Finite Group Databasebeta

← #27489502

Subgroups — #27489502

Order 13568.

1326 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810947C211yesmaximal, characteristic, complemented
2C2×#16810762C211yesmaximal, characteristic, complemented
3#20280439C211yesmaximal, characteristic
4#20275915C212yesmaximal, complemented
5#20275908C212yesmaximal, complemented
6#20275716C212yesmaximal, complemented
7#20275716C212yesmaximal, complemented
8#20274810C211yesmaximal, characteristic, complemented
9#20274650C211yesmaximal, characteristic, complemented
10#20274627C211yesmaximal, characteristic, complemented
11C2×C53×#536C211yesmaximal, characteristic, complemented
12C22×#11719924C2212yes—
13C2×#13133191C2211yescharacteristic
14C2×#13133191C2212yes—
15#16810947—24no—
16#16810762—24no—
17C2×#13133150C2211yescharacteristic
18C2×#13133148C2211yescharacteristic
19C2×#13133141C2212yes—
20C2×#13133141C2212yes—
21C2×#13133141C2212yes—
22C2×#13133137C2211yescharacteristic
23C2×#13133137C2211yescharacteristic
24C2×C22⋊#88322C2211yescharacteristic
25C2×C22⋊#88322C2211yescharacteristic
26C2×C22⋊#88322C2212yes—
27C2×C22⋊#88322C2212yes—
28#16810534—21no—
29#16810531—22no—
30#16810531—21no—
31#16810483C2211yescharacteristic
32#16810483C2212yes—
33#16810385C2212yes—
34#16810385C2212yes—
35C22×#41×C53C2211yescharacteristic
36C23×Q8×C53C2211yescharacteristic
37C53×#536—24no—
38C53×#536—24no—
39C2×C4⋊Q8×C53—21no—
40C2×#124×C53—21no—
41C2×#123×C53—22no—
42C2×C22⋊Q8×C53C2212yes—
43C2×C22⋊Q8×C53C2212yes—
44C2×#117×C53C2211yescharacteristic
45C22×D212C2311yescharacteristic
46C22×C22⋊D53—22no—
47C22×C22⋊D53—21no—
48C2×#11719924—24no—
49C2×#11719924—42no—
50C2×#11719924—42no—
51C22×#88322C2311yescharacteristic
52C22×#88322C2311yescharacteristic
53C22×#88322C2312yes—
54#13133191—22no—
55#13133191—22no—
56#13133191—42no—
57#13133191—42no—
58C2×C4⋊Dic53C2312yes—
59C2×C4⋊Dic53C2311yescharacteristic
60C2×C22⋊Dic53—22no—
61C2×C22⋊Dic53—21no—
62C2×#11719921—22no—
63C2×#11719921—21no—
64#13133150—22no—
65#13133148—22no—
66#13133141—24no—
67#13133141—42no—
68#13133141—42no—
69#13133137—22no—
70#13133137—22no—
71C22⋊#88322—22no—
72C22⋊#88322—22no—
73C22⋊#88322—42no—
74C22⋊#88322—42no—
75#13133124—22no—
76#13133124—21no—
77C2×#41×C53—22no—
78C2×#41×C53—22no—
79C2×#41×C53—42no—
80C2×#41×C53—42no—
81C22×Q8×C53—21no—
82C22×Q8×C53—22no—
83C22×Q8×C53—42no—
84C22×Q8×C53—42no—
85C22×Q8×C53D411yescharacteristic
86C22×Q8×C53D414yescomplemented
87C22×Q8×C53C2312yes—
88C22×D4×C53D412yes—
89C22×D4×C53D411yescharacteristic
90C23×C4×C53—21no—
91C23×C4×C53C2312yes—
92C23×C4×C53C2311yescharacteristic
93C4⋊Q8×C53—28no—
94#124×C53—28no—
95#123×C53—216no—
96C22⋊Q8×C53—24no—
97C22⋊Q8×C53—24no—
98C22⋊Q8×C53—24no—
99C22⋊Q8×C53—24no—
100#117×C53—22no—
101#117×C53D414yescomplemented
102C2×C4⋊C4×C53—22no—
103C2×C4⋊C4×C53—22no—
104C2×C4⋊C4×C53C2311yescharacteristic
105C2×C4⋊C4×C53C2311yescharacteristic
106C2×C22⋊C4×C53—21no—
107C2×C22⋊C4×C53—22no—
108C2×C22⋊C4×C53—22no—
109C2×C42×C53—21no—
110C24×C53C2×D412yes—
111C24×C53C2×D411yescharacteristic
112C23×D53—21no—
113#41×C53—48no—
114#41×C53—44no—
115#41×C53—44no—
116C2×Q8×C53—24no—
117C2×Q8×C53—22no—
118C2×Q8×C53—22no—
119C2×Q8×C53—22no—
120C2×Q8×C53—22no—
121C2×Q8×C53—22no—
122C2×Q8×C53—22no—
123C2×Q8×C53—42no—
124C2×Q8×C53—48no—
125C2×Q8×C53—44no—
126C2×Q8×C53—44no—
127C2×Q8×C53—44no—
128C2×Q8×C53C2×D414yes—
129C2×D4×C53—24no—
130C2×D4×C53—22no—
131C2×D4×C53—42no—
132C2×D4×C53—42no—
133C2×D4×C53—42no—
134C22×C4×C53—22no—
135C22×C4×C53—22no—
136C22×C4×C53—24no—
137C22×C4×C53—21no—
138C22×C4×C53—21no—
139C22×C4×C53—22no—
140C22×C4×C53—22no—
141C22×C4×C53—22no—
142C22×C4×C53—22no—
143C22×C4×C53—22no—
144C22×C4×C53—22no—
145C22×C4×C53—22no—
146C22×C4×C53—44no—
147C22×C4×C53C2×D412yes—
148C22×C4×C53C2×D411yescharacteristic
149C22×C4×C53C2×D412yes—
150C22×C4×C53C2×D412yes—
151C22×C4×C53C2411yescharacteristic
152C2×C22⋊D53—44no—
153C2×C22⋊D53—42no—
154C2×C22⋊D53—82no—
155C2×C22⋊D53—82no—
156C2×C22⋊D53—82no—
157C22×Dic53—21no—
158C22×Dic53—22no—
159C2×D212—21no—
160C2×D212—21no—
161C2×D212—22no—
162C2×D212—22no—
163C4⋊C4×C53—28no—
164C4⋊C4×C53—28no—
165C4⋊C4×C53—21no—
166C4⋊C4×C53—21no—
167C4⋊C4×C53C2×D412yes—
168C4⋊C4×C53C2×D412yes—
169C22⋊C4×C53—28no—
170C22⋊C4×C53—28no—
171C22⋊C4×C53—24no—
172C42×C53—24no—
173C22⋊Dic53—81no—
174C22⋊Dic53—82no—
175#11719924—48no—
176#11719921—44no—
177#11719921—42no—
178C4⋊Dic53—21no—
179C4⋊Dic53—21no—
180C4⋊Dic53—22no—
181C4⋊Dic53—22no—
182C2×#88322—21no—
183C2×#88322—21no—
184C2×#88322—22no—
185C2×#88322—22no—
186C2×#88322—22no—
187C2×#88322—22no—
188C2×#88322—22no—
189C23×C53—22no—
190C23×C53—22no—
191C23×C53—21no—
192C23×C53—21no—
193C23×C53—22no—
194C23×C53—22no—
195C23×C53—22no—
196C23×C53—22no—
197C23×C53—22no—
198C23×C53—22no—
199C23×C53—22no—
200C23×C53#12012yes—
201C23×C53C22×D411yescharacteristic
202C22×D53—41no—
203C22×D53—42no—
204C22×D53—41no—
205C22×D53—42no—
206C22×D53—41no—
207Q8×C53—24no—
208Q8×C53—24no—
209Q8×C53—48no—
210Q8×C53—44no—
211Q8×C53—42no—
212Q8×C53—42no—
213D4×C53—44no—
214D4×C53—48no—
215C2×C4×C53—22no—
216C2×C4×C53—24no—
217C2×C4×C53—22no—
218C2×C4×C53—22no—
219C2×C4×C53—24no—
220C2×C4×C53—22no—
221C2×C4×C53—22no—
222C2×C4×C53—22no—
223C2×C4×C53—22no—
224C2×C4×C53—22no—
225C2×C4×C53—48no—
226C2×C4×C53—42no—
227C2×C4×C53—42no—
228C2×C4×C53—42no—
229C2×C4×C53—41no—
230C2×C4×C53—41no—
231C2×C4×C53—42no—
232C2×C4×C53—42no—
233C2×C4×C53—42no—
234C2×C4×C53—42no—
235C2×C4×C53#12012yes—
236C2×C4×C53C22×D411yescharacteristic
237C2×C4×C53C22×D411yescharacteristic
238C2×C4×C53H(5,𝔽2)12yes—
239C2×C4×C53H(5,𝔽2)12yes—
240C22⋊D53—84no—
241C22⋊D53—88no—
242C2×Dic53—41no—
243C2×Dic53—42no—
244C2×Dic53—42no—
245C2×Dic53—42no—
246C2×Dic53—42no—
247D212—42no—
248D212—41no—
249D212—41no—
250#88322—42no—
251#88322—41no—
252#88322—41no—
253#29603—531nomaximal, Sylow
254C22×C53—22no—
255C22×C53—22no—
256C22×C53—22no—
257C22×C53—42no—
258C22×C53—42no—
259C22×C53—42no—
260C22×C53—42no—
261C22×C53—42no—
262C22×C53—42no—
263C22×C53—42no—
264C22×C53—42no—
265C22×C53—42no—
266C22×C53C2×#12011yescharacteristic
267C22×C53#53412yes—
268C22×C53#57711yescharacteristic
269C22×C53#57712yes—
270C22×C53#57811yescharacteristic
271C2×D53—81no—
272C2×D53—82no—
273C2×D53—82no—
274C2×D53—81no—
275C2×D53—81no—
276C4×C53—22no—
277C4×C53—42no—
278C4×C53—44no—
279C4×C53—42no—
280C4×C53—42no—
281C4×C53#53412yes—
282C4×C53#53511yescharacteristic
283C4×C53#53511yescharacteristic
284Dic53—82no—
285Dic53—81no—
286C2×#536D5311yescharacteristic, complemented
287C2×#482—531no—
288C2×#448—531no—
289#2615—531no—
290#2036—532no—
291#2035—532no—
292#2000—532no—
293#2000—532no—
294#1825—531no—
295#1787—531no—
296#1780—531no—
297C2×C53—42no—
298C2×C53—82no—
299C2×C53#261311yescharacteristic
300C2×C53#301011yescharacteristic
301C2×C53#301311yescharacteristic
302C2×C53#320012yes—
303C2×C53#320112yes—
304D53—161no—
305C22×#41C2×D5311yescharacteristic
306C23×Q8C2×D5311yescharacteristic
307C22×SD16—532no—
308#536—24no—
309#536—24no—
310C2×C4⋊Q8—21no—
311C2×#124—21no—
312C2×#123—22no—
313C2×C22⋊Q8C2×D5312yes—
314C2×C22⋊Q8C2×D5312yes—
315C2×#121—531no—
316C2×#121—532no—
317C2×#117C2×D5311yescharacteristic
318#482—1064no—
319#448—1064no—
320C2×#107—531no—
321C2×#106—531no—
322C2×#103—532no—
323C2×#103—532no—
324C2×#103—532no—
325C2×#102—531no—
326C2×#102—531no—
327C2×C22⋊C8—531no—
328C2×C22⋊C8—531no—
329C2×C22⋊C8—532no—
330C2×C22⋊C8—532no—
331#393—1061no—
332#392—1062no—
333#392—1061no—
334#380—531no—
335#380—532no—
336#340—532no—
337#340—532no—
338C53#2960311yescharacteristic, complemented, Sylow
339C2×#41—22no—
340C2×#41—22no—
341C2×#41—42no—
342C2×#41—42no—
343C22×Q8—21no—
344C22×Q8—22no—
345C22×Q8—42no—
346C22×Q8—42no—
347C22×Q8C22⋊D5311yescharacteristic
348C22×Q8C22⋊D5314yescomplemented
349C22×Q8C22×D5312yes—
350C22×D4—531no—
351C22×D4—1062no—
352C22×D4—1061no—
353C22×D4C22⋊D5312yes—
354C22×D4C22⋊D5311yescharacteristic
355C23×C4—21no—
356C23×C4C22×D5312yes—
357C23×C4C22×D5311yescharacteristic
358C2×SD16—1064no—
359C2×SD16—2122no—
360C2×SD16—2122no—
361C22×C8—532no—
362C22×C8—531no—
363C22×C8—531no—
364C4⋊Q8—28no—
365#124—28no—
366#123—216no—
367C22⋊Q8—24no—
368C22⋊Q8—24no—
369C22⋊Q8—24no—
370C22⋊Q8—24no—
371#121—1062no—
372#121—1062no—
373#121—2122no—
374#121—2122no—
375#117—22no—
376#117C22⋊D5314yescomplemented
377C2×C4⋊C4—22no—
378C2×C4⋊C4—22no—
379C2×C4⋊C4—532no—
380C2×C4⋊C4—531no—
381C2×C4⋊C4C22×D5311yescharacteristic
382C2×C4⋊C4C22×D5311yescharacteristic
383C2×C22⋊C4—21no—
384C2×C22⋊C4—22no—
385C2×C22⋊C4—22no—
386C2×C22⋊C4—1062no—
387C2×C22⋊C4—1061no—
388C2×C22⋊C4—1062no—
389C2×C22⋊C4—1061no—
390C2×C42—21no—
391#107—1062no—
392#106—1062no—
393#103—1064no—
394#103—2122no—
395#103—2122no—
396#102—1062no—
397#102—1062no—
398C22⋊C8—1062no—
399C22⋊C8—1062no—
400C22⋊C8—2122no—
401C22⋊C8—2122no—
402#95—1062no—
403#95—1061no—
404C24—1061no—
405C24C2×C22⋊D5312yes—
406C24C2×C22⋊D5311yescharacteristic
407#41—48no—
408#41—44no—
409#41—44no—
410C2×Q8—24no—
411C2×Q8—22no—
412C2×Q8—22no—
413C2×Q8—22no—
414C2×Q8—22no—
415C2×Q8—22no—
416C2×Q8—22no—
417C2×Q8—42no—
418C2×Q8—48no—
419C2×Q8—44no—
420C2×Q8—44no—
421C2×Q8—44no—
422C2×Q8C2×C22⋊D5314yes—
423C2×D4—24no—
424C2×D4—22no—
425C2×D4—42no—
426C2×D4—42no—
427C2×D4—42no—
428C2×D4—1061no—
429C2×D4—1061no—
430C2×D4—1062no—
431C2×D4—1062no—
432C2×D4—4242no—
433C2×D4—4242no—
434C2×D4—4242no—
435C2×D4—2124no—
436C2×D4—2122no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—24no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—44no—
450C22×C4—1061no—
451C22×C4—1062no—
452C22×C4C2×C22⋊D5312yes—
453C22×C4C2×C22⋊D5311yescharacteristic
454C22×C4C2×C22⋊D5312yes—
455C22×C4C2×C22⋊D5312yes—
456C22×C4C23×D5311yescharacteristic
457SD16—2128no—
458C2×C8—1061no—
459C2×C8—1061no—
460C2×C8—1062no—
461C2×C8—1062no—
462C2×C8—1062no—
463C2×C8—1062no—
464C2×C8—1062no—
465C4⋊C4—28no—
466C4⋊C4—28no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—1061no—
470C4⋊C4—1061no—
471C4⋊C4—1062no—
472C4⋊C4—1062no—
473C4⋊C4C2×C22⋊D5312yes—
474C4⋊C4C2×C22⋊D5312yes—
475C22⋊C4—28no—
476C22⋊C4—28no—
477C22⋊C4—24no—
478C22⋊C4—4241no—
479C22⋊C4—4242no—
480C22⋊C4—2124no—
481C22⋊C4—2122no—
482C42—24no—
483C23—21no—
484C23—22no—
485C23—22no—
486C23—21no—
487C23—22no—
488C23—22no—
489C23—22no—
490C23—22no—
491C23—22no—
492C23—22no—
493C23—22no—
494C23—2121no—
495C23—2122no—
496C23—2121no—
497C23—2122no—
498C23—2121no—
499C23#1313318612yes—
500C23C22×C22⋊D5311yescharacteristic
501Q8—24no—
502Q8—24no—
503Q8—48no—
504Q8—44no—
505Q8—42no—
506Q8—42no—
507D4—44no—
508D4—48no—
509D4—4244no—
510D4—4248no—
511D4—2122no—
512D4—2121no—
513D4—2121no—
514C2×C4—22no—
515C2×C4—24no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—24no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—48no—
525C2×C4—42no—
526C2×C4—41no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—41no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—2121no—
535C2×C4—2122no—
536C2×C4—2122no—
537C2×C4—2122no—
538C2×C4—2122no—
539C2×C4#1313318612yes—
540C2×C4C22×C22⋊D5311yescharacteristic
541C2×C4C22×C22⋊D5311yescharacteristic
542C2×C4#1313326212yes—
543C2×C4#1313326212yes—
544C8—2122no—
545C8—2121no—
546C8—2121no—
547C22—22no—
548C22—22no—
549C22—22no—
550C22—42no—
551C22—42no—
552C22—42no—
553C22—42no—
554C22—42no—
555C22—42no—
556C22—42no—
557C22—42no—
558C22—42no—
559C22—4241no—
560C22—4241no—
561C22—4241no—
562C22—4242no—
563C22—4242no—
564C22C2×#1313318611yescharacteristic
565C22#1681118212yes—
566C22#1681143112yes—
567C22#1681143111yescharacteristic
568C22#1681143811yescharacteristic
569C4—22no—
570C4—42no—
571C4—42no—
572C4—44no—
573C4—42no—
574C4—4241no—
575C4—4242no—
576C4#1681118212yes—
577C4#1681119011yescharacteristic
578C4#1681119011yescharacteristic
579C2—42no—
580C2—82no—
581C2—8481no—
582C2#2028042711yescharacteristic
583C2#2028400111yescharacteristic
584C2#2028403011yescharacteristic
585C2#2028558212yes—
586C2#2028559712yes—
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC2×C53×#536C2
socleC23×C53C22×D4
prime radical#20280439C2
hypercenterC2×#536D53
hypocenterC53#29603
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C22×#41 ◁ C2×#536

Complements

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

NH ≅ G/Ntype
1. C2×#16810947C2semidirect
2. C2×#16810762C2semidirect
4. #20275915C2semidirect
5. #20275908C2semidirect
6. #20275716C2semidirect
7. #20275716C2semidirect
8. #20274810C2semidirect
9. #20274650C2semidirect
10. #20274627C2semidirect
11. C2×C53×#536C2semidirect
86. C22×Q8×C53D4semidirect
101. #117×C53D4semidirect
286. C2×#536D53semidirect
338. C53#29603semidirect
348. C22×Q8C22⋊D53semidirect
376. #117C22⋊D53semidirect

Maximal subgroups

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

subgroupindex
C2×#168109472
C2×#168107622
#202804392
#202759152
#202759082
#202757162
#202748102
#202746502
#202746272
C2×C53×#5362
#2960353