Finite Group Databasebeta

← #27393911

Subgroups — #27393911

Order 13568.

1254 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811213C211yesmaximal, characteristic
2#20277328C211yesmaximal, characteristic, complemented
3#20277310C211yesmaximal, characteristic, complemented
4#20275701C211yesmaximal, characteristic, complemented
5#20275551C211yesmaximal, characteristic, complemented
6#20275550C211yesmaximal, characteristic, complemented
7#20275167C214yesmaximal, complemented
8#20274702C214yesmaximal, complemented
9C53×#1996C211yesmaximal, characteristic, complemented
10#16811213—22no—
11#16811213—22no—
12#16811213—22no—
13#16811213—22no—
14C2×#13133212C412yescomplemented
15C2×#13133206C412yescomplemented
16C2×#13133202C412yescomplemented
17C2×#13133192C412yescomplemented
18C2×#13133191C2211yescharacteristic
19C2×#13133187C2212yes—
20C2×#13133177C2212yes—
21C22×#11719921C2211yescharacteristic
22C2×C22⋊C4×D53C2212yes—
23C2×#13133131C2214yes—
24C2×#13133131C2214yes—
25#16810514—22no—
26#16810512—22no—
27#16810497—22no—
28#16810494—22no—
29#16810483C2211yescharacteristic
30#16810481C2212yes—
31#16810481C2212yes—
32#16810478C2212yes—
33C4×C22⋊Dic53C2211yescharacteristic
34#16810390C2214yes—
35C2×C22⋊Q8×C53C2211yescharacteristic
36C53×#380C2211yescharacteristic
37C4×C22⋊C4×C53C2211yescharacteristic
38C53×#342C2214yes—
39C24×D53C2311yescharacteristic
40C22×D212C2×C412yes—
41C22×D212C2×C412yes—
42C22×D212C2×C411yescharacteristic, complemented
43C22×C22⋊D53C2312yes—
44C23×Dic53C2311yescharacteristic
45C22×C4×D53—22no—
46C22×C4×D53C2×C412yescomplemented
47#13133212—24no—
48#13133212—24no—
49#13133206—24no—
50#13133206—24no—
51#13133202—24no—
52#13133202—24no—
53#13133192—24no—
54#13133192—24no—
55#13133191—21no—
56#13133191—21no—
57#13133191—21no—
58#13133191—21no—
59#13133187—24no—
60#13133187—24no—
61#13133177—24no—
62#13133177—24no—
63C2×C4⋊Dic53—21no—
64C2×C4⋊Dic53C2×C411yescharacteristic, complemented
65C2×#11719919—22no—
66C2×#11719919—22no—
67C2×#11719919C2×C412yescomplemented
68C2×C22⋊Dic53C2312yes—
69C2×C22⋊Dic53C2312yes—
70C2×#11719921—22no—
71C2×#11719921—22no—
72C2×#11719921—22no—
73C2×#11719921C2×C412yes—
74C2×#11719921C2×C412yes—
75C2×#11719921C2×C412yes—
76C2×#11719921C2×C412yes—
77C2×#11719921C2×C412yescomplemented
78C2×#11719921C2×C412yescomplemented
79C2×#11719921D412yes—
80C2×#11719921D412yes—
81C2×#11719921D412yes—
82C2×#11719921D412yes—
83C22⋊C4×D53—44no—
84C2×C4×Dic53—21no—
85C2×C4×Dic53C2×C412yescomplemented
86#13133131—44no—
87#13133131—44no—
88#13133124—21no—
89#13133124—21no—
90#13133124—22no—
91#13133124—22no—
92C22×Q8×C53C2×C411yescharacteristic, complemented
93C23×C4×C53C2311yescharacteristic
94C22⋊Q8×C53—21no—
95C22⋊Q8×C53—21no—
96C22⋊Q8×C53—21no—
97C22⋊Q8×C53—21no—
98C2×C4⋊C4×C53—21no—
99C2×C4⋊C4×C53C2×C412yescomplemented
100C2×C4⋊C4×C53C2×C411yescharacteristic, complemented
101C2×C22⋊C4×C53C2312yes—
102C2×C22⋊C4×C53C2312yes—
103C2×C22⋊C4×C53C2312yes—
104C2×C42×C53—21no—
105#95×C53—21no—
106#95×C53—21no—
107C24×C53C2411yescharacteristic
108C23×D53—22no—
109C23×D53—22no—
110C23×D53—81no—
111C23×D53—81no—
112C23×D53C22×C412yes—
113C23×D53C22×C412yes—
114C23×D53C2×D412yes—
115C23×D53C2×D412yes—
116C2×Q8×C53—21no—
117C2×Q8×C53—21no—
118C2×Q8×C53—81no—
119C22×C4×C53—22no—
120C22×C4×C53—22no—
121C22×C4×C53—81no—
122C22×C4×C53C22×C412yes—
123C22×C4×C53C22×C412yes—
124C22×C4×C53C22×C411yescharacteristic
125C22×C4×C53C22×C411yescharacteristic
126C22×C4×C53C2×D411yescharacteristic
127C22×C4×C53C2×D411yescharacteristic
128C22×C4×C53C2×D411yescharacteristic
129C22×C4×C53C2×D411yescharacteristic
130C2×C22⋊D53—22no—
131C2×C22⋊D53—22no—
132C2×C22⋊D53—24no—
133C2×C22⋊D53—24no—
134C22×Dic53—22no—
135C22×Dic53—22no—
136C22×Dic53—81no—
137C22×Dic53C22×C412yes—
138C22×Dic53C22×C412yes—
139C22×Dic53C2×D412yes—
140C22×Dic53C2×D412yes—
141C2×D212—24no—
142C2×D212—24no—
143C2×D212—21no—
144C2×D212—21no—
145C2×D212—82no—
146C2×D212—82no—
147C2×D212—81no—
148C2×C4×D53—24no—
149C2×C4×D53—44no—
150C2×C4×D53—82no—
151C2×C4×D53—82no—
152C2×C4×D53—82no—
153C4⋊C4×C53—22no—
154C4⋊C4×C53—22no—
155C4⋊C4×C53—21no—
156C4⋊C4×C53—21no—
157C4⋊C4×C53—81no—
158C22⋊C4×C53—22no—
159C22⋊C4×C53—22no—
160C22⋊C4×C53—22no—
161C22⋊C4×C53—24no—
162C22⋊C4×C53—22no—
163C42×C53—81no—
164C22⋊Dic53—22no—
165C22⋊Dic53—24no—
166C22⋊Dic53—22no—
167#11719921—24no—
168#11719921—24no—
169#11719921—22no—
170#11719921—22no—
171#11719921—24no—
172#11719921—44no—
173#11719921—82no—
174#11719921—82no—
175#11719921C22⋊C414yes—
176#11719921C22⋊C414yes—
177#11719921C22⋊C414yes—
178#11719921C22⋊C414yes—
179C4⋊Dic53—21no—
180C4⋊Dic53—21no—
181C4⋊Dic53—81no—
182#11719919—24no—
183#11719919—82no—
184#11719919—82no—
185C4×Dic53—24no—
186C4×Dic53—81no—
187C23×C53—41no—
188C23×C53—41no—
189C23×C53C23×C411yescharacteristic
190C23×C53C22×D411yescharacteristic
191C23×C53C22×D411yescharacteristic
192C23×C53H(5,𝔽2)12yes—
193C23×C532−1+412yes—
194C22×D53—22no—
195C22×D53—22no—
196C22×D53—24no—
197C22×D53—24no—
198C22×D53—22no—
199C22×D53—24no—
200C22×D53—42no—
201C22×D53—42no—
202C22×D53—42no—
203C22×D53—42no—
204C22×D53—42no—
205C22×D53—42no—
206C22×D53—42no—
207C22×D53—42no—
208C22×D53—84no—
209C22×D53—84no—
210C22×D53C2×C22⋊C414yes—
211Q8×C53—81no—
212Q8×C53—81no—
213C2×C4×C53—22no—
214C2×C4×C53—22no—
215C2×C4×C53—41no—
216C2×C4×C53—41no—
217C2×C4×C53—41no—
218C2×C4×C53—41no—
219C2×C4×C53—41no—
220C2×C4×C53—41no—
221C2×C4×C53—42no—
222C2×C4×C53—42no—
223C2×C4×C53—42no—
224C2×C4×C53—42no—
225C2×C4×C53—42no—
226C2×C4×C53—42no—
227C2×C4×C53—42no—
228C2×C4×C53—42no—
229C2×C4×C53C2×C22⋊C411yescharacteristic
230C2×C4×C53C2×C22⋊C411yescharacteristic
231C2×C4×C53C2×C22⋊C411yescharacteristic
232C2×C4×C53C2×C22⋊C411yescharacteristic
233C22⋊D53—42no—
234C22⋊D53—42no—
235C22⋊D53—44no—
236C2×Dic53—22no—
237C2×Dic53—24no—
238C2×Dic53—42no—
239C2×Dic53—42no—
240C2×Dic53—42no—
241C2×Dic53—42no—
242C2×Dic53—42no—
243C2×Dic53—42no—
244C2×Dic53—42no—
245C2×Dic53—42no—
246C2×Dic53—42no—
247C2×Dic53C2×C22⋊C414yes—
248D212—81no—
249D212—81no—
250D212—84no—
251D212—84no—
252C4×D53—84no—
253C4×D53—88no—
254#21876—531nomaximal, Sylow
255C22×C53—21no—
256C22×C53—21no—
257C22×C53—21no—
258C22×C53—21no—
259C22×C53—21no—
260C22×C53—21no—
261C22×C53—42no—
262C22×C53—42no—
263C22×C53C22×C22⋊C411yescharacteristic
264C22×C53#52012yes—
265C22×C53#53511yescharacteristic
266C22×C53#53611yescharacteristic
267C22×C53#53712yes—
268C2×D53—24no—
269C2×D53—24no—
270C2×D53—24no—
271C2×D53—44no—
272C2×D53—44no—
273C2×D53—42no—
274C2×D53—42no—
275C2×D53—42no—
276C2×D53—84no—
277C2×D53—84no—
278C2×D53—84no—
279C4×C53—21no—
280C4×C53—21no—
281C4×C53—21no—
282C4×C53—21no—
283C4×C53—42no—
284C4×C53—42no—
285C4×C53—82no—
286C4×C53—82no—
287Dic53—24no—
288Dic53—42no—
289Dic53—82no—
290Dic53—82no—
291C2×#538—531no—
292#2315—531no—
293#2312—531no—
294#1996—531no—
295#1996D5311yescharacteristic, complemented
296#1964—531no—
297#1964—531no—
298#1898—534no—
299#1799—534no—
300C2×C53—21no—
301C2×C53—22no—
302C2×C53—21no—
303C2×C53—22no—
304C2×C53#231111yescharacteristic
305C2×C53#288311yescharacteristic
306C2×C53#288311yescharacteristic
307D53—44no—
308D53—82no—
309#538—1062no—
310#538—1062no—
311#538—1062no—
312#538—1062no—
313C2×#126—532no—
314C2×#124—532no—
315C2×#123—532no—
316C2×C22⋊Q8C2×D5311yescharacteristic
317C2×#121—531no—
318C2×#121—532no—
319C2×#120—532no—
320C2×C4×D4—532no—
321C22×C22⋊C4—531no—
322C22×C22⋊C4—532no—
323C2×#99—534no—
324C2×#99—534no—
325#388—1062no—
326#388—1062no—
327#385—1062no—
328#385—1062no—
329#380—531no—
330#380—532no—
331#380—532no—
332#380C2×D5311yescharacteristic
333#379—532no—
334C4×C22⋊C4—531no—
335C4×C22⋊C4C2×D5311yescharacteristic
336#342—534no—
337#342C2×D5314yes—
338C53#2187611yescharacteristic, complemented, Sylow
339C25—531no—
340C22×Q8C4×D5311yescharacteristic, complemented
341C22×D4—532no—
342C22×D4—532no—
343C22×D4—532no—
344C22×D4—531no—
345C23×C4—531no—
346C23×C4—532no—
347C23×C4—1062no—
348C23×C4C22×D5311yescharacteristic
349#126—1064no—
350#126—1064no—
351#124—1064no—
352#124—1064no—
353#123—1064no—
354#123—1064no—
355C22⋊Q8—21no—
356C22⋊Q8—21no—
357C22⋊Q8—21no—
358C22⋊Q8—21no—
359#121—1064no—
360#121—1064no—
361#121—1061no—
362#121—1061no—
363#121—1061no—
364#121—1061no—
365#120—1064no—
366#120—1064no—
367C4×D4—1064no—
368C4×D4—1064no—
369C2×C4⋊C4—21no—
370C2×C4⋊C4—532no—
371C2×C4⋊C4—531no—
372C2×C4⋊C4—1062no—
373C2×C4⋊C4—1062no—
374C2×C4⋊C4—1061no—
375C2×C4⋊C4C4×D5312yescomplemented
376C2×C4⋊C4C4×D5311yescharacteristic, complemented
377C2×C22⋊C4—532no—
378C2×C22⋊C4—532no—
379C2×C22⋊C4—532no—
380C2×C22⋊C4—532no—
381C2×C22⋊C4—532no—
382C2×C22⋊C4—532no—
383C2×C22⋊C4—532no—
384C2×C22⋊C4—532no—
385C2×C22⋊C4—532no—
386C2×C22⋊C4—532no—
387C2×C22⋊C4—532no—
388C2×C22⋊C4—532no—
389C2×C22⋊C4—1062no—
390C2×C22⋊C4—1062no—
391C2×C22⋊C4—1062no—
392C2×C22⋊C4—2124no—
393C2×C22⋊C4C22×D5312yes—
394C2×C22⋊C4C22×D5312yes—
395C2×C22⋊C4C22×D5312yes—
396C2×C42—21no—
397C2×C42—532no—
398C2×C42—1061no—
399#99—2124no—
400#99—2124no—
401#95—21no—
402#95—21no—
403#95—1062no—
404#95—1062no—
405#95—1061no—
406#95—1061no—
407C24—532no—
408C24—532no—
409C24—532no—
410C24—532no—
411C24—1062no—
412C24—1062no—
413C24—4241no—
414C24—4241no—
415C24C23×D5311yescharacteristic
416C2×Q8—21no—
417C2×Q8—21no—
418C2×Q8—81no—
419C2×D4—1064no—
420C2×D4—1064no—
421C2×D4—1062no—
422C2×D4—1062no—
423C2×D4—1064no—
424C2×D4—1064no—
425C2×D4—1061no—
426C2×D4—1061no—
427C2×D4—4241no—
428C2×D4—4242no—
429C2×D4—4242no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—81no—
433C22×C4—532no—
434C22×C4—532no—
435C22×C4—532no—
436C22×C4—532no—
437C22×C4—1064no—
438C22×C4—1062no—
439C22×C4—1062no—
440C22×C4—4242no—
441C22×C4—4242no—
442C22×C4—4242no—
443C22×C4—4241no—
444C22×C4—2124no—
445C22×C4C2×C4×D5312yes—
446C22×C4C2×C4×D5312yes—
447C22×C4C2×C4×D5311yescharacteristic
448C22×C4C2×C4×D5311yescharacteristic
449C22×C4D4×D5311yescharacteristic
450C22×C4D4×D5311yescharacteristic
451C22×C4D4×D5311yescharacteristic
452C22×C4D4×D5311yescharacteristic
453C4⋊C4—22no—
454C4⋊C4—22no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—81no—
458C4⋊C4—1064no—
459C4⋊C4—1061no—
460C4⋊C4—1061no—
461C4⋊C4—4242no—
462C4⋊C4—4242no—
463C4⋊C4—4241no—
464C22⋊C4—22no—
465C22⋊C4—22no—
466C22⋊C4—22no—
467C22⋊C4—24no—
468C22⋊C4—22no—
469C22⋊C4—534no—
470C22⋊C4—534no—
471C22⋊C4—534no—
472C22⋊C4—534no—
473C22⋊C4—1064no—
474C22⋊C4—1064no—
475C22⋊C4—1064no—
476C22⋊C4—1062no—
477C22⋊C4—1062no—
478C22⋊C4—1062no—
479C22⋊C4—1064no—
480C22⋊C4—1062no—
481C22⋊C4—4242no—
482C22⋊C4—4242no—
483C22⋊C4—2124no—
484C42—81no—
485C42—1064no—
486C42—4241no—
487C23—41no—
488C23—41no—
489C23—534no—
490C23—1064no—
491C23—1064no—
492C23—1062no—
493C23—1062no—
494C23—1064no—
495C23—1062no—
496C23—4244no—
497C23—4244no—
498C23—2122no—
499C23—2122no—
500C23—2122no—
501C23—2122no—
502C23—2122no—
503C23—2122no—
504C23—2122no—
505C23—2122no—
506C23C22×C4×D5311yescharacteristic
507C23C2×D4×D5311yescharacteristic
508C23C2×D4×D5311yescharacteristic
509C23#1313326312yes—
510C23#1313326512yes—
511Q8—81no—
512Q8—81no—
513D4—4241no—
514D4—4241no—
515D4—4244no—
516D4—4244no—
517D4—2122no—
518D4—2122no—
519D4—2124no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—534no—
537C2×C4—1062no—
538C2×C4—1064no—
539C2×C4—4244no—
540C2×C4—4248no—
541C2×C4—2122no—
542C2×C4—2122no—
543C2×C4—2122no—
544C2×C4—2122no—
545C2×C4—2122no—
546C2×C4—2122no—
547C2×C4—2122no—
548C2×C4—2122no—
549C2×C4—2122no—
550C2×C4C22⋊C4×D5311yescharacteristic
551C2×C4C22⋊C4×D5311yescharacteristic
552C2×C4C22⋊C4×D5311yescharacteristic
553C2×C4C22⋊C4×D5311yescharacteristic
554C22—21no—
555C22—21no—
556C22—21no—
557C22—21no—
558C22—21no—
559C22—21no—
560C22—42no—
561C22—42no—
562C22—1064no—
563C22—1064no—
564C22—1064no—
565C22—4244no—
566C22—4244no—
567C22—4244no—
568C22—2124no—
569C22—2124no—
570C22—2122no—
571C22—2122no—
572C22—2122no—
573C22C2×C22⋊C4×D5311yescharacteristic
574C22#1681110512yes—
575C22#1681119111yescharacteristic
576C22#1681119811yescharacteristic
577C22#1681120312yes—
578C4—21no—
579C4—21no—
580C4—21no—
581C4—21no—
582C4—42no—
583C4—42no—
584C4—82no—
585C4—82no—
586C4—1064no—
587C4—4242no—
588C4—4242no—
589C4—2122no—
590C2—21no—
591C2—22no—
592C2—22no—
593C2—21no—
594C2—4242no—
595C2—2124no—
596C2#2027730611yescharacteristic
597C2#2028317311yescharacteristic
598C2#2028317411yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D53
commutatorC23×C53C23×C4
FrattiniC24C23×D53
FittingC53×#1996C2
socleC22×C53C22×C22⋊C4
prime radicalC2×#16811213C2
cosocleC24×C53C24
hypercenter#1996D53
hypocenterC53#21876
LCSG ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1996

Complements

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

NH ≅ G/Ntype
2. #20277328C2semidirect
3. #20277310C2semidirect
4. #20275701C2semidirect
5. #20275551C2semidirect
6. #20275550C2semidirect
7. #20275167C2semidirect
8. #20274702C2semidirect
9. C53×#1996C2semidirect
14. C2×#13133212C4semidirect
15. C2×#13133206C4semidirect
16. C2×#13133202C4semidirect
17. C2×#13133192C4semidirect
42. C22×D212C2×C4semidirect
46. C22×C4×D53C2×C4semidirect
64. C2×C4⋊Dic53C2×C4semidirect
67. C2×#11719919C2×C4semidirect
77. C2×#11719921C2×C4semidirect
78. C2×#11719921C2×C4semidirect
85. C2×C4×Dic53C2×C4semidirect
92. C22×Q8×C53C2×C4semidirect
99. C2×C4⋊C4×C53C2×C4semidirect
100. C2×C4⋊C4×C53C2×C4semidirect
295. #1996D53semidirect
338. C53#21876semidirect
340. C22×Q8C4×D53semidirect
375. C2×C4⋊C4C4×D53semidirect
376. C2×C4⋊C4C4×D53semidirect

Maximal subgroups

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

subgroupindex
C2×#168112132
#202773282
#202773102
#202757012
#202755512
#202755502
#202751672
#202747022
C53×#19962
#2187653