Finite Group Databasebeta

← #27313041

Subgroups — #27313041

Order 13568.

1918 conjugacy classes of subgroups.

The subgroup lattice has 680 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810525C211yesmaximal, characteristic
2C2×#16810501C212yesmaximal
3#20272967C212yesmaximal, complemented
4#20272959C212yesmaximal, complemented
5#20272959C212yesmaximal, complemented
6#20272942C211yesmaximal, characteristic, complemented
7#20272936C212yesmaximal, complemented
8#20272936C212yesmaximal, complemented
9C53×#1434C211yesmaximal, characteristic, complemented
10C23×D212C2211yescharacteristic
11C23×C4×D53C2212yes—
12C22×C4⋊Dic53C2211yescharacteristic
13C22×#11719919C2212yes—
14C22×#11719921C2212yes—
15C22×#11719921C2212yes—
16C22×#11719921C2212yes—
17C22×#11719921C2212yes—
18C22×#11719921C2212yes—
19#16810525C418yescomplemented
20#16810501—28no—
21C2×#13133124C2212yes—
22C2×#13133124C2212yes—
23C2×#13133124C2211yescharacteristic
24C2×#13133124C2212yes—
25C2×#13133124C2212yes—
26C2×#13133124C2212yes—
27C2×#13133124C2211yescharacteristic
28C22×C42×C53C2211yescharacteristic
29C2×#95×C53C2212yes—
30C2×#95×C53C2212yes—
31C2×#95×C53C2211yescharacteristic
32C2×#95×C53C2211yescharacteristic
33C24×D53C2312yes—
34C22×D212—41no—
35C22×D212—41no—
36C22×D212—41no—
37C22×D212—41no—
38C22×D212C2×C414yes—
39C22×D212D414yes—
40C22×D212Q814yes—
41C23×Dic53C2312yes—
42C23×Dic53C2312yes—
43C23×Dic53C2312yes—
44C22×C4×D53—24no—
45C22×C4×D53—24no—
46C22×C4×D53—24no—
47C22×C4×D53—42no—
48C22×C4×D53—42no—
49C22×C4×D53—42no—
50C22×C4×D53—42no—
51C2×C4⋊Dic53—41no—
52C2×C4⋊Dic53—41no—
53C2×C4⋊Dic53C2×C414yescomplemented
54C2×#11719919—42no—
55C2×#11719919—42no—
56C2×#11719919—42no—
57C2×#11719921—24no—
58C2×#11719921—24no—
59C2×#11719921—24no—
60C2×#11719921—24no—
61C2×#11719921—24no—
62C2×#11719921—42no—
63C2×#11719921—42no—
64C2×#11719921—42no—
65C2×#11719921—42no—
66C2×#11719921—42no—
67C2×#11719921—42no—
68C2×#11719921—42no—
69C2×#11719921—42no—
70C2×#11719921C2×C418yescomplemented
71C2×#11719921C2×C418yescomplemented
72#13133124—24no—
73#13133124—24no—
74#13133124—42no—
75#13133124—42no—
76#13133124—42no—
77#13133124—41no—
78#13133124—41no—
79C23×C4×C53C2312yes—
80C23×C4×C53C2311yescharacteristic
81C23×C4×C53C2311yescharacteristic
82C23×C4×C53C2312yes—
83C23×C4×C53C2311yescharacteristic
84C2×C42×C53—41no—
85C2×C42×C53—41no—
86C2×C42×C53C2×C414yescomplemented
87#95×C53—24no—
88#95×C53—24no—
89#95×C53—41no—
90#95×C53—41no—
91C24×C53C2411yescharacteristic
92C23×D53—22no—
93C23×D53—22no—
94C23×D53—22no—
95C23×D53—22no—
96C23×D53—22no—
97C23×D53—22no—
98C23×D53—22no—
99C23×D53—22no—
100C23×D53—22no—
101C23×D53—22no—
102C23×D53—22no—
103C23×D53—22no—
104C23×D53C22×C414yes—
105C23×D53C2×D414yes—
106C23×D53C2×Q814yes—
107C22×C4×C53—21no—
108C22×C4×C53—21no—
109C22×C4×C53—21no—
110C22×C4×C53—21no—
111C22×C4×C53—22no—
112C22×C4×C53—22no—
113C22×C4×C53—22no—
114C22×C4×C53—22no—
115C22×C4×C53—21no—
116C22×C4×C53—21no—
117C22×C4×C53—22no—
118C22×C4×C53—21no—
119C22×C4×C53—21no—
120C22×C4×C53—22no—
121C22×C4×C53—22no—
122C22×C4×C53—22no—
123C22×C4×C53—22no—
124C22×C4×C53—22no—
125C22×C4×C53—22no—
126C22×C4×C53—22no—
127C22×C4×C53—22no—
128C22×C4×C53—22no—
129C22×C4×C53—22no—
130C22×C4×C53—22no—
131C22×C4×C53—22no—
132C22×C4×C53C22×C412yes—
133C22×C4×C53C22×C412yes—
134C22×C4×C53C22×C412yes—
135C22×C4×C53C2×D412yes—
136C22×C4×C53C2×Q812yes—
137C22×C4×C53#4114yes—
138C22×Dic53—22no—
139C22×Dic53—22no—
140C22×Dic53—22no—
141C22×Dic53—22no—
142C22×Dic53—22no—
143C22×Dic53—22no—
144C22×Dic53—22no—
145C22×Dic53—22no—
146C22×Dic53—22no—
147C22×Dic53—22no—
148C22×Dic53—22no—
149C22×Dic53—22no—
150C22×Dic53—22no—
151C22×Dic53—22no—
152C22×Dic53—22no—
153C22×Dic53—22no—
154C22×Dic53—22no—
155C22×Dic53—22no—
156C22×Dic53—22no—
157C22×Dic53C22×C414yes—
158C22×Dic53#4114yes—
159C2×D212—44no—
160C2×D212—44no—
161C2×D212—44no—
162C2×D212—44no—
163C2×D212—44no—
164C2×D212—44no—
165C2×D212C4⋊C4116yescomplemented
166C2×C4×D53—48no—
167C2×C4×D53—48no—
168C2×C4×D53—48no—
169C2×C4×D53—48no—
170C2×C4×D53—48no—
171C2×C4×D53—48no—
172C2×C4×D53—48no—
173C42×C53—44no—
174#11719921—48no—
175#11719921—48no—
176#11719921—48no—
177#11719921—48no—
178#11719921—48no—
179C4⋊Dic53—44no—
180#11719919—48no—
181C23×C53C23×C411yescharacteristic
182C23×C53C22×D411yescharacteristic
183C23×C53C22×Q811yescharacteristic
184C23×C53C2×#4112yes—
185C23×C53C2×#4112yes—
186C23×C53H(5,𝔽2)11yescharacteristic
187C23×C53H(5,𝔽2)11yescharacteristic
188C23×C53H(5,𝔽2)11yescharacteristic
189C23×C53H(5,𝔽2)11yescharacteristic
190C23×C53H(5,𝔽2)12yes—
191C23×C53H(5,𝔽2)12yes—
192C22×D53—24no—
193C22×D53—24no—
194C22×D53—24no—
195C22×D53—24no—
196C22×D53—24no—
197C22×D53—24no—
198C22×D53—24no—
199C22×D53—24no—
200C22×D53—24no—
201C22×D53—24no—
202C22×D53—24no—
203C22×D53—24no—
204C22×D53—24no—
205C22×D53—24no—
206C22×D53—24no—
207C22×D53—24no—
208C22×D53—24no—
209C22×D53—24no—
210C22×D53—42no—
211C22×D53—42no—
212C22×D53—42no—
213C22×D53—42no—
214C22×D53—42no—
215C22×D53—42no—
216C22×D53—42no—
217C22×D53—42no—
218C22×D53—42no—
219C22×D53—42no—
220C22×D53—42no—
221C22×D53—42no—
222C22×D53—42no—
223C22×D53—42no—
224C22×D53—42no—
225C22×D53—42no—
226C22×D53C2×C4⋊C418yes—
227C2×C4×C53—22no—
228C2×C4×C53—22no—
229C2×C4×C53—22no—
230C2×C4×C53—22no—
231C2×C4×C53—22no—
232C2×C4×C53—22no—
233C2×C4×C53—22no—
234C2×C4×C53—22no—
235C2×C4×C53—22no—
236C2×C4×C53—22no—
237C2×C4×C53—22no—
238C2×C4×C53—22no—
239C2×C4×C53—24no—
240C2×C4×C53—24no—
241C2×C4×C53—24no—
242C2×C4×C53—42no—
243C2×C4×C53—42no—
244C2×C4×C53—42no—
245C2×C4×C53—42no—
246C2×C4×C53—42no—
247C2×C4×C53—42no—
248C2×C4×C53—42no—
249C2×C4×C53—42no—
250C2×C4×C53—42no—
251C2×C4×C53—42no—
252C2×C4×C53—42no—
253C2×C4×C53—42no—
254C2×C4×C53—42no—
255C2×C4×C53—42no—
256C2×C4×C53—42no—
257C2×C4×C53C2×C4⋊C414yes—
258C2×Dic53—24no—
259C2×Dic53—24no—
260C2×Dic53—24no—
261C2×Dic53—24no—
262C2×Dic53—24no—
263C2×Dic53—24no—
264C2×Dic53—42no—
265C2×Dic53—42no—
266C2×Dic53—42no—
267C2×Dic53—42no—
268C2×Dic53—42no—
269C2×Dic53—42no—
270C2×Dic53—42no—
271C2×Dic53—42no—
272C2×Dic53—42no—
273C2×Dic53—42no—
274C2×Dic53—42no—
275C2×Dic53—42no—
276C2×Dic53—42no—
277C2×Dic53—42no—
278C2×Dic53—42no—
279D212—416no—
280C4×D53—432no—
281#14486—531nomaximal, Sylow
282C22×C53C22×C4⋊C411yescharacteristic
283C22×C53C4×#4111yescharacteristic
284C22×C53C4×#4111yescharacteristic
285C22×C53#51811yescharacteristic
286C22×C53#51811yescharacteristic
287C22×C53#51811yescharacteristic
288C22×C53#51811yescharacteristic
289C22×C53#53411yescharacteristic
290C22×C53#53411yescharacteristic
291C22×C53#53812yes—
292C22×C53#53812yes—
293C22×C53#54012yes—
294C22×C53#54012yes—
295C22×C53#54311yescharacteristic
296C22×C53#54311yescharacteristic
297C22×C53#54612yes—
298C22×C53#54612yes—
299C22×C53#55112yes—
300C22×C53#55412yes—
301C22×C53#55412yes—
302C22×C53#55512yes—
303C22×C53#56012yes—
304C22×C53#56012yes—
305C2×D53—28no—
306C2×D53—28no—
307C2×D53—28no—
308C2×D53—44no—
309C2×D53—44no—
310C2×D53—44no—
311C2×D53—44no—
312C2×D53—44no—
313C2×D53—44no—
314C2×D53—44no—
315C2×D53—44no—
316C2×D53—44no—
317C2×D53—44no—
318C2×D53—44no—
319C2×D53—44no—
320C4×C53—24no—
321C4×C53—42no—
322C4×C53—42no—
323C4×C53—44no—
324C4×C53—82no—
325Dic53—44no—
326Dic53—44no—
327Dic53—82no—
328C2×#390—531no—
329C2×#386—532no—
330#1441—532no—
331#1439—532no—
332#1439—532no—
333#1434—531no—
334#1434D5311yescharacteristic, complemented
335C22⋊#95—532no—
336C22⋊#95—532no—
337C2×C53#231511yescharacteristic
338C2×C53#233311yescharacteristic
339C2×C53#233311yescharacteristic
340C2×C53#234911yescharacteristic
341C2×C53#234911yescharacteristic
342C2×C53#237211yescharacteristic
343C2×C53#238011yescharacteristic
344C2×C53#269712yes—
345C2×C53#269912yes—
346C2×C53#274812yes—
347C2×C53#278412yes—
348D53—48no—
349C23×D4—531no—
350C24×C4—532no—
351C22×C4⋊C4—531no—
352C22×C4⋊C4—532no—
353C22×C22⋊C4—532no—
354C22×C22⋊C4—532no—
355C22×C22⋊C4—532no—
356C22×C22⋊C4—532no—
357C22×C22⋊C4—532no—
358C22×C42C2×D5311yescharacteristic
359#390—538no—
360#386—1068no—
361C2×#95—532no—
362C2×#95—532no—
363C2×#95—531no—
364C2×#95—532no—
365C2×#95—532no—
366C2×#95—532no—
367C2×#95—531no—
368C2×#95C2×D5312yes—
369C2×#95C2×D5312yes—
370C2×#95C2×D5311yescharacteristic
371C2×#95C2×D5311yescharacteristic
372C53#1448611yescharacteristic, complemented, Sylow
373C25—532no—
374C22×D4—534no—
375C22×D4—534no—
376C22×D4—534no—
377C22×D4—2121no—
378C22×D4—2121no—
379C22×D4—2121no—
380C22×D4—2121no—
381C23×C4—532no—
382C23×C4—532no—
383C23×C4—532no—
384C23×C4—1064no—
385C23×C4—1064no—
386C23×C4—1064no—
387C23×C4—2122no—
388C23×C4—2122no—
389C23×C4—2122no—
390C23×C4—2122no—
391C23×C4C22×D5312yes—
392C23×C4C22×D5311yescharacteristic
393C23×C4C22×D5311yescharacteristic
394C23×C4C22×D5312yes—
395C23×C4C22×D5311yescharacteristic
396C2×C4⋊C4—534no—
397C2×C4⋊C4—2122no—
398C2×C4⋊C4—2122no—
399C2×C4⋊C4—2122no—
400C2×C4⋊C4—2121no—
401C2×C4⋊C4—2121no—
402C2×C22⋊C4—538no—
403C2×C22⋊C4—538no—
404C2×C22⋊C4—1064no—
405C2×C22⋊C4—1064no—
406C2×C22⋊C4—1064no—
407C2×C22⋊C4—1064no—
408C2×C22⋊C4—1064no—
409C2×C22⋊C4—2122no—
410C2×C22⋊C4—2122no—
411C2×C22⋊C4—2122no—
412C2×C22⋊C4—2122no—
413C2×C22⋊C4—2122no—
414C2×C22⋊C4—2122no—
415C2×C22⋊C4—2122no—
416C2×C22⋊C4—2122no—
417C2×C42—41no—
418C2×C42—41no—
419C2×C42C4×D5314yescomplemented
420#95—24no—
421#95—24no—
422#95—41no—
423#95—41no—
424#95—1064no—
425#95—1064no—
426#95—2121no—
427#95—2121no—
428#95—2122no—
429#95—2122no—
430#95—2122no—
431C24—534no—
432C24—534no—
433C24—534no—
434C24—1062no—
435C24—1062no—
436C24—1062no—
437C24—1062no—
438C24—1062no—
439C24—1062no—
440C24—1062no—
441C24—1062no—
442C24—1062no—
443C24—1062no—
444C24—1062no—
445C24—1062no—
446C24C23×D5311yescharacteristic
447C2×D4—5316no—
448C2×D4—2124no—
449C2×D4—2124no—
450C2×D4—2124no—
451C2×D4—2124no—
452C2×D4—2124no—
453C2×D4—2124no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—22no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—534no—
480C22×C4—534no—
481C22×C4—1062no—
482C22×C4—1062no—
483C22×C4—1062no—
484C22×C4—1062no—
485C22×C4—1062no—
486C22×C4—1062no—
487C22×C4—1062no—
488C22×C4—1062no—
489C22×C4—1062no—
490C22×C4—1062no—
491C22×C4—1062no—
492C22×C4—1062no—
493C22×C4—1062no—
494C22×C4—1062no—
495C22×C4—1062no—
496C22×C4—1062no—
497C22×C4—1062no—
498C22×C4—1062no—
499C22×C4—1062no—
500C22×C4—2128no—
501C22×C4—2128no—
502C22×C4—2128no—
503C22×C4—2128no—
504C22×C4—2128no—
505C22×C4—2128no—
506C22×C4—2128no—
507C22×C4C2×C4×D5312yes—
508C22×C4C2×C4×D5312yes—
509C22×C4C2×C4×D5312yes—
510C22×C4#1171994514yes—
511C22×C4D4×D5312yes—
512C22×C4Q8×D5312yes—
513C4⋊C4—2124no—
514C4⋊C4—2128no—
515C22⋊C4—2128no—
516C22⋊C4—2128no—
517C22⋊C4—2128no—
518C22⋊C4—2128no—
519C22⋊C4—2128no—
520C42—44no—
521C23—538no—
522C23—1064no—
523C23—1064no—
524C23—1064no—
525C23—1064no—
526C23—1064no—
527C23—1064no—
528C23—1064no—
529C23—1064no—
530C23—1064no—
531C23—1064no—
532C23—1064no—
533C23—1064no—
534C23—1064no—
535C23—1064no—
536C23—1064no—
537C23—1064no—
538C23—1064no—
539C23—1064no—
540C23—2122no—
541C23—2122no—
542C23—2122no—
543C23—2122no—
544C23—2122no—
545C23—2122no—
546C23—2122no—
547C23—2122no—
548C23—2122no—
549C23—2122no—
550C23—2122no—
551C23—2122no—
552C23—2122no—
553C23—2122no—
554C23—2122no—
555C23—2122no—
556C23C22×C4×D5311yescharacteristic
557C23C2×D4×D5311yescharacteristic
558C23C2×Q8×D5311yescharacteristic
559C23#41×D5312yes—
560C23C2×#1171994512yes—
561C23#1313326211yescharacteristic
562C23#1313326211yescharacteristic
563C23#1313326312yes—
564C23#1313326312yes—
565C23#1313326311yescharacteristic
566C23#1313326311yescharacteristic
567D4—21216no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—24no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—1064no—
599C2×C4—1064no—
600C2×C4—1064no—
601C2×C4—1064no—
602C2×C4—1064no—
603C2×C4—1064no—
604C2×C4—21232no—
605C2×C4—2122no—
606C2×C4—2122no—
607C2×C4—2122no—
608C2×C4—2122no—
609C2×C4—2122no—
610C2×C4—2122no—
611C2×C4—2122no—
612C2×C4—2122no—
613C2×C4—2122no—
614C2×C4—2122no—
615C2×C4—2122no—
616C2×C4—2122no—
617C2×C4—2122no—
618C2×C4—2122no—
619C2×C4—2122no—
620C2×C4C4⋊C4×D5314yes—
621C22—1068no—
622C22—1068no—
623C22—1068no—
624C22—2124no—
625C22—2124no—
626C22—2124no—
627C22—2124no—
628C22—2124no—
629C22—2124no—
630C22—2124no—
631C22—2124no—
632C22—2124no—
633C22—2124no—
634C22—2124no—
635C22—2124no—
636C22C2×C4⋊C4×D5311yescharacteristic
637C22#1681109111yescharacteristic
638C22C4×#1171994511yescharacteristic
639C22#1681109611yescharacteristic
640C22#1681109711yescharacteristic
641C22#1681109711yescharacteristic
642C22#1681109711yescharacteristic
643C22#1681118111yescharacteristic
644C22#1681118111yescharacteristic
645C22#1681120612yes—
646C22#1681120912yes—
647C22#1681122412yes—
648C22#1681122512yes—
649C22#1681124811yescharacteristic
650C22#1681124811yescharacteristic
651C22#1681126612yes—
652C22#1681126912yes—
653C22#1681129712yes—
654C22#1681132912yes—
655C22#1681133312yes—
656C22#1681133612yes—
657C22#1681136412yes—
658C22#1681136412yes—
659C4—24no—
660C4—44no—
661C4—42no—
662C4—42no—
663C4—82no—
664C4—4242no—
665C4—2124no—
666C4—2124no—
667C2—2128no—
668C2#2027732811yescharacteristic
669C2#2027747711yescharacteristic
670C2#2027748011yescharacteristic
671C2#2027762311yescharacteristic
672C2#2027762611yescharacteristic
673C2#2027782511yescharacteristic
674C2#2027789111yescharacteristic
675C2#2028127512yes—
676C2#2028129912yes—
677C2#2028192312yes—
678C2#2028241112yes—
subgroupG/H
center, FrattiniC24C23×D53
commutatorC23×C53C23×C4
FittingC53×#1434C2
socle, cosocleC24×C53C24
prime radicalC23×D212C22
hypercenter#1434D53
hypocenterC53#14486
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C24 ◁ #1434

Complements

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

NH ≅ G/Ntype
3. #20272967C2semidirect
4. #20272959C2semidirect
5. #20272959C2semidirect
6. #20272942C2semidirect
7. #20272936C2semidirect
8. #20272936C2semidirect
9. C53×#1434C2semidirect
19. #16810525C4semidirect
53. C2×C4⋊Dic53C2×C4semidirect
70. C2×#11719921C2×C4semidirect
71. C2×#11719921C2×C4semidirect
86. C2×C42×C53C2×C4semidirect
165. C2×D212C4⋊C4semidirect
334. #1434D53semidirect
372. C53#14486semidirect
419. C2×C42C4×D53semidirect

Maximal subgroups

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

subgroupindex
C2×#168105252
C2×#168105012
#202729672
#202729592
#202729422
#202729362
C53×#14342
#1448653