Finite Group Databasebeta

← #27312489

Subgroups — #27312489

Order 13568.

2018 conjugacy classes of subgroups.

The subgroup lattice has 668 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810525C211yesmaximal, characteristic
2C2×#16810501C212yesmaximal
3C2×#16810497C212yesmaximal, complemented
4#20272967C212yesmaximal, complemented
5#20272961C212yesmaximal, complemented
6#20272959C212yesmaximal, complemented
7#20272945C211yesmaximal, characteristic, complemented
8#20272936C212yesmaximal, complemented
9C53×#1434C211yesmaximal, characteristic, complemented
10C23×D212C2211yescharacteristic
11C23×C4×D53C2212yes—
12C22×C4⋊Dic53C2212yes—
13C22×C4⋊Dic53C2211yescharacteristic
14C22×#11719919C2212yes—
15C22×#11719921C2212yes—
16C22×#11719921C2212yes—
17C22×#11719921C2212yes—
18C22×#11719921C2212yes—
19C22×#11719921C2212yes—
20#16810525C418yescomplemented
21#16810501—28no—
22#16810497—28no—
23C2×#13133124C2212yes—
24C2×#13133124C2212yes—
25C2×#13133124C2211yescharacteristic
26C2×#13133124C2212yes—
27C2×#13133124C2211yescharacteristic
28C2×#13133124C2212yes—
29C22×C42×C53C2211yescharacteristic
30C2×#95×C53C2211yescharacteristic
31C2×#95×C53C2212yes—
32C2×#95×C53C2212yes—
33C2×#95×C53C2211yescharacteristic
34C24×D53C2312yes—
35C22×D212—42no—
36C22×D212—42no—
37C22×D212C2×C414yes—
38C22×D212D414yes—
39C22×D212Q814yes—
40C23×Dic53C2312yes—
41C23×Dic53C2312yes—
42C23×Dic53C2312yes—
43C22×C4×D53—24no—
44C22×C4×D53—24no—
45C22×C4×D53—24no—
46C22×C4×D53—42no—
47C22×C4×D53—42no—
48C22×C4×D53—42no—
49C22×C4×D53—42no—
50C2×C4⋊Dic53—24no—
51C2×C4⋊Dic53—24no—
52C2×C4⋊Dic53—42no—
53C2×C4⋊Dic53—42no—
54C2×C4⋊Dic53C2×C414yescomplemented
55C2×#11719919—24no—
56C2×#11719919—42no—
57C2×#11719919—42no—
58C2×#11719921—24no—
59C2×#11719921—24no—
60C2×#11719921—24no—
61C2×#11719921—24no—
62C2×#11719921—24no—
63C2×#11719921—24no—
64C2×#11719921—24no—
65C2×#11719921—42no—
66C2×#11719921—42no—
67C2×#11719921—42no—
68C2×#11719921—42no—
69C2×#11719921—42no—
70C2×#11719921—42no—
71C2×#11719921C2×C418yescomplemented
72C2×#11719921C2×C418yescomplemented
73#13133124—24no—
74#13133124—24no—
75#13133124—42no—
76#13133124—42no—
77#13133124—41no—
78#13133124—41no—
79C23×C4×C53C2312yes—
80C23×C4×C53C2311yescharacteristic
81C23×C4×C53C2312yes—
82C23×C4×C53C2311yescharacteristic
83C23×C4×C53C2311yescharacteristic
84C2×C42×C53—42no—
85C2×C42×C53C2×C414yescomplemented
86#95×C53—24no—
87#95×C53—22no—
88#95×C53—42no—
89#95×C53D414yescomplemented
90C24×C53C2411yescharacteristic
91C23×D53—22no—
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×D53C22×C414yes—
104C23×D53C2×D414yes—
105C23×D53C2×Q814yes—
106C22×C4×C53—22no—
107C22×C4×C53—22no—
108C22×C4×C53—22no—
109C22×C4×C53—22no—
110C22×C4×C53—22no—
111C22×C4×C53—22no—
112C22×C4×C53—22no—
113C22×C4×C53—22no—
114C22×C4×C53—21no—
115C22×C4×C53—22no—
116C22×C4×C53—22no—
117C22×C4×C53—22no—
118C22×C4×C53—22no—
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×C53C22×C412yes—
128C22×C4×C53C22×C412yes—
129C22×C4×C53C22×C412yes—
130C22×C4×C53C2×D414yes—
131C22×C4×C53C2×D412yes—
132C22×C4×C53C2×D412yes—
133C22×C4×C53C2×Q812yes—
134C22×C4×C53#4112yes—
135C22×Dic53—22no—
136C22×Dic53—22no—
137C22×Dic53—22no—
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×Dic53C22×C414yes—
155C22×Dic53#4114yes—
156C2×D212—44no—
157C2×D212—44no—
158C2×D212—48no—
159C2×D212—48no—
160C2×D212C4⋊C4116yescomplemented
161C2×C4×D53—48no—
162C2×C4×D53—48no—
163C2×C4×D53—48no—
164C2×C4×D53—48no—
165C2×C4×D53—48no—
166C2×C4×D53—48no—
167C2×C4×D53—48no—
168C42×C53—44no—
169#11719921—216no—
170#11719921—48no—
171#11719921—48no—
172#11719921—48no—
173#11719921—48no—
174C4⋊Dic53—44no—
175C4⋊Dic53—48no—
176#11719919—48no—
177C23×C53C23×C411yescharacteristic
178C23×C53C22×D411yescharacteristic
179C23×C53C22×D411yescharacteristic
180C23×C53C22×Q811yescharacteristic
181C23×C53C2×#4112yes—
182C23×C53C2×#4111yescharacteristic
183C23×C53H(5,𝔽2)12yes—
184C23×C53H(5,𝔽2)12yes—
185C23×C53H(5,𝔽2)12yes—
186C23×C53H(5,𝔽2)11yescharacteristic
187C23×C532−1+411yescharacteristic
188C22×D53—24no—
189C22×D53—24no—
190C22×D53—24no—
191C22×D53—24no—
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—42no—
207C22×D53—42no—
208C22×D53—42no—
209C22×D53—42no—
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×D53C2×C4⋊C418yes—
223C2×C4×C53—22no—
224C2×C4×C53—22no—
225C2×C4×C53—22no—
226C2×C4×C53—22no—
227C2×C4×C53—24no—
228C2×C4×C53—24no—
229C2×C4×C53—24no—
230C2×C4×C53—24no—
231C2×C4×C53—22no—
232C2×C4×C53—22no—
233C2×C4×C53—22no—
234C2×C4×C53—24no—
235C2×C4×C53—24no—
236C2×C4×C53—24no—
237C2×C4×C53—42no—
238C2×C4×C53—42no—
239C2×C4×C53—42no—
240C2×C4×C53—42no—
241C2×C4×C53—42no—
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×C53C2×C4⋊C414yes—
251C2×C4×C53C4×D414yes—
252C2×Dic53—24no—
253C2×Dic53—24no—
254C2×Dic53—24no—
255C2×Dic53—24no—
256C2×Dic53—24no—
257C2×Dic53—24no—
258C2×Dic53—42no—
259C2×Dic53—42no—
260C2×Dic53—42no—
261C2×Dic53—42no—
262C2×Dic53—42no—
263C2×Dic53—42no—
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—
273D212—416no—
274C4×D53—432no—
275#14438—531nomaximal, Sylow
276C22×C53C22×C4⋊C411yescharacteristic
277C22×C53C2×C4×D411yescharacteristic
278C22×C53C4×#4111yescharacteristic
279C22×C53#51811yescharacteristic
280C22×C53#51811yescharacteristic
281C22×C53#51811yescharacteristic
282C22×C53#52011yescharacteristic
283C22×C53#53412yes—
284C22×C53#53512yes—
285C22×C53#53812yes—
286C22×C53#54012yes—
287C22×C53#54212yes—
288C22×C53#54312yes—
289C22×C53D4211yescharacteristic
290C22×C53#54612yes—
291C22×C53#54612yes—
292C22×C53#54711yescharacteristic
293C22×C53D4×Q811yescharacteristic
294C22×C53#55312yes—
295C22×C53#55412yes—
296C22×C53#55411yescharacteristic
297C22×C53#56012yes—
298C22×C53#56212yes—
299C2×D53—28no—
300C2×D53—28no—
301C2×D53—28no—
302C2×D53—44no—
303C2×D53—44no—
304C2×D53—44no—
305C2×D53—44no—
306C2×D53—44no—
307C2×D53—44no—
308C2×D53—44no—
309C2×D53—44no—
310C2×D53—44no—
311C2×D53—44no—
312C2×D53—44no—
313C2×D53—44no—
314C4×C53—24no—
315C4×C53—24no—
316C4×C53—42no—
317C4×C53—44no—
318C4×C53—82no—
319Dic53—44no—
320Dic53—44no—
321Dic53—82no—
322C2×#390—531no—
323C2×#386—532no—
324C2×#385—532no—
325#1441—532no—
326#1439—532no—
327#1439—532no—
328C4⋊#95—531no—
329#1434D5311yescharacteristic, complemented
330C22⋊#95—532no—
331C2×C53#231511yescharacteristic
332C2×C53#232811yescharacteristic
333C2×C53#233711yescharacteristic
334C2×C53D4×C4⋊C411yescharacteristic
335C2×C53#234911yescharacteristic
336C2×C53#236411yescharacteristic
337C2×C53#237711yescharacteristic
338C2×C53#267812yes—
339C2×C53#269212yes—
340C2×C53#270112yes—
341C2×C53#278012yes—
342D53—48no—
343C23×D4—531no—
344C24×C4—532no—
345C22×C4⋊C4—531no—
346C22×C4⋊C4—532no—
347C22×C4⋊C4—532no—
348C22×C22⋊C4—532no—
349C22×C22⋊C4—532no—
350C22×C22⋊C4—532no—
351C22×C22⋊C4—532no—
352C22×C22⋊C4—532no—
353C22×C42C2×D5311yescharacteristic
354#390—538no—
355#386—1068no—
356#385—1068no—
357C2×#95—532no—
358C2×#95—532no—
359C2×#95—531no—
360C2×#95—532no—
361C2×#95—531no—
362C2×#95—532no—
363C2×#95C2×D5311yescharacteristic
364C2×#95C2×D5312yes—
365C2×#95C2×D5312yes—
366C2×#95C2×D5311yescharacteristic
367C53#1443811yescharacteristic, complemented, Sylow
368C25—532no—
369C22×D4—534no—
370C22×D4—534no—
371C22×D4—534no—
372C22×D4—2122no—
373C22×D4—2122no—
374C23×C4—532no—
375C23×C4—532no—
376C23×C4—532no—
377C23×C4—1064no—
378C23×C4—1064no—
379C23×C4—1064no—
380C23×C4—2122no—
381C23×C4—2122no—
382C23×C4—2122no—
383C23×C4—2122no—
384C23×C4C22×D5312yes—
385C23×C4C22×D5311yescharacteristic
386C23×C4C22×D5312yes—
387C23×C4C22×D5311yescharacteristic
388C23×C4C22×D5311yescharacteristic
389C2×C4⋊C4—534no—
390C2×C4⋊C4—1064no—
391C2×C4⋊C4—1064no—
392C2×C4⋊C4—1064no—
393C2×C4⋊C4—2122no—
394C2×C4⋊C4—2122no—
395C2×C4⋊C4—2122no—
396C2×C4⋊C4—2122no—
397C2×C22⋊C4—538no—
398C2×C22⋊C4—538no—
399C2×C22⋊C4—1064no—
400C2×C22⋊C4—1064no—
401C2×C22⋊C4—1064no—
402C2×C22⋊C4—1064no—
403C2×C22⋊C4—1064no—
404C2×C22⋊C4—1064no—
405C2×C22⋊C4—1064no—
406C2×C22⋊C4—2122no—
407C2×C22⋊C4—2122no—
408C2×C22⋊C4—2122no—
409C2×C22⋊C4—2122no—
410C2×C22⋊C4—2122no—
411C2×C22⋊C4—2122no—
412C2×C42—42no—
413C2×C42C4×D5314yescomplemented
414#95—22no—
415#95—24no—
416#95—42no—
417#95—1064no—
418#95—1064no—
419#95—2121no—
420#95—2121no—
421#95—2122no—
422#95—2122no—
423#95D21214yescomplemented
424C24—534no—
425C24—534no—
426C24—534no—
427C24—1062no—
428C24—1062no—
429C24—1062no—
430C24—1062no—
431C24—1062no—
432C24—1062no—
433C24—1062no—
434C24—1062no—
435C24—1062no—
436C24—1062no—
437C24—1062no—
438C24—1062no—
439C24C23×D5311yescharacteristic
440C2×D4—5316no—
441C2×D4—2124no—
442C2×D4—2124no—
443C2×D4—2128no—
444C2×D4—2128no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—21no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—21no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—534no—
467C22×C4—534no—
468C22×C4—1062no—
469C22×C4—1062no—
470C22×C4—1062no—
471C22×C4—1062no—
472C22×C4—1062no—
473C22×C4—1062no—
474C22×C4—1062no—
475C22×C4—1062no—
476C22×C4—1062no—
477C22×C4—1062no—
478C22×C4—1062no—
479C22×C4—1062no—
480C22×C4—1062no—
481C22×C4—1062no—
482C22×C4—1062no—
483C22×C4—1062no—
484C22×C4—1062no—
485C22×C4—1062no—
486C22×C4—1062no—
487C22×C4—2128no—
488C22×C4—2128no—
489C22×C4—2128no—
490C22×C4—2128no—
491C22×C4—2128no—
492C22×C4—2128no—
493C22×C4—2128no—
494C22×C4C2×C4×D5312yes—
495C22×C4C2×C4×D5312yes—
496C22×C4C2×C4×D5312yes—
497C22×C4C2×D21214yes—
498C22×C4C2×D21212yes—
499C22×C4#1171994512yes—
500C22×C4D4×D5312yes—
501C22×C4Q8×D5312yes—
502C4⋊C4—2124no—
503C4⋊C4—2128no—
504C4⋊C4—2128no—
505C22⋊C4—10616no—
506C22⋊C4—2128no—
507C22⋊C4—2128no—
508C22⋊C4—2128no—
509C22⋊C4—2128no—
510C42—44no—
511C23—538no—
512C23—1064no—
513C23—1064no—
514C23—1064no—
515C23—1064no—
516C23—1064no—
517C23—1064no—
518C23—1064no—
519C23—1064no—
520C23—1064no—
521C23—1064no—
522C23—1064no—
523C23—1064no—
524C23—1064no—
525C23—1064no—
526C23—1064no—
527C23—1064no—
528C23—1064no—
529C23—1064no—
530C23—2122no—
531C23—2122no—
532C23—2122no—
533C23—2122no—
534C23—2122no—
535C23—2122no—
536C23—2122no—
537C23—2122no—
538C23—2122no—
539C23—2122no—
540C23—2122no—
541C23—2122no—
542C23—2122no—
543C23—2122no—
544C23—2122no—
545C23—2122no—
546C23C22×C4×D5311yescharacteristic
547C23C2×D4×D5311yescharacteristic
548C23C22×D21211yescharacteristic
549C23C2×Q8×D5311yescharacteristic
550C23#41×D5312yes—
551C23C2×#1171994511yescharacteristic
552C23#1313326212yes—
553C23#1313326312yes—
554C23#1313326311yescharacteristic
555C23#1313326312yes—
556C23#1313326411yescharacteristic
557D4—21216no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—1064no—
586C2×C4—1064no—
587C2×C4—1064no—
588C2×C4—1064no—
589C2×C4—1064no—
590C2×C4—1064no—
591C2×C4—21232no—
592C2×C4—2122no—
593C2×C4—2122no—
594C2×C4—2122no—
595C2×C4—2122no—
596C2×C4—2122no—
597C2×C4—2122no—
598C2×C4—2122no—
599C2×C4—2122no—
600C2×C4—2122no—
601C2×C4—2122no—
602C2×C4—2122no—
603C2×C4—2122no—
604C2×C4—2122no—
605C2×C4—2122no—
606C2×C4—2122no—
607C2×C4C4⋊C4×D5314yes—
608C2×C4C4×D21214yes—
609C22—1068no—
610C22—1068no—
611C22—1068no—
612C22—2124no—
613C22—2124no—
614C22—2124no—
615C22—2124no—
616C22—2124no—
617C22—2124no—
618C22—2124no—
619C22—2124no—
620C22—2124no—
621C22—2124no—
622C22—2124no—
623C22—2124no—
624C22C2×C4⋊C4×D5311yescharacteristic
625C22C2×C4×D21211yescharacteristic
626C22#1681109111yescharacteristic
627C22#1681109611yescharacteristic
628C22#1681109711yescharacteristic
629C22#1681109711yescharacteristic
630C22#1681110311yescharacteristic
631C22#1681118112yes—
632C22#1681119212yes—
633C22#1681121212yes—
634C22#1681122812yes—
635C22#1681123912yes—
636C22#1681124812yes—
637C22D4×D21211yescharacteristic
638C22#1681126612yes—
639C22#1681126812yes—
640C22#1681127811yescharacteristic
641C22Q8×D21211yescharacteristic
642C22#1681132212yes—
643C22#1681132912yes—
644C22#1681133311yescharacteristic
645C22#1681136312yes—
646C22#1681137012yes—
647C4—24no—
648C4—24no—
649C4—42no—
650C4—44no—
651C4—82no—
652C4—4242no—
653C4—2124no—
654C4—2124no—
655C2—2128no—
656C2#2027732811yescharacteristic
657C2#2027744011yescharacteristic
658C2#2027752511yescharacteristic
659C2C4⋊C4×D21211yescharacteristic
660C2#2027762311yescharacteristic
661C2#2027776511yescharacteristic
662C2#2027787411yescharacteristic
663C2#2028103812yes—
664C2#2028120912yes—
665C2#2028132512yes—
666C2#2028236612yes—
subgroupG/H
center, FrattiniC24C23×D53
commutatorC23×C53C23×C4
FittingC53×#1434C2
socle, cosocleC24×C53C24
prime radicalC23×D212C22
hypercenter#1434D53
hypocenterC53#14438
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C24 ◁ #1434

Complements

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

NH ≅ G/Ntype
3. C2×#16810497C2semidirect
4. #20272967C2semidirect
5. #20272961C2semidirect
6. #20272959C2semidirect
7. #20272945C2semidirect
8. #20272936C2semidirect
9. C53×#1434C2semidirect
20. #16810525C4semidirect
54. C2×C4⋊Dic53C2×C4semidirect
71. C2×#11719921C2×C4semidirect
72. C2×#11719921C2×C4semidirect
85. C2×C42×C53C2×C4semidirect
89. #95×C53D4semidirect
160. C2×D212C4⋊C4semidirect
329. #1434D53semidirect
367. C53#14438semidirect
413. C2×C42C4×D53semidirect
423. #95D212semidirect

Maximal subgroups

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

subgroupindex
C2×#168105252
C2×#168105012
C2×#168104972
#202729672
#202729612
#202729592
#202729452
#202729362
C53×#14342
#1443853