Finite Group Databasebeta

← #27275547

Subgroups — #27275547

Order 13568.

1296 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20274625C211yesmaximal, characteristic, complemented
2C2×#16810360C212yesmaximal, complemented
3C2×#16810357C212yesmaximal, complemented
4#20272938C211yesmaximal, characteristic, complemented
5C2×C53×#525C211yesmaximal, characteristic
6C22×C22⋊Dic53—21no—
7C2×C22⋊#88322—21no—
8C2×C22⋊#88322—21no—
9C2×C22⋊#88322C2212yes—
10C2×#13133124—21no—
11C2×#13133124C2212yes—
12#16810360—24no—
13#16810357—24no—
14C22×#41×C53C411yescharacteristic, complemented
15C24×C4×C53C2211yescharacteristic
16C53×#525—24no—
17C53×#525—24no—
18C2×#123×C53—21no—
19C2×C22⋊Q8×C53C411yescharacteristic, complemented
20C2×C22⋊Q8×C53C2211yescharacteristic
21C2×#121×C53C411yescharacteristic, complemented
22C2×#121×C53C2211yescharacteristic
23C2×#120×C53—21no—
24C2×C4×D4×C53—21no—
25C2×C4×D4×C53—21no—
26C2×#117×C53C411yescharacteristic, complemented
27C23×Dic53—22no—
28C22×#88322—22no—
29C2×C22⋊Dic53—24no—
30C2×C22⋊Dic53—42no—
31C2×C22⋊Dic53—42no—
32C22⋊#88322—22no—
33C22⋊#88322—22no—
34C22⋊#88322—42no—
35C22⋊#88322—42no—
36#13133124—22no—
37#13133124—22no—
38#13133124—42no—
39C25×C53D411yescharacteristic
40C2×#41×C53—22no—
41C2×#41×C53—22no—
42C2×#41×C53—42no—
43C2×#41×C53—42no—
44C22×Q8×C53C2×C411yescharacteristic
45C22×D4×C53—21no—
46C22×D4×C53—21no—
47C22×D4×C53—21no—
48C22×D4×C53C2×C411yescharacteristic
49C23×C4×C53—21no—
50C23×C4×C53—22no—
51C23×C4×C53—22no—
52C23×C4×C53—42no—
53C23×C4×C53—42no—
54C23×C4×C53C2×C411yescharacteristic
55C23×C4×C53C2×C411yescharacteristic
56C23×C4×C53D411yescharacteristic
57C23×C4×C53D411yescharacteristic
58C23×C4×C53D411yescharacteristic
59C23×C4×C53C2311yescharacteristic
60#123×C53—28no—
61C22⋊Q8×C53—22no—
62C22⋊Q8×C53—22no—
63C22⋊Q8×C53—22no—
64C22⋊Q8×C53D414yes—
65#121×C53—22no—
66#121×C53—22no—
67#121×C53—22no—
68#121×C53D414yes—
69#120×C53—28no—
70C4×D4×C53—28no—
71C4×D4×C53—28no—
72#117×C53—22no—
73#117×C53—22no—
74C2×C4⋊C4×C53—21no—
75C2×C4⋊C4×C53—21no—
76C2×C4⋊C4×C53C2×C411yescharacteristic
77C2×C4⋊C4×C53C2×C411yescharacteristic
78C2×C22⋊C4×C53—21no—
79C2×C22⋊C4×C53—21no—
80C2×C22⋊C4×C53—21no—
81C2×C22⋊C4×C53—21no—
82C2×C22⋊C4×C53—21no—
83C2×C42×C53—21no—
84C24×C53—22no—
85C24×C53—22no—
86C24×C53—21no—
87C24×C53—21no—
88C24×C53—21no—
89C24×C53—22no—
90C24×C53—22no—
91C24×C53—41no—
92C24×C53—41no—
93C24×C53C22⋊C411yescharacteristic
94C24×C53C22⋊C411yescharacteristic
95C24×C53C2×D411yescharacteristic
96#41×C53—48no—
97#41×C53—48no—
98C2×Q8×C53—21no—
99C2×Q8×C53—41no—
100C2×Q8×C53—41no—
101C2×Q8×C53C22⋊C412yescomplemented
102C2×D4×C53—24no—
103C2×D4×C53—24no—
104C2×D4×C53—24no—
105C2×D4×C53—21no—
106C2×D4×C53—41no—
107C2×D4×C53—41no—
108C2×D4×C53—42no—
109C2×D4×C53—42no—
110C2×D4×C53—42no—
111C2×D4×C53—42no—
112C2×D4×C53—42no—
113C2×D4×C53—41no—
114C2×D4×C53—41no—
115C2×D4×C53C22⋊C412yescomplemented
116C22×C4×C53—22no—
117C22×C4×C53—22no—
118C22×C4×C53—21no—
119C22×C4×C53—21no—
120C22×C4×C53—24no—
121C22×C4×C53—22no—
122C22×C4×C53—21no—
123C22×C4×C53—21no—
124C22×C4×C53—21no—
125C22×C4×C53—22no—
126C22×C4×C53—22no—
127C22×C4×C53—22no—
128C22×C4×C53—21no—
129C22×C4×C53—21no—
130C22×C4×C53—22no—
131C22×C4×C53—22no—
132C22×C4×C53—41no—
133C22×C4×C53—41no—
134C22×C4×C53—41no—
135C22×C4×C53—41no—
136C22×C4×C53—41no—
137C22×C4×C53—41no—
138C22×C4×C53—41no—
139C22×C4×C53—48no—
140C22×C4×C53—48no—
141C22×C4×C53—42no—
142C22×C4×C53—42no—
143C22×C4×C53C22⋊C411yescharacteristic
144C22×C4×C53C22⋊C411yescharacteristic
145C22×C4×C53C22×C411yescharacteristic
146C22×C4×C53C2×D411yescharacteristic
147C22×C4×C53C2×D412yes—
148C22×C4×C53C2×D412yes—
149C22×Dic53—24no—
150C22×Dic53—42no—
151C22×Dic53—42no—
152C22×Dic53—42no—
153C22×Dic53—42no—
154C22×Dic53—42no—
155C22×Dic53—42no—
156C4⋊C4×C53—24no—
157C4⋊C4×C53—21no—
158C4⋊C4×C53—21no—
159C4⋊C4×C53—24no—
160C4⋊C4×C53C22⋊C412yescomplemented
161C4⋊C4×C53C22⋊C412yescomplemented
162C22⋊C4×C53—24no—
163C22⋊C4×C53—24no—
164C22⋊C4×C53—24no—
165C22⋊C4×C53—24no—
166C22⋊C4×C53—24no—
167C42×C53—24no—
168C22⋊Dic53—48no—
169C2×#88322—42no—
170C2×#88322—42no—
171C2×#88322—42no—
172C23×C53—21no—
173C23×C53—21no—
174C23×C53—21no—
175C23×C53—21no—
176C23×C53—24no—
177C23×C53—24no—
178C23×C53—21no—
179C23×C53—21no—
180C23×C53—21no—
181C23×C53—21no—
182C23×C53—21no—
183C23×C53—21no—
184C23×C53—21no—
185C23×C53—21no—
186C23×C53—21no—
187C23×C53—21no—
188C23×C53—21no—
189C23×C53—21no—
190C23×C53—22no—
191C23×C53—44no—
192C23×C53—44no—
193C23×C53—44no—
194C23×C53—44no—
195C23×C53—42no—
196C23×C53—42no—
197C23×C53—42no—
198C23×C53—42no—
199C23×C53—41no—
200C23×C53—41no—
201C23×C53—41no—
202C23×C53—41no—
203C23×C53—41no—
204C23×C53—41no—
205C23×C53#9912yes—
206C23×C53#10412yes—
207C23×C53C2×C22⋊C411yescharacteristic
208C23×C53#12012yes—
209C23×C53#13512yes—
210C23×C53AGL(1,ℤ/8)11yescharacteristic
211C23×C53#13711yescharacteristic
212Q8×C53—42no—
213Q8×C53—42no—
214D4×C53—42no—
215D4×C53—42no—
216D4×C53—48no—
217D4×C53—48no—
218D4×C53—44no—
219D4×C53—44no—
220C2×C4×C53—24no—
221C2×C4×C53—24no—
222C2×C4×C53—21no—
223C2×C4×C53—21no—
224C2×C4×C53—21no—
225C2×C4×C53—21no—
226C2×C4×C53—21no—
227C2×C4×C53—21no—
228C2×C4×C53—21no—
229C2×C4×C53—22no—
230C2×C4×C53—22no—
231C2×C4×C53—22no—
232C2×C4×C53—21no—
233C2×C4×C53—21no—
234C2×C4×C53—42no—
235C2×C4×C53—42no—
236C2×C4×C53—44no—
237C2×C4×C53—44no—
238C2×C4×C53—44no—
239C2×C4×C53—42no—
240C2×C4×C53—42no—
241C2×C4×C53—48no—
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#9912yes—
249C2×C4×C53C2×C22⋊C411yescharacteristic
250C2×C4×C53C2×C22⋊C411yescharacteristic
251C2×C4×C53#12012yes—
252C2×Dic53—44no—
253C2×Dic53—44no—
254C2×Dic53—44no—
255C2×Dic53—82no—
256C2×Dic53—82no—
257C2×Dic53—82no—
258C2×Dic53—82no—
259#88322—82no—
260#8331—531nomaximal, Sylow
261C22×C53—22no—
262C22×C53—21no—
263C22×C53—21no—
264C22×C53—21no—
265C22×C53—21no—
266C22×C53—22no—
267C22×C53—22no—
268C22×C53—22no—
269C22×C53—22no—
270C22×C53—22no—
271C22×C53—22no—
272C22×C53—22no—
273C22×C53—22no—
274C22×C53—22no—
275C22×C53—48no—
276C22×C53—48no—
277C22×C53—41no—
278C22×C53—41no—
279C22×C53—41no—
280C22×C53—41no—
281C22×C53—41no—
282C22×C53—41no—
283C22×C53—41no—
284C22×C53—41no—
285C22×C53—42no—
286C22×C53—42no—
287C22×C53—42no—
288C22×C53—81no—
289C22×C53—81no—
290C22×C53—81no—
291C22×C53—81no—
292C22×C53#37911yescharacteristic
293C22×C53C2×#9911yescharacteristic
294C22×C53C2×#9911yescharacteristic
295C22×C53#41611yescharacteristic
296C22×C53#41711yescharacteristic
297C22×C53C2×#10411yescharacteristic
298C22×C53#42111yescharacteristic
299C22×C53#44912yes—
300C22×C53#45212yes—
301C4×C53—22no—
302C4×C53—22no—
303C4×C53—21no—
304C4×C53—21no—
305C4×C53—41no—
306C4×C53—42no—
307C4×C53—42no—
308C4×C53—42no—
309Dic53—84no—
310C2×#525D5311yescharacteristic
311#1779—531no—
312C2×#328—532no—
313C2×#327—532no—
314C22⋊#95—531no—
315C2×C53—22no—
316C2×C53—22no—
317C2×C53—42no—
318C2×C53—42no—
319C2×C53—82no—
320C2×C53#150111yescharacteristic
321C2×C53#150411yescharacteristic
322C2×C53#151311yescharacteristic
323C2×C53#151611yescharacteristic
324C2×C53#178111yescharacteristic
325C2×C53#178811yescharacteristic
326C2×C53#178911yescharacteristic
327C22×#41Dic5311yescharacteristic, complemented
328C24×C4C2×D5311yescharacteristic
329#525—24no—
330#525—24no—
331C2×#123—21no—
332C2×C22⋊Q8Dic5311yescharacteristic, complemented
333C2×C22⋊Q8C2×D5311yescharacteristic
334C2×#121Dic5311yescharacteristic, complemented
335C2×#121C2×D5311yescharacteristic
336C2×#120—21no—
337C2×C4×D4—21no—
338C2×C4×D4—21no—
339C2×#117Dic5311yescharacteristic, complemented
340C22×C22⋊C4—1061no—
341C2×C22⋊C8—532no—
342C2×C22⋊C8—1061no—
343C2×C22⋊C8—1061no—
344C2×#95—532no—
345C2×#95—1061no—
346#328—1064no—
347#327—1064no—
348C53#833111yescharacteristic, complemented, Sylow
349C25C22⋊D5311yescharacteristic
350C2×#41—22no—
351C2×#41—22no—
352C2×#41—42no—
353C2×#41—42no—
354C22×Q8C2×Dic5311yescharacteristic
355C22×D4—21no—
356C22×D4—21no—
357C22×D4—21no—
358C22×D4C2×Dic5311yescharacteristic
359C23×C4—22no—
360C23×C4—22no—
361C23×C4—21no—
362C23×C4—42no—
363C23×C4—42no—
364C23×C4—1062no—
365C23×C4C2×Dic5311yescharacteristic
366C23×C4C2×Dic5311yescharacteristic
367C23×C4C22⋊D5311yescharacteristic
368C23×C4C22⋊D5311yescharacteristic
369C23×C4C22⋊D5311yescharacteristic
370C23×C4C22×D5311yescharacteristic
371C22×C8—1062no—
372#123—28no—
373C22⋊Q8—22no—
374C22⋊Q8—22no—
375C22⋊Q8—22no—
376C22⋊Q8C22⋊D5314yes—
377#121—22no—
378#121—22no—
379#121—22no—
380#121C22⋊D5314yes—
381#120—28no—
382C4×D4—28no—
383C4×D4—28no—
384#117—22no—
385#117—22no—
386C2×C4⋊C4—21no—
387C2×C4⋊C4—21no—
388C2×C4⋊C4C2×Dic5311yescharacteristic
389C2×C4⋊C4C2×Dic5311yescharacteristic
390C2×C22⋊C4—21no—
391C2×C22⋊C4—21no—
392C2×C22⋊C4—21no—
393C2×C22⋊C4—21no—
394C2×C22⋊C4—21no—
395C2×C22⋊C4—1064no—
396C2×C22⋊C4—2122no—
397C2×C22⋊C4—2122no—
398C2×C42—21no—
399C22⋊C8—1062no—
400C22⋊C8—1062no—
401C22⋊C8—2122no—
402C22⋊C8—2122no—
403#95—1062no—
404#95—1062no—
405#95—2122no—
406C24—22no—
407C24—22no—
408C24—21no—
409C24—21no—
410C24—21no—
411C24—22no—
412C24—22no—
413C24—41no—
414C24—41no—
415C24C22⋊Dic5311yescharacteristic
416C24C22⋊Dic5311yescharacteristic
417C24C2×C22⋊D5311yescharacteristic
418#41—48no—
419#41—48no—
420C2×Q8—21no—
421C2×Q8—41no—
422C2×Q8—41no—
423C2×Q8C22⋊Dic5312yescomplemented
424C2×D4—24no—
425C2×D4—24no—
426C2×D4—24no—
427C2×D4—21no—
428C2×D4—41no—
429C2×D4—41no—
430C2×D4—42no—
431C2×D4—42no—
432C2×D4—42no—
433C2×D4—42no—
434C2×D4—42no—
435C2×D4—41no—
436C2×D4—41no—
437C2×D4C22⋊Dic5312yescomplemented
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—24no—
443C22×C4—22no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—41no—
455C22×C4—41no—
456C22×C4—41no—
457C22×C4—41no—
458C22×C4—41no—
459C22×C4—41no—
460C22×C4—41no—
461C22×C4—48no—
462C22×C4—48no—
463C22×C4—42no—
464C22×C4—42no—
465C22×C4—1064no—
466C22×C4—2122no—
467C22×C4—2122no—
468C22×C4—2122no—
469C22×C4—2122no—
470C22×C4—2122no—
471C22×C4—2122no—
472C22×C4C22⋊Dic5311yescharacteristic
473C22×C4C22⋊Dic5311yescharacteristic
474C22×C4C22×Dic5311yescharacteristic
475C22×C4C2×C22⋊D5311yescharacteristic
476C22×C4C2×C22⋊D5312yes—
477C22×C4C2×C22⋊D5312yes—
478C2×C8—2122no—
479C2×C8—2122no—
480C2×C8—2122no—
481C4⋊C4—24no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—24no—
485C4⋊C4C22⋊Dic5312yescomplemented
486C4⋊C4C22⋊Dic5312yescomplemented
487C22⋊C4—24no—
488C22⋊C4—24no—
489C22⋊C4—24no—
490C22⋊C4—24no—
491C22⋊C4—24no—
492C22⋊C4—2128no—
493C42—24no—
494C23—21no—
495C23—21no—
496C23—21no—
497C23—21no—
498C23—24no—
499C23—24no—
500C23—21no—
501C23—21no—
502C23—21no—
503C23—21no—
504C23—21no—
505C23—21no—
506C23—21no—
507C23—21no—
508C23—21no—
509C23—21no—
510C23—21no—
511C23—21no—
512C23—22no—
513C23—44no—
514C23—44no—
515C23—44no—
516C23—44no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23—41no—
522C23—41no—
523C23—41no—
524C23—41no—
525C23—41no—
526C23—41no—
527C23#1313313212yes—
528C23#1313314412yes—
529C23C2×C22⋊Dic5311yescharacteristic
530C23#1313318612yes—
531C23#1313323712yes—
532C23#1313324211yescharacteristic
533C23#1313324711yescharacteristic
534Q8—42no—
535Q8—42no—
536D4—42no—
537D4—42no—
538D4—48no—
539D4—48no—
540D4—44no—
541D4—44no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—21no—
555C2×C4—22no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—48no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—4242no—
571C2×C4—4242no—
572C2×C4—4242no—
573C2×C4—4242no—
574C2×C4—2124no—
575C2×C4—2124no—
576C2×C4—2124no—
577C2×C4#1313313212yes—
578C2×C4C2×C22⋊Dic5311yescharacteristic
579C2×C4C2×C22⋊Dic5311yescharacteristic
580C2×C4#1313318612yes—
581C8—4242no—
582C22—22no—
583C22—21no—
584C22—21no—
585C22—21no—
586C22—21no—
587C22—22no—
588C22—22no—
589C22—22no—
590C22—22no—
591C22—22no—
592C22—22no—
593C22—22no—
594C22—22no—
595C22—22no—
596C22—48no—
597C22—48no—
598C22—41no—
599C22—41no—
600C22—41no—
601C22—41no—
602C22—41no—
603C22—41no—
604C22—41no—
605C22—41no—
606C22—42no—
607C22—42no—
608C22—42no—
609C22—81no—
610C22—81no—
611C22—81no—
612C22—81no—
613C22#1681047911yescharacteristic
614C22C2×#1313313211yescharacteristic
615C22C2×#1313313211yescharacteristic
616C22#1681061911yescharacteristic
617C22#1681062411yescharacteristic
618C22C2×#1313314411yescharacteristic
619C22#1681064011yescharacteristic
620C22#1681076812yes—
621C22#1681078512yes—
622C4—22no—
623C4—22no—
624C4—21no—
625C4—21no—
626C4—41no—
627C4—42no—
628C4—42no—
629C4—42no—
630C4—4244no—
631C2—22no—
632C2—22no—
633C2—42no—
634C2—42no—
635C2—82no—
636C2#2027320611yescharacteristic
637C2#2027321911yescharacteristic
638C2#2027325811yescharacteristic
639C2#2027327111yescharacteristic
640C2#2027463011yescharacteristic
641C2#2027465511yescharacteristic
642C2#2027465911yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic53
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
Fitting, prime radicalC2×C53×#525C2
socleC23×C53C2×C22⋊C4
cosocleC23×C4×C53C23
hypercenterC2×#525D53
hypocenterC53#8331
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C24×C4 ◁ C2×#525

Complements

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

NH ≅ G/Ntype
1. #20274625C2semidirect
2. C2×#16810360C2semidirect
3. C2×#16810357C2semidirect
4. #20272938C2semidirect
14. C22×#41×C53C4semidirect
19. C2×C22⋊Q8×C53C4semidirect
21. C2×#121×C53C4semidirect
26. C2×#117×C53C4semidirect
101. C2×Q8×C53C22⋊C4semidirect
115. C2×D4×C53C22⋊C4semidirect
160. C4⋊C4×C53C22⋊C4semidirect
161. C4⋊C4×C53C22⋊C4semidirect
327. C22×#41Dic53semidirect
332. C2×C22⋊Q8Dic53semidirect
334. C2×#121Dic53semidirect
339. C2×#117Dic53semidirect
348. C53#8331semidirect
423. C2×Q8C22⋊Dic53semidirect
437. C2×D4C22⋊Dic53semidirect
485. C4⋊C4C22⋊Dic53semidirect
486. C4⋊C4C22⋊Dic53semidirect

Maximal subgroups

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

subgroupindex
#202746252
C2×#168103602
C2×#168103572
#202729382
C2×C53×#5252
#833153