Finite Group Databasebeta

← #27385096

Subgroups — #27385096

Order 13568.

1458 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×#16810663C211yesmaximal, characteristic, complemented
2C2×#16810624C211yesmaximal, characteristic
3C2×#16810619C211yesmaximal, characteristic
4#20277331C211yesmaximal, characteristic, complemented
5#20274983C212yesmaximal, complemented
6#20274976C212yesmaximal, complemented
7#20274969C212yesmaximal, complemented
8#20274962C212yesmaximal, complemented
9C2×#16810385C211yesmaximal, characteristic, complemented
10#20274412C211yesmaximal, characteristic, complemented
11C2×C53×#520C211yesmaximal, characteristic
12C22×#11719917C2211yescharacteristic
13C23×#88322C2211yescharacteristic
14C2×#13133176C2211yescharacteristic
15C22×C4⋊Dic53C2211yescharacteristic
16#16810663—24no—
17C2×#13133150C2212yes—
18C2×#13133148C2212yes—
19#16810624C418yescomplemented
20#16810619—24no—
21C2×#13133141C2211yescharacteristic
22C2×#13133141C2211yescharacteristic
23C2×#13133141C2212yes—
24C2×#13133138C2211yescharacteristic
25C2×#13133138C2211yescharacteristic
26C2×#13133138C2212yes—
27#16810521C2212yes—
28#16810521C2212yes—
29#16810519—21no—
30#16810519—21no—
31#16810481—22no—
32#16810385C2212yes—
33#16810385C2212yes—
34#16810378C2212yes—
35#16810378C2212yes—
36C22×#41×C53C2211yescharacteristic
37C53×#520—24no—
38C53×#520—24no—
39C2×C4×Q8×C53C2212yes—
40C2×C4×D4×C53—22no—
41C2×C4×D4×C53C2212yes—
42C2×#117×C53—21no—
43C2×#117×C53C2211yescharacteristic
44C22×C4⋊C4×C53—21no—
45C22×C4⋊C4×C53C2211yescharacteristic
46C23×Dic53—21no—
47C2×#11719917—22no—
48C2×#11719917—22no—
49C2×#11719917C2×C414yescomplemented
50C22×#88322—22no—
51C22×#88322—22no—
52C22×#88322—22no—
53C22×#88322C2311yescharacteristic
54C22×#88322C2311yescharacteristic
55C22×#88322C2312yes—
56#13133176—22no—
57#13133176—22no—
58C2×C4⋊Dic53—21no—
59C2×C4⋊Dic53—22no—
60C2×C4⋊Dic53C2×C414yescomplemented
61C2×C4⋊Dic53C2311yescharacteristic
62C2×C4⋊Dic53C2311yescharacteristic
63C2×C4⋊Dic53C2312yes—
64C2×#11719919—22no—
65C2×#11719919—22no—
66C2×C22⋊Dic53—21no—
67C2×C4×Dic53—21no—
68#13133150—24no—
69#13133148—24no—
70#13133141—22no—
71#13133141—22no—
72#13133141C2×C418yescomplemented
73#13133138—22no—
74#13133138—22no—
75#13133138C2×C418yescomplemented
76#13133124—22no—
77#13133124—22no—
78C2×#41×C53—22no—
79C2×#41×C53—22no—
80C2×#41×C53—22no—
81C2×#41×C53C2×C414yes—
82C2×#41×C53D414yes—
83C2×#41×C53Q814yes—
84C22×Q8×C53C2311yescharacteristic
85C22×D4×C53—21no—
86C22×D4×C53C2311yescharacteristic
87C23×C4×C53—21no—
88C23×C4×C53—21no—
89C23×C4×C53—22no—
90C23×C4×C53C2311yescharacteristic
91C4×Q8×C53—24no—
92C4×Q8×C53—24no—
93C4×D4×C53—216no—
94C4×D4×C53—24no—
95C4×D4×C53—24no—
96#117×C53—28no—
97#117×C53—22no—
98#117×C53—22no—
99C2×C4⋊C4×C53—28no—
100C2×C4⋊C4×C53—21no—
101C2×C4⋊C4×C53—22no—
102C2×C4⋊C4×C53—22no—
103C2×C4⋊C4×C53D414yes—
104C2×C4⋊C4×C53D411yescharacteristic
105C2×C4⋊C4×C53D411yescharacteristic
106C2×C4⋊C4×C53C2312yes—
107C2×C22⋊C4×C53—22no—
108C2×C22⋊C4×C53D412yes—
109C2×C42×C53—22no—
110C2×C42×C53C2312yes—
111C24×C53—21no—
112C24×C53C2×D411yescharacteristic
113#41×C53—28no—
114#41×C53—28no—
115#41×C53—28no—
116#41×C53C4⋊C4116yescomplemented
117C2×Q8×C53—21no—
118C2×Q8×C53—21no—
119C2×Q8×C53—21no—
120C2×Q8×C53C22×C412yes—
121C2×Q8×C53C2×D412yes—
122C2×Q8×C53C2×Q812yes—
123C2×D4×C53—24no—
124C2×D4×C53—24no—
125C2×D4×C53—24no—
126C2×D4×C53—21no—
127C2×D4×C53—21no—
128C2×D4×C53—21no—
129C2×D4×C53C22×C412yes—
130C2×D4×C53C2×D412yes—
131C2×D4×C53C2×Q812yes—
132C22×C4×C53—21no—
133C22×C4×C53—24no—
134C22×C4×C53—24no—
135C22×C4×C53—24no—
136C22×C4×C53—22no—
137C22×C4×C53—28no—
138C22×C4×C53—24no—
139C22×C4×C53—82no—
140C22×C4×C53—82no—
141C22×C4×C53—81no—
142C22×C4×C53—81no—
143C22×C4×C53C22×C412yes—
144C22×C4×C53C2×D412yes—
145C22×C4×C53C2×D412yes—
146C22×C4×C53C2×D412yes—
147C22×C4×C53C2×D411yescharacteristic
148C22×C4×C53C2×D412yes—
149C22×C4×C53C2×Q812yes—
150C22×C4×C53#4112yes—
151C22×C4×C53#4112yes—
152C22×C4×C53C2411yescharacteristic
153C22×Dic53—22no—
154C22×Dic53—24no—
155C22×Dic53—21no—
156C22×Dic53—21no—
157C22×Dic53—81no—
158C22×Dic53—81no—
159C4⋊C4×C53—28no—
160C4⋊C4×C53—24no—
161C4⋊C4×C53—22no—
162C4⋊C4×C53—22no—
163C4⋊C4×C53—22no—
164C4⋊C4×C53C2×D414yes—
165C22⋊C4×C53—28no—
166C22⋊C4×C53—24no—
167C42×C53—28no—
168C42×C53—22no—
169C42×C53—22no—
170C22⋊Dic53—42no—
171C4⋊Dic53—24no—
172C4⋊Dic53—22no—
173C4⋊Dic53—21no—
174C4⋊Dic53—21no—
175C4⋊Dic53—21no—
176C4⋊Dic53—21no—
177C4⋊Dic53C22×C414yes—
178#11719919—82no—
179#11719919—82no—
180C4×Dic53—42no—
181#11719917—28no—
182C2×#88322—22no—
183C2×#88322—22no—
184C2×#88322—21no—
185C2×#88322—21no—
186C2×#88322—21no—
187C2×#88322—21no—
188C2×#88322—21no—
189C2×#88322—21no—
190C2×#88322—44no—
191C2×#88322C22×C414yes—
192C23×C53—21no—
193C23×C53—21no—
194C23×C53—21no—
195C23×C53—21no—
196C23×C53—22no—
197C23×C53—22no—
198C23×C53—22no—
199C23×C53—41no—
200C23×C53—41no—
201C23×C53—41no—
202C23×C53—41no—
203C23×C53C4×D412yes—
204C23×C53#12012yes—
205C23×C53C22⋊Q812yes—
206C23×C53C22×D411yescharacteristic
207Q8×C53—22no—
208Q8×C53—22no—
209Q8×C53—22no—
210Q8×C53C2×C4⋊C414yes—
211D4×C53—216no—
212D4×C53—22no—
213D4×C53—22no—
214D4×C53—22no—
215D4×C53C2×C4⋊C414yes—
216C2×C4×C53—22no—
217C2×C4×C53—22no—
218C2×C4×C53—22no—
219C2×C4×C53—216no—
220C2×C4×C53—24no—
221C2×C4×C53—22no—
222C2×C4×C53—22no—
223C2×C4×C53—22no—
224C2×C4×C53—22no—
225C2×C4×C53—22no—
226C2×C4×C53—22no—
227C2×C4×C53—42no—
228C2×C4×C53—42no—
229C2×C4×C53—84no—
230C2×C4×C53—88no—
231C2×C4×C53C2×C4⋊C414yes—
232C2×C4×C53C4×D414yes—
233C2×C4×C53C4×D412yes—
234C2×C4×C53#12012yes—
235C2×C4×C53C22⋊Q814yes—
236C2×C4×C53C22⋊Q812yes—
237C2×C4×C53#12314yes—
238C2×C4×C53C23×C411yescharacteristic
239C2×C4×C53C22×D411yescharacteristic
240C2×C4×C53C22×D411yescharacteristic
241C2×C4×C53C22×Q811yescharacteristic
242C2×C4×C53C2×#4111yescharacteristic
243C2×C4×C53C2×#4111yescharacteristic
244C2×Dic53—24no—
245C2×Dic53—42no—
246C2×Dic53—42no—
247C2×Dic53—41no—
248C2×Dic53—41no—
249C2×Dic53—41no—
250C2×Dic53—41no—
251C2×Dic53—41no—
252C2×Dic53—41no—
253C2×Dic53—84no—
254#88322—24no—
255#88322—41no—
256#88322—41no—
257#20878—531nomaximal, Sylow
258C22×C53—22no—
259C22×C53—22no—
260C22×C53—22no—
261C22×C53—22no—
262C22×C53—22no—
263C22×C53—22no—
264C22×C53—24no—
265C22×C53—42no—
266C22×C53—42no—
267C22×C53—42no—
268C22×C53—42no—
269C22×C53—42no—
270C22×C53—42no—
271C22×C53#38514yes—
272C22×C53C2×C4×D411yescharacteristic
273C22×C53C2×#12011yescharacteristic
274C22×C53C2×C22⋊Q811yescharacteristic
275C22×C53#57611yescharacteristic
276C22×C53#57711yescharacteristic
277C22×C53#57711yescharacteristic
278C22×C53#57811yescharacteristic
279C4×C53—24no—
280C4×C53—42no—
281C4×C53—44no—
282C4×C53—42no—
283C4×C53#38514yes—
284C4×C53C22×C4⋊C411yescharacteristic
285C4×C53C2×C4×D411yescharacteristic
286C4×C53C2×C22⋊Q811yescharacteristic
287C4×C53C2×#12311yescharacteristic
288Dic53—44no—
289Dic53—81no—
290Dic53—81no—
291C2×#520D5311yescharacteristic
292C2×#428—531no—
293C2×#417—531no—
294C2×#416—531no—
295#2316—531no—
296#1865—532no—
297#1864—532no—
298#1863—532no—
299#1862—532no—
300C2×#340—531no—
301#1729—531no—
302C2×C53—24no—
303C2×C53—44no—
304C2×C53C2×#38511yescharacteristic
305C2×C53#294611yescharacteristic
306C2×C53#294711yescharacteristic
307C2×C53#301211yescharacteristic
308C2×C53#301311yescharacteristic
309C2×C53#308311yescharacteristic
310C2×C53#308411yescharacteristic
311C22×#41C2×D5311yescharacteristic
312C22×M16—531no—
313C23×C8—531no—
314#520—24no—
315#520—24no—
316C2×C4×Q8C2×D5312yes—
317C2×C4×D4—22no—
318C2×C4×D4C2×D5312yes—
319C2×#117—21no—
320C2×#117—531no—
321C2×#117C2×D5311yescharacteristic
322C22×C4⋊C4—21no—
323C22×C4⋊C4—531no—
324C22×C4⋊C4C2×D5311yescharacteristic
325#428—1064no—
326C2×#107—532no—
327C2×#106—532no—
328#417—538no—
329#416—1064no—
330C2×#103—532no—
331C2×#103—531no—
332C2×#103—531no—
333C2×#102—532no—
334C2×#102—531no—
335C2×#102—531no—
336#389—532no—
337#389—532no—
338#389—1061no—
339#389—1061no—
340#380—1062no—
341#340—532no—
342#340—532no—
343#336—532no—
344#336—532no—
345C53#2087811yescharacteristic, complemented, Sylow
346C2×#41—22no—
347C2×#41—22no—
348C2×#41—22no—
349C2×#41Dic10614yes—
350C2×#41C4×D5314yes—
351C2×#41C22⋊D5314yes—
352C22×Q8C22×D5311yescharacteristic
353C22×D4—21no—
354C22×D4C22×D5311yescharacteristic
355C23×C4—21no—
356C23×C4—21no—
357C23×C4—22no—
358C23×C4—1061no—
359C23×C4C22×D5311yescharacteristic
360C2×M16—534no—
361C2×M16—1062no—
362C2×M16—1062no—
363C22×C8—532no—
364C22×C8—531no—
365C22×C8—531no—
366C22×C8—1062no—
367C22×C8—1062no—
368C22×C8—1062no—
369C4×Q8—24no—
370C4×Q8—24no—
371C4×D4—216no—
372C4×D4—24no—
373C4×D4—24no—
374#117—28no—
375#117—22no—
376#117—22no—
377#117—1062no—
378#117—1062no—
379C2×C4⋊C4—28no—
380C2×C4⋊C4—21no—
381C2×C4⋊C4—22no—
382C2×C4⋊C4—22no—
383C2×C4⋊C4—534no—
384C2×C4⋊C4—532no—
385C2×C4⋊C4—531no—
386C2×C4⋊C4—531no—
387C2×C4⋊C4—1062no—
388C2×C4⋊C4—1061no—
389C2×C4⋊C4—1062no—
390C2×C4⋊C4—1062no—
391C2×C4⋊C4C22⋊D5311yescharacteristic
392C2×C4⋊C4C22⋊D5311yescharacteristic
393C2×C4⋊C4C22⋊D5314yes—
394C2×C4⋊C4C22×D5312yes—
395C2×C22⋊C4—22no—
396C2×C22⋊C4—1061no—
397C2×C22⋊C4C22⋊D5312yes—
398C2×C42—22no—
399C2×C42—1061no—
400C2×C42C22×D5312yes—
401#107—1064no—
402#106—1064no—
403#103—538no—
404#103—1062no—
405#103—1062no—
406#102—538no—
407#102—1062no—
408#102—1062no—
409#95—1062no—
410#95—1062no—
411C24—21no—
412C24C2×C22⋊D5311yescharacteristic
413#41—28no—
414#41—28no—
415#41—28no—
416#41#11719919116yescomplemented
417C2×Q8—21no—
418C2×Q8—21no—
419C2×Q8—21no—
420C2×Q8C2×Dic10612yes—
421C2×Q8C2×C4×D5312yes—
422C2×Q8C2×C22⋊D5312yes—
423C2×D4—24no—
424C2×D4—24no—
425C2×D4—24no—
426C2×D4—21no—
427C2×D4—21no—
428C2×D4—21no—
429C2×D4C2×Dic10612yes—
430C2×D4C2×C4×D5312yes—
431C2×D4C2×C22⋊D5312yes—
432C22×C4—21no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—22no—
437C22×C4—28no—
438C22×C4—24no—
439C22×C4—82no—
440C22×C4—82no—
441C22×C4—81no—
442C22×C4—81no—
443C22×C4—1064no—
444C22×C4—1062no—
445C22×C4—1061no—
446C22×C4—1061no—
447C22×C4—4241no—
448C22×C4—4241no—
449C22×C4C2×Dic10612yes—
450C22×C4C2×C4×D5312yes—
451C22×C4#1171994512yes—
452C22×C4#1171994512yes—
453C22×C4C2×C22⋊D5312yes—
454C22×C4C2×C22⋊D5312yes—
455C22×C4C2×C22⋊D5312yes—
456C22×C4C2×C22⋊D5311yescharacteristic
457C22×C4C2×C22⋊D5312yes—
458C22×C4C23×D5311yescharacteristic
459M16—1068no—
460C2×C8—534no—
461C2×C8—1062no—
462C2×C8—1062no—
463C2×C8—1061no—
464C2×C8—1061no—
465C2×C8—1061no—
466C2×C8—1061no—
467C2×C8—1061no—
468C2×C8—1061no—
469C2×C8—2124no—
470C4⋊C4—28no—
471C4⋊C4—22no—
472C4⋊C4—24no—
473C4⋊C4—22no—
474C4⋊C4—22no—
475C4⋊C4—534no—
476C4⋊C4—1064no—
477C4⋊C4—1062no—
478C4⋊C4—1061no—
479C4⋊C4—1061no—
480C4⋊C4—1061no—
481C4⋊C4—1061no—
482C4⋊C4—4242no—
483C4⋊C4—4242no—
484C4⋊C4C2×C22⋊D5314yes—
485C22⋊C4—28no—
486C22⋊C4—24no—
487C22⋊C4—2122no—
488C42—28no—
489C42—22no—
490C42—22no—
491C42—2122no—
492C23—21no—
493C23—21no—
494C23—21no—
495C23—21no—
496C23—22no—
497C23—22no—
498C23—22no—
499C23—41no—
500C23—41no—
501C23—41no—
502C23—41no—
503C23C4×C22⋊D5312yes—
504C23#1313318612yes—
505C23#1313319612yes—
506C23C22×C22⋊D5311yescharacteristic
507Q8—22no—
508Q8—22no—
509Q8—22no—
510Q8C2×#1171991914yes—
511D4—216no—
512D4—22no—
513D4—22no—
514D4—22no—
515D4C2×#1171991914yes—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—216no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—24no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—84no—
530C2×C4—88no—
531C2×C4—1064no—
532C2×C4—4244no—
533C2×C4—2121no—
534C2×C4—2121no—
535C2×C4—2121no—
536C2×C4—2121no—
537C2×C4—2121no—
538C2×C4—2121no—
539C2×C4—2122no—
540C2×C4—2122no—
541C2×C4C2×#1171991914yes—
542C2×C4C4×C22⋊D5312yes—
543C2×C4C4×C22⋊D5314yes—
544C2×C4#1313318612yes—
545C2×C4#1313319612yes—
546C2×C4#1313319614yes—
547C2×C4#1313320014yes—
548C2×C4C22×C4×D5311yescharacteristic
549C2×C4C22×C22⋊D5311yescharacteristic
550C2×C4C22×C22⋊D5311yescharacteristic
551C2×C4C22×Dic10611yescharacteristic
552C2×C4C2×#1171994511yescharacteristic
553C2×C4C2×#1171994511yescharacteristic
554C8—1064no—
555C8—2121no—
556C8—2121no—
557C22—22no—
558C22—22no—
559C22—22no—
560C22—22no—
561C22—22no—
562C22—22no—
563C22—24no—
564C22—42no—
565C22—42no—
566C22—42no—
567C22—42no—
568C22—42no—
569C22—42no—
570C22#1681049614yes—
571C22C2×C4×C22⋊D5311yescharacteristic
572C22C2×#1313318611yescharacteristic
573C22C2×#1313319611yescharacteristic
574C22#1681142611yescharacteristic
575C22#1681143111yescharacteristic
576C22#1681143411yescharacteristic
577C22#1681143811yescharacteristic
578C4—24no—
579C4—42no—
580C4—44no—
581C4—42no—
582C4—4241no—
583C4—4241no—
584C4—2124no—
585C4#1681049614yes—
586C4C22×#1171991911yescharacteristic
587C4C2×C4×C22⋊D5311yescharacteristic
588C4C2×#1313319611yescharacteristic
589C4C2×#1313320011yescharacteristic
590C2—24no—
591C2—44no—
592C2C2×#1681049611yescharacteristic
593C2#2028354311yescharacteristic
594C2#2028355411yescharacteristic
595C2#2028401911yescharacteristic
596C2#2028403011yescharacteristic
597C2#2028456611yescharacteristic
598C2#2028457711yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC2×C53×#520C2
socleC23×C53C22×D4
prime radicalC22×#41×C53C22
cosocleC22×C4×C53C24
hypercenterC2×#520D53
hypocenterC53#20878
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×#117 ◁ C2×#520

Complements

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

NH ≅ G/Ntype
1. C2×#16810663C2semidirect
4. #20277331C2semidirect
5. #20274983C2semidirect
6. #20274976C2semidirect
7. #20274969C2semidirect
8. #20274962C2semidirect
9. C2×#16810385C2semidirect
10. #20274412C2semidirect
19. #16810624C4semidirect
49. C2×#11719917C2×C4semidirect
60. C2×C4⋊Dic53C2×C4semidirect
72. #13133141C2×C4semidirect
75. #13133138C2×C4semidirect
116. #41×C53C4⋊C4semidirect
345. C53#20878semidirect
416. #41#11719919semidirect

Maximal subgroups

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

subgroupindex
C2×#168106632
C2×#168106242
C2×#168106192
#202773312
#202749832
#202749762
#202749692
#202749622
C2×#168103852
#202744122
C2×C53×#5202
#2087853