Finite Group Databasebeta

← #27416796

Subgroups — #27416796

Order 13568.

1772 conjugacy classes of subgroups.

The subgroup lattice has 708 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13133191C211yesmaximal, characteristic
2C2×#16810947C212yesmaximal, complemented
3#20275904C212yesmaximal, complemented
4#20275897C212yesmaximal, complemented
5#20275037C212yesmaximal, complemented
6#20274907C211yesmaximal, characteristic, complemented
7#20274902C211yesmaximal, characteristic, complemented
8#20274645C211yesmaximal, characteristic, complemented
9#20274645C211yesmaximal, characteristic, complemented
10#20274625C211yesmaximal, characteristic, complemented
11C53×#2280C211yesmaximal, characteristic, complemented
12C23×D212C2211yescharacteristic
13C23×C22⋊D53—21no—
14C2×#13133191—28no—
15C2×#13133191—21no—
16C2×#13133191—21no—
17C2×#13133191C2212yes—
18C2×#13133191C2212yes—
19C22×C4⋊Dic53C2211yescharacteristic
20C22×#11719921—21no—
21#16810947—28no—
22C2×#13133150C2212yescomplemented
23C2×#13133148C2212yescomplemented
24C2×#13133137C2212yescomplemented
25C2×#13133137C2212yescomplemented
26C2×#13133137C2212yescomplemented
27C2×#13133137C2212yescomplemented
28C2×C22⋊#88322C2211yescharacteristic
29C2×C22⋊#88322C2211yescharacteristic
30C2×C22⋊#88322C2212yescomplemented
31C2×C22⋊#88322C2212yescomplemented
32#16810385C2212yescomplemented
33#16810385C2212yescomplemented
34C24×C4×C53C2211yescharacteristic
35C2×#117×C53C2211yescharacteristic, complemented
36C2×#117×C53C2211yescharacteristic, complemented
37C53×#381—21no—
38C53×#380C2211yescharacteristic
39C53×#380C2211yescharacteristic
40C53×#380C2211yescharacteristic, complemented
41C53×#380C2211yescharacteristic, complemented
42C53×#379—21no—
43C4×C22⋊C4×C53—21no—
44C4×C22⋊C4×C53—21no—
45C24×D53—21no—
46C22×D212—21no—
47C22×D212—24no—
48C22×D212—22no—
49C22×D212—22no—
50C22×D212—42no—
51C22×D212C2312yes—
52C22×C22⋊D53—28no—
53C22×C22⋊D53—22no—
54C22×C22⋊D53—22no—
55C22×C22⋊D53—44no—
56C22×C22⋊D53—44no—
57C23×Dic53—21no—
58C22×#88322C2312yes—
59C22×#88322C2312yes—
60#13133191—28no—
61#13133191—28no—
62#13133191—24no—
63#13133191—24no—
64#13133191—24no—
65#13133191D418yes—
66C2×C4⋊Dic53—24no—
67C2×C4⋊Dic53—21no—
68C2×C4⋊Dic53C2312yes—
69C2×#11719921—28no—
70C2×#11719921—22no—
71C2×#11719921—22no—
72#13133150—24no—
73#13133148—24no—
74#13133137—24no—
75#13133137—24no—
76#13133137—42no—
77#13133137—42no—
78C22⋊#88322—24no—
79C22⋊#88322—22no—
80C22⋊#88322—22no—
81C22⋊#88322—42no—
82C25×C53D411yescharacteristic
83C23×C4×C53—24no—
84C23×C4×C53—22no—
85C23×C4×C53—22no—
86C23×C4×C53—42no—
87C23×C4×C53D411yescharacteristic
88C23×C4×C53D411yescharacteristic
89C23×C4×C53D411yescharacteristic
90C23×C4×C53C2311yescharacteristic
91C23×C4×C53C2311yescharacteristic
92C23×C4×C53C2311yescharacteristic
93#117×C53—24no—
94#117×C53—42no—
95C2×C4⋊C4×C53C2311yescharacteristic
96C2×C4⋊C4×C53C2311yescharacteristic
97C2×C4⋊C4×C53C2311yescharacteristic
98C2×C4⋊C4×C53C2311yescharacteristic
99C2×C22⋊C4×C53—21no—
100C2×C22⋊C4×C53—21no—
101C2×C22⋊C4×C53—21no—
102C2×C22⋊C4×C53—21no—
103C2×C22⋊C4×C53—21no—
104C2×C22⋊C4×C53—21no—
105C2×C42×C53—21no—
106C2×C42×C53—21no—
107#95×C53—21no—
108#95×C53—21no—
109#95×C53—21no—
110#95×C53—21no—
111C24×C53—22no—
112C24×C53—22no—
113C24×C53—22no—
114C24×C53—22no—
115C24×C53—22no—
116C24×C53—41no—
117C24×C53—41no—
118C24×C53C2×D411yescharacteristic
119C24×C53C2×D411yescharacteristic
120C24×C53C2×D411yescharacteristic
121C23×D53—24no—
122C23×D53—22no—
123C23×D53—42no—
124C23×D53—42no—
125C23×D53—42no—
126C23×D53—42no—
127C23×D53—42no—
128C23×D53—42no—
129C22×C4×C53—22no—
130C22×C4×C53—21no—
131C22×C4×C53—21no—
132C22×C4×C53—22no—
133C22×C4×C53—22no—
134C22×C4×C53—22no—
135C22×C4×C53—22no—
136C22×C4×C53—22no—
137C22×C4×C53—22no—
138C22×C4×C53—22no—
139C22×C4×C53—22no—
140C22×C4×C53—22no—
141C22×C4×C53—21no—
142C22×C4×C53—21no—
143C22×C4×C53—22no—
144C22×C4×C53—21no—
145C22×C4×C53—21no—
146C22×C4×C53—21no—
147C22×C4×C53—21no—
148C22×C4×C53—41no—
149C22×C4×C53—41no—
150C22×C4×C53—41no—
151C22×C4×C53—48no—
152C22×C4×C53—48no—
153C22×C4×C53C2×D411yescharacteristic
154C22×C4×C53C2×D411yescharacteristic
155C22×C4×C53C2×D414yes—
156C22×C4×C53C2×D411yescharacteristic
157C22×C4×C53#4112yes—
158C22×C4×C53#4112yes—
159C22×C4×C53C2411yescharacteristic
160C2×C22⋊D53—28no—
161C2×C22⋊D53—28no—
162C2×C22⋊D53—432no—
163C2×C22⋊D53—44no—
164C2×C22⋊D53—44no—
165C2×C22⋊D53—44no—
166C2×C22⋊D53—44no—
167C22×Dic53—24no—
168C22×Dic53—22no—
169C22×Dic53—42no—
170C22×Dic53—42no—
171C2×D212—22no—
172C2×D212—22no—
173C2×D212—22no—
174C2×D212—22no—
175C2×D212—24no—
176C2×D212—22no—
177C2×D212—48no—
178C2×D212—48no—
179C2×D212—42no—
180C2×D212—42no—
181C2×D212C2×D414yes—
182C4⋊C4×C53—22no—
183C4⋊C4×C53—22no—
184C4⋊C4×C53—22no—
185C4⋊C4×C53—22no—
186C22⋊C4×C53—42no—
187C22⋊C4×C53—42no—
188C22⋊C4×C53—42no—
189C22⋊C4×C53—42no—
190C22⋊C4×C53—42no—
191C22⋊C4×C53—42no—
192C42×C53—42no—
193C42×C53—42no—
194#11719921—28no—
195#11719921—28no—
196C4⋊Dic53—24no—
197C4⋊Dic53—22no—
198C4⋊Dic53C2×D414yes—
199C2×#88322—22no—
200C2×#88322—22no—
201C2×#88322—22no—
202C2×#88322—22no—
203C2×#88322—22no—
204C2×#88322—22no—
205C23×C53—21no—
206C23×C53—21no—
207C23×C53—21no—
208C23×C53—21no—
209C23×C53—28no—
210C23×C53—21no—
211C23×C53—21no—
212C23×C53—21no—
213C23×C53—21no—
214C23×C53—21no—
215C23×C53—21no—
216C23×C53—21no—
217C23×C53—21no—
218C23×C53—21no—
219C23×C53—21no—
220C23×C53—21no—
221C23×C53—21no—
222C23×C53—44no—
223C23×C53—44no—
224C23×C53—44no—
225C23×C53—44no—
226C23×C53—44no—
227C23×C53—44no—
228C23×C53#12012yes—
229C23×C53#12312yes—
230C23×C53#13512yes—
231C23×C53AGL(1,ℤ/8)12yes—
232C23×C53AGL(1,ℤ/8)12yes—
233C23×C53C22×D411yescharacteristic
234C22×D53—24no—
235C22×D53—48no—
236C22×D53—48no—
237C22×D53—48no—
238C22×D53—42no—
239C22×D53—42no—
240C22×D53—42no—
241C22×D53—42no—
242C22×D53—42no—
243C22×D53—42no—
244C22×D53—84no—
245C22×D53—84no—
246C22×D53—84no—
247C22×D53—84no—
248C2×C4×C53—21no—
249C2×C4×C53—21no—
250C2×C4×C53—21no—
251C2×C4×C53—21no—
252C2×C4×C53—24no—
253C2×C4×C53—24no—
254C2×C4×C53—24no—
255C2×C4×C53—24no—
256C2×C4×C53—21no—
257C2×C4×C53—21no—
258C2×C4×C53—21no—
259C2×C4×C53—44no—
260C2×C4×C53—44no—
261C2×C4×C53—44no—
262C2×C4×C53—42no—
263C2×C4×C53—42no—
264C2×C4×C53—41no—
265C2×C4×C53—41no—
266C2×C4×C53—41no—
267C2×C4×C53—41no—
268C2×C4×C53—41no—
269C2×C4×C53—41no—
270C2×C4×C53—41no—
271C2×C4×C53—41no—
272C2×C4×C53—41no—
273C2×C4×C53—41no—
274C2×C4×C53—41no—
275C2×C4×C53—41no—
276C2×C4×C53#12012yes—
277C2×C4×C53#12312yes—
278C2×C4×C53C22×D412yes—
279C2×C4×C53C2×#4111yescharacteristic
280C2×C4×C53C2×#4111yescharacteristic
281C2×C4×C53H(5,𝔽2)12yes—
282C22⋊D53—416no—
283C22⋊D53—416no—
284C2×Dic53—24no—
285C2×Dic53—48no—
286C2×Dic53—42no—
287C2×Dic53—42no—
288D212—24no—
289D212—24no—
290D212—48no—
291D212—42no—
292D212—42no—
293#88322—42no—
294#88322—42no—
295#24009—531nomaximal, Sylow
296C22×C53—22no—
297C22×C53—22no—
298C22×C53—22no—
299C22×C53—22no—
300C22×C53—22no—
301C22×C53—22no—
302C22×C53—22no—
303C22×C53—22no—
304C22×C53—22no—
305C22×C53—22no—
306C22×C53—22no—
307C22×C53—22no—
308C22×C53—416no—
309C22×C53—41no—
310C22×C53—41no—
311C22×C53—41no—
312C22×C53—41no—
313C22×C53—41no—
314C22×C53—41no—
315C22×C53—41no—
316C22×C53—41no—
317C22×C53#48214yescomplemented
318C22×C53C2×#12011yescharacteristic
319C22×C53C2×#12311yescharacteristic
320C22×C53#53411yescharacteristic
321C22×C53C2×#13511yescharacteristic
322C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
323C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
324C22×C53#57511yescharacteristic
325C2×D53—44no—
326C2×D53—44no—
327C2×D53—44no—
328C2×D53—816no—
329C2×D53—82no—
330C2×D53—82no—
331C2×D53—82no—
332C2×D53—82no—
333C4×C53—22no—
334C4×C53—22no—
335C4×C53—41no—
336C4×C53—81no—
337C4×C53—81no—
338C4×C53—81no—
339C4×C53—81no—
340C4×C53#54612yes—
341C4×C53#54612yes—
342Dic53—44no—
343C22×#121—531no—
344C2×#482—532no—
345#2280D5311yescharacteristic, complemented
346#2034—532no—
347#2033—532no—
348#1875—532no—
349#1853—531no—
350#1852—531no—
351#1786—531no—
352#1786—531no—
353#1779—531no—
354C2×C53—24no—
355C2×C53—42no—
356C2×C53—42no—
357C2×C53#240511yescharacteristic
358C2×C53#300211yescharacteristic
359C2×C53#300311yescharacteristic
360C2×C53C2×#48211yescharacteristic
361C2×C53#309411yescharacteristic
362C2×C53#318811yescharacteristic
363C2×C53#319411yescharacteristic
364D53—84no—
365C23×D4—531no—
366C23×D4—1061no—
367C24×C4C2×D5311yescharacteristic
368C2×#121—532no—
369C2×#121—532no—
370C2×#121—1068no—
371C2×#121—1061no—
372C2×#121—1061no—
373C2×#117C2×D5311yescharacteristic, complemented
374C2×#117C2×D5311yescharacteristic, complemented
375C22×C4⋊C4—531no—
376C22×C22⋊C4—1061no—
377#482—1068no—
378C2×#107—532no—
379C2×#106—532no—
380C2×#102—532no—
381C2×#102—532no—
382C2×#102—532no—
383C2×#102—532no—
384C2×C22⋊C8—532no—
385C2×C22⋊C8—531no—
386C2×C22⋊C8—531no—
387C2×C22⋊C8—532no—
388#381—21no—
389#380C2×D5311yescharacteristic
390#380C2×D5311yescharacteristic
391#380C2×D5311yescharacteristic, complemented
392#380C2×D5311yescharacteristic, complemented
393#379—21no—
394C4×C22⋊C4—21no—
395C4×C22⋊C4—21no—
396#340—532no—
397#340—532no—
398C53#2400911yescharacteristic, complemented, Sylow
399C25—1061no—
400C25C22⋊D5311yescharacteristic
401C22×D4—532no—
402C22×D4—1064no—
403C22×D4—1068no—
404C22×D4—1062no—
405C22×D4—1062no—
406C22×D4—1062no—
407C22×D4—1062no—
408C22×D4—1061no—
409C22×D4—2122no—
410C22×D4—2124no—
411C22×D4—2124no—
412C23×C4—22no—
413C23×C4—22no—
414C23×C4—24no—
415C23×C4—42no—
416C23×C4—1061no—
417C23×C4C22⋊D5311yescharacteristic
418C23×C4C22⋊D5311yescharacteristic
419C23×C4C22⋊D5311yescharacteristic
420C23×C4C22×D5311yescharacteristic
421C23×C4C22×D5311yescharacteristic
422C23×C4C22×D5311yescharacteristic
423C22×C8—532no—
424C22×C8—532no—
425#121—538no—
426#121—1064no—
427#121—1064no—
428#121—1068no—
429#121—1068no—
430#121—1064no—
431#117—24no—
432#117—42no—
433C2×C4⋊C4—532no—
434C2×C4⋊C4—1064no—
435C2×C4⋊C4—1061no—
436C2×C4⋊C4C22×D5311yescharacteristic
437C2×C4⋊C4C22×D5311yescharacteristic
438C2×C4⋊C4C22×D5311yescharacteristic
439C2×C4⋊C4C22×D5311yescharacteristic
440C2×C22⋊C4—21no—
441C2×C22⋊C4—21no—
442C2×C22⋊C4—21no—
443C2×C22⋊C4—21no—
444C2×C22⋊C4—21no—
445C2×C22⋊C4—21no—
446C2×C22⋊C4—1068no—
447C2×C22⋊C4—1062no—
448C2×C22⋊C4—1062no—
449C2×C42—21no—
450C2×C42—21no—
451#107—1064no—
452#106—1064no—
453#102—1064no—
454#102—1064no—
455#102—2122no—
456#102—2122no—
457C22⋊C8—1062no—
458C22⋊C8—1062no—
459C22⋊C8—1064no—
460C22⋊C8—2122no—
461#95—21no—
462#95—21no—
463#95—21no—
464#95—21no—
465C24—22no—
466C24—22no—
467C24—22no—
468C24—22no—
469C24—22no—
470C24—41no—
471C24—41no—
472C24—1064no—
473C24—1062no—
474C24—2122no—
475C24—2122no—
476C24—2122no—
477C24—2122no—
478C24—2122no—
479C24—2122no—
480C24C2×C22⋊D5311yescharacteristic
481C24C2×C22⋊D5311yescharacteristic
482C24C2×C22⋊D5311yescharacteristic
483C2×D4—534no—
484C2×D4—1064no—
485C2×D4—1068no—
486C2×D4—1068no—
487C2×D4—1062no—
488C2×D4—1062no—
489C2×D4—1062no—
490C2×D4—1062no—
491C2×D4—1062no—
492C2×D4—2128no—
493C2×D4—2128no—
494C2×D4—2122no—
495C2×D4—21232no—
496C2×D4—2124no—
497C2×D4—2124no—
498C2×D4—2124no—
499C2×D4—2124no—
500C2×D4—2122no—
501C22×C4—22no—
502C22×C4—21no—
503C22×C4—21no—
504C22×C4—22no—
505C22×C4—22no—
506C22×C4—22no—
507C22×C4—22no—
508C22×C4—22no—
509C22×C4—22no—
510C22×C4—22no—
511C22×C4—22no—
512C22×C4—22no—
513C22×C4—21no—
514C22×C4—21no—
515C22×C4—22no—
516C22×C4—21no—
517C22×C4—21no—
518C22×C4—21no—
519C22×C4—21no—
520C22×C4—41no—
521C22×C4—41no—
522C22×C4—41no—
523C22×C4—48no—
524C22×C4—48no—
525C22×C4—1064no—
526C22×C4—1062no—
527C22×C4—2122no—
528C22×C4—2122no—
529C22×C4#1171994512yes—
530C22×C4#1171994512yes—
531C22×C4D4×D5314yes—
532C22×C4C2×C22⋊D5311yescharacteristic
533C22×C4C2×C22⋊D5311yescharacteristic
534C22×C4C2×C22⋊D5311yescharacteristic
535C22×C4C23×D5311yescharacteristic
536C2×C8—1062no—
537C2×C8—1062no—
538C2×C8—1062no—
539C2×C8—1062no—
540C2×C8—1062no—
541C2×C8—1062no—
542C4⋊C4—22no—
543C4⋊C4—22no—
544C4⋊C4—22no—
545C4⋊C4—22no—
546C4⋊C4—534no—
547C4⋊C4—1064no—
548C4⋊C4—1062no—
549C22⋊C4—42no—
550C22⋊C4—42no—
551C22⋊C4—42no—
552C22⋊C4—42no—
553C22⋊C4—42no—
554C22⋊C4—42no—
555C22⋊C4—1068no—
556C22⋊C4—1068no—
557C42—42no—
558C42—42no—
559C23—21no—
560C23—21no—
561C23—21no—
562C23—21no—
563C23—28no—
564C23—21no—
565C23—21no—
566C23—21no—
567C23—21no—
568C23—21no—
569C23—21no—
570C23—21no—
571C23—21no—
572C23—21no—
573C23—21no—
574C23—21no—
575C23—21no—
576C23—44no—
577C23—44no—
578C23—44no—
579C23—44no—
580C23—44no—
581C23—44no—
582C23—1064no—
583C23—4244no—
584C23—4244no—
585C23—4244no—
586C23—4244no—
587C23—2128no—
588C23—2128no—
589C23—2128no—
590C23—2122no—
591C23—2122no—
592C23—2122no—
593C23—2122no—
594C23—2122no—
595C23—2122no—
596C23#1313318512yes—
597C23#1313320012yes—
598C23#1313323712yes—
599C23#1313324212yes—
600C23#1313324212yes—
601C23C22×C22⋊D5311yescharacteristic
602D4—1064no—
603D4—1064no—
604D4—2128no—
605D4—21216no—
606D4—21216no—
607D4—2122no—
608D4—2122no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—21no—
612C2×C4—21no—
613C2×C4—24no—
614C2×C4—24no—
615C2×C4—24no—
616C2×C4—24no—
617C2×C4—21no—
618C2×C4—21no—
619C2×C4—21no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—44no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—41no—
626C2×C4—41no—
627C2×C4—41no—
628C2×C4—41no—
629C2×C4—41no—
630C2×C4—41no—
631C2×C4—41no—
632C2×C4—41no—
633C2×C4—41no—
634C2×C4—41no—
635C2×C4—41no—
636C2×C4—41no—
637C2×C4—1064no—
638C2×C4—2128no—
639C2×C4—2122no—
640C2×C4—2122no—
641C2×C4#1313318512yes—
642C2×C4#1313320012yes—
643C2×C4C2×D4×D5312yes—
644C2×C4C2×#1171994511yescharacteristic
645C2×C4C2×#1171994511yescharacteristic
646C2×C4#1313326212yes—
647C8—2122no—
648C8—2122no—
649C22—22no—
650C22—22no—
651C22—22no—
652C22—22no—
653C22—22no—
654C22—22no—
655C22—22no—
656C22—22no—
657C22—22no—
658C22—22no—
659C22—22no—
660C22—22no—
661C22—416no—
662C22—41no—
663C22—41no—
664C22—41no—
665C22—41no—
666C22—41no—
667C22—41no—
668C22—41no—
669C22—41no—
670C22—42416no—
671C22—4242no—
672C22—4242no—
673C22—4242no—
674C22—4242no—
675C22—2124no—
676C22—2124no—
677C22—2124no—
678C22#1681094714yescomplemented
679C22C2×#1313318511yescharacteristic
680C22C2×#1313320011yescharacteristic
681C22#1681118211yescharacteristic
682C22C2×#1313323711yescharacteristic
683C22C2×#1313324211yescharacteristic
684C22C2×#1313324211yescharacteristic
685C22#1681142111yescharacteristic
686C4—22no—
687C4—22no—
688C4—41no—
689C4—81no—
690C4—81no—
691C4—81no—
692C4—81no—
693C4—2124no—
694C4#1681126312yes—
695C4#1681126312yes—
696C2—24no—
697C2—42no—
698C2—42no—
699C2—4244no—
700C2#2027804211yescharacteristic
701C2#2028393111yescharacteristic
702C2#2028394111yescharacteristic
703C2C2×#1681094711yescharacteristic
704C2#2028465511yescharacteristic
705C2#2028547611yescharacteristic
706C2#2028552511yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2280C2
socleC23×C53C22×D4
prime radicalC22×#13133191C2
hypercenter#2280D53
hypocenterC53#24009
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C24×C4 ◁ #2280

Complements

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

NH ≅ G/Ntype
2. C2×#16810947C2semidirect
3. #20275904C2semidirect
4. #20275897C2semidirect
5. #20275037C2semidirect
6. #20274907C2semidirect
7. #20274902C2semidirect
8. #20274645C2semidirect
9. #20274645C2semidirect
10. #20274625C2semidirect
11. C53×#2280C2semidirect
22. C2×#13133150C22semidirect
23. C2×#13133148C22semidirect
24. C2×#13133137C22semidirect
25. C2×#13133137C22semidirect
26. C2×#13133137C22semidirect
27. C2×#13133137C22semidirect
30. C2×C22⋊#88322C22semidirect
31. C2×C22⋊#88322C22semidirect
32. #16810385C22semidirect
33. #16810385C22semidirect
35. C2×#117×C53C22semidirect
36. C2×#117×C53C22semidirect
40. C53×#380C22semidirect
41. C53×#380C22semidirect
317. C22×C53#482semidirect
345. #2280D53semidirect
373. C2×#117C2×D53semidirect
374. C2×#117C2×D53semidirect
391. #380C2×D53semidirect
392. #380C2×D53semidirect
398. C53#24009semidirect
678. C22#16810947semidirect

Maximal subgroups

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

subgroupindex
C22×#131331912
C2×#168109472
#202759042
#202758972
#202750372
#202749072
#202749022
#202746452
#202746252
C53×#22802
#2400953