Finite Group Databasebeta

← #27425355

Subgroups — #27425355

Order 13568.

868 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810923C211yesmaximal, characteristic, complemented
2C2×#16810680C211yesmaximal, characteristic, complemented
3#20283025C211yesmaximal, characteristic, complemented
4#20276151C211yesmaximal, characteristic, complemented
5#20276141C211yesmaximal, characteristic, complemented
6#20275901C211yesmaximal, characteristic
7#20275894C211yesmaximal, characteristic
8#20275672C211yesmaximal, characteristic, complemented
9#20275483C211yesmaximal, characteristic, complemented
10#20275455C211yesmaximal, characteristic, complemented
11#20275450C211yesmaximal, characteristic, complemented
12#20275415C211yesmaximal, characteristic, complemented
13#20275415C211yesmaximal, characteristic, complemented
14#20274725C211yesmaximal, characteristic, complemented
15C4×#121×C53C211yesmaximal, characteristic
16C2×#13133209C2211yescharacteristic
17#16810923—24no—
18#16810680—24no—
19C2×#13133150C2211yescharacteristic
20C2×#13133150C2211yescharacteristic
21C2×#13133148C2211yescharacteristic
22C2×#13133148C2211yescharacteristic
23C2×C4⋊#88322C2211yescharacteristic
24C2×C4⋊#88322C2211yescharacteristic
25C2×#13133138C2211yescharacteristic
26C2×#13133138C2211yescharacteristic
27C2×#13133138C2211yescharacteristic
28C2×#13133138C2211yescharacteristic
29C2×C22⋊#88322C2211yescharacteristic
30C2×C22⋊#88322C2211yescharacteristic
31C2×#13133128C2211yescharacteristic
32C2×#13133128C2211yescharacteristic
33#16810521C2211yescharacteristic
34#16810521C2211yescharacteristic
35#16810507—21no—
36#16810507—21no—
37#16810496—21no—
38#16810496—21no—
39#16810489C2211yescharacteristic
40#16810483C2211yescharacteristic
41#16810483C2211yescharacteristic
42#16810385C2211yescharacteristic
43#16810385C2211yescharacteristic
44#16810385C2211yescharacteristic
45#16810385C2211yescharacteristic
46#16810385C2211yescharacteristic
47#16810385C2211yescharacteristic
48#16810385C2211yescharacteristic
49#16810385C2211yescharacteristic
50C2×#121×C53C2211yescharacteristic
51C2×C4×D4×C53—21no—
52C2×C4×D4×C53C2211yescharacteristic
53C22×C42×C53C2211yescharacteristic
54C53×#390C2211yescharacteristic
55C53×#389C2211yescharacteristic
56C53×#388—21no—
57C53×#386—21no—
58C53×#380C2211yescharacteristic
59C4×C4⋊C4×C53C2211yescharacteristic
60C4×C22⋊C4×C53—21no—
61C22×#88322C2311yescharacteristic
62C22×#88322C2311yescharacteristic
63C22×#88322C2311yescharacteristic
64C22×#88322C2311yescharacteristic
65#13133209—22no—
66#13133209—22no—
67C2×C4⋊Dic53C2311yescharacteristic
68C2×C4⋊Dic53C2311yescharacteristic
69C2×C4⋊Dic53C2311yescharacteristic
70C2×C4⋊Dic53C2311yescharacteristic
71C2×#11719919—21no—
72C2×#11719919—21no—
73C2×#11719919—21no—
74C2×#11719919—21no—
75C2×C22⋊Dic53—21no—
76C2×C22⋊Dic53—21no—
77#13133150—22no—
78#13133150—22no—
79#13133148—22no—
80#13133148—22no—
81C4⋊#88322—22no—
82C4⋊#88322D414yescomplemented
83#13133138—22no—
84#13133138—22no—
85#13133138—22no—
86#13133138—22no—
87C22⋊#88322—22no—
88C22⋊#88322—22no—
89#13133128—22no—
90#13133128—22no—
91#13133124—21no—
92#13133124—21no—
93#13133124—21no—
94#13133124—21no—
95#13133124—21no—
96#13133124—21no—
97C22×D4×C53—21no—
98C22×D4×C53C2311yescharacteristic
99C23×C4×C53—21no—
100C23×C4×C53D411yescharacteristic
101C23×C4×C53D411yescharacteristic
102C23×C4×C53C2311yescharacteristic
103#121×C53—22no—
104#121×C53—22no—
105C4×D4×C53—22no—
106C4×D4×C53—22no—
107C4×D4×C53—44no—
108C2×C4⋊C4×C53—21no—
109C2×C4⋊C4×C53C2311yescharacteristic
110C2×C4⋊C4×C53C2311yescharacteristic
111C2×C4⋊C4×C53C2311yescharacteristic
112C2×C22⋊C4×C53—21no—
113C2×C22⋊C4×C53—21no—
114C2×C22⋊C4×C53—21no—
115C2×C22⋊C4×C53—21no—
116C2×C22⋊C4×C53—21no—
117C2×C42×C53—22no—
118C2×C42×C53—22no—
119C2×C42×C53—21no—
120C2×C42×C53D411yescharacteristic
121C2×C42×C53D411yescharacteristic
122C2×C42×C53C2311yescharacteristic
123C2×C42×C53C2311yescharacteristic
124#95×C53—21no—
125#95×C53—21no—
126#95×C53—21no—
127C24×C53—21no—
128C24×C53C2×D411yescharacteristic
129C2×D4×C53—24no—
130C2×D4×C53—21no—
131C2×D4×C53—21no—
132C2×D4×C53—21no—
133C2×D4×C53—21no—
134C2×D4×C53—21no—
135C2×D4×C53—42no—
136C2×D4×C53—42no—
137C2×D4×C53#4112yes—
138C22×C4×C53—22no—
139C22×C4×C53—21no—
140C22×C4×C53—22no—
141C22×C4×C53—22no—
142C22×C4×C53—21no—
143C22×C4×C53—21no—
144C22×C4×C53—21no—
145C22×C4×C53—21no—
146C22×C4×C53—21no—
147C22×C4×C53—21no—
148C22×C4×C53—21no—
149C22×C4×C53—21no—
150C22×C4×C53—41no—
151C22×C4×C53—41no—
152C22×C4×C53—41no—
153C22×C4×C53—42no—
154C22×C4×C53—42no—
155C22×C4×C53—42no—
156C22×C4×C53C2×D411yescharacteristic
157C22×C4×C53C2×D411yescharacteristic
158C22×C4×C53C2×D411yescharacteristic
159C22×C4×C53C2×D411yescharacteristic
160C22×C4×C53C2×D411yescharacteristic
161C22×C4×C53C2411yescharacteristic
162C22×Dic53—21no—
163C22×Dic53—21no—
164C22×Dic53—21no—
165C22×Dic53—21no—
166C4⋊C4×C53—21no—
167C4⋊C4×C53—21no—
168C4⋊C4×C53—21no—
169C4⋊C4×C53—21no—
170C4⋊C4×C53—21no—
171C4⋊C4×C53—42no—
172C4⋊C4×C53#4112yes—
173C22⋊C4×C53—24no—
174C22⋊C4×C53—42no—
175C22⋊C4×C53—42no—
176C22⋊C4×C53—42no—
177C22⋊C4×C53—42no—
178C42×C53—22no—
179C42×C53—22no—
180C42×C53—21no—
181C42×C53—21no—
182C42×C53—42no—
183C42×C53C2×D412yes—
184C42×C53#4112yes—
185C22⋊Dic53—42no—
186C22⋊Dic53—42no—
187C4⋊Dic53—21no—
188C4⋊Dic53—21no—
189C4⋊Dic53—21no—
190C4⋊Dic53—21no—
191C4⋊Dic53—21no—
192C4⋊Dic53—21no—
193C4⋊Dic53—21no—
194C4⋊Dic53—21no—
195#11719919—42no—
196#11719919—42no—
197#11719919—42no—
198#11719919—42no—
199C2×#88322—21no—
200C2×#88322—21no—
201C2×#88322—21no—
202C2×#88322—21no—
203C2×#88322—21no—
204C2×#88322—21no—
205C2×#88322—21no—
206C2×#88322—21no—
207C2×#88322—21no—
208C2×#88322—21no—
209C2×#88322C2×D412yes—
210C2×#88322C2×D412yes—
211C23×C53—21no—
212C23×C53—21no—
213C23×C53—21no—
214C23×C53—21no—
215C23×C53—21no—
216C23×C53—21no—
217C23×C53—22no—
218C23×C53—41no—
219C23×C53—41no—
220C23×C53—41no—
221C23×C53—41no—
222C23×C53—41no—
223C23×C53—41no—
224C23×C53AGL(1,ℤ/8)12yes—
225C23×C53C22×D411yescharacteristic
226D4×C53—22no—
227D4×C53—22no—
228D4×C53—48no—
229D4×C53—41no—
230D4×C53—41no—
231C2×C4×C53—21no—
232C2×C4×C53—21no—
233C2×C4×C53—21no—
234C2×C4×C53—21no—
235C2×C4×C53—21no—
236C2×C4×C53—21no—
237C2×C4×C53—21no—
238C2×C4×C53—21no—
239C2×C4×C53—22no—
240C2×C4×C53—21no—
241C2×C4×C53—21no—
242C2×C4×C53—21no—
243C2×C4×C53—21no—
244C2×C4×C53—21no—
245C2×C4×C53—44no—
246C2×C4×C53—44no—
247C2×C4×C53—41no—
248C2×C4×C53—41no—
249C2×C4×C53—41no—
250C2×C4×C53—41no—
251C2×C4×C53—41no—
252C2×C4×C53—41no—
253C2×C4×C53—41no—
254C2×C4×C53—41no—
255C2×C4×C53—41no—
256C2×C4×C53—41no—
257C2×C4×C53—41no—
258C2×C4×C53—41no—
259C2×C4×C53—84no—
260C2×C4×C53#12112yes—
261C2×C4×C53#12112yes—
262C2×C4×C53C22×D411yescharacteristic
263C2×C4×C53C2×#4111yescharacteristic
264C2×C4×C53C2×#4111yescharacteristic
265C2×C4×C53C2×#4111yescharacteristic
266C2×C4×C532−1+411yescharacteristic
267C2×C4×C532−1+411yescharacteristic
268C2×Dic53—41no—
269C2×Dic53—41no—
270C2×Dic53—41no—
271C2×Dic53—41no—
272C2×Dic53—41no—
273C2×Dic53—41no—
274C2×Dic53—41no—
275C2×Dic53—41no—
276C2×Dic53—41no—
277C2×Dic53—41no—
278C2×Dic53—41no—
279C2×Dic53—41no—
280#88322—22no—
281#88322—22no—
282#88322—41no—
283#88322—41no—
284#24706—531nomaximal, Sylow
285C22×C53—22no—
286C22×C53—22no—
287C22×C53—22no—
288C22×C53—41no—
289C22×C53—41no—
290C22×C53—41no—
291C22×C53—41no—
292C22×C53—42no—
293C22×C53—42no—
294C22×C53—42no—
295C22×C53—81no—
296C22×C53—81no—
297C22×C53—81no—
298C22×C53—81no—
299C22×C53C2×#12111yescharacteristic
300C22×C53#52511yescharacteristic
301C22×C53#53611yescharacteristic
302C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
303C22×C53#57511yescharacteristic
304C22×C53#57511yescharacteristic
305C22×C53#57511yescharacteristic
306C4×C53—22no—
307C4×C53—22no—
308C4×C53—42no—
309C4×C53—41no—
310C4×C53—41no—
311C4×C53—41no—
312C4×C53—81no—
313C4×C53—81no—
314C4×C53#54711yescharacteristic
315C4×C53#54811yescharacteristic
316C4×C53#55211yescharacteristic
317C4×C53#55611yescharacteristic
318Dic53—81no—
319Dic53—81no—
320Dic53—81no—
321Dic53—81no—
322C2×#478—531no—
323C2×#432—531no—
324#2851—531no—
325C4×#121D5311yescharacteristic
326#2077—531no—
327#2075—531no—
328#2034—531no—
329#2033—531no—
330#1989—531no—
331#1949—531no—
332#1944—531no—
333#1943—531no—
334#1936—531no—
335#1936—531no—
336#1804—531no—
337C2×C53—42no—
338C2×C53—82no—
339C2×C53#241511yescharacteristic
340C2×C53#305411yescharacteristic
341C2×C53#305511yescharacteristic
342C2×C53#311111yescharacteristic
343C2×C53#311411yescharacteristic
344C2×C53#323411yescharacteristic
345C2×C53#324011yescharacteristic
346C2×#125—531no—
347C2×#121C2×D5311yescharacteristic
348C2×C4×D4—21no—
349C2×C4×D4C2×D5311yescharacteristic
350C22×C42C2×D5311yescharacteristic
351#478—1064no—
352#432—1064no—
353C2×#107—531no—
354C2×#107—531no—
355C2×#106—531no—
356C2×#106—531no—
357C2×C4⋊C8—531no—
358C2×C4⋊C8—531no—
359C2×#102—531no—
360C2×#102—531no—
361C2×#102—531no—
362C2×#102—531no—
363C2×C22⋊C8—531no—
364C2×C22⋊C8—531no—
365C2×#97—531no—
366C2×#97—531no—
367#390C2×D5311yescharacteristic
368#389—531no—
369#389—531no—
370#389C2×D5311yescharacteristic
371#388—21no—
372#387—1061no—
373#387—1061no—
374#386—21no—
375#385—1061no—
376#385—1061no—
377#382—531no—
378#380—531no—
379#380—531no—
380#380C2×D5311yescharacteristic
381C4×C4⋊C4C2×D5311yescharacteristic
382C4×C22⋊C4—21no—
383#340—531no—
384#340—531no—
385#340—531no—
386#340—531no—
387#340—531no—
388#340—531no—
389#340—531no—
390#340—531no—
391C53#2470611yescharacteristic, complemented, Sylow
392C22×D4—21no—
393C22×D4C22×D5311yescharacteristic
394C23×C4—21no—
395C23×C4C22⋊D5311yescharacteristic
396C23×C4C22⋊D5311yescharacteristic
397C23×C4C22×D5311yescharacteristic
398C22×C8—531no—
399C22×C8—531no—
400C22×C8—531no—
401C22×C8—531no—
402#125—1062no—
403#125—1062no—
404#121—22no—
405#121—22no—
406C4×D4—22no—
407C4×D4—22no—
408C4×D4—44no—
409C2×C4⋊C4—21no—
410C2×C4⋊C4—531no—
411C2×C4⋊C4—531no—
412C2×C4⋊C4—531no—
413C2×C4⋊C4—531no—
414C2×C4⋊C4—1061no—
415C2×C4⋊C4—1061no—
416C2×C4⋊C4—1061no—
417C2×C4⋊C4—1061no—
418C2×C4⋊C4C22×D5311yescharacteristic
419C2×C4⋊C4C22×D5311yescharacteristic
420C2×C4⋊C4C22×D5311yescharacteristic
421C2×C22⋊C4—21no—
422C2×C22⋊C4—21no—
423C2×C22⋊C4—21no—
424C2×C22⋊C4—21no—
425C2×C22⋊C4—21no—
426C2×C22⋊C4—1061no—
427C2×C22⋊C4—1061no—
428C2×C42—22no—
429C2×C42—22no—
430C2×C42—21no—
431C2×C42C22⋊D5311yescharacteristic
432C2×C42C22⋊D5311yescharacteristic
433C2×C42C22×D5311yescharacteristic
434C2×C42C22×D5311yescharacteristic
435#107—1062no—
436#107—1062no—
437#106—1062no—
438#106—1062no—
439C4⋊C8—534no—
440C4⋊C8—1062no—
441#102—1062no—
442#102—1062no—
443#102—1062no—
444#102—1062no—
445C22⋊C8—1062no—
446C22⋊C8—1062no—
447#97—1062no—
448#97—1062no—
449#95—21no—
450#95—21no—
451#95—21no—
452#95—1061no—
453#95—1061no—
454#95—1061no—
455#95—1061no—
456#95—1061no—
457#95—1061no—
458C24—21no—
459C24C2×C22⋊D5311yescharacteristic
460C2×D4—24no—
461C2×D4—21no—
462C2×D4—21no—
463C2×D4—21no—
464C2×D4—21no—
465C2×D4—21no—
466C2×D4—42no—
467C2×D4—42no—
468C2×D4#1171994512yes—
469C22×C4—22no—
470C22×C4—21no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—41no—
482C22×C4—41no—
483C22×C4—41no—
484C22×C4—42no—
485C22×C4—42no—
486C22×C4—42no—
487C22×C4—1061no—
488C22×C4—1061no—
489C22×C4—1061no—
490C22×C4—1061no—
491C22×C4C2×C22⋊D5311yescharacteristic
492C22×C4C2×C22⋊D5311yescharacteristic
493C22×C4C2×C22⋊D5311yescharacteristic
494C22×C4C2×C22⋊D5311yescharacteristic
495C22×C4C2×C22⋊D5311yescharacteristic
496C22×C4C23×D5311yescharacteristic
497C2×C8—532no—
498C2×C8—532no—
499C2×C8—1061no—
500C2×C8—1061no—
501C2×C8—1061no—
502C2×C8—1061no—
503C2×C8—1061no—
504C2×C8—1061no—
505C2×C8—1061no—
506C2×C8—1061no—
507C2×C8—1061no—
508C2×C8—1061no—
509C4⋊C4—21no—
510C4⋊C4—21no—
511C4⋊C4—21no—
512C4⋊C4—21no—
513C4⋊C4—21no—
514C4⋊C4—42no—
515C4⋊C4—1061no—
516C4⋊C4—1061no—
517C4⋊C4—1061no—
518C4⋊C4—1061no—
519C4⋊C4—1061no—
520C4⋊C4—1061no—
521C4⋊C4—1061no—
522C4⋊C4—1061no—
523C4⋊C4—2122no—
524C4⋊C4—2122no—
525C4⋊C4—2122no—
526C4⋊C4—2122no—
527C4⋊C4#1171994512yes—
528C22⋊C4—24no—
529C22⋊C4—42no—
530C22⋊C4—42no—
531C22⋊C4—42no—
532C22⋊C4—42no—
533C22⋊C4—2122no—
534C22⋊C4—2122no—
535C42—22no—
536C42—22no—
537C42—21no—
538C42—21no—
539C42—42no—
540C42D4×D5312yes—
541C42#1171994712yes—
542C23—21no—
543C23—21no—
544C23—21no—
545C23—21no—
546C23—21no—
547C23—21no—
548C23—22no—
549C23—41no—
550C23—41no—
551C23—41no—
552C23—41no—
553C23—41no—
554C23—41no—
555C23#1313324312yes—
556C23C22×C22⋊D5311yescharacteristic
557D4—22no—
558D4—22no—
559D4—48no—
560D4—41no—
561D4—41no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—22no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—84no—
591C2×C4—2121no—
592C2×C4—2121no—
593C2×C4—2121no—
594C2×C4—2121no—
595C2×C4—2121no—
596C2×C4—2121no—
597C2×C4—2121no—
598C2×C4—2121no—
599C2×C4—2121no—
600C2×C4—2121no—
601C2×C4—2121no—
602C2×C4—2121no—
603C2×C4#1313318912yes—
604C2×C4#1313318912yes—
605C2×C4C2×D4×D5311yescharacteristic
606C2×C4C2×#1171994711yescharacteristic
607C2×C4C2×#1171994511yescharacteristic
608C2×C4C2×#1171994511yescharacteristic
609C2×C4#1313326411yescharacteristic
610C2×C4#1313326411yescharacteristic
611C8—1062no—
612C8—1062no—
613C8—2121no—
614C8—2121no—
615C22—22no—
616C22—22no—
617C22—22no—
618C22—41no—
619C22—41no—
620C22—41no—
621C22—41no—
622C22—42no—
623C22—42no—
624C22—42no—
625C22—81no—
626C22—81no—
627C22—81no—
628C22—81no—
629C22C2×#1313318911yescharacteristic
630C22#1681113411yescharacteristic
631C22#1681119411yescharacteristic
632C22C2×#1313324311yescharacteristic
633C22#1681142111yescharacteristic
634C22#1681142111yescharacteristic
635C22#1681142111yescharacteristic
636C4—22no—
637C4—22no—
638C4—41no—
639C4—41no—
640C4—41no—
641C4—42no—
642C4—81no—
643C4—81no—
644C4—4241no—
645C4—4241no—
646C4—4241no—
647C4—4241no—
648C4#1681127811yescharacteristic
649C4#1681128211yescharacteristic
650C4#1681131311yescharacteristic
651C4#1681134311yescharacteristic
652C2—42no—
653C2—82no—
654C2#2027814411yescharacteristic
655C2#2028432111yescharacteristic
656C2#2028433011yescharacteristic
657C2#2028482411yescharacteristic
658C2#2028485311yescharacteristic
659C2#2028590111yescharacteristic
660C2#2028595111yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC4×#121×C53C2
socleC23×C53C22×D4
prime radicalC2×#121×C53C22
hypercenterC4×#121D53
hypocenterC53#24706
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C22×C42 ◁ C4×#121

Complements

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

NH ≅ G/Ntype
1. C2×#16810923C2semidirect
2. C2×#16810680C2semidirect
3. #20283025C2semidirect
4. #20276151C2semidirect
5. #20276141C2semidirect
8. #20275672C2semidirect
9. #20275483C2semidirect
10. #20275455C2semidirect
11. #20275450C2semidirect
12. #20275415C2semidirect
13. #20275415C2semidirect
14. #20274725C2semidirect
82. C4⋊#88322D4semidirect
391. C53#24706semidirect

Maximal subgroups

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

subgroupindex
C2×#168109232
C2×#168106802
#202830252
#202761512
#202761412
#202759012
#202758942
#202756722
#202754832
#202754552
#202754502
#202754152
#202747252
C4×#121×C532
#2470653