Finite Group Databasebeta

← #27306304

Subgroups — #27306304

Order 13568.

1818 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810491C211yesmaximal, characteristic, complemented
2C2×C4×#11719921C211yesmaximal, characteristic
3C2×#95×D53C211yesmaximal, characteristic
4#20272976C212yesmaximal, complemented
5#20272975C212yesmaximal, complemented
6#20272941C211yesmaximal, characteristic, complemented
7#20272940C211yesmaximal, characteristic, complemented
8#20272936C211yesmaximal, characteristic
9#20272936C212yesmaximal
10#20272936C212yesmaximal
11C53×#1434C211yesmaximal, characteristic, complemented
12C23×C4×D53C2212yes—
13C23×C4×D53C2211yescharacteristic
14C22×#11719921C2212yes—
15C22×#11719921C2211yescharacteristic
16C22×#11719921C2211yescharacteristic
17C22×C4×Dic53C2211yescharacteristic
18#16810491—24no—
19C4×#11719921C418yescomplemented
20C2×#13133124C2212yes—
21C2×#13133124C2211yescharacteristic
22C2×#13133124C2212yes—
23C2×#13133124C2212yes—
24C2×#13133124C2212yes—
25C2×#13133124C2212yes—
26C2×#13133124C2212yes—
27C2×#13133124C2212yes—
28C2×#13133124C2212yes—
29C2×#13133124C2211yescharacteristic
30C2×#13133124C2211yescharacteristic
31C2×#13133124C2211yescharacteristic
32#95×D53—24no—
33C22×C42×C53C2211yescharacteristic
34C2×#95×C53C2211yescharacteristic
35C2×#95×C53C2212yes—
36C2×#95×C53C2211yescharacteristic
37C2×#95×C53C2212yes—
38C24×D53C2311yescharacteristic
39C23×Dic53C2312yes—
40C23×Dic53C2311yescharacteristic
41C23×Dic53C2311yescharacteristic
42C23×Dic53C2311yescharacteristic
43C23×Dic53C2312yes—
44C22×C4×D53—24no—
45C22×C4×D53—24no—
46C22×C4×D53—22no—
47C22×C4×D53—24no—
48C22×C4×D53—22no—
49C22×C4×D53—24no—
50C22×C4×D53—24no—
51C22×C4×D53—24no—
52C22×C4×D53—42no—
53C22×C4×D53—42no—
54C22×C4×D53—42no—
55C22×C4×D53C2×C414yes—
56C2×#11719921—22no—
57C2×#11719921—24no—
58C2×#11719921—22no—
59C2×#11719921—24no—
60C2×#11719921—24no—
61C2×#11719921—24no—
62C2×#11719921C2×C414yes—
63C2×#11719921C2×C414yes—
64C2×C4×Dic53—42no—
65C2×C4×Dic53C2×C414yescomplemented
66#13133124—22no—
67#13133124—24no—
68#13133124—24no—
69#13133124—24no—
70#13133124—24no—
71#13133124—42no—
72#13133124—42no—
73#13133124—42no—
74#13133124—42no—
75#13133124C2×C414yescomplemented
76#13133124C2×C414yescomplemented
77#13133124D414yes—
78C23×C4×C53C2312yes—
79C23×C4×C53C2311yescharacteristic
80C23×C4×C53C2311yescharacteristic
81C23×C4×C53C2312yes—
82C23×C4×C53C2311yescharacteristic
83C2×C42×C53—22no—
84C2×C42×C53—22no—
85C2×C42×C53C2×C414yescomplemented
86#95×C53—22no—
87#95×C53—24no—
88#95×C53—22no—
89#95×C53—42no—
90C24×C53C2411yescharacteristic
91C23×D53—21no—
92C23×D53—21no—
93C23×D53—22no—
94C23×D53—22no—
95C23×D53—21no—
96C23×D53—21no—
97C23×D53C22×C412yes—
98C23×D53#4112yes—
99C23×D53#4114yes—
100C23×D53#4114yes—
101C23×D53#4112yes—
102C22×C4×C53—22no—
103C22×C4×C53—22no—
104C22×C4×C53—22no—
105C22×C4×C53—21no—
106C22×C4×C53—21no—
107C22×C4×C53—22no—
108C22×C4×C53—21no—
109C22×C4×C53—21no—
110C22×C4×C53—22no—
111C22×C4×C53—21no—
112C22×C4×C53—21no—
113C22×C4×C53—22no—
114C22×C4×C53—22no—
115C22×C4×C53—22no—
116C22×C4×C53—21no—
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×D412yes—
131C22×C4×C53#4114yes—
132C22×C4×C53#4112yes—
133C22×Dic53—21no—
134C22×Dic53—22no—
135C22×Dic53—22no—
136C22×Dic53—21no—
137C22×Dic53—21no—
138C22×Dic53—22no—
139C22×Dic53—21no—
140C22×Dic53—22no—
141C22×Dic53—22no—
142C22×Dic53—22no—
143C22×Dic53—22no—
144C22×Dic53—21no—
145C22×Dic53—22no—
146C22×Dic53—22no—
147C22×Dic53—22no—
148C22×Dic53—22no—
149C22×Dic53—22no—
150C22×Dic53—21no—
151C22×Dic53—22no—
152C22×Dic53—22no—
153C22×Dic53—22no—
154C22×Dic53—22no—
155C22×Dic53—22no—
156C22×Dic53—22no—
157C22×Dic53—22no—
158C22×Dic53C22×C412yes—
159C22×Dic53C22×C412yes—
160C22×Dic53C22×C412yes—
161C22×Dic53C2×D414yes—
162C2×C4×D53—216no—
163C2×C4×D53—28no—
164C2×C4×D53—28no—
165C2×C4×D53—44no—
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—
173C2×C4×D53—48no—
174C42×C53—28no—
175#11719921—28no—
176#11719921—28no—
177#11719921—48no—
178C4×Dic53—44no—
179C23×C53C23×C411yescharacteristic
180C23×C53C22×D411yescharacteristic
181C23×C53C2×#4112yes—
182C23×C53C2×#4112yes—
183C23×C53C2×#4111yescharacteristic
184C23×C53C2×#4111yescharacteristic
185C23×C53C2×#4111yescharacteristic
186C23×C53C2×#4112yes—
187C23×C53H(5,𝔽2)11yescharacteristic
188C23×C53H(5,𝔽2)12yes—
189C23×C532−1+411yescharacteristic
190C22×D53—22no—
191C22×D53—22no—
192C22×D53—22no—
193C22×D53—22no—
194C22×D53—24no—
195C22×D53—24no—
196C22×D53—24no—
197C22×D53—24no—
198C22×D53—24no—
199C22×D53—22no—
200C22×D53—22no—
201C22×D53—24no—
202C22×D53—24no—
203C22×D53—24no—
204C22×D53—22no—
205C22×D53—24no—
206C22×D53—24no—
207C22×D53—22no—
208C22×D53#11714yes—
209C22×D53#11714yes—
210C22×D53#11714yes—
211C22×D53#12618yes—
212C22×D53#12618yes—
213C2×C4×C53—22no—
214C2×C4×C53—22no—
215C2×C4×C53—24no—
216C2×C4×C53—24no—
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—24no—
224C2×C4×C53—24no—
225C2×C4×C53—24no—
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—41no—
233C2×C4×C53—41no—
234C2×C4×C53—41no—
235C2×C4×C53—41no—
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×C53C4×D414yes—
244C2×Dic53—22no—
245C2×Dic53—22no—
246C2×Dic53—22no—
247C2×Dic53—22no—
248C2×Dic53—22no—
249C2×Dic53—22no—
250C2×Dic53—22no—
251C2×Dic53—22no—
252C2×Dic53—22no—
253C2×Dic53—24no—
254C2×Dic53—24no—
255C2×Dic53—24no—
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—
270C2×Dic53—42no—
271C2×Dic53—42no—
272C2×Dic53—42no—
273C4×D53—416no—
274C4×D53—432no—
275#13901—531nomaximal, Sylow
276C22×C53C2×#11711yescharacteristic
277C22×C53C2×#11711yescharacteristic
278C22×C53C2×#11711yescharacteristic
279C22×C53C2×C4×D411yescharacteristic
280C22×C53C4×#4111yescharacteristic
281C22×C53#51811yescharacteristic
282C22×C53#52011yescharacteristic
283C22×C53#52512yes—
284C22×C53C2×#12612yes—
285C22×C53C2×#12612yes—
286C22×C53#52912yes—
287C22×C53#53812yes—
288C22×C53#53912yes—
289C22×C53#54612yes—
290C22×C53#54611yescharacteristic
291C22×C53#54811yescharacteristic
292C22×C53#55112yes—
293C22×C53#55112yes—
294C22×C53#55112yes—
295C22×C53#55211yescharacteristic
296C22×C53#55212yes—
297C22×C53#55312yes—
298C22×C53#55311yescharacteristic
299C2×D53—24no—
300C2×D53—24no—
301C2×D53—24no—
302C2×D53—24no—
303C2×D53—24no—
304C2×D53—28no—
305C2×D53—28no—
306C2×D53—28no—
307C2×D53—28no—
308C2×D53—24no—
309C2×D53#38318yescomplemented
310C4×C53—24no—
311C4×C53—42no—
312C4×C53—44no—
313C4×C53—42no—
314C4×C53—82no—
315Dic53—42no—
316Dic53—42no—
317Dic53—42no—
318Dic53—44no—
319Dic53—82no—
320C2×#383—531no—
321C2×C4×C22⋊C4—531no—
322C22×#95—531no—
323#1443—532no—
324#1443—532no—
325#1434—531no—
326#1434—531no—
327#1434D5311yescharacteristic, complemented
328C22⋊#95—532no—
329C22⋊#95—532no—
330C22⋊#95—531no—
331C2×C53C2×#38311yescharacteristic
332C2×C53#229011yescharacteristic
333C2×C53#231811yescharacteristic
334C2×C53#234211yescharacteristic
335C2×C53#234311yescharacteristic
336C2×C53#235211yescharacteristic
337C2×C53#235311yescharacteristic
338C2×C53#248012yes—
339C2×C53#248112yes—
340C2×C53#249312yes—
341C2×C53#251012yes—
342D53—28no—
343C24×C4—531no—
344C24×C4—532no—
345C22×C22⋊C4—531no—
346C22×C22⋊C4—531no—
347C22×C22⋊C4—532no—
348C22×C42—531no—
349C22×C42C2×D5311yescharacteristic
350#383—1064no—
351C4×C22⋊C4—538no—
352C2×#95—531no—
353C2×#95—531no—
354C2×#95—531no—
355C2×#95—532no—
356C2×#95—531no—
357C2×#95—532no—
358C2×#95—532no—
359C2×#95—532no—
360C2×#95—532no—
361C2×#95—532no—
362C2×#95—532no—
363C2×#95—532no—
364C2×#95—1064no—
365C2×#95C2×D5311yescharacteristic
366C2×#95C2×D5312yes—
367C2×#95C2×D5311yescharacteristic
368C2×#95C2×D5312yes—
369C53#1390111yescharacteristic, complemented, Sylow
370C25—531no—
371C23×C4—531no—
372C23×C4—531no—
373C23×C4—531no—
374C23×C4—532no—
375C23×C4—534no—
376C23×C4—532no—
377C23×C4—1064no—
378C23×C4—1064no—
379C23×C4—1064no—
380C23×C4—1064no—
381C23×C4—1064no—
382C23×C4—1064no—
383C23×C4—1062no—
384C23×C4—1062no—
385C23×C4—2122no—
386C23×C4—2122no—
387C23×C4—2122no—
388C23×C4C22×D5312yes—
389C23×C4C22×D5311yescharacteristic
390C23×C4C22×D5311yescharacteristic
391C23×C4C22×D5312yes—
392C23×C4C22×D5311yescharacteristic
393C2×C22⋊C4—534no—
394C2×C22⋊C4—534no—
395C2×C22⋊C4—1062no—
396C2×C22⋊C4—1064no—
397C2×C22⋊C4—1062no—
398C2×C22⋊C4—1064no—
399C2×C22⋊C4—1064no—
400C2×C22⋊C4—1064no—
401C2×C42—22no—
402C2×C42—22no—
403C2×C42—534no—
404C2×C42—2122no—
405C2×C42C4×D5314yescomplemented
406#95—24no—
407#95—22no—
408#95—22no—
409#95—42no—
410#95—534no—
411#95—534no—
412#95—534no—
413#95—1064no—
414#95—1064no—
415#95—1064no—
416#95—1064no—
417#95—1062no—
418#95—2122no—
419#95—2122no—
420#95—2122no—
421#95—2122no—
422C24—532no—
423C24—532no—
424C24—534no—
425C24—532no—
426C24—534no—
427C24—1061no—
428C24—1061no—
429C24—1062no—
430C24—1061no—
431C24—1062no—
432C24—1061no—
433C24C23×D5311yescharacteristic
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
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—22no—
450C22×C4—22no—
451C22×C4—21no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—532no—
460C22×C4—532no—
461C22×C4—532no—
462C22×C4—534no—
463C22×C4—1068no—
464C22×C4—10616no—
465C22×C4—1061no—
466C22×C4—1061no—
467C22×C4—1061no—
468C22×C4—1062no—
469C22×C4—1062no—
470C22×C4—1062no—
471C22×C4—1062no—
472C22×C4—1062no—
473C22×C4—1061no—
474C22×C4—1062no—
475C22×C4—1062no—
476C22×C4—1061no—
477C22×C4—1062no—
478C22×C4—1062no—
479C22×C4—1062no—
480C22×C4—1062no—
481C22×C4—1062no—
482C22×C4—1061no—
483C22×C4—1068no—
484C22×C4—1062no—
485C22×C4—1062no—
486C22×C4—1062no—
487C22×C4—1062no—
488C22×C4—1062no—
489C22×C4—1062no—
490C22×C4—1062no—
491C22×C4—2124no—
492C22×C4—2128no—
493C22×C4—2128no—
494C22×C4—2128no—
495C22×C4—2128no—
496C22×C4—2128no—
497C22×C4—2128no—
498C22×C4—2128no—
499C22×C4—2128no—
500C22×C4C2×C4×D5312yes—
501C22×C4C2×C4×D5312yes—
502C22×C4C2×C4×D5312yes—
503C22×C4#1171994514yes—
504C22×C4D4×D5312yes—
505C22×C4#1171994712yes—
506C22⋊C4—1068no—
507C22⋊C4—1068no—
508C22⋊C4—2128no—
509C42—28no—
510C42—2124no—
511C23—534no—
512C23—534no—
513C23—534no—
514C23—538no—
515C23—538no—
516C23—1064no—
517C23—1062no—
518C23—1062no—
519C23—1064no—
520C23—1064no—
521C23—1064no—
522C23—1062no—
523C23—1064no—
524C23—1064no—
525C23—1062no—
526C23—1062no—
527C23—1062no—
528C23—1062no—
529C23—1062no—
530C23—1064no—
531C23—1064no—
532C23—1064no—
533C23—1064no—
534C23C22×C4×D5311yescharacteristic
535C23C2×D4×D5311yescharacteristic
536C23#41×D5312yes—
537C23#41×D5311yescharacteristic
538C23#41×D5311yescharacteristic
539C23#41×D5312yes—
540C23C2×#1171994711yescharacteristic
541C23C2×#1171994512yes—
542C23#1313326211yescharacteristic
543C23#1313326312yes—
544C23#1313326511yescharacteristic
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—24no—
558C2×C4—42no—
559C2×C4—41no—
560C2×C4—42no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
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—1062no—
578C2×C4—1062no—
579C2×C4—1062no—
580C2×C4—1062no—
581C2×C4—1062no—
582C2×C4—1062no—
583C2×C4—1062no—
584C2×C4—1062no—
585C2×C4—1062no—
586C2×C4—1064no—
587C2×C4—21216no—
588C2×C4—21232no—
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—2122no—
600C2×C4—2122no—
601C2×C4—2122no—
602C2×C4—2122no—
603C2×C4—2122no—
604C2×C4—2122no—
605C2×C4—2122no—
606C2×C4#1313317714yes—
607C22—538no—
608C22—1064no—
609C22—1068no—
610C22—1064no—
611C22—1064no—
612C22—1064no—
613C22—1064no—
614C22—1064no—
615C22—1068no—
616C22—1068no—
617C22—1068no—
618C22#117×D5311yescharacteristic
619C22#117×D5311yescharacteristic
620C22#117×D5311yescharacteristic
621C22C2×#1313317711yescharacteristic
622C22C4×#1171994511yescharacteristic
623C22#1681109711yescharacteristic
624C22#1681110411yescharacteristic
625C22#1681113212yes—
626C22#126×D5312yes—
627C22#126×D5312yes—
628C22#1681115512yes—
629C22#1681120712yes—
630C22#1681122112yes—
631C22#1681126512yes—
632C22#1681127011yescharacteristic
633C22#1681128311yescharacteristic
634C22#1681129612yes—
635C22#1681129712yes—
636C22#1681130212yes—
637C22#1681130711yescharacteristic
638C22#1681131412yes—
639C22#1681131712yes—
640C22#1681132011yescharacteristic
641C4—24no—
642C4—42no—
643C4—42no—
644C4—44no—
645C4—82no—
646C4—4242no—
647C4—2122no—
648C4—2122no—
649C4—2122no—
650C4—2124no—
651C2—1068no—
652C2#383×D5311yescharacteristic
653C2#2027716111yescharacteristic
654C2#2027735111yescharacteristic
655C2#2027755711yescharacteristic
656C2#2027757811yescharacteristic
657C2#2027763711yescharacteristic
658C2#2027765611yescharacteristic
659C2#2027896512yes—
660C2#2027898012yes—
661C2#2027909912yes—
662C2#2027928012yes—
subgroupG/H
center, FrattiniC24C23×D53
commutatorC23×C53C23×C4
FittingC53×#1434C2
socle, cosocleC24×C53C24
prime radicalC24×D53C23
hypercenter#1434D53
hypocenterC53#13901
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C24 ◁ #1434

Complements

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

NH ≅ G/Ntype
1. C2×#16810491C2semidirect
4. #20272976C2semidirect
5. #20272975C2semidirect
6. #20272941C2semidirect
7. #20272940C2semidirect
11. C53×#1434C2semidirect
19. C4×#11719921C4semidirect
65. C2×C4×Dic53C2×C4semidirect
75. #13133124C2×C4semidirect
76. #13133124C2×C4semidirect
85. C2×C42×C53C2×C4semidirect
309. C2×D53#383semidirect
327. #1434D53semidirect
369. C53#13901semidirect
405. C2×C42C4×D53semidirect

Maximal subgroups

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

subgroupindex
C2×#168104912
C2×C4×#117199212
C2×#95×D532
#202729762
#202729752
#202729412
#202729402
#202729362
C53×#14342
#1390153