Finite Group Databasebeta

← #27425474

Subgroups — #27425474

Order 13568.

738 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810930C211yesmaximal, characteristic, complemented
2C2×#16810684C211yesmaximal, characteristic
3#20280545C211yesmaximal, characteristic
4#20276152C211yesmaximal, characteristic
5#20276141C211yesmaximal, characteristic
6#20275915C211yesmaximal, characteristic
7#20275915C211yesmaximal, characteristic
8#20275672C211yesmaximal, characteristic, complemented
9#20275473C211yesmaximal, characteristic, complemented
10#20275450C211yesmaximal, characteristic, complemented
11#20275445C211yesmaximal, characteristic, complemented
12#20275408C211yesmaximal, characteristic, complemented
13#20275394C211yesmaximal, characteristic, complemented
14#20274720C211yesmaximal, characteristic, complemented
15C53×#2323C211yesmaximal, characteristic
16C2×#13133209C2211yescharacteristic
17#16810930—24no—
18#16810684—24no—
19C2×#13133150C2211yescharacteristic
20C2×#13133150C2211yescharacteristic
21C2×#13133150C2211yescharacteristic
22C2×#13133148C2211yescharacteristic
23C2×C4⋊#88322C2211yescharacteristic
24C2×C4⋊#88322C2211yescharacteristic
25C2×#13133141C2211yescharacteristic
26C2×#13133141C2211yescharacteristic
27C2×#13133141C2211yescharacteristic
28C2×#13133141C2211yescharacteristic
29C2×C22⋊#88322C2211yescharacteristic
30C2×C22⋊#88322C2211yescharacteristic
31C2×#13133128C2211yescharacteristic
32C2×C4×#88322C2211yescharacteristic
33#16810556—21no—
34#16810556—21no—
35#16810546—21no—
36#16810546—21no—
37#16810521C2211yescharacteristic
38#16810521C2211yescharacteristic
39#16810493C2211yescharacteristic
40#16810483C2211yescharacteristic
41#16810483C2211yescharacteristic
42#16810385C2211yescharacteristic
43#16810385C2211yescharacteristic
44#16810385C2211yescharacteristic
45#16810385C2211yescharacteristic
46#16810385C2211yescharacteristic
47#16810385C2211yescharacteristic
48#16810385C2211yescharacteristic
49#16810385C2211yescharacteristic
50C2×C22⋊Q8×C53C2211yescharacteristic
51C2×C4×Q8×C53C2211yescharacteristic
52C2×#117×C53C2211yescharacteristic
53C53×#391C2211yescharacteristic
54C53×#389—21no—
55C53×#389C2211yescharacteristic
56C53×#388—21no—
57C53×#388—21no—
58C53×#387—21no—
59C53×#380C2211yescharacteristic
60C4×C4⋊C4×C53C2211yescharacteristic
61C22×#88322C2311yescharacteristic
62C22×#88322C2311yescharacteristic
63C22×#88322C2311yescharacteristic
64C22×#88322C2311yescharacteristic
65#13133209—22no—
66#13133209—22no—
67C2×C4⋊Dic53—21no—
68C2×C4⋊Dic53—21no—
69C2×C4⋊Dic53C2311yescharacteristic
70C2×C4⋊Dic53C2311yescharacteristic
71C2×C4⋊Dic53C2311yescharacteristic
72C2×C4⋊Dic53C2311yescharacteristic
73C2×#11719919—21no—
74C2×#11719919—21no—
75C2×#11719919—21no—
76C2×#11719919—21no—
77C2×C22⋊Dic53—21no—
78C2×C22⋊Dic53—21no—
79#13133150—22no—
80#13133150—22no—
81#13133150—22no—
82#13133148—22no—
83C4⋊#88322—22no—
84C4⋊#88322D414yes—
85#13133141—22no—
86#13133141—22no—
87#13133141—22no—
88#13133141—22no—
89C22⋊#88322—22no—
90C22⋊#88322—22no—
91#13133128—22no—
92C4×#88322—22no—
93#13133124—21no—
94#13133124—21no—
95#13133124—21no—
96#13133124—21no—
97C22×Q8×C53C2311yescharacteristic
98C23×C4×C53C2311yescharacteristic
99C22⋊Q8×C53—22no—
100C22⋊Q8×C53—22no—
101C4×Q8×C53—22no—
102C4×Q8×C53—22no—
103#117×C53—22no—
104#117×C53—22no—
105C2×C4⋊C4×C53—21no—
106C2×C4⋊C4×C53—21no—
107C2×C4⋊C4×C53—21no—
108C2×C4⋊C4×C53D411yescharacteristic
109C2×C4⋊C4×C53D411yescharacteristic
110C2×C4⋊C4×C53C2311yescharacteristic
111C2×C4⋊C4×C53C2311yescharacteristic
112C2×C4⋊C4×C53C2311yescharacteristic
113C2×C22⋊C4×C53—21no—
114C2×C22⋊C4×C53—21no—
115C2×C22⋊C4×C53D411yescharacteristic
116C2×C22⋊C4×C53D411yescharacteristic
117C2×C42×C53—21no—
118C2×C42×C53—21no—
119C2×C42×C53C2311yescharacteristic
120C2×C42×C53C2311yescharacteristic
121#95×C53—21no—
122#95×C53—21no—
123#95×C53—21no—
124#95×C53—21no—
125#95×C53—21no—
126C24×C53C2×D411yescharacteristic
127C2×Q8×C53—21no—
128C2×Q8×C53—21no—
129C2×Q8×C53—21no—
130C2×Q8×C53—21no—
131C2×Q8×C53—21no—
132C2×Q8×C53#4112yes—
133C22×C4×C53—21no—
134C22×C4×C53—21no—
135C22×C4×C53—21no—
136C22×C4×C53—21no—
137C22×C4×C53—21no—
138C22×C4×C53—21no—
139C22×C4×C53—21no—
140C22×C4×C53—21no—
141C22×C4×C53—21no—
142C22×C4×C53—21no—
143C22×C4×C53C2×D411yescharacteristic
144C22×C4×C53C2×D411yescharacteristic
145C22×C4×C53C2×D411yescharacteristic
146C22×C4×C53C2×D411yescharacteristic
147C22×C4×C53C2×D411yescharacteristic
148C22×C4×C53C2411yescharacteristic
149C22×Dic53—21no—
150C22×Dic53—21no—
151C22×Dic53—21no—
152C22×Dic53—21no—
153C4⋊C4×C53—24no—
154C4⋊C4×C53—22no—
155C4⋊C4×C53—22no—
156C4⋊C4×C53—21no—
157C4⋊C4×C53—21no—
158C4⋊C4×C53—21no—
159C4⋊C4×C53—21no—
160C4⋊C4×C53—21no—
161C4⋊C4×C53—42no—
162C4⋊C4×C53—81no—
163C4⋊C4×C53#4112yes—
164C22⋊C4×C53—24no—
165C22⋊C4×C53—22no—
166C22⋊C4×C53—22no—
167C22⋊C4×C53—81no—
168C42×C53—21no—
169C42×C53—21no—
170C42×C53—42no—
171C42×C53—81no—
172C42×C53C2×D412yes—
173C42×C53#4112yes—
174C22⋊Dic53—81no—
175C22⋊Dic53—81no—
176C4⋊Dic53—21no—
177C4⋊Dic53—21no—
178C4⋊Dic53—21no—
179C4⋊Dic53—21no—
180C4⋊Dic53—21no—
181C4⋊Dic53—21no—
182C4⋊Dic53—21no—
183C4⋊Dic53—21no—
184C4⋊Dic53—81no—
185C4⋊Dic53—81no—
186#11719919—42no—
187#11719919—42no—
188#11719919—81no—
189#11719919—81no—
190C2×#88322—21no—
191C2×#88322—21no—
192C2×#88322—21no—
193C2×#88322—21no—
194C2×#88322—21no—
195C2×#88322—21no—
196C2×#88322—21no—
197C2×#88322—21no—
198C2×#88322—21no—
199C2×#88322—21no—
200C2×#88322C2×D412yes—
201C2×#88322C2×D412yes—
202C23×C53—21no—
203C23×C53—21no—
204C23×C53—21no—
205C23×C53—21no—
206C23×C53—21no—
207C23×C53—21no—
208C23×C53—21no—
209C23×C53C22×D411yescharacteristic
210Q8×C53—22no—
211Q8×C53—22no—
212Q8×C53—41no—
213Q8×C53—41no—
214C2×C4×C53—21no—
215C2×C4×C53—21no—
216C2×C4×C53—21no—
217C2×C4×C53—21no—
218C2×C4×C53—21no—
219C2×C4×C53—21no—
220C2×C4×C53—21no—
221C2×C4×C53—21no—
222C2×C4×C53—22no—
223C2×C4×C53—22no—
224C2×C4×C53—21no—
225C2×C4×C53—21no—
226C2×C4×C53—21no—
227C2×C4×C53—21no—
228C2×C4×C53—21no—
229C2×C4×C53—41no—
230C2×C4×C53—41no—
231C2×C4×C53—41no—
232C2×C4×C53—41no—
233C2×C4×C53—41no—
234C2×C4×C53—41no—
235C2×C4×C53—41no—
236C2×C4×C53—41no—
237C2×C4×C53—41no—
238C2×C4×C53—41no—
239C2×C4×C53—41no—
240C2×C4×C53—41no—
241C2×C4×C53—41no—
242C2×C4×C53—41no—
243C2×C4×C53#12112yes—
244C2×C4×C53#12112yes—
245C2×C4×C53C22×D411yescharacteristic
246C2×C4×C53C2×#4111yescharacteristic
247C2×C4×C53C2×#4111yescharacteristic
248C2×C4×C53C2×#4111yescharacteristic
249C2×C4×C532−1+411yescharacteristic
250C2×C4×C532−1+411yescharacteristic
251C2×Dic53—41no—
252C2×Dic53—41no—
253C2×Dic53—41no—
254C2×Dic53—41no—
255C2×Dic53—41no—
256C2×Dic53—41no—
257C2×Dic53—41no—
258C2×Dic53—41no—
259C2×Dic53—41no—
260C2×Dic53—41no—
261C2×Dic53—41no—
262C2×Dic53—41no—
263#88322—22no—
264#88322—22no—
265#88322—41no—
266#88322—41no—
267#24715—531nomaximal, Sylow
268C22×C53—41no—
269C22×C53—41no—
270C22×C53—41no—
271C22×C53—41no—
272C22×C53—41no—
273C22×C53—41no—
274C22×C53—41no—
275C22×C53C2×#12111yescharacteristic
276C22×C53#52511yescharacteristic
277C22×C53#53611yescharacteristic
278C22×C53#57611yescharacteristic
279C22×C53#57711yescharacteristic
280C22×C53#57811yescharacteristic
281C22×C53#57811yescharacteristic
282C4×C53—22no—
283C4×C53—22no—
284C4×C53—42no—
285C4×C53—41no—
286C4×C53—42no—
287C4×C53—41no—
288C4×C53—41no—
289C4×C53—81no—
290C4×C53—81no—
291C4×C53#54711yescharacteristic
292C4×C53#54811yescharacteristic
293C4×C53#55211yescharacteristic
294C4×C53#55611yescharacteristic
295Dic53—81no—
296Dic53—81no—
297Dic53—81no—
298Dic53—81no—
299C2×#479—531no—
300C2×#433—531no—
301#2629—531no—
302#2323D5311yescharacteristic
303#2077—531no—
304#2075—531no—
305#2036—531no—
306#2036—531no—
307#1989—531no—
308#1947—531no—
309#1943—531no—
310#1942—531no—
311#1935—531no—
312#1933—531no—
313#1803—531no—
314C2×C53—81no—
315C2×C53#241511yescharacteristic
316C2×C53#306111yescharacteristic
317C2×C53#306311yescharacteristic
318C2×C53#311911yescharacteristic
319C2×C53#312711yescharacteristic
320C2×C53#324211yescharacteristic
321C2×C53#324311yescharacteristic
322C2×#125—531no—
323C2×C22⋊Q8C2×D5311yescharacteristic
324C2×C4×Q8C2×D5311yescharacteristic
325C2×#117C2×D5311yescharacteristic
326#479—1064no—
327#433—1064no—
328C2×#107—531no—
329C2×#107—531no—
330C2×#107—531no—
331C2×#106—531no—
332C2×C4⋊C8—531no—
333C2×C4⋊C8—531no—
334C2×#103—531no—
335C2×#103—531no—
336C2×#103—531no—
337C2×#103—531no—
338C2×C22⋊C8—531no—
339C2×C22⋊C8—531no—
340C2×#97—531no—
341C2×C4×C8—531no—
342#398—1061no—
343#398—1061no—
344#396—1061no—
345#396—1061no—
346#391C2×D5311yescharacteristic
347#389—21no—
348#389—531no—
349#389—531no—
350#389C2×D5311yescharacteristic
351#388—21no—
352#388—21no—
353#387—21no—
354#384—531no—
355#380—531no—
356#380—531no—
357#380C2×D5311yescharacteristic
358C4×C4⋊C4C2×D5311yescharacteristic
359#340—531no—
360#340—531no—
361#340—531no—
362#340—531no—
363#340—531no—
364#340—531no—
365#340—531no—
366#340—531no—
367C53#2471511yescharacteristic, complemented, Sylow
368C22×Q8C22×D5311yescharacteristic
369C23×C4C22×D5311yescharacteristic
370C22×C8—531no—
371C22×C8—531no—
372C22×C8—531no—
373C22×C8—531no—
374#125—1062no—
375#125—1062no—
376C22⋊Q8—22no—
377C22⋊Q8—22no—
378C4×Q8—22no—
379C4×Q8—22no—
380#117—22no—
381#117—22no—
382C2×C4⋊C4—21no—
383C2×C4⋊C4—21no—
384C2×C4⋊C4—21no—
385C2×C4⋊C4—531no—
386C2×C4⋊C4—531no—
387C2×C4⋊C4—531no—
388C2×C4⋊C4—531no—
389C2×C4⋊C4—1061no—
390C2×C4⋊C4—1061no—
391C2×C4⋊C4—1061no—
392C2×C4⋊C4—1061no—
393C2×C4⋊C4—1061no—
394C2×C4⋊C4—1061no—
395C2×C4⋊C4C22⋊D5311yescharacteristic
396C2×C4⋊C4C22⋊D5311yescharacteristic
397C2×C4⋊C4C22×D5311yescharacteristic
398C2×C4⋊C4C22×D5311yescharacteristic
399C2×C4⋊C4C22×D5311yescharacteristic
400C2×C22⋊C4—21no—
401C2×C22⋊C4—21no—
402C2×C22⋊C4—1061no—
403C2×C22⋊C4—1061no—
404C2×C22⋊C4C22⋊D5311yescharacteristic
405C2×C22⋊C4C22⋊D5311yescharacteristic
406C2×C42—21no—
407C2×C42—21no—
408C2×C42C22×D5311yescharacteristic
409C2×C42C22×D5311yescharacteristic
410#107—1062no—
411#107—1062no—
412#107—1062no—
413#106—1062no—
414C4⋊C8—534no—
415C4⋊C8—1062no—
416#103—1062no—
417#103—1062no—
418#103—1062no—
419#103—1062no—
420C22⋊C8—1062no—
421C22⋊C8—1062no—
422#97—1062no—
423C4×C8—1062no—
424#95—21no—
425#95—21no—
426#95—21no—
427#95—21no—
428#95—21no—
429#95—1061no—
430#95—1061no—
431#95—1061no—
432#95—1061no—
433C24C2×C22⋊D5311yescharacteristic
434C2×Q8—21no—
435C2×Q8—21no—
436C2×Q8—21no—
437C2×Q8—21no—
438C2×Q8—21no—
439C2×Q8#1171994512yes—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—1061no—
451C22×C4—1061no—
452C22×C4—1061no—
453C22×C4—1061no—
454C22×C4C2×C22⋊D5311yescharacteristic
455C22×C4C2×C22⋊D5311yescharacteristic
456C22×C4C2×C22⋊D5311yescharacteristic
457C22×C4C2×C22⋊D5311yescharacteristic
458C22×C4C2×C22⋊D5311yescharacteristic
459C22×C4C23×D5311yescharacteristic
460C2×C8—532no—
461C2×C8—532no—
462C2×C8—1061no—
463C2×C8—1061no—
464C2×C8—1061no—
465C2×C8—1061no—
466C2×C8—1061no—
467C2×C8—1061no—
468C2×C8—1061no—
469C2×C8—1061no—
470C2×C8—1061no—
471C2×C8—1061no—
472C4⋊C4—24no—
473C4⋊C4—22no—
474C4⋊C4—22no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—42no—
481C4⋊C4—81no—
482C4⋊C4—1061no—
483C4⋊C4—1061no—
484C4⋊C4—1061no—
485C4⋊C4—1061no—
486C4⋊C4—1061no—
487C4⋊C4—1061no—
488C4⋊C4—1061no—
489C4⋊C4—1061no—
490C4⋊C4—4241no—
491C4⋊C4—4241no—
492C4⋊C4—4241no—
493C4⋊C4—4241no—
494C4⋊C4—2122no—
495C4⋊C4—2122no—
496C4⋊C4#1171994512yes—
497C22⋊C4—24no—
498C22⋊C4—22no—
499C22⋊C4—22no—
500C22⋊C4—81no—
501C22⋊C4—4241no—
502C22⋊C4—4241no—
503C42—21no—
504C42—21no—
505C42—42no—
506C42—81no—
507C42D4×D5312yes—
508C42#1171994712yes—
509C23—21no—
510C23—21no—
511C23—21no—
512C23—21no—
513C23—21no—
514C23—21no—
515C23—21no—
516C23C22×C22⋊D5311yescharacteristic
517Q8—22no—
518Q8—22no—
519Q8—41no—
520Q8—41no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—2121no—
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×C4—2121no—
561C2×C4—2121no—
562C2×C4#1313318912yes—
563C2×C4#1313318912yes—
564C2×C4C2×D4×D5311yescharacteristic
565C2×C4C2×#1171994711yescharacteristic
566C2×C4C2×#1171994511yescharacteristic
567C2×C4C2×#1171994511yescharacteristic
568C2×C4#1313326411yescharacteristic
569C2×C4#1313326411yescharacteristic
570C8—1062no—
571C8—1062no—
572C8—2121no—
573C8—2121no—
574C22—41no—
575C22—41no—
576C22—41no—
577C22—41no—
578C22—41no—
579C22—41no—
580C22—41no—
581C22C2×#1313318911yescharacteristic
582C22#1681113411yescharacteristic
583C22#1681119411yescharacteristic
584C22#1681142611yescharacteristic
585C22#1681143111yescharacteristic
586C22#1681143811yescharacteristic
587C22#1681143811yescharacteristic
588C4—22no—
589C4—22no—
590C4—41no—
591C4—42no—
592C4—41no—
593C4—41no—
594C4—42no—
595C4—81no—
596C4—81no—
597C4—4241no—
598C4—4241no—
599C4—4241no—
600C4—4241no—
601C4#1681127811yescharacteristic
602C4#1681128211yescharacteristic
603C4#1681131311yescharacteristic
604C4#1681134311yescharacteristic
605C2—81no—
606C2#2027814411yescharacteristic
607C2#2028438211yescharacteristic
608C2#2028440211yescharacteristic
609C2#2028491411yescharacteristic
610C2#2028501511yescharacteristic
611C2#2028597711yescharacteristic
612C2#2028599211yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2323C2
socleC23×C53C22×D4
prime radicalC24×C53C2×D4
hypercenter#2323D53
hypocenterC53#24715
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×#117 ◁ #2323

Complements

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

NH ≅ G/Ntype
1. C2×#16810930C2semidirect
8. #20275672C2semidirect
9. #20275473C2semidirect
10. #20275450C2semidirect
11. #20275445C2semidirect
12. #20275408C2semidirect
13. #20275394C2semidirect
14. #20274720C2semidirect
367. C53#24715semidirect

Maximal subgroups

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

subgroupindex
C2×#168109302
C2×#168106842
#202805452
#202761522
#202761412
#202759152
#202756722
#202754732
#202754502
#202754452
#202754082
#202753942
#202747202
C53×#23232
#2471553