Finite Group Databasebeta

← #27313239

Subgroups — #27313239

Order 13568.

1558 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810525C211yesmaximal, characteristic
2#20272963C212yesmaximal, complemented
3#20272963C212yesmaximal, complemented
4#20272955C212yesmaximal
5#20272951C212yesmaximal, complemented
6#20272951C212yesmaximal, complemented
7#20272949C212yesmaximal, complemented
8#20272942C211yesmaximal, characteristic, complemented
9C53×#1434C211yesmaximal, characteristic, complemented
10C23×D212C2211yescharacteristic
11C22×C4⋊Dic53C2211yescharacteristic
12C22×#11719921C2212yes—
13C22×#11719921C2212yes—
14C22×#11719921C2212yes—
15C22×#11719921C2212yes—
16C22×#11719921C2212yes—
17C22×#11719921C2212yes—
18C22×C4×Dic53C2212yes—
19#16810525C418yescomplemented
20C2×#13133124C2212yes—
21C2×#13133124C2212yes—
22C2×#13133124C2211yescharacteristic
23C2×#13133124C2211yescharacteristic
24C2×#13133124C2212yes—
25C2×#13133124C2212yes—
26C2×#13133124C2212yes—
27C22×C42×C53C2211yescharacteristic
28C2×#95×C53C2212yes—
29C2×#95×C53C2212yes—
30C2×#95×C53C2211yescharacteristic
31C2×#95×C53C2211yescharacteristic
32C24×D53C2312yes—
33C22×D212—22no—
34C22×D212—22no—
35C22×D212—41no—
36C22×D212—41no—
37C22×D212—41no—
38C22×D212—41no—
39C22×D212C2×C414yes—
40C23×Dic53C2312yes—
41C23×Dic53C2312yes—
42C23×Dic53C2312yes—
43C2×C4⋊Dic53—41no—
44C2×C4⋊Dic53—41no—
45C2×C4⋊Dic53C2×C414yes—
46C2×#11719921—24no—
47C2×#11719921—24no—
48C2×#11719921—24no—
49C2×#11719921—24no—
50C2×#11719921—42no—
51C2×#11719921—42no—
52C2×#11719921—42no—
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×#11719921C2×C418yes—
63C2×#11719921C2×C418yes—
64C2×C4×Dic53—42no—
65C2×C4×Dic53—42no—
66C2×C4×Dic53—42no—
67#13133124—24no—
68#13133124—24no—
69#13133124—42no—
70#13133124—41no—
71#13133124—42no—
72#13133124—42no—
73#13133124—41no—
74C23×C4×C53C2312yes—
75C23×C4×C53C2311yescharacteristic
76C23×C4×C53C2311yescharacteristic
77C23×C4×C53C2312yes—
78C23×C4×C53C2311yescharacteristic
79C2×C42×C53—41no—
80C2×C42×C53—41no—
81C2×C42×C53C2×C414yes—
82#95×C53—24no—
83#95×C53—24no—
84#95×C53—41no—
85#95×C53—41no—
86C24×C53C2411yescharacteristic
87C23×D53—22no—
88C23×D53—22no—
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×D53C22×C414yes—
102C22×C4×C53—21no—
103C22×C4×C53—21no—
104C22×C4×C53—21no—
105C22×C4×C53—21no—
106C22×C4×C53—22no—
107C22×C4×C53—22no—
108C22×C4×C53—22no—
109C22×C4×C53—22no—
110C22×C4×C53—21no—
111C22×C4×C53—21no—
112C22×C4×C53—22no—
113C22×C4×C53—21no—
114C22×C4×C53—21no—
115C22×C4×C53—22no—
116C22×C4×C53—22no—
117C22×C4×C53—22no—
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×C53#4114yes—
131C22×C4×C53#4112yes—
132C22×C4×C53#4112yes—
133C22×Dic53—22no—
134C22×Dic53—22no—
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×Dic53C22×C414yes—
153C22×Dic53#4114yes—
154C2×D212—28no—
155C2×D212—44no—
156C2×D212—44no—
157C2×D212—44no—
158C2×D212—44no—
159C2×D212—44no—
160C2×D212—44no—
161C42×C53—44no—
162#11719921—48no—
163#11719921—48no—
164#11719921—48no—
165#11719921—48no—
166#11719921—48no—
167#11719921—48no—
168C4⋊Dic53—44no—
169C4×Dic53—48no—
170C23×C53C23×C411yescharacteristic
171C23×C53C2×#4112yes—
172C23×C53C2×#4111yescharacteristic
173C23×C53C2×#4112yes—
174C23×C53C2×#4111yescharacteristic
175C23×C53H(5,𝔽2)11yescharacteristic
176C23×C53H(5,𝔽2)11yescharacteristic
177C23×C53H(5,𝔽2)11yescharacteristic
178C23×C53H(5,𝔽2)11yescharacteristic
179C23×C53H(5,𝔽2)12yes—
180C23×C53H(5,𝔽2)12yes—
181C22×D53—24no—
182C22×D53—24no—
183C22×D53—24no—
184C22×D53—24no—
185C22×D53—24no—
186C22×D53—24no—
187C22×D53—24no—
188C22×D53—42no—
189C22×D53—42no—
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—
216C2×C4×C53—22no—
217C2×C4×C53—22no—
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—22no—
225C2×C4×C53—22no—
226C2×C4×C53—22no—
227C2×C4×C53—22no—
228C2×C4×C53—24no—
229C2×C4×C53—24no—
230C2×C4×C53—24no—
231C2×C4×C53—42no—
232C2×C4×C53—42no—
233C2×C4×C53—42no—
234C2×C4×C53—42no—
235C2×C4×C53—42no—
236C2×C4×C53—42no—
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#11714yes—
247C2×Dic53—24no—
248C2×Dic53—24no—
249C2×Dic53—24no—
250C2×Dic53—24no—
251C2×Dic53—24no—
252C2×Dic53—24no—
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—
268D212—416no—
269#14503—531nomaximal, Sylow
270C22×C53C2×#11711yescharacteristic
271C22×C53C4×#4111yescharacteristic
272C22×C53C4×#4111yescharacteristic
273C22×C53#51811yescharacteristic
274C22×C53#51811yescharacteristic
275C22×C53#51811yescharacteristic
276C22×C53#51811yescharacteristic
277C22×C53#53812yes—
278C22×C53#53812yes—
279C22×C53#53811yescharacteristic
280C22×C53#53811yescharacteristic
281C22×C53#53811yescharacteristic
282C22×C53#53811yescharacteristic
283C22×C53#54012yes—
284C22×C53#54012yes—
285C22×C53#55112yes—
286C22×C53#55312yes—
287C22×C53#55312yes—
288C22×C53#55512yes—
289C22×C53#55912yes—
290C22×C53#55912yes—
291C22×C53#56012yes—
292C22×C53#56012yes—
293C2×D53—44no—
294C2×D53—44no—
295C2×D53—44no—
296C2×D53—44no—
297C2×D53—44no—
298C2×D53—44no—
299C2×D53—44no—
300C2×D53—82no—
301C2×D53—82no—
302C2×D53—82no—
303C2×D53—82no—
304C2×D53—82no—
305C2×D53—82no—
306C2×D53—82no—
307C2×D53—82no—
308C4×C53—24no—
309C4×C53—42no—
310C4×C53—42no—
311C4×C53—44no—
312C4×C53—82no—
313Dic53—44no—
314Dic53—44no—
315Dic53—82no—
316C2×#390—531no—
317#1440—532no—
318#1440—532no—
319#1438—532no—
320#1437—532no—
321#1437—532no—
322#1436—532no—
323#1434—531no—
324#1434D5311yescharacteristic, complemented
325C2×C53#231211yescharacteristic
326C2×C53#233311yescharacteristic
327C2×C53#233311yescharacteristic
328C2×C53#235711yescharacteristic
329C2×C53#235711yescharacteristic
330C2×C53#237211yescharacteristic
331C2×C53#238011yescharacteristic
332C2×C53#279712yes—
333C2×C53#279712yes—
334C2×C53#280312yes—
335C2×C53#280312yes—
336D53—84no—
337C23×D4—531no—
338C22×C4⋊C4—531no—
339C22×C22⋊C4—532no—
340C22×C22⋊C4—532no—
341C22×C22⋊C4—532no—
342C22×C22⋊C4—532no—
343C22×C22⋊C4—532no—
344C22×C22⋊C4—532no—
345C22×C42—532no—
346C22×C42C2×D5311yescharacteristic
347#390—538no—
348C2×#95—532no—
349C2×#95—532no—
350C2×#95—531no—
351C2×#95—531no—
352C2×#95—532no—
353C2×#95—532no—
354C2×#95—532no—
355C2×#95C2×D5312yes—
356C2×#95C2×D5312yes—
357C2×#95C2×D5311yescharacteristic
358C2×#95C2×D5311yescharacteristic
359C53#1450311yescharacteristic, complemented, Sylow
360C25—532no—
361C22×D4—534no—
362C22×D4—1062no—
363C22×D4—1062no—
364C22×D4—2121no—
365C22×D4—2121no—
366C22×D4—2121no—
367C22×D4—2121no—
368C23×C4—532no—
369C23×C4—532no—
370C23×C4—532no—
371C23×C4C22×D5312yes—
372C23×C4C22×D5311yescharacteristic
373C23×C4C22×D5311yescharacteristic
374C23×C4C22×D5312yes—
375C23×C4C22×D5311yescharacteristic
376C2×C4⋊C4—534no—
377C2×C4⋊C4—2121no—
378C2×C4⋊C4—2121no—
379C2×C22⋊C4—538no—
380C2×C22⋊C4—538no—
381C2×C22⋊C4—1064no—
382C2×C22⋊C4—1064no—
383C2×C22⋊C4—1064no—
384C2×C22⋊C4—1064no—
385C2×C22⋊C4—2122no—
386C2×C22⋊C4—2122no—
387C2×C22⋊C4—2122no—
388C2×C22⋊C4—2122no—
389C2×C22⋊C4—2122no—
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×C42—41no—
398C2×C42—41no—
399C2×C42—2122no—
400C2×C42—2122no—
401C2×C42—2122no—
402C2×C42C4×D5314yes—
403#95—24no—
404#95—24no—
405#95—41no—
406#95—41no—
407#95—1064no—
408#95—1064no—
409#95—2121no—
410#95—2121no—
411#95—2122no—
412#95—2122no—
413#95—2122no—
414C24—534no—
415C24—1062no—
416C24—1062no—
417C24—1062no—
418C24—1062no—
419C24—1062no—
420C24—1062no—
421C24—1062no—
422C24—1062no—
423C24—1062no—
424C24—1062no—
425C24—1062no—
426C24—1062no—
427C24—1062no—
428C24—1062no—
429C24C23×D5311yescharacteristic
430C2×D4—1068no—
431C2×D4—2124no—
432C2×D4—2124no—
433C2×D4—2124no—
434C2×D4—2124no—
435C2×D4—2124no—
436C2×D4—2124no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—22no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
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—534no—
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×C4#1171994514yes—
487C22×C4#1171994712yes—
488C22×C4#1171994912yes—
489C4⋊C4—2124no—
490C22⋊C4—2128no—
491C22⋊C4—2128no—
492C22⋊C4—2128no—
493C22⋊C4—2128no—
494C22⋊C4—2128no—
495C22⋊C4—2128no—
496C42—44no—
497C42—2128no—
498C23—1064no—
499C23—1064no—
500C23—1064no—
501C23—1064no—
502C23—1064no—
503C23—1064no—
504C23—1064no—
505C23—2122no—
506C23—2122no—
507C23—2122no—
508C23—2122no—
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—
533C23C22×C4×D5311yescharacteristic
534C23#41×D5312yes—
535C23C2×#1171994711yescharacteristic
536C23C2×#1171994512yes—
537C23C2×#1171994911yescharacteristic
538C23#1313326211yescharacteristic
539C23#1313326211yescharacteristic
540C23#1313326311yescharacteristic
541C23#1313326312yes—
542C23#1313326312yes—
543C23#1313326311yescharacteristic
544D4—21216no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—24no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—1064no—
576C2×C4—1064no—
577C2×C4—1064no—
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#1313317514yes—
597C22—4242no—
598C22—4242no—
599C22—4242no—
600C22—4242no—
601C22—4242no—
602C22—4242no—
603C22—4242no—
604C22—4242no—
605C22—2124no—
606C22—2124no—
607C22—2124no—
608C22—2124no—
609C22—2124no—
610C22—2124no—
611C22—2124no—
612C22C2×#1313317511yescharacteristic
613C22#1681109111yescharacteristic
614C22C4×#1171994511yescharacteristic
615C22#1681109611yescharacteristic
616C22#1681109711yescharacteristic
617C22#1681109711yescharacteristic
618C22#1681109711yescharacteristic
619C22#1681120612yes—
620C22#1681120912yes—
621C22#1681121011yescharacteristic
622C22#1681121011yescharacteristic
623C22#1681121311yescharacteristic
624C22#1681121311yescharacteristic
625C22#1681122412yes—
626C22#1681122512yes—
627C22#1681129712yes—
628C22#1681131912yes—
629C22#1681132612yes—
630C22#1681133612yes—
631C22#1681135512yes—
632C22#1681135812yes—
633C22#1681136412yes—
634C22#1681136412yes—
635C4—24no—
636C4—44no—
637C4—42no—
638C4—42no—
639C4—82no—
640C4—4242no—
641C4—2124no—
642C4—2124no—
643C2—4244no—
644C2#2027731211yescharacteristic
645C2#2027747711yescharacteristic
646C2#2027748011yescharacteristic
647C2#2027768611yescharacteristic
648C2#2027769011yescharacteristic
649C2#2027782511yescharacteristic
650C2#2027789111yescharacteristic
651C2#2028256712yes—
652C2#2028257012yes—
653C2#2028264812yes—
654C2#2028265212yes—
subgroupG/H
center, FrattiniC24C23×D53
commutatorC23×C53C23×C4
FittingC53×#1434C2
socle, cosocleC24×C53C24
prime radicalC23×D212C22
hypercenter#1434D53
hypocenterC53#14503
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C24 ◁ #1434

Complements

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

NH ≅ G/Ntype
2. #20272963C2semidirect
3. #20272963C2semidirect
5. #20272951C2semidirect
6. #20272951C2semidirect
7. #20272949C2semidirect
8. #20272942C2semidirect
9. C53×#1434C2semidirect
19. #16810525C4semidirect
324. #1434D53semidirect
359. C53#14503semidirect

Maximal subgroups

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

subgroupindex
C2×#168105252
#202729632
#202729552
#202729512
#202729492
#202729422
C53×#14342
#1450353