Finite Group Databasebeta

← #27389677

Subgroups — #27389677

Order 13568.

1880 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13133179C211yesmaximal, characteristic
2C2×#16810773C211yesmaximal, characteristic
3C2×#16810708C212yesmaximal, complemented
4C22⋊Q8×Dic53C211yesmaximal, characteristic, complemented
5#20275046C212yesmaximal, complemented
6#20274977C212yesmaximal, complemented
7#20274908C211yesmaximal, characteristic, complemented
8#20274647C211yesmaximal, characteristic
9#20274529C212yesmaximal, complemented
10C22⋊C8×Dic53C211yesmaximal, characteristic, complemented
11C53×#1872C211yesmaximal, characteristic, complemented
12C23×D212C2211yescharacteristic
13C23×C4×D53—21no—
14C22×#11719924C2212yes—
15C2×Q8×Dic53C2211yescharacteristic, complemented
16C2×#13133179—28no—
17C2×#13133179—21no—
18C2×#13133179—21no—
19C2×#13133179C2212yes—
20C2×#13133179C2212yes—
21C22×#11719921—21no—
22C22×C4×Dic53C2211yescharacteristic
23#16810773C418yescomplemented
24#16810708—28no—
25C2×#13133148C2212yescomplemented
26C2×#13133139C2212yes—
27C2×#13133139C2212yes—
28C2×#13133137C2212yes—
29C2×C22⋊#88322C2211yescharacteristic
30C2×C8×Dic53C2212yescomplemented
31#16810530C2211yescharacteristic, complemented
32#16810522C2211yescharacteristic, complemented
33#16810509—21no—
34#16810498—21no—
35#16810483C2211yescharacteristic
36C4⋊C4×Dic53—21no—
37C4⋊C4×Dic53C2211yescharacteristic, complemented
38C22⋊C4×Dic53—21no—
39#16810385C2212yescomplemented
40#16810377C2212yescomplemented
41C2×C22⋊Q8×C53C2211yescharacteristic
42C22×C4⋊C4×C53C2211yescharacteristic
43C2×#103×C53C2212yescomplemented
44C2×C22⋊C8×C53C2211yescharacteristic
45C53×#340C2212yescomplemented
46C24×D53—21no—
47C22×D212—21no—
48C22×D212—22no—
49C22×D212—24no—
50C22×D212—24no—
51C22×D212C2×C414yes—
52C22×D212C2312yes—
53C23×Dic53D411yescharacteristic
54C23×Dic53D411yescharacteristic
55C22×C4×D53—28no—
56C22×C4×D53—22no—
57C22×C4×D53—22no—
58C22×C4×D53—44no—
59C22×C4×D53—44no—
60C2×#11719924—24no—
61C2×#11719924—24no—
62C2×#11719924C2×C418yescomplemented
63C22×#88322C2312yes—
64Q8×Dic53—24no—
65#13133179—28no—
66#13133179—28no—
67#13133179—24no—
68#13133179—24no—
69#13133179—24no—
70#13133179D418yes—
71C2×C4⋊Dic53—21no—
72C2×C4⋊Dic53—21no—
73C2×C4⋊Dic53C2311yescharacteristic
74C2×C4⋊Dic53C2311yescharacteristic
75C2×C22⋊Dic53—21no—
76C2×#11719921—28no—
77C2×#11719921—22no—
78C2×#11719921—22no—
79C2×C4×Dic53—24no—
80C2×C4×Dic53—21no—
81C2×C4×Dic53—21no—
82C2×C4×Dic53D411yescharacteristic
83C2×C4×Dic53D411yescharacteristic
84C2×C4×Dic53C2311yescharacteristic
85C2×C4×Dic53C2311yescharacteristic
86#13133148—24no—
87#13133139—24no—
88#13133139—24no—
89#13133137C2×C418yescomplemented
90C22⋊#88322C2×C414yescomplemented
91C8×Dic53—24no—
92#13133124—21no—
93#13133124—21no—
94#13133124—21no—
95#13133124—21no—
96#13133124—21no—
97C22×Q8×C53C2311yescharacteristic
98C23×C4×C53—21no—
99C23×C4×C53C2311yescharacteristic
100C22×C8×C53C2312yes—
101C22⋊Q8×C53—22no—
102C22⋊Q8×C53C2×C414yescomplemented
103C2×C4⋊C4×C53—24no—
104C2×C4⋊C4×C53—21no—
105C2×C4⋊C4×C53—21no—
106C2×C4⋊C4×C53C2311yescharacteristic
107C2×C4⋊C4×C53C2312yes—
108C2×C22⋊C4×C53—21no—
109#103×C53—24no—
110C22⋊C8×C53—22no—
111C24×C53C2×D411yescharacteristic
112C23×D53—24no—
113C23×D53—24no—
114C23×D53—24no—
115C23×D53—22no—
116C23×D53—42no—
117C23×D53—42no—
118C23×D53—42no—
119C23×D53—42no—
120C2×Q8×C53—21no—
121C2×Q8×C53—22no—
122C2×Q8×C53—22no—
123C2×Q8×C53C22×C412yescomplemented
124C22×C4×C53—22no—
125C22×C4×C53—22no—
126C22×C4×C53—24no—
127C22×C4×C53—21no—
128C22×C4×C53—21no—
129C22×C4×C53—22no—
130C22×C4×C53—42no—
131C22×C4×C53—42no—
132C22×C4×C53C22×C412yes—
133C22×C4×C53C2×D411yescharacteristic
134C22×C4×C53#4112yes—
135C22×C4×C53C2411yescharacteristic
136C22×Dic53—22no—
137C22×Dic53—22no—
138C22×Dic53—22no—
139C22×Dic53—22no—
140C22×Dic53—21no—
141C22×Dic53—21no—
142C22×Dic53—41no—
143C22×Dic53SD1614yescomplemented
144C22×Dic53C2×D411yescharacteristic
145C22×Dic53C2×D411yescharacteristic
146C22×Dic53C2×D411yescharacteristic
147C22×Dic53C2×D411yescharacteristic
148C2×D212—24no—
149C2×D212—24no—
150C2×D212—216no—
151C2×D212—24no—
152C2×D212—22no—
153C2×D212—22no—
154C2×D212—22no—
155C2×D212—216no—
156C2×D212C22×C414yes—
157C2×D212C2×D414yes—
158C2×D212#4114yes—
159C2×C4×D53—28no—
160C2×C4×D53—28no—
161C2×C4×D53—44no—
162C2×C4×D53—432no—
163C2×C4×D53—44no—
164C2×C4×D53—44no—
165C2×C4×D53—44no—
166C2×C8×C53—22no—
167C2×C8×C53—22no—
168C2×C8×C53—22no—
169C4⋊C4×C53—24no—
170C4⋊C4×C53—24no—
171C4⋊C4×C53—22no—
172C4⋊C4×C53—21no—
173C4⋊C4×C53C22×C412yescomplemented
174C4⋊C4×C53C2×D414yes—
175C22⋊C4×C53—24no—
176C22⋊Dic53—42no—
177#11719924—216no—
178#11719921—28no—
179#11719921—28no—
180C4⋊Dic53—22no—
181C4⋊Dic53—22no—
182C4⋊Dic53—42no—
183C4⋊Dic53—42no—
184C4×Dic53—22no—
185C4×Dic53—22no—
186C4×Dic53—22no—
187C4×Dic53—42no—
188C4×Dic53—42no—
189C4×Dic53C2×D414yes—
190C2×#88322—22no—
191C2×#88322—22no—
192C2×#88322C22×C414yescomplemented
193C23×C53—21no—
194C23×C53—21no—
195C23×C53—21no—
196C23×C53—21no—
197C23×C53C4×D412yes—
198C23×C53C2×SD1612yes—
199C23×C53#13512yes—
200C23×C53C22×D411yescharacteristic
201C22×D53—216no—
202C22×D53—24no—
203C22×D53—24no—
204C22×D53—24no—
205C22×D53—48no—
206C22×D53—48no—
207C22×D53—48no—
208C22×D53—48no—
209C22×D53—48no—
210C22×D53—42no—
211C22×D53—42no—
212C22×D53—42no—
213C22×D53—42no—
214C22×D53—48no—
215Q8×C53—24no—
216Q8×C53—42no—
217C2×C4×C53—21no—
218C2×C4×C53—21no—
219C2×C4×C53—24no—
220C2×C4×C53—24no—
221C2×C4×C53—22no—
222C2×C4×C53—22no—
223C2×C4×C53—24no—
224C2×C4×C53—48no—
225C2×C4×C53—41no—
226C2×C4×C53—41no—
227C2×C4×C53—42no—
228C2×C4×C53—42no—
229C2×C4×C53—41no—
230C2×C4×C53—41no—
231C2×C4×C53C4×D412yes—
232C2×C4×C53C23×C411yescharacteristic
233C2×C4×C53C22×D412yes—
234C2×C4×C53C2×#4112yes—
235C2×C4×C53C2×#4111yescharacteristic
236C2×Dic53—21no—
237C2×Dic53—21no—
238C2×Dic53—21no—
239C2×Dic53—21no—
240C2×Dic53—28no—
241C2×Dic53—21no—
242C2×Dic53—21no—
243C2×Dic53—21no—
244C2×Dic53—44no—
245C2×Dic53—41no—
246C2×Dic53—41no—
247C2×Dic53—41no—
248C2×Dic53—41no—
249C2×Dic53—41no—
250C2×Dic53—41no—
251C2×Dic53#12012yes—
252C2×Dic53#12012yes—
253C2×Dic53C2×SD1612yes—
254C2×Dic53C2×SD1612yes—
255C2×Dic53AGL(1,ℤ/8)12yes—
256D212—216no—
257D212—24no—
258D212—24no—
259D212—24no—
260D212C4×D418yes—
261C4×D53—416no—
262C4×D53—416no—
263C8×C53—42no—
264#88322—24no—
265C4×#450—531nomaximal, Sylow
266C22×C53—22no—
267C22×C53—22no—
268C22×C53—22no—
269C22×C53—22no—
270C22×C53—22no—
271C22×C53—22no—
272C22×C53C4×SD1614yescomplemented
273C22×C53C2×C4×D411yescharacteristic
274C22×C53C2×#12011yescharacteristic
275C22×C53#52511yescharacteristic
276C22×C53C22×SD1611yescharacteristic
277C22×C53C2×#13511yescharacteristic
278C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
279C22×C53#57511yescharacteristic
280C2×D53—28no—
281C2×D53—432no—
282C2×D53—44no—
283C2×D53—44no—
284C2×D53—44no—
285C2×D53—44no—
286C2×D53—44no—
287C2×D53—44no—
288C4×C53—22no—
289C4×C53—42no—
290C4×C53—44no—
291C4×C53—42no—
292C4×C53C2×C4×D412yes—
293C4×C53#52512yes—
294Dic53—22no—
295Dic53—22no—
296Dic53—41no—
297Dic53—81no—
298Dic53—81no—
299Dic53#45014yescomplemented
300C22×C4×D4—531no—
301C2×#450—531no—
302C2×C4×SD16—532no—
303C4×C22⋊Q8—531no—
304#1877—532no—
305#1872D5311yescharacteristic, complemented
306#1864—532no—
307#1853—531no—
308#1786—531no—
309C4×#102—532no—
310C4×C22⋊C8—531no—
311C2×C53—24no—
312C2×C53C4×#12011yescharacteristic
313C2×C53C2×C4×SD1611yescharacteristic
314C2×C53C4×AGL(1,ℤ/8)11yescharacteristic
315C2×C53C2×#45011yescharacteristic
316C2×C53#300211yescharacteristic
317C2×C53#310111yescharacteristic
318C2×C53#310511yescharacteristic
319D53—48no—
320C23×D4—531no—
321C24×C4—1061no—
322C22×SD16—532no—
323C2×C22⋊Q8C2×D5311yescharacteristic
324C2×C4×Q8—531no—
325C2×C4×D4—532no—
326C2×C4×D4—532no—
327C2×C4×D4—1061no—
328C2×C4×D4—1061no—
329C2×C4×D4—1068no—
330C22×C4⋊C4C2×D5311yescharacteristic
331C22×C22⋊C4—1061no—
332C22×C42—531no—
333#450—538no—
334C4×SD16—1068no—
335C2×#106—532no—
336C2×#103C2×D5312yescomplemented
337C2×#102—532no—
338C2×#102—532no—
339C2×#102—532no—
340C2×C22⋊C8—531no—
341C2×C22⋊C8C2×D5311yescharacteristic
342C2×C4×C8—532no—
343#391—531no—
344#389—531no—
345#387—1061no—
346#385—1061no—
347#380—531no—
348C4×C4⋊C4—531no—
349C4×C4⋊C4—1061no—
350C4×C22⋊C4—1061no—
351#340—532no—
352#340C2×D5312yescomplemented
353#336—532no—
354C53C4×#45011yescharacteristic, complemented, Sylow
355C25—1061no—
356C22×Q8C22×D5311yescharacteristic
357C22×D4—534no—
358C22×D4—532no—
359C22×D4—1064no—
360C22×D4—1064no—
361C22×D4—1061no—
362C22×D4—1062no—
363C23×C4—21no—
364C23×C4—531no—
365C23×C4—531no—
366C23×C4—1062no—
367C23×C4—1062no—
368C23×C4—1068no—
369C23×C4—2124no—
370C23×C4—2124no—
371C23×C4C22×D5311yescharacteristic
372C2×SD16—538no—
373C2×SD16—1064no—
374C2×SD16—1064no—
375C22×C8—532no—
376C22×C8C22×D5312yes—
377C22⋊Q8—22no—
378C22⋊Q8C4×D5314yescomplemented
379C4×Q8—1064no—
380C4×D4—538no—
381C4×D4—1064no—
382C4×D4—1064no—
383C4×D4—1068no—
384C4×D4—1068no—
385C4×D4—1064no—
386C2×C4⋊C4—21no—
387C2×C4⋊C4—21no—
388C2×C4⋊C4—24no—
389C2×C4⋊C4—531no—
390C2×C4⋊C4—531no—
391C2×C4⋊C4—1061no—
392C2×C4⋊C4—1061no—
393C2×C4⋊C4C22×D5311yescharacteristic
394C2×C4⋊C4C22×D5312yes—
395C2×C22⋊C4—21no—
396C2×C22⋊C4—1062no—
397C2×C22⋊C4—1062no—
398C2×C22⋊C4—1068no—
399C2×C22⋊C4—1061no—
400C2×C42—531no—
401C2×C42—531no—
402C2×C42—531no—
403C2×C42—531no—
404C2×C42—1064no—
405C2×C42—1061no—
406C2×C42—1061no—
407#106—1064no—
408#103—24no—
409#102—538no—
410#102—1064no—
411#102—1064no—
412C22⋊C8—22no—
413C22⋊C8—534no—
414C4×C8—1064no—
415#95—1061no—
416#95—1061no—
417#95—1061no—
418#95—1061no—
419#95—1061no—
420C24—1064no—
421C24—1064no—
422C24—1064no—
423C24—1062no—
424C24—2122no—
425C24—2122no—
426C24—2122no—
427C24—2122no—
428C24D4×D5311yescharacteristic
429C2×Q8—21no—
430C2×Q8—22no—
431C2×Q8—22no—
432C2×Q8C2×C4×D5312yescomplemented
433C2×D4—534no—
434C2×D4—534no—
435C2×D4—534no—
436C2×D4—10616no—
437C2×D4—1064no—
438C2×D4—1064no—
439C2×D4—10616no—
440C2×D4—1064no—
441C2×D4—1062no—
442C2×D4—1062no—
443C2×D4—1062no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—24no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—22no—
450C22×C4—42no—
451C22×C4—42no—
452C22×C4—531no—
453C22×C4—534no—
454C22×C4—531no—
455C22×C4—531no—
456C22×C4—531no—
457C22×C4—1068no—
458C22×C4—1068no—
459C22×C4—1062no—
460C22×C4—1062no—
461C22×C4—1062no—
462C22×C4—1062no—
463C22×C4—1061no—
464C22×C4—1061no—
465C22×C4—2124no—
466C22×C4—2124no—
467C22×C4—2124no—
468C22×C4—21232no—
469C22×C4—2124no—
470C22×C4—2121no—
471C22×C4C2×C4×D5312yes—
472C22×C4D4×D5311yescharacteristic
473C22×C4#1171994712yes—
474C22×C4C23×D5311yescharacteristic
475SD16—10616no—
476C2×C8—22no—
477C2×C8—22no—
478C2×C8—22no—
479C2×C8—534no—
480C2×C8—1062no—
481C2×C8—1062no—
482C4⋊C4—24no—
483C4⋊C4—24no—
484C4⋊C4—22no—
485C4⋊C4—21no—
486C4⋊C4—1062no—
487C4⋊C4—1062no—
488C4⋊C4—2122no—
489C4⋊C4—2122no—
490C4⋊C4C2×C4×D5312yescomplemented
491C4⋊C4D4×D5314yes—
492C22⋊C4—24no—
493C22⋊C4—1068no—
494C22⋊C4—1068no—
495C22⋊C4—2122no—
496C42—534no—
497C42—1062no—
498C42—1062no—
499C42—1062no—
500C42—2122no—
501C42—2122no—
502C23—21no—
503C23—21no—
504C23—21no—
505C23—21no—
506C23—10616no—
507C23—1064no—
508C23—1064no—
509C23—1064no—
510C23—2128no—
511C23—2128no—
512C23—2128no—
513C23—2128no—
514C23—2128no—
515C23—2122no—
516C23—2122no—
517C23—2122no—
518C23—2122no—
519C23—2128no—
520C23#1313317712yes—
521C23SD16×D5312yes—
522C23#1313324012yes—
523C23C2×D4×D5311yescharacteristic
524Q8—24no—
525Q8—42no—
526D4—538no—
527D4—10616no—
528D4—1064no—
529D4—1064no—
530D4—1064no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—24no—
538C2×C4—48no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—42no—
542C2×C4—41no—
543C2×C4—42no—
544C2×C4—41no—
545C2×C4—532no—
546C2×C4—532no—
547C2×C4—532no—
548C2×C4—532no—
549C2×C4—532no—
550C2×C4—1068no—
551C2×C4—1061no—
552C2×C4—1061no—
553C2×C4—1061no—
554C2×C4—1061no—
555C2×C4—1061no—
556C2×C4—1061no—
557C2×C4—1061no—
558C2×C4—21216no—
559C2×C4—21216no—
560C2×C4—2124no—
561C2×C4—2121no—
562C2×C4—2121no—
563C2×C4—2121no—
564C2×C4—2121no—
565C2×C4—2121no—
566C2×C4—2121no—
567C2×C4#1313317712yes—
568C2×C4C22×C4×D5311yescharacteristic
569C2×C4C2×D4×D5312yes—
570C2×C4#41×D5312yes—
571C2×C4C2×#1171994711yescharacteristic
572C8—42no—
573C8—1064no—
574C22—22no—
575C22—22no—
576C22—22no—
577C22—22no—
578C22—22no—
579C22—22no—
580C22—1068no—
581C22—21232no—
582C22—2124no—
583C22—2124no—
584C22—2124no—
585C22—2124no—
586C22—2124no—
587C22—2124no—
588C22#1681070814yescomplemented
589C22C2×#1313317711yescharacteristic
590C22#120×D5311yescharacteristic
591C22#1681113111yescharacteristic
592C22C2×SD16×D5311yescharacteristic
593C22C2×#1313324011yescharacteristic
594C22AGL(1,ℤ/8)×D5311yescharacteristic
595C22#1681142211yescharacteristic
596C4—22no—
597C4—42no—
598C4—42no—
599C4—44no—
600C4—534no—
601C4—1062no—
602C4—1062no—
603C4—4241no—
604C4—4241no—
605C4—2121no—
606C4C4×D4×D5312yes—
607C4#1681113312yes—
608C2—24no—
609C2—2128no—
610C2#2027722011yescharacteristic
611C2C2×#1681070811yescharacteristic
612C2#2028352811yescharacteristic
613C2#450×D5311yescharacteristic
614C2#2028393811yescharacteristic
615C2#2028472011yescharacteristic
616C2#2028476511yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#1872C2
socleC23×C53C22×D4
prime radicalC23×D212C22
cosocleC22×C4×C53C24
hypercenter#1872D53
hypocenterC53C4×#450
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1872

Complements

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

NH ≅ G/Ntype
3. C2×#16810708C2semidirect
4. C22⋊Q8×Dic53C2semidirect
5. #20275046C2semidirect
6. #20274977C2semidirect
7. #20274908C2semidirect
9. #20274529C2semidirect
10. C22⋊C8×Dic53C2semidirect
11. C53×#1872C2semidirect
15. C2×Q8×Dic53C22semidirect
23. #16810773C4semidirect
25. C2×#13133148C22semidirect
30. C2×C8×Dic53C22semidirect
31. #16810530C22semidirect
32. #16810522C22semidirect
37. C4⋊C4×Dic53C22semidirect
39. #16810385C22semidirect
40. #16810377C22semidirect
43. C2×#103×C53C22semidirect
45. C53×#340C22semidirect
62. C2×#11719924C2×C4semidirect
89. #13133137C2×C4semidirect
90. C22⋊#88322C2×C4semidirect
102. C22⋊Q8×C53C2×C4semidirect
123. C2×Q8×C53C22×C4semidirect
143. C22×Dic53SD16semidirect
173. C4⋊C4×C53C22×C4semidirect
192. C2×#88322C22×C4semidirect
272. C22×C53C4×SD16semidirect
299. Dic53#450semidirect
305. #1872D53semidirect
336. C2×#103C2×D53semidirect
352. #340C2×D53semidirect
354. C53C4×#450semidirect
378. C22⋊Q8C4×D53semidirect
432. C2×Q8C2×C4×D53semidirect
490. C4⋊C4C2×C4×D53semidirect
588. C22#16810708semidirect

Maximal subgroups

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

subgroupindex
C22×#131331792
C2×#168107732
C2×#168107082
C22⋊Q8×Dic532
#202750462
#202749772
#202749082
#202746472
#202745292
C22⋊C8×Dic532
C53×#18722
C4×#45053