Finite Group Databasebeta

← #27418443

Subgroups — #27418443

Order 13568.

1394 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811173C211yesmaximal, characteristic
2C2×#16810908C211yesmaximal, characteristic, complemented
3C2×#16810823C211yesmaximal, characteristic, complemented
4#20275904C212yesmaximal, complemented
5#20275716C212yesmaximal, complemented
6#20275667C211yesmaximal, characteristic, complemented
7#20275642C211yesmaximal, characteristic, complemented
8#20275052C212yesmaximal, complemented
9#20274975C212yesmaximal, complemented
10#20274695C211yesmaximal, characteristic, complemented
11C4×C22⋊Q8×C53C211yesmaximal, characteristic, complemented
12C22×#11719945—21no—
13C22×#11719924C2212yes—
14#16811173—24no—
15#16811173—24no—
16C2×#13133217C2211yescharacteristic
17C2×#13133214C2211yescharacteristic
18C2×#13133205—21no—
19C2×#13133191—21no—
20C2×#13133191C2212yes—
21C2×C4×D212—21no—
22C2×C4×D212C2212yes—
23#16810908—24no—
24#16810823—24no—
25C2×#13133148C2212yes—
26C2×C4⋊#88322C2211yescharacteristic
27C2×C4⋊#88322C2211yescharacteristic
28C2×#13133141C2212yes—
29C2×#13133137C2212yes—
30C2×#13133137C2212yes—
31C2×#13133137C2212yes—
32C2×C22⋊#88322C2212yes—
33#16810385C2212yes—
34#16810385C2212yes—
35#16810378C2212yes—
36C2×C22⋊Q8×C53C2211yescharacteristic
37C2×C4×Q8×C53C2211yescharacteristic
38C22×C42×C53C2211yescharacteristic
39C53×#391C2211yescharacteristic
40C53×#389C2211yescharacteristic
41C53×#387—21no—
42C53×#385—21no—
43C53×#380C2211yescharacteristic
44C4×C4⋊C4×C53—21no—
45C4×C4⋊C4×C53C2211yescharacteristic
46C4×C22⋊C4×C53—21no—
47C2×#11719945—28no—
48C2×#11719945—44no—
49C2×#11719945—44no—
50C22×Dic106—21no—
51C22×D212—21no—
52C22×D212—21no—
53C22×D212C2312yes—
54C22×C22⋊D53—22no—
55C22×C4×D53—22no—
56C2×#11719924—24no—
57C2×#11719924—24no—
58C2×#11719924—42no—
59C22×#88322C2312yes—
60C22×#88322C2312yes—
61#13133217—22no—
62#13133217D414yes—
63#13133214—22no—
64#13133214D414yes—
65#13133205—28no—
66#13133191—28no—
67#13133191—24no—
68#13133191—24no—
69C4×D212—28no—
70C4×D212—24no—
71C4×D212—24no—
72C2×C4⋊Dic53—21no—
73C2×C4⋊Dic53C2312yes—
74C2×#11719921—22no—
75C2×#11719921—22no—
76#13133148—24no—
77C4⋊#88322—22no—
78C4⋊#88322—22no—
79#13133141—24no—
80#13133137—24no—
81#13133137—24no—
82#13133137—42no—
83C22⋊#88322—42no—
84C22×Q8×C53C2311yescharacteristic
85C23×C4×C53D411yescharacteristic
86C23×C4×C53D411yescharacteristic
87C23×C4×C53C2311yescharacteristic
88C22⋊Q8×C53—22no—
89C22⋊Q8×C53—22no—
90C4×Q8×C53—22no—
91C4×Q8×C53—22no—
92C2×C4⋊C4×C53—21no—
93C2×C4⋊C4×C53—21no—
94C2×C4⋊C4×C53—21no—
95C2×C4⋊C4×C53C2311yescharacteristic
96C2×C4⋊C4×C53C2311yescharacteristic
97C2×C4⋊C4×C53C2311yescharacteristic
98C2×C22⋊C4×C53—21no—
99C2×C22⋊C4×C53—21no—
100C2×C42×C53—22no—
101C2×C42×C53—22no—
102C2×C42×C53—21no—
103C2×C42×C53—21no—
104C2×C42×C53D411yescharacteristic
105C2×C42×C53D411yescharacteristic
106C2×C42×C53C2311yescharacteristic
107C2×C42×C53C2311yescharacteristic
108#95×C53—21no—
109#95×C53—21no—
110#95×C53—21no—
111#95×C53—21no—
112#95×C53—21no—
113C24×C53C2×D411yescharacteristic
114C23×D53—22no—
115C2×Q8×C53—21no—
116C2×Q8×C53—21no—
117C2×Q8×C53—21no—
118C2×Q8×C53—21no—
119C2×Q8×C53—21no—
120C2×Q8×C53#4112yescomplemented
121C22×C4×C53—22no—
122C22×C4×C53—22no—
123C22×C4×C53—21no—
124C22×C4×C53—21no—
125C22×C4×C53—22no—
126C22×C4×C53—21no—
127C22×C4×C53—21no—
128C22×C4×C53—21no—
129C22×C4×C53—21no—
130C22×C4×C53—21no—
131C22×C4×C53—21no—
132C22×C4×C53—21no—
133C22×C4×C53—21no—
134C22×C4×C53—41no—
135C22×C4×C53—41no—
136C22×C4×C53SD1614yescomplemented
137C22×C4×C53C2×D411yescharacteristic
138C22×C4×C53C2×D411yescharacteristic
139C22×C4×C53C2×D411yescharacteristic
140C22×C4×C53C2×D411yescharacteristic
141C22×C4×C53C2×D411yescharacteristic
142C22×C4×C53C2411yescharacteristic
143C2×C22⋊D53—28no—
144C2×C22⋊D53—44no—
145C2×C22⋊D53—44no—
146C22×Dic53—22no—
147#11719945—432no—
148C2×D212—22no—
149C2×D212—22no—
150C2×D212—22no—
151C2×D212—22no—
152C2×D212—24no—
153C2×D212—24no—
154C2×D212—22no—
155C2×D212—42no—
156C2×D212—42no—
157C2×D212—42no—
158C2×D212—42no—
159C2×D212C2×D414yes—
160C2×C4×D53—28no—
161C2×C4×D53—44no—
162C2×C4×D53—44no—
163C2×Dic106—24no—
164C2×Dic106—42no—
165C2×Dic106—42no—
166C4⋊C4×C53—24no—
167C4⋊C4×C53—21no—
168C4⋊C4×C53—21no—
169C4⋊C4×C53—21no—
170C4⋊C4×C53—21no—
171C4⋊C4×C53—21no—
172C4⋊C4×C53—42no—
173C4⋊C4×C53—42no—
174C4⋊C4×C53#4112yescomplemented
175C22⋊C4×C53—24no—
176C22⋊C4×C53—42no—
177C42×C53—22no—
178C42×C53—22no—
179C42×C53—21no—
180C42×C53—21no—
181C42×C53—42no—
182C42×C53—42no—
183C42×C53C2×D412yes—
184C42×C53C2×D412yes—
185#11719924—48no—
186#11719921—28no—
187#11719921—28no—
188C4⋊Dic53—24no—
189C4⋊Dic53—22no—
190C4⋊Dic53C2×D414yes—
191C2×#88322—22no—
192C2×#88322—22no—
193C2×#88322—22no—
194C2×#88322—22no—
195C2×#88322—22no—
196C2×#88322—22no—
197C23×C53—21no—
198C23×C53—21no—
199C23×C53—21no—
200C23×C53—21no—
201C23×C53—21no—
202C23×C53—21no—
203C23×C53C2×SD1612yes—
204C23×C53C22×D411yescharacteristic
205C22×D53—24no—
206C22×D53—42no—
207C22×D53—42no—
208C22×D53—42no—
209C22×D53—42no—
210C22×D53—42no—
211C22×D53—42no—
212Q8×C53—22no—
213Q8×C53—22no—
214Q8×C53—41no—
215Q8×C53—41no—
216C2×C4×C53—21no—
217C2×C4×C53—21no—
218C2×C4×C53—21no—
219C2×C4×C53—21no—
220C2×C4×C53—28no—
221C2×C4×C53—21no—
222C2×C4×C53—21no—
223C2×C4×C53—21no—
224C2×C4×C53—22no—
225C2×C4×C53—22no—
226C2×C4×C53—21no—
227C2×C4×C53—21no—
228C2×C4×C53—21no—
229C2×C4×C53—44no—
230C2×C4×C53—41no—
231C2×C4×C53—41no—
232C2×C4×C53—41no—
233C2×C4×C53—41no—
234C2×C4×C53—41no—
235C2×C4×C53—41no—
236C2×C4×C53—41no—
237C2×C4×C53—41no—
238C2×C4×C53—41no—
239C2×C4×C53—41no—
240C2×C4×C53—41no—
241C2×C4×C53—41no—
242C2×C4×C53—41no—
243C2×C4×C53—41no—
244C2×C4×C53#12012yes—
245C2×C4×C53#12012yes—
246C2×C4×C53C2×SD1612yes—
247C2×C4×C53C2×SD1612yes—
248C2×C4×C53AGL(1,ℤ/8)12yes—
249C2×C4×C53C22×D411yescharacteristic
250C2×C4×C53C22×D411yescharacteristic
251C2×C4×C53C2×#4111yescharacteristic
252C2×C4×C53C2×#4111yescharacteristic
253C2×C4×C53H(5,𝔽2)11yescharacteristic
254C2×C4×C532−1+411yescharacteristic
255C22⋊D53—416no—
256C2×Dic53—24no—
257C2×Dic53—42no—
258C2×Dic53—42no—
259D212—24no—
260D212—24no—
261D212—48no—
262D212—42no—
263D212—42no—
264D212—48no—
265C4×D53—416no—
266Dic106—48no—
267#88322—42no—
268#88322—42no—
269#24154—531nomaximal, Sylow
270C22×C53—22no—
271C22×C53—22no—
272C22×C53—22no—
273C22×C53—41no—
274C22×C53—41no—
275C22×C53—41no—
276C22×C53—41no—
277C22×C53C2×#12011yescharacteristic
278C22×C53#52511yescharacteristic
279C22×C53#53711yescharacteristic
280C22×C53C22×SD1611yescharacteristic
281C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
282C22×C53#57511yescharacteristic
283C22×C53#57511yescharacteristic
284C2×D53—44no—
285C2×D53—44no—
286C2×D53—44no—
287C2×D53—82no—
288C2×D53—82no—
289C2×D53—82no—
290C2×D53—82no—
291C4×C53—22no—
292C4×C53—22no—
293C4×C53—42no—
294C4×C53—41no—
295C4×C53—42no—
296C4×C53—41no—
297C4×C53—81no—
298C4×C53—81no—
299C4×C53#45014yes—
300C4×C53#54611yescharacteristic
301C4×C53#54711yescharacteristic
302C4×C53#54811yescharacteristic
303C4×C53#55011yescharacteristic
304Dic53—44no—
305C2×#532—531no—
306C2×#475—531no—
307C2×#460—531no—
308C4×C22⋊Q8D5311yescharacteristic, complemented
309#2034—532no—
310#2000—532no—
311#1988—531no—
312#1982—531no—
313#1877—532no—
314#1864—532no—
315#1797—531no—
316C2×C53—42no—
317C2×C53#240711yescharacteristic
318C2×C53C2×#45011yescharacteristic
319C2×C53#300511yescharacteristic
320C2×C53#310311yescharacteristic
321C2×C53#311511yescharacteristic
322C2×C53#324611yescharacteristic
323C2×C53#325111yescharacteristic
324D53—84no—
325C22×#41—1061no—
326C22×SD16—532no—
327#532—1064no—
328#532—1064no—
329C2×C4⋊Q8—531no—
330C2×#127—531no—
331C2×#124—1061no—
332C2×C22⋊Q8C2×D5311yescharacteristic
333C2×#121—532no—
334C2×#121—1061no—
335C2×C4×Q8C2×D5311yescharacteristic
336C2×C4×D4—532no—
337C2×C4×D4—1061no—
338C22×C42C2×D5311yescharacteristic
339#475—1064no—
340#460—1064no—
341C2×#106—532no—
342C2×C4⋊C8—531no—
343C2×C4⋊C8—531no—
344C2×#103—532no—
345C2×#102—532no—
346C2×#102—532no—
347C2×#102—532no—
348C2×C22⋊C8—532no—
349#391C2×D5311yescharacteristic
350#389C2×D5311yescharacteristic
351#387—21no—
352#385—21no—
353#380C2×D5311yescharacteristic
354C4×C4⋊C4—21no—
355C4×C4⋊C4C2×D5311yescharacteristic
356C4×C22⋊C4—21no—
357#340—532no—
358#340—532no—
359#336—532no—
360C53#2415411yescharacteristic, complemented, Sylow
361C2×#41—1068no—
362C2×#41—2124no—
363C2×#41—2124no—
364C22×Q8—1061no—
365C22×Q8C22×D5311yescharacteristic
366C22×D4—532no—
367C22×D4—1062no—
368C22×D4—1061no—
369C22×D4—1061no—
370C23×C4—1062no—
371C23×C4C22⋊D5311yescharacteristic
372C23×C4C22⋊D5311yescharacteristic
373C23×C4C22×D5311yescharacteristic
374C2×SD16—1064no—
375C2×SD16—1064no—
376C2×SD16—2122no—
377C22×C8—532no—
378C22×C8—532no—
379C4⋊Q8—534no—
380C4⋊Q8—1062no—
381#127—534no—
382#127—1062no—
383#124—1068no—
384C22⋊Q8—22no—
385C22⋊Q8—22no—
386#121—1064no—
387#121—1064no—
388#121—1068no—
389C4×Q8—22no—
390C4×Q8—22no—
391C4×D4—1064no—
392C4×D4—1064no—
393C4×D4—1068no—
394C2×C4⋊C4—21no—
395C2×C4⋊C4—21no—
396C2×C4⋊C4—21no—
397C2×C4⋊C4—532no—
398C2×C4⋊C4—1061no—
399C2×C4⋊C4C22×D5311yescharacteristic
400C2×C4⋊C4C22×D5311yescharacteristic
401C2×C4⋊C4C22×D5311yescharacteristic
402C2×C22⋊C4—21no—
403C2×C22⋊C4—21no—
404C2×C22⋊C4—1062no—
405C2×C22⋊C4—1062no—
406C2×C42—22no—
407C2×C42—22no—
408C2×C42—21no—
409C2×C42—21no—
410C2×C42C22⋊D5311yescharacteristic
411C2×C42C22⋊D5311yescharacteristic
412C2×C42C22×D5311yescharacteristic
413C2×C42C22×D5311yescharacteristic
414#106—1064no—
415C4⋊C8—1062no—
416C4⋊C8—1062no—
417#103—1064no—
418#102—1064no—
419#102—1064no—
420#102—2122no—
421C22⋊C8—2122no—
422#95—21no—
423#95—21no—
424#95—21no—
425#95—21no—
426#95—21no—
427C24—1062no—
428C24C2×C22⋊D5311yescharacteristic
429#41—21232no—
430C2×Q8—21no—
431C2×Q8—21no—
432C2×Q8—21no—
433C2×Q8—21no—
434C2×Q8—21no—
435C2×Q8—1064no—
436C2×Q8—2122no—
437C2×Q8—2122no—
438C2×Q8#1171994512yescomplemented
439C2×D4—534no—
440C2×D4—1068no—
441C2×D4—1064no—
442C2×D4—1064no—
443C2×D4—1062no—
444C2×D4—1062no—
445C2×D4—1062no—
446C2×D4—1062no—
447C2×D4—1062no—
448C2×D4—2122no—
449C2×D4—2122no—
450C2×D4—2124no—
451C2×D4—2124no—
452C2×D4—2122no—
453C2×D4—2122no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—22no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—41no—
468C22×C4—41no—
469C22×C4—1068no—
470C22×C4—1062no—
471C22×C4—2124no—
472C22×C4—2124no—
473C22×C4#1171992414yescomplemented
474C22×C4C2×C22⋊D5311yescharacteristic
475C22×C4C2×C22⋊D5311yescharacteristic
476C22×C4C2×C22⋊D5311yescharacteristic
477C22×C4C2×C22⋊D5311yescharacteristic
478C22×C4C2×C22⋊D5311yescharacteristic
479C22×C4C23×D5311yescharacteristic
480SD16—2128no—
481C2×C8—1062no—
482C2×C8—1062no—
483C2×C8—1062no—
484C2×C8—1062no—
485C2×C8—1062no—
486C2×C8—1062no—
487C4⋊C4—24no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—42no—
494C4⋊C4—42no—
495C4⋊C4—534no—
496C4⋊C4—1064no—
497C4⋊C4—1062no—
498C4⋊C4#1171994512yescomplemented
499C22⋊C4—24no—
500C22⋊C4—42no—
501C22⋊C4—1068no—
502C22⋊C4—1068no—
503C42—22no—
504C42—22no—
505C42—21no—
506C42—21no—
507C42—42no—
508C42—42no—
509C42D4×D5312yes—
510C42D4×D5312yes—
511C23—21no—
512C23—21no—
513C23—21no—
514C23—21no—
515C23—21no—
516C23—21no—
517C23—1064no—
518C23—2122no—
519C23—2122no—
520C23—2122no—
521C23—2122no—
522C23—2122no—
523C23—2122no—
524C23C2×#1171992412yes—
525C23C22×C22⋊D5311yescharacteristic
526Q8—22no—
527Q8—22no—
528Q8—41no—
529Q8—41no—
530Q8—2128no—
531D4—1064no—
532D4—1064no—
533D4—2128no—
534D4—21216no—
535D4—2122no—
536D4—2128no—
537D4—2122no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—28no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—44no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
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—1064no—
567C2×C4—21216no—
568C2×C4—2122no—
569C2×C4—2122no—
570C2×C4#1313318512yes—
571C2×C4#1313318512yes—
572C2×C4C2×#1171992412yes—
573C2×C4C2×#1171992412yes—
574C2×C4#1313324212yes—
575C2×C4C2×D4×D5311yescharacteristic
576C2×C4C2×D4×D5311yescharacteristic
577C2×C4C2×#1171994511yescharacteristic
578C2×C4C2×#1171994511yescharacteristic
579C2×C4#1313326311yescharacteristic
580C2×C4#1313326411yescharacteristic
581C8—2122no—
582C8—2122no—
583C22—22no—
584C22—22no—
585C22—22no—
586C22—41no—
587C22—41no—
588C22—41no—
589C22—41no—
590C22—4242no—
591C22—4242no—
592C22—4242no—
593C22—4242no—
594C22—2124no—
595C22—2124no—
596C22—2124no—
597C22C2×#1313318511yescharacteristic
598C22#1681113411yescharacteristic
599C22#1681120111yescharacteristic
600C22C22×#1171992411yescharacteristic
601C22C2×#1313324211yescharacteristic
602C22#1681142111yescharacteristic
603C22#1681142111yescharacteristic
604C4—22no—
605C4—22no—
606C4—41no—
607C4—42no—
608C4—41no—
609C4—42no—
610C4—81no—
611C4—81no—
612C4—2124no—
613C4#1681077314yes—
614C4#1681126911yescharacteristic
615C4#1681127811yescharacteristic
616C4#1681128211yescharacteristic
617C4#1681129411yescharacteristic
618C2—42no—
619C2—4244no—
620C2#2027806011yescharacteristic
621C2C2×#1681077311yescharacteristic
622C2#2028395511yescharacteristic
623C2#2028474611yescharacteristic
624C2#2028486611yescharacteristic
625C2#2028602711yescharacteristic
626C2#2028607711yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC4×C22⋊Q8×C53C2
socleC23×C53C22×D4
prime radicalC2×#16811173C2
hypercenterC4×C22⋊Q8D53
hypocenterC53#24154
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C22×C42 ◁ C4×C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. C2×#16810908C2semidirect
3. C2×#16810823C2semidirect
4. #20275904C2semidirect
5. #20275716C2semidirect
6. #20275667C2semidirect
7. #20275642C2semidirect
8. #20275052C2semidirect
9. #20274975C2semidirect
10. #20274695C2semidirect
11. C4×C22⋊Q8×C53C2semidirect
120. C2×Q8×C53#41semidirect
136. C22×C4×C53SD16semidirect
174. C4⋊C4×C53#41semidirect
308. C4×C22⋊Q8D53semidirect
360. C53#24154semidirect
438. C2×Q8#11719945semidirect
473. C22×C4#11719924semidirect
498. C4⋊C4#11719945semidirect

Maximal subgroups

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

subgroupindex
C2×#168111732
C2×#168109082
C2×#168108232
#202759042
#202757162
#202756672
#202756422
#202750522
#202749752
#202746952
C4×C22⋊Q8×C532
#2415453