Finite Group Databasebeta

← #27354732

Subgroups — #27354732

Order 13568.

582 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285319C211yesmaximal, characteristic
2#20283867C211yesmaximal, characteristic
3#20283811C211yesmaximal, characteristic, complemented
4#20273622C211yesmaximal, characteristic, complemented
5#20273615C211yesmaximal, characteristic, complemented
6#20273607C211yesmaximal, characteristic, complemented
7#20273534C211yesmaximal, characteristic, complemented
8#20273526C211yesmaximal, characteristic, complemented
9#20273519C211yesmaximal, characteristic, complemented
10#20273512C211yesmaximal, characteristic, complemented
11#20273481C211yesmaximal, characteristic, complemented
12#20273174C211yesmaximal, characteristic
13#20273168C211yesmaximal, characteristic, complemented
14C8×#11719912C211yesmaximal, characteristic, complemented
15C53×#1591C211yesmaximal, characteristic, complemented
16#16811177C2211yescharacteristic
17#16811160C2211yescharacteristic
18#16811031C411yescharacteristic
19#16811016C411yescharacteristic
20#16810990C411yescharacteristic
21#16810984C411yescharacteristic
22#16810880C411yescharacteristic
23#16810878C411yescharacteristic
24#16810872C411yescharacteristic
25#16810870C411yescharacteristic
26#16810753C2211yescharacteristic
27#16810751—21no—
28#16810750—21no—
29Q8×#88322C2211yescharacteristic
30Q8×#88322C2211yescharacteristic
31#16810747C2211yescharacteristic
32C4×#11719925C2211yescharacteristic
33C4×#11719913C2211yescharacteristic
34C4×#11719924C2211yescharacteristic
35#16810702C2211yescharacteristic
36#16810700—21no—
37#16810696—21no—
38#16810688C2211yescharacteristic
39#16810685—21no—
40#16810685—21no—
41#16810683—21no—
42#16810679C2211yescharacteristic
43#16810674C2211yescharacteristic
44#16810648—21no—
45#16810645C2211yescharacteristic
46C4×#11719912C2211yescharacteristic
47C4×C8×D53C2211yescharacteristic
48#16810375C2211yescharacteristic
49#16810374C2211yescharacteristic
50#16810373C2211yescharacteristic
51#16810355C2211yescharacteristic
52Q8⋊#88322C2211yescharacteristic
53Q8⋊#88322C2211yescharacteristic
54#16810353C2211yescharacteristic
55#16810352C2211yescharacteristic
56#16810350C2211yescharacteristic
57#16810345C2211yescharacteristic
58#16810344C2211yescharacteristic
59C8×#88322C2211yescharacteristic
60C53×#446C2211yescharacteristic
61C53×#446C2211yescharacteristic
62C4×Q16×C53C2211yescharacteristic
63C53×#335C2211yescharacteristic
64C53×Q8⋊C8C2211yescharacteristic
65C53×Q8⋊C8C2211yescharacteristic
66C82×C53C2211yescharacteristic
67C2×#11719925C2×C411yescharacteristic
68C2×#11719913C2×C411yescharacteristic
69C2×#11719924C2×C411yescharacteristic
70C2×#11719912—21no—
71C2×#11719912—21no—
72C2×#11719912C2×C411yescharacteristic
73C2×#11719912C2×C411yescharacteristic
74C2×C8×D53—21no—
75C2×C8×D53—21no—
76#13133216C2×C411yescharacteristic
77#13133215C2×C411yescharacteristic
78#13133212—21no—
79#13133208C2×C411yescharacteristic
80#13133207—21no—
81#13133206C2×C411yescharacteristic
82#13133202—21no—
83#13133197C2×C411yescharacteristic
84#13133195—21no—
85#13133193C2×C411yescharacteristic
86#13133193C2×C411yescharacteristic
87#13133192C2×C411yescharacteristic
88C4×Dic106—21no—
89C4×Dic106C2311yescharacteristic
90C4×D212—21no—
91C4×D212C2311yescharacteristic
92#13133174—21no—
93#13133174—21no—
94C42×D53C2311yescharacteristic
95#13133150C2×C411yescharacteristic
96#13133149C2×C411yescharacteristic
97#13133148C2×C411yescharacteristic
98C4⋊#88322—21no—
99C4⋊#88322—21no—
100C4⋊#88322C2311yescharacteristic
101C4⋊#88322C2311yescharacteristic
102#13133146—21no—
103#13133146—21no—
104#13133146—21no—
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#13133129D411yescharacteristic
117#13133129D411yescharacteristic
118#13133127—21no—
119#13133127—21no—
120#13133127C2×C411yescharacteristic
121#13133127C2×C411yescharacteristic
122C4×#88322—21no—
123C4×#88322—21no—
124C4×#88322C2311yescharacteristic
125C4×#88322C2311yescharacteristic
126C4×#88322C2311yescharacteristic
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—
140#97×C53—21no—
141C4×C8×C53C2311yescharacteristic
142C4×C8×C53C2311yescharacteristic
143C4×C8×C53C2311yescharacteristic
144C2×Q8×C53C22×C411yescharacteristic
145C2×Q8×C53C22×C411yescharacteristic
146C2×D212—21no—
147C2×D212C22×C411yescharacteristic
148C2×C4×D53—21no—
149C2×C4×D53C22×C411yescharacteristic
150C2×C4×D53C2×D411yescharacteristic
151C2×C4×D53#4111yescharacteristic
152C2×Dic106—21no—
153C2×Dic106C22×C411yescharacteristic
154Q16×C53—22no—
155C2×C8×C53—21no—
156C2×C8×C53—21no—
157C2×C8×C53—21no—
158C2×C8×C53C22×C411yescharacteristic
159C2×C8×C53C22×C411yescharacteristic
160C2×C8×C53C2×D411yescharacteristic
161C2×C8×C53C2×D411yescharacteristic
162C4⋊C4×C53—21no—
163C4⋊C4×C53—21no—
164C4⋊C4×C53C22×C411yescharacteristic
165C4⋊C4×C53C22×C411yescharacteristic
166C42×C53—21no—
167C42×C53—21no—
168C42×C53C2411yescharacteristic
169#11719925—22no—
170#11719924—22no—
171#11719921—21no—
172#11719921—21no—
173#11719921—21no—
174#11719921—21no—
175#11719921—21no—
176#11719921—21no—
177#11719921—21no—
178C4⋊Dic53—21no—
179C4⋊Dic53—21no—
180C4⋊Dic53C22×C411yescharacteristic
181C4⋊Dic53C22×C411yescharacteristic
182#11719919—21no—
183#11719919—21no—
184#11719919—21no—
185#11719919—21no—
186#11719919—21no—
187#11719919—21no—
188#11719919—21no—
189C4×Dic53—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
200#11719913—22no—
201#11719912—22no—
202#11719912—22no—
203#11719912—81no—
204#11719912—81no—
205C8×D53—42no—
206C8×D53—81no—
207C22×D53—21no—
208C22×D53C4×D411yescharacteristic
209Q8×C53—21no—
210Q8×C53—21no—
211Q8×C53—21no—
212Q8×C53—21no—
213C2×C4×C53—21no—
214C2×C4×C53—21no—
215C2×C4×C53—21no—
216C2×C4×C53—21no—
217C2×C4×C53C23×C411yescharacteristic
218C2×C4×C53C22×D411yescharacteristic
219C2×C4×C53C2×#4111yescharacteristic
220C2×Dic53—21no—
221C2×Dic53—21no—
222C2×Dic53—21no—
223C2×Dic53C4×D411yescharacteristic
224C2×Dic53C4×D411yescharacteristic
225C2×Dic53C4×D411yescharacteristic
226D212—21no—
227D212—21no—
228D212—81no—
229C4×D53—21no—
230C4×D53—21no—
231C4×D53—21no—
232C4×D53—21no—
233C4×D53—41no—
234C4×D53—41no—
235C4×D53—41no—
236Dic106—21no—
237Dic106—21no—
238Dic106—81no—
239C8×C53—22no—
240C8×C53—21no—
241C8×C53—21no—
242C8×C53—21no—
243C8×C53—41no—
244C8×C53—41no—
245C8×C53#13112yes—
246#88322—21no—
247#88322—21no—
248#88322—21no—
249#88322—41no—
250#88322—41no—
251#88322—41no—
252#88322#13112yes—
253#17969—531nomaximal, Sylow
254C22×C53C2×C4×D411yescharacteristic
255C2×D53—21no—
256C2×D53—21no—
257C2×D53—21no—
258C2×D53—41no—
259C2×D53—41no—
260C2×D53—41no—
261C4×C53—21no—
262C4×C53—41no—
263C4×C53—41no—
264C4×C53—41no—
265C4×C53—41no—
266C4×C53C2×#13111yescharacteristic
267C4×C53C2×#13111yescharacteristic
268C4×C53#57711yescharacteristic
269C4×C53#57811yescharacteristic
270Dic53—21no—
271Dic53—21no—
272Dic53—21no—
273Dic53—41no—
274Dic53—41no—
275Dic53—41no—
276#3172—531no—
277#2993—531no—
278#2985—531no—
279#1593—531no—
280#1592—531no—
281#1591D5311yescharacteristic, complemented
282#1590—531no—
283#1579—531no—
284SD16⋊C8—531no—
285C8×Q16—531no—
286C8×SD16—531no—
287#1569—531no—
288#1494—531no—
289#1493—531no—
290C8×M16—531no—
291C2×C53#293111yescharacteristic
292C2×C53#294711yescharacteristic
293C2×C53#295611yescharacteristic
294D53—41no—
295D53—81no—
296#533—531no—
297#529—531no—
298#501—531no—
299#497—531no—
300#491—531no—
301#489—531no—
302#470—531no—
303#470—531no—
304#468—531no—
305#468—531no—
306#446—531no—
307#446—1061no—
308#446—1061no—
309#446C2×D5311yescharacteristic
310#446C2×D5311yescharacteristic
311C8×Q8—531no—
312C8×Q8—531no—
313C8×Q8—531no—
314C4×Q16—531no—
315C4×Q16C2×D5311yescharacteristic
316C4×SD16—531no—
317C4×SD16—531no—
318#436—531no—
319#436—1061no—
320#435—1061no—
321C8×D4—531no—
322#433—1061no—
323#433—1061no—
324#433—1061no—
325#432—531no—
326#431—531no—
327#424—1061no—
328#423—531no—
329C4×M16—531no—
330C2×C4×C8—531no—
331#335—531no—
332#335C2×D5311yescharacteristic
333#334—531no—
334#334—531no—
335Q8⋊C8—531no—
336Q8⋊C8—531no—
337Q8⋊C8—531no—
338Q8⋊C8—531no—
339Q8⋊C8C2×D5311yescharacteristic
340Q8⋊C8C2×D5311yescharacteristic
341D4⋊C8—531no—
342D4⋊C8—531no—
343#322—531no—
344#322—531no—
345C82—531no—
346C82C2×D5311yescharacteristic
347C53#1796911yescharacteristic, complemented, Sylow
348C2×Q16—531no—
349C2×Q16C4×D5311yescharacteristic
350C2×SD16—531no—
351C2×SD16—531no—
352C2×M16—531no—
353C2×M16—531no—
354C2×M16—1061no—
355C2×M16—1061no—
356C22×C8—1061no—
357C22×C8—1061no—
358C4⋊Q8—531no—
359C4⋊Q8—531no—
360#126—1061no—
361#125—531no—
362#125—1061no—
363#124—531no—
364#123—1061no—
365C22⋊Q8—531no—
366C22⋊Q8—531no—
367C22⋊Q8—531no—
368C22⋊Q8—1061no—
369#121—531no—
370C4×Q8—531no—
371C4×Q8—1061no—
372C4×Q8C22×D5311yescharacteristic
373C4×Q8C22×D5311yescharacteristic
374C4×D4—531no—
375C4×D4—1061no—
376#117—1061no—
377#117—1061no—
378C2×C42—531no—
379#107—531no—
380#107C4×D5311yescharacteristic
381#106—531no—
382#106—531no—
383C4⋊C8—21no—
384C4⋊C8—21no—
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⋊C8C22×D5311yescharacteristic
395C4⋊C8C22×D5311yescharacteristic
396#103—531no—
397#103—531no—
398#103—531no—
399#103—531no—
400#103C4×D5311yescharacteristic
401#103C4×D5311yescharacteristic
402#102—531no—
403#102—531no—
404C22⋊C8—531no—
405C22⋊C8—531no—
406C22⋊C8—1061no—
407C22⋊C8—1061no—
408C22⋊C8—1061no—
409#97—21no—
410#97—21no—
411#97—531no—
412#97—531no—
413#97—1061no—
414#97—1061no—
415C4×C8—531no—
416C4×C8—531no—
417C4×C8—531no—
418C4×C8—1061no—
419C4×C8—1061no—
420C4×C8—1061no—
421C4×C8—1061no—
422C4×C8C22×D5311yescharacteristic
423C4×C8C22×D5311yescharacteristic
424C4×C8C22×D5311yescharacteristic
425C2×Q8—531no—
426C2×Q8—1061no—
427C2×Q8C2×C4×D5311yescharacteristic
428C2×Q8C2×C4×D5311yescharacteristic
429C2×D4—531no—
430C2×D4—1061no—
431C22×C4—531no—
432C22×C4—531no—
433C22×C4—531no—
434C22×C4—1061no—
435Q16—22no—
436Q16—1062no—
437SD16—1062no—
438SD16—1062no—
439M16—1062no—
440M16—1062no—
441M16—4241no—
442M16—4241no—
443C2×C8—21no—
444C2×C8—21no—
445C2×C8—21no—
446C2×C8—531no—
447C2×C8—531no—
448C2×C8—531no—
449C2×C8—531no—
450C2×C8—1061no—
451C2×C8—1061no—
452C2×C8—1061no—
453C2×C8—4241no—
454C2×C8—2122no—
455C2×C8C2×C4×D5311yescharacteristic
456C2×C8C2×C4×D5311yescharacteristic
457C2×C8D4×D5311yescharacteristic
458C2×C8D4×D5311yescharacteristic
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—531no—
462C4⋊C4—531no—
463C4⋊C4—1061no—
464C4⋊C4—1061no—
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—
479C22⋊C4—1061no—
480C22⋊C4—1061no—
481C42—21no—
482C42—21no—
483C42—531no—
484C42—531no—
485C42—531no—
486C42—1061no—
487C42C23×D5311yescharacteristic
488C23—531no—
489C23—1061no—
490Q8—21no—
491Q8—21no—
492Q8—21no—
493Q8—21no—
494Q8—1061no—
495Q8—1061no—
496Q8—4241no—
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—1061no—
508C2×C4—1061no—
509C2×C4—1061no—
510C2×C4—1061no—
511C2×C4—1061no—
512C2×C4—1061no—
513C2×C4—1061no—
514C2×C4—2121no—
515C2×C4—2121no—
516C2×C4—2121no—
517C2×C4C22×C4×D5311yescharacteristic
518C2×C4C2×D4×D5311yescharacteristic
519C2×C4#41×D5311yescharacteristic
520C8—22no—
521C8—21no—
522C8—21no—
523C8—21no—
524C8—41no—
525C8—41no—
526C8—532no—
527C8—1061no—
528C8—1061no—
529C8—1061no—
530C8—2121no—
531C8—2121no—
532C8—2121no—
533C8#1313322512yes—
534C22—1061no—
535C22—1061no—
536C22—1061no—
537C22—2121no—
538C22—2121no—
539C22—2121no—
540C22C4×D4×D5311yescharacteristic
541C4—21no—
542C4—41no—
543C4—41no—
544C4—41no—
545C4—41no—
546C4—1061no—
547C4—1061no—
548C4—1061no—
549C4—2121no—
550C4—2121no—
551C4—2121no—
552C4#131×D5311yescharacteristic
553C4C2×#1313322511yescharacteristic
554C4#1681143011yescharacteristic
555C4#1681143611yescharacteristic
556C2—4241no—
557C2—2121no—
558C2#2028342411yescharacteristic
559C2#2028355311yescharacteristic
560C2#2028362011yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC42C23×D53
FittingC53×#1591C2
socleC22×C53C2×C4×D4
prime radical#13133192C2×C4
cosocleC42×C53C24
hypercenter#1591D53
hypocenterC53#17969
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ #1591

Complements

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

NH ≅ G/Ntype
3. #20283811C2semidirect
4. #20273622C2semidirect
5. #20273615C2semidirect
6. #20273607C2semidirect
7. #20273534C2semidirect
8. #20273526C2semidirect
9. #20273519C2semidirect
10. #20273512C2semidirect
11. #20273481C2semidirect
13. #20273168C2semidirect
14. C8×#11719912C2semidirect
15. C53×#1591C2semidirect
281. #1591D53semidirect
347. C53#17969semidirect

Maximal subgroups

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

subgroupindex
#202853192
#202838672
#202838112
#202736222
#202736152
#202736072
#202735342
#202735262
#202735192
#202735122
#202734812
#202731742
#202731682
C8×#117199122
C53×#15912
#1796953