Finite Group Databasebeta

← #27403345

Subgroups — #27403345

Order 13568.

886 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810714C211yesmaximal, characteristic
2#20277663C211yesmaximal, characteristic
3#20277629C211yesmaximal, characteristic, complemented
4#20275400C211yesmaximal, characteristic
5#20275303C211yesmaximal, characteristic, complemented
6#20275297C211yesmaximal, characteristic, complemented
7#20275073C211yesmaximal, characteristic
8#20275067C211yesmaximal, characteristic
9#20274841C211yesmaximal, characteristic, complemented
10#20274719C211yesmaximal, characteristic, complemented
11#20274713C211yesmaximal, characteristic
12C8×#11719919C211yesmaximal, characteristic, complemented
13C4×#13133142C211yesmaximal, characteristic, complemented
14#20274534C211yesmaximal, characteristic, complemented
15C53×#1934C211yesmaximal, characteristic
16C22×#11719923C2211yescharacteristic
17C2×C4×Dic106C2211yescharacteristic
18C2×#13133182C2211yescharacteristic
19C2×#13133182C2211yescharacteristic
20C2×D4×Dic53C2211yescharacteristic
21C2×#13133173—21no—
22#16810714C418yescomplemented
23C2×#13133148C2211yescharacteristic
24C2×#13133146C2211yescharacteristic
25C2×#13133142C2211yescharacteristic
26C2×#13133142C2211yescharacteristic
27C2×#13133140C2211yescharacteristic
28C2×#13133138C2211yescharacteristic
29C2×C8×Dic53C2211yescharacteristic
30C2×C8×Dic53C2211yescharacteristic
31#16810526C2211yescharacteristic
32#16810522C2211yescharacteristic
33#16810520—21no—
34#16810518—21no—
35#16810517—21no—
36#16810511—21no—
37#16810492—21no—
38#16810492C2211yescharacteristic
39#16810492C2211yescharacteristic
40#16810481—21no—
41C4⋊C4×Dic53C2211yescharacteristic
42C4×#11719919C2211yescharacteristic
43C4×C4⋊Dic53C2211yescharacteristic
44#16810474—21no—
45#16810386C2211yescharacteristic
46#16810386C2211yescharacteristic
47#16810385C2211yescharacteristic
48#16810385C2211yescharacteristic
49#16810385C2211yescharacteristic
50#16810385C2211yescharacteristic
51#16810378C2211yescharacteristic
52#16810377C2211yescharacteristic
53C2×#102×C53C2211yescharacteristic
54C2×#102×C53C2211yescharacteristic
55C2×C4×C8×C53C2211yescharacteristic
56C53×#390C2211yescharacteristic
57C53×#389C2211yescharacteristic
58C53×#340C2211yescharacteristic
59C53×#340C2211yescharacteristic
60C22×Dic106C2311yescharacteristic
61C23×Dic53—21no—
62C2×#11719923—22no—
63C2×#11719923—22no—
64C2×#11719923C2×C414yes—
65C22×#88322C2311yescharacteristic
66C22×#88322C2311yescharacteristic
67C4×Dic106—22no—
68C4×Dic106—41no—
69#13133182—22no—
70#13133182—22no—
71#13133182—41no—
72#13133182C2×C414yes—
73D4×Dic53—22no—
74D4×Dic53C2×C414yes—
75#13133173—44no—
76C2×C4⋊Dic53—21no—
77C2×C4⋊Dic53C2311yescharacteristic
78C2×C4⋊Dic53C2311yescharacteristic
79C2×C4⋊Dic53C2311yescharacteristic
80C2×#11719919—21no—
81C2×#11719919—21no—
82C2×#11719919—21no—
83C2×#11719919—21no—
84C2×#11719919—21no—
85C2×#11719919—21no—
86C2×#11719919—21no—
87C2×#11719919D411yescharacteristic
88C2×#11719919D411yescharacteristic
89C2×C22⋊Dic53—21no—
90C2×C22⋊Dic53—21no—
91C2×C22⋊Dic53—21no—
92C2×C4×Dic53—21no—
93C2×C4×Dic53—21no—
94C2×C4×Dic53—21no—
95C2×C4×Dic53—21no—
96C2×C4×Dic53—21no—
97C2×C4×Dic53C2311yescharacteristic
98C2×C4×Dic53C2311yescharacteristic
99#13133148C2×C414yes—
100#13133146—22no—
101#13133142—22no—
102#13133142C2×C414yes—
103#13133140—22no—
104#13133138—41no—
105C8×Dic53—41no—
106C8×Dic53C2×C414yes—
107#13133124—21no—
108#13133124—21no—
109#13133124—21no—
110#13133124D411yescharacteristic
111#13133124D411yescharacteristic
112C22×D4×C53C2311yescharacteristic
113C22×C8×C53C2311yescharacteristic
114C22×C8×C53C2311yescharacteristic
115C2×C4⋊C4×C53—21no—
116C2×C4⋊C4×C53C2311yescharacteristic
117C2×C4⋊C4×C53C2311yescharacteristic
118C2×C4⋊C4×C53C2311yescharacteristic
119C2×C22⋊C4×C53—21no—
120C2×C22⋊C4×C53—21no—
121C2×C42×C53C2311yescharacteristic
122#102×C53—41no—
123#102×C53C2×C414yes—
124C4×C8×C53—41no—
125#95×C53—21no—
126C24×C53—21no—
127C2×D4×C53—21no—
128C2×D4×C53—21no—
129C2×D4×C53—21no—
130C2×D4×C53—21no—
131C2×D4×C53—21no—
132C2×D4×C53C22×C412yes—
133C22×C4×C53—21no—
134C22×C4×C53—21no—
135C22×C4×C53—21no—
136C22×C4×C53C2×D411yescharacteristic
137C22×C4×C53C2×D411yescharacteristic
138C22×C4×C53C2411yescharacteristic
139C22×Dic53—21no—
140C22×Dic53—21no—
141C22×Dic53—24no—
142C22×Dic53—21no—
143C22×Dic53—21no—
144C22×Dic53—42no—
145C22×Dic53—42no—
146C22×Dic53C2×D411yescharacteristic
147C22×Dic53C2×D411yescharacteristic
148C22×Dic53C2×D411yescharacteristic
149C22×Dic53C2×D411yescharacteristic
150C2×Dic106—21no—
151C2×Dic106—21no—
152C2×Dic106—21no—
153C2×Dic106C22×C412yes—
154C2×Dic106#4112yes—
155C2×Dic106#4112yes—
156C2×C8×C53—21no—
157C2×C8×C53—21no—
158C2×C8×C53—21no—
159C2×C8×C53—21no—
160C2×C8×C53—21no—
161C2×C8×C53C22×C412yes—
162C4⋊C4×C53—21no—
163C4⋊C4×C53—21no—
164C4⋊C4×C53—21no—
165C4⋊C4×C53—21no—
166C4⋊C4×C53—21no—
167C4⋊C4×C53—81no—
168C4⋊C4×C53C22×C412yes—
169C22⋊C4×C53—42no—
170C22⋊C4×C53—81no—
171C42×C53—21no—
172C42×C53—21no—
173C22⋊Dic53—24no—
174C22⋊Dic53—42no—
175C22⋊Dic53—81no—
176#11719923—28no—
177C4⋊Dic53—21no—
178C4⋊Dic53—21no—
179C4⋊Dic53—21no—
180C4⋊Dic53—21no—
181C4⋊Dic53—21no—
182C4⋊Dic53—81no—
183C4⋊Dic53C22×C412yes—
184#11719919—24no—
185#11719919—22no—
186#11719919—42no—
187#11719919—42no—
188#11719919—42no—
189#11719919—42no—
190#11719919—81no—
191#11719919—81no—
192#11719919SD1614yescomplemented
193C4×Dic53—21no—
194C4×Dic53—21no—
195C4×Dic53—24no—
196C4×Dic53—42no—
197C4×Dic53—42no—
198C4×Dic53—42no—
199C4×Dic53—42no—
200C4×Dic53C22×C412yes—
201C4×Dic53#4112yes—
202C2×#88322—21no—
203C2×#88322—21no—
204C2×#88322—21no—
205C2×#88322—21no—
206C2×#88322—21no—
207C2×#88322C22×C412yes—
208C23×C53—22no—
209C23×C53—41no—
210C23×C53—41no—
211C23×C53—41no—
212C23×C53—41no—
213C23×C53—41no—
214C23×C53—41no—
215C23×C53C22×D411yescharacteristic
216D4×C53—22no—
217D4×C53—22no—
218D4×C53—41no—
219D4×C53—41no—
220C2×C4×C53—21no—
221C2×C4×C53—21no—
222C2×C4×C53—22no—
223C2×C4×C53—21no—
224C2×C4×C53—21no—
225C2×C4×C53—21no—
226C2×C4×C53—41no—
227C2×C4×C53—41no—
228C2×C4×C53—41no—
229C2×C4×C53—41no—
230C2×C4×C53—41no—
231C2×C4×C53—41no—
232C2×C4×C53—41no—
233C2×C4×C53—41no—
234C2×C4×C53C2×SD1612yes—
235C2×C4×C53C23×C411yescharacteristic
236C2×C4×C53C2×#4111yescharacteristic
237C2×C4×C53C2×#4111yescharacteristic
238C2×C4×C53C2×#4111yescharacteristic
239C2×C4×C53H(5,𝔽2)11yescharacteristic
240C2×C4×C532−1+411yescharacteristic
241C2×Dic53—22no—
242C2×Dic53—22no—
243C2×Dic53—21no—
244C2×Dic53—22no—
245C2×Dic53—21no—
246C2×Dic53—21no—
247C2×Dic53—21no—
248C2×Dic53—22no—
249C2×Dic53—21no—
250C2×Dic53—21no—
251C2×Dic53—21no—
252C2×Dic53—41no—
253C2×Dic53—41no—
254C2×Dic53—48no—
255C2×Dic53—41no—
256C2×Dic53—41no—
257C2×Dic53—41no—
258C2×Dic53—41no—
259C2×Dic53—41no—
260C2×Dic53—41no—
261C2×Dic53C4×D412yes—
262C2×Dic53C4×D412yes—
263C2×Dic53C2×SD1612yes—
264C2×Dic53C2×SD1612yes—
265C2×Dic53#13512yes—
266Dic106—22no—
267Dic106—22no—
268Dic106—22no—
269Dic106#11714yescomplemented
270C8×C53—22no—
271C8×C53—41no—
272#88322—22no—
273#88322—41no—
274#22767—531nomaximal, Sylow
275C22×C53—42no—
276C22×C53—42no—
277C22×C53—42no—
278C22×C53—81no—
279C22×C53—81no—
280C22×C53—81no—
281C22×C53—81no—
282C22×C53C2×C4×D411yescharacteristic
283C22×C53#54611yescharacteristic
284C22×C53#54811yescharacteristic
285C22×C53C22×SD1611yescharacteristic
286C22×C53C2×#13511yescharacteristic
287C22×C53#57611yescharacteristic
288C22×C53#57811yescharacteristic
289C4×C53—22no—
290C4×C53—42no—
291C4×C53—41no—
292C4×C53—81no—
293C4×C53—81no—
294C4×C53C2×#11711yescharacteristic
295C4×C53#52011yescharacteristic
296C4×C53#55211yescharacteristic
297C4×C53#55311yescharacteristic
298Dic53—24no—
299Dic53—22no—
300Dic53—22no—
301Dic53—42no—
302Dic53—41no—
303Dic53—81no—
304Dic53—81no—
305Dic53C4×SD1614yes—
306C2×C4×SD16—531no—
307#2354—531no—
308Q8×C4⋊C4—531no—
309#1934—531no—
310#1934D5311yescharacteristic
311#1920—531no—
312#1920—531no—
313#1880—531no—
314#1880—531no—
315#1834—531no—
316#1803—531no—
317#1802—531no—
318C8×C4⋊C4—531no—
319C4×#103—531no—
320C4×#103—531no—
321C2×C53—82no—
322C2×C53#235211yescharacteristic
323C2×C53C2×C4×SD1611yescharacteristic
324C2×C53#295111yescharacteristic
325C2×C53#329811yescharacteristic
326C2×C53#330511yescharacteristic
327C2×C53#336211yescharacteristic
328C2×C53#336411yescharacteristic
329C22×SD16—531no—
330C2×C4×Q8—531no—
331C2×C4×Q8—531no—
332C2×C4×Q8—531no—
333C2×C4×D4—531no—
334C2×#117—1061no—
335C4×SD16—538no—
336C2×#106—531no—
337C2×C4⋊C8—531no—
338C2×#103—531no—
339C2×#103—531no—
340C2×#103—531no—
341C2×#102—531no—
342C2×#102C2×D5311yescharacteristic
343C2×#102C2×D5311yescharacteristic
344C2×C4×C8—531no—
345C2×C4×C8—531no—
346C2×C4×C8C2×D5311yescharacteristic
347#390—531no—
348#390C2×D5311yescharacteristic
349#389—531no—
350#389—1061no—
351#389—1061no—
352#389C2×D5311yescharacteristic
353#388—1061no—
354#388—1061no—
355#384—531no—
356#384—531no—
357#384—1061no—
358#380—1061no—
359C4×C4⋊C4—531no—
360C4×C4⋊C4—531no—
361C4×C4⋊C4—531no—
362C4×C4⋊C4—1061no—
363#340—531no—
364#340—531no—
365#340—531no—
366#340—531no—
367#340—531no—
368#340—531no—
369#340C2×D5311yescharacteristic
370#340C2×D5311yescharacteristic
371#336—531no—
372#336—531no—
373C53#2276711yescharacteristic, complemented, Sylow
374C22×Q8—531no—
375C22×D4C22×D5311yescharacteristic
376C23×C4—1061no—
377C2×SD16—534no—
378C2×SD16—1062no—
379C2×SD16—1062no—
380C22×C8—531no—
381C22×C8—531no—
382C22×C8C22×D5311yescharacteristic
383C22×C8C22×D5311yescharacteristic
384C4×Q8—534no—
385C4×Q8—1062no—
386C4×Q8—1062no—
387C4×Q8—1062no—
388C4×Q8—2121no—
389C4×Q8—2121no—
390C4×D4—534no—
391C4×D4—1062no—
392#117—2124no—
393C2×C4⋊C4—21no—
394C2×C4⋊C4—531no—
395C2×C4⋊C4—531no—
396C2×C4⋊C4—531no—
397C2×C4⋊C4—531no—
398C2×C4⋊C4—531no—
399C2×C4⋊C4—1061no—
400C2×C4⋊C4—1061no—
401C2×C4⋊C4—1061no—
402C2×C4⋊C4—1061no—
403C2×C4⋊C4—1061no—
404C2×C4⋊C4—1061no—
405C2×C4⋊C4—1061no—
406C2×C4⋊C4—1061no—
407C2×C4⋊C4C22×D5311yescharacteristic
408C2×C4⋊C4C22×D5311yescharacteristic
409C2×C4⋊C4C22×D5311yescharacteristic
410C2×C22⋊C4—21no—
411C2×C22⋊C4—21no—
412C2×C22⋊C4—1061no—
413C2×C22⋊C4—1061no—
414C2×C22⋊C4—1061no—
415C2×C42—531no—
416C2×C42—531no—
417C2×C42—1061no—
418C2×C42—1061no—
419C2×C42—1061no—
420C2×C42—1061no—
421C2×C42—1061no—
422C2×C42C22×D5311yescharacteristic
423#106—534no—
424C4⋊C8—1062no—
425#103—534no—
426#103—1062no—
427#103—1062no—
428#102—41no—
429#102—2121no—
430#102C4×D5314yes—
431C4×C8—41no—
432C4×C8—534no—
433C4×C8—2121no—
434#95—21no—
435#95—531no—
436#95—531no—
437#95—1061no—
438#95—1061no—
439#95—1061no—
440C24—21no—
441C2×Q8—532no—
442C2×Q8—532no—
443C2×Q8—532no—
444C2×Q8—1061no—
445C2×Q8—1061no—
446C2×Q8—1061no—
447C2×D4—21no—
448C2×D4—21no—
449C2×D4—21no—
450C2×D4—21no—
451C2×D4—21no—
452C2×D4C2×C4×D5312yes—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—531no—
457C22×C4—531no—
458C22×C4—531no—
459C22×C4—531no—
460C22×C4—1064no—
461C22×C4—1061no—
462C22×C4—1061no—
463C22×C4—1061no—
464C22×C4—1061no—
465C22×C4—2122no—
466C22×C4—2122no—
467C22×C4D4×D5311yescharacteristic
468C22×C4D4×D5311yescharacteristic
469C22×C4C23×D5311yescharacteristic
470SD16—1068no—
471C2×C8—21no—
472C2×C8—21no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—532no—
477C2×C8—1061no—
478C2×C8—1061no—
479C2×C8—1061no—
480C2×C8—1061no—
481C2×C8—1061no—
482C2×C8C2×C4×D5312yes—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—81no—
489C4⋊C4—534no—
490C4⋊C4—532no—
491C4⋊C4—1062no—
492C4⋊C4—1064no—
493C4⋊C4—1061no—
494C4⋊C4—1061no—
495C4⋊C4—1061no—
496C4⋊C4—1061no—
497C4⋊C4—1061no—
498C4⋊C4—4241no—
499C4⋊C4—4241no—
500C4⋊C4—4241no—
501C4⋊C4—2122no—
502C4⋊C4—2122no—
503C4⋊C4—2122no—
504C4⋊C4—2122no—
505C4⋊C4C2×C4×D5312yes—
506C22⋊C4—42no—
507C22⋊C4—81no—
508C22⋊C4—1064no—
509C22⋊C4—4241no—
510C22⋊C4—2122no—
511C42—21no—
512C42—21no—
513C42—532no—
514C42—532no—
515C42—1064no—
516C42—1061no—
517C42—1061no—
518C42—2122no—
519C42—2122no—
520C42—2122no—
521C42—2122no—
522C23—22no—
523C23—41no—
524C23—41no—
525C23—41no—
526C23—41no—
527C23—41no—
528C23—41no—
529C23C2×D4×D5311yescharacteristic
530Q8—534no—
531Q8—1062no—
532Q8—1062no—
533Q8—1062no—
534D4—22no—
535D4—22no—
536D4—41no—
537D4—41no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—22no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—532no—
553C2×C4—532no—
554C2×C4—532no—
555C2×C4—532no—
556C2×C4—532no—
557C2×C4—1062no—
558C2×C4—1062no—
559C2×C4—1061no—
560C2×C4—1061no—
561C2×C4—1061no—
562C2×C4—1061no—
563C2×C4—1062no—
564C2×C4—1062no—
565C2×C4—1061no—
566C2×C4—1061no—
567C2×C4—1061no—
568C2×C4—2128no—
569C2×C4—2121no—
570C2×C4—2121no—
571C2×C4—2121no—
572C2×C4—2121no—
573C2×C4—2121no—
574C2×C4—2121no—
575C2×C4—2121no—
576C2×C4—2121no—
577C2×C4SD16×D5312yes—
578C2×C4C22×C4×D5311yescharacteristic
579C2×C4#41×D5311yescharacteristic
580C2×C4#41×D5311yescharacteristic
581C2×C4#41×D5311yescharacteristic
582C2×C4#1313326211yescharacteristic
583C2×C4#1313326411yescharacteristic
584C8—22no—
585C8—41no—
586C8—1062no—
587C8—2121no—
588C22—42no—
589C22—42no—
590C22—42no—
591C22—81no—
592C22—81no—
593C22—81no—
594C22—81no—
595C22C4×D4×D5311yescharacteristic
596C22#1681126211yescharacteristic
597C22#1681128111yescharacteristic
598C22C2×SD16×D5311yescharacteristic
599C22#135×D5311yescharacteristic
600C22#1681142411yescharacteristic
601C22#1681143711yescharacteristic
602C4—22no—
603C4—42no—
604C4—41no—
605C4—81no—
606C4—81no—
607C4—534no—
608C4—1062no—
609C4—1064no—
610C4—1062no—
611C4—4241no—
612C4—4241no—
613C4—2122no—
614C4—2121no—
615C4#117×D5311yescharacteristic
616C4#1681110211yescharacteristic
617C4#1681130911yescharacteristic
618C4#1681131811yescharacteristic
619C2—82no—
620C2#2027764311yescharacteristic
621C2C4×SD16×D5311yescharacteristic
622C2#2028358511yescharacteristic
623C2#2028657511yescharacteristic
624C2#2028666011yescharacteristic
625C2#2028738311yescharacteristic
626C2#2028739711yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#1934C2
socleC23×C53C22×D4
prime radicalC22×D4×C53C23
cosocleC22×C4×C53C24
hypercenter#1934D53
hypocenterC53#22767
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1934

Complements

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

NH ≅ G/Ntype
3. #20277629C2semidirect
5. #20275303C2semidirect
6. #20275297C2semidirect
9. #20274841C2semidirect
10. #20274719C2semidirect
12. C8×#11719919C2semidirect
13. C4×#13133142C2semidirect
14. #20274534C2semidirect
22. #16810714C4semidirect
192. #11719919SD16semidirect
269. Dic106#117semidirect
373. C53#22767semidirect

Maximal subgroups

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

subgroupindex
C2×#168107142
#202776632
#202776292
#202754002
#202753032
#202752972
#202750732
#202750672
#202748412
#202747192
#202747132
C8×#117199192
C4×#131331422
#202745342
C53×#19342
#2276753