Finite Group Databasebeta

← #27276128

Subgroups — #27276128

Order 13568.

792 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20274735C211yesmaximal, characteristic, complemented
2C4×#13133138C211yesmaximal, characteristic, complemented
3#20273214C211yesmaximal, characteristic, complemented
4#20273170C211yesmaximal, characteristic, complemented
5#20273140C211yesmaximal, characteristic, complemented
6#20273097C211yesmaximal, characteristic, complemented
7C53×#3475C211yesmaximal, characteristic
8#16810676C2211yescharacteristic
9#16810647C2211yescharacteristic
10C2×#13133144—21no—
11C2×#13133138—21no—
12C2×C4×#88322C2211yescharacteristic
13C4×C4⋊Dic53C2211yescharacteristic
14#16810385—21no—
15#16810382—21no—
16#16810360—22no—
17Q8⋊#88322—22no—
18Q8⋊#88322—21no—
19D4⋊#88322—21no—
20D4⋊#88322—21no—
21D4⋊#88322—21no—
22C53×#556C411yescharacteristic, complemented
23C53×#555C411yescharacteristic, complemented
24C53×#554C411yescharacteristic, complemented
25C53×#553—21no—
26C53×#552—21no—
27C53×#552—21no—
28C53×#551—22no—
29C53×#547C411yescharacteristic, complemented
30C53×#546—21no—
31C53×#533C2211yescharacteristic
32C53×#529—22no—
33C53×#529—21no—
34C53×#525—21no—
35C4×#41×C53—21no—
36C4×#41×C53C2211yescharacteristic
37C2×C4×D4×C53C2211yescharacteristic
38C2×#117×C53—21no—
39C2×#11719917—21no—
40C22×#88322—21no—
41C2×C4⋊Dic53—21no—
42C2×C4×Dic53—21no—
43C4⋊#88322—21no—
44C4⋊#88322—21no—
45C4⋊#88322—21no—
46#13133144—42no—
47#13133138—24no—
48C22⋊#88322—21no—
49C4×#88322—21no—
50C4×#88322—21no—
51C4×#88322—21no—
52#13133124—21no—
53C2×#41×C53—21no—
54C2×#41×C53D411yescharacteristic
55C22×D4×C53D411yescharacteristic
56C23×C4×C53—21no—
57C4⋊Q8×C53C2×C411yescharacteristic
58C4⋊Q8×C53C2×C411yescharacteristic
59#126×C53—22no—
60#126×C53—22no—
61#126×C53—22no—
62#125×C53—22no—
63#125×C53—21no—
64#125×C53—21no—
65#124×C53—22no—
66#124×C53—21no—
67#123×C53—22no—
68#123×C53—22no—
69#123×C53—22no—
70#123×C53—22no—
71C22⋊Q8×C53—21no—
72C22⋊Q8×C53—21no—
73C22⋊Q8×C53—21no—
74C22⋊Q8×C53—22no—
75C22⋊Q8×C53—21no—
76C22⋊Q8×C53D412yes—
77#121×C53—22no—
78#121×C53—21no—
79#121×C53—21no—
80#121×C53—21no—
81#120×C53—22no—
82C4×Q8×C53—21no—
83C4×Q8×C53—21no—
84C4×Q8×C53C2×C411yescharacteristic
85C4×Q8×C53D411yescharacteristic
86C4×Q8×C53D412yes—
87C4×D4×C53—22no—
88C4×D4×C53—22no—
89C4×D4×C53—22no—
90C4×D4×C53—21no—
91C4×D4×C53—21no—
92C4×D4×C53—21no—
93C4×D4×C53—21no—
94C4×D4×C53—21no—
95C4×D4×C53C2×C411yescharacteristic
96C4×D4×C53C2×C411yescharacteristic
97C4×D4×C53C2×C411yescharacteristic
98C4×D4×C53D411yescharacteristic
99C4×D4×C53D411yescharacteristic
100C4×D4×C53D411yescharacteristic
101#117×C53—22no—
102#117×C53—22no—
103#117×C53—22no—
104#117×C53—21no—
105#117×C53—21no—
106#117×C53—21no—
107#117×C53—21no—
108#117×C53—21no—
109#117×C53D411yescharacteristic
110C2×C4⋊C4×C53—21no—
111C2×C4⋊C4×C53—21no—
112C2×C4⋊C4×C53D411yescharacteristic
113C2×C22⋊C4×C53—22no—
114C2×C22⋊C4×C53—21no—
115C2×C42×C53—22no—
116C2×C42×C53—21no—
117C2×C42×C53C2311yescharacteristic
118C24×C53—21no—
119#41×C53—21no—
120#41×C53—21no—
121#41×C53—21no—
122#41×C53—21no—
123#41×C53—42no—
124#41×C53—42no—
125C2×Q8×C53—21no—
126C2×Q8×C53C22⋊C411yescharacteristic
127C2×Q8×C53C22⋊C412yes—
128C2×D4×C53—22no—
129C2×D4×C53—22no—
130C2×D4×C53—21no—
131C2×D4×C53—22no—
132C2×D4×C53—21no—
133C2×D4×C53—41no—
134C2×D4×C53C22⋊C411yescharacteristic
135C2×D4×C53C22⋊C411yescharacteristic
136C2×D4×C53C22⋊C411yescharacteristic
137C22×C4×C53—22no—
138C22×C4×C53—21no—
139C22×C4×C53—22no—
140C22×C4×C53—21no—
141C22×C4×C53—21no—
142C22×C4×C53—21no—
143C22×C4×C53—21no—
144C22×C4×C53—21no—
145C22×C4×C53—21no—
146C22×C4×C53—21no—
147C22×C4×C53—41no—
148C22×C4×C53—41no—
149C22×C4×C53—41no—
150C22×C4×C53—41no—
151C22×C4×C53C2×D411yescharacteristic
152C22×C4×C53C2×D411yescharacteristic
153C22×C4×C53C2×D411yescharacteristic
154C22×Dic53—41no—
155C4⋊C4×C53—22no—
156C4⋊C4×C53—22no—
157C4⋊C4×C53—22no—
158C4⋊C4×C53—22no—
159C4⋊C4×C53—21no—
160C4⋊C4×C53—21no—
161C4⋊C4×C53—21no—
162C4⋊C4×C53—21no—
163C4⋊C4×C53—21no—
164C4⋊C4×C53—22no—
165C4⋊C4×C53—21no—
166C4⋊C4×C53—22no—
167C4⋊C4×C53C22⋊C411yescharacteristic
168C4⋊C4×C53C22⋊C411yescharacteristic
169C4⋊C4×C53C22⋊C411yescharacteristic
170C4⋊C4×C53C22⋊C411yescharacteristic
171C4⋊C4×C53C22⋊C412yes—
172C22⋊C4×C53—22no—
173C22⋊C4×C53—22no—
174C22⋊C4×C53—22no—
175C22⋊C4×C53—22no—
176C22⋊C4×C53—22no—
177C22⋊C4×C53—22no—
178C22⋊C4×C53—21no—
179C22⋊C4×C53—22no—
180C22⋊C4×C53—21no—
181C22⋊C4×C53—21no—
182C22⋊C4×C53—21no—
183C22⋊C4×C53—21no—
184C22⋊C4×C53—21no—
185C22⋊C4×C53—21no—
186C42×C53—22no—
187C42×C53—22no—
188C42×C53—21no—
189C42×C53—21no—
190C42×C53—22no—
191C42×C53—21no—
192C42×C53C22×C411yescharacteristic
193C42×C53C2×D411yescharacteristic
194C42×C53C2×D411yescharacteristic
195C42×C53C2×D411yescharacteristic
196C4⋊Dic53—22no—
197C4⋊Dic53—41no—
198C4×Dic53—41no—
199C4×Dic53—41no—
200#11719917—41no—
201#11719917—41no—
202C2×#88322—22no—
203C2×#88322—41no—
204C2×#88322—41no—
205C2×#88322—41no—
206C2×#88322—41no—
207C23×C53—21no—
208C23×C53—21no—
209C23×C53—21no—
210C23×C53—42no—
211C23×C53—41no—
212C23×C53—41no—
213C23×C53—81no—
214C23×C53#12011yescharacteristic
215Q8×C53—21no—
216Q8×C53—21no—
217Q8×C53—42no—
218Q8×C53—41no—
219Q8×C53—41no—
220D4×C53—21no—
221D4×C53—21no—
222D4×C53—21no—
223D4×C53—21no—
224D4×C53—21no—
225D4×C53—42no—
226D4×C53—44no—
227D4×C53—41no—
228D4×C53—41no—
229D4×C53—82no—
230D4×C53#10412yescomplemented
231C2×C4×C53—21no—
232C2×C4×C53—21no—
233C2×C4×C53—21no—
234C2×C4×C53—21no—
235C2×C4×C53—21no—
236C2×C4×C53—21no—
237C2×C4×C53—22no—
238C2×C4×C53—22no—
239C2×C4×C53—21no—
240C2×C4×C53—41no—
241C2×C4×C53—44no—
242C2×C4×C53—41no—
243C2×C4×C53—41no—
244C2×C4×C53—41no—
245C2×C4×C53—41no—
246C2×C4×C53—41no—
247C2×C4×C53—41no—
248C2×C4×C53—41no—
249C2×C4×C53—41no—
250C2×C4×C53—41no—
251C2×C4×C53—41no—
252C2×C4×C53—41no—
253C2×C4×C53—41no—
254C2×C4×C53—41no—
255C2×C4×C53—41no—
256C2×C4×C53—81no—
257C2×C4×C53—82no—
258C2×C4×C53C2×C22⋊C411yescharacteristic
259C2×C4×C53C2×C22⋊C411yescharacteristic
260C2×C4×C53C2×C22⋊C411yescharacteristic
261C2×C4×C53#12011yescharacteristic
262C2×C4×C53#12011yescharacteristic
263C2×C4×C53#12011yescharacteristic
264C2×C4×C53#13511yescharacteristic
265C2×C4×C53#13511yescharacteristic
266C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
267C2×C4×C53#13711yescharacteristic
268C2×Dic53—42no—
269C2×Dic53—81no—
270C2×Dic53—81no—
271#88322—42no—
272#88322—81no—
273#88322—81no—
274#8438—531nomaximal, Sylow
275C22×C53—21no—
276C22×C53—21no—
277C22×C53—22no—
278C22×C53—41no—
279C22×C53—41no—
280C22×C53—41no—
281C22×C53—41no—
282C22×C53—41no—
283C22×C53—41no—
284C22×C53—41no—
285C22×C53—41no—
286C22×C53—82no—
287C22×C53—82no—
288C22×C53#37911yescharacteristic
289C22×C53#44911yescharacteristic
290C22×C53#45211yescharacteristic
291C4×C53—21no—
292C4×C53—21no—
293C4×C53—21no—
294C4×C53—41no—
295C4×C53—42no—
296C4×C53—41no—
297C4×C53—41no—
298C4×C53—41no—
299C4×C53—41no—
300C4×C53—42no—
301C4×C53#41611yescharacteristic
302C4×C53#41711yescharacteristic
303C4×C53C2×#10411yescharacteristic
304C4×C53#42111yescharacteristic
305C4×C53#44911yescharacteristic
306C4×C53#45211yescharacteristic
307Dic53—82no—
308#3475D5311yescharacteristic
309#1806—531no—
310C4×#102—531no—
311#1503—531no—
312#1493—531no—
313#1486—531no—
314#1476—531no—
315C2×C53—21no—
316C2×C53—21no—
317C2×C53—42no—
318C2×C53—81no—
319C2×C53—81no—
320C2×C53#178811yescharacteristic
321C2×C53#178911yescharacteristic
322C2×C53#179411yescharacteristic
323#556Dic5311yescharacteristic, complemented
324#555Dic5311yescharacteristic, complemented
325#554Dic5311yescharacteristic, complemented
326#553—21no—
327#552—21no—
328#552—21no—
329#551—22no—
330#547Dic5311yescharacteristic, complemented
331#546—21no—
332#533C2×D5311yescharacteristic
333#529—22no—
334#529—21no—
335#525—21no—
336C4×#41—21no—
337C4×#41C2×D5311yescharacteristic
338C2×C4×D4C2×D5311yescharacteristic
339C2×#117—21no—
340#431—531no—
341#423—531no—
342C2×#104—1061no—
343C2×#102—1061no—
344C2×C4×C8—531no—
345C4×C4⋊C4—531no—
346#340—1061no—
347#339—1061no—
348#328—1062no—
349Q8⋊C8—1061no—
350Q8⋊C8—1062no—
351D4⋊C8—1061no—
352D4⋊C8—1061no—
353D4⋊C8—1061no—
354C53#843811yescharacteristic, complemented, Sylow
355C2×#41—21no—
356C2×#41C22⋊D5311yescharacteristic
357C22×D4C22⋊D5311yescharacteristic
358C23×C4—21no—
359C2×M16—1061no—
360C22×C8—1061no—
361C4⋊Q8C2×Dic5311yescharacteristic
362C4⋊Q8C2×Dic5311yescharacteristic
363#126—22no—
364#126—22no—
365#126—22no—
366#125—22no—
367#125—21no—
368#125—21no—
369#124—22no—
370#124—21no—
371#123—22no—
372#123—22no—
373#123—22no—
374#123—22no—
375C22⋊Q8—21no—
376C22⋊Q8—21no—
377C22⋊Q8—22no—
378C22⋊Q8—21no—
379C22⋊Q8—21no—
380C22⋊Q8C22⋊D5312yes—
381#121—22no—
382#121—21no—
383#121—21no—
384#121—21no—
385#120—22no—
386C4×Q8—21no—
387C4×Q8—21no—
388C4×Q8C2×Dic5311yescharacteristic
389C4×Q8C22⋊D5311yescharacteristic
390C4×Q8C22⋊D5312yes—
391C4×D4—22no—
392C4×D4—22no—
393C4×D4—22no—
394C4×D4—21no—
395C4×D4—21no—
396C4×D4—21no—
397C4×D4—21no—
398C4×D4—21no—
399C4×D4C2×Dic5311yescharacteristic
400C4×D4C2×Dic5311yescharacteristic
401C4×D4C2×Dic5311yescharacteristic
402C4×D4C22⋊D5311yescharacteristic
403C4×D4C22⋊D5311yescharacteristic
404C4×D4C22⋊D5311yescharacteristic
405#117—22no—
406#117—22no—
407#117—22no—
408#117—21no—
409#117—21no—
410#117—21no—
411#117—21no—
412#117—21no—
413#117C22⋊D5311yescharacteristic
414C2×C4⋊C4—21no—
415C2×C4⋊C4—21no—
416C2×C4⋊C4—1061no—
417C2×C4⋊C4C22⋊D5311yescharacteristic
418C2×C22⋊C4—22no—
419C2×C22⋊C4—21no—
420C2×C42—22no—
421C2×C42—21no—
422C2×C42—1061no—
423C2×C42C22×D5311yescharacteristic
424C4⋊C8—1061no—
425C4⋊C8—1061no—
426C4⋊C8—1061no—
427#104—2122no—
428#102—1064no—
429C22⋊C8—1061no—
430C4×C8—1061no—
431C4×C8—1061no—
432C4×C8—1061no—
433#95—1061no—
434C24—21no—
435#41—21no—
436#41—21no—
437#41—21no—
438#41—21no—
439#41—42no—
440#41—42no—
441C2×Q8—21no—
442C2×Q8C22⋊Dic5311yescharacteristic
443C2×Q8C22⋊Dic5312yes—
444C2×D4—22no—
445C2×D4—22no—
446C2×D4—21no—
447C2×D4—22no—
448C2×D4—21no—
449C2×D4—41no—
450C2×D4C22⋊Dic5311yescharacteristic
451C2×D4C22⋊Dic5311yescharacteristic
452C2×D4C22⋊Dic5311yescharacteristic
453C22×C4—22no—
454C22×C4—21no—
455C22×C4—22no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—41no—
464C22×C4—41no—
465C22×C4—41no—
466C22×C4—41no—
467C22×C4—2121no—
468C22×C4C2×C22⋊D5311yescharacteristic
469C22×C4C2×C22⋊D5311yescharacteristic
470C22×C4C2×C22⋊D5311yescharacteristic
471M16—2121no—
472M16—2121no—
473C2×C8—1062no—
474C2×C8—2121no—
475C2×C8—2121no—
476C2×C8—2121no—
477C2×C8—2121no—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—22no—
488C4⋊C4—21no—
489C4⋊C4—22no—
490C4⋊C4—1062no—
491C4⋊C4—2121no—
492C4⋊C4C22⋊Dic5311yescharacteristic
493C4⋊C4C22⋊Dic5311yescharacteristic
494C4⋊C4C22⋊Dic5311yescharacteristic
495C4⋊C4C22⋊Dic5311yescharacteristic
496C4⋊C4C22⋊Dic5312yes—
497C22⋊C4—22no—
498C22⋊C4—22no—
499C22⋊C4—22no—
500C22⋊C4—22no—
501C22⋊C4—22no—
502C22⋊C4—22no—
503C22⋊C4—21no—
504C22⋊C4—22no—
505C22⋊C4—21no—
506C22⋊C4—21no—
507C22⋊C4—21no—
508C22⋊C4—21no—
509C22⋊C4—21no—
510C22⋊C4—21no—
511C42—22no—
512C42—22no—
513C42—21no—
514C42—21no—
515C42—21no—
516C42—22no—
517C42—2121no—
518C42—2121no—
519C42C22×Dic5311yescharacteristic
520C42C2×C22⋊D5311yescharacteristic
521C42C2×C22⋊D5311yescharacteristic
522C42C2×C22⋊D5311yescharacteristic
523C23—21no—
524C23—21no—
525C23—21no—
526C23—42no—
527C23—41no—
528C23—41no—
529C23—81no—
530C23#1313318611yescharacteristic
531Q8—21no—
532Q8—21no—
533Q8—42no—
534Q8—41no—
535Q8—41no—
536D4—21no—
537D4—21no—
538D4—21no—
539D4—21no—
540D4—21no—
541D4—42no—
542D4—44no—
543D4—41no—
544D4—41no—
545D4—82no—
546D4#1313314412yescomplemented
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—21no—
556C2×C4—41no—
557C2×C4—44no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—81no—
573C2×C4—82no—
574C2×C4—4241no—
575C2×C4—4241no—
576C2×C4—2122no—
577C2×C4C2×C22⋊Dic5311yescharacteristic
578C2×C4C2×C22⋊Dic5311yescharacteristic
579C2×C4C2×C22⋊Dic5311yescharacteristic
580C2×C4#1313318611yescharacteristic
581C2×C4#1313318611yescharacteristic
582C2×C4#1313318611yescharacteristic
583C2×C4#1313323711yescharacteristic
584C2×C4#1313323711yescharacteristic
585C2×C4#1313324211yescharacteristic
586C2×C4#1313324711yescharacteristic
587C8—4241no—
588C8—4241no—
589C8—2122no—
590C22—21no—
591C22—21no—
592C22—22no—
593C22—41no—
594C22—41no—
595C22—41no—
596C22—41no—
597C22—41no—
598C22—41no—
599C22—41no—
600C22—41no—
601C22—82no—
602C22—82no—
603C22#1681047911yescharacteristic
604C22#1681076811yescharacteristic
605C22#1681078511yescharacteristic
606C4—21no—
607C4—21no—
608C4—21no—
609C4—41no—
610C4—41no—
611C4—41no—
612C4—41no—
613C4—42no—
614C4—41no—
615C4—42no—
616C4—4242no—
617C4#1681061911yescharacteristic
618C4#1681062411yescharacteristic
619C4C2×#1313314411yescharacteristic
620C4#1681064011yescharacteristic
621C4#1681076811yescharacteristic
622C4#1681078511yescharacteristic
623C2—21no—
624C2—21no—
625C2—42no—
626C2—81no—
627C2—81no—
628C2#2027465511yescharacteristic
629C2#2027465911yescharacteristic
630C2#2027467911yescharacteristic
subgroupG/H
centerC2×C4#13133186
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
Fitting, prime radicalC53×#3475C2
socleC22×C53#379
cosocleC2×C42×C53C23
hypercenter#3475D53
hypocenterC53#8438
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3475

Complements

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

NH ≅ G/Ntype
1. #20274735C2semidirect
2. C4×#13133138C2semidirect
3. #20273214C2semidirect
4. #20273170C2semidirect
5. #20273140C2semidirect
6. #20273097C2semidirect
22. C53×#556C4semidirect
23. C53×#555C4semidirect
24. C53×#554C4semidirect
29. C53×#547C4semidirect
230. D4×C53#104semidirect
323. #556Dic53semidirect
324. #555Dic53semidirect
325. #554Dic53semidirect
330. #547Dic53semidirect
354. C53#8438semidirect
546. D4#13133144semidirect

Maximal subgroups

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

subgroupindex
#202747352
C4×#131331382
#202732142
#202731702
#202731402
#202730972
C53×#34752
#843853