Finite Group Databasebeta

← #27305483

Subgroups — #27305483

Order 13568.

1958 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#11719921C211yesmaximal, characteristic
2C2×#16810470C211yesmaximal, characteristic
3C2×#95×D53C211yesmaximal, characteristic
4#20272942C211yesmaximal, characteristic, complemented
5#20272941C212yesmaximal, complemented
6#20272940C211yesmaximal, characteristic, complemented
7#20272940C211yesmaximal, characteristic, complemented
8#20272939C212yesmaximal, complemented
9#20272936C212yesmaximal
10#20272936C212yesmaximal
11C53×#1434C211yesmaximal, characteristic, complemented
12C23×C4×D53C2212yes—
13C23×C4×D53C2211yescharacteristic
14C22×#11719921C2212yes—
15C22×#11719921C2211yescharacteristic
16C22×#11719921C2211yescharacteristic
17C22×C4×Dic53C2212yes—
18C22×C4×Dic53C2211yescharacteristic
19C4×#11719921C418yescomplemented
20#16810470C418yescomplemented
21C2×#13133124C2212yes—
22C2×#13133124C2212yes—
23C2×#13133124C2212yes—
24C2×#13133124C2212yes—
25C2×#13133124C2212yes—
26C2×#13133124C2212yes—
27C2×#13133124C2212yes—
28C2×#13133124C2211yescharacteristic
29C2×#13133124C2211yescharacteristic
30C2×#13133124C2211yescharacteristic
31C2×#13133124C2211yescharacteristic
32#95×D53C418yescomplemented
33C22×C42×C53C2211yescharacteristic
34C2×#95×C53C2212yes—
35C2×#95×C53C2212yes—
36C2×#95×C53C2211yescharacteristic
37C2×#95×C53C2211yescharacteristic
38C24×D53C2311yescharacteristic
39C23×Dic53C2312yes—
40C23×Dic53C2311yescharacteristic
41C23×Dic53C2311yescharacteristic
42C23×Dic53C2312yes—
43C23×Dic53C2311yescharacteristic
44C22×C4×D53—24no—
45C22×C4×D53—24no—
46C22×C4×D53—42no—
47C22×C4×D53—42no—
48C22×C4×D53—42no—
49C22×C4×D53—42no—
50C22×C4×D53—42no—
51C22×C4×D53—42no—
52C22×C4×D53C2×C418yes—
53C22×C4×D53C2×C414yes—
54C22×C4×D53C2×C414yes—
55C22×C4×D53C2×C414yes—
56C2×#11719921—24no—
57C2×#11719921—24no—
58C2×#11719921—24no—
59C2×#11719921—24no—
60C2×#11719921C2×C418yes—
61C2×#11719921C2×C414yes—
62C2×#11719921C2×C414yes—
63C2×C4×Dic53—42no—
64C2×C4×Dic53—42no—
65C2×C4×Dic53—42no—
66C2×C4×Dic53C2×C418yescomplemented
67C2×C4×Dic53C2×C414yescomplemented
68#13133124—42no—
69#13133124—42no—
70#13133124—42no—
71#13133124—42no—
72#13133124—42no—
73#13133124—42no—
74#13133124C2×C414yes—
75#13133124C2×C418yes—
76#13133124C2×C414yescomplemented
77#13133124C2×C414yescomplemented
78#13133124C2×C414yescomplemented
79C23×C4×C53C2311yescharacteristic
80C23×C4×C53C2311yescharacteristic
81C23×C4×C53C2312yes—
82C23×C4×C53C2312yes—
83C23×C4×C53C2311yescharacteristic
84C2×C42×C53—42no—
85C2×C42×C53C2×C414yescomplemented
86#95×C53—42no—
87#95×C53—42no—
88#95×C53C2×C414yes—
89#95×C53C2×C414yescomplemented
90C24×C53C2411yescharacteristic
91C23×D53—22no—
92C23×D53—22no—
93C23×D53—21no—
94C23×D53—21no—
95C23×D53—21no—
96C23×D53—21no—
97C23×D53C22×C412yes—
98C23×D53C22×C412yes—
99C23×D53C22×C412yes—
100C23×D53#4114yes—
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—22no—
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—21no—
118C22×C4×C53—22no—
119C22×C4×C53—22no—
120C22×C4×C53—22no—
121C22×C4×C53—21no—
122C22×C4×C53—22no—
123C22×C4×C53—21no—
124C22×C4×C53C22×C414yes—
125C22×C4×C53C22×C414yes—
126C22×C4×C53C22×C412yes—
127C22×C4×C53C22×C412yes—
128C22×C4×C53C22×C412yes—
129C22×C4×C53C22×C412yes—
130C22×C4×C53C22×C412yes—
131C22×Dic53—22no—
132C22×Dic53—22no—
133C22×Dic53—22no—
134C22×Dic53—22no—
135C22×Dic53—22no—
136C22×Dic53—22no—
137C22×Dic53—22no—
138C22×Dic53—22no—
139C22×Dic53—21no—
140C22×Dic53—21no—
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—21no—
151C22×Dic53—21no—
152C22×Dic53—22no—
153C22×Dic53C22×C414yes—
154C22×Dic53C22×C414yes—
155C22×Dic53C22×C412yes—
156C22×Dic53C22×C412yes—
157C22×Dic53C22×C412yes—
158C22×Dic53C22×C412yes—
159C22×Dic53C22×C412yes—
160C2×C4×D53—216no—
161C2×C4×D53—48no—
162C2×C4×D53—44no—
163C2×C4×D53—44no—
164C2×C4×D53—48no—
165C2×C4×D53—48no—
166C2×C4×D53—48no—
167C2×C4×D53—48no—
168C2×C4×D53—48no—
169C2×C4×D53—48no—
170C2×C4×D53—48no—
171C2×C4×D53C42116yes—
172C42×C53—44no—
173#11719921—216no—
174#11719921—28no—
175#11719921—28no—
176C4×Dic53—48no—
177C4×Dic53—44no—
178C23×C53C23×C411yescharacteristic
179C23×C53C23×C411yescharacteristic
180C23×C53C23×C411yescharacteristic
181C23×C53C2×#4112yes—
182C23×C53C2×#4112yes—
183C23×C53H(5,𝔽2)12yes—
184C23×C53H(5,𝔽2)11yescharacteristic
185C23×C53H(5,𝔽2)12yes—
186C23×C53H(5,𝔽2)11yescharacteristic
187C23×C532−1+411yescharacteristic
188C23×C532−1+411yescharacteristic
189C22×D53—22no—
190C22×D53—24no—
191C22×D53—24no—
192C22×D53—22no—
193C22×D53—22no—
194C22×D53—22no—
195C22×D53—22no—
196C22×D53—22no—
197C22×D53—22no—
198C22×D53—22no—
199C22×D53—24no—
200C22×D53—24no—
201C22×D53—24no—
202C22×D53—24no—
203C22×D53—24no—
204C22×D53—24no—
205C22×D53—24no—
206C22×D53—24no—
207C22×D53C2×C4214yes—
208C22×D53#11718yes—
209C22×D53#11714yes—
210C22×D53#11718yes—
211C22×D53#11714yes—
212C2×C4×C53—24no—
213C2×C4×C53—24no—
214C2×C4×C53—22no—
215C2×C4×C53—22no—
216C2×C4×C53—24no—
217C2×C4×C53—24no—
218C2×C4×C53—22no—
219C2×C4×C53—22no—
220C2×C4×C53—24no—
221C2×C4×C53—24no—
222C2×C4×C53—24no—
223C2×C4×C53—22no—
224C2×C4×C53—22no—
225C2×C4×C53—24no—
226C2×C4×C53—22no—
227C2×C4×C53—22no—
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×C53—42no—
238C2×C4×C53—42no—
239C2×C4×C53—42no—
240C2×C4×C53C2×C4214yes—
241C2×Dic53—24no—
242C2×Dic53—22no—
243C2×Dic53—22no—
244C2×Dic53—24no—
245C2×Dic53—24no—
246C2×Dic53—22no—
247C2×Dic53—24no—
248C2×Dic53—22no—
249C2×Dic53—24no—
250C2×Dic53—24no—
251C2×Dic53—24no—
252C2×Dic53—22no—
253C2×Dic53—22no—
254C2×Dic53—22no—
255C2×Dic53—22no—
256C2×Dic53—24no—
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×Dic53C2×C4214yes—
270C4×D53—416no—
271C4×D53—432no—
272#13796—531nomaximal, Sylow
273C22×C53C22×C4211yescharacteristic
274C22×C53C2×#11712yes—
275C22×C53C2×#11711yescharacteristic
276C22×C53C2×#11712yes—
277C22×C53C2×#11711yescharacteristic
278C22×C53#51811yescharacteristic
279C22×C53#51812yes—
280C22×C53#51812yes—
281C22×C53#51811yescharacteristic
282C22×C53#51811yescharacteristic
283C22×C53#51911yescharacteristic
284C22×C53#52011yescharacteristic
285C22×C53#52011yescharacteristic
286C22×C53#52011yescharacteristic
287C22×C53#52011yescharacteristic
288C22×C53#53812yes—
289C22×C53#53812yes—
290C22×C53#53812yes—
291C22×C53#53812yes—
292C22×C53#53912yes—
293C22×C53#53912yes—
294C22×C53#53912yes—
295C22×C53#53912yes—
296C2×D53—28no—
297C2×D53—28no—
298C2×D53—24no—
299C2×D53—28no—
300C2×D53—24no—
301C2×D53—28no—
302C2×D53—24no—
303C2×D53—24no—
304C2×D53—24no—
305C2×D53—24no—
306C2×D53#37618yes—
307C4×C53—24no—
308C4×C53—44no—
309C4×C53—42no—
310C4×C53—44no—
311C4×C53—42no—
312Dic53—24no—
313Dic53—44no—
314Dic53—42no—
315Dic53—44no—
316Dic53—42no—
317C2×C4×C22⋊C4—531no—
318C2×C4×C22⋊C4—531no—
319C22×#95—531no—
320#1434—531no—
321#1434—532no—
322#1434—531no—
323#1434—531no—
324#1434—532no—
325#1434D5311yescharacteristic, complemented
326C22⋊#95—532no—
327C22⋊#95—532no—
328C2×C53C2×#37611yescharacteristic
329C2×C53#227311yescharacteristic
330C2×C53#227411yescharacteristic
331C2×C53#231212yes—
332C2×C53#231312yes—
333C2×C53#231811yescharacteristic
334C2×C53#231812yes—
335C2×C53#231811yescharacteristic
336C2×C53#231812yes—
337C2×C53#231811yescharacteristic
338C2×C53#231811yescharacteristic
339D53—28no—
340C24×C4—531no—
341C24×C4—532no—
342C22×C22⋊C4—531no—
343C22×C22⋊C4—531no—
344C22×C22⋊C4—532no—
345C22×C42—532no—
346C22×C42—531no—
347C22×C42C2×D5311yescharacteristic
348C4×C22⋊C4—538no—
349C4×C22⋊C4—538no—
350C2×#95—532no—
351C2×#95—531no—
352C2×#95—531no—
353C2×#95—531no—
354C2×#95—531no—
355C2×#95—538no—
356C2×#95—532no—
357C2×#95—532no—
358C2×#95—532no—
359C2×#95—532no—
360C2×#95—532no—
361C2×#95—532no—
362C2×#95C2×D5312yes—
363C2×#95C2×D5312yes—
364C2×#95C2×D5311yescharacteristic
365C2×#95C2×D5311yescharacteristic
366C53#1379611yescharacteristic, complemented, Sylow
367C25—531no—
368C23×C4—531no—
369C23×C4—534no—
370C23×C4—534no—
371C23×C4—534no—
372C23×C4—531no—
373C23×C4—532no—
374C23×C4—531no—
375C23×C4—532no—
376C23×C4—538no—
377C23×C4—1064no—
378C23×C4—1064no—
379C23×C4—2122no—
380C23×C4—2122no—
381C23×C4—2122no—
382C23×C4—2122no—
383C23×C4—2122no—
384C23×C4—2122no—
385C23×C4C22×D5311yescharacteristic
386C23×C4C22×D5311yescharacteristic
387C23×C4C22×D5312yes—
388C23×C4C22×D5312yes—
389C23×C4C22×D5311yescharacteristic
390C2×C22⋊C4—534no—
391C2×C22⋊C4—534no—
392C2×C22⋊C4—538no—
393C2×C22⋊C4—1064no—
394C2×C22⋊C4—1064no—
395C2×C22⋊C4—1064no—
396C2×C22⋊C4—1064no—
397C2×C42—42no—
398C2×C42—538no—
399C2×C42—534no—
400C2×C42—2122no—
401C2×C42—2122no—
402C2×C42—2122no—
403C2×C42C4×D5314yescomplemented
404#95—42no—
405#95—42no—
406#95—538no—
407#95—534no—
408#95—534no—
409#95—534no—
410#95—534no—
411#95—2122no—
412#95—2122no—
413#95—2122no—
414#95—2122no—
415#95—2122no—
416#95—2122no—
417#95C4×D5314yes—
418#95C4×D5314yescomplemented
419C24—532no—
420C24—532no—
421C24—532no—
422C24—534no—
423C24—534no—
424C24—1062no—
425C24—1061no—
426C24—1062no—
427C24—1061no—
428C24—1061no—
429C24—1061no—
430C24C23×D5311yescharacteristic
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—21no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—21no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—532no—
454C22×C4—532no—
455C22×C4—532no—
456C22×C4—5316no—
457C22×C4—534no—
458C22×C4—534no—
459C22×C4—532no—
460C22×C4—532no—
461C22×C4—10616no—
462C22×C4—1062no—
463C22×C4—1062no—
464C22×C4—1062no—
465C22×C4—1062no—
466C22×C4—1062no—
467C22×C4—1062no—
468C22×C4—1062no—
469C22×C4—1062no—
470C22×C4—1061no—
471C22×C4—1062no—
472C22×C4—1062no—
473C22×C4—1062no—
474C22×C4—1061no—
475C22×C4—1062no—
476C22×C4—1062no—
477C22×C4—1062no—
478C22×C4—1062no—
479C22×C4—1062no—
480C22×C4—1062no—
481C22×C4—1061no—
482C22×C4—1061no—
483C22×C4—1062no—
484C22×C4—2128no—
485C22×C4—2128no—
486C22×C4—2124no—
487C22×C4—2124no—
488C22×C4—2128no—
489C22×C4—2128no—
490C22×C4—2128no—
491C22×C4—2128no—
492C22×C4—2128no—
493C22×C4—2128no—
494C22×C4C2×C4×D5314yes—
495C22×C4C2×C4×D5312yes—
496C22×C4C2×C4×D5312yes—
497C22×C4C2×C4×D5314yes—
498C22×C4C2×C4×D5312yes—
499C22×C4C2×C4×D5312yes—
500C22×C4C2×C4×D5312yes—
501C22⋊C4—1068no—
502C22⋊C4—10616no—
503C22⋊C4—1068no—
504C42—44no—
505C42—2124no—
506C42—2128no—
507C23—534no—
508C23—534no—
509C23—538no—
510C23—534no—
511C23—538no—
512C23—1062no—
513C23—1064no—
514C23—1064no—
515C23—1062no—
516C23—1062no—
517C23—1062no—
518C23—1062no—
519C23—1062no—
520C23—1062no—
521C23—1062no—
522C23—1064no—
523C23—1064no—
524C23—1064no—
525C23—1064no—
526C23—1064no—
527C23—1064no—
528C23—1064no—
529C23—1064no—
530C23C22×C4×D5311yescharacteristic
531C23C22×C4×D5311yescharacteristic
532C23C22×C4×D5311yescharacteristic
533C23#41×D5312yes—
534C23#41×D5312yes—
535C23#1313326212yes—
536C23#1313326211yescharacteristic
537C23#1313326312yes—
538C23#1313326311yescharacteristic
539C23#1313326411yescharacteristic
540C23#1313326511yescharacteristic
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—24no—
547C2×C4—22no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—24no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
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—534no—
570C2×C4—1064no—
571C2×C4—1064no—
572C2×C4—1062no—
573C2×C4—1062no—
574C2×C4—1062no—
575C2×C4—1064no—
576C2×C4—1062no—
577C2×C4—1064no—
578C2×C4—1064no—
579C2×C4—1064no—
580C2×C4—1064no—
581C2×C4—1062no—
582C2×C4—1062no—
583C2×C4—1062no—
584C2×C4—1062no—
585C2×C4—1064no—
586C2×C4—21216no—
587C2×C4—21232no—
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—2122no—
597C2×C4—2122no—
598C2×C4—2122no—
599C2×C4—2122no—
600C2×C4C42×D5314yes—
601C22—538no—
602C22—1068no—
603C22—1068no—
604C22—1064no—
605C22—1064no—
606C22—1064no—
607C22—1068no—
608C22—1064no—
609C22—1068no—
610C22—1064no—
611C22—1064no—
612C22C2×C42×D5311yescharacteristic
613C22#117×D5312yes—
614C22#117×D5311yescharacteristic
615C22#117×D5312yes—
616C22#117×D5311yescharacteristic
617C22#1681109511yescharacteristic
618C22#1681109512yes—
619C22#1681109512yes—
620C22#1681109611yescharacteristic
621C22#1681109711yescharacteristic
622C22#1681109911yescharacteristic
623C22#1681110211yescharacteristic
624C22#1681110311yescharacteristic
625C22#1681110411yescharacteristic
626C22#1681110511yescharacteristic
627C22#1681120512yes—
628C22#1681120712yes—
629C22#1681120912yes—
630C22#1681121212yes—
631C22#1681121412yes—
632C22#1681121612yes—
633C22#1681121912yes—
634C22#1681122112yes—
635C4—24no—
636C4—44no—
637C4—42no—
638C4—44no—
639C4—42no—
640C4—1064no—
641C4—2124no—
642C4—2122no—
643C4—2124no—
644C4—2122no—
645C2—1068no—
646C2#376×D5311yescharacteristic
647C2#2027706711yescharacteristic
648C2#2027706911yescharacteristic
649C2#2027730912yes—
650C2#2027731412yes—
651C2#2027734411yescharacteristic
652C2#2027734512yes—
653C2#2027734611yescharacteristic
654C2#2027734712yes—
655C2#2027735011yescharacteristic
656C2#2027735111yescharacteristic
subgroupG/H
center, FrattiniC24C23×D53
commutatorC22×C53C22×C42
FittingC53×#1434C2
socle, cosocleC24×C53C24
prime radicalC24×D53C23
hypercenter#1434D53
hypocenterC53#13796
LCSG ▷ C22×C53 ▷ C53
UCSC1 ◁ C24 ◁ #1434

Complements

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

NH ≅ G/Ntype
4. #20272942C2semidirect
5. #20272941C2semidirect
6. #20272940C2semidirect
7. #20272940C2semidirect
8. #20272939C2semidirect
11. C53×#1434C2semidirect
19. C4×#11719921C4semidirect
20. #16810470C4semidirect
32. #95×D53C4semidirect
66. C2×C4×Dic53C2×C4semidirect
67. C2×C4×Dic53C2×C4semidirect
76. #13133124C2×C4semidirect
77. #13133124C2×C4semidirect
78. #13133124C2×C4semidirect
85. C2×C42×C53C2×C4semidirect
89. #95×C53C2×C4semidirect
325. #1434D53semidirect
366. C53#13796semidirect
403. C2×C42C4×D53semidirect
418. #95C4×D53semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#117199212
C2×#168104702
C2×#95×D532
#202729422
#202729412
#202729402
#202729392
#202729362
C53×#14342
#1379653