Finite Group Databasebeta

← #27354166

Subgroups — #27354166

Order 13568.

596 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20284799C211yesmaximal, characteristic
2#20283867C211yesmaximal, characteristic
3#20283794C211yesmaximal, characteristic, complemented
4#20273595C211yesmaximal, characteristic, complemented
5#20273563C211yesmaximal, characteristic, complemented
6#20273558C211yesmaximal, characteristic, complemented
7#20273531C211yesmaximal, characteristic, complemented
8#20273530C211yesmaximal, characteristic, complemented
9Q16×#88322C211yesmaximal, characteristic, complemented
10#20273514C211yesmaximal, characteristic, complemented
11#20273502C211yesmaximal, characteristic, complemented
12#20273142C211yesmaximal, characteristic, complemented
13#20273122C211yesmaximal, characteristic
14#20273016C211yesmaximal, characteristic, complemented
15C53×#1587C211yesmaximal, characteristic, complemented
16#16811160C2211yescharacteristic
17C4×Q8×D53C2211yescharacteristic
18#16810952C411yescharacteristic
19#16810938C411yescharacteristic
20#16810929C411yescharacteristic
21#16810928C411yescharacteristic
22#16810848C411yescharacteristic
23#16810847C411yescharacteristic
24#16810777C411yescharacteristic
25#16810763C411yescharacteristic
26Q8×#88322C2211yescharacteristic
27Q8×#88322C2211yescharacteristic
28C8×Dic106—21no—
29C8×Dic106C2211yescharacteristic
30#16810747—21no—
31#16810747C2211yescharacteristic
32C4×#11719925C2211yescharacteristic
33C4×#11719913C2211yescharacteristic
34C4×#11719924C2211yescharacteristic
35#16810702C2211yescharacteristic
36#16810700—21no—
37#16810696—21no—
38#16810688C2211yescharacteristic
39#16810685—21no—
40#16810675—21no—
41#16810675C2211yescharacteristic
42#16810674—21no—
43#16810674C2211yescharacteristic
44#16810674C2211yescharacteristic
45#16810674C2211yescharacteristic
46#16810648—21no—
47#16810645C2211yescharacteristic
48#16810376C2211yescharacteristic
49#16810375C2211yescharacteristic
50#16810373C2211yescharacteristic
51#16810355C2211yescharacteristic
52Q8⋊#88322C2211yescharacteristic
53Q8⋊#88322C2211yescharacteristic
54#16810353C2211yescharacteristic
55#16810352C2211yescharacteristic
56#16810350C2211yescharacteristic
57#16810345C2211yescharacteristic
58#16810344C2211yescharacteristic
59C8×#88322C2211yescharacteristic
60C53×#446C2211yescharacteristic
61C8×Q8×C53C2211yescharacteristic
62C4×Q16×C53C2211yescharacteristic
63C53×#334C2211yescharacteristic
64C53×Q8⋊C8C2211yescharacteristic
65C53×Q8⋊C8C2211yescharacteristic
66C53×#322C2211yescharacteristic
67C2×Q8×D53C2×C411yescharacteristic
68C2×#11719925C2×C411yescharacteristic
69C2×#11719913C2×C411yescharacteristic
70C2×#11719924C2×C411yescharacteristic
71C2×#11719912—21no—
72C2×#11719912—21no—
73C2×C8×D53—21no—
74#13133212—21no—
75#13133208C2×C411yescharacteristic
76#13133206C2×C411yescharacteristic
77#13133202—21no—
78#13133193C2×C411yescharacteristic
79#13133192C2×C411yescharacteristic
80Q8×Dic53C2×C411yescharacteristic
81C4×Dic106—21no—
82C4×Dic106C2311yescharacteristic
83#13133182—21no—
84#13133182C2×C411yescharacteristic
85C4×D212—21no—
86C4×D212C2311yescharacteristic
87#13133174—21no—
88C4⋊C4×D53—21no—
89C4⋊C4×D53C2×C411yescharacteristic
90C42×D53—21no—
91C42×D53C2311yescharacteristic
92#13133150C2×C411yescharacteristic
93#13133149C2×C411yescharacteristic
94#13133148C2×C411yescharacteristic
95C4⋊#88322—21no—
96C4⋊#88322—21no—
97C4⋊#88322C2311yescharacteristic
98C4⋊#88322C2311yescharacteristic
99#13133146—21no—
100#13133146—21no—
101#13133146—21no—
102#13133146—21no—
103#13133146C2×C411yescharacteristic
104#13133146C2×C411yescharacteristic
105#13133146D411yescharacteristic
106#13133146D411yescharacteristic
107#13133142C2×C411yescharacteristic
108#13133141C2×C411yescharacteristic
109#13133141C2×C411yescharacteristic
110#13133140C2×C411yescharacteristic
111#13133139C2×C411yescharacteristic
112#13133137C2×C411yescharacteristic
113#13133129—21no—
114#13133129—21no—
115#13133129—21no—
116#13133129—21no—
117#13133129C2×C411yescharacteristic
118#13133129C2×C411yescharacteristic
119#13133129D411yescharacteristic
120#13133129D411yescharacteristic
121C4×#88322—21no—
122C4×#88322—21no—
123C4×#88322C2311yescharacteristic
124C4×#88322C2311yescharacteristic
125C4×#88322C2311yescharacteristic
126C8×Dic53—21no—
127C8×Dic53—21no—
128C8×Dic53—21no—
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
146Q8×D53—22no—
147Q8×D53—22no—
148C2×D212—21no—
149C2×D212C22×C411yescharacteristic
150C2×C4×D53—21no—
151C2×C4×D53—21no—
152C2×C4×D53—21no—
153C2×C4×D53C22×C411yescharacteristic
154C2×C4×D53C2×D411yescharacteristic
155C2×C4×D53#4111yescharacteristic
156C2×Dic106—21no—
157C2×Dic106C22×C411yescharacteristic
158Q16×C53—22no—
159C2×C8×C53—21no—
160C2×C8×C53—21no—
161C2×C8×C53—21no—
162C2×C8×C53C22×C411yescharacteristic
163C2×C8×C53C22×C411yescharacteristic
164C2×C8×C53C2×D411yescharacteristic
165C2×C8×C53C2×D411yescharacteristic
166C4⋊C4×C53—21no—
167C4⋊C4×C53—21no—
168C4⋊C4×C53C22×C411yescharacteristic
169C4⋊C4×C53C22×C411yescharacteristic
170C42×C53—21no—
171C42×C53—21no—
172C42×C53C2411yescharacteristic
173#11719925—22no—
174#11719924—22no—
175#11719921—21no—
176#11719921—21no—
177#11719921—21no—
178#11719921—21no—
179#11719921—21no—
180C4⋊Dic53—21no—
181C4⋊Dic53—21no—
182C4⋊Dic53C22×C411yescharacteristic
183C4⋊Dic53C22×C411yescharacteristic
184#11719919—21no—
185#11719919—21no—
186#11719919—21no—
187#11719919—21no—
188#11719919—21no—
189C4×Dic53—21no—
190C4×Dic53—21no—
191C4×Dic53—21no—
192C4×Dic53C22×C411yescharacteristic
193C4×Dic53C2×D411yescharacteristic
194C4×Dic53#4111yescharacteristic
195C2×#88322—21no—
196C2×#88322—21no—
197C2×#88322—21no—
198C2×#88322C22×C411yescharacteristic
199C2×#88322C22×C411yescharacteristic
200C2×#88322C2×D411yescharacteristic
201C2×#88322C2×D411yescharacteristic
202#11719913—22no—
203#11719912—81no—
204#11719912—81no—
205C8×D53—81no—
206C22×D53—21no—
207C22×D53C4×D411yescharacteristic
208Q8×C53—21no—
209Q8×C53—21no—
210Q8×C53—21no—
211Q8×C53#13112yes—
212C2×C4×C53—21no—
213C2×C4×C53—21no—
214C2×C4×C53—21no—
215C2×C4×C53—21no—
216C2×C4×C53C23×C411yescharacteristic
217C2×C4×C53C22×D411yescharacteristic
218C2×C4×C53C2×#4111yescharacteristic
219C2×Dic53—21no—
220C2×Dic53—21no—
221C2×Dic53—21no—
222C2×Dic53C4×D411yescharacteristic
223C2×Dic53C4×D411yescharacteristic
224C2×Dic53C4×D411yescharacteristic
225D212—21no—
226D212—21no—
227D212—81no—
228C4×D53—24no—
229C4×D53—21no—
230C4×D53—21no—
231C4×D53—21no—
232C4×D53—21no—
233C4×D53—41no—
234C4×D53—41no—
235C4×D53—41no—
236C4×D53—81no—
237Dic106—24no—
238Dic106—21no—
239Dic106#13112yes—
240C8×C53—21no—
241C8×C53—21no—
242C8×C53—21no—
243C8×C53—21no—
244C8×C53—41no—
245C8×C53—41no—
246C8×C53—41no—
247#88322—21no—
248#88322—21no—
249#88322—21no—
250#88322—21no—
251#88322—41no—
252#88322—41no—
253#88322—41no—
254#17919—531nomaximal, Sylow
255C22×C53C2×C4×D411yescharacteristic
256C2×D53—21no—
257C2×D53—21no—
258C2×D53—21no—
259C2×D53—41no—
260C2×D53—41no—
261C2×D53—41no—
262C4×C53—22no—
263C4×C53—21no—
264C4×C53—41no—
265C4×C53—41no—
266C4×C53—41no—
267C4×C53C2×#13111yescharacteristic
268C4×C53C2×#13111yescharacteristic
269C4×C53#57711yescharacteristic
270C4×C53#57811yescharacteristic
271Dic53—21no—
272Dic53—22no—
273Dic53—21no—
274Dic53—21no—
275Dic53—41no—
276Dic53—41no—
277#3108—531no—
278#2993—531no—
279#2982—531no—
280#1588—531no—
281#1587D5311yescharacteristic, complemented
282#1584—531no—
283#1583—531no—
284#1579—531no—
285SD16⋊C8—531no—
286C8×Q16—531no—
287C8×SD16—531no—
288#1574—531no—
289#1487—531no—
290#1482—531no—
291#1454—531no—
292C2×C53#292911yescharacteristic
293C2×C53#294711yescharacteristic
294C2×C53#295711yescharacteristic
295D53—41no—
296D53—81no—
297#529—531no—
298C2×C4×Q8—531no—
299#483—531no—
300#480—531no—
301#479—531no—
302#479—531no—
303#464—531no—
304#464—531no—
305#451—531no—
306#448—531no—
307#446C2×D5311yescharacteristic
308C8×Q8—531no—
309C8×Q8—531no—
310C8×Q8—531no—
311C8×Q8—531no—
312C8×Q8—1061no—
313C8×Q8—1061no—
314C8×Q8C2×D5311yescharacteristic
315C4×Q16—531no—
316C4×Q16C2×D5311yescharacteristic
317C4×SD16—531no—
318C4×SD16—531no—
319#436—531no—
320#436—1061no—
321#435—1061no—
322C8×D4—531no—
323#433—1061no—
324#431—531no—
325#431—531no—
326#431—531no—
327#431—531no—
328#431—1061no—
329#431—1061no—
330#424—1061no—
331#423—531no—
332#335—531no—
333#335—531no—
334#334—531no—
335#334C2×D5311yescharacteristic
336Q8⋊C8—531no—
337Q8⋊C8—531no—
338Q8⋊C8—531no—
339Q8⋊C8—531no—
340Q8⋊C8C2×D5311yescharacteristic
341Q8⋊C8C2×D5311yescharacteristic
342D4⋊C8—531no—
343D4⋊C8—531no—
344#322—531no—
345#322—531no—
346#322C2×D5311yescharacteristic
347C82—531no—
348C53#1791911yescharacteristic, complemented, Sylow
349C22×Q8—531no—
350C2×Q16—531no—
351C2×Q16C4×D5311yescharacteristic
352C2×SD16—531no—
353C2×SD16—531no—
354C2×M16—1061no—
355C2×M16—1061no—
356C22×C8—1061no—
357#126—1061no—
358#125—531no—
359#124—531no—
360#123—1061no—
361C22⋊Q8—531no—
362#121—531no—
363C4×Q8—531no—
364C4×Q8—531no—
365C4×Q8—531no—
366C4×Q8—1061no—
367C4×Q8—1061no—
368C4×Q8C22×D5311yescharacteristic
369C4×Q8C22×D5311yescharacteristic
370C4×D4—531no—
371C4×D4—1061no—
372#117—1061no—
373C2×C4⋊C4—531no—
374C2×C4⋊C4—1061no—
375C2×C42—531no—
376C2×C42—1061no—
377#107—531no—
378#107C4×D5311yescharacteristic
379#106—531no—
380#106—531no—
381C4⋊C8—21no—
382C4⋊C8—21no—
383C4⋊C8—531no—
384C4⋊C8—531no—
385C4⋊C8—531no—
386C4⋊C8—531no—
387C4⋊C8—531no—
388C4⋊C8—531no—
389C4⋊C8—1061no—
390C4⋊C8—1061no—
391C4⋊C8—1061no—
392C4⋊C8—1061no—
393C4⋊C8—1061no—
394C4⋊C8—1061no—
395C4⋊C8C22×D5311yescharacteristic
396C4⋊C8C22×D5311yescharacteristic
397#103—531no—
398#103—531no—
399#103—531no—
400#103—531no—
401#103C4×D5311yescharacteristic
402#103C4×D5311yescharacteristic
403#102—531no—
404#102—531no—
405C22⋊C8—531no—
406C22⋊C8—531no—
407C22⋊C8—531no—
408C22⋊C8—531no—
409C22⋊C8—1061no—
410C22⋊C8—1061no—
411C22⋊C8—1061no—
412C22⋊C8—1061no—
413#97—21no—
414C4×C8—21no—
415C4×C8—531no—
416C4×C8—531no—
417C4×C8—531no—
418C4×C8—1061no—
419C4×C8—1061no—
420C4×C8—1061no—
421C4×C8—1061no—
422C4×C8—1061no—
423C4×C8C22×D5311yescharacteristic
424C4×C8C22×D5311yescharacteristic
425C4×C8C22×D5311yescharacteristic
426C2×Q8—531no—
427C2×Q8—1061no—
428C2×Q8—1062no—
429C2×Q8—1062no—
430C2×Q8C2×C4×D5311yescharacteristic
431C2×Q8C2×C4×D5311yescharacteristic
432C2×D4—531no—
433C2×D4—1061no—
434C22×C4—531no—
435C22×C4—531no—
436C22×C4—531no—
437C22×C4—1061no—
438C22×C4—1061no—
439C22×C4—1061no—
440Q16—22no—
441Q16—1062no—
442SD16—1062no—
443SD16—1062no—
444M16—4241no—
445M16—4241no—
446C2×C8—21no—
447C2×C8—21no—
448C2×C8—21no—
449C2×C8—531no—
450C2×C8—531no—
451C2×C8—531no—
452C2×C8—531no—
453C2×C8—1061no—
454C2×C8—1061no—
455C2×C8—1061no—
456C2×C8—4241no—
457C2×C8C2×C4×D5311yescharacteristic
458C2×C8C2×C4×D5311yescharacteristic
459C2×C8D4×D5311yescharacteristic
460C2×C8D4×D5311yescharacteristic
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—531no—
464C4⋊C4—531no—
465C4⋊C4—1061no—
466C4⋊C4—1061no—
467C4⋊C4—1061no—
468C4⋊C4—1061no—
469C4⋊C4—1061no—
470C4⋊C4—1061no—
471C4⋊C4—1061no—
472C4⋊C4C2×C4×D5311yescharacteristic
473C4⋊C4C2×C4×D5311yescharacteristic
474C22⋊C4—1061no—
475C22⋊C4—1061no—
476C22⋊C4—1061no—
477C22⋊C4—1061no—
478C22⋊C4—1061no—
479C42—21no—
480C42—21no—
481C42—531no—
482C42—531no—
483C42—531no—
484C42—1061no—
485C42—1061no—
486C42—1061no—
487C42C23×D5311yescharacteristic
488C23—531no—
489C23—1061no—
490Q8—21no—
491Q8—21no—
492Q8—21no—
493Q8—532no—
494Q8—1064no—
495Q8—1061no—
496Q8#1313322512yes—
497D4—1061no—
498D4—1061no—
499D4—4241no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—531no—
505C2×C4—531no—
506C2×C4—531no—
507C2×C4—1064no—
508C2×C4—1061no—
509C2×C4—1061no—
510C2×C4—1061no—
511C2×C4—1061no—
512C2×C4—1061no—
513C2×C4—1061no—
514C2×C4—1061no—
515C2×C4—4241no—
516C2×C4—2121no—
517C2×C4—2121no—
518C2×C4—2121no—
519C2×C4C22×C4×D5311yescharacteristic
520C2×C4C2×D4×D5311yescharacteristic
521C2×C4#41×D5311yescharacteristic
522C8—21no—
523C8—21no—
524C8—21no—
525C8—21no—
526C8—41no—
527C8—41no—
528C8—41no—
529C8—1061no—
530C8—1061no—
531C8—1061no—
532C8—1061no—
533C8—2121no—
534C8—2121no—
535C8—2121no—
536C22—1061no—
537C22—1061no—
538C22—1061no—
539C22—2121no—
540C22—2121no—
541C22—2121no—
542C22C4×D4×D5311yescharacteristic
543C4—22no—
544C4—21no—
545C4—41no—
546C4—41no—
547C4—41no—
548C4—1061no—
549C4—1062no—
550C4—1061no—
551C4—1061no—
552C4—2121no—
553C4—2121no—
554C4#131×D5311yescharacteristic
555C4C2×#1313322511yescharacteristic
556C4#1681143211yescharacteristic
557C4#1681143911yescharacteristic
558C2—4241no—
559C2—2121no—
560C2#2028340611yescharacteristic
561C2#2028355711yescharacteristic
562C2#2028362711yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC42C23×D53
FittingC53×#1587C2
socleC22×C53C2×C4×D4
prime radical#13133192C2×C4
cosocleC42×C53C24
hypercenter#1587D53
hypocenterC53#17919
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ #1587

Complements

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

NH ≅ G/Ntype
3. #20283794C2semidirect
4. #20273595C2semidirect
5. #20273563C2semidirect
6. #20273558C2semidirect
7. #20273531C2semidirect
8. #20273530C2semidirect
9. Q16×#88322C2semidirect
10. #20273514C2semidirect
11. #20273502C2semidirect
12. #20273142C2semidirect
14. #20273016C2semidirect
15. C53×#1587C2semidirect
281. #1587D53semidirect
348. C53#17919semidirect

Maximal subgroups

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

subgroupindex
#202847992
#202838672
#202837942
#202735952
#202735632
#202735582
#202735312
#202735302
Q16×#883222
#202735142
#202735022
#202731422
#202731222
#202730162
C53×#15872
#1791953