Finite Group Databasebeta

← #27306151

Subgroups — #27306151

Order 13568.

1602 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810485C211yesmaximal, characteristic
2C2×#16810474C211yesmaximal, characteristic, complemented
3C2×C4×#11719921C211yesmaximal, characteristic
4#20272973C212yesmaximal, complemented
5#20272959C212yesmaximal
6#20272951C212yesmaximal
7#20272942C211yesmaximal, characteristic, complemented
8#20272941C212yesmaximal, complemented
9#20272939C211yesmaximal, characteristic, complemented
10#20272936C211yesmaximal, characteristic
11C53×#1434C211yesmaximal, characteristic, complemented
12C23×C4×D53C2211yescharacteristic
13C22×#11719919C2212yes—
14C22×#11719921C2212yes—
15C22×#11719921C2212yes—
16C22×#11719921C2211yescharacteristic
17C22×#11719921C2211yescharacteristic
18C22×C4×Dic53C2212yes—
19C22×C4×Dic53C2211yescharacteristic
20#16810485—24no—
21#16810474—24no—
22C4×#11719921C418yescomplemented
23C2×#13133124C2211yescharacteristic
24C2×#13133124C2212yes—
25C2×#13133124C2212yes—
26C2×#13133124C2212yes—
27C2×#13133124C2212yes—
28C2×#13133124C2212yes—
29C2×#13133124C2212yes—
30C2×#13133124C2211yescharacteristic
31C2×#13133124C2211yescharacteristic
32C2×#13133124C2211yescharacteristic
33C22×C42×C53C2211yescharacteristic
34C2×#95×C53C2211yescharacteristic
35C2×#95×C53C2211yescharacteristic
36C2×#95×C53C2212yes—
37C2×#95×C53C2212yes—
38C24×D53C2311yescharacteristic
39C23×Dic53C2312yes—
40C23×Dic53C2311yescharacteristic
41C23×Dic53C2312yes—
42C23×Dic53C2311yescharacteristic
43C23×Dic53C2311yescharacteristic
44C22×C4×D53—24no—
45C22×C4×D53—22no—
46C22×C4×D53—22no—
47C22×C4×D53—42no—
48C22×C4×D53C2×C414yes—
49C2×#11719919—24no—
50C2×#11719919—24no—
51C2×#11719919—24no—
52C2×#11719921—24no—
53C2×#11719921—24no—
54C2×#11719921—24no—
55C2×#11719921—24no—
56C2×#11719921—22no—
57C2×#11719921—24no—
58C2×#11719921—22no—
59C2×#11719921—42no—
60C2×#11719921—42no—
61C2×#11719921C2×C414yes—
62C2×#11719921C2×C414yes—
63C2×C4×Dic53—22no—
64C2×C4×Dic53—22no—
65C2×C4×Dic53—24no—
66C2×C4×Dic53—24no—
67C2×C4×Dic53—42no—
68C2×C4×Dic53C2×C414yes—
69#13133124—22no—
70#13133124—24no—
71#13133124—24no—
72#13133124—24no—
73#13133124—42no—
74#13133124—42no—
75#13133124—42no—
76#13133124C2×C414yes—
77#13133124C2×C414yes—
78#13133124D414yes—
79C23×C4×C53C2312yes—
80C23×C4×C53C2311yescharacteristic
81C23×C4×C53C2311yescharacteristic
82C23×C4×C53C2312yes—
83C23×C4×C53C2311yescharacteristic
84C2×C42×C53—42no—
85C2×C42×C53C2×C414yes—
86#95×C53—22no—
87#95×C53—22no—
88#95×C53—24no—
89#95×C53—42no—
90C24×C53C2411yescharacteristic
91C23×D53—21no—
92C23×D53—22no—
93C23×D53—21no—
94C23×D53—21no—
95C23×D53—22no—
96C23×D53—21no—
97C23×D53—21no—
98C23×D53—22no—
99C23×D53—21no—
100C23×D53C22×C412yes—
101C23×D53#4114yes—
102C22×C4×C53—22no—
103C22×C4×C53—22no—
104C22×C4×C53—22no—
105C22×C4×C53—22no—
106C22×C4×C53—22no—
107C22×C4×C53—22no—
108C22×C4×C53—21no—
109C22×C4×C53—21no—
110C22×C4×C53—22no—
111C22×C4×C53—22no—
112C22×C4×C53—22no—
113C22×C4×C53—22no—
114C22×C4×C53—22no—
115C22×C4×C53—21no—
116C22×C4×C53—22no—
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×C53C22×C412yes—
126C22×C4×C53C22×C412yes—
127C22×C4×C53C22×C412yes—
128C22×C4×C53C2×D412yes—
129C22×C4×C53#4114yes—
130C22×C4×C53#4112yes—
131C22×Dic53—21no—
132C22×Dic53—21no—
133C22×Dic53—21no—
134C22×Dic53—21no—
135C22×Dic53—22no—
136C22×Dic53—21no—
137C22×Dic53—21no—
138C22×Dic53—21no—
139C22×Dic53—22no—
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—22no—
151C22×Dic53—22no—
152C22×Dic53—22no—
153C22×Dic53—22no—
154C22×Dic53—22no—
155C22×Dic53C22×C412yes—
156C22×Dic53C22×C412yes—
157C22×Dic53C22×C412yes—
158C22×Dic53C2×D414yes—
159C22×Dic53#4112yes—
160C22×Dic53#4114yes—
161C22×Dic53#4112yes—
162C2×C4×D53—28no—
163C2×C4×D53—28no—
164C2×C4×D53—44no—
165C2×C4×D53—48no—
166C2×C4×D53—48no—
167C42×C53—44no—
168#11719921—28no—
169#11719921—28no—
170#11719921—48no—
171#11719921—48no—
172#11719919—216no—
173C4×Dic53—28no—
174C4×Dic53—48no—
175C23×C53C23×C411yescharacteristic
176C23×C53C22×D411yescharacteristic
177C23×C53C2×#4112yes—
178C23×C53C2×#4112yes—
179C23×C53C2×#4111yescharacteristic
180C23×C53C2×#4111yescharacteristic
181C23×C53C2×#4112yes—
182C23×C53C2×#4111yescharacteristic
183C23×C53H(5,𝔽2)12yes—
184C23×C53H(5,𝔽2)11yescharacteristic
185C23×C532−1+411yescharacteristic
186C22×D53—22no—
187C22×D53—22no—
188C22×D53—22no—
189C22×D53—22no—
190C22×D53—22no—
191C22×D53—22no—
192C22×D53—24no—
193C22×D53—24no—
194C22×D53—24no—
195C22×D53—24no—
196C22×D53—24no—
197C22×D53—24no—
198C22×D53—42no—
199C22×D53—42no—
200C22×D53—42no—
201C22×D53—42no—
202C22×D53—41no—
203C22×D53—41no—
204C22×D53—41no—
205C22×D53—42no—
206C22×D53—41no—
207C22×D53—42no—
208C22×D53#11714yes—
209C2×C4×C53—22no—
210C2×C4×C53—22no—
211C2×C4×C53—24no—
212C2×C4×C53—24no—
213C2×C4×C53—22no—
214C2×C4×C53—22no—
215C2×C4×C53—22no—
216C2×C4×C53—22no—
217C2×C4×C53—22no—
218C2×C4×C53—22no—
219C2×C4×C53—24no—
220C2×C4×C53—24no—
221C2×C4×C53—24no—
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×C53—42no—
236C2×C4×C53—42no—
237C2×C4×C53C4×D414yes—
238C2×Dic53—22no—
239C2×Dic53—22no—
240C2×Dic53—22no—
241C2×Dic53—22no—
242C2×Dic53—24no—
243C2×Dic53—24no—
244C2×Dic53—22no—
245C2×Dic53—22no—
246C2×Dic53—22no—
247C2×Dic53—22no—
248C2×Dic53—22no—
249C2×Dic53—22no—
250C2×Dic53—22no—
251C2×Dic53—24no—
252C2×Dic53—24no—
253C2×Dic53—24no—
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#11714yes—
268C2×Dic53#11714yes—
269C4×D53—416no—
270#13887—531nomaximal, Sylow
271C22×C53C2×#11711yescharacteristic
272C22×C53C2×#11711yescharacteristic
273C22×C53C2×#11711yescharacteristic
274C22×C53C2×C4×D411yescharacteristic
275C22×C53C4×#4111yescharacteristic
276C22×C53#51811yescharacteristic
277C22×C53#52011yescharacteristic
278C22×C53#52512yes—
279C22×C53#52912yes—
280C22×C53#52912yes—
281C22×C53#52912yes—
282C22×C53#54012yes—
283C22×C53#54212yes—
284C22×C53#54611yescharacteristic
285C22×C53#54612yes—
286C22×C53#54811yescharacteristic
287C22×C53#55112yes—
288C22×C53#55112yes—
289C22×C53#55211yescharacteristic
290C22×C53#55212yes—
291C22×C53#55312yes—
292C22×C53#55311yescharacteristic
293C22×C53#55912yes—
294C2×D53—24no—
295C2×D53—24no—
296C2×D53—24no—
297C2×D53—42no—
298C2×D53—42no—
299C2×D53—42no—
300C2×D53—44no—
301C2×D53—44no—
302C2×D53—44no—
303C2×D53—44no—
304C2×D53—42no—
305C4×C53—24no—
306C4×C53—42no—
307C4×C53—42no—
308C4×C53—44no—
309C4×C53—82no—
310Dic53—24no—
311Dic53—24no—
312Dic53—42no—
313Dic53—44no—
314Dic53—82no—
315C2×#381—531no—
316C2×C4×C4⋊C4—531no—
317C2×C4×C22⋊C4—531no—
318#1442—532no—
319#1439—532no—
320#1437—532no—
321#1434—531no—
322#1434—531no—
323#1434—532no—
324#1434D5311yescharacteristic, complemented
325C22⋊#95—531no—
326C2×C53#228311yescharacteristic
327C2×C53#229011yescharacteristic
328C2×C53#231911yescharacteristic
329C2×C53#234211yescharacteristic
330C2×C53#234311yescharacteristic
331C2×C53#235211yescharacteristic
332C2×C53#235711yescharacteristic
333C2×C53#245712yes—
334C2×C53#245812yes—
335C2×C53#251512yes—
336C2×C53#251612yes—
337D53—44no—
338C24×C4—531no—
339C22×C4⋊C4—532no—
340C22×C22⋊C4—531no—
341C22×C22⋊C4—531no—
342C22×C22⋊C4—532no—
343C22×C22⋊C4—532no—
344C22×C42—531no—
345C22×C42—532no—
346C22×C42C2×D5311yescharacteristic
347#381—1064no—
348C4×C4⋊C4—1064no—
349C4×C22⋊C4—538no—
350C2×#95—531no—
351C2×#95—531no—
352C2×#95—531no—
353C2×#95—531no—
354C2×#95—532no—
355C2×#95—532no—
356C2×#95—532no—
357C2×#95—532no—
358C2×#95—532no—
359C2×#95—532no—
360C2×#95C2×D5311yescharacteristic
361C2×#95C2×D5311yescharacteristic
362C2×#95C2×D5312yes—
363C2×#95C2×D5312yes—
364C53#1388711yescharacteristic, complemented, Sylow
365C25—531no—
366C23×C4—531no—
367C23×C4—531no—
368C23×C4—532no—
369C23×C4—531no—
370C23×C4—534no—
371C23×C4—532no—
372C23×C4—1064no—
373C23×C4—1062no—
374C23×C4—1062no—
375C23×C4—2122no—
376C23×C4C22×D5312yes—
377C23×C4C22×D5311yescharacteristic
378C23×C4C22×D5311yescharacteristic
379C23×C4C22×D5312yes—
380C23×C4C22×D5311yescharacteristic
381C2×C4⋊C4—1064no—
382C2×C4⋊C4—1064no—
383C2×C4⋊C4—1064no—
384C2×C22⋊C4—534no—
385C2×C22⋊C4—534no—
386C2×C22⋊C4—1062no—
387C2×C22⋊C4—1064no—
388C2×C22⋊C4—1062no—
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×C42—42no—
396C2×C42—534no—
397C2×C42—1062no—
398C2×C42—1062no—
399C2×C42—1064no—
400C2×C42—1064no—
401C2×C42—2122no—
402C2×C42C4×D5314yes—
403#95—24no—
404#95—22no—
405#95—22no—
406#95—42no—
407#95—534no—
408#95—534no—
409#95—534no—
410#95—1064no—
411#95—1064no—
412#95—1064no—
413#95—1062no—
414#95—2122no—
415#95—2122no—
416#95—2122no—
417C24—532no—
418C24—534no—
419C24—1061no—
420C24—1061no—
421C24—1061no—
422C24—1062no—
423C24—1062no—
424C24—1061no—
425C24—1062no—
426C24—1061no—
427C24—1061no—
428C24C23×D5311yescharacteristic
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—21no—
443C22×C4—22no—
444C22×C4—21no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—532no—
453C22×C4—532no—
454C22×C4—532no—
455C22×C4—534no—
456C22×C4—532no—
457C22×C4—532no—
458C22×C4—534no—
459C22×C4—1068no—
460C22×C4—1061no—
461C22×C4—1061no—
462C22×C4—1062no—
463C22×C4—1061no—
464C22×C4—1061no—
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—1062no—
477C22×C4—1068no—
478C22×C4—1062no—
479C22×C4—1062no—
480C22×C4—1062no—
481C22×C4—1062no—
482C22×C4—1062no—
483C22×C4—1062no—
484C22×C4—1062no—
485C22×C4—2124no—
486C22×C4—2128no—
487C22×C4—2128no—
488C22×C4C2×C4×D5312yes—
489C22×C4C2×C4×D5312yes—
490C22×C4C2×C4×D5312yes—
491C22×C4#1171994514yes—
492C22×C4D4×D5312yes—
493C22×C4#1171994912yes—
494C4⋊C4—10616no—
495C22⋊C4—1068no—
496C22⋊C4—1068no—
497C22⋊C4—2128no—
498C22⋊C4—2128no—
499C42—44no—
500C42—1068no—
501C42—2128no—
502C23—534no—
503C23—1064no—
504C23—1064no—
505C23—1064no—
506C23—1062no—
507C23—1062no—
508C23—1062no—
509C23—1062no—
510C23—1062no—
511C23—1062no—
512C23—1064no—
513C23—1064no—
514C23—1064no—
515C23—2121no—
516C23—2121no—
517C23—2121no—
518C23—2122no—
519C23—2121no—
520C23—2122no—
521C23—2122no—
522C23—2122no—
523C23—2122no—
524C23—2122no—
525C23C22×C4×D5311yescharacteristic
526C23C2×D4×D5311yescharacteristic
527C23#41×D5311yescharacteristic
528C23#41×D5311yescharacteristic
529C23#41×D5312yes—
530C23#41×D5312yes—
531C23C2×#1171994512yes—
532C23C2×#1171994911yescharacteristic
533C23#1313326212yes—
534C23#1313326311yescharacteristic
535C23#1313326411yescharacteristic
536C2×C4—24no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—24no—
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—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—534no—
565C2×C4—534no—
566C2×C4—1062no—
567C2×C4—1062no—
568C2×C4—1062no—
569C2×C4—1062no—
570C2×C4—1062no—
571C2×C4—1062no—
572C2×C4—1062no—
573C2×C4—1062no—
574C2×C4—1064no—
575C2×C4—1064no—
576C2×C4—1064no—
577C2×C4—1062no—
578C2×C4—1062no—
579C2×C4—1062no—
580C2×C4—1064no—
581C2×C4—1064no—
582C2×C4—21216no—
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#1313317914yes—
597C22—1064no—
598C22—1064no—
599C22—1064no—
600C22—2124no—
601C22—2124no—
602C22—2124no—
603C22—2122no—
604C22—2124no—
605C22—2122no—
606C22—2122no—
607C22—2122no—
608C22#117×D5311yescharacteristic
609C22#117×D5311yescharacteristic
610C22#117×D5311yescharacteristic
611C22C2×#1313317911yescharacteristic
612C22C4×#1171994511yescharacteristic
613C22#1681109611yescharacteristic
614C22#1681110311yescharacteristic
615C22#1681113212yes—
616C22#1681115612yes—
617C22#1681115612yes—
618C22#1681116012yes—
619C22#1681122512yes—
620C22#1681124112yes—
621C22#1681126212yes—
622C22#1681126611yescharacteristic
623C22#1681128111yescharacteristic
624C22#1681129712yes—
625C22#1681130012yes—
626C22#1681131111yescharacteristic
627C22#1681131212yes—
628C22#1681132212yes—
629C22#1681132611yescharacteristic
630C22#1681135612yes—
631C4—24no—
632C4—44no—
633C4—42no—
634C4—42no—
635C4—82no—
636C4—1064no—
637C4—1064no—
638C4—4242no—
639C4—2122no—
640C4—2124no—
641C2—2124no—
642C2#2027711211yescharacteristic
643C2#2027715611yescharacteristic
644C2#2027735511yescharacteristic
645C2#2027756111yescharacteristic
646C2#2027757611yescharacteristic
647C2#2027764111yescharacteristic
648C2#2027768911yescharacteristic
649C2#2027862812yes—
650C2#2027864312yes—
651C2#2027934012yes—
652C2#2027935512yes—
subgroupG/H
center, FrattiniC24C23×D53
commutatorC23×C53C23×C4
FittingC53×#1434C2
socle, cosocleC24×C53C24
prime radicalC24×D53C23
hypercenter#1434D53
hypocenterC53#13887
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C24 ◁ #1434

Complements

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

NH ≅ G/Ntype
2. C2×#16810474C2semidirect
4. #20272973C2semidirect
7. #20272942C2semidirect
8. #20272941C2semidirect
9. #20272939C2semidirect
11. C53×#1434C2semidirect
22. C4×#11719921C4semidirect
324. #1434D53semidirect
364. C53#13887semidirect

Maximal subgroups

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

subgroupindex
C2×#168104852
C2×#168104742
C2×C4×#117199212
#202729732
#202729592
#202729512
#202729422
#202729412
#202729392
#202729362
C53×#14342
#1388753