Finite Group Databasebeta

← #27403340

Subgroups — #27403340

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×#16810708C211yesmaximal, characteristic
2#20277660C211yesmaximal, characteristic
3Q8×#11719919C211yesmaximal, characteristic, complemented
4#20275398C211yesmaximal, characteristic
5#20275396C211yesmaximal, characteristic
6#20275300C211yesmaximal, characteristic, complemented
7#20275298C211yesmaximal, characteristic, complemented
8#20275070C211yesmaximal, characteristic
9#20275068C211yesmaximal, characteristic
10#20274841C211yesmaximal, characteristic, complemented
11#20274718C211yesmaximal, characteristic, complemented
12#20274714C211yesmaximal, characteristic
13C8×#11719919C211yesmaximal, characteristic, complemented
14#103×Dic53C211yesmaximal, characteristic, complemented
15C4×#103×C53C211yesmaximal, characteristic, complemented
16C22×#11719924C2211yescharacteristic
17C2×Q8×Dic53C2211yescharacteristic
18C2×Q8×Dic53C2211yescharacteristic
19C2×#13133179C2211yescharacteristic
20C2×#13133174—21no—
21#16810708C418yescomplemented
22C2×#13133148C2211yescharacteristic
23C2×#13133146C2211yescharacteristic
24C2×#13133141C2211yescharacteristic
25C2×#13133139C2211yescharacteristic
26C2×#13133139C2211yescharacteristic
27C2×#13133137C2211yescharacteristic
28C2×C8×Dic53C2211yescharacteristic
29C2×C8×Dic53C2211yescharacteristic
30#16810525C2211yescharacteristic
31#16810524C2211yescharacteristic
32#16810521—21no—
33#16810521C2211yescharacteristic
34#16810519—21no—
35#16810519C2211yescharacteristic
36#16810514—21no—
37#16810512—21no—
38#16810492—21no—
39#16810492C2211yescharacteristic
40#16810492C2211yescharacteristic
41#16810482—21no—
42C4⋊C4×Dic53C2211yescharacteristic
43C4×#11719919—21no—
44C4×#11719919C2211yescharacteristic
45#16810386C2211yescharacteristic
46#16810386C2211yescharacteristic
47#16810385C2211yescharacteristic
48#16810385C2211yescharacteristic
49#16810385C2211yescharacteristic
50#16810385C2211yescharacteristic
51#16810378C2211yescharacteristic
52#16810377C2211yescharacteristic
53C2×C4×Q8×C53C2211yescharacteristic
54C2×#103×C53C2211yescharacteristic
55C2×#103×C53C2211yescharacteristic
56C2×C4×C8×C53C2211yescharacteristic
57C4×C4⋊C4×C53C2211yescharacteristic
58C53×#340C2211yescharacteristic
59C53×#340C2211yescharacteristic
60C22×D212C2311yescharacteristic
61C22×C4×D53—21no—
62C2×#11719924—22no—
63C2×#11719924—22no—
64C2×#11719924C2×C414yes—
65C22×#88322C2311yescharacteristic
66C22×#88322C2311yescharacteristic
67Q8×Dic53—22no—
68Q8×Dic53—22no—
69Q8×Dic53—41no—
70Q8×Dic53C2×C414yes—
71#13133179—22no—
72#13133179C2×C414yes—
73#13133174—44no—
74C2×C4⋊Dic53—21no—
75C2×C4⋊Dic53—21no—
76C2×C4⋊Dic53C2311yescharacteristic
77C2×C4⋊Dic53C2311yescharacteristic
78C2×C4⋊Dic53C2311yescharacteristic
79C2×#11719919—21no—
80C2×#11719919—21no—
81C2×#11719919—21no—
82C2×#11719919—21no—
83C2×#11719919—21no—
84C2×#11719919—21no—
85C2×#11719919D411yescharacteristic
86C2×#11719919D411yescharacteristic
87C2×#11719921—21no—
88C2×#11719921—21no—
89C2×#11719921—21no—
90C2×#11719921—21no—
91C2×#11719921—21no—
92C2×C4×Dic53—21no—
93C2×C4×Dic53—21no—
94C2×C4×Dic53—21no—
95C2×C4×Dic53C2311yescharacteristic
96C2×C4×Dic53C2311yescharacteristic
97#13133148C2×C414yes—
98#13133146—22no—
99#13133141—22no—
100#13133139—41no—
101#13133139C2×C414yes—
102#13133137—41no—
103C8×Dic53—41no—
104C8×Dic53C2×C414yes—
105#13133124—21no—
106#13133124—21no—
107#13133124—21no—
108#13133124D411yescharacteristic
109#13133124D411yescharacteristic
110C22×Q8×C53C2311yescharacteristic
111C22×C8×C53C2311yescharacteristic
112C22×C8×C53C2311yescharacteristic
113C4×Q8×C53—22no—
114C4×Q8×C53—41no—
115C2×C4⋊C4×C53—21no—
116C2×C4⋊C4×C53C2311yescharacteristic
117C2×C4⋊C4×C53C2311yescharacteristic
118C2×C4⋊C4×C53C2311yescharacteristic
119C2×C42×C53—21no—
120C2×C42×C53—21no—
121C2×C42×C53C2311yescharacteristic
122#103×C53—22no—
123#103×C53C2×C414yes—
124C4×C8×C53—41no—
125#95×C53—21no—
126C23×D53—21no—
127C2×Q8×C53—21no—
128C2×Q8×C53—21no—
129C2×Q8×C53—21no—
130C2×Q8×C53C22×C412yes—
131C2×Q8×C53#4112yes—
132C2×Q8×C53#4112yes—
133C22×C4×C53—21no—
134C22×C4×C53—21no—
135C22×C4×C53—21no—
136C22×C4×C53—21no—
137C22×C4×C53C2×D411yescharacteristic
138C22×C4×C53C2×D411yescharacteristic
139C22×C4×C53C2411yescharacteristic
140C22×Dic53—21no—
141C22×Dic53—21no—
142C22×Dic53—21no—
143C22×Dic53C2×D411yescharacteristic
144C22×Dic53C2×D411yescharacteristic
145C22×Dic53C2×D411yescharacteristic
146C22×Dic53C2×D411yescharacteristic
147C2×D212—21no—
148C2×D212—21no—
149C2×D212—21no—
150C2×D212—21no—
151C2×D212—21no—
152C2×D212C22×C412yes—
153C2×C4×D53—24no—
154C2×C4×D53—42no—
155C2×C4×D53—42no—
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—42no—
168C4⋊C4×C53C22×C412yes—
169C42×C53—21no—
170C42×C53—21no—
171C42×C53—42no—
172C42×C53—42no—
173#11719924—28no—
174#11719921—24no—
175#11719921—42no—
176#11719921—42no—
177#11719921—81no—
178#11719921—81no—
179C4⋊Dic53—21no—
180C4⋊Dic53—24no—
181C4⋊Dic53—21no—
182C4⋊Dic53—21no—
183C4⋊Dic53—21no—
184C4⋊Dic53—21no—
185C4⋊Dic53—42no—
186C4⋊Dic53C22×C412yes—
187#11719919—22no—
188#11719919—42no—
189#11719919—42no—
190#11719919—81no—
191#11719919—81no—
192#11719919—81no—
193#11719919—81no—
194#11719919SD1614yescomplemented
195C4×Dic53—21no—
196C4×Dic53—21no—
197C4×Dic53—24no—
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×C53C22×D411yescharacteristic
209C22×D53—22no—
210C22×D53—41no—
211C22×D53—41no—
212C22×D53—41no—
213C22×D53—41no—
214C22×D53—41no—
215C22×D53—41no—
216Q8×C53—22no—
217Q8×C53—22no—
218Q8×C53—22no—
219Q8×C53#11714yescomplemented
220C2×C4×C53—22no—
221C2×C4×C53—22no—
222C2×C4×C53—21no—
223C2×C4×C53—22no—
224C2×C4×C53—21no—
225C2×C4×C53—22no—
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×C53C2×SD1612yes—
238C2×C4×C53C23×C411yescharacteristic
239C2×C4×C53C2×#4111yescharacteristic
240C2×C4×C53C2×#4111yescharacteristic
241C2×C4×C53C2×#4111yescharacteristic
242C2×C4×C53H(5,𝔽2)11yescharacteristic
243C2×C4×C532−1+411yescharacteristic
244C2×Dic53—21no—
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—41no—
255C2×Dic53—41no—
256C2×Dic53—41no—
257C2×Dic53—41no—
258C2×Dic53—41no—
259C2×Dic53—41no—
260C2×Dic53C4×D412yes—
261C2×Dic53C4×D412yes—
262C2×Dic53C2×SD1612yes—
263C2×Dic53C2×SD1612yes—
264C2×Dic53#13512yes—
265D212—22no—
266D212—22no—
267D212—41no—
268D212—41no—
269C4×D53—48no—
270C8×C53—22no—
271C8×C53—41no—
272#88322—22no—
273#88322—41no—
274#22767—531nomaximal, Sylow
275C22×C53C2×C4×D411yescharacteristic
276C22×C53#54611yescharacteristic
277C22×C53#54811yescharacteristic
278C22×C53C22×SD1611yescharacteristic
279C22×C53C2×#13511yescharacteristic
280C22×C53#57611yescharacteristic
281C22×C53#57811yescharacteristic
282C2×D53—42no—
283C2×D53—42no—
284C2×D53—42no—
285C2×D53—81no—
286C2×D53—81no—
287C2×D53—81no—
288C2×D53—81no—
289C4×C53—24no—
290C4×C53—22no—
291C4×C53—42no—
292C4×C53—41no—
293C4×C53—81no—
294C4×C53—81no—
295C4×C53C2×#11711yescharacteristic
296C4×C53#52011yescharacteristic
297C4×C53#55211yescharacteristic
298C4×C53#55311yescharacteristic
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#1934—531no—
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×#103D5311yescharacteristic, complemented
321C2×C53#235211yescharacteristic
322C2×C53C2×C4×SD1611yescharacteristic
323C2×C53#295111yescharacteristic
324C2×C53#329811yescharacteristic
325C2×C53#330511yescharacteristic
326C2×C53#336211yescharacteristic
327C2×C53#336411yescharacteristic
328D53—82no—
329C22×SD16—531no—
330C2×C4×Q8—531no—
331C2×C4×Q8—531no—
332C2×C4×Q8C2×D5311yescharacteristic
333C2×C4×D4—531no—
334C2×#117—1061no—
335C4×SD16—538no—
336C2×#106—531no—
337C2×C4⋊C8—531no—
338C2×#103—531no—
339C2×#103C2×D5311yescharacteristic
340C2×#103C2×D5311yescharacteristic
341C2×#102—531no—
342C2×#102—531no—
343C2×#102—531no—
344C2×C4×C8—531no—
345C2×C4×C8—531no—
346C2×C4×C8C2×D5311yescharacteristic
347#390—531no—
348#390—531no—
349#389—531no—
350#389—531no—
351#389—1061no—
352#389—1061no—
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—1061no—
362C4×C4⋊C4C2×D5311yescharacteristic
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×Q8C22×D5311yescharacteristic
375C22×D4—531no—
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—22no—
385C4×Q8—41no—
386C4×Q8—534no—
387C4×Q8—1062no—
388C4×Q8—1062no—
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—1061no—
411C2×C22⋊C4—1061no—
412C2×C22⋊C4—1061no—
413C2×C22⋊C4—1061no—
414C2×C22⋊C4—1061no—
415C2×C42—21no—
416C2×C42—21no—
417C2×C42—531no—
418C2×C42—531no—
419C2×C42—1061no—
420C2×C42—1061no—
421C2×C42—1061no—
422C2×C42C22×D5311yescharacteristic
423#106—534no—
424C4⋊C8—1062no—
425#103—22no—
426#103—1062no—
427#103C4×D5314yes—
428#102—534no—
429#102—2121no—
430#102—2121no—
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—1061no—
441C2×Q8—21no—
442C2×Q8—21no—
443C2×Q8—21no—
444C2×Q8C2×C4×D5312yes—
445C2×Q8#1171994512yes—
446C2×Q8#1171994512yes—
447C2×D4—532no—
448C2×D4—1061no—
449C2×D4—1061no—
450C2×D4—1061no—
451C2×D4—1061no—
452C2×D4—1061no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—531no—
458C22×C4—531no—
459C22×C4—531no—
460C22×C4—531no—
461C22×C4—1064no—
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—42no—
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—4241no—
502C4⋊C4—2122no—
503C4⋊C4—2122no—
504C4⋊C4—2122no—
505C4⋊C4C2×C4×D5312yes—
506C22⋊C4—1064no—
507C22⋊C4—4241no—
508C22⋊C4—4241no—
509C22⋊C4—2122no—
510C22⋊C4—2122no—
511C42—21no—
512C42—21no—
513C42—42no—
514C42—42no—
515C42—532no—
516C42—532no—
517C42—1064no—
518C42—1061no—
519C42—1061no—
520C42—2122no—
521C42—2122no—
522C23—1062no—
523C23—2121no—
524C23—2121no—
525C23—2121no—
526C23—2121no—
527C23—2121no—
528C23—2121no—
529C23C2×D4×D5311yescharacteristic
530Q8—22no—
531Q8—22no—
532Q8—22no—
533Q8#1313317414yescomplemented
534D4—1062no—
535D4—1062no—
536D4—2121no—
537D4—2121no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—532no—
556C2×C4—532no—
557C2×C4—532no—
558C2×C4—532no—
559C2×C4—532no—
560C2×C4—1061no—
561C2×C4—1061no—
562C2×C4—1061no—
563C2×C4—1061no—
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×C4C2×#1171994511yescharacteristic
581C2×C4C2×#1171994511yescharacteristic
582C2×C4#1313326211yescharacteristic
583C2×C4#1313326411yescharacteristic
584C8—22no—
585C8—41no—
586C8—1062no—
587C8—2121no—
588C22—4241no—
589C22—4241no—
590C22—4241no—
591C22—4241no—
592C22—2122no—
593C22—2122no—
594C22—2122no—
595C22C4×D4×D5311yescharacteristic
596C22#1681126311yescharacteristic
597C22#1681128211yescharacteristic
598C22C2×SD16×D5311yescharacteristic
599C22#135×D5311yescharacteristic
600C22#1681142511yescharacteristic
601C22#1681143611yescharacteristic
602C4—22no—
603C4—24no—
604C4—42no—
605C4—41no—
606C4—81no—
607C4—81no—
608C4—534no—
609C4—1062no—
610C4—1062no—
611C4—4241no—
612C4—4241no—
613C4—2121no—
614C4—2122no—
615C4C2×#1313317411yescharacteristic
616C4#1681110211yescharacteristic
617C4#1681130611yescharacteristic
618C4#1681132211yescharacteristic
619C2—4242no—
620C2#2027763811yescharacteristic
621C2C4×SD16×D5311yescharacteristic
622C2#2028359011yescharacteristic
623C2#2028657611yescharacteristic
624C2#2028666111yescharacteristic
625C2#2028738411yescharacteristic
626C2#2028740111yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC4×#103×C53C2
socleC23×C53C22×D4
prime radicalC22×D212C23
cosocleC22×C4×C53C24
hypercenterC4×#103D53
hypocenterC53#22767
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ C4×#103

Complements

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

NH ≅ G/Ntype
3. Q8×#11719919C2semidirect
6. #20275300C2semidirect
7. #20275298C2semidirect
10. #20274841C2semidirect
11. #20274718C2semidirect
13. C8×#11719919C2semidirect
14. #103×Dic53C2semidirect
15. C4×#103×C53C2semidirect
21. #16810708C4semidirect
194. #11719919SD16semidirect
219. Q8×C53#117semidirect
320. C4×#103D53semidirect
373. C53#22767semidirect
533. Q8#13133174semidirect

Maximal subgroups

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

subgroupindex
C2×#168107082
#202776602
Q8×#117199192
#202753982
#202753962
#202753002
#202752982
#202750702
#202750682
#202748412
#202747182
#202747142
C8×#117199192
#103×Dic532
C4×#103×C532
#2276753