Finite Group Databasebeta

← #27310650

Subgroups — #27310650

Order 13568.

1658 conjugacy classes of subgroups.

The subgroup lattice has 660 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810525C211yesmaximal, characteristic
2C2×#16810514C212yesmaximal, complemented
3#20272967C212yesmaximal, complemented
4#20272963C212yesmaximal, complemented
5#20272959C212yesmaximal, complemented
6#20272955C212yesmaximal
7#20272953C212yesmaximal, complemented
8#20272942C211yesmaximal, characteristic, complemented
9C4×#95×C53C211yesmaximal, characteristic, complemented
10C23×D212C2211yescharacteristic
11C22×C4⋊Dic53C2211yescharacteristic
12C22×#11719919C2212yes—
13C22×#11719921C2212yes—
14C22×#11719921C2212yes—
15C22×#11719921C2212yes—
16C22×#11719921C2212yes—
17C22×#11719921C2212yes—
18C22×#11719921C2212yes—
19#16810525C418yescomplemented
20#16810514—28no—
21C2×#13133124C2212yes—
22C2×#13133124C2212yes—
23C2×#13133124C2212yes—
24C2×#13133124C2211yescharacteristic
25C2×#13133124C2211yescharacteristic
26C2×#13133124C2212yes—
27C2×#13133124C2212yes—
28C22×C42×C53C2211yescharacteristic
29C22×C42×C53C2211yescharacteristic
30C22×C42×C53C2211yescharacteristic
31C2×#95×C53C2212yes—
32C2×#95×C53C2212yes—
33C24×D53C2312yes—
34C22×D212—24no—
35C22×D212—41no—
36C22×D212—41no—
37C22×D212—42no—
38C22×D212C2×C414yes—
39C23×Dic53C2312yes—
40C23×Dic53C2312yes—
41C23×Dic53C2312yes—
42C2×C4⋊Dic53—42no—
43C2×C4⋊Dic53C2×C414yes—
44C2×#11719919—24no—
45C2×#11719919—42no—
46C2×#11719919—42no—
47C2×#11719921—24no—
48C2×#11719921—24no—
49C2×#11719921—24no—
50C2×#11719921—24no—
51C2×#11719921—24no—
52C2×#11719921—24no—
53C2×#11719921—42no—
54C2×#11719921—42no—
55C2×#11719921—42no—
56C2×#11719921—42no—
57C2×#11719921—42no—
58C2×#11719921—42no—
59C2×#11719921—42no—
60C2×#11719921—42no—
61C2×#11719921—42no—
62C2×#11719921—42no—
63C2×#11719921C2×C418yes—
64C2×#11719921C2×C418yes—
65#13133124—24no—
66#13133124—24no—
67#13133124—24no—
68#13133124—42no—
69#13133124—42no—
70#13133124—41no—
71#13133124—41no—
72C23×C4×C53C2311yescharacteristic
73C23×C4×C53C2312yes—
74C23×C4×C53C2312yes—
75C23×C4×C53C2311yescharacteristic
76C23×C4×C53C2311yescharacteristic
77C2×C42×C53—22no—
78C2×C42×C53—41no—
79C2×C42×C53—41no—
80C2×C42×C53—41no—
81C2×C42×C53—41no—
82C2×C42×C53—41no—
83C2×C42×C53—41no—
84C2×C42×C53C2×C414yes—
85C2×C42×C53D414yescomplemented
86#95×C53—24no—
87#95×C53—24no—
88C24×C53C2411yescharacteristic
89C23×D53—22no—
90C23×D53—22no—
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—
104C22×C4×C53—21no—
105C22×C4×C53—21no—
106C22×C4×C53—21no—
107C22×C4×C53—21no—
108C22×C4×C53—21no—
109C22×C4×C53—22no—
110C22×C4×C53—21no—
111C22×C4×C53—22no—
112C22×C4×C53—22no—
113C22×C4×C53—22no—
114C22×C4×C53—22no—
115C22×C4×C53—22no—
116C22×C4×C53—21no—
117C22×C4×C53—21no—
118C22×C4×C53—22no—
119C22×C4×C53—22no—
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×C53#4114yes—
133C22×C4×C53#4114yes—
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×Dic53—22no—
155C22×Dic53C22×C414yes—
156C2×D212—28no—
157C2×D212—44no—
158C2×D212—44no—
159C2×D212—48no—
160C2×D212—48no—
161C42×C53—44no—
162C42×C53—44no—
163C42×C53—44no—
164#11719921—216no—
165#11719921—48no—
166#11719921—48no—
167#11719921—48no—
168#11719921—48no—
169#11719921—48no—
170C4⋊Dic53—44no—
171#11719919—48no—
172C23×C53C23×C411yescharacteristic
173C23×C53C22×D411yescharacteristic
174C23×C53C2×#4112yes—
175C23×C53C2×#4111yescharacteristic
176C23×C53C2×#4112yes—
177C23×C53H(5,𝔽2)11yescharacteristic
178C23×C53H(5,𝔽2)11yescharacteristic
179C23×C53H(5,𝔽2)11yescharacteristic
180C23×C53H(5,𝔽2)12yes—
181C23×C53H(5,𝔽2)12yes—
182C23×C532−1+411yescharacteristic
183C22×D53—24no—
184C22×D53—24no—
185C22×D53—24no—
186C22×D53—24no—
187C22×D53—24no—
188C22×D53—24no—
189C22×D53—24no—
190C22×D53—42no—
191C22×D53—42no—
192C22×D53—42no—
193C22×D53—42no—
194C22×D53—42no—
195C22×D53—42no—
196C22×D53—42no—
197C22×D53—42no—
198C22×D53—42no—
199C22×D53—42no—
200C22×D53—42no—
201C22×D53—42no—
202C22×D53—42no—
203C22×D53—42no—
204C22×D53—42no—
205C22×D53—42no—
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—
218C2×C4×C53—22no—
219C2×C4×C53—22no—
220C2×C4×C53—22no—
221C2×C4×C53—22no—
222C2×C4×C53—22no—
223C2×C4×C53—22no—
224C2×C4×C53—24no—
225C2×C4×C53—24no—
226C2×C4×C53—22no—
227C2×C4×C53—22no—
228C2×C4×C53—22no—
229C2×C4×C53—22no—
230C2×C4×C53—22no—
231C2×C4×C53—24no—
232C2×C4×C53—24no—
233C2×C4×C53—24no—
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#11714yes—
248C2×C4×C53C4×D414yes—
249C2×Dic53—24no—
250C2×Dic53—24no—
251C2×Dic53—24no—
252C2×Dic53—42no—
253C2×Dic53—42no—
254C2×Dic53—42no—
255C2×Dic53—42no—
256C2×Dic53—42no—
257C2×Dic53—42no—
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—
270D212—416no—
271#14278—531nomaximal, Sylow
272C22×C53C2×#11711yescharacteristic
273C22×C53C2×C4×D411yescharacteristic
274C22×C53C4×#4111yescharacteristic
275C22×C53#51811yescharacteristic
276C22×C53#51811yescharacteristic
277C22×C53#51811yescharacteristic
278C22×C53#52011yescharacteristic
279C22×C53#52912yes—
280C22×C53#53212yes—
281C22×C53#53411yescharacteristic
282C22×C53#53512yes—
283C22×C53#53711yescharacteristic
284C22×C53#53812yes—
285C22×C53#53812yes—
286C22×C53#53812yes—
287C22×C53#53812yes—
288C22×C53#53811yescharacteristic
289C22×C53#53911yescharacteristic
290C22×C53#54012yes—
291C22×C53#54012yes—
292C22×C53#54012yes—
293C22×C53#54212yes—
294C22×C53#54212yes—
295C2×D53—44no—
296C2×D53—44no—
297C2×D53—44no—
298C2×D53—44no—
299C2×D53—44no—
300C2×D53—44no—
301C2×D53—44no—
302C2×D53—82no—
303C2×D53—82no—
304C2×D53—82no—
305C2×D53—82no—
306C2×D53—82no—
307C2×D53—82no—
308C2×D53—82no—
309C2×D53—82no—
310C4×C53—24no—
311C4×C53—24no—
312C4×C53—42no—
313C4×C53—44no—
314C4×C53—44no—
315Dic53—44no—
316Dic53—82no—
317Dic53—82no—
318C2×#390—531no—
319C2×#388—532no—
320#1441—532no—
321#1440—532no—
322#1439—532no—
323#1438—532no—
324#1437—532no—
325#1434—531no—
326C4×#95D5311yescharacteristic, complemented
327C2×C53#229311yescharacteristic
328C2×C53#231211yescharacteristic
329C2×C53#231911yescharacteristic
330C2×C53#232511yescharacteristic
331C2×C53#232811yescharacteristic
332C2×C53#233311yescharacteristic
333C2×C53#233711yescharacteristic
334C2×C53#264812yes—
335C2×C53#265112yes—
336C2×C53#266112yes—
337C2×C53#266212yes—
338D53—84no—
339C23×D4—531no—
340C22×C4⋊C4—531no—
341C22×C4⋊C4—532no—
342C22×C22⋊C4—532no—
343C22×C22⋊C4—532no—
344C22×C22⋊C4—532no—
345C22×C22⋊C4—532no—
346C22×C22⋊C4—532no—
347C22×C22⋊C4—532no—
348C22×C42C2×D5311yescharacteristic
349C22×C42C2×D5311yescharacteristic
350C22×C42C2×D5311yescharacteristic
351#390—538no—
352#388—1068no—
353C2×#95—532no—
354C2×#95—532no—
355C2×#95—532no—
356C2×#95—531no—
357C2×#95—531no—
358C2×#95—532no—
359C2×#95—532no—
360C2×#95C2×D5312yes—
361C2×#95C2×D5312yes—
362C53#1427811yescharacteristic, complemented, Sylow
363C25—532no—
364C22×D4—534no—
365C22×D4—1064no—
366C22×D4—2121no—
367C22×D4—2121no—
368C22×D4—2122no—
369C23×C4—532no—
370C23×C4—532no—
371C23×C4—532no—
372C23×C4C22×D5311yescharacteristic
373C23×C4C22×D5312yes—
374C23×C4C22×D5312yes—
375C23×C4C22×D5311yescharacteristic
376C23×C4C22×D5311yescharacteristic
377C2×C4⋊C4—534no—
378C2×C4⋊C4—1064no—
379C2×C4⋊C4—2122no—
380C2×C4⋊C4—2122no—
381C2×C4⋊C4—2122no—
382C2×C22⋊C4—538no—
383C2×C22⋊C4—538no—
384C2×C22⋊C4—1064no—
385C2×C22⋊C4—1064no—
386C2×C22⋊C4—1064no—
387C2×C22⋊C4—1064no—
388C2×C22⋊C4—1064no—
389C2×C22⋊C4—1064no—
390C2×C22⋊C4—2122no—
391C2×C22⋊C4—2122no—
392C2×C22⋊C4—2122no—
393C2×C22⋊C4—2122no—
394C2×C22⋊C4—2122no—
395C2×C22⋊C4—2122no—
396C2×C22⋊C4—2122no—
397C2×C22⋊C4—2122no—
398C2×C22⋊C4—2122no—
399C2×C22⋊C4—2122no—
400C2×C42—22no—
401C2×C42—41no—
402C2×C42—41no—
403C2×C42—41no—
404C2×C42—41no—
405C2×C42—41no—
406C2×C42—41no—
407C2×C42C4×D5314yes—
408C2×C42D21214yescomplemented
409#95—24no—
410#95—24no—
411#95—1064no—
412#95—1064no—
413#95—1064no—
414#95—2121no—
415#95—2121no—
416#95—2122no—
417#95—2122no—
418C24—534no—
419C24—1062no—
420C24—1062no—
421C24—1062no—
422C24—1062no—
423C24—1062no—
424C24—1062no—
425C24—1062no—
426C24—1062no—
427C24—1062no—
428C24—1062no—
429C24—1062no—
430C24—1062no—
431C24—1062no—
432C24—1062no—
433C24C23×D5311yescharacteristic
434C2×D4—1068no—
435C2×D4—2124no—
436C2×D4—2124no—
437C2×D4—2128no—
438C2×D4—2128no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—22no—
445C22×C4—21no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—534no—
463C22×C4—1062no—
464C22×C4—1062no—
465C22×C4—1062no—
466C22×C4—1062no—
467C22×C4—1062no—
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×C4C2×C4×D5312yes—
484C22×C4C2×C4×D5312yes—
485C22×C4C2×C4×D5312yes—
486C22×C4C2×D21214yes—
487C22×C4C2×D21212yes—
488C22×C4#1171994514yes—
489C22×C4#1171994514yes—
490C22×C4#1171994512yes—
491C4⋊C4—2124no—
492C4⋊C4—2128no—
493C22⋊C4—10616no—
494C22⋊C4—2128no—
495C22⋊C4—2128no—
496C22⋊C4—2128no—
497C22⋊C4—2128no—
498C22⋊C4—2128no—
499C42—44no—
500C42—44no—
501C42—44no—
502C23—1064no—
503C23—1064no—
504C23—1064no—
505C23—1064no—
506C23—1064no—
507C23—1064no—
508C23—1064no—
509C23—2122no—
510C23—2122no—
511C23—2122no—
512C23—2122no—
513C23—2122no—
514C23—2122no—
515C23—2122no—
516C23—2122no—
517C23—2122no—
518C23—2122no—
519C23—2122no—
520C23—2122no—
521C23—2122no—
522C23—2122no—
523C23—2122no—
524C23—2122no—
525C23—2122no—
526C23—2122no—
527C23—2122no—
528C23—2122no—
529C23—2122no—
530C23—2122no—
531C23—2122no—
532C23—2122no—
533C23—2122no—
534C23—2122no—
535C23—2122no—
536C23—2122no—
537C23C22×C4×D5311yescharacteristic
538C23C22×D21211yescharacteristic
539C23C2×#1171994512yes—
540C23C2×#1171994511yescharacteristic
541C23C2×#1171994512yes—
542C23#1313326211yescharacteristic
543C23#1313326211yescharacteristic
544C23#1313326211yescharacteristic
545C23#1313326312yes—
546C23#1313326312yes—
547C23#1313326511yescharacteristic
548D4—21216no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—1064no—
579C2×C4—1064no—
580C2×C4—1064no—
581C2×C4—2122no—
582C2×C4—2122no—
583C2×C4—2122no—
584C2×C4—2122no—
585C2×C4—2122no—
586C2×C4—2122no—
587C2×C4—2122no—
588C2×C4—2122no—
589C2×C4—2122no—
590C2×C4—2122no—
591C2×C4—2122no—
592C2×C4—2122no—
593C2×C4—2122no—
594C2×C4—2122no—
595C2×C4—2122no—
596C2×C4—2122no—
597C2×C4—2122no—
598C2×C4—2122no—
599C2×C4#1313317414yes—
600C2×C4C4×D21214yes—
601C22—4242no—
602C22—4242no—
603C22—4242no—
604C22—4242no—
605C22—4242no—
606C22—4242no—
607C22—4242no—
608C22—4242no—
609C22—2124no—
610C22—2124no—
611C22—2124no—
612C22—2124no—
613C22—2124no—
614C22—2124no—
615C22—2124no—
616C22C2×#1313317411yescharacteristic
617C22C2×C4×D21211yescharacteristic
618C22C4×#1171994511yescharacteristic
619C22#1681109611yescharacteristic
620C22#1681109711yescharacteristic
621C22#1681109711yescharacteristic
622C22#1681110411yescharacteristic
623C22#1681115712yes—
624C22#1681117312yes—
625C22#1681118511yescharacteristic
626C22#1681119212yes—
627C22#1681120411yescharacteristic
628C22#1681120612yes—
629C22#1681120612yes—
630C22#1681120612yes—
631C22#1681120612yes—
632C22#1681121111yescharacteristic
633C22#1681122011yescharacteristic
634C22#1681122412yes—
635C22#1681122412yes—
636C22#1681122812yes—
637C22#1681124512yes—
638C22#1681124512yes—
639C4—24no—
640C4—24no—
641C4—44no—
642C4—44no—
643C4—42no—
644C4—4242no—
645C4—4242no—
646C4—2124no—
647C2—4244no—
648C2#2027717911yescharacteristic
649C2#2027731011yescharacteristic
650C2#2027735311yescharacteristic
651C2#2027741811yescharacteristic
652C2#2027744011yescharacteristic
653C2#2027748011yescharacteristic
654C2#2027752811yescharacteristic
655C2#2028069712yes—
656C2#2028074212yes—
657C2#2028084112yes—
658C2#2028085212yes—
subgroupG/H
center, FrattiniC24C23×D53
commutatorC23×C53C23×C4
FittingC4×#95×C53C2
socle, cosocleC24×C53C24
prime radicalC23×D212C22
hypercenterC4×#95D53
hypocenterC53#14278
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C24 ◁ C4×#95

Complements

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

NH ≅ G/Ntype
2. C2×#16810514C2semidirect
3. #20272967C2semidirect
4. #20272963C2semidirect
5. #20272959C2semidirect
7. #20272953C2semidirect
8. #20272942C2semidirect
9. C4×#95×C53C2semidirect
19. #16810525C4semidirect
85. C2×C42×C53D4semidirect
326. C4×#95D53semidirect
362. C53#14278semidirect
408. C2×C42D212semidirect

Maximal subgroups

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

subgroupindex
C2×#168105252
C2×#168105142
#202729672
#202729632
#202729592
#202729552
#202729532
#202729422
C4×#95×C532
#1427853