Finite Group Databasebeta

← #27392157

Subgroups — #27392157

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×#16810981C211yesmaximal, characteristic
2#20277286C211yesmaximal, characteristic, complemented
3#20277262C211yesmaximal, characteristic
4#20275686C211yesmaximal, characteristic, complemented
5#20275530C211yesmaximal, characteristic
6#20275415C211yesmaximal, characteristic
7#20275405C211yesmaximal, characteristic, complemented
8#20275404C211yesmaximal, characteristic, complemented
9#20274847C211yesmaximal, characteristic, complemented
10#20274534C211yesmaximal, characteristic, complemented
11#20274534C211yesmaximal, characteristic, complemented
12#102×Dic53C211yesmaximal, characteristic
13#102×Dic53C211yesmaximal, characteristic
14#97×Dic53C211yesmaximal, characteristic, complemented
15C53×#1936C211yesmaximal, characteristic
16C2×#13133216C2211yescharacteristic
17C2×#13133204C2211yescharacteristic
18C2×#13133182—21no—
19C2×#13133182C2211yescharacteristic
20C2×D4×Dic53C2211yescharacteristic
21#16810981C418yescomplemented
22C2×#13133142C2211yescharacteristic
23C2×#13133142C2211yescharacteristic
24C2×#13133140C2211yescharacteristic
25C2×#13133140C2211yescharacteristic
26C2×#13133138C2211yescharacteristic
27C2×#13133138C2211yescharacteristic
28C2×#13133128C2211yescharacteristic
29C2×#13133127C2211yescharacteristic
30C2×#13133127C2211yescharacteristic
31C2×C8×Dic53C2211yescharacteristic
32C2×C8×Dic53C2211yescharacteristic
33#16810529C2211yescharacteristic
34#16810527—21no—
35#16810521C2211yescharacteristic
36#16810518—21no—
37#16810511—21no—
38#16810511—21no—
39#16810492C2211yescharacteristic
40#16810488C2211yescharacteristic
41C4⋊C4×Dic53C2211yescharacteristic
42C4⋊C4×Dic53C2211yescharacteristic
43#16810474—21no—
44C22⋊C4×Dic53—21no—
45C22⋊C4×Dic53—21no—
46C42×Dic53C2211yescharacteristic
47#16810386C2211yescharacteristic
48#16810386C2211yescharacteristic
49#16810385C2211yescharacteristic
50#16810385C2211yescharacteristic
51#16810385C2211yescharacteristic
52#16810385C2211yescharacteristic
53C2×#102×C53C2211yescharacteristic
54C2×#102×C53C2211yescharacteristic
55C2×#97×C53C2211yescharacteristic
56C53×#390C2211yescharacteristic
57C53×#389C2211yescharacteristic
58C53×#340C2211yescharacteristic
59C53×#340C2211yescharacteristic
60C22×Dic106—21no—
61C22×Dic106C2311yescharacteristic
62C23×Dic53—21no—
63C22×#88322C2311yescharacteristic
64C22×#88322C2311yescharacteristic
65#13133216—22no—
66#13133216C2×C414yescomplemented
67#13133204—22no—
68#13133204C2×C414yes—
69#13133182—22no—
70#13133182—22no—
71#13133182—44no—
72D4×Dic53—22no—
73D4×Dic53—22no—
74C2×C4⋊Dic53C2311yescharacteristic
75C2×C4⋊Dic53C2311yescharacteristic
76C2×C4⋊Dic53C2311yescharacteristic
77C2×#11719919—21no—
78C2×#11719919—21no—
79C2×#11719919—21no—
80C2×#11719919—21no—
81C2×#11719919—21no—
82C2×C22⋊Dic53—21no—
83C2×C22⋊Dic53—21no—
84C2×C22⋊Dic53—21no—
85C2×C4×Dic53—21no—
86C2×C4×Dic53—21no—
87C2×C4×Dic53—21no—
88C2×C4×Dic53D411yescharacteristic
89C2×C4×Dic53D411yescharacteristic
90C2×C4×Dic53D411yescharacteristic
91C2×C4×Dic53D411yescharacteristic
92C2×C4×Dic53C2311yescharacteristic
93C2×C4×Dic53C2311yescharacteristic
94#13133142—22no—
95#13133142C2×C414yescomplemented
96#13133140—22no—
97#13133140C2×C414yescomplemented
98#13133138—22no—
99#13133138C2×C414yes—
100#13133128—22no—
101#13133127—22no—
102#13133127C2×C414yescomplemented
103C8×Dic53—22no—
104C8×Dic53—22no—
105#13133124—21no—
106#13133124—21no—
107#13133124—21no—
108#13133124—21no—
109#13133124—21no—
110#13133124—21no—
111#13133124—21no—
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—22no—
123#102×C53C2×C414yes—
124#97×C53—22no—
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—21no—
142C22×Dic53—21no—
143C22×Dic53—42no—
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×Dic106—21no—
154C2×Dic106—21no—
155C2×Dic106—24no—
156C2×Dic106—42no—
157C2×Dic106—42no—
158C2×Dic106C22×C412yes—
159C2×C8×C53—21no—
160C2×C8×C53—21no—
161C2×C8×C53—21no—
162C2×C8×C53—21no—
163C2×C8×C53C22×C412yes—
164C2×C8×C53#4112yes—
165C4⋊C4×C53—21no—
166C4⋊C4×C53—21no—
167C4⋊C4×C53—21no—
168C4⋊C4×C53—21no—
169C4⋊C4×C53—21no—
170C4⋊C4×C53—42no—
171C4⋊C4×C53C22×C412yes—
172C22⋊C4×C53—42no—
173C22⋊C4×C53—42no—
174C42×C53—21no—
175C42×C53—21no—
176C22⋊Dic53—24no—
177C22⋊Dic53—24no—
178C22⋊Dic53—42no—
179C4⋊Dic53—21no—
180C4⋊Dic53—21no—
181C4⋊Dic53—21no—
182C4⋊Dic53—21no—
183C4⋊Dic53—21no—
184C4⋊Dic53C22×C412yes—
185#11719919—24no—
186#11719919—42no—
187#11719919—42no—
188#11719919—42no—
189#11719919—42no—
190C4×Dic53—22no—
191C4×Dic53—22no—
192C4×Dic53—41no—
193C4×Dic53—41no—
194C4×Dic53—42no—
195C4×Dic53—42no—
196C4×Dic53—42no—
197C4×Dic53C22×C412yes—
198C4×Dic53#4112yes—
199C4×Dic53#4112yes—
200C4×Dic53#4112yes—
201C2×#88322—21no—
202C2×#88322—21no—
203C2×#88322—21no—
204C2×#88322—21no—
205C2×#88322C22×C412yes—
206C2×#88322#4112yes—
207C23×C53—22no—
208C23×C53—41no—
209C23×C53—41no—
210C23×C53—41no—
211C23×C53—41no—
212C23×C53—41no—
213C23×C53—41no—
214C23×C53C22×D411yescharacteristic
215D4×C53—22no—
216D4×C53—22no—
217D4×C53—41no—
218D4×C53—41no—
219C2×C4×C53—22no—
220C2×C4×C53—21no—
221C2×C4×C53—21no—
222C2×C4×C53—21no—
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×C53C23×C411yescharacteristic
235C2×C4×C53C2×#4111yescharacteristic
236C2×C4×C53C2×#4111yescharacteristic
237C2×C4×C53C2×#4111yescharacteristic
238C2×C4×C53C2×#4111yescharacteristic
239C2×C4×C53C2×#4111yescharacteristic
240C2×Dic53—21no—
241C2×Dic53—21no—
242C2×Dic53—21no—
243C2×Dic53—21no—
244C2×Dic53—21no—
245C2×Dic53—21no—
246C2×Dic53—22no—
247C2×Dic53—22no—
248C2×Dic53—41no—
249C2×Dic53—41no—
250C2×Dic53—41no—
251C2×Dic53—41no—
252C2×Dic53—41no—
253C2×Dic53—41no—
254C2×Dic53—41no—
255C2×Dic53—41no—
256C2×Dic53—41no—
257C2×Dic53—41no—
258C2×Dic53—84no—
259C2×Dic53C4×D412yes—
260C2×Dic53C4×D412yes—
261C2×Dic53#12412yes—
262C2×Dic53#12412yes—
263C2×Dic53AGL(1,ℤ/8)12yes—
264C2×Dic53#13712yes—
265Dic106—22no—
266Dic106—22no—
267Dic106—48no—
268Dic106—41no—
269Dic106—41no—
270C8×C53—22no—
271C8×C53—22no—
272#88322—22no—
273#88322—22no—
274C4×#489—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×C53C2×#12411yescharacteristic
284C22×C53#53211yescharacteristic
285C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
286C22×C53C2×#13711yescharacteristic
287C22×C53#57511yescharacteristic
288C22×C53#57511yescharacteristic
289C4×C53—41no—
290C4×C53—41no—
291C4×C53—42no—
292C4×C53—81no—
293C4×C53—81no—
294C4×C53C4×#4111yescharacteristic
295C4×C53C4×#4111yescharacteristic
296C4×C53#52911yescharacteristic
297C4×C53#52911yescharacteristic
298Dic53—22no—
299Dic53—22no—
300Dic53—22no—
301Dic53—42no—
302Dic53—42no—
303Dic53—81no—
304Dic53—81no—
305Dic53#48914yescomplemented
306C2×#489—531no—
307C4×C4⋊Q8—531no—
308C4×#124—531no—
309#1993—531no—
310#1959—531no—
311#1936—531no—
312#1936D5311yescharacteristic
313#1935—531no—
314#1935—531no—
315#1836—531no—
316C4×#103—531no—
317C4×#103—531no—
318C4×#102—531no—
319C4×#102—531no—
320C4×#97—531no—
321C2×C53—82no—
322C2×C53C4×#12411yescharacteristic
323C2×C53C4×AGL(1,ℤ/8)11yescharacteristic
324C2×C53C4×#13711yescharacteristic
325C2×C53C2×#48911yescharacteristic
326C2×C53#313511yescharacteristic
327C2×C53#317111yescharacteristic
328C2×C53#317211yescharacteristic
329C2×C4⋊Q8—531no—
330C2×#124—531no—
331C2×C4×Q8—531no—
332C2×C4×Q8—1061no—
333C2×C4×D4—531no—
334#489—538no—
335C2×#103—531no—
336C2×#103—531no—
337C2×#103—531no—
338C2×#103—531no—
339C2×#102—531no—
340C2×#102—531no—
341C2×#102C2×D5311yescharacteristic
342C2×#102C2×D5311yescharacteristic
343C2×#97—531no—
344C2×#97—531no—
345C2×#97—531no—
346C2×#97C2×D5311yescharacteristic
347C2×C4×C8—531no—
348C2×C4×C8—531no—
349#391—531no—
350#391—1061no—
351#390C2×D5311yescharacteristic
352#389—531no—
353#389—1061no—
354#389C2×D5311yescharacteristic
355#388—1061no—
356#388—1061no—
357#384—531no—
358#382—531no—
359C4×C4⋊C4—531no—
360C4×C4⋊C4—531no—
361C4×C4⋊C4—1061no—
362C4×C22⋊C4—1061no—
363C4×C22⋊C4—1061no—
364C43—531no—
365#340—531no—
366#340—531no—
367#340—531no—
368#340—531no—
369#340—531no—
370#340—531no—
371#340C2×D5311yescharacteristic
372#340C2×D5311yescharacteristic
373C53C4×#48911yescharacteristic, complemented, Sylow
374C22×Q8—531no—
375C22×Q8—1061no—
376C22×D4C22×D5311yescharacteristic
377C23×C4—1061no—
378C22×C8—531no—
379C22×C8—531no—
380C22×C8C22×D5311yescharacteristic
381C22×C8C22×D5311yescharacteristic
382C4⋊Q8—534no—
383C4⋊Q8—1062no—
384#124—534no—
385#124—1062no—
386C4×Q8—1062no—
387C4×Q8—1062no—
388C4×Q8—2124no—
389C4×D4—1062no—
390C4×D4—1062no—
391C2×C4⋊C4—21no—
392C2×C4⋊C4—531no—
393C2×C4⋊C4—531no—
394C2×C4⋊C4—531no—
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×C22⋊C4—21no—
404C2×C22⋊C4—21no—
405C2×C22⋊C4—1061no—
406C2×C22⋊C4—1061no—
407C2×C22⋊C4—1061no—
408C2×C42—531no—
409C2×C42—531no—
410C2×C42—531no—
411C2×C42—531no—
412C2×C42—531no—
413C2×C42—531no—
414C2×C42—1061no—
415C2×C42—1061no—
416C2×C42—1061no—
417C2×C42C22×D5311yescharacteristic
418#103—534no—
419#103—534no—
420#103—1062no—
421#103—1062no—
422#102—22no—
423#102—534no—
424#102—1062no—
425#102C4×D5314yes—
426#97—22no—
427#97—534no—
428#97—1062no—
429#97—1062no—
430C4×C8—1062no—
431C4×C8—1062no—
432#95—21no—
433#95—1061no—
434#95—1061no—
435#95—1061no—
436#95—1061no—
437#95—1061no—
438#95—1061no—
439#95—1061no—
440C24—21no—
441C2×Q8—532no—
442C2×Q8—1064no—
443C2×Q8—1061no—
444C2×Q8—1061no—
445C2×Q8—1061no—
446C2×Q8—1061no—
447C2×Q8—1061no—
448C2×Q8—2122no—
449C2×Q8—2122no—
450C2×D4—21no—
451C2×D4—21no—
452C2×D4—21no—
453C2×D4—21no—
454C2×D4—21no—
455C2×D4C2×C4×D5312yes—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—531no—
460C22×C4—531no—
461C22×C4—531no—
462C22×C4—531no—
463C22×C4—1061no—
464C22×C4—1061no—
465C22×C4—1061no—
466C22×C4—1061no—
467C22×C4—2122no—
468C22×C4—2122no—
469C22×C4—2122no—
470C22×C4D4×D5311yescharacteristic
471C22×C4D4×D5311yescharacteristic
472C22×C4C23×D5311yescharacteristic
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—532no—
478C2×C8—532no—
479C2×C8—1061no—
480C2×C8—1061no—
481C2×C8—1061no—
482C2×C8—1061no—
483C2×C8C2×C4×D5312yes—
484C2×C8#1171994712yes—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—42no—
491C4⋊C4—532no—
492C4⋊C4—1064no—
493C4⋊C4—1061no—
494C4⋊C4—1061no—
495C4⋊C4—1061no—
496C4⋊C4—1061no—
497C4⋊C4—1061no—
498C4⋊C4—2122no—
499C4⋊C4—2122no—
500C4⋊C4—2122no—
501C4⋊C4—2122no—
502C4⋊C4C2×C4×D5312yes—
503C22⋊C4—42no—
504C22⋊C4—42no—
505C22⋊C4—1064no—
506C22⋊C4—1064no—
507C22⋊C4—2122no—
508C42—21no—
509C42—21no—
510C42—532no—
511C42—532no—
512C42—532no—
513C42—532no—
514C42—1062no—
515C42—1062no—
516C42—2121no—
517C42—2121no—
518C42—2122no—
519C42—2122no—
520C42—2122no—
521C23—22no—
522C23—41no—
523C23—41no—
524C23—41no—
525C23—41no—
526C23—41no—
527C23—41no—
528C23C2×D4×D5311yescharacteristic
529Q8—1062no—
530Q8—1062no—
531Q8—2128no—
532Q8—2121no—
533Q8—2121no—
534D4—22no—
535D4—22no—
536D4—41no—
537D4—41no—
538C2×C4—22no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—532no—
554C2×C4—532no—
555C2×C4—532no—
556C2×C4—532no—
557C2×C4—532no—
558C2×C4—532no—
559C2×C4—1061no—
560C2×C4—1061no—
561C2×C4—1061no—
562C2×C4—1061no—
563C2×C4—1061no—
564C2×C4—1061no—
565C2×C4—1062no—
566C2×C4—1062no—
567C2×C4—4244no—
568C2×C4—2121no—
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×C4—2121no—
578C2×C4C22×C4×D5311yescharacteristic
579C2×C4#41×D5311yescharacteristic
580C2×C4#41×D5311yescharacteristic
581C2×C4#41×D5311yescharacteristic
582C2×C4#41×D5311yescharacteristic
583C2×C4C2×#1171994711yescharacteristic
584C8—22no—
585C8—22no—
586C8—1062no—
587C8—1062no—
588C22—42no—
589C22—42no—
590C22—42no—
591C22—81no—
592C22—81no—
593C22—81no—
594C22—81no—
595C22C4×D4×D5311yescharacteristic
596C22#124×D5311yescharacteristic
597C22#1681116811yescharacteristic
598C22AGL(1,ℤ/8)×D5311yescharacteristic
599C22#137×D5311yescharacteristic
600C22#1681141911yescharacteristic
601C22#1681141911yescharacteristic
602C4—41no—
603C4—41no—
604C4—42no—
605C4—81no—
606C4—81no—
607C4—534no—
608C4—1062no—
609C4—1062no—
610C4—1062no—
611C4—4241no—
612C4—4241no—
613C4—2122no—
614C4—2122no—
615C4C4×#1171994711yescharacteristic
616C4#1681109111yescharacteristic
617C4#1681115011yescharacteristic
618C4#1681115611yescharacteristic
619C2—82no—
620C2#2027725611yescharacteristic
621C2#2028352611yescharacteristic
622C2#2028353711yescharacteristic
623C2#489×D5311yescharacteristic
624C2#2028505911yescharacteristic
625C2#2028530511yescharacteristic
626C2#2028531411yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#1936C2
socleC23×C53C22×D4
prime radicalC22×D4×C53C23
cosocleC22×C4×C53C24
hypercenter#1936D53
hypocenterC53C4×#489
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1936

Complements

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

NH ≅ G/Ntype
2. #20277286C2semidirect
4. #20275686C2semidirect
7. #20275405C2semidirect
8. #20275404C2semidirect
9. #20274847C2semidirect
10. #20274534C2semidirect
11. #20274534C2semidirect
14. #97×Dic53C2semidirect
21. #16810981C4semidirect
66. #13133216C2×C4semidirect
95. #13133142C2×C4semidirect
97. #13133140C2×C4semidirect
102. #13133127C2×C4semidirect
305. Dic53#489semidirect
373. C53C4×#489semidirect

Maximal subgroups

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

subgroupindex
C2×#168109812
#202772862
#202772622
#202756862
#202755302
#202754152
#202754052
#202754042
#202748472
#202745342
#102×Dic532
#97×Dic532
C53×#19362
C4×#48953