Finite Group Databasebeta

← #27386325

Subgroups — #27386325

Order 13568.

898 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810658C211yesmaximal, characteristic
2#20277337C211yesmaximal, characteristic
3#20277069C211yesmaximal, characteristic
4#20275310C211yesmaximal, characteristic, complemented
5#20275304C211yesmaximal, characteristic, complemented
6#20275301C211yesmaximal, characteristic, complemented
7#20275299C211yesmaximal, characteristic, complemented
8#20275295C211yesmaximal, characteristic, complemented
9#20275286C211yesmaximal, characteristic, complemented
10#20275284C211yesmaximal, characteristic, complemented
11#20274433C211yesmaximal, characteristic
12#20274433C211yesmaximal, characteristic
13#340×D53C211yesmaximal, characteristic
14#20274410C211yesmaximal, characteristic
15C53×#1919C211yesmaximal, characteristic, complemented
16C22×#11719912C2211yescharacteristic
17C22×C8×D53C2211yescharacteristic
18C2×#13133175C2211yescharacteristic
19C2×#13133175C2211yescharacteristic
20C2×#13133174C2211yescharacteristic
21C2×C4⋊C4×D53C2211yescharacteristic
22C2×C4⋊C4×D53C2211yescharacteristic
23#16810658C418yescomplemented
24C2×#13133150C2211yescharacteristic
25C2×#13133150C2211yescharacteristic
26C2×#13133149C2211yescharacteristic
27#16810522C2211yescharacteristic
28#16810521C2211yescharacteristic
29#16810519C2211yescharacteristic
30#16810497—21no—
31#16810494—21no—
32#16810492—21no—
33#16810487—21no—
34C4⋊C4×Dic53C2211yescharacteristic
35C4⋊C4×Dic53C2211yescharacteristic
36C4×#11719919—21no—
37C4×C4⋊Dic53C2211yescharacteristic
38#16810474—21no—
39C4×#11719921—21no—
40#16810470—21no—
41#16810386C2211yescharacteristic
42#16810386C2211yescharacteristic
43#16810386C2211yescharacteristic
44#16810385C2211yescharacteristic
45#16810385C2211yescharacteristic
46#16810385C2211yescharacteristic
47#16810385C2211yescharacteristic
48#16810385C2211yescharacteristic
49#16810385C2211yescharacteristic
50#16810378C2211yescharacteristic
51#16810377C2211yescharacteristic
52#16810377C2211yescharacteristic
53C2×#106×C53C2211yescharacteristic
54C53×#389C2211yescharacteristic
55C4×C4⋊C4×C53C2211yescharacteristic
56C53×#340C2211yescharacteristic
57C53×#340C2211yescharacteristic
58C53×#340C2211yescharacteristic
59C53×#336C2211yescharacteristic
60C22×C4×D53—21no—
61C22×C4×D53—21no—
62C22×C4×D53C2311yescharacteristic
63C2×#11719912—22no—
64C2×#11719912—22no—
65C2×#11719912—22no—
66C22×#88322C2311yescharacteristic
67C22×#88322C2311yescharacteristic
68C2×C8×D53—22no—
69C2×C8×D53—22no—
70C2×C8×D53C2×C414yes—
71#13133175—22no—
72#13133175—22no—
73#13133175C2×C414yes—
74#13133175C2×C414yes—
75#13133174—22no—
76#13133174—22no—
77C2×C4⋊Dic53—21no—
78C2×C4⋊Dic53C2311yescharacteristic
79C2×C4⋊Dic53C2311yescharacteristic
80C2×C4⋊Dic53C2311yescharacteristic
81C2×C4⋊Dic53C2311yescharacteristic
82C2×#11719919—21no—
83C2×#11719919—21no—
84C2×#11719919—21no—
85C2×#11719919—21no—
86C2×#11719919D411yescharacteristic
87C2×#11719919D411yescharacteristic
88C4⋊C4×D53—22no—
89C4⋊C4×D53—22no—
90C4⋊C4×D53—22no—
91C4⋊C4×D53Q814yes—
92C2×#11719921—21no—
93C2×#11719921—21no—
94C2×#11719921D411yescharacteristic
95C2×#11719921D411yescharacteristic
96C2×C4×Dic53—21no—
97C2×C4×Dic53—21no—
98C2×C4×Dic53—21no—
99C2×C4×Dic53—21no—
100C2×C4×Dic53—21no—
101C2×C4×Dic53C2311yescharacteristic
102#13133150C2×C414yescomplemented
103#13133150C2×C414yescomplemented
104#13133149C2×C414yescomplemented
105#13133124—21no—
106#13133124—21no—
107#13133124—21no—
108#13133124—21no—
109#13133124—21no—
110#13133124—21no—
111C22×C8×C53C2311yescharacteristic
112C22×C8×C53C2311yescharacteristic
113C2×C4⋊C4×C53—21no—
114C2×C4⋊C4×C53C2311yescharacteristic
115C2×C4⋊C4×C53C2311yescharacteristic
116C2×C4⋊C4×C53C2311yescharacteristic
117C2×C4⋊C4×C53C2311yescharacteristic
118C2×C42×C53—21no—
119C2×C42×C53C2311yescharacteristic
120#106×C53C2×C414yescomplemented
121#95×C53—21no—
122#95×C53—21no—
123C23×D53C2×D411yescharacteristic
124C22×C4×C53—21no—
125C22×C4×C53—21no—
126C22×C4×C53—21no—
127C22×C4×C53—21no—
128C22×C4×C53C2×D411yescharacteristic
129C22×C4×C53C2×D411yescharacteristic
130C22×C4×C53C2411yescharacteristic
131C22×Dic53—21no—
132C22×Dic53—21no—
133C22×Dic53—21no—
134C22×Dic53—21no—
135C22×Dic53C2×D411yescharacteristic
136C22×Dic53C2×D411yescharacteristic
137C22×Dic53C2×D411yescharacteristic
138C2×C4×D53—24no—
139C2×C4×D53—42no—
140C2×C4×D53—81no—
141C2×C4×D53—81no—
142C2×C4×D53—81no—
143C2×C4×D53—81no—
144C2×C4×D53C22×C412yes—
145C2×C4×D53C2×Q812yes—
146C2×C4×D53#4112yes—
147C2×C4×D53#4112yes—
148C2×C4×D53#4112yes—
149C2×C4×D53#4112yes—
150C2×C8×C53—21no—
151C2×C8×C53—21no—
152C2×C8×C53—21no—
153C2×C8×C53—21no—
154C2×C8×C53—21no—
155C2×C8×C53C22×C412yes—
156C4⋊C4×C53—21no—
157C4⋊C4×C53—21no—
158C4⋊C4×C53—21no—
159C4⋊C4×C53—21no—
160C4⋊C4×C53—21no—
161C4⋊C4×C53—81no—
162C4⋊C4×C53C22×C412yes—
163C4⋊C4×C53C22×C412yes—
164C4⋊C4×C53C2×Q812yes—
165C42×C53—21no—
166C42×C53—21no—
167C42×C53—81no—
168#11719921—24no—
169#11719921—24no—
170#11719921—22no—
171#11719921—22no—
172C4⋊Dic53—21no—
173C4⋊Dic53—21no—
174C4⋊Dic53—21no—
175C4⋊Dic53—21no—
176C4⋊Dic53—21no—
177C4⋊Dic53—81no—
178C4⋊Dic53C22×C412yes—
179C4⋊Dic53C22×C412yes—
180C4⋊Dic53C2×Q812yes—
181#11719919—24no—
182#11719919—22no—
183#11719919—22no—
184#11719919—42no—
185#11719919—81no—
186#11719919—81no—
187C4×Dic53—21no—
188C4×Dic53—21no—
189C4×Dic53—24no—
190C4×Dic53—24no—
191C4×Dic53—81no—
192C4×Dic53—81no—
193C4×Dic53—81no—
194C2×#88322—21no—
195C2×#88322—21no—
196C2×#88322—21no—
197C2×#88322—21no—
198C2×#88322—21no—
199C2×#88322C22×C412yes—
200#11719912—44no—
201C8×D53—28no—
202C23×C53C22×D411yescharacteristic
203C22×D53—21no—
204C22×D53—21no—
205C22×D53—21no—
206C22×D53—21no—
207C22×D53C4×D412yes—
208C22×D53C22⋊Q812yes—
209C22×D53#12312yes—
210C2×C4×C53—22no—
211C2×C4×C53—22no—
212C2×C4×C53—22no—
213C2×C4×C53—21no—
214C2×C4×C53—21no—
215C2×C4×C53—21no—
216C2×C4×C53—21no—
217C2×C4×C53—21no—
218C2×C4×C53—21no—
219C2×C4×C53—41no—
220C2×C4×C53—41no—
221C2×C4×C53—41no—
222C2×C4×C53—41no—
223C2×C4×C53—41no—
224C2×C4×C53—41no—
225C2×C4×C53—41no—
226C2×C4×C53—41no—
227C2×C4×C53—41no—
228C2×C4×C53C23×C411yescharacteristic
229C2×C4×C53C22×Q811yescharacteristic
230C2×C4×C53C2×#4111yescharacteristic
231C2×C4×C53C2×#4111yescharacteristic
232C2×C4×C53C2×#4111yescharacteristic
233C2×C4×C53C2×#4111yescharacteristic
234C2×Dic53—22no—
235C2×Dic53—22no—
236C2×Dic53—22no—
237C2×Dic53—21no—
238C2×Dic53—21no—
239C2×Dic53—21no—
240C2×Dic53—21no—
241C2×Dic53—21no—
242C2×Dic53—21no—
243C2×Dic53—41no—
244C2×Dic53—41no—
245C2×Dic53—41no—
246C2×Dic53—41no—
247C2×Dic53—41no—
248C2×Dic53—41no—
249C2×Dic53—41no—
250C2×Dic53—41no—
251C2×Dic53—41no—
252C2×Dic53C4×D412yes—
253C2×Dic53C22⋊Q812yes—
254C2×Dic53#12312yes—
255C4×D53—84no—
256C4×D53—84no—
257C4×D53#11714yes—
258C4×D53C4×Q814yes—
259C4×D53#12514yes—
260C4×D53#12614yes—
261C8×C53—22no—
262C8×C53—41no—
263#88322—22no—
264#88322—41no—
265#21039—531nomaximal, Sylow
266C22×C53C2×C4×D411yescharacteristic
267C22×C53C2×C22⋊Q811yescharacteristic
268C22×C53C2×#12311yescharacteristic
269C22×C53#57611yescharacteristic
270C22×C53#57711yescharacteristic
271C22×C53#57711yescharacteristic
272C22×C53#57811yescharacteristic
273C2×D53—22no—
274C2×D53—22no—
275C2×D53—22no—
276C2×D53—22no—
277C2×D53—22no—
278C2×D53—22no—
279C2×D53#38714yes—
280C4×C53—42no—
281C4×C53—41no—
282C4×C53—41no—
283C4×C53—42no—
284C4×C53—42no—
285C4×C53—81no—
286C4×C53C2×#11711yescharacteristic
287C4×C53C2×C4×Q811yescharacteristic
288C4×C53C2×#12511yescharacteristic
289C4×C53C2×#12611yescharacteristic
290Dic53—42no—
291Dic53—41no—
292Dic53—41no—
293Dic53—42no—
294Dic53—42no—
295Dic53—81no—
296Dic53#38714yes—
297C2×#427—531no—
298#2317—531no—
299#2274—531no—
300#1921—531no—
301#1921—531no—
302#1920—531no—
303#1920—531no—
304#1919—531no—
305#1919D5311yescharacteristic, complemented
306#1918—531no—
307#1918—531no—
308#1736—531no—
309#1736—531no—
310C2×#340—531no—
311#1729—531no—
312C2×C53C2×#38711yescharacteristic
313C2×C53#295111yescharacteristic
314C2×C53#295211yescharacteristic
315C2×C53#308111yescharacteristic
316C2×C53#308211yescharacteristic
317C2×C53#309611yescharacteristic
318C2×C53#309711yescharacteristic
319D53—24no—
320C22×M16—531no—
321C23×C8—531no—
322C2×#117—531no—
323C2×#117—531no—
324C2×#117—531no—
325C22×C4⋊C4—531no—
326C22×C4⋊C4—531no—
327#427—538no—
328C2×#107—531no—
329C2×#107—531no—
330C2×#106—531no—
331C2×#106C2×D5311yescharacteristic
332#389—531no—
333#389—531no—
334#389—531no—
335#389C2×D5311yescharacteristic
336#385—1061no—
337#385—1061no—
338#384—1061no—
339#382—1061no—
340C4×C4⋊C4—531no—
341C4×C4⋊C4—531no—
342C4×C4⋊C4—531no—
343C4×C4⋊C4—1061no—
344C4×C4⋊C4—1061no—
345C4×C4⋊C4C2×D5311yescharacteristic
346C4×C22⋊C4—1061no—
347C4×C22⋊C4—1061no—
348#340—531no—
349#340—531no—
350#340—531no—
351#340—531no—
352#340—531no—
353#340—531no—
354#340—531no—
355#340—531no—
356#340—531no—
357#340C2×D5311yescharacteristic
358#340C2×D5311yescharacteristic
359#340C2×D5311yescharacteristic
360#336—531no—
361#336—531no—
362#336—531no—
363#336C2×D5311yescharacteristic
364C53#2103911yescharacteristic, complemented, Sylow
365C23×C4—531no—
366C23×C4—1061no—
367C23×C4—1061no—
368C2×M16—1062no—
369C2×M16—1062no—
370C2×M16—1062no—
371C22×C8—531no—
372C22×C8—534no—
373C22×C8—531no—
374C22×C8—1062no—
375C22×C8—1062no—
376C22×C8C22×D5311yescharacteristic
377C22×C8C22×D5311yescharacteristic
378#117—534no—
379#117—534no—
380#117—1062no—
381#117—1062no—
382#117—1062no—
383#117—1062no—
384C2×C4⋊C4—21no—
385C2×C4⋊C4—531no—
386C2×C4⋊C4—531no—
387C2×C4⋊C4—531no—
388C2×C4⋊C4—534no—
389C2×C4⋊C4—531no—
390C2×C4⋊C4—531no—
391C2×C4⋊C4—531no—
392C2×C4⋊C4—1062no—
393C2×C4⋊C4—1062no—
394C2×C4⋊C4—1062no—
395C2×C4⋊C4—1061no—
396C2×C4⋊C4—1061no—
397C2×C4⋊C4—1061no—
398C2×C4⋊C4—1061no—
399C2×C4⋊C4—1061no—
400C2×C4⋊C4C22×D5311yescharacteristic
401C2×C4⋊C4C22×D5311yescharacteristic
402C2×C4⋊C4C22×D5311yescharacteristic
403C2×C4⋊C4C22×D5311yescharacteristic
404C2×C22⋊C4—531no—
405C2×C22⋊C4—531no—
406C2×C22⋊C4—1061no—
407C2×C22⋊C4—1061no—
408C2×C42—21no—
409C2×C42—531no—
410C2×C42—1061no—
411C2×C42—1061no—
412C2×C42—1061no—
413C2×C42—1061no—
414C2×C42—1061no—
415C2×C42C22×D5311yescharacteristic
416#107—534no—
417#107—534no—
418#106—534no—
419#106C4×D5314yescomplemented
420#95—21no—
421#95—21no—
422#95—1061no—
423#95—1061no—
424#95—1061no—
425#95—1061no—
426#95—1061no—
427#95—1061no—
428C24—531no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—531no—
434C22×C4—532no—
435C22×C4—532no—
436C22×C4—532no—
437C22×C4—532no—
438C22×C4—532no—
439C22×C4—532no—
440C22×C4—531no—
441C22×C4—531no—
442C22×C4—1064no—
443C22×C4—1061no—
444C22×C4—1061no—
445C22×C4—1061no—
446C22×C4—1061no—
447C22×C4—4241no—
448C22×C4—4241no—
449C22×C4—4241no—
450C22×C4—4241no—
451C22×C4—2122no—
452C22×C4D4×D5311yescharacteristic
453C22×C4D4×D5311yescharacteristic
454C22×C4C23×D5311yescharacteristic
455M16—2124no—
456C2×C8—21no—
457C2×C8—21no—
458C2×C8—21no—
459C2×C8—21no—
460C2×C8—21no—
461C2×C8—532no—
462C2×C8—1068no—
463C2×C8—1061no—
464C2×C8—1061no—
465C2×C8—1061no—
466C2×C8—1061no—
467C2×C8—1061no—
468C2×C8C2×C4×D5312yes—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—81no—
475C4⋊C4—532no—
476C4⋊C4—532no—
477C4⋊C4—532no—
478C4⋊C4—1064no—
479C4⋊C4—1062no—
480C4⋊C4—1062no—
481C4⋊C4—1061no—
482C4⋊C4—1061no—
483C4⋊C4—1061no—
484C4⋊C4—1061no—
485C4⋊C4—1061no—
486C4⋊C4—4241no—
487C4⋊C4—4241no—
488C4⋊C4—4241no—
489C4⋊C4—2122no—
490C4⋊C4C2×C4×D5312yes—
491C4⋊C4C2×C4×D5312yes—
492C4⋊C4Q8×D5312yes—
493C22⋊C4—1062no—
494C22⋊C4—1062no—
495C22⋊C4—1064no—
496C22⋊C4—1064no—
497C42—21no—
498C42—21no—
499C42—81no—
500C42—1064no—
501C42—1064no—
502C42—1061no—
503C42—1061no—
504C42—4241no—
505C42—4241no—
506C42—4241no—
507C23—532no—
508C23—532no—
509C23—532no—
510C23—1061no—
511C23—1061no—
512C23—1061no—
513C23—1061no—
514C23C2×D4×D5311yescharacteristic
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—532no—
534C2×C4—532no—
535C2×C4—532no—
536C2×C4—534no—
537C2×C4—534no—
538C2×C4—534no—
539C2×C4—534no—
540C2×C4—1062no—
541C2×C4—1062no—
542C2×C4—1062no—
543C2×C4—1061no—
544C2×C4—1061no—
545C2×C4—1061no—
546C2×C4—1061no—
547C2×C4—1061no—
548C2×C4—1061no—
549C2×C4—4244no—
550C2×C4—4244no—
551C2×C4—2121no—
552C2×C4—2121no—
553C2×C4—2121no—
554C2×C4—2121no—
555C2×C4—2121no—
556C2×C4—2121no—
557C2×C4—2121no—
558C2×C4—2121no—
559C2×C4—2121no—
560C2×C4C22×C4×D5311yescharacteristic
561C2×C4C2×Q8×D5311yescharacteristic
562C2×C4#41×D5311yescharacteristic
563C2×C4#41×D5311yescharacteristic
564C2×C4#41×D5311yescharacteristic
565C2×C4#41×D5311yescharacteristic
566C8—22no—
567C8—41no—
568C8—1062no—
569C8—2121no—
570C22—534no—
571C22—1062no—
572C22—1062no—
573C22—1062no—
574C22—1062no—
575C22—1062no—
576C22—1062no—
577C22C4×D4×D5311yescharacteristic
578C22C22⋊Q8×D5311yescharacteristic
579C22#123×D5311yescharacteristic
580C22#1681142711yescharacteristic
581C22#1681142911yescharacteristic
582C22#1681143011yescharacteristic
583C22#1681143911yescharacteristic
584C4—41no—
585C4—41no—
586C4—42no—
587C4—42no—
588C4—42no—
589C4—81no—
590C4—534no—
591C4—4241no—
592C4—2121no—
593C4—2121no—
594C4—2122no—
595C4—2122no—
596C4—2122no—
597C4#117×D5311yescharacteristic
598C4C4×Q8×D5311yescharacteristic
599C4#125×D5311yescharacteristic
600C4#126×D5311yescharacteristic
601C2—1064no—
602C2#387×D5311yescharacteristic
603C2#2028358711yescharacteristic
604C2#2028359311yescharacteristic
605C2#2028454511yescharacteristic
606C2#2028455011yescharacteristic
607C2#2028467211yescharacteristic
608C2#2028468111yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#1919C2
socleC23×C53C22×D4
prime radicalC23×D53C2×D4
cosocleC22×C4×C53C24
hypercenter#1919D53
hypocenterC53#21039
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1919

Complements

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

NH ≅ G/Ntype
4. #20275310C2semidirect
5. #20275304C2semidirect
6. #20275301C2semidirect
7. #20275299C2semidirect
8. #20275295C2semidirect
9. #20275286C2semidirect
10. #20275284C2semidirect
15. C53×#1919C2semidirect
23. #16810658C4semidirect
102. #13133150C2×C4semidirect
103. #13133150C2×C4semidirect
104. #13133149C2×C4semidirect
120. #106×C53C2×C4semidirect
305. #1919D53semidirect
364. C53#21039semidirect
419. #106C4×D53semidirect

Maximal subgroups

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

subgroupindex
C2×#168106582
#202773372
#202770692
#202753102
#202753042
#202753012
#202752992
#202752952
#202752862
#202752842
#202744332
#340×D532
#202744102
C53×#19192
#2103953