Finite Group Databasebeta

← #27399747

Subgroups — #27399747

Order 13568.

1530 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811276C211yesmaximal, characteristic
2C2×#16810700C211yesmaximal, characteristic, complemented
3C2×#16810623C212yesmaximal, complemented
4C4⋊C4×D212C211yesmaximal, characteristic, complemented
5#20275518C211yesmaximal, characteristic, complemented
6#20275145C212yesmaximal, complemented
7#20275019C212yesmaximal, complemented
8#20274716C211yesmaximal, characteristic, complemented
9#20274654C212yesmaximal, complemented
10C4×#13133137C211yesmaximal, characteristic, complemented
11C53×#1981C211yesmaximal, characteristic, complemented
12C22×#11719949C2212yes—
13C22×#11719912C2212yes—
14#16811276—24no—
15#16811276C418yescomplemented
16C2×#13133202—22no—
17C2×#13133193C2212yes—
18C2×#13133192—21no—
19C2×C4×D212—21no—
20C2×C4×D212C2211yescharacteristic
21C2×C4×D212C2211yescharacteristic
22C2×C4⋊C4×D53—21no—
23C2×C4⋊C4×D53C2212yes—
24#16810700—24no—
25#16810623—28no—
26C2×#13133141C2212yes—
27C2×#13133141C2212yes—
28C2×#13133139C2212yes—
29C2×#13133137C2211yescharacteristic
30C2×#13133137C2211yescharacteristic
31C2×#13133129C2212yes—
32C2×C4×#88322C2211yescharacteristic
33#16810521—21no—
34#16810497—22no—
35#16810493C2211yescharacteristic
36#16810480C2212yes—
37#16810386C2212yes—
38#16810385C2211yescharacteristic
39#16810385C2211yescharacteristic
40C2×C4⋊Q8×C53C2211yescharacteristic
41C2×C4⋊C8×C53C2211yescharacteristic
42C2×#103×C53C2212yes—
43C2×#103×C53C2212yes—
44C4×C4⋊C4×C53C2211yescharacteristic
45C2×#11719949—24no—
46C2×#11719949—24no—
47C2×#11719949—24no—
48C2×#11719949—42no—
49C2×#11719949—42no—
50C2×#11719949C2×C418yescomplemented
51C22×D212—22no—
52C22×D212D411yescharacteristic
53C22×D212C2311yescharacteristic
54C22×C4×D53—21no—
55C22×C4×D53—22no—
56C22×C4×D53—22no—
57C22×C4×D53C2312yes—
58C2×#11719912—24no—
59C2×#11719912—24no—
60C2×#11719912—42no—
61C22×#88322C2311yescharacteristic
62C22×#88322C2311yescharacteristic
63#13133202—216no—
64#13133193—24no—
65#13133193C2×C418yescomplemented
66#13133192—28no—
67C4×D212—22no—
68C4×D212—44no—
69C4×D212C2×C414yes—
70C4×D212C2×C414yescomplemented
71C4×D212D414yes—
72C2×C4⋊Dic53—21no—
73C2×C4⋊Dic53—21no—
74C2×C4⋊Dic53D411yescharacteristic
75C2×C4⋊Dic53C2312yes—
76C2×C4⋊Dic53C2311yescharacteristic
77C2×#11719919—22no—
78C2×#11719919—22no—
79C4⋊C4×D53—28no—
80C4⋊C4×D53—24no—
81C4⋊C4×D53—42no—
82C2×#11719921—22no—
83C2×#11719921—22no—
84C2×#11719921—21no—
85C2×#11719921D412yes—
86#13133141—24no—
87#13133141—42no—
88#13133139—24no—
89#13133137—22no—
90#13133137D414yescomplemented
91#13133129—24no—
92C4×#88322—22no—
93#13133124—22no—
94C22×Q8×C53C2312yes—
95C22×C8×C53C2312yes—
96C4⋊Q8×C53—22no—
97C4⋊Q8×C53C2×C414yescomplemented
98C2×C4⋊C4×C53—21no—
99C2×C4⋊C4×C53—21no—
100C2×C4⋊C4×C53C2311yescharacteristic
101C2×C4⋊C4×C53C2311yescharacteristic
102C2×C42×C53—21no—
103C2×C42×C53C2311yescharacteristic
104C4⋊C8×C53—22no—
105#103×C53—24no—
106#103×C53—42no—
107#95×C53—21no—
108C23×D53—21no—
109C23×D53C2×D412yes—
110C2×Q8×C53—22no—
111C2×Q8×C53—22no—
112C2×Q8×C53—22no—
113C2×Q8×C53—22no—
114C2×Q8×C53—22no—
115C2×Q8×C53C22×C414yes—
116C22×C4×C53—21no—
117C22×C4×C53—22no—
118C22×C4×C53—21no—
119C22×C4×C53C2×D411yescharacteristic
120C22×C4×C53C2×D411yescharacteristic
121C22×C4×C53C2411yescharacteristic
122C22×Dic53—21no—
123C22×Dic53—22no—
124C22×Dic53C2×D412yes—
125#11719949—216no—
126#11719949—48no—
127#11719949—48no—
128#11719949—48no—
129C2×D212—22no—
130C2×D212—22no—
131C2×D212—21no—
132C2×D212—21no—
133C2×D212—21no—
134C2×D212—28no—
135C2×D212—44no—
136C2×D212—44no—
137C2×D212C22⋊C414yes—
138C2×D212C22×C412yes—
139C2×D212C2×D412yes—
140C2×D212#4112yes—
141C2×C4×D53—22no—
142C2×C4×D53—22no—
143C2×C4×D53—22no—
144C2×C4×D53—22no—
145C2×C4×D53—24no—
146C2×C4×D53—28no—
147C2×C4×D53—44no—
148C2×C4×D53—44no—
149C2×C4×D53—42no—
150C2×C4×D53—42no—
151C2×C4×D53—44no—
152C2×C4×D53—44no—
153C2×C4×D53—82no—
154C2×C4×D53C22×C414yes—
155C2×C4×D53C2×D414yes—
156C2×C8×C53—22no—
157C2×C8×C53—22no—
158C2×C8×C53—22no—
159C4⋊C4×C53—24no—
160C4⋊C4×C53—24no—
161C4⋊C4×C53—21no—
162C4⋊C4×C53—21no—
163C4⋊C4×C53—21no—
164C4⋊C4×C53C2×D412yes—
165C42×C53—42no—
166C42×C53C22×C412yes—
167C42×C53C2×D412yes—
168#11719921—28no—
169#11719921—24no—
170#11719921—44no—
171#11719921C22⋊C418yes—
172C4⋊Dic53—21no—
173C4⋊Dic53—22no—
174C4⋊Dic53—22no—
175C4⋊Dic53—42no—
176C4⋊Dic53—81no—
177C4⋊Dic53C22⋊C414yes—
178C4⋊Dic53C22×C412yes—
179#11719919—28no—
180#11719919—44no—
181C2×#88322—21no—
182C2×#88322—21no—
183C2×#88322—21no—
184C2×#88322—21no—
185C2×#88322—21no—
186C2×#88322C2×D412yes—
187#11719912—48no—
188C23×C53C22×D411yescharacteristic
189C22×D53—22no—
190C22×D53—22no—
191C22×D53—22no—
192C22×D53—22no—
193C22×D53—22no—
194C22×D53—22no—
195C22×D53—22no—
196C22×D53—22no—
197C22×D53—22no—
198C22×D53—41no—
199C22×D53—41no—
200C22×D53—41no—
201C22×D53—41no—
202C22×D53C2×C22⋊C414yes—
203Q8×C53—24no—
204Q8×C53—24no—
205Q8×C53—42no—
206Q8×C53—42no—
207C2×C4×C53—21no—
208C2×C4×C53—21no—
209C2×C4×C53—24no—
210C2×C4×C53—22no—
211C2×C4×C53—42no—
212C2×C4×C53—42no—
213C2×C4×C53—41no—
214C2×C4×C53—41no—
215C2×C4×C53—41no—
216C2×C4×C53—41no—
217C2×C4×C53—41no—
218C2×C4×C53C2×C22⋊C412yes—
219C2×C4×C53C2×C22⋊C412yes—
220C2×C4×C53AGL(1,ℤ/8)12yes—
221C2×C4×C53#13712yes—
222C2×C4×C53C23×C411yescharacteristic
223C2×C4×C53C22×D411yescharacteristic
224C2×C4×C53C22×D411yescharacteristic
225C2×C4×C53C2×#4111yescharacteristic
226C2×C4×C53H(5,𝔽2)11yescharacteristic
227C2×C4×C53H(5,𝔽2)11yescharacteristic
228C2×Dic53—22no—
229C2×Dic53—22no—
230C2×Dic53—22no—
231C2×Dic53—41no—
232C2×Dic53—41no—
233C2×Dic53—42no—
234C2×Dic53—42no—
235C2×Dic53—42no—
236C2×Dic53C2×C22⋊C414yes—
237D212—22no—
238D212—22no—
239D212—22no—
240D212—28no—
241D212—416no—
242D212C4×D414yescomplemented
243C4×D53—24no—
244C4×D53—24no—
245C4×D53—24no—
246C4×D53—24no—
247C4×D53—416no—
248C4×D53—48no—
249C4×D53—88no—
250C8×C53—42no—
251#88322—22no—
252#88322—41no—
253#22431—531nomaximal, Sylow
254C22×C53C22×C22⋊C411yescharacteristic
255C22×C53D4211yescharacteristic
256C22×C53#54611yescharacteristic
257C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
258C22×C53C2×#13711yescharacteristic
259C22×C53#57711yescharacteristic
260C22×C53#57811yescharacteristic
261C2×D53—24no—
262C2×D53—24no—
263C2×D53—24no—
264C2×D53—24no—
265C2×D53—42no—
266C2×D53—42no—
267C2×D53—42no—
268C2×D53—42no—
269C2×D53—42no—
270C2×D53—42no—
271C2×D53—42no—
272C2×D53—42no—
273C2×D53—42no—
274C2×D53—42no—
275C4×C53—22no—
276C4×C53—44no—
277C4×C53—42no—
278C4×C53—81no—
279C4×C53#41714yescomplemented
280C4×C53C2×C4×D411yescharacteristic
281C4×C53#51811yescharacteristic
282C4×C53D4211yescharacteristic
283C4×C53#54611yescharacteristic
284Dic53—24no—
285Dic53—42no—
286Dic53—82no—
287C2×#547—531no—
288C2×#436—531no—
289C2×#417—532no—
290D4×C4⋊C4—531no—
291#1981D5311yescharacteristic, complemented
292#1956—531no—
293#1894—532no—
294#1872—532no—
295#1802—531no—
296#1788—532no—
297C4×#102—531no—
298C2×C53D4×C22⋊C411yescharacteristic
299C2×C53C2×#41711yescharacteristic
300C2×C53#289811yescharacteristic
301C2×C53#327211yescharacteristic
302C2×C53#327611yescharacteristic
303C2×C53#333011yescharacteristic
304C2×C53#333111yescharacteristic
305D53—44no—
306D53—44no—
307C22×#41—532no—
308C22×M16—532no—
309#547—538no—
310#547—1064no—
311C2×C4⋊Q8C2×D5311yescharacteristic
312C2×#123—1062no—
313C2×C22⋊Q8—532no—
314C2×#121—1061no—
315C2×C4×D4—531no—
316C2×C4×D4—531no—
317C2×C4×D4—1061no—
318C22×C4⋊C4—532no—
319C22×C4⋊C4—1061no—
320#436—1064no—
321C2×C4⋊C8C2×D5311yescharacteristic
322#417—1068no—
323C2×#103—532no—
324C2×#103—532no—
325C2×#103C2×D5312yes—
326C2×#103C2×D5312yes—
327C2×#102—531no—
328C2×#102—532no—
329C2×#102—531no—
330C2×C22⋊C8—532no—
331C2×C4×C8—531no—
332#389—1061no—
333#385—1062no—
334#384—531no—
335#380—532no—
336C4×C4⋊C4C2×D5311yescharacteristic
337#340—532no—
338#340—531no—
339#340—531no—
340C53#2243111yescharacteristic, complemented, Sylow
341C2×#41—538no—
342C2×#41—1064no—
343C2×#41—1064no—
344C2×#41—1064no—
345C2×#41—2122no—
346C2×#41—2122no—
347C22×Q8C22×D5312yes—
348C22×D4—531no—
349C22×D4—531no—
350C22×D4—1062no—
351C23×C4—532no—
352C23×C4—1062no—
353C23×C4—1062no—
354C23×C4—1061no—
355C2×M16—1064no—
356C2×M16—1064no—
357C2×M16—2122no—
358C22×C8—531no—
359C22×C8—531no—
360C22×C8C22×D5312yes—
361C4⋊Q8—22no—
362C4⋊Q8C4×D5314yescomplemented
363#123—10616no—
364C22⋊Q8—538no—
365C22⋊Q8—1064no—
366#121—1068no—
367C4×D4—534no—
368C4×D4—534no—
369C4×D4—534no—
370C4×D4—1062no—
371C4×D4—2124no—
372C2×C4⋊C4—21no—
373C2×C4⋊C4—21no—
374C2×C4⋊C4—531no—
375C2×C4⋊C4—531no—
376C2×C4⋊C4—532no—
377C2×C4⋊C4—1068no—
378C2×C4⋊C4—1062no—
379C2×C4⋊C4—1064no—
380C2×C4⋊C4—1061no—
381C2×C4⋊C4—1062no—
382C2×C4⋊C4—1061no—
383C2×C4⋊C4—2122no—
384C2×C4⋊C4C22×D5311yescharacteristic
385C2×C4⋊C4C22×D5311yescharacteristic
386C2×C22⋊C4—532no—
387C2×C22⋊C4—1062no—
388C2×C22⋊C4—1061no—
389C2×C22⋊C4—1062no—
390C2×C42—21no—
391C2×C42C22×D5311yescharacteristic
392C4⋊C8—22no—
393#103—24no—
394#103—42no—
395#103—1064no—
396#103—2122no—
397#102—534no—
398#102—1062no—
399#102—1064no—
400C22⋊C8—1064no—
401C4×C8—1062no—
402#95—21no—
403#95—1062no—
404C24—532no—
405C24—1061no—
406#41—10616no—
407#41—2128no—
408#41—2128no—
409#41—2128no—
410C2×Q8—22no—
411C2×Q8—22no—
412C2×Q8—22no—
413C2×Q8—22no—
414C2×Q8—22no—
415C2×Q8C2×C4×D5314yes—
416C2×D4—534no—
417C2×D4—532no—
418C2×D4—532no—
419C2×D4—532no—
420C2×D4—1068no—
421C2×D4—1062no—
422C2×D4—1062no—
423C2×D4—1061no—
424C2×D4—1061no—
425C2×D4—1061no—
426C2×D4—2124no—
427C2×D4—2124no—
428C22×C4—21no—
429C22×C4—22no—
430C22×C4—21no—
431C22×C4—532no—
432C22×C4—534no—
433C22×C4—534no—
434C22×C4—1068no—
435C22×C4—1062no—
436C22×C4—1062no—
437C22×C4—1062no—
438C22×C4—1062no—
439C22×C4—1061no—
440C22×C4—1064no—
441C22×C4—1062no—
442C22×C4—4242no—
443C22×C4—2124no—
444C22×C4—2124no—
445C22×C4—2124no—
446C22×C4—2124no—
447C22×C4—2122no—
448C22×C4—2122no—
449C22×C4D4×D5311yescharacteristic
450C22×C4D4×D5311yescharacteristic
451C22×C4C23×D5311yescharacteristic
452M16—2128no—
453C2×C8—22no—
454C2×C8—22no—
455C2×C8—22no—
456C2×C8—532no—
457C2×C8—1061no—
458C2×C8—1061no—
459C2×C8—1061no—
460C2×C8—1061no—
461C2×C8—1061no—
462C4⋊C4—24no—
463C4⋊C4—24no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—534no—
468C4⋊C4—532no—
469C4⋊C4—1068no—
470C4⋊C4—1061no—
471C4⋊C4—1062no—
472C4⋊C4—1062no—
473C4⋊C4—4241no—
474C4⋊C4—2124no—
475C4⋊C4—2122no—
476C4⋊C4D4×D5312yes—
477C22⋊C4—538no—
478C22⋊C4—1068no—
479C22⋊C4—1064no—
480C22⋊C4—2124no—
481C42—42no—
482C42C2×C4×D5312yes—
483C42D4×D5312yes—
484C23—534no—
485C23—1062no—
486C23—1062no—
487C23—1062no—
488C23—1062no—
489C23—1062no—
490C23—1062no—
491C23—1062no—
492C23—1062no—
493C23—1062no—
494C23—2121no—
495C23—2121no—
496C23—2121no—
497C23—2121no—
498C23C2×D4×D5311yescharacteristic
499Q8—24no—
500Q8—24no—
501Q8—42no—
502Q8—42no—
503D4—534no—
504D4—1062no—
505D4—1062no—
506D4—1068no—
507D4—1062no—
508D4—21216no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—24no—
512C2×C4—22no—
513C2×C4—42no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—42no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—534no—
521C2×C4—1062no—
522C2×C4—1062no—
523C2×C4—1064no—
524C2×C4—1064no—
525C2×C4—1064no—
526C2×C4—1064no—
527C2×C4—1062no—
528C2×C4—4248no—
529C2×C4—21216no—
530C2×C4—2122no—
531C2×C4—2122no—
532C2×C4—2122no—
533C2×C4—2121no—
534C2×C4—2121no—
535C2×C4—2128no—
536C2×C4C22⋊C4×D5312yes—
537C2×C4C22⋊C4×D5312yes—
538C2×C4#1313324412yes—
539C2×C4#1313324912yes—
540C2×C4C22×C4×D5311yescharacteristic
541C2×C4C2×D4×D5311yescharacteristic
542C2×C4C2×D4×D5311yescharacteristic
543C2×C4#41×D5311yescharacteristic
544C2×C4#1313326211yescharacteristic
545C2×C4#1313326311yescharacteristic
546C8—42no—
547C8—1062no—
548C8—2121no—
549C22—1064no—
550C22—1064no—
551C22—1064no—
552C22—1064no—
553C22—2122no—
554C22—2122no—
555C22—2122no—
556C22—2122no—
557C22—2122no—
558C22—2122no—
559C22—2122no—
560C22—2122no—
561C22—2122no—
562C22—2122no—
563C22C2×C22⋊C4×D5311yescharacteristic
564C22#1681125711yescharacteristic
565C22#1681127011yescharacteristic
566C22C2×#1313324411yescharacteristic
567C22C2×#1313324911yescharacteristic
568C22#1681143211yescharacteristic
569C22#1681143911yescharacteristic
570C4—22no—
571C4—44no—
572C4—42no—
573C4—81no—
574C4—1064no—
575C4—4242no—
576C4—2122no—
577C4#1681062314yescomplemented
578C4C4×D4×D5311yescharacteristic
579C4#1681109511yescharacteristic
580C4#1681125811yescharacteristic
581C4#1681126611yescharacteristic
582C2—2124no—
583C2—2124no—
584C2#2027753611yescharacteristic
585C2C2×#1681062311yescharacteristic
586C2#2028324711yescharacteristic
587C2#2028631811yescharacteristic
588C2#2028635611yescharacteristic
589C2#2028702311yescharacteristic
590C2#2028703411yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#1981C2
socleC23×C53C22×D4
prime radicalC2×#16811276C2
cosocleC22×C4×C53C24
hypercenter#1981D53
hypocenterC53#22431
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1981

Complements

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

NH ≅ G/Ntype
2. C2×#16810700C2semidirect
3. C2×#16810623C2semidirect
4. C4⋊C4×D212C2semidirect
5. #20275518C2semidirect
6. #20275145C2semidirect
7. #20275019C2semidirect
8. #20274716C2semidirect
9. #20274654C2semidirect
10. C4×#13133137C2semidirect
11. C53×#1981C2semidirect
15. #16811276C4semidirect
50. C2×#11719949C2×C4semidirect
65. #13133193C2×C4semidirect
70. C4×D212C2×C4semidirect
90. #13133137D4semidirect
97. C4⋊Q8×C53C2×C4semidirect
242. D212C4×D4semidirect
279. C4×C53#417semidirect
291. #1981D53semidirect
340. C53#22431semidirect
362. C4⋊Q8C4×D53semidirect
577. C4#16810623semidirect

Maximal subgroups

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

subgroupindex
C2×#168112762
C2×#168107002
C2×#168106232
C4⋊C4×D2122
#202755182
#202751452
#202750192
#202747162
#202746542
C4×#131331372
C53×#19812
#2243153