Finite Group Databasebeta

← #27306243

Subgroups — #27306243

Order 13568.

1962 conjugacy classes of subgroups.

The subgroup lattice has 678 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810494C212yesmaximal
2C2×C4×#11719921C211yesmaximal, characteristic
3C2×#95×D53C211yesmaximal, characteristic
4#20272974C212yesmaximal, complemented
5#20272970C212yesmaximal, complemented
6#20272943C211yesmaximal, characteristic, complemented
7#20272942C211yesmaximal, characteristic, complemented
8#20272940C211yesmaximal, characteristic, complemented
9#20272936C211yesmaximal, characteristic
10#20272936C212yesmaximal
11C2×C53×#382C211yesmaximal, characteristic, complemented
12C23×C4×D53C2212yes—
13C23×C4×D53C2211yescharacteristic
14C22×#11719919C2212yes—
15C22×#11719921C2212yes—
16C22×#11719921C2211yescharacteristic
17C22×#11719921C2211yescharacteristic
18C22×C4×Dic53C2211yescharacteristic
19#16810494—28no—
20C4×#11719921C418yescomplemented
21C2×#13133124C2211yescharacteristic
22C2×#13133124C2212yes—
23C2×#13133124C2212yes—
24C2×#13133124C2211yescharacteristic
25C2×#13133124C2212yes—
26C2×#13133124C2212yes—
27C2×#13133124C2212yes—
28C2×#13133124C2212yes—
29C2×#13133124C2212yes—
30C2×#13133124C2211yescharacteristic
31C2×#13133124C2211yescharacteristic
32#95×D53—24no—
33C22×C4⋊C4×C53C2212yes—
34C22×C42×C53C2211yescharacteristic
35C53×#382—24no—
36C2×#95×C53C2211yescharacteristic
37C2×#95×C53C2212yes—
38C2×#95×C53C2211yescharacteristic
39C24×D53C2311yescharacteristic
40C23×Dic53C2312yes—
41C23×Dic53C2311yescharacteristic
42C23×Dic53C2311yescharacteristic
43C23×Dic53C2312yes—
44C23×Dic53C2311yescharacteristic
45C22×C4×D53—24no—
46C22×C4×D53—24no—
47C22×C4×D53—24no—
48C22×C4×D53—22no—
49C22×C4×D53—22no—
50C22×C4×D53—24no—
51C22×C4×D53—24no—
52C22×C4×D53—24no—
53C22×C4×D53—42no—
54C22×C4×D53—42no—
55C22×C4×D53—42no—
56C22×C4×D53C2×C414yes—
57C2×#11719919—24no—
58C2×#11719919—24no—
59C2×#11719919—42no—
60C2×#11719921—24no—
61C2×#11719921—24no—
62C2×#11719921—24no—
63C2×#11719921—24no—
64C2×#11719921C2×C414yes—
65C2×#11719921C2×C414yes—
66C2×#11719921D414yes—
67C2×#11719921Q814yes—
68C2×C4×Dic53—42no—
69C2×C4×Dic53C2×C414yescomplemented
70#13133124—24no—
71#13133124—22no—
72#13133124—24no—
73#13133124—22no—
74#13133124—24no—
75#13133124—24no—
76#13133124—42no—
77#13133124—42no—
78#13133124—42no—
79#13133124C2×C414yescomplemented
80#13133124C2×C414yescomplemented
81C23×C4×C53C2312yes—
82C23×C4×C53C2311yescharacteristic
83C23×C4×C53C2312yes—
84C23×C4×C53C2311yescharacteristic
85C23×C4×C53C2311yescharacteristic
86C2×C4⋊C4×C53—24no—
87C2×C4⋊C4×C53—24no—
88C2×C4⋊C4×C53—42no—
89C2×C42×C53—22no—
90C2×C42×C53—22no—
91C2×C42×C53C2×C414yescomplemented
92#95×C53—24no—
93#95×C53—22no—
94#95×C53D414yes—
95C24×C53C2411yescharacteristic
96C23×D53—21no—
97C23×D53—21no—
98C23×D53—22no—
99C23×D53—22no—
100C23×D53—21no—
101C23×D53—21no—
102C23×D53C22×C412yes—
103C23×D53C2×D412yes—
104C23×D53C2×Q812yes—
105C23×D53#4114yes—
106C23×D53#4114yes—
107C22×C4×C53—21no—
108C22×C4×C53—21no—
109C22×C4×C53—22no—
110C22×C4×C53—21no—
111C22×C4×C53—22no—
112C22×C4×C53—21no—
113C22×C4×C53—21no—
114C22×C4×C53—22no—
115C22×C4×C53—21no—
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×C53—22no—
128C22×C4×C53—22no—
129C22×C4×C53—22no—
130C22×C4×C53C22×C412yes—
131C22×C4×C53C22×C412yes—
132C22×C4×C53C22×C412yes—
133C22×C4×C53C2×D414yes—
134C22×C4×C53C2×D412yes—
135C22×C4×C53C2×D412yes—
136C22×C4×C53C2×Q812yes—
137C22×C4×C53#4112yes—
138C22×Dic53—22no—
139C22×Dic53—22no—
140C22×Dic53—21no—
141C22×Dic53—22no—
142C22×Dic53—22no—
143C22×Dic53—22no—
144C22×Dic53—22no—
145C22×Dic53—21no—
146C22×Dic53—22no—
147C22×Dic53—22no—
148C22×Dic53—21no—
149C22×Dic53—22no—
150C22×Dic53—22no—
151C22×Dic53—22no—
152C22×Dic53—22no—
153C22×Dic53—21no—
154C22×Dic53—22no—
155C22×Dic53—22no—
156C22×Dic53—22no—
157C22×Dic53—22no—
158C22×Dic53—22no—
159C22×Dic53—22no—
160C22×Dic53—22no—
161C22×Dic53C22×C412yes—
162C22×Dic53C22×C412yes—
163C22×Dic53C22×C412yes—
164C22×Dic53C2×D412yes—
165C22×Dic53C2×Q812yes—
166C22×Dic53#4114yes—
167C2×C4×D53—216no—
168C2×C4×D53—28no—
169C2×C4×D53—28no—
170C2×C4×D53—44no—
171C2×C4×D53—48no—
172C2×C4×D53—48no—
173C2×C4×D53—48no—
174C2×C4×D53—48no—
175C2×C4×D53—48no—
176C2×C4×D53—48no—
177C2×C4×D53—48no—
178C2×C4×D53—48no—
179C4⋊C4×C53—48no—
180C42×C53—28no—
181#11719921—28no—
182#11719921—48no—
183#11719921C4⋊C4116yes—
184#11719919—48no—
185C4×Dic53—44no—
186C23×C53C23×C411yescharacteristic
187C23×C53C22×D411yescharacteristic
188C23×C53C22×D411yescharacteristic
189C23×C53C22×Q811yescharacteristic
190C23×C53C2×#4112yes—
191C23×C53C2×#4112yes—
192C23×C53C2×#4112yes—
193C23×C53C2×#4111yescharacteristic
194C23×C53H(5,𝔽2)11yescharacteristic
195C23×C53H(5,𝔽2)12yes—
196C23×C532−1+411yescharacteristic
197C22×D53—22no—
198C22×D53—22no—
199C22×D53—22no—
200C22×D53—22no—
201C22×D53—24no—
202C22×D53—24no—
203C22×D53—24no—
204C22×D53—24no—
205C22×D53—22no—
206C22×D53—24no—
207C22×D53—24no—
208C22×D53—22no—
209C22×D53—22no—
210C22×D53—24no—
211C22×D53—24no—
212C22×D53—24no—
213C22×D53—22no—
214C22×D53—24no—
215C22×D53C2×C4⋊C414yes—
216C22×D53#11714yes—
217C22×D53#11714yes—
218C22×D53#12418yes—
219C22×D53#12518yes—
220C2×C4×C53—22no—
221C2×C4×C53—22no—
222C2×C4×C53—22no—
223C2×C4×C53—22no—
224C2×C4×C53—22no—
225C2×C4×C53—24no—
226C2×C4×C53—22no—
227C2×C4×C53—22no—
228C2×C4×C53—24no—
229C2×C4×C53—22no—
230C2×C4×C53—22no—
231C2×C4×C53—22no—
232C2×C4×C53—22no—
233C2×C4×C53—24no—
234C2×C4×C53—24no—
235C2×C4×C53—24no—
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—42no—
247C2×C4×C53—42no—
248C2×C4×C53—42no—
249C2×C4×C53C2×C4⋊C414yes—
250C2×C4×C53C4×D414yes—
251C2×Dic53—22no—
252C2×Dic53—22no—
253C2×Dic53—24no—
254C2×Dic53—24no—
255C2×Dic53—22no—
256C2×Dic53—22no—
257C2×Dic53—22no—
258C2×Dic53—22no—
259C2×Dic53—22no—
260C2×Dic53—22no—
261C2×Dic53—24no—
262C2×Dic53—24no—
263C2×Dic53—24no—
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—
273C2×Dic53—42no—
274C2×Dic53—42no—
275C2×Dic53—42no—
276C2×Dic53—42no—
277C2×Dic53—42no—
278C2×Dic53—42no—
279C2×Dic53C2×C4⋊C414yes—
280C4×D53—416no—
281C4×D53—432no—
282#13896—531nomaximal, Sylow
283C22×C53C22×C4⋊C411yescharacteristic
284C22×C53C2×#11711yescharacteristic
285C22×C53C2×#11711yescharacteristic
286C22×C53C2×C4×D411yescharacteristic
287C22×C53C4×#4111yescharacteristic
288C22×C53#51811yescharacteristic
289C22×C53#52011yescharacteristic
290C22×C53#52512yes—
291C22×C53C2×#12412yes—
292C22×C53C2×#12512yes—
293C22×C53#52912yes—
294C22×C53#53812yes—
295C22×C53#53912yes—
296C22×C53D4211yescharacteristic
297C22×C53#54612yes—
298C22×C53#54612yes—
299C22×C53#54711yescharacteristic
300C22×C53D4×Q811yescharacteristic
301C22×C53#55112yes—
302C22×C53#55112yes—
303C22×C53#55212yes—
304C22×C53#55411yescharacteristic
305C22×C53#55412yes—
306C2×D53—24no—
307C2×D53—24no—
308C2×D53—24no—
309C2×D53—24no—
310C2×D53—24no—
311C2×D53—28no—
312C2×D53—28no—
313C2×D53—28no—
314C2×D53—28no—
315C2×D53—24no—
316C2×D53#38218yescomplemented
317C4×C53—24no—
318C4×C53—24no—
319C4×C53—44no—
320C4×C53—42no—
321C4×C53—82no—
322Dic53—24no—
323Dic53—42no—
324Dic53—42no—
325Dic53—44no—
326Dic53—82no—
327C2×#385—532no—
328C2×#382D5311yescharacteristic, complemented
329C2×C4×C22⋊C4—531no—
330C22×#95—531no—
331#1442—532no—
332#1442—532no—
333C4⋊#95—531no—
334#1434—531no—
335#1434—531no—
336C22⋊#95—532no—
337C22⋊#95—531no—
338C2×C53C2×#38211yescharacteristic
339C2×C53#229011yescharacteristic
340C2×C53#231811yescharacteristic
341C2×C53#234211yescharacteristic
342C2×C53#234311yescharacteristic
343C2×C53D4×C4⋊C411yescharacteristic
344C2×C53#234911yescharacteristic
345C2×C53#247212yes—
346C2×C53#247312yes—
347C2×C53#248812yes—
348C2×C53#250512yes—
349D53—28no—
350C24×C4—531no—
351C24×C4—532no—
352C22×C4⋊C4—532no—
353C22×C4⋊C4C2×D5312yes—
354C22×C22⋊C4—531no—
355C22×C22⋊C4—531no—
356C22×C22⋊C4—532no—
357C22×C42—531no—
358C22×C42C2×D5311yescharacteristic
359#385—1068no—
360#382—24no—
361C4×C22⋊C4—538no—
362C2×#95—531no—
363C2×#95—531no—
364C2×#95—531no—
365C2×#95—532no—
366C2×#95—531no—
367C2×#95—532no—
368C2×#95—532no—
369C2×#95—532no—
370C2×#95—532no—
371C2×#95—532no—
372C2×#95—532no—
373C2×#95—1064no—
374C2×#95C2×D5311yescharacteristic
375C2×#95C2×D5312yes—
376C2×#95C2×D5311yescharacteristic
377C53#1389611yescharacteristic, complemented, Sylow
378C25—531no—
379C23×C4—531no—
380C23×C4—531no—
381C23×C4—532no—
382C23×C4—531no—
383C23×C4—534no—
384C23×C4—532no—
385C23×C4—1064no—
386C23×C4—1064no—
387C23×C4—1064no—
388C23×C4—1064no—
389C23×C4—1064no—
390C23×C4—1062no—
391C23×C4—1064no—
392C23×C4—1062no—
393C23×C4—2122no—
394C23×C4—2122no—
395C23×C4—2122no—
396C23×C4C22×D5312yes—
397C23×C4C22×D5311yescharacteristic
398C23×C4C22×D5312yes—
399C23×C4C22×D5311yescharacteristic
400C23×C4C22×D5311yescharacteristic
401C2×C4⋊C4—24no—
402C2×C4⋊C4—24no—
403C2×C4⋊C4—42no—
404C2×C4⋊C4—1064no—
405C2×C4⋊C4—1064no—
406C2×C4⋊C4—2122no—
407C2×C22⋊C4—534no—
408C2×C22⋊C4—534no—
409C2×C22⋊C4—534no—
410C2×C22⋊C4—534no—
411C2×C22⋊C4—1064no—
412C2×C22⋊C4—1064no—
413C2×C22⋊C4—1064no—
414C2×C22⋊C4—1064no—
415C2×C42—22no—
416C2×C42—22no—
417C2×C42—534no—
418C2×C42—2122no—
419C2×C42C4×D5314yescomplemented
420#95—24no—
421#95—22no—
422#95—534no—
423#95—534no—
424#95—1064no—
425#95—1064no—
426#95—1062no—
427#95—1064no—
428#95—1064no—
429#95—1062no—
430#95—2122no—
431#95—2122no—
432#95—2122no—
433#95C22⋊D5314yes—
434C24—532no—
435C24—532no—
436C24—532no—
437C24—534no—
438C24—534no—
439C24—1061no—
440C24—1062no—
441C24—1061no—
442C24—1061no—
443C24—1062no—
444C24—1061no—
445C24C23×D5311yescharacteristic
446C22×C4—21no—
447C22×C4—22no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—22no—
451C22×C4—21no—
452C22×C4—22no—
453C22×C4—21no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—21no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—532no—
470C22×C4—532no—
471C22×C4—532no—
472C22×C4—532no—
473C22×C4—532no—
474C22×C4—534no—
475C22×C4—1068no—
476C22×C4—10616no—
477C22×C4—1062no—
478C22×C4—1062no—
479C22×C4—1062no—
480C22×C4—1061no—
481C22×C4—1062no—
482C22×C4—1062no—
483C22×C4—1062no—
484C22×C4—1062no—
485C22×C4—1062no—
486C22×C4—1061no—
487C22×C4—1061no—
488C22×C4—1062no—
489C22×C4—1062no—
490C22×C4—1062no—
491C22×C4—1062no—
492C22×C4—1061no—
493C22×C4—1068no—
494C22×C4—1062no—
495C22×C4—1062no—
496C22×C4—1062no—
497C22×C4—1062no—
498C22×C4—1062no—
499C22×C4—1062no—
500C22×C4—1062no—
501C22×C4—2124no—
502C22×C4—2128no—
503C22×C4—2128no—
504C22×C4—2128no—
505C22×C4—2128no—
506C22×C4—2128no—
507C22×C4—2128no—
508C22×C4—2128no—
509C22×C4—2128no—
510C22×C4C2×C4×D5312yes—
511C22×C4C2×C4×D5312yes—
512C22×C4C2×C4×D5312yes—
513C22×C4#1171994512yes—
514C22×C4D4×D5312yes—
515C22×C4Q8×D5312yes—
516C22×C4C2×C22⋊D5314yes—
517C22×C4C2×C22⋊D5312yes—
518C4⋊C4—48no—
519C4⋊C4—2128no—
520C22⋊C4—5316no—
521C22⋊C4—1068no—
522C22⋊C4—2128no—
523C42—28no—
524C42—2124no—
525C23—534no—
526C23—534no—
527C23—534no—
528C23—538no—
529C23—538no—
530C23—1064no—
531C23—1062no—
532C23—1062no—
533C23—1064no—
534C23—1064no—
535C23—1064no—
536C23—1062no—
537C23—1064no—
538C23—1062no—
539C23—1062no—
540C23—1062no—
541C23—1062no—
542C23—1064no—
543C23—1064no—
544C23—1064no—
545C23—1064no—
546C23—1062no—
547C23—1064no—
548C23C22×C4×D5311yescharacteristic
549C23C2×D4×D5311yescharacteristic
550C23C22×C22⋊D5311yescharacteristic
551C23C2×Q8×D5311yescharacteristic
552C23#41×D5312yes—
553C23#41×D5312yes—
554C23#41×D5312yes—
555C23C2×#1171994511yescharacteristic
556C23#1313326211yescharacteristic
557C23#1313326312yes—
558C23#1313326511yescharacteristic
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—24no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—24no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—534no—
589C2×C4—1064no—
590C2×C4—1064no—
591C2×C4—1062no—
592C2×C4—1062no—
593C2×C4—1064no—
594C2×C4—1064no—
595C2×C4—1062no—
596C2×C4—1062no—
597C2×C4—1062no—
598C2×C4—1062no—
599C2×C4—1062no—
600C2×C4—1062no—
601C2×C4—1064no—
602C2×C4—21216no—
603C2×C4—21232no—
604C2×C4—2122no—
605C2×C4—2122no—
606C2×C4—2122no—
607C2×C4—2122no—
608C2×C4—2122no—
609C2×C4—2122no—
610C2×C4—2122no—
611C2×C4—2122no—
612C2×C4—2122no—
613C2×C4—2122no—
614C2×C4—2122no—
615C2×C4—2122no—
616C2×C4—2122no—
617C2×C4—2122no—
618C2×C4—2122no—
619C2×C4C4⋊C4×D5314yes—
620C2×C4C4×C22⋊D5314yes—
621C22—538no—
622C22—1068no—
623C22—1064no—
624C22—1064no—
625C22—1068no—
626C22—1064no—
627C22—1064no—
628C22—1064no—
629C22—1064no—
630C22—1068no—
631C22—1068no—
632C22C2×C4⋊C4×D5311yescharacteristic
633C22#117×D5311yescharacteristic
634C22#117×D5311yescharacteristic
635C22C2×C4×C22⋊D5311yescharacteristic
636C22#1681109111yescharacteristic
637C22#1681109711yescharacteristic
638C22#1681110411yescharacteristic
639C22#1681113012yes—
640C22#124×D5312yes—
641C22#125×D5312yes—
642C22#1681115112yes—
643C22#1681120712yes—
644C22#1681122112yes—
645C22D4×C22⋊D5311yescharacteristic
646C22#1681126112yes—
647C22#1681126612yes—
648C22#1681127411yescharacteristic
649C22Q8×C22⋊D5311yescharacteristic
650C22#1681129712yes—
651C22#1681130312yes—
652C22#1681130512yes—
653C22#1681132811yescharacteristic
654C22#1681132912yes—
655C4—24no—
656C4—24no—
657C4—44no—
658C4—42no—
659C4—82no—
660C4—1064no—
661C4—4242no—
662C4—2124no—
663C4—2122no—
664C4—2122no—
665C2—1068no—
666C2#382×D5311yescharacteristic
667C2#2027715211yescharacteristic
668C2#2027735111yescharacteristic
669C2#2027756511yescharacteristic
670C2#2027756811yescharacteristic
671C2C4⋊C4×C22⋊D5311yescharacteristic
672C2#2027762311yescharacteristic
673C2#2027883712yes—
674C2#2027885212yes—
675C2#2027903812yes—
676C2#2027921512yes—
subgroupG/H
center, FrattiniC24C23×D53
commutatorC23×C53C23×C4
FittingC2×C53×#382C2
socle, cosocleC24×C53C24
prime radicalC24×D53C23
hypercenterC2×#382D53
hypocenterC53#13896
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C24 ◁ C2×#382

Complements

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

NH ≅ G/Ntype
4. #20272974C2semidirect
5. #20272970C2semidirect
6. #20272943C2semidirect
7. #20272942C2semidirect
8. #20272940C2semidirect
11. C2×C53×#382C2semidirect
20. C4×#11719921C4semidirect
69. C2×C4×Dic53C2×C4semidirect
79. #13133124C2×C4semidirect
80. #13133124C2×C4semidirect
91. C2×C42×C53C2×C4semidirect
316. C2×D53#382semidirect
328. C2×#382D53semidirect
377. C53#13896semidirect
419. C2×C42C4×D53semidirect

Maximal subgroups

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

subgroupindex
C2×#168104942
C2×C4×#117199212
C2×#95×D532
#202729742
#202729702
#202729432
#202729422
#202729402
#202729362
C2×C53×#3822
#1389653