Finite Group Databasebeta

← #27311361

Subgroups — #27311361

Order 13568.

1566 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810485C211yesmaximal, characteristic
2C2×#16810482C211yesmaximal, characteristic
3C2×C4×#11719921C211yesmaximal, characteristic
4#20272978C212yesmaximal, complemented
5#20272970C212yesmaximal, complemented
6#20272959C212yesmaximal
7#20272957C212yesmaximal
8#20272943C211yesmaximal, characteristic, complemented
9#20272942C211yesmaximal, characteristic, complemented
10#20272941C211yesmaximal, characteristic, complemented
11C53×C4⋊#95C211yesmaximal, characteristic, complemented
12C23×C4×D53C2211yescharacteristic
13C22×#11719919C2212yes—
14C22×#11719919C2212yes—
15C22×#11719921C2211yescharacteristic
16C22×#11719921C2211yescharacteristic
17C22×#11719921C2212yes—
18C22×#11719921C2212yes—
19C22×C4×Dic53C2211yescharacteristic
20#16810485—24no—
21#16810482—24no—
22C4×#11719921C418yescomplemented
23C2×#13133124C2211yescharacteristic
24C2×#13133124C2212yes—
25C2×#13133124C2211yescharacteristic
26C2×#13133124C2212yes—
27C2×#13133124C2212yes—
28C2×#13133124C2212yes—
29C2×#13133124C2212yes—
30C2×#13133124C2212yes—
31C2×#13133124C2211yescharacteristic
32C2×#13133124C2211yescharacteristic
33C22×C4⋊C4×C53C2212yes—
34C22×C42×C53C2211yescharacteristic
35C2×#95×C53C2211yescharacteristic
36C2×#95×C53C2211yescharacteristic
37C2×#95×C53C2212yes—
38C24×D53C2311yescharacteristic
39C23×Dic53C2312yes—
40C23×Dic53C2312yes—
41C23×Dic53C2311yescharacteristic
42C23×Dic53C2311yescharacteristic
43C23×Dic53C2311yescharacteristic
44C22×C4×D53—42no—
45C22×C4×D53—42no—
46C22×C4×D53C2×C414yes—
47C22×C4×D53D414yes—
48C22×C4×D53Q814yes—
49C2×#11719919—24no—
50C2×#11719919—24no—
51C2×#11719919—42no—
52C2×#11719919—42no—
53C2×#11719919—42no—
54C2×#11719919—42no—
55C2×#11719921—24no—
56C2×#11719921—24no—
57C2×#11719921—24no—
58C2×#11719921—24no—
59C2×#11719921—42no—
60C2×#11719921—42no—
61C2×#11719921—42no—
62C2×#11719921—42no—
63C2×#11719921C2×C414yes—
64C2×#11719921C2×C414yes—
65C2×C4×Dic53—42no—
66C2×C4×Dic53C2×C414yes—
67#13133124—22no—
68#13133124—22no—
69#13133124—24no—
70#13133124—24no—
71#13133124—24no—
72#13133124—42no—
73#13133124—42no—
74#13133124—42no—
75#13133124C2×C414yes—
76#13133124C2×C414yes—
77C23×C4×C53C2312yes—
78C23×C4×C53C2312yes—
79C23×C4×C53C2311yescharacteristic
80C23×C4×C53C2311yescharacteristic
81C23×C4×C53C2311yescharacteristic
82C2×C4⋊C4×C53—24no—
83C2×C4⋊C4×C53—42no—
84C2×C4⋊C4×C53—42no—
85C2×C42×C53—42no—
86C2×C42×C53C2×C414yes—
87#95×C53—22no—
88#95×C53—22no—
89#95×C53—42no—
90C24×C53C2411yescharacteristic
91C23×D53—22no—
92C23×D53—22no—
93C23×D53—21no—
94C23×D53—21no—
95C23×D53—22no—
96C23×D53—22no—
97C23×D53—21no—
98C23×D53—21no—
99C23×D53C22×C412yes—
100C23×D53C2×D412yes—
101C23×D53C2×Q812yes—
102C22×C4×C53—22no—
103C22×C4×C53—22no—
104C22×C4×C53—21no—
105C22×C4×C53—22no—
106C22×C4×C53—21no—
107C22×C4×C53—22no—
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—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×C53C22×C412yes—
126C22×C4×C53C22×C412yes—
127C22×C4×C53C22×C412yes—
128C22×C4×C53C2×D412yes—
129C22×C4×C53C2×Q812yes—
130C22×C4×C53#4114yes—
131C22×Dic53—22no—
132C22×Dic53—21no—
133C22×Dic53—22no—
134C22×Dic53—22no—
135C22×Dic53—21no—
136C22×Dic53—22no—
137C22×Dic53—21no—
138C22×Dic53—22no—
139C22×Dic53—21no—
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×C412yes—
155C22×Dic53C22×C412yes—
156C22×Dic53C22×C412yes—
157C22×Dic53C2×D412yes—
158C22×Dic53C2×Q812yes—
159C22×Dic53#4114yes—
160C2×C4×D53—44no—
161C2×C4×D53—44no—
162C2×C4×D53—48no—
163C2×C4×D53—48no—
164C2×C4×D53C4⋊C4116yescomplemented
165C4⋊C4×C53—48no—
166C42×C53—44no—
167#11719921—28no—
168#11719921—28no—
169#11719921—48no—
170#11719921—48no—
171#11719919—48no—
172#11719919—48no—
173C4×Dic53—44no—
174C23×C53C23×C411yescharacteristic
175C23×C53C22×D411yescharacteristic
176C23×C53C22×Q811yescharacteristic
177C23×C53C2×#4112yes—
178C23×C53C2×#4112yes—
179C23×C53H(5,𝔽2)12yes—
180C23×C53H(5,𝔽2)11yescharacteristic
181C23×C53H(5,𝔽2)11yescharacteristic
182C23×C53H(5,𝔽2)12yes—
183C23×C532−1+411yescharacteristic
184C23×C532−1+411yescharacteristic
185C22×D53—22no—
186C22×D53—22no—
187C22×D53—22no—
188C22×D53—22no—
189C22×D53—22no—
190C22×D53—22no—
191C22×D53—24no—
192C22×D53—24no—
193C22×D53—24no—
194C22×D53—22no—
195C22×D53—22no—
196C22×D53—24no—
197C22×D53—22no—
198C22×D53—22no—
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×D53C2×C4⋊C414yes—
208C2×C4×C53—22no—
209C2×C4×C53—22no—
210C2×C4×C53—22no—
211C2×C4×C53—22no—
212C2×C4×C53—24no—
213C2×C4×C53—24no—
214C2×C4×C53—24no—
215C2×C4×C53—24no—
216C2×C4×C53—22no—
217C2×C4×C53—22no—
218C2×C4×C53—22no—
219C2×C4×C53—42no—
220C2×C4×C53—42no—
221C2×C4×C53—42no—
222C2×C4×C53—42no—
223C2×C4×C53—42no—
224C2×C4×C53—42no—
225C2×C4×C53—42no—
226C2×C4×C53—42no—
227C2×C4×C53—42no—
228C2×C4×C53—42no—
229C2×C4×C53—42no—
230C2×C4×C53—42no—
231C2×C4×C53—42no—
232C2×C4×C53—42no—
233C2×C4×C53—42no—
234C2×C4×C53—42no—
235C2×C4×C53C2×C4⋊C414yes—
236C2×C4×C53#11714yes—
237C2×Dic53—22no—
238C2×Dic53—22no—
239C2×Dic53—22no—
240C2×Dic53—22no—
241C2×Dic53—24no—
242C2×Dic53—24no—
243C2×Dic53—24no—
244C2×Dic53—24no—
245C2×Dic53—22no—
246C2×Dic53—22no—
247C2×Dic53—22no—
248C2×Dic53—42no—
249C2×Dic53—42no—
250C2×Dic53—42no—
251C2×Dic53—42no—
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×Dic53C2×C4⋊C414yes—
265C2×Dic53#11714yes—
266C4×D53—416no—
267#14355—531nomaximal, Sylow
268C22×C53C22×C4⋊C411yescharacteristic
269C22×C53C2×#11711yescharacteristic
270C22×C53C2×#11711yescharacteristic
271C22×C53#51811yescharacteristic
272C22×C53#51811yescharacteristic
273C22×C53#52011yescharacteristic
274C22×C53#52011yescharacteristic
275C22×C53#53212yes—
276C22×C53#53312yes—
277C22×C53#53412yes—
278C22×C53#53511yescharacteristic
279C22×C53#53611yescharacteristic
280C22×C53#53812yes—
281C22×C53#53812yes—
282C22×C53#53912yes—
283C22×C53#53912yes—
284C22×C53#54012yes—
285C22×C53#54012yes—
286C22×C53#54212yes—
287C22×C53#54212yes—
288C22×C53#54312yes—
289C22×C53#54411yescharacteristic
290C22×C53#54411yescharacteristic
291C2×D53—24no—
292C2×D53—24no—
293C2×D53—24no—
294C2×D53—42no—
295C2×D53—42no—
296C2×D53—42no—
297C2×D53—42no—
298C2×D53—44no—
299C2×D53—44no—
300C2×D53—44no—
301C2×D53—44no—
302C4×C53—24no—
303C4×C53—24no—
304C4×C53—42no—
305C4×C53—82no—
306C4×C53—82no—
307Dic53—24no—
308Dic53—24no—
309Dic53—42no—
310Dic53—82no—
311Dic53—82no—
312C2×#381—531no—
313C2×#380—531no—
314C2×C4×C22⋊C4—531no—
315#1444—532no—
316#1442—532no—
317#1439—532no—
318#1439—532no—
319C4⋊#95—531no—
320C4⋊#95D5311yescharacteristic, complemented
321#1434—531no—
322#1434—531no—
323C2×C53#228511yescharacteristic
324C2×C53#231511yescharacteristic
325C2×C53#231711yescharacteristic
326C2×C53#231811yescharacteristic
327C2×C53#231811yescharacteristic
328C2×C53#231911yescharacteristic
329C2×C53#231911yescharacteristic
330C2×C53#266112yes—
331C2×C53#266312yes—
332C2×C53#266512yes—
333C2×C53#266712yes—
334D53—44no—
335C24×C4—531no—
336C22×C4⋊C4—532no—
337C22×C4⋊C4—532no—
338C22×C4⋊C4C2×D5312yes—
339C22×C22⋊C4—531no—
340C22×C22⋊C4—531no—
341C22×C22⋊C4—532no—
342C22×C22⋊C4—532no—
343C22×C42—531no—
344C22×C42C2×D5311yescharacteristic
345#381—1064no—
346#380—1064no—
347C4×C22⋊C4—538no—
348C2×#95—531no—
349C2×#95—531no—
350C2×#95—531no—
351C2×#95—531no—
352C2×#95—532no—
353C2×#95—532no—
354C2×#95—532no—
355C2×#95—532no—
356C2×#95—532no—
357C2×#95—532no—
358C2×#95C2×D5311yescharacteristic
359C2×#95C2×D5311yescharacteristic
360C2×#95C2×D5312yes—
361C53#1435511yescharacteristic, complemented, Sylow
362C25—531no—
363C23×C4—531no—
364C23×C4—531no—
365C23×C4—534no—
366C23×C4—534no—
367C23×C4—534no—
368C23×C4—531no—
369C23×C4—532no—
370C23×C4—532no—
371C23×C4—2122no—
372C23×C4—2122no—
373C23×C4C22×D5312yes—
374C23×C4C22×D5312yes—
375C23×C4C22×D5311yescharacteristic
376C23×C4C22×D5311yescharacteristic
377C23×C4C22×D5311yescharacteristic
378C2×C4⋊C4—24no—
379C2×C4⋊C4—42no—
380C2×C4⋊C4—42no—
381C2×C4⋊C4—1064no—
382C2×C4⋊C4—1064no—
383C2×C4⋊C4—2122no—
384C2×C4⋊C4—2122no—
385C2×C4⋊C4—2122no—
386C2×C4⋊C4—2122no—
387C2×C22⋊C4—534no—
388C2×C22⋊C4—534no—
389C2×C22⋊C4—1064no—
390C2×C22⋊C4—1064no—
391C2×C22⋊C4—1064no—
392C2×C22⋊C4—1064no—
393C2×C22⋊C4—2122no—
394C2×C22⋊C4—2122no—
395C2×C22⋊C4—2122no—
396C2×C22⋊C4—2122no—
397C2×C42—42no—
398C2×C42—534no—
399C2×C42—2122no—
400C2×C42C4×D5314yes—
401#95—22no—
402#95—22no—
403#95—42no—
404#95—534no—
405#95—534no—
406#95—1064no—
407#95—1064no—
408#95—1064no—
409#95—1062no—
410#95—1062no—
411#95—2122no—
412#95—2122no—
413#95—2122no—
414C24—532no—
415C24—532no—
416C24—532no—
417C24—1062no—
418C24—1062no—
419C24—1061no—
420C24—1061no—
421C24—1062no—
422C24—1062no—
423C24—1061no—
424C24—1061no—
425C24C23×D5311yescharacteristic
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—21no—
429C22×C4—22no—
430C22×C4—21no—
431C22×C4—22no—
432C22×C4—21no—
433C22×C4—22no—
434C22×C4—21no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—532no—
450C22×C4—532no—
451C22×C4—532no—
452C22×C4—532no—
453C22×C4—534no—
454C22×C4—5316no—
455C22×C4—532no—
456C22×C4—1061no—
457C22×C4—1062no—
458C22×C4—1062no—
459C22×C4—1062no—
460C22×C4—1061no—
461C22×C4—1062no—
462C22×C4—1061no—
463C22×C4—1062no—
464C22×C4—1061no—
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—2124no—
480C22×C4—2124no—
481C22×C4—2128no—
482C22×C4—2128no—
483C22×C4C2×C4×D5312yes—
484C22×C4C2×C4×D5312yes—
485C22×C4C2×C4×D5312yes—
486C22×C4#1171994514yes—
487C22×C4D4×D5312yes—
488C22×C4Q8×D5312yes—
489C4⋊C4—48no—
490C4⋊C4—2128no—
491C4⋊C4—2128no—
492C22⋊C4—1068no—
493C22⋊C4—1068no—
494C22⋊C4—2128no—
495C22⋊C4—2128no—
496C42—44no—
497C42—2124no—
498C23—534no—
499C23—1062no—
500C23—1062no—
501C23—1062no—
502C23—1062no—
503C23—1062no—
504C23—1062no—
505C23—1064no—
506C23—1064no—
507C23—1064no—
508C23—1062no—
509C23—1062no—
510C23—1064no—
511C23—1062no—
512C23—1062no—
513C23—2122no—
514C23—2122no—
515C23—2122no—
516C23—2122no—
517C23—2122no—
518C23—2122no—
519C23—2122no—
520C23—2122no—
521C23C22×C4×D5311yescharacteristic
522C23C2×D4×D5311yescharacteristic
523C23C2×Q8×D5311yescharacteristic
524C23#41×D5312yes—
525C23C2×#1171994512yes—
526C23#1313326212yes—
527C23#1313326211yescharacteristic
528C23#1313326211yescharacteristic
529C23#1313326312yes—
530C23#1313326511yescharacteristic
531C23#1313326511yescharacteristic
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—534no—
560C2×C4—534no—
561C2×C4—1062no—
562C2×C4—1062no—
563C2×C4—1062no—
564C2×C4—1062no—
565C2×C4—1064no—
566C2×C4—1064no—
567C2×C4—1064no—
568C2×C4—1064no—
569C2×C4—1062no—
570C2×C4—1062no—
571C2×C4—1062no—
572C2×C4—21216no—
573C2×C4—2122no—
574C2×C4—2122no—
575C2×C4—2122no—
576C2×C4—2122no—
577C2×C4—2122no—
578C2×C4—2122no—
579C2×C4—2122no—
580C2×C4—2122no—
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×C4C4⋊C4×D5314yes—
590C2×C4#1313317414yes—
591C22—1064no—
592C22—1064no—
593C22—1064no—
594C22—2124no—
595C22—2124no—
596C22—2124no—
597C22—2124no—
598C22—2122no—
599C22—2122no—
600C22—2122no—
601C22—2122no—
602C22C2×C4⋊C4×D5311yescharacteristic
603C22#117×D5311yescharacteristic
604C22C2×#1313317411yescharacteristic
605C22#1681109611yescharacteristic
606C22#1681109711yescharacteristic
607C22#1681110411yescharacteristic
608C22#1681110411yescharacteristic
609C22#1681117012yes—
610C22#1681117712yes—
611C22#1681118112yes—
612C22#1681118611yescharacteristic
613C22#1681119811yescharacteristic
614C22#1681120712yes—
615C22#1681121112yes—
616C22#1681122012yes—
617C22#1681122112yes—
618C22#1681122312yes—
619C22#1681122412yes—
620C22#1681124312yes—
621C22#1681124512yes—
622C22#1681124812yes—
623C22#1681125011yescharacteristic
624C22#1681125011yescharacteristic
625C4—24no—
626C4—24no—
627C4—42no—
628C4—82no—
629C4—82no—
630C4—1064no—
631C4—1064no—
632C4—4242no—
633C4—4242no—
634C4—2122no—
635C2—2124no—
636C2#2027711911yescharacteristic
637C2#2027732811yescharacteristic
638C2#2027733611yescharacteristic
639C2#2027734911yescharacteristic
640C2#2027735111yescharacteristic
641C2#2027735311yescharacteristic
642C2#2027735711yescharacteristic
643C2#2028084512yes—
644C2#2028086912yes—
645C2#2028088912yes—
646C2#2028091312yes—
subgroupG/H
center, FrattiniC24C23×D53
commutatorC23×C53C23×C4
FittingC53×C4⋊#95C2
socle, cosocleC24×C53C24
prime radicalC24×D53C23
hypercenterC4⋊#95D53
hypocenterC53#14355
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C24 ◁ C4⋊#95

Complements

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

NH ≅ G/Ntype
4. #20272978C2semidirect
5. #20272970C2semidirect
8. #20272943C2semidirect
9. #20272942C2semidirect
10. #20272941C2semidirect
11. C53×C4⋊#95C2semidirect
22. C4×#11719921C4semidirect
164. C2×C4×D53C4⋊C4semidirect
320. C4⋊#95D53semidirect
361. C53#14355semidirect

Maximal subgroups

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

subgroupindex
C2×#168104852
C2×#168104822
C2×C4×#117199212
#202729782
#202729702
#202729592
#202729572
#202729432
#202729422
#202729412
C53×C4⋊#952
#1435553