Finite Group Databasebeta

← #27356137

Subgroups — #27356137

Order 13568.

598 conjugacy classes of subgroups.

The subgroup lattice has 578 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285288C211yesmaximal, characteristic
2#20283878C211yesmaximal, characteristic, complemented
3#20283874C211yesmaximal, characteristic, complemented
4#20273620C211yesmaximal, characteristic, complemented
5#20273615C211yesmaximal, characteristic, complemented
6#20273566C211yesmaximal, characteristic, complemented
7#20273546C211yesmaximal, characteristic, complemented
8#20273543C211yesmaximal, characteristic, complemented
9#20273530C211yesmaximal, characteristic, complemented
10#20273525C211yesmaximal, characteristic, complemented
11#335×D53C211yesmaximal, characteristic, complemented
12#20273127C211yesmaximal, characteristic, complemented
13#20273127C211yesmaximal, characteristic, complemented
14#20273003C211yesmaximal, characteristic, complemented
15C53×#1584C211yesmaximal, characteristic, complemented
16#16811160C2211yescharacteristic
17#16811160C2211yescharacteristic
18#16811024C411yescharacteristic
19#16811008C411yescharacteristic
20#16810979C411yescharacteristic
21#16810979C411yescharacteristic
22#16810861C411yescharacteristic
23#16810860C411yescharacteristic
24#16810855C411yescharacteristic
25#16810855C411yescharacteristic
26#16810754C2211yescharacteristic
27Q8×#88322C2211yescharacteristic
28C4×#11719924C2211yescharacteristic
29C4×#11719924C2211yescharacteristic
30C4×D424C2211yescharacteristic
31#16810702C2211yescharacteristic
32#16810700C2211yescharacteristic
33#16810696—21no—
34#16810693—21no—
35C8×D212—21no—
36C8×D212C2211yescharacteristic
37#16810688—21no—
38#16810688C2211yescharacteristic
39#16810685—21no—
40#16810682—21no—
41#16810674C2211yescharacteristic
42#16810648—21no—
43#16810648—21no—
44C4⋊C8×D53C2211yescharacteristic
45C4⋊C8×D53C2211yescharacteristic
46C4×#11719912C2211yescharacteristic
47C4×C8×D53C2211yescharacteristic
48#16810376C2211yescharacteristic
49#16810375C2211yescharacteristic
50#16810375C2211yescharacteristic
51Q8⋊#88322C2211yescharacteristic
52Q8⋊#88322C2211yescharacteristic
53#16810352C2211yescharacteristic
54#16810352C2211yescharacteristic
55#16810350C2211yescharacteristic
56#16810350C2211yescharacteristic
57#16810345C2211yescharacteristic
58#16810345C2211yescharacteristic
59#16810345C2211yescharacteristic
60C53×#446C2211yescharacteristic
61C8×Q8×C53C2211yescharacteristic
62C4×Q16×C53C2211yescharacteristic
63C53×#335C2211yescharacteristic
64C53×Q8⋊C8C2211yescharacteristic
65C53×Q8⋊C8C2211yescharacteristic
66C53×#322C2211yescharacteristic
67C2×#11719924C2×C411yescharacteristic
68C2×#11719924C2×C411yescharacteristic
69C2×D424C2×C411yescharacteristic
70C2×#11719912—21no—
71C2×#11719912—21no—
72C2×#11719912—21no—
73C2×C8×D53—21no—
74C2×C8×D53—21no—
75C2×C8×D53—21no—
76C2×C8×D53—21no—
77C2×C8×D53C2×C411yescharacteristic
78C2×C8×D53C2×C411yescharacteristic
79#13133212—21no—
80#13133212—21no—
81#13133208C2×C411yescharacteristic
82#13133208C2×C411yescharacteristic
83#13133206C2×C411yescharacteristic
84#13133206C2×C411yescharacteristic
85#13133202—21no—
86#13133202—21no—
87#13133193C2×C411yescharacteristic
88#13133193C2×C411yescharacteristic
89#13133192C2×C411yescharacteristic
90#13133192C2×C411yescharacteristic
91C4×D212—21no—
92C4×D212—21no—
93C4×D212C2311yescharacteristic
94C4×D212C2311yescharacteristic
95#13133174—21no—
96#13133174—21no—
97C42×D53C2311yescharacteristic
98#13133151C2×C411yescharacteristic
99#13133148C2×C411yescharacteristic
100#13133148C2×C411yescharacteristic
101C4⋊#88322—21no—
102C4⋊#88322—21no—
103C4⋊#88322C2311yescharacteristic
104C4⋊#88322C2311yescharacteristic
105#13133146—21no—
106#13133146—21no—
107#13133146D411yescharacteristic
108#13133146D411yescharacteristic
109#13133141C2×C411yescharacteristic
110#13133141C2×C411yescharacteristic
111#13133139C2×C411yescharacteristic
112#13133139C2×C411yescharacteristic
113#13133137C2×C411yescharacteristic
114#13133137C2×C411yescharacteristic
115#13133129—21no—
116#13133129—21no—
117#13133129—21no—
118#13133129—21no—
119#13133129D411yescharacteristic
120#13133129D411yescharacteristic
121#13133128—21no—
122#13133127—21no—
123C4×#88322—21no—
124C4×#88322C2311yescharacteristic
125C4×#88322C2311yescharacteristic
126C4×#88322C2311yescharacteristic
127C8×Dic53C2×C411yescharacteristic
128C8×Dic53C2×C411yescharacteristic
129C2×Q16×C53C2×C411yescharacteristic
130C4×Q8×C53C2311yescharacteristic
131C4×Q8×C53C2311yescharacteristic
132#107×C53C2×C411yescharacteristic
133C4⋊C8×C53—21no—
134C4⋊C8×C53—21no—
135C4⋊C8×C53C2311yescharacteristic
136C4⋊C8×C53C2311yescharacteristic
137#103×C53C2×C411yescharacteristic
138#103×C53C2×C411yescharacteristic
139#97×C53—21no—
140C4×C8×C53—21no—
141C4×C8×C53C2311yescharacteristic
142C4×C8×C53C2311yescharacteristic
143C4×C8×C53C2311yescharacteristic
144C2×Q8×C53C22×C411yescharacteristic
145C2×Q8×C53C22×C411yescharacteristic
146C2×D212—21no—
147C2×D212—21no—
148C2×D212C22×C411yescharacteristic
149C2×D212C22×C411yescharacteristic
150C2×C4×D53—21no—
151C2×C4×D53—21no—
152C2×C4×D53C22×C411yescharacteristic
153C2×C4×D53C2×D411yescharacteristic
154C2×C4×D53#4111yescharacteristic
155Q16×C53—41no—
156C2×C8×C53—21no—
157C2×C8×C53—21no—
158C2×C8×C53—21no—
159C2×C8×C53C22×C411yescharacteristic
160C2×C8×C53C22×C411yescharacteristic
161C2×C8×C53C2×D411yescharacteristic
162C2×C8×C53C2×D411yescharacteristic
163C4⋊C4×C53—21no—
164C4⋊C4×C53—21no—
165C4⋊C4×C53C22×C411yescharacteristic
166C4⋊C4×C53C22×C411yescharacteristic
167C42×C53—21no—
168C42×C53—21no—
169C42×C53C2411yescharacteristic
170#11719924—22no—
171#11719924—22no—
172#11719921—21no—
173#11719921—21no—
174#11719921—21no—
175#11719921—21no—
176#11719921—21no—
177#11719921—21no—
178#11719921—21no—
179#11719921—21no—
180#11719921—21no—
181#11719921—21no—
182C4⋊Dic53—21no—
183C4⋊Dic53—21no—
184C4⋊Dic53C22×C411yescharacteristic
185C4⋊Dic53C22×C411yescharacteristic
186#11719919—21no—
187#11719919—21no—
188#11719919—21no—
189#11719919—21no—
190C4×Dic53C22×C411yescharacteristic
191C4×Dic53C2×D411yescharacteristic
192C4×Dic53#4111yescharacteristic
193C2×#88322—21no—
194C2×#88322—21no—
195C2×#88322—21no—
196C2×#88322C22×C411yescharacteristic
197C2×#88322C22×C411yescharacteristic
198C2×#88322C2×D411yescharacteristic
199C2×#88322C2×D411yescharacteristic
200D424—41no—
201#11719912—42no—
202#11719912—81no—
203#11719912—81no—
204C8×D53—22no—
205C8×D53—41no—
206C8×D53—81no—
207C8×D53—81no—
208C8×D53—81no—
209C8×D53—81no—
210C22×D53—21no—
211C22×D53—21no—
212C22×D53C4×D411yescharacteristic
213Q8×C53—21no—
214Q8×C53—21no—
215Q8×C53—21no—
216Q8×C53—21no—
217C2×C4×C53—21no—
218C2×C4×C53—21no—
219C2×C4×C53—21no—
220C2×C4×C53—21no—
221C2×C4×C53C23×C411yescharacteristic
222C2×C4×C53C22×D411yescharacteristic
223C2×C4×C53C2×#4111yescharacteristic
224C2×Dic53—21no—
225C2×Dic53—21no—
226C2×Dic53C4×D411yescharacteristic
227C2×Dic53C4×D411yescharacteristic
228C2×Dic53C4×D411yescharacteristic
229D212—21no—
230D212—21no—
231D212—21no—
232D212—21no—
233D212—42no—
234D212—42no—
235C4×D53—22no—
236C4×D53—21no—
237C4×D53—21no—
238C4×D53—21no—
239C4×D53—21no—
240C4×D53—41no—
241C4×D53—41no—
242C4×D53—41no—
243C4×D53—41no—
244C8×C53—21no—
245C8×C53—21no—
246C8×C53—21no—
247C8×C53—21no—
248C8×C53—41no—
249C8×C53—41no—
250C8×C53—41no—
251#88322—21no—
252#88322—21no—
253#88322—21no—
254#88322—41no—
255#88322—41no—
256#88322—41no—
257#88322#13112yes—
258#18092—531nomaximal, Sylow
259C22×C53C2×C4×D411yescharacteristic
260C2×D53—21no—
261C2×D53—21no—
262C2×D53—41no—
263C2×D53—41no—
264C2×D53—41no—
265C2×D53—41no—
266C2×D53—41no—
267C2×D53—41no—
268C2×D53#44412yes—
269C4×C53—21no—
270C4×C53—41no—
271C4×C53—41no—
272C4×C53—41no—
273C4×C53—41no—
274C4×C53C2×#13111yescharacteristic
275C4×C53#56811yescharacteristic
276C4×C53#57611yescharacteristic
277C4×C53#57811yescharacteristic
278Dic53—21no—
279Dic53—21no—
280Dic53—21no—
281Dic53—41no—
282Dic53—41no—
283#3168—531no—
284#2994—531no—
285#2994—531no—
286#1592—531no—
287#1592—531no—
288#1584—531no—
289#1584D5311yescharacteristic, complemented
290#1581—531no—
291#1581—531no—
292SD16⋊C8—531no—
293SD16⋊C8—531no—
294C2×#335—531no—
295#1483—531no—
296#1483—531no—
297#1451—531no—
298C2×C53#293511yescharacteristic
299C2×C53#295111yescharacteristic
300C2×C53C2×#44411yescharacteristic
301D53—22no—
302D53—81no—
303D53—81no—
304#529—531no—
305#529—531no—
306#499—531no—
307#495—531no—
308#488—531no—
309#488—531no—
310#467—531no—
311#466—531no—
312#465—531no—
313#465—531no—
314#446—531no—
315#446C2×D5311yescharacteristic
316C8×Q8—531no—
317C8×Q8C2×D5311yescharacteristic
318C4×Q16C2×D5311yescharacteristic
319C4×SD16—531no—
320C4×SD16—531no—
321C4×D8—531no—
322#436—531no—
323#436—531no—
324#435—1061no—
325#435—1061no—
326C8×D4—531no—
327C8×D4—531no—
328C8×D4—1061no—
329C8×D4—1061no—
330#433—1061no—
331#433—1061no—
332#431—531no—
333#424—1061no—
334#424—1061no—
335C2×C4⋊C8—531no—
336C2×C4⋊C8—531no—
337C4×M16—531no—
338C2×C4×C8—531no—
339#335—531no—
340#335—531no—
341#335—531no—
342#335C2×D5311yescharacteristic
343Q8⋊C8—531no—
344Q8⋊C8—531no—
345Q8⋊C8C2×D5311yescharacteristic
346Q8⋊C8C2×D5311yescharacteristic
347D4⋊C8—531no—
348D4⋊C8—531no—
349D4⋊C8—531no—
350D4⋊C8—531no—
351#322—531no—
352#322—531no—
353#322—531no—
354#322C2×D5311yescharacteristic
355C53#1809211yescharacteristic, complemented, Sylow
356C2×Q16C4×D5311yescharacteristic
357C2×SD16—531no—
358C2×SD16—531no—
359C2×D8—531no—
360C2×M16—1061no—
361C2×M16—1061no—
362C2×M16—1061no—
363C22×C8—531no—
364C22×C8—531no—
365C22×C8—1061no—
366C22×C8—1061no—
367C22×C8—1061no—
368C22×C8—1061no—
369#126—1061no—
370#126—1061no—
371#125—531no—
372#125—531no—
373#124—531no—
374#124—531no—
375#123—1061no—
376#123—1061no—
377C22⋊Q8—531no—
378C22⋊Q8—531no—
379#121—531no—
380#121—531no—
381C4×Q8C22×D5311yescharacteristic
382C4×Q8C22×D5311yescharacteristic
383C4×D4—531no—
384C4×D4—531no—
385C4×D4—1061no—
386C4×D4—1061no—
387#117—1061no—
388#117—1061no—
389C2×C42—531no—
390#107—531no—
391#107C4×D5311yescharacteristic
392#106—531no—
393#106—531no—
394C4⋊C8—21no—
395C4⋊C8—21no—
396C4⋊C8—531no—
397C4⋊C8—531no—
398C4⋊C8—531no—
399C4⋊C8—531no—
400C4⋊C8—1061no—
401C4⋊C8—1061no—
402C4⋊C8—1061no—
403C4⋊C8—1061no—
404C4⋊C8C22×D5311yescharacteristic
405C4⋊C8C22×D5311yescharacteristic
406#103—531no—
407#103—531no—
408#103C4×D5311yescharacteristic
409#103C4×D5311yescharacteristic
410#102—531no—
411#102—531no—
412#102—531no—
413#102—531no—
414C22⋊C8—531no—
415C22⋊C8—531no—
416C22⋊C8—1061no—
417C22⋊C8—1061no—
418C22⋊C8—1061no—
419C22⋊C8—1061no—
420#97—21no—
421#97—1061no—
422#97—1061no—
423C4×C8—21no—
424C4×C8—531no—
425C4×C8—531no—
426C4×C8—531no—
427C4×C8—531no—
428C4×C8—531no—
429C4×C8—1061no—
430C4×C8C22×D5311yescharacteristic
431C4×C8C22×D5311yescharacteristic
432C4×C8C22×D5311yescharacteristic
433C2×Q8C2×C4×D5311yescharacteristic
434C2×Q8C2×C4×D5311yescharacteristic
435C2×D4—531no—
436C2×D4—531no—
437C2×D4—1061no—
438C2×D4—1061no—
439C22×C4—531no—
440C22×C4—531no—
441C22×C4—531no—
442C22×C4—1061no—
443C22×C4—1061no—
444Q16—41no—
445SD16—1062no—
446SD16—1062no—
447D8—2121no—
448M16—4241no—
449M16—4241no—
450M16—2122no—
451C2×C8—21no—
452C2×C8—21no—
453C2×C8—21no—
454C2×C8—531no—
455C2×C8—531no—
456C2×C8—531no—
457C2×C8—531no—
458C2×C8—1062no—
459C2×C8—1061no—
460C2×C8—1061no—
461C2×C8—1061no—
462C2×C8—4241no—
463C2×C8—4241no—
464C2×C8—4241no—
465C2×C8—4241no—
466C2×C8—2121no—
467C2×C8C2×C4×D5311yescharacteristic
468C2×C8C2×C4×D5311yescharacteristic
469C2×C8D4×D5311yescharacteristic
470C2×C8D4×D5311yescharacteristic
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—531no—
474C4⋊C4—531no—
475C4⋊C4—1061no—
476C4⋊C4—1061no—
477C4⋊C4—1061no—
478C4⋊C4—1061no—
479C4⋊C4—1061no—
480C4⋊C4—1061no—
481C4⋊C4C2×C4×D5311yescharacteristic
482C4⋊C4C2×C4×D5311yescharacteristic
483C22⋊C4—1061no—
484C22⋊C4—1061no—
485C22⋊C4—1061no—
486C22⋊C4—1061no—
487C22⋊C4—1061no—
488C22⋊C4—1061no—
489C22⋊C4—1061no—
490C22⋊C4—1061no—
491C22⋊C4—1061no—
492C22⋊C4—1061no—
493C42—21no—
494C42—21no—
495C42—531no—
496C42—531no—
497C42—531no—
498C42C23×D5311yescharacteristic
499C23—531no—
500C23—1061no—
501C23—1061no—
502Q8—21no—
503Q8—21no—
504Q8—21no—
505Q8—21no—
506D4—1061no—
507D4—1061no—
508D4—1061no—
509D4—1061no—
510D4—2122no—
511D4—2122no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—531no—
517C2×C4—531no—
518C2×C4—531no—
519C2×C4—1062no—
520C2×C4—1061no—
521C2×C4—1061no—
522C2×C4—1061no—
523C2×C4—1061no—
524C2×C4—1061no—
525C2×C4—1061no—
526C2×C4—2121no—
527C2×C4—2121no—
528C2×C4—2121no—
529C2×C4—2121no—
530C2×C4C22×C4×D5311yescharacteristic
531C2×C4C2×D4×D5311yescharacteristic
532C2×C4#41×D5311yescharacteristic
533C8—21no—
534C8—21no—
535C8—21no—
536C8—21no—
537C8—41no—
538C8—41no—
539C8—41no—
540C8—532no—
541C8—1061no—
542C8—1061no—
543C8—1061no—
544C8—2121no—
545C8—2121no—
546C8—2121no—
547C22—532no—
548C22—1061no—
549C22—1061no—
550C22—2121no—
551C22—2121no—
552C22—2121no—
553C22—2121no—
554C22—2121no—
555C22—2121no—
556C22C4×D4×D5311yescharacteristic
557C4—21no—
558C4—41no—
559C4—41no—
560C4—41no—
561C4—41no—
562C4—1061no—
563C4—1061no—
564C4—1061no—
565C4—2121no—
566C4—2121no—
567C4#131×D5311yescharacteristic
568C4#1681138911yescharacteristic
569C4#1681142511yescharacteristic
570C4#1681143611yescharacteristic
571C2—1062no—
572C2—4241no—
573C2—4241no—
574C2#2028345611yescharacteristic
575C2#2028359011yescharacteristic
576C2#444×D5311yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC42C23×D53
FittingC53×#1584C2
socleC22×C53C2×C4×D4
prime radical#16810860C4
cosocleC42×C53C24
hypercenter#1584D53
hypocenterC53#18092
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ #1584

Complements

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

NH ≅ G/Ntype
2. #20283878C2semidirect
3. #20283874C2semidirect
4. #20273620C2semidirect
5. #20273615C2semidirect
6. #20273566C2semidirect
7. #20273546C2semidirect
8. #20273543C2semidirect
9. #20273530C2semidirect
10. #20273525C2semidirect
11. #335×D53C2semidirect
12. #20273127C2semidirect
13. #20273127C2semidirect
14. #20273003C2semidirect
15. C53×#1584C2semidirect
289. #1584D53semidirect
355. C53#18092semidirect

Maximal subgroups

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

subgroupindex
#202852882
#202838782
#202838742
#202736202
#202736152
#202735662
#202735462
#202735432
#202735302
#202735252
#335×D532
#202731272
#202730032
C53×#15842
#1809253