Finite Group Databasebeta

← #27354216

Subgroups — #27354216

Order 13568.

626 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20284744C211yesmaximal, characteristic
2#20283794C211yesmaximal, characteristic, complemented
3#20283789C211yesmaximal, characteristic
4#20273596C211yesmaximal, characteristic, complemented
5#20273590C211yesmaximal, characteristic, complemented
6#20273559C211yesmaximal, characteristic, complemented
7#20273557C211yesmaximal, characteristic, complemented
8#20273527C211yesmaximal, characteristic, complemented
9#20273525C211yesmaximal, characteristic, complemented
10C8×#11719913C211yesmaximal, characteristic, complemented
11#20273492C211yesmaximal, characteristic, complemented
12#20273160C211yesmaximal, characteristic, complemented
13#20273130C211yesmaximal, characteristic
14#20273009C211yesmaximal, characteristic, complemented
15C8×SD16×C53C211yesmaximal, characteristic, complemented
16#16811170C2211yescharacteristic
17C4×Q8×D53C2211yescharacteristic
18#16810943C411yescharacteristic
19#16810940C411yescharacteristic
20#16810907C411yescharacteristic
21#16810906C411yescharacteristic
22#16810822C411yescharacteristic
23#16810821C411yescharacteristic
24#16810763C411yescharacteristic
25#16810760C411yescharacteristic
26Q8×#88322C2211yescharacteristic
27C8×Dic106—21no—
28C8×Dic106C2211yescharacteristic
29#16810747—21no—
30#16810747C2211yescharacteristic
31C4×#11719913C2211yescharacteristic
32C4×#11719924C2211yescharacteristic
33C4×#11719923C2211yescharacteristic
34#16810701C2211yescharacteristic
35#16810700C2211yescharacteristic
36#16810697—21no—
37#16810694—21no—
38C8×D212C2211yescharacteristic
39#16810679C2211yescharacteristic
40#16810675—21no—
41#16810675C2211yescharacteristic
42#16810674—21no—
43#16810674C2211yescharacteristic
44#16810674C2211yescharacteristic
45#16810589—21no—
46#16810588—21no—
47C4×#11719912C2211yescharacteristic
48#16810375C2211yescharacteristic
49#16810375C2211yescharacteristic
50#16810374C2211yescharacteristic
51#16810355C2211yescharacteristic
52Q8⋊#88322C2211yescharacteristic
53#16810353C2211yescharacteristic
54#16810352C2211yescharacteristic
55D4⋊#88322C2211yescharacteristic
56#16810350C2211yescharacteristic
57#16810345C2211yescharacteristic
58#16810345C2211yescharacteristic
59#16810344C2211yescharacteristic
60C8×Q8×C53C2211yescharacteristic
61C4×SD16×C53C2211yescharacteristic
62C8×D4×C53C2211yescharacteristic
63C53×#334C2211yescharacteristic
64C53×Q8⋊C8C2211yescharacteristic
65C53×D4⋊C8C2211yescharacteristic
66C82×C53C2211yescharacteristic
67C2×#11719945—21no—
68C2×Q8×D53C2×C411yescharacteristic
69C2×#11719913C2×C411yescharacteristic
70C2×#11719923C2×C411yescharacteristic
71C2×#11719924C2×C411yescharacteristic
72C2×#11719917—21no—
73C2×#11719912—21no—
74C2×#11719912—21no—
75C2×C8×D53—21no—
76#13133216C2×C411yescharacteristic
77#13133213C2×C411yescharacteristic
78#13133203—21no—
79#13133192C2×C411yescharacteristic
80#13133190—21no—
81#13133188C2×C411yescharacteristic
82Q8×Dic53C2×C411yescharacteristic
83C4×Dic106—21no—
84C4×Dic106C2311yescharacteristic
85#13133182—21no—
86#13133182C2×C411yescharacteristic
87C4×D212C2311yescharacteristic
88C4×C22⋊D53—21no—
89C4⋊C4×D53—21no—
90C4⋊C4×D53C2×C411yescharacteristic
91C42×D53—21no—
92C42×D53C2311yescharacteristic
93#13133149C2×C411yescharacteristic
94#13133148C2×C411yescharacteristic
95#13133148C2×C411yescharacteristic
96C4⋊#88322—21no—
97C4⋊#88322C2311yescharacteristic
98C4⋊#88322C2311yescharacteristic
99#13133146—21no—
100#13133146—21no—
101#13133146C2×C411yescharacteristic
102#13133146C2×C411yescharacteristic
103#13133146D411yescharacteristic
104#13133146D411yescharacteristic
105#13133142C2×C411yescharacteristic
106#13133141C2×C411yescharacteristic
107#13133140C2×C411yescharacteristic
108#13133139C2×C411yescharacteristic
109#13133138C2×C411yescharacteristic
110#13133137C2×C411yescharacteristic
111C22⋊#88322—21no—
112#13133129—21no—
113#13133129—21no—
114#13133129—21no—
115#13133129—21no—
116#13133129C2×C411yescharacteristic
117#13133129C2×C411yescharacteristic
118#13133129D411yescharacteristic
119#13133129D411yescharacteristic
120#13133127—21no—
121#13133127—21no—
122C4×#88322—21no—
123C4×#88322C2311yescharacteristic
124C4×#88322C2311yescharacteristic
125C4×#88322C2311yescharacteristic
126C8×Dic53—21no—
127C8×Dic53—21no—
128C8×Dic53—21no—
129C2×SD16×C53C2×C411yescharacteristic
130C22×C8×C53—21no—
131C4×Q8×C53C2311yescharacteristic
132C4×D4×C53C2311yescharacteristic
133#106×C53C2×C411yescharacteristic
134C4⋊C8×C53—21no—
135C4⋊C8×C53C2311yescharacteristic
136C4⋊C8×C53C2311yescharacteristic
137#103×C53C2×C411yescharacteristic
138#102×C53C2×C411yescharacteristic
139C22⋊C8×C53—21no—
140C4×C8×C53—21no—
141C4×C8×C53C2311yescharacteristic
142C4×C8×C53C2311yescharacteristic
143C4×C8×C53C2311yescharacteristic
144C2×Q8×C53C22×C411yescharacteristic
145C2×D4×C53C22×C411yescharacteristic
146C22×C4×C53—21no—
147C2×C22⋊D53—21no—
148C2×C22⋊D53—21no—
149C2×C22⋊D53—21no—
150Q8×D53—22no—
151Q8×D53—22no—
152#11719945—81no—
153#11719945—81no—
154C2×D212—21no—
155C2×D212C22×C411yescharacteristic
156C2×C4×D53—21no—
157C2×C4×D53—21no—
158C2×C4×D53—21no—
159C2×C4×D53C22×C411yescharacteristic
160C2×C4×D53C2×D411yescharacteristic
161C2×C4×D53#4111yescharacteristic
162C2×Dic106—21no—
163C2×Dic106—21no—
164C2×Dic106C22×C411yescharacteristic
165SD16×C53—22no—
166C2×C8×C53—21no—
167C2×C8×C53—21no—
168C2×C8×C53—21no—
169C2×C8×C53—81no—
170C2×C8×C53C22×C411yescharacteristic
171C2×C8×C53C22×C411yescharacteristic
172C2×C8×C53C2×D411yescharacteristic
173C2×C8×C53C2×D411yescharacteristic
174C4⋊C4×C53—21no—
175C4⋊C4×C53C22×C411yescharacteristic
176C4⋊C4×C53C22×C411yescharacteristic
177C22⋊C4×C53—21no—
178C42×C53—21no—
179C42×C53C2411yescharacteristic
180C22⋊Dic53—21no—
181#11719924—22no—
182#11719923—22no—
183#11719921—21no—
184#11719921—21no—
185C4⋊Dic53—21no—
186C4⋊Dic53C22×C411yescharacteristic
187C4⋊Dic53C22×C411yescharacteristic
188#11719919—21no—
189#11719919—21no—
190#11719919—21no—
191#11719919—21no—
192C4×Dic53—21no—
193C4×Dic53—21no—
194C4×Dic53—21no—
195C4×Dic53C22×C411yescharacteristic
196C4×Dic53C2×D411yescharacteristic
197C4×Dic53#4111yescharacteristic
198#11719917—81no—
199C2×#88322—21no—
200C2×#88322—21no—
201C2×#88322—21no—
202C2×#88322C22×C411yescharacteristic
203C2×#88322C22×C411yescharacteristic
204C2×#88322C2×D411yescharacteristic
205C2×#88322C2×D411yescharacteristic
206#11719913—22no—
207#11719912—42no—
208#11719912—81no—
209C8×D53—81no—
210C23×C53—21no—
211C22×D53—21no—
212C22×D53C4×D411yescharacteristic
213Q8×C53—21no—
214Q8×C53#13112yes—
215D4×C53—21no—
216D4×C53—21no—
217C2×C4×C53—21no—
218C2×C4×C53—21no—
219C2×C4×C53—21no—
220C2×C4×C53—41no—
221C2×C4×C53—41no—
222C2×C4×C53C23×C411yescharacteristic
223C2×C4×C53C22×D411yescharacteristic
224C2×C4×C53C2×#4111yescharacteristic
225C22⋊D53—81no—
226C22⋊D53—81no—
227C22⋊D53—81no—
228C2×Dic53—21no—
229C2×Dic53—21no—
230C2×Dic53—21no—
231C2×Dic53C4×D411yescharacteristic
232C2×Dic53C4×D411yescharacteristic
233C2×Dic53C4×D411yescharacteristic
234D212—21no—
235D212—21no—
236D212—81no—
237C4×D53—24no—
238C4×D53—21no—
239C4×D53—21no—
240C4×D53—21no—
241C4×D53—21no—
242C4×D53—41no—
243C4×D53—41no—
244C4×D53—41no—
245C4×D53—81no—
246Dic106—24no—
247Dic106—21no—
248Dic106—81no—
249Dic106#13112yes—
250C8×C53—22no—
251C8×C53—21no—
252C8×C53—21no—
253C8×C53—21no—
254C8×C53—21no—
255C8×C53—41no—
256C8×C53—41no—
257#88322—21no—
258#88322—21no—
259#88322—21no—
260#88322—21no—
261#88322—41no—
262#88322—41no—
263#88322—41no—
264#17923—531nomaximal, Sylow
265C22×C53—41no—
266C22×C53—41no—
267C22×C53—41no—
268C22×C53C2×C4×D411yescharacteristic
269C2×D53—21no—
270C2×D53—21no—
271C2×D53—21no—
272C2×D53—41no—
273C2×D53—41no—
274C2×D53—41no—
275C4×C53—22no—
276C4×C53—21no—
277C4×C53—41no—
278C4×C53—41no—
279C4×C53C2×#13111yescharacteristic
280C4×C53C2×#13111yescharacteristic
281C4×C53#57711yescharacteristic
282C4×C53#57711yescharacteristic
283Dic53—21no—
284Dic53—22no—
285Dic53—21no—
286Dic53—21no—
287Dic53—41no—
288Dic53—41no—
289#3103—531no—
290#2982—531no—
291#2981—531no—
292#1588—531no—
293#1588—531no—
294#1583—531no—
295#1583—531no—
296SD16⋊C8—531no—
297SD16⋊C8—531no—
298C8×SD16—531no—
299C8×SD16D5311yescharacteristic, complemented
300#1572—531no—
301#1491—531no—
302#1484—531no—
303#1453—531no—
304C2×C53—81no—
305C2×C53#292911yescharacteristic
306C2×C53#294911yescharacteristic
307C2×C53#295611yescharacteristic
308D53—41no—
309D53—81no—
310#532—531no—
311C2×C4×Q8—531no—
312#481—531no—
313#481—531no—
314#475—531no—
315#475—531no—
316#460—531no—
317#460—531no—
318#448—531no—
319#448—531no—
320C8×Q8—531no—
321C8×Q8—531no—
322C8×Q8—531no—
323C8×Q8—1061no—
324C8×Q8—1061no—
325C8×Q8C2×D5311yescharacteristic
326C4×SD16—531no—
327C4×SD16—531no—
328C4×SD16—531no—
329C4×SD16C2×D5311yescharacteristic
330#436—531no—
331#436—531no—
332#435—1061no—
333#435—1061no—
334C8×D4—531no—
335C8×D4C2×D5311yescharacteristic
336#432—531no—
337#431—531no—
338#431—531no—
339#431—531no—
340#431—1061no—
341#431—1061no—
342#408—1061no—
343#408—1061no—
344C4×M16—531no—
345#335—531no—
346#335—531no—
347#334—531no—
348#334C2×D5311yescharacteristic
349Q8⋊C8—531no—
350Q8⋊C8—531no—
351Q8⋊C8—531no—
352Q8⋊C8C2×D5311yescharacteristic
353D4⋊C8—531no—
354D4⋊C8—531no—
355D4⋊C8—531no—
356D4⋊C8C2×D5311yescharacteristic
357#322—531no—
358#322—531no—
359#322—531no—
360C82C2×D5311yescharacteristic
361C53#1792311yescharacteristic, complemented, Sylow
362C2×#41—1061no—
363C22×Q8—531no—
364C2×SD16—531no—
365C2×SD16—531no—
366C2×SD16—531no—
367C2×SD16C4×D5311yescharacteristic
368C2×M16—1061no—
369C2×M16—1061no—
370C2×M16—1061no—
371C22×C8—21no—
372C22×C8—1061no—
373C4⋊Q8—531no—
374#127—531no—
375#124—1061no—
376#121—531no—
377#121—531no—
378#121—1061no—
379C4×Q8—531no—
380C4×Q8—531no—
381C4×Q8—531no—
382C4×Q8—1061no—
383C4×Q8—1061no—
384C4×Q8C22×D5311yescharacteristic
385C4×D4—531no—
386C4×D4—1061no—
387C4×D4C22×D5311yescharacteristic
388C2×C4⋊C4—531no—
389C2×C4⋊C4—1061no—
390C2×C42—531no—
391C2×C42—1061no—
392#106—531no—
393#106—531no—
394#106—531no—
395#106C4×D5311yescharacteristic
396C4⋊C8—21no—
397C4⋊C8—531no—
398C4⋊C8—531no—
399C4⋊C8—531no—
400C4⋊C8—531no—
401C4⋊C8—531no—
402C4⋊C8—531no—
403C4⋊C8—1061no—
404C4⋊C8—1061no—
405C4⋊C8—1061no—
406C4⋊C8C22×D5311yescharacteristic
407C4⋊C8C22×D5311yescharacteristic
408#103—531no—
409#103—531no—
410#103—531no—
411#103C4×D5311yescharacteristic
412#102—531no—
413#102—531no—
414#102—531no—
415#102C4×D5311yescharacteristic
416C22⋊C8—21no—
417C22⋊C8—531no—
418C22⋊C8—531no—
419C22⋊C8—531no—
420C22⋊C8—531no—
421C22⋊C8—1061no—
422C22⋊C8—1061no—
423C22⋊C8—1061no—
424C22⋊C8—1061no—
425C22⋊C8—1061no—
426#97—1061no—
427#97—1061no—
428C4×C8—21no—
429C4×C8—531no—
430C4×C8—531no—
431C4×C8—531no—
432C4×C8—1061no—
433C4×C8—1061no—
434C4×C8—1061no—
435C4×C8—1061no—
436C4×C8C22×D5311yescharacteristic
437C4×C8C22×D5311yescharacteristic
438C4×C8C22×D5311yescharacteristic
439#41—4241no—
440#41—4241no—
441C2×Q8—531no—
442C2×Q8—1061no—
443C2×Q8—1061no—
444C2×Q8—1062no—
445C2×Q8—1062no—
446C2×Q8C2×C4×D5311yescharacteristic
447C2×D4—531no—
448C2×D4—1061no—
449C2×D4—1061no—
450C2×D4—1061no—
451C2×D4—1061no—
452C2×D4C2×C4×D5311yescharacteristic
453C22×C4—21no—
454C22×C4—531no—
455C22×C4—531no—
456C22×C4—531no—
457C22×C4—1061no—
458C22×C4—1061no—
459C22×C4—1061no—
460SD16—22no—
461SD16—1062no—
462SD16—1062no—
463SD16—1062no—
464M16—4241no—
465M16—4241no—
466M16—2122no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—81no—
471C2×C8—531no—
472C2×C8—531no—
473C2×C8—531no—
474C2×C8—531no—
475C2×C8—1061no—
476C2×C8—1061no—
477C2×C8—1061no—
478C2×C8—4241no—
479C2×C8C2×C4×D5311yescharacteristic
480C2×C8C2×C4×D5311yescharacteristic
481C2×C8D4×D5311yescharacteristic
482C2×C8D4×D5311yescharacteristic
483C4⋊C4—21no—
484C4⋊C4—531no—
485C4⋊C4—531no—
486C4⋊C4—1061no—
487C4⋊C4—1061no—
488C4⋊C4—1061no—
489C4⋊C4—1061no—
490C4⋊C4—1061no—
491C4⋊C4C2×C4×D5311yescharacteristic
492C4⋊C4C2×C4×D5311yescharacteristic
493C22⋊C4—21no—
494C22⋊C4—1061no—
495C22⋊C4—1061no—
496C22⋊C4—1061no—
497C42—21no—
498C42—531no—
499C42—531no—
500C42—531no—
501C42—1061no—
502C42—1061no—
503C42—1061no—
504C42C23×D5311yescharacteristic
505C23—21no—
506C23—531no—
507C23—1061no—
508Q8—21no—
509Q8—532no—
510Q8—1064no—
511Q8—1061no—
512Q8—4241no—
513Q8#1313322312yes—
514D4—21no—
515D4—21no—
516D4—1061no—
517D4—1061no—
518D4—4241no—
519D4—4241no—
520D4—4241no—
521D4—4241no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—531no—
528C2×C4—531no—
529C2×C4—531no—
530C2×C4—1064no—
531C2×C4—1061no—
532C2×C4—1061no—
533C2×C4—1061no—
534C2×C4—1061no—
535C2×C4—1061no—
536C2×C4—1061no—
537C2×C4—1061no—
538C2×C4—4241no—
539C2×C4—2121no—
540C2×C4—2121no—
541C2×C4—2121no—
542C2×C4C22×C4×D5311yescharacteristic
543C2×C4C2×D4×D5311yescharacteristic
544C2×C4#41×D5311yescharacteristic
545C8—22no—
546C8—21no—
547C8—21no—
548C8—21no—
549C8—21no—
550C8—41no—
551C8—41no—
552C8—1061no—
553C8—1061no—
554C8—1061no—
555C8—1061no—
556C8—2121no—
557C8—2121no—
558C8—2121no—
559C22—41no—
560C22—41no—
561C22—41no—
562C22—1061no—
563C22—1061no—
564C22—1061no—
565C22—2121no—
566C22—2121no—
567C22—2121no—
568C22C4×D4×D5311yescharacteristic
569C4—22no—
570C4—21no—
571C4—41no—
572C4—41no—
573C4—1061no—
574C4—1062no—
575C4—1061no—
576C4—1061no—
577C4—2121no—
578C4—2121no—
579C4#131×D5311yescharacteristic
580C4C2×#1313322311yescharacteristic
581C4#1681142911yescharacteristic
582C4#1681143011yescharacteristic
583C2—81no—
584C2—4241no—
585C2—2121no—
586C2#2028340411yescharacteristic
587C2#2028356911yescharacteristic
588C2#2028362011yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC42C23×D53
FittingC8×SD16×C53C2
socleC22×C53C2×C4×D4
prime radical#16811170C22
cosocleC42×C53C24
hypercenterC8×SD16D53
hypocenterC53#17923
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ C8×SD16

Complements

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

NH ≅ G/Ntype
2. #20283794C2semidirect
4. #20273596C2semidirect
5. #20273590C2semidirect
6. #20273559C2semidirect
7. #20273557C2semidirect
8. #20273527C2semidirect
9. #20273525C2semidirect
10. C8×#11719913C2semidirect
11. #20273492C2semidirect
12. #20273160C2semidirect
14. #20273009C2semidirect
15. C8×SD16×C53C2semidirect
299. C8×SD16D53semidirect
361. C53#17923semidirect

Maximal subgroups

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

subgroupindex
#202847442
#202837942
#202837892
#202735962
#202735902
#202735592
#202735572
#202735272
#202735252
C8×#117199132
#202734922
#202731602
#202731302
#202730092
C8×SD16×C532
#1792353