Finite Group Databasebeta

← #27437440

Subgroups — #27437440

Order 13568.

1262 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810989C211yesmaximal, characteristic, complemented
2C2×#16810823C211yesmaximal, characteristic, complemented
3#20279661C211yesmaximal, characteristic
4#20275716C212yesmaximal, complemented
5#20275716C212yesmaximal, complemented
6#20275673C211yesmaximal, characteristic, complemented
7#20275514C211yesmaximal, characteristic, complemented
8#20275037C212yesmaximal, complemented
9#20275032C212yesmaximal, complemented
10#20274545C211yesmaximal, characteristic, complemented
11C2×C53×#552C211yesmaximal, characteristic, complemented
12C22×#11719924C2212yes—
13C2×#13133214C2211yescharacteristic
14C2×#13133213—21no—
15C2×#13133191C2212yes—
16C2×#13133191C2212yes—
17#16810989—24no—
18#16810823—24no—
19C2×C4⋊#88322C2211yescharacteristic
20C2×#13133141C2212yes—
21C2×#13133137C2212yes—
22C2×#13133137C2212yes—
23C2×#13133137C2212yes—
24C2×C22⋊#88322C2212yes—
25C2×C22⋊#88322C2212yes—
26C2×#13133128C2211yescharacteristic
27#16810532—21no—
28#16810532—21no—
29#16810524—21no—
30C4×C4⋊Dic53C2211yescharacteristic
31#16810385C2212yes—
32#16810385C2212yes—
33#16810385C2212yes—
34C53×#552—24no—
35C53×#552—24no—
36C2×#126×C53—21no—
37C2×#125×C53—21no—
38C2×#125×C53C2211yescharacteristic
39C2×#123×C53—21no—
40C2×C22⋊Q8×C53C2211yescharacteristic
41C2×C22⋊Q8×C53C2211yescharacteristic
42C2×C4×Q8×C53C2211yescharacteristic
43C2×C4×D4×C53C2211yescharacteristic
44C2×#117×C53—21no—
45C2×#117×C53C2211yescharacteristic
46C22×C4⋊C4×C53C2211yescharacteristic
47C22×D212—21no—
48C22×D212—21no—
49C22×D212C2312yes—
50C22×C22⋊D53—22no—
51C22×C22⋊D53—22no—
52C2×#11719924—24no—
53C2×#11719924—42no—
54C2×#11719924—42no—
55C22×#88322C2312yes—
56C22×#88322C2312yes—
57#13133214—22no—
58#13133214—22no—
59#13133213—44no—
60#13133191—24no—
61#13133191—24no—
62#13133191—42no—
63#13133191—42no—
64C2×C4⋊Dic53C2312yes—
65C2×#11719921—22no—
66C2×#11719921—22no—
67C2×C4×Dic53—21no—
68C4⋊#88322—22no—
69#13133141—24no—
70#13133137—24no—
71#13133137—24no—
72#13133137—24no—
73C22⋊#88322—42no—
74C22⋊#88322—42no—
75#13133128—22no—
76#13133124—21no—
77C22×Q8×C53C2311yescharacteristic
78C22×D4×C53D411yescharacteristic
79C23×C4×C53—21no—
80C23×C4×C53C2311yescharacteristic
81C23×C4×C53C2311yescharacteristic
82#126×C53—28no—
83#125×C53—28no—
84#125×C53—22no—
85#125×C53D414yescomplemented
86#123×C53—28no—
87C22⋊Q8×C53—22no—
88C22⋊Q8×C53—22no—
89C22⋊Q8×C53—22no—
90C22⋊Q8×C53—22no—
91C4×Q8×C53—22no—
92C4×Q8×C53D414yescomplemented
93C4×D4×C53—22no—
94C4×D4×C53—22no—
95#117×C53—28no—
96#117×C53—22no—
97#117×C53—22no—
98C2×C4⋊C4×C53—22no—
99C2×C4⋊C4×C53—22no—
100C2×C4⋊C4×C53—21no—
101C2×C4⋊C4×C53—21no—
102C2×C4⋊C4×C53—21no—
103C2×C4⋊C4×C53—21no—
104C2×C4⋊C4×C53—21no—
105C2×C4⋊C4×C53—21no—
106C2×C4⋊C4×C53D411yescharacteristic
107C2×C4⋊C4×C53C2311yescharacteristic
108C2×C4⋊C4×C53C2311yescharacteristic
109C2×C4⋊C4×C53C2311yescharacteristic
110C2×C22⋊C4×C53—21no—
111C2×C22⋊C4×C53—21no—
112C2×C22⋊C4×C53—21no—
113C2×C22⋊C4×C53D411yescharacteristic
114C2×C22⋊C4×C53D411yescharacteristic
115C2×C42×C53—21no—
116C2×C42×C53—21no—
117C2×C42×C53C2311yescharacteristic
118C24×C53C2×D411yescharacteristic
119C24×C53C2×D411yescharacteristic
120C23×D53—22no—
121C2×Q8×C53—21no—
122C2×Q8×C53—22no—
123C2×Q8×C53—22no—
124C2×Q8×C53C2×D412yes—
125C2×D4×C53—22no—
126C2×D4×C53—22no—
127C2×D4×C53—41no—
128C22×C4×C53—24no—
129C22×C4×C53—21no—
130C22×C4×C53—21no—
131C22×C4×C53—21no—
132C22×C4×C53—21no—
133C22×C4×C53—21no—
134C22×C4×C53—21no—
135C22×C4×C53—21no—
136C22×C4×C53—21no—
137C22×C4×C53—22no—
138C22×C4×C53—22no—
139C22×C4×C53—22no—
140C22×C4×C53—22no—
141C22×C4×C53—42no—
142C22×C4×C53—42no—
143C22×C4×C53C2×D411yescharacteristic
144C22×C4×C53C2×D411yescharacteristic
145C22×C4×C53C2×D411yescharacteristic
146C22×C4×C53C2×D411yescharacteristic
147C22×C4×C53C2411yescharacteristic
148C2×C22⋊D53—44no—
149C2×C22⋊D53—44no—
150C2×C22⋊D53—44no—
151C2×C22⋊D53—44no—
152C2×C22⋊D53—82no—
153C2×C22⋊D53—82no—
154C22×Dic53—22no—
155C2×D212—22no—
156C2×D212—22no—
157C2×D212—22no—
158C2×D212—22no—
159C2×D212—22no—
160C2×D212—22no—
161C2×D212—42no—
162C2×D212—42no—
163C2×D212—42no—
164C2×D212—42no—
165C2×D212—82no—
166C4⋊C4×C53—24no—
167C4⋊C4×C53—24no—
168C4⋊C4×C53—24no—
169C4⋊C4×C53—24no—
170C4⋊C4×C53—24no—
171C4⋊C4×C53—24no—
172C4⋊C4×C53—22no—
173C4⋊C4×C53—21no—
174C4⋊C4×C53—21no—
175C4⋊C4×C53—21no—
176C4⋊C4×C53C2×D412yes—
177C4⋊C4×C53C2×D412yes—
178C4⋊C4×C53C2×D412yes—
179C22⋊C4×C53—24no—
180C22⋊C4×C53—24no—
181C22⋊C4×C53—24no—
182C22⋊C4×C53—22no—
183C22⋊C4×C53—22no—
184C42×C53—24no—
185C42×C53—24no—
186C42×C53C2×D412yes—
187C42×C53C2×D412yes—
188#11719924—48no—
189#11719921—44no—
190#11719921—44no—
191C4⋊Dic53—22no—
192C4⋊Dic53#4114yes—
193C4×Dic53—42no—
194C2×#88322—22no—
195C2×#88322—22no—
196C2×#88322—22no—
197C2×#88322—22no—
198C2×#88322—22no—
199C2×#88322—22no—
200C23×C53—21no—
201C23×C53—22no—
202C23×C53—21no—
203C23×C53—22no—
204C23×C53—21no—
205C23×C53—22no—
206C23×C53—21no—
207C23×C53—21no—
208C23×C53—22no—
209C23×C53AGL(1,ℤ/8)12yes—
210C23×C53C22×D411yescharacteristic
211C22×D53—42no—
212C22×D53—42no—
213C22×D53—42no—
214C22×D53—42no—
215C22×D53—42no—
216C22×D53—42no—
217C22×D53—42no—
218Q8×C53—22no—
219Q8×C53—22no—
220Q8×C53—42no—
221D4×C53—44no—
222C2×C4×C53—21no—
223C2×C4×C53—21no—
224C2×C4×C53—22no—
225C2×C4×C53—22no—
226C2×C4×C53—22no—
227C2×C4×C53—21no—
228C2×C4×C53—21no—
229C2×C4×C53—22no—
230C2×C4×C53—22no—
231C2×C4×C53—22no—
232C2×C4×C53—22no—
233C2×C4×C53—48no—
234C2×C4×C53—42no—
235C2×C4×C53—41no—
236C2×C4×C53—41no—
237C2×C4×C53—41no—
238C2×C4×C53—41no—
239C2×C4×C53—42no—
240C2×C4×C53—41no—
241C2×C4×C53—41no—
242C2×C4×C53—41no—
243C2×C4×C53—41no—
244C2×C4×C53—42no—
245C2×C4×C53—42no—
246C2×C4×C53#12012yes—
247C2×C4×C53#12012yes—
248C2×C4×C53C22×D411yescharacteristic
249C2×C4×C53C22×D411yescharacteristic
250C2×C4×C53C2×#4112yes—
251C2×C4×C53H(5,𝔽2)12yes—
252C22⋊D53—88no—
253C22⋊D53—88no—
254C2×Dic53—24no—
255C2×Dic53—42no—
256C2×Dic53—42no—
257D212—42no—
258D212—42no—
259D212—42no—
260D212—42no—
261D212—84no—
262D212—84no—
263#88322—42no—
264#88322—42no—
265#25583—531nomaximal, Sylow
266C22×C53—22no—
267C22×C53—24no—
268C22×C53—41no—
269C22×C53—41no—
270C22×C53—42no—
271C22×C53—41no—
272C22×C53—41no—
273C22×C53—41no—
274C22×C53—42no—
275C22×C53—42no—
276C22×C53C2×#12011yescharacteristic
277C22×C53#54612yes—
278C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
279C22×C53#57511yescharacteristic
280C22×C53#57712yes—
281C2×D53—82no—
282C2×D53—82no—
283C2×D53—82no—
284C2×D53—82no—
285C2×D53—82no—
286C2×D53—82no—
287C2×D53—82no—
288C4×C53—22no—
289C4×C53—22no—
290C4×C53—41no—
291C4×C53—41no—
292C4×C53—42no—
293C4×C53—42no—
294C4×C53—42no—
295C4×C53—42no—
296C4×C53#53211yescharacteristic
297C4×C53#53511yescharacteristic
298C4×C53#55012yes—
299Dic53—44no—
300C2×#552D5311yescharacteristic, complemented
301C2×#490—531no—
302C2×#460—531no—
303#2541—531no—
304#2000—532no—
305#2000—532no—
306#1989—531no—
307#1955—531no—
308#1875—532no—
309#1874—532no—
310#1763—531no—
311C2×C53—42no—
312C2×C53—81no—
313C2×C53#243111yescharacteristic
314C2×C53#300611yescharacteristic
315C2×C53#301011yescharacteristic
316C2×C53#331012yes—
317C2×C53#331712yes—
318D53—162no—
319C22×SD16—532no—
320#552—24no—
321#552—24no—
322C2×#127—531no—
323C2×#127—1061no—
324C2×#126—21no—
325C2×#125—21no—
326C2×#125C2×D5311yescharacteristic
327C2×#123—21no—
328C2×C22⋊Q8C2×D5311yescharacteristic
329C2×C22⋊Q8C2×D5311yescharacteristic
330C2×#121—532no—
331C2×#121—532no—
332C2×C4×Q8C2×D5311yescharacteristic
333C2×C4×D4C2×D5311yescharacteristic
334C2×#117—21no—
335C2×#117C2×D5311yescharacteristic
336C22×C4⋊C4C2×D5311yescharacteristic
337#490—1064no—
338#460—1064no—
339C2×C4⋊C8—531no—
340C2×#103—532no—
341C2×#102—532no—
342C2×#102—532no—
343C2×#102—532no—
344C2×C22⋊C8—532no—
345C2×C22⋊C8—532no—
346C2×#97—531no—
347#392—1061no—
348#392—1061no—
349#390—1061no—
350C4×C4⋊C4—531no—
351#340—532no—
352#340—532no—
353#340—532no—
354C53#2558311yescharacteristic, complemented, Sylow
355C22×Q8C22×D5311yescharacteristic
356C22×D4—532no—
357C22×D4—1062no—
358C22×D4—1062no—
359C22×D4—1061no—
360C22×D4—1061no—
361C22×D4C22⋊D5311yescharacteristic
362C23×C4—21no—
363C23×C4C22×D5311yescharacteristic
364C23×C4C22×D5311yescharacteristic
365C2×SD16—1064no—
366C2×SD16—2122no—
367C2×SD16—2122no—
368C22×C8—532no—
369C22×C8—532no—
370#127—1062no—
371#127—1062no—
372#127—2124no—
373#126—28no—
374#125—28no—
375#125—22no—
376#125C22⋊D5314yescomplemented
377#123—28no—
378C22⋊Q8—22no—
379C22⋊Q8—22no—
380C22⋊Q8—22no—
381C22⋊Q8—22no—
382#121—1064no—
383#121—1064no—
384#121—2122no—
385#121—2122no—
386C4×Q8—22no—
387C4×Q8C22⋊D5314yescomplemented
388C4×D4—22no—
389C4×D4—22no—
390#117—28no—
391#117—22no—
392#117—22no—
393C2×C4⋊C4—21no—
394C2×C4⋊C4—21no—
395C2×C4⋊C4—21no—
396C2×C4⋊C4—21no—
397C2×C4⋊C4—21no—
398C2×C4⋊C4—21no—
399C2×C4⋊C4—22no—
400C2×C4⋊C4—22no—
401C2×C4⋊C4—532no—
402C2×C4⋊C4C22⋊D5311yescharacteristic
403C2×C4⋊C4C22×D5311yescharacteristic
404C2×C4⋊C4C22×D5311yescharacteristic
405C2×C4⋊C4C22×D5311yescharacteristic
406C2×C22⋊C4—21no—
407C2×C22⋊C4—21no—
408C2×C22⋊C4—21no—
409C2×C22⋊C4—1062no—
410C2×C22⋊C4—1062no—
411C2×C22⋊C4C22⋊D5311yescharacteristic
412C2×C22⋊C4C22⋊D5311yescharacteristic
413C2×C42—21no—
414C2×C42—21no—
415C2×C42—1061no—
416C2×C42C22×D5311yescharacteristic
417C4⋊C8—1062no—
418#103—1064no—
419#102—1064no—
420#102—1064no—
421#102—1064no—
422C22⋊C8—2122no—
423C22⋊C8—2122no—
424#97—1062no—
425#95—1061no—
426C24—1062no—
427C24C2×C22⋊D5311yescharacteristic
428C24C2×C22⋊D5311yescharacteristic
429C2×Q8—21no—
430C2×Q8—22no—
431C2×Q8—22no—
432C2×Q8C2×C22⋊D5312yes—
433C2×D4—22no—
434C2×D4—22no—
435C2×D4—41no—
436C2×D4—1062no—
437C2×D4—1062no—
438C2×D4—1062no—
439C2×D4—1062no—
440C2×D4—1062no—
441C2×D4—1062no—
442C2×D4—4242no—
443C2×D4—4242no—
444C2×D4—4242no—
445C2×D4—2122no—
446C2×D4—2122no—
447C2×D4—2124no—
448C2×D4—2124no—
449C2×D4—2124no—
450C2×D4—2124no—
451C2×D4—2122no—
452C2×D4—2122no—
453C22×C4—24no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—42no—
467C22×C4—42no—
468C22×C4—1062no—
469C22×C4C2×C22⋊D5311yescharacteristic
470C22×C4C2×C22⋊D5311yescharacteristic
471C22×C4C2×C22⋊D5311yescharacteristic
472C22×C4C2×C22⋊D5311yescharacteristic
473C22×C4C23×D5311yescharacteristic
474SD16—2128no—
475C2×C8—1062no—
476C2×C8—1062no—
477C2×C8—1062no—
478C2×C8—1062no—
479C2×C8—1062no—
480C2×C8—1062no—
481C4⋊C4—24no—
482C4⋊C4—24no—
483C4⋊C4—24no—
484C4⋊C4—24no—
485C4⋊C4—24no—
486C4⋊C4—24no—
487C4⋊C4—22no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—534no—
492C4⋊C4—1062no—
493C4⋊C4C2×C22⋊D5312yes—
494C4⋊C4C2×C22⋊D5312yes—
495C4⋊C4C2×C22⋊D5312yes—
496C22⋊C4—24no—
497C22⋊C4—24no—
498C22⋊C4—24no—
499C22⋊C4—22no—
500C22⋊C4—22no—
501C22⋊C4—2124no—
502C22⋊C4—2124no—
503C42—24no—
504C42—24no—
505C42—2122no—
506C42C2×C22⋊D5312yes—
507C42C2×C22⋊D5312yes—
508C23—21no—
509C23—21no—
510C23—22no—
511C23—22no—
512C23—21no—
513C23—21no—
514C23—21no—
515C23—22no—
516C23—22no—
517C23—2122no—
518C23—2122no—
519C23—2122no—
520C23—2122no—
521C23—2122no—
522C23—2122no—
523C23—2122no—
524C23#1313324212yes—
525C23C22×C22⋊D5311yescharacteristic
526Q8—22no—
527Q8—22no—
528Q8—42no—
529D4—44no—
530D4—4244no—
531D4—4244no—
532D4—4248no—
533D4—4248no—
534D4—2122no—
535D4—2122no—
536D4—2122no—
537D4—2122no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—22no—
541C2×C4—21no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—21no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—48no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—1064no—
563C2×C4—2122no—
564C2×C4—2122no—
565C2×C4#1313318612yes—
566C2×C4#1313318612yes—
567C2×C4C22×C22⋊D5311yescharacteristic
568C2×C4C22×C22⋊D5311yescharacteristic
569C2×C4#41×D5312yes—
570C2×C4#1313326312yes—
571C8—2122no—
572C8—2122no—
573C22—24no—
574C22—22no—
575C22—41no—
576C22—41no—
577C22—42no—
578C22—42no—
579C22—41no—
580C22—41no—
581C22—41no—
582C22—42no—
583C22—4242no—
584C22—4242no—
585C22—4242no—
586C22—4242no—
587C22—4242no—
588C22—4242no—
589C22—4242no—
590C22C2×#1313318611yescharacteristic
591C22#1681126112yes—
592C22C2×#1313324211yescharacteristic
593C22#1681142111yescharacteristic
594C22#1681143112yes—
595C4—22no—
596C4—22no—
597C4—41no—
598C4—41no—
599C4—42no—
600C4—42no—
601C4—42no—
602C4—42no—
603C4—2124no—
604C4#1681116911yescharacteristic
605C4#1681118711yescharacteristic
606C4#1681129212yes—
607C2—42no—
608C2—81no—
609C2—8482no—
610C2#2027833011yescharacteristic
611C2#2028396711yescharacteristic
612C2#2028400111yescharacteristic
613C2#2028673012yes—
614C2#2028683512yes—
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC2×C53×#552C2
socleC23×C53C22×D4
prime radical#20279661C2
hypercenterC2×#552D53
hypocenterC53#25583
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×#552

Complements

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

NH ≅ G/Ntype
1. C2×#16810989C2semidirect
2. C2×#16810823C2semidirect
4. #20275716C2semidirect
5. #20275716C2semidirect
6. #20275673C2semidirect
7. #20275514C2semidirect
8. #20275037C2semidirect
9. #20275032C2semidirect
10. #20274545C2semidirect
11. C2×C53×#552C2semidirect
85. #125×C53D4semidirect
92. C4×Q8×C53D4semidirect
300. C2×#552D53semidirect
354. C53#25583semidirect
376. #125C22⋊D53semidirect
387. C4×Q8C22⋊D53semidirect

Maximal subgroups

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

subgroupindex
C2×#168109892
C2×#168108232
#202796612
#202757162
#202756732
#202755142
#202750372
#202750322
#202745452
C2×C53×#5522
#2558353