C2×C53#301212yes—
304C2×C53#309612yes—
305C2×C53#319911yescharacteristic
306C2×C53#320011yescharacteristic
307C22×#41C2×D5311yescharacteristic
308C23×D4C2×D5311yescharacteristic
309#535—28no—
310C2×#127—21no—
311C2×#124—21no—
312C2×#121C2×D5312yes—
313C2×#121C2×D5312yes—
314C2×#120—21no—
315C2×#120—21no—
316C2×#117—531no—
317C2×#117C2×D5311yescharacteristic
318C22×C4⋊C4—531no—
319#482—1068no—
320C2×#107—532no—
321C2×#106—532no—
322C2×#102—532no—
323C2×#102—532no—
324C2×#102—532no—
325C2×#102—532no—
326C2×C22⋊C8—532no—
327C2×C22⋊C8—532no—
328C2×C22⋊C8—531no—
329C2×C22⋊C8—531no—
330#391—1061no—
331#390—1061no—
332#386—1061no—
333#386—1061no—
334#380—532no—
335#380—532no—
336#340—532no—
337#340—532no—
338C53#2401511yescharacteristic, complemented, Sylow
339C25—21no—
340C2×#41—24no—
341C2×#41—24no—
342C2×#41—41no—
343C2×#41—41no—
344C22×Q8C22⋊D5311yescharacteristic
345C22×D4—22no—
346C22×D4—22no—
347C22×D4—24no—
348C22×D4—24no—
349C22×D4—21no—
350C22×D4—41no—
351C22×D4—41no—
352C22×D4C22⋊D5311yescharacteristic
353C22×D4C22⋊D5311yescharacteristic
354C22×D4C22⋊D5311yescharacteristic
355C22×D4C22×D5312yes—
356C23×C4—1061no—
357C23×C4C22×D5311yescharacteristic
358C23×C4C22×D5311yescharacteristic
359C23×C4C22×D5311yescharacteristic
360C22×C8—532no—
361C22×C8—532no—
362#127—28no—
363#124—28no—
364#121—24no—
365#121—24no—
366#121—24no—
367#121C22⋊D5318yes—
368#120—28no—
369#120—28no—
370#117—24no—
371#117—1064no—
372C2×C4⋊C4—532no—
373C2×C4⋊C4—532no—
374C2×C4⋊C4—1061no—
375C2×C4⋊C4—2121no—
376C2×C4⋊C4—2121no—
377C2×C4⋊C4C22×D5312yes—
378C2×C22⋊C4—21no—
379C2×C22⋊C4—22no—
380C2×C22⋊C4—22no—
381C2×C22⋊C4—1062no—
382C2×C22⋊C4—1062no—
383C2×C22⋊C4—1061no—
384C2×C42—21no—
385C2×C42—1061no—
386#107—1064no—
387#106—1064no—
388#102—1064no—
389#102—1064no—
390#102—2122no—
391#102—2122no—
392C22⋊C8—1064no—
393C22⋊C8—1064no—
394C22⋊C8—2121no—
395C22⋊C8—2121no—
396#95—1062no—
397#95—1062no—
398C24—24no—
399C24—22no—
400C24—42no—
401C24—42no—
402C24—82no—
403C24—82no—
404C24C2×C22⋊D5311yescharacteristic
405C24C2×C22⋊D5311yescharacteristic
406C24C2×C22⋊D5311yescharacteristic
407#41—48no—
408#41—48no—
409C2×Q8—22no—
410C2×Q8—22no—
411C2×Q8—41no—
412C2×D4—24no—
413C2×D4—22no—
414C2×D4—22no—
415C2×D4—22no—
416C2×D4—22no—
417C2×D4—28no—
418C2×D4—28no—
419C2×D4—22no—
420C2×D4—22no—
421C2×D4—22no—
422C2×D4—22no—
423C2×D4—22no—
424C2×D4—22no—
425C2×D4—22no—
426C2×D4—48no—
427C2×D4—48no—
428C2×D4—41no—
429C2×D4—41no—
430C2×D4—41no—
431C2×D4—42no—
432C2×D4—42no—
433C2×D4—82no—
434C2×D4—82no—
435C2×D4—82no—
436C2×D4—82no—
437C2×D4C2×C22⋊D5314yes—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—1062no—
450C22×C4—1062no—
451C22×C4—4242no—
452C22×C4—2122no—
453C22×C4#1171994714yes—
454C22×C4C2×C22⋊D5311yescharacteristic
455C22×C4C2×C22⋊D5311yescharacteristic
456C22×C4C2×C22⋊D5314yes—
457C22×C4C2×C22⋊D5311yescharacteristic
458C22×C4C23×D5311yescharacteristic
459C2×C8—1062no—
460C2×C8—1062no—
461C2×C8—1062no—
462C2×C8—1062no—
463C2×C8—1062no—
464C2×C8—1062no—
465C4⋊C4—22no—
466C4⋊C4—1062no—
467C4⋊C4—1062no—
468C4⋊C4—1062no—
469C4⋊C4—1062no—
470C4⋊C4—2122no—
471C4⋊C4C2×C22⋊D5314yes—
472C22⋊C4—24no—
473C22⋊C4—28no—
474C22⋊C4—28no—
475C22⋊C4—4242no—
476C22⋊C4—4242no—
477C22⋊C4—2122no—
478C42—24no—
479C42—2122no—
480C23—21no—
481C23—21no—
482C23—22no—
483C23—22no—
484C23—22no—
485C23—22no—
486C23—21no—
487C23—21no—
488C23—24no—
489C23—21no—
490C23—21no—
491C23—22no—
492C23—22no—
493C23—21no—
494C23—48no—
495C23—42no—
496C23—42no—
497C23—42no—
498C23—42no—
499C23—42no—
500C23—42no—
501C23—88no—
502C23—88no—
503C23—84no—
504C23—84no—
505C23#1313318612yes—
506C23#1313319812yes—
507C23C22×C22⋊D5311yescharacteristic
508Q8—44no—
509D4—24no—
510D4—24no—
511D4—44no—
512D4—44no—
513D4—48no—
514D4—44no—
515D4—42no—
516D4—42no—
517D4—88no—
518D4—88no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—4244no—
536C2×C4—2122no—
537C2×C4—2122no—
538C2×C4—2122no—
539C2×C4—2122no—
540C2×C4—2122no—
541C2×C4—2122no—
542C2×C4#1313318612yes—
543C2×C4#1313319812yes—
544C2×C4C22×C22⋊D5312yes—
545C2×C4C2×#1171994712yes—
546C2×C4#1313326211yescharacteristic
547C2×C4#1313326211yescharacteristic
548C8—2122no—
549C8—2122no—
550C22—22no—
551C22—22no—
552C22—24no—
553C22—24no—
554C22—42no—
555C22—41no—
556C22—41no—
557C22—41no—
558C22—41no—
559C22—42no—
560C22—41no—
561C22—44no—
562C22—42no—
563C22—44no—
564C22—44no—
565C22—41no—
566C22—41no—
567C22—42no—
568C22—816no—
569C22—82no—
570C22—82no—
571C22—82no—
572C22—82no—
573C22C2×#1313318611yescharacteristic
574C22C2×#1313319811yescharacteristic
575C22#1681118211yescharacteristic
576C22#1681142612yes—
577C22#1681143412yes—
578C4—22no—
579C4—22no—
580C4—41no—
581C4—44no—
582C4—4242no—
583C4—4242no—
584C4#1681126712yes—
585C4#1681126712yes—
586C2—42no—
587C2—42no—
588C2—81no—
589C2—84no—
590C2#2027803811yescharacteristic
591C2#2028401912yes—
592C2#2028466712yes—
593C2#2028557611yescharacteristic
594C2#2028558711yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
Fitting, prime radicalC2×C53×#535C2
socleC23×C53C22×D4
hypercenterC2×#535D53
hypocenterC53#24015
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C22×#41 ◁ C2×#535

Complements

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

NH ≅ G/Ntype
1. C2×#16810944C2semidirect
2. #20277307C2semidirect
3. #20275901C2semidirect
4. #20275894C2semidirect
5. #20275033C2semidirect
6. #20274914C2semidirect
7. #20274646C2semidirect
8. #20274627C2semidirect
10. C2×#13133176C22semidirect
11. C22×C4⋊Dic53C22semidirect
13. C2×#13133150C22semidirect
14. C2×#13133148C22semidirect
20. C2×C22⋊#88322C22semidirect
21. C2×C22⋊#88322C22semidirect
22. C2×C22⋊#88322C22semidirect
28. #16810483C22semidirect
29. #16810385C22semidirect
30. #16810385C22semidirect
338. C53#24015semidirect

Maximal subgroups

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

subgroupindex
C2×#168109442
#202773072
#202759012
#202758942
#202750332
#202749142
#202746462
#202746272
C2×C53×#5352
#2401553