Finite Group Databasebeta

← #27422814

Subgroups — #27422814

Order 13568.

1188 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810774C211yesmaximal, characteristic
2C2×#16810680C211yesmaximal, characteristic, complemented
3#20283054C211yesmaximal, characteristic, complemented
4#20276103C212yesmaximal, complemented
5#20276032C212yesmaximal, complemented
6#20275595C212yesmaximal, complemented
7#20275415C212yesmaximal, complemented
8#20275022C211yesmaximal, characteristic, complemented
9#20274903C211yesmaximal, characteristic, complemented
10#20274646C211yesmaximal, characteristic
11C2×C53×#390C211yesmaximal, characteristic
12C22×#11719923C2212yes—
13C2×#13133196C2211yescharacteristic
14#16810774—24no—
15#16810680—24no—
16C2×#13133150C2212yescomplemented
17C2×C4⋊#88322C2212yescomplemented
18C2×#13133140C2212yescomplemented
19C2×#13133138C2212yes—
20C2×#13133138C2212yes—
21C2×#13133138C2212yes—
22C2×C22⋊#88322C2211yescharacteristic
23C2×C22⋊#88322C2211yescharacteristic
24C2×#13133128C2212yescomplemented
25#16810529C2211yescharacteristic, complemented
26#16810529C2211yescharacteristic, complemented
27#16810521C2211yescharacteristic, complemented
28#16810521C2211yescharacteristic, complemented
29#16810507—21no—
30#16810507—21no—
31#16810496—21no—
32#16810486—21no—
33#16810483C2211yescharacteristic
34#16810385C2212yescomplemented
35#16810385C2212yescomplemented
36C23×D4×C53C2211yescharacteristic
37C22×C4⋊C4×C53C2211yescharacteristic
38C22×C22⋊C4×C53—21no—
39C22×C22⋊C4×C53—21no—
40C22×C42×C53C2211yescharacteristic
41C53×#390—21no—
42C53×#390—21no—
43C53×#390C2212yes—
44C53×#390C2212yes—
45C22×Dic106C2311yescharacteristic
46C2×#11719923—24no—
47C2×#11719923—24no—
48C2×#11719923—42no—
49C22×#88322C2312yes—
50C22×#88322C2312yes—
51#13133196—22no—
52#13133196—22no—
53C2×C4⋊Dic53C2311yescharacteristic
54C2×C4⋊Dic53C2311yescharacteristic
55C2×C4⋊Dic53C2311yescharacteristic
56C2×#11719919—21no—
57C2×#11719919—21no—
58C2×#11719919—21no—
59C2×C22⋊Dic53—21no—
60C2×C22⋊Dic53—21no—
61#13133150—42no—
62C4⋊#88322—24no—
63#13133140—42no—
64#13133138—24no—
65#13133138—24no—
66#13133138—42no—
67C22⋊#88322—22no—
68C22⋊#88322D414yes—
69#13133128—24no—
70#13133124—21no—
71#13133124—21no—
72#13133124—21no—
73#13133124—21no—
74#13133124—21no—
75#13133124—21no—
76#13133124—21no—
77C25×C53—21no—
78C22×D4×C53—24no—
79C22×D4×C53—21no—
80C22×D4×C53—22no—
81C22×D4×C53—22no—
82C22×D4×C53—42no—
83C22×D4×C53C2312yes—
84C23×C4×C53—21no—
85C23×C4×C53D411yescharacteristic
86C23×C4×C53D411yescharacteristic
87C23×C4×C53C2311yescharacteristic
88C2×C4⋊C4×C53—21no—
89C2×C4⋊C4×C53—42no—
90C2×C4⋊C4×C53C2312yes—
91C2×C22⋊C4×C53—22no—
92C2×C22⋊C4×C53—22no—
93C2×C22⋊C4×C53—22no—
94C2×C22⋊C4×C53—22no—
95C2×C22⋊C4×C53—44no—
96C2×C22⋊C4×C53—44no—
97C2×C42×C53—22no—
98C2×C42×C53—22no—
99C2×C42×C53D411yescharacteristic
100C2×C42×C53D411yescharacteristic
101C2×C42×C53C2311yescharacteristic
102C2×C42×C53C2311yescharacteristic
103C24×C53—24no—
104C24×C53—22no—
105C24×C53—42no—
106C24×C53—42no—
107C24×C53—42no—
108C24×C53—42no—
109C24×C53—42no—
110C24×C53—42no—
111C24×C53C2×D411yescharacteristic
112C2×D4×C53—24no—
113C2×D4×C53—22no—
114C2×D4×C53—22no—
115C2×D4×C53—22no—
116C2×D4×C53—22no—
117C2×D4×C53—22no—
118C2×D4×C53—48no—
119C2×D4×C53—48no—
120C2×D4×C53—42no—
121C2×D4×C53—42no—
122C2×D4×C53#4114yes—
123C22×C4×C53—22no—
124C22×C4×C53—22no—
125C22×C4×C53—22no—
126C22×C4×C53—22no—
127C22×C4×C53—22no—
128C22×C4×C53—22no—
129C22×C4×C53—22no—
130C22×C4×C53—22no—
131C22×C4×C53—41no—
132C22×C4×C53—42no—
133C22×C4×C53—42no—
134C22×C4×C53—42no—
135C22×C4×C53C2×D411yescharacteristic
136C22×C4×C53C2×D411yescharacteristic
137C22×C4×C53C2×D411yescharacteristic
138C22×C4×C53C2×D411yescharacteristic
139C22×C4×C53C2×D412yes—
140C22×C4×C53C2×D411yescharacteristic
141C22×C4×C53#4112yes—
142C22×C4×C53C2411yescharacteristic
143C22×Dic53—21no—
144C22×Dic53—21no—
145C22×Dic53—21no—
146C22×Dic53—21no—
147C2×Dic106—21no—
148C2×Dic106—21no—
149C2×Dic106—22no—
150C2×Dic106—22no—
151C4⋊C4×C53—22no—
152C4⋊C4×C53—22no—
153C4⋊C4×C53—42no—
154C22⋊C4×C53—44no—
155C22⋊C4×C53—44no—
156C22⋊C4×C53—44no—
157C22⋊C4×C53—44no—
158C42×C53—22no—
159C42×C53—22no—
160C42×C53—21no—
161C42×C53—21no—
162C42×C53—21no—
163C42×C53—21no—
164C22⋊Dic53—42no—
165C22⋊Dic53—42no—
166#11719923—48no—
167C4⋊Dic53—21no—
168C4⋊Dic53—22no—
169C4⋊Dic53—21no—
170C4⋊Dic53—22no—
171#11719919—42no—
172#11719919—42no—
173#11719919—42no—
174C2×#88322—22no—
175C2×#88322—22no—
176C2×#88322—22no—
177C2×#88322—22no—
178C2×#88322—22no—
179C2×#88322C2×D414yes—
180C23×C53—21no—
181C23×C53—21no—
182C23×C53—21no—
183C23×C53—21no—
184C23×C53—24no—
185C23×C53—48no—
186C23×C53—48no—
187C23×C53—48no—
188C23×C53—42no—
189C23×C53—42no—
190C23×C53—42no—
191C23×C53—42no—
192C23×C53—42no—
193C23×C53—42no—
194C23×C53—84no—
195C23×C53—84no—
196C23×C53—84no—
197C23×C53—84no—
198C23×C53#12112yes—
199C23×C53AGL(1,ℤ/8)12yes—
200C23×C53#13712yes—
201C23×C53C22×D411yescharacteristic
202D4×C53—24no—
203D4×C53—24no—
204D4×C53—48no—
205D4×C53—42no—
206D4×C53—42no—
207C2×C4×C53—21no—
208C2×C4×C53—21no—
209C2×C4×C53—21no—
210C2×C4×C53—21no—
211C2×C4×C53—24no—
212C2×C4×C53—24no—
213C2×C4×C53—21no—
214C2×C4×C53—21no—
215C2×C4×C53—21no—
216C2×C4×C53—21no—
217C2×C4×C53—21no—
218C2×C4×C53—21no—
219C2×C4×C53—21no—
220C2×C4×C53—21no—
221C2×C4×C53—21no—
222C2×C4×C53—44no—
223C2×C4×C53—44no—
224C2×C4×C53—42no—
225C2×C4×C53—42no—
226C2×C4×C53—42no—
227C2×C4×C53—84no—
228C2×C4×C53#12112yes—
229C2×C4×C53AGL(1,ℤ/8)12yes—
230C2×C4×C53C22×D411yescharacteristic
231C2×C4×C53C2×#4111yescharacteristic
232C2×C4×C53C2×#4112yes—
233C2×C4×C53H(5,𝔽2)12yes—
234C2×Dic53—41no—
235C2×Dic53—41no—
236C2×Dic53—41no—
237C2×Dic53—41no—
238C2×Dic53—41no—
239C2×Dic53—41no—
240C2×Dic53—41no—
241C2×Dic53—41no—
242C2×Dic53—41no—
243C2×Dic53—41no—
244C2×Dic53—41no—
245C2×Dic53—41no—
246Dic106—42no—
247Dic106—42no—
248#88322—24no—
249#88322—42no—
250#24507—531nomaximal, Sylow
251C22×C53—22no—
252C22×C53—22no—
253C22×C53—22no—
254C22×C53—22no—
255C22×C53—22no—
256C22×C53—22no—
257C22×C53—44no—
258C22×C53—44no—
259C22×C53—44no—
260C22×C53—816no—
261C22×C53—82no—
262C22×C53—82no—
263C22×C53—82no—
264C22×C53—82no—
265C22×C53#46814yes—
266C22×C53C2×#12111yescharacteristic
267C22×C53#52511yescharacteristic
268C22×C53#53511yescharacteristic
269C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
270C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
271C22×C53C2×#13711yescharacteristic
272C22×C53#57511yescharacteristic
273C4×C53—22no—
274C4×C53—22no—
275C4×C53—41no—
276C4×C53—41no—
277C4×C53—41no—
278C4×C53—82no—
279C4×C53#54612yes—
280C4×C53#55112yes—
281Dic53—81no—
282Dic53—81no—
283Dic53—81no—
284Dic53—81no—
285C2×#450—531no—
286C2×#432—531no—
287#2855—531no—
288C2×#390D5311yescharacteristic
289#2067—532no—
290#2055—532no—
291#1973—532no—
292#1936—532no—
293#1872—531no—
294#1852—531no—
295#1786—531no—
296C2×C53—24no—
297C2×C53—84no—
298C2×C53#241111yescharacteristic
299C2×C53C2×#46811yescharacteristic
300C2×C53#305511yescharacteristic
301C2×C53#311011yescharacteristic
302C2×C53#311411yescharacteristic
303C2×C53#321511yescharacteristic
304C2×C53#321711yescharacteristic
305C23×D4C2×D5311yescharacteristic
306C22×SD16—532no—
307C2×C22⋊Q8—531no—
308C22×C4⋊C4C2×D5311yescharacteristic
309C22×C22⋊C4—21no—
310C22×C22⋊C4—21no—
311C22×C42C2×D5311yescharacteristic
312#450—1064no—
313#432—1064no—
314C2×#107—532no—
315C2×C4⋊C8—532no—
316C2×#103—532no—
317C2×#102—532no—
318C2×#102—532no—
319C2×#102—532no—
320C2×C22⋊C8—531no—
321C2×C22⋊C8—531no—
322C2×#97—532no—
323#391—531no—
324#391—531no—
325#390—21no—
326#390—21no—
327#390C2×D5312yes—
328#390C2×D5312yes—
329#389—531no—
330#389—531no—
331#387—1061no—
332#387—1061no—
333#385—1061no—
334#381—1061no—
335#380—531no—
336#340—532no—
337#340—532no—
338C53#2450711yescharacteristic, complemented, Sylow
339C25—21no—
340C22×Q8—531no—
341C22×D4—24no—
342C22×D4—22no—
343C22×D4—22no—
344C22×D4—21no—
345C22×D4—42no—
346C22×D4C22×D5312yes—
347C23×C4—21no—
348C23×C4C22⋊D5311yescharacteristic
349C23×C4C22⋊D5311yescharacteristic
350C23×C4C22×D5311yescharacteristic
351C2×SD16—1064no—
352C2×SD16—1064no—
353C2×SD16—2122no—
354C22×C8—532no—
355C22×C8—532no—
356C22⋊Q8—1062no—
357C22⋊Q8—1062no—
358C2×C4⋊C4—21no—
359C2×C4⋊C4—42no—
360C2×C4⋊C4—531no—
361C2×C4⋊C4—531no—
362C2×C4⋊C4—531no—
363C2×C4⋊C4—1061no—
364C2×C4⋊C4—1061no—
365C2×C4⋊C4—1061no—
366C2×C4⋊C4C22×D5312yes—
367C2×C22⋊C4—22no—
368C2×C22⋊C4—22no—
369C2×C22⋊C4—22no—
370C2×C22⋊C4—22no—
371C2×C22⋊C4—44no—
372C2×C22⋊C4—44no—
373C2×C22⋊C4—1061no—
374C2×C22⋊C4—1061no—
375C2×C42—22no—
376C2×C42—22no—
377C2×C42C22⋊D5311yescharacteristic
378C2×C42C22⋊D5311yescharacteristic
379C2×C42C22×D5311yescharacteristic
380C2×C42C22×D5311yescharacteristic
381#107—2122no—
382C4⋊C8—1064no—
383#103—2122no—
384#102—1064no—
385#102—1064no—
386#102—2122no—
387C22⋊C8—534no—
388C22⋊C8—1062no—
389#97—1064no—
390#95—1061no—
391#95—1061no—
392#95—1061no—
393#95—1061no—
394#95—1061no—
395#95—1061no—
396#95—1061no—
397C24—24no—
398C24—22no—
399C24—42no—
400C24—42no—
401C24—42no—
402C24—42no—
403C24—42no—
404C24—42no—
405C24C2×C22⋊D5311yescharacteristic
406C2×Q8—1061no—
407C2×Q8—1061no—
408C2×Q8—1062no—
409C2×Q8—1062no—
410C2×D4—24no—
411C2×D4—22no—
412C2×D4—22no—
413C2×D4—22no—
414C2×D4—22no—
415C2×D4—22no—
416C2×D4—48no—
417C2×D4—48no—
418C2×D4—42no—
419C2×D4—42no—
420C2×D4#1171994514yes—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—41no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—1061no—
434C22×C4—1061no—
435C22×C4—1061no—
436C22×C4—1061no—
437C22×C4D4×D5312yes—
438C22×C4#1171994712yes—
439C22×C4C2×C22⋊D5311yescharacteristic
440C22×C4C2×C22⋊D5311yescharacteristic
441C22×C4C2×C22⋊D5311yescharacteristic
442C22×C4C2×C22⋊D5311yescharacteristic
443C22×C4C2×C22⋊D5311yescharacteristic
444C22×C4C23×D5311yescharacteristic
445SD16—2128no—
446C2×C8—534no—
447C2×C8—1062no—
448C2×C8—1062no—
449C2×C8—1062no—
450C2×C8—1062no—
451C2×C8—1062no—
452C4⋊C4—22no—
453C4⋊C4—22no—
454C4⋊C4—42no—
455C4⋊C4—1061no—
456C4⋊C4—1061no—
457C4⋊C4—1062no—
458C4⋊C4—1062no—
459C4⋊C4—2122no—
460C4⋊C4—2122no—
461C4⋊C4—2122no—
462C22⋊C4—44no—
463C22⋊C4—44no—
464C22⋊C4—44no—
465C22⋊C4—44no—
466C22⋊C4—2122no—
467C22⋊C4—2122no—
468C42—22no—
469C42—22no—
470C42—21no—
471C42—21no—
472C42—21no—
473C42—21no—
474C23—21no—
475C23—21no—
476C23—21no—
477C23—21no—
478C23—24no—
479C23—48no—
480C23—48no—
481C23—48no—
482C23—42no—
483C23—42no—
484C23—42no—
485C23—42no—
486C23—42no—
487C23—42no—
488C23—84no—
489C23—84no—
490C23—84no—
491C23—84no—
492C23#1313318912yes—
493C23#1313324312yes—
494C23#1313324712yes—
495C23C22×C22⋊D5311yescharacteristic
496Q8—2122no—
497Q8—2122no—
498D4—24no—
499D4—24no—
500D4—48no—
501D4—42no—
502D4—42no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—44no—
519C2×C4—44no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—84no—
524C2×C4—2121no—
525C2×C4—2121no—
526C2×C4—2121no—
527C2×C4—2121no—
528C2×C4—2121no—
529C2×C4—2121no—
530C2×C4—2121no—
531C2×C4—2121no—
532C2×C4—2121no—
533C2×C4—2121no—
534C2×C4—2121no—
535C2×C4—2121no—
536C2×C4#1313318912yes—
537C2×C4#1313324312yes—
538C2×C4C2×D4×D5311yescharacteristic
539C2×C4C2×#1171994711yescharacteristic
540C2×C4C2×#1171994512yes—
541C2×C4#1313326212yes—
542C8—1064no—
543C8—2122no—
544C22—22no—
545C22—22no—
546C22—22no—
547C22—22no—
548C22—22no—
549C22—22no—
550C22—44no—
551C22—44no—
552C22—44no—
553C22—816no—
554C22—82no—
555C22—82no—
556C22—82no—
557C22—82no—
558C22#1681086714yes—
559C22C2×#1313318911yescharacteristic
560C22#1681113411yescharacteristic
561C22#1681119011yescharacteristic
562C22C2×#1313324311yescharacteristic
563C22C2×#1313324311yescharacteristic
564C22C2×#1313324711yescharacteristic
565C22#1681142111yescharacteristic
566C4—22no—
567C4—22no—
568C4—41no—
569C4—41no—
570C4—41no—
571C4—82no—
572C4—4241no—
573C4—4241no—
574C4—4241no—
575C4—4241no—
576C4#1681126312yes—
577C4#1681129912yes—
578C2—24no—
579C2—84no—
580C2#2027810011yescharacteristic
581C2C2×#1681086711yescharacteristic
582C2#2028433011yescharacteristic
583C2#2028481411yescharacteristic
584C2#2028485311yescharacteristic
585C2#2028571811yescharacteristic
586C2#2028573311yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC2×C53×#390C2
socleC23×C53C22×D4
prime radicalC23×D4×C53C22
hypercenterC2×#390D53
hypocenterC53#24507
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#390

Complements

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

NH ≅ G/Ntype
2. C2×#16810680C2semidirect
3. #20283054C2semidirect
4. #20276103C2semidirect
5. #20276032C2semidirect
6. #20275595C2semidirect
7. #20275415C2semidirect
8. #20275022C2semidirect
9. #20274903C2semidirect
16. C2×#13133150C22semidirect
17. C2×C4⋊#88322C22semidirect
18. C2×#13133140C22semidirect
24. C2×#13133128C22semidirect
25. #16810529C22semidirect
26. #16810529C22semidirect
27. #16810521C22semidirect
28. #16810521C22semidirect
34. #16810385C22semidirect
35. #16810385C22semidirect
338. C53#24507semidirect

Maximal subgroups

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

subgroupindex
C2×#168107742
C2×#168106802
#202830542
#202761032
#202760322
#202755952
#202754152
#202750222
#202749032
#202746462
C2×C53×#3902
#2450753