Finite Group Databasebeta

← #27391265

Subgroups — #27391265

Order 13568.

888 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810962C211yesmaximal, characteristic
2#20277253C211yesmaximal, characteristic
3C4×#13133175C211yesmaximal, characteristic
4#20275353C211yesmaximal, characteristic, complemented
5#20275294C211yesmaximal, characteristic, complemented
6#20275286C211yesmaximal, characteristic, complemented
7#20275024C211yesmaximal, characteristic
8#20274913C211yesmaximal, characteristic
9#20274808C211yesmaximal, characteristic
10#20274607C211yesmaximal, characteristic, complemented
11#106×Dic53C211yesmaximal, characteristic, complemented
12#103×Dic53C211yesmaximal, characteristic, complemented
13#103×Dic53C211yesmaximal, characteristic, complemented
14#20274472C211yesmaximal, characteristic
15C53×#1927C211yesmaximal, characteristic, complemented
16C2×#13133193C2211yescharacteristic
17C2×Q8×Dic53C2211yescharacteristic
18C2×#13133175C2211yescharacteristic
19C2×#13133175C2211yescharacteristic
20C2×C42×D53C2211yescharacteristic
21#16810962C418yescomplemented
22C2×#13133150C2211yescharacteristic
23C2×#13133150C2211yescharacteristic
24C2×#13133149C2211yescharacteristic
25C2×#13133141C2211yescharacteristic
26C2×#13133141C2211yescharacteristic
27C2×#13133129C2211yescharacteristic
28C2×#13133129C2211yescharacteristic
29C2×C8×Dic53C2211yescharacteristic
30C2×C8×Dic53C2211yescharacteristic
31#16810521C2211yescharacteristic
32#16810510—21no—
33#16810497—21no—
34#16810480C2211yescharacteristic
35C4⋊C4×Dic53C2211yescharacteristic
36C4⋊C4×Dic53C2211yescharacteristic
37C4⋊C4×Dic53C2211yescharacteristic
38C4×C4⋊Dic53C2211yescharacteristic
39#16810474—21no—
40C4×#11719921—21no—
41#16810470—21no—
42#16810470—21no—
43#16810468—21no—
44C42×Dic53—21no—
45#16810386C2211yescharacteristic
46#16810386C2211yescharacteristic
47#16810386C2211yescharacteristic
48#16810385C2211yescharacteristic
49#16810385C2211yescharacteristic
50#16810385C2211yescharacteristic
51#16810385C2211yescharacteristic
52#16810378C2211yescharacteristic
53C2×#106×C53C2211yescharacteristic
54C2×#103×C53C2211yescharacteristic
55C2×#103×C53C2211yescharacteristic
56C53×#391C2211yescharacteristic
57C4×C4⋊C4×C53C2211yescharacteristic
58C53×#340C2211yescharacteristic
59C53×#336C2211yescharacteristic
60C22×C4×D53D411yescharacteristic
61C22×C4×D53D411yescharacteristic
62C22×C4×D53C2311yescharacteristic
63C22×#88322C2311yescharacteristic
64C22×#88322C2311yescharacteristic
65#13133193—22no—
66#13133193C2×C414yes—
67Q8×Dic53—22no—
68Q8×Dic53—22no—
69#13133175—22no—
70#13133175—22no—
71#13133175—22no—
72#13133175C2×C414yes—
73C2×C4⋊Dic53—21no—
74C2×C4⋊Dic53C2311yescharacteristic
75C2×C4⋊Dic53C2311yescharacteristic
76C2×C4⋊Dic53C2311yescharacteristic
77C2×C4⋊Dic53C2311yescharacteristic
78C2×#11719919—21no—
79C2×#11719919—21no—
80C2×#11719921—21no—
81C2×#11719921—21no—
82C2×#11719921—21no—
83C2×#11719921—21no—
84C2×C4×Dic53—21no—
85C2×C4×Dic53—21no—
86C2×C4×Dic53—21no—
87C2×C4×Dic53—21no—
88C2×C4×Dic53—21no—
89C2×C4×Dic53—21no—
90C2×C4×Dic53—21no—
91C2×C4×Dic53D411yescharacteristic
92C2×C4×Dic53D411yescharacteristic
93C2×C4×Dic53C2311yescharacteristic
94C42×D53—22no—
95C42×D53—22no—
96#13133150—22no—
97#13133150C2×C414yescomplemented
98#13133149C2×C414yescomplemented
99#13133141—22no—
100#13133141C2×C414yescomplemented
101#13133129—22no—
102#13133129C2×C414yes—
103C8×Dic53—22no—
104C8×Dic53—22no—
105#13133124—21no—
106#13133124—21no—
107#13133124—21no—
108#13133124—21no—
109#13133124—21no—
110#13133124—21no—
111C22×Q8×C53C2311yescharacteristic
112C22×C8×C53C2311yescharacteristic
113C22×C8×C53C2311yescharacteristic
114C2×C4⋊C4×C53C2311yescharacteristic
115C2×C4⋊C4×C53C2311yescharacteristic
116C2×C4⋊C4×C53C2311yescharacteristic
117C2×C42×C53—21no—
118C2×C42×C53C2311yescharacteristic
119#106×C53—22no—
120#103×C53—22no—
121#103×C53C2×C414yescomplemented
122#95×C53—21no—
123#95×C53—21no—
124#95×C53—21no—
125C23×D53C2×D411yescharacteristic
126C2×Q8×C53—21no—
127C2×Q8×C53—21no—
128C2×Q8×C53—21no—
129C2×Q8×C53—21no—
130C2×Q8×C53—21no—
131C2×Q8×C53C22×C412yes—
132C22×C4×C53—21no—
133C22×C4×C53—21no—
134C22×C4×C53—21no—
135C22×C4×C53—21no—
136C22×C4×C53C2×D411yescharacteristic
137C22×C4×C53C2×D411yescharacteristic
138C22×C4×C53C2411yescharacteristic
139C22×Dic53—21no—
140C22×Dic53—21no—
141C22×Dic53—21no—
142C22×Dic53—21no—
143C22×Dic53C2×D411yescharacteristic
144C22×Dic53C2×D411yescharacteristic
145C22×Dic53C2×D411yescharacteristic
146C2×C4×D53—21no—
147C2×C4×D53—21no—
148C2×C4×D53—21no—
149C2×C4×D53—21no—
150C2×C4×D53—22no—
151C2×C4×D53—22no—
152C2×C4×D53—22no—
153C2×C4×D53—22no—
154C2×C4×D53—22no—
155C2×C4×D53—41no—
156C2×C4×D53C22×C412yes—
157C2×C4×D53#4112yes—
158C2×C8×C53—21no—
159C2×C8×C53—21no—
160C2×C8×C53—21no—
161C2×C8×C53—21no—
162C2×C8×C53—21no—
163C2×C8×C53C22×C412yes—
164C4⋊C4×C53—21no—
165C4⋊C4×C53—21no—
166C4⋊C4×C53—21no—
167C4⋊C4×C53—21no—
168C4⋊C4×C53C22×C412yes—
169C4⋊C4×C53#4112yes—
170C42×C53—21no—
171C42×C53—21no—
172C42×C53—42no—
173#11719921—24no—
174#11719921—24no—
175#11719921—24no—
176#11719921—42no—
177C4⋊Dic53—21no—
178C4⋊Dic53—21no—
179C4⋊Dic53—21no—
180C4⋊Dic53—21no—
181C4⋊Dic53—21no—
182C4⋊Dic53—42no—
183C4⋊Dic53C22×C412yes—
184C4⋊Dic53C22×C412yes—
185C4⋊Dic53#4112yes—
186#11719919—24no—
187#11719919—42no—
188C4×Dic53—24no—
189C4×Dic53—24no—
190C4×Dic53—22no—
191C4×Dic53—22no—
192C4×Dic53—42no—
193C4×Dic53—42no—
194C4×Dic53—42no—
195C4×Dic53—42no—
196C4×Dic53—42no—
197C4×Dic53#4112yes—
198C4×Dic53#4112yes—
199C2×#88322—21no—
200C2×#88322—21no—
201C2×#88322—21no—
202C2×#88322—21no—
203C2×#88322—21no—
204C2×#88322C22×C412yes—
205C23×C53C22×D411yescharacteristic
206C22×D53—21no—
207C22×D53—21no—
208C22×D53—21no—
209C22×D53—21no—
210C22×D53—21no—
211C22×D53—21no—
212C22×D53C4×D412yes—
213Q8×C53—22no—
214Q8×C53—22no—
215Q8×C53—41no—
216Q8×C53—41no—
217C2×C4×C53—21no—
218C2×C4×C53—21no—
219C2×C4×C53—22no—
220C2×C4×C53—22no—
221C2×C4×C53—21no—
222C2×C4×C53—22no—
223C2×C4×C53—21no—
224C2×C4×C53—21no—
225C2×C4×C53—41no—
226C2×C4×C53—41no—
227C2×C4×C53—41no—
228C2×C4×C53—41no—
229C2×C4×C53—41no—
230C2×C4×C53—41no—
231C2×C4×C53—41no—
232C2×C4×C53—41no—
233C2×C4×C53—41no—
234C2×C4×C53#13512yes—
235C2×C4×C53C23×C411yescharacteristic
236C2×C4×C53C2×#4111yescharacteristic
237C2×C4×C53C2×#4111yescharacteristic
238C2×C4×C53C2×#4111yescharacteristic
239C2×C4×C53C2×#4111yescharacteristic
240C2×C4×C53C2×#4111yescharacteristic
241C2×Dic53—21no—
242C2×Dic53—21no—
243C2×Dic53—21no—
244C2×Dic53—21no—
245C2×Dic53—22no—
246C2×Dic53—22no—
247C2×Dic53—22no—
248C2×Dic53—41no—
249C2×Dic53—41no—
250C2×Dic53—41no—
251C2×Dic53—41no—
252C2×Dic53—41no—
253C2×Dic53—41no—
254C2×Dic53—41no—
255C2×Dic53—41no—
256C2×Dic53—41no—
257C2×Dic53C4×D412yes—
258C2×Dic53#12312yes—
259C2×Dic53#12312yes—
260C2×Dic53#13512yes—
261C2×Dic53#13712yes—
262C4×D53—22no—
263C4×D53—22no—
264C4×D53—22no—
265C4×D53—22no—
266C4×D53—44no—
267C4×D53—44no—
268C8×C53—22no—
269C8×C53—41no—
270#88322—22no—
271#88322—41no—
272C4×#485—531nomaximal, Sylow
273C22×C53C2×C4×D411yescharacteristic
274C22×C53C2×#12311yescharacteristic
275C22×C53#52511yescharacteristic
276C22×C53C2×#13511yescharacteristic
277C22×C53C2×#13511yescharacteristic
278C22×C53C2×#13711yescharacteristic
279C22×C53#57511yescharacteristic
280C2×D53—22no—
281C2×D53—22no—
282C2×D53—22no—
283C2×D53—41no—
284C2×D53—41no—
285C2×D53—41no—
286C2×D53—41no—
287C4×C53—22no—
288C4×C53—41no—
289C4×C53—42no—
290C4×C53—42no—
291C4×C53—42no—
292C4×C53—81no—
293C4×C53C4×#4111yescharacteristic
294C4×C53C4×#4111yescharacteristic
295C4×C53#52911yescharacteristic
296C4×C53#52911yescharacteristic
297Dic53—22no—
298Dic53—22no—
299Dic53—42no—
300Dic53—42no—
301Dic53—42no—
302Dic53—81no—
303Dic53#48514yescomplemented
304C2×#485—531no—
305C4×C22⋊Q8—531no—
306C4×#117—531no—
307#1928—531no—
308#1927D5311yescharacteristic, complemented
309#1919—531no—
310#1918—531no—
311#1873—531no—
312#1854—531no—
313#1825—531no—
314C4×#107—531no—
315C4×#106—531no—
316C4×#103—531no—
317C4×#103—531no—
318C4×C22⋊C8—531no—
319C2×C53C4×#12311yescharacteristic
320C2×C53C4×#13511yescharacteristic
321C2×C53C4×#13711yescharacteristic
322C2×C53C2×#48511yescharacteristic
323C2×C53#309111yescharacteristic
324C2×C53#310211yescharacteristic
325C2×C53#310711yescharacteristic
326D53—42no—
327C2×C22⋊Q8—531no—
328C2×C4×Q8—531no—
329C2×#117—531no—
330C2×#117—531no—
331C22×C42—531no—
332#485—538no—
333C2×#107—531no—
334C2×#107—531no—
335C2×#106—531no—
336C2×#106C2×D5311yescharacteristic
337C2×#103—531no—
338C2×#103—531no—
339C2×#103C2×D5311yescharacteristic
340C2×#103C2×D5311yescharacteristic
341C2×C22⋊C8—531no—
342C2×C22⋊C8—531no—
343C2×C4×C8—531no—
344C2×C4×C8—531no—
345#391C2×D5311yescharacteristic
346#389—531no—
347#387—1061no—
348#385—1061no—
349#380—531no—
350C4×C4⋊C4—531no—
351C4×C4⋊C4—531no—
352C4×C4⋊C4—531no—
353C4×C4⋊C4—531no—
354C4×C4⋊C4—1061no—
355C4×C4⋊C4C2×D5311yescharacteristic
356C4×C22⋊C4—1061no—
357C4×C22⋊C4—1061no—
358C4×C22⋊C4—1061no—
359#376—1061no—
360C43—1061no—
361#340—531no—
362#340—531no—
363#340—531no—
364#340—531no—
365#340—531no—
366#340—531no—
367#340—531no—
368#340C2×D5311yescharacteristic
369#336—531no—
370#336C2×D5311yescharacteristic
371C53C4×#48511yescharacteristic, complemented, Sylow
372C22×Q8C22×D5311yescharacteristic
373C23×C4—531no—
374C23×C4—531no—
375C23×C4—531no—
376C22×C8—531no—
377C22×C8—531no—
378C22×C8C22×D5311yescharacteristic
379C22×C8C22×D5311yescharacteristic
380C22⋊Q8—534no—
381C22⋊Q8—1062no—
382C4×Q8—1062no—
383C4×Q8—1062no—
384#117—534no—
385#117—1062no—
386#117—1062no—
387#117—1062no—
388C2×C4⋊C4—531no—
389C2×C4⋊C4—531no—
390C2×C4⋊C4—531no—
391C2×C4⋊C4—531no—
392C2×C4⋊C4—1061no—
393C2×C4⋊C4—1061no—
394C2×C4⋊C4—1061no—
395C2×C4⋊C4C22×D5311yescharacteristic
396C2×C4⋊C4C22×D5311yescharacteristic
397C2×C4⋊C4C22×D5311yescharacteristic
398C2×C22⋊C4—1061no—
399C2×C22⋊C4—1061no—
400C2×C22⋊C4—1061no—
401C2×C22⋊C4—1061no—
402C2×C42—21no—
403C2×C42—531no—
404C2×C42—531no—
405C2×C42—531no—
406C2×C42—1062no—
407C2×C42—1062no—
408C2×C42—1061no—
409C2×C42—1061no—
410C2×C42—1061no—
411C2×C42—1061no—
412C2×C42—1061no—
413C2×C42—1061no—
414C2×C42—1061no—
415C2×C42C22×D5311yescharacteristic
416#107—534no—
417#107—1062no—
418#106—22no—
419#106—534no—
420#103—22no—
421#103—534no—
422#103—1062no—
423#103C4×D5314yescomplemented
424C22⋊C8—534no—
425C22⋊C8—1062no—
426C4×C8—1062no—
427C4×C8—1062no—
428#95—21no—
429#95—21no—
430#95—21no—
431#95—1061no—
432#95—1061no—
433#95—1061no—
434#95—1061no—
435#95—1061no—
436#95—1061no—
437C24—531no—
438C2×Q8—21no—
439C2×Q8—21no—
440C2×Q8—21no—
441C2×Q8—21no—
442C2×Q8—21no—
443C2×Q8C2×C4×D5312yes—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—531no—
449C22×C4—531no—
450C22×C4—531no—
451C22×C4—532no—
452C22×C4—532no—
453C22×C4—1062no—
454C22×C4—1062no—
455C22×C4—1062no—
456C22×C4—1062no—
457C22×C4—1061no—
458C22×C4—1061no—
459C22×C4—1061no—
460C22×C4—1061no—
461C22×C4—1061no—
462C22×C4—1061no—
463C22×C4—1061no—
464C22×C4—1062no—
465C22×C4—1061no—
466C22×C4—2121no—
467C22×C4D4×D5311yescharacteristic
468C22×C4D4×D5311yescharacteristic
469C22×C4C23×D5311yescharacteristic
470C2×C8—21no—
471C2×C8—21no—
472C2×C8—21no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—532no—
476C2×C8—1061no—
477C2×C8—1061no—
478C2×C8—1061no—
479C2×C8—1061no—
480C2×C8—1061no—
481C2×C8C2×C4×D5312yes—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—532no—
487C4⋊C4—532no—
488C4⋊C4—532no—
489C4⋊C4—1064no—
490C4⋊C4—1061no—
491C4⋊C4—1061no—
492C4⋊C4—1061no—
493C4⋊C4—1061no—
494C4⋊C4—1061no—
495C4⋊C4—2122no—
496C4⋊C4—2122no—
497C4⋊C4C2×C4×D5312yes—
498C4⋊C4#1171994712yes—
499C22⋊C4—1064no—
500C22⋊C4—1064no—
501C22⋊C4—1064no—
502C22⋊C4—2122no—
503C42—21no—
504C42—21no—
505C42—42no—
506C42—532no—
507C42—532no—
508C42—1062no—
509C42—1062no—
510C42—1064no—
511C42—1064no—
512C42—2122no—
513C42—2122no—
514C42—2122no—
515C42—2122no—
516C42—2122no—
517C23—532no—
518C23—1061no—
519C23—1061no—
520C23—1061no—
521C23—1061no—
522C23—1061no—
523C23—1061no—
524C23C2×D4×D5311yescharacteristic
525Q8—22no—
526Q8—22no—
527Q8—41no—
528Q8—41no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—21no—
534C2×C4—22no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—532no—
547C2×C4—532no—
548C2×C4—532no—
549C2×C4—532no—
550C2×C4—532no—
551C2×C4—1061no—
552C2×C4—1061no—
553C2×C4—1061no—
554C2×C4—1061no—
555C2×C4—1062no—
556C2×C4—1062no—
557C2×C4—1062no—
558C2×C4—1062no—
559C2×C4—1062no—
560C2×C4—1062no—
561C2×C4—1062no—
562C2×C4—2124no—
563C2×C4—2124no—
564C2×C4—2121no—
565C2×C4—2121no—
566C2×C4—2121no—
567C2×C4—2121no—
568C2×C4—2121no—
569C2×C4—2121no—
570C2×C4—2121no—
571C2×C4—2121no—
572C2×C4—2121no—
573C2×C4#1313323812yes—
574C2×C4C22×C4×D5311yescharacteristic
575C2×C4#41×D5311yescharacteristic
576C2×C4#41×D5311yescharacteristic
577C2×C4#41×D5311yescharacteristic
578C2×C4#41×D5311yescharacteristic
579C2×C4C2×#1171994711yescharacteristic
580C8—22no—
581C8—41no—
582C8—1062no—
583C8—2121no—
584C22—1062no—
585C22—1062no—
586C22—1062no—
587C22—2121no—
588C22—2121no—
589C22—2121no—
590C22—2121no—
591C22C4×D4×D5311yescharacteristic
592C22#123×D5311yescharacteristic
593C22#1681113311yescharacteristic
594C22#135×D5311yescharacteristic
595C22C2×#1313323811yescharacteristic
596C22#137×D5311yescharacteristic
597C22#1681142211yescharacteristic
598C4—22no—
599C4—42no—
600C4—42no—
601C4—41no—
602C4—42no—
603C4—81no—
604C4—534no—
605C4—1062no—
606C4—1062no—
607C4—4241no—
608C4—2122no—
609C4—2122no—
610C4—2122no—
611C4C4×#1171994711yescharacteristic
612C4#1681109111yescharacteristic
613C4#1681115211yescharacteristic
614C4#1681115611yescharacteristic
615C2—2122no—
616C2#2027724211yescharacteristic
617C2C4×#1313323811yescharacteristic
618C2#2028353111yescharacteristic
619C2#485×D5311yescharacteristic
620C2#2028462911yescharacteristic
621C2#2028473211yescharacteristic
622C2#2028478311yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#1927C2
socleC23×C53C22×D4
prime radicalC23×D53C2×D4
cosocleC22×C4×C53C24
hypercenter#1927D53
hypocenterC53C4×#485
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1927

Complements

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

NH ≅ G/Ntype
4. #20275353C2semidirect
5. #20275294C2semidirect
6. #20275286C2semidirect
10. #20274607C2semidirect
11. #106×Dic53C2semidirect
12. #103×Dic53C2semidirect
13. #103×Dic53C2semidirect
15. C53×#1927C2semidirect
21. #16810962C4semidirect
97. #13133150C2×C4semidirect
98. #13133149C2×C4semidirect
100. #13133141C2×C4semidirect
121. #103×C53C2×C4semidirect
303. Dic53#485semidirect
308. #1927D53semidirect
371. C53C4×#485semidirect
423. #103C4×D53semidirect

Maximal subgroups

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

subgroupindex
C2×#168109622
#202772532
C4×#131331752
#202753532
#202752942
#202752862
#202750242
#202749132
#202748082
#202746072
#106×Dic532
#103×Dic532
#202744722
C53×#19272
C4×#48553