Finite Group Databasebeta

← #27366508

Subgroups — #27366508

Order 13568.

672 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20284497C211yesmaximal, characteristic, complemented
2#20284164C211yesmaximal, characteristic
3C8×#11719945C211yesmaximal, characteristic, complemented
4#20274142C211yesmaximal, characteristic, complemented
5#20274135C211yesmaximal, characteristic, complemented
6#20274128C211yesmaximal, characteristic, complemented
7#20274121C211yesmaximal, characteristic, complemented
8#20274094C211yesmaximal, characteristic, complemented
9#20274087C211yesmaximal, characteristic, complemented
10#20274060C211yesmaximal, characteristic, complemented
11#20274053C211yesmaximal, characteristic, complemented
12C2×#16810374C211yesmaximal, characteristic, complemented
13#20273372C211yesmaximal, characteristic, complemented
14#20273364C211yesmaximal, characteristic, complemented
15C53×#3167C211yesmaximal, characteristic, complemented
16C2×#13133237—21no—
17C4×#11719945C2211yescharacteristic
18#16810927—21no—
19#16810920—21no—
20#16810915C2211yescharacteristic
21#16810910C2211yescharacteristic
22#16810905C2211yescharacteristic
23#16810900C2211yescharacteristic
24#16810842—21no—
25#16810835—21no—
26#16810830C2211yescharacteristic
27#16810825C2211yescharacteristic
28#16810820C2211yescharacteristic
29#16810815C2211yescharacteristic
30C8×Dic106C2211yescharacteristic
31C8×Dic106C2211yescharacteristic
32C8×D212C2211yescharacteristic
33C8×D212C2211yescharacteristic
34C8×C22⋊D53—21no—
35C8×C22⋊D53—21no—
36#16810676C2211yescharacteristic
37#16810674—21no—
38#16810659—21no—
39C2×C4⋊#88322C2211yescharacteristic
40C2×C4⋊#88322C2211yescharacteristic
41#16810619—21no—
42#16810617—21no—
43C4×C8×D53—21no—
44#16810374C2211yescharacteristic
45#16810374C2211yescharacteristic
46#16810374C2211yescharacteristic
47#16810374C2211yescharacteristic
48#16810372C2211yescharacteristic
49#16810372C2211yescharacteristic
50#16810369C2211yescharacteristic
51#16810369C2211yescharacteristic
52#16810369C2211yescharacteristic
53#16810369C2211yescharacteristic
54#16810367C2211yescharacteristic
55#16810367C2211yescharacteristic
56C53×#533C2211yescharacteristic
57C53×#532C2211yescharacteristic
58C53×#498C2211yescharacteristic
59C53×#492C2211yescharacteristic
60C53×#487C2211yescharacteristic
61C53×#486C2211yescharacteristic
62C53×#465—21no—
63C53×#465—21no—
64C4×SD16×C53—21no—
65C4×SD16×C53—21no—
66C2×C4×C8×C53C2211yescharacteristic
67C2×#11719945D411yescharacteristic
68#13133237—44no—
69C2×#11719925—21no—
70C2×#11719923—21no—
71C2×#11719924—21no—
72C2×#11719922—21no—
73C22×#88322—21no—
74C22×#88322—21no—
75C2×C8×D53—21no—
76C2×C8×D53—21no—
77C4×Dic106D411yescharacteristic
78C4×Dic106C2311yescharacteristic
79C4×D212D411yescharacteristic
80C4×D212C2311yescharacteristic
81C4×C22⋊D53—21no—
82C4×C22⋊D53—21no—
83#13133176D411yescharacteristic
84#13133174—21no—
85C42×D53—21no—
86#13133150—21no—
87#13133150—21no—
88#13133148—21no—
89#13133148—21no—
90C4⋊#88322—21no—
91C4⋊#88322—21no—
92C4⋊#88322C2311yescharacteristic
93C4⋊#88322C2311yescharacteristic
94C4⋊#88322C2311yescharacteristic
95C4⋊#88322C2311yescharacteristic
96C4⋊#88322C2311yescharacteristic
97C4⋊#88322C2311yescharacteristic
98#13133146—21no—
99#13133146—21no—
100#13133141—21no—
101#13133141—21no—
102#13133140—21no—
103#13133140—21no—
104#13133138—21no—
105#13133138—21no—
106#13133137—21no—
107#13133137—21no—
108C22⋊#88322—21no—
109#13133129—21no—
110#13133129—21no—
111C4×#88322—21no—
112C8×Dic53—21no—
113C8×Dic53—21no—
114C2×#41×C53—21no—
115C2×SD16×C53—21no—
116C2×SD16×C53—21no—
117C22×C8×C53D411yescharacteristic
118C22×C8×C53D411yescharacteristic
119C4⋊Q8×C53C2311yescharacteristic
120C4⋊Q8×C53C2311yescharacteristic
121C4⋊Q8×C53C2311yescharacteristic
122#127×C53C2311yescharacteristic
123#125×C53—21no—
124#124×C53—21no—
125C22⋊Q8×C53—21no—
126C22⋊Q8×C53—21no—
127#121×C53—21no—
128#121×C53—21no—
129C4×Q8×C53—21no—
130C4×Q8×C53—21no—
131C4×D4×C53—21no—
132C4×D4×C53—21no—
133#117×C53—21no—
134C2×C42×C53C2311yescharacteristic
135#106×C53—21no—
136#106×C53—21no—
137#103×C53—21no—
138#103×C53—21no—
139#102×C53—21no—
140#102×C53—21no—
141C4×C8×C53D411yescharacteristic
142C4×C8×C53D411yescharacteristic
143C4×C8×C53C2311yescharacteristic
144C4×C8×C53C2311yescharacteristic
145#41×C53—42no—
146#41×C53—42no—
147C2×Q8×C53—21no—
148C2×Q8×C53—21no—
149C2×Q8×C53—21no—
150C2×D4×C53—21no—
151C2×D4×C53—21no—
152C2×D4×C53—21no—
153C2×D4×C53—41no—
154C22×C4×C53—41no—
155C22×C4×C53C2×D411yescharacteristic
156C22×C4×C53C2×D411yescharacteristic
157C22×C4×C53#4111yescharacteristic
158C2×C22⋊D53—21no—
159#11719945—22no—
160#11719945SD1614yescomplemented
161C2×D212C2×D411yescharacteristic
162C2×C4×D53—21no—
163C2×C4×D53—21no—
164C2×C4×D53—21no—
165C2×Dic106C2×D411yescharacteristic
166SD16×C53—42no—
167SD16×C53—42no—
168C2×C8×C53—22no—
169C2×C8×C53—41no—
170C2×C8×C53C2×D411yescharacteristic
171C2×C8×C53C2×D411yescharacteristic
172C2×C8×C53C2×D411yescharacteristic
173C2×C8×C53C2×D411yescharacteristic
174C4⋊C4×C53—21no—
175C4⋊C4×C53—21no—
176C4⋊C4×C53—21no—
177C4⋊C4×C53—21no—
178C4⋊C4×C53—21no—
179C4⋊C4×C53—21no—
180C4⋊C4×C53—41no—
181C4⋊C4×C53—41no—
182C22⋊C4×C53—41no—
183C22⋊C4×C53—41no—
184C22⋊C4×C53—41no—
185C42×C53—41no—
186C42×C53C2×D411yescharacteristic
187C42×C53C2×D411yescharacteristic
188C42×C53#4111yescharacteristic
189C42×C53C2411yescharacteristic
190C22⋊Dic53—21no—
191#11719925—42no—
192#11719924—42no—
193#11719923—42no—
194#11719922—42no—
195#11719921—21no—
196#11719921—21no—
197C4⋊Dic53C2×D411yescharacteristic
198C4⋊Dic53C2×D411yescharacteristic
199#11719919—21no—
200#11719919—21no—
201C4×Dic53—21no—
202C4×Dic53—21no—
203C4×Dic53—21no—
204C2×#88322—21no—
205C2×#88322—21no—
206C2×#88322—21no—
207C2×#88322—21no—
208C2×#88322—21no—
209C2×#88322—21no—
210C2×#88322—42no—
211C2×#88322—42no—
212C8×D53—42no—
213C8×D53—42no—
214C23×C53—41no—
215C23×C53#12111yescharacteristic
216C22×D53—21no—
217Q8×C53—41no—
218Q8×C53—41no—
219Q8×C53—41no—
220Q8×C53—41no—
221Q8×C53—81no—
222D4×C53—41no—
223D4×C53—41no—
224D4×C53—41no—
225D4×C53—41no—
226D4×C53—81no—
227D4×C53—82no—
228C2×C4×C53—21no—
229C2×C4×C53—21no—
230C2×C4×C53—41no—
231C2×C4×C53—41no—
232C2×C4×C53—41no—
233C2×C4×C53—41no—
234C2×C4×C53—82no—
235C2×C4×C53#12111yescharacteristic
236C2×C4×C53#12111yescharacteristic
237C2×C4×C53#12111yescharacteristic
238C2×C4×C53C2×SD1612yes—
239C2×C4×C53#13512yes—
240C2×C4×C53C22×D411yescharacteristic
241C2×C4×C53C22×D411yescharacteristic
242C2×C4×C53C2×#4111yescharacteristic
243C22⋊D53—24no—
244C2×Dic53—21no—
245C2×Dic53—21no—
246C2×Dic53—21no—
247D212—21no—
248D212C2×SD1612yes—
249C4×D53—24no—
250C4×D53—42no—
251C4×D53—41no—
252C4×D53—41no—
253Dic106—21no—
254Dic106C2×SD1612yes—
255C8×C53—21no—
256C8×C53—21no—
257C8×C53—21no—
258C8×C53#13512yescomplemented
259#88322—41no—
260#88322—41no—
261#88322—41no—
262#88322—41no—
263#88322—41no—
264#88322—41no—
265#19124—531nomaximal, Sylow
266C22×C53—21no—
267C22×C53—21no—
268C22×C53—81no—
269C22×C53—81no—
270C22×C53—81no—
271C22×C53#46512yes—
272C22×C53C2×#12111yescharacteristic
273C2×D53—22no—
274C2×D53—41no—
275C2×D53—41no—
276C4×C53—21no—
277C4×C53—21no—
278C4×C53—21no—
279C4×C53—81no—
280C4×C53—81no—
281C4×C53—81no—
282C4×C53#46512yes—
283C4×C53C22×SD1611yescharacteristic
284C4×C53C2×#13511yescharacteristic
285C4×C53C2×#13511yescharacteristic
286C4×C53#57511yescharacteristic
287Dic53—22no—
288Dic53—41no—
289Dic53—41no—
290#3167D5311yescharacteristic, complemented
291#3076—531no—
292#3035—531no—
293C8×#41—531no—
294#1680—531no—
295#1679—531no—
296#1678—531no—
297#1677—531no—
298#1672—531no—
299#1671—531no—
300#1666—531no—
301#1665—531no—
302C2×#334—531no—
303#1542—531no—
304#1540—531no—
305C2×C53—22no—
306C2×C53—161no—
307C2×C53#303511yescharacteristic
308C2×C53C2×#46511yescharacteristic
309C2×C53#306711yescharacteristic
310D53—42no—
311C2×#135—1061no—
312#533C2×D5311yescharacteristic
313#532C2×D5311yescharacteristic
314C4×#41—531no—
315#498C2×D5311yescharacteristic
316#492C2×D5311yescharacteristic
317#487C2×D5311yescharacteristic
318#486C2×D5311yescharacteristic
319#479—1061no—
320#478—1061no—
321#477—531no—
322#476—531no—
323#475—531no—
324#474—531no—
325#465—21no—
326#465—21no—
327#464—1061no—
328#463—1061no—
329#462—531no—
330#461—531no—
331#460—531no—
332#459—531no—
333C8×Q8—531no—
334C8×Q8—531no—
335C4×SD16—21no—
336C4×SD16—21no—
337C8×D4—531no—
338C8×D4—531no—
339C8×D4—1061no—
340C8×D4—1061no—
341#431—531no—
342#431—1061no—
343#427—1061no—
344C2×C4⋊C8—531no—
345C2×C4⋊C8—531no—
346#416—1061no—
347#416—1061no—
348C2×C4×C8—1061no—
349C2×C4×C8C2×D5311yescharacteristic
350#334—531no—
351#334—531no—
352#334—531no—
353#334—531no—
354#333—531no—
355#333—531no—
356#332—531no—
357#332—531no—
358#332—531no—
359#332—531no—
360#331—531no—
361#331—531no—
362C53#1912411yescharacteristic, complemented, Sylow
363C2×#41—21no—
364C2×#41—531no—
365#135—2124no—
366C2×Q16—1061no—
367C2×SD16—21no—
368C2×SD16—21no—
369C2×SD16—1061no—
370C2×SD16—1061no—
371C2×D8—1061no—
372C22×C8—1061no—
373C22×C8—1061no—
374C22×C8—1061no—
375C22×C8—1061no—
376C22×C8C22⋊D5311yescharacteristic
377C22×C8C22⋊D5311yescharacteristic
378C4⋊Q8C22×D5311yescharacteristic
379C4⋊Q8C22×D5311yescharacteristic
380C4⋊Q8C22×D5311yescharacteristic
381#127C22×D5311yescharacteristic
382#125—21no—
383#124—21no—
384C22⋊Q8—21no—
385C22⋊Q8—21no—
386#121—21no—
387#121—21no—
388C4×Q8—21no—
389C4×Q8—21no—
390C4×Q8—531no—
391C4×Q8—531no—
392C4×D4—21no—
393C4×D4—21no—
394C4×D4—531no—
395C4×D4—531no—
396C4×D4—1061no—
397C4×D4—1061no—
398#117—21no—
399#117—531no—
400#117—1061no—
401C2×C42—1061no—
402C2×C42C22×D5311yescharacteristic
403#107—1061no—
404#107—1061no—
405#106—21no—
406#106—21no—
407#106—1061no—
408#106—1061no—
409C4⋊C8—531no—
410C4⋊C8—531no—
411C4⋊C8—531no—
412C4⋊C8—531no—
413C4⋊C8—531no—
414C4⋊C8—531no—
415C4⋊C8—1061no—
416C4⋊C8—1061no—
417C4⋊C8—1061no—
418C4⋊C8—1061no—
419#103—21no—
420#103—21no—
421#103—1061no—
422#103—1061no—
423#103—1061no—
424#103—1061no—
425#102—21no—
426#102—21no—
427#102—1061no—
428#102—1061no—
429#102—1061no—
430#102—1061no—
431C22⋊C8—1061no—
432C22⋊C8—1061no—
433C22⋊C8—1061no—
434C4×C8—1061no—
435C4×C8—1061no—
436C4×C8—1061no—
437C4×C8C22⋊D5311yescharacteristic
438C4×C8C22⋊D5311yescharacteristic
439C4×C8C22×D5311yescharacteristic
440C4×C8C22×D5311yescharacteristic
441#41—42no—
442#41—42no—
443#41—534no—
444#41—1062no—
445C2×Q8—21no—
446C2×Q8—21no—
447C2×Q8—21no—
448C2×Q8—531no—
449C2×D4—21no—
450C2×D4—21no—
451C2×D4—21no—
452C2×D4—41no—
453C2×D4—531no—
454C2×D4—1061no—
455C22×C4—41no—
456C22×C4—1061no—
457C22×C4—1061no—
458C22×C4—1061no—
459C22×C4D4×D5311yescharacteristic
460C22×C4#1171994711yescharacteristic
461C22×C4C2×C22⋊D5311yescharacteristic
462Q16—2122no—
463SD16—42no—
464SD16—42no—
465SD16—2122no—
466SD16—2122no—
467D8—2122no—
468C2×C8—22no—
469C2×C8—41no—
470C2×C8—1061no—
471C2×C8—1061no—
472C2×C8—1061no—
473C2×C8—1061no—
474C2×C8—1061no—
475C2×C8—1061no—
476C2×C8—2122no—
477C2×C8—2122no—
478C2×C8—2122no—
479C2×C8—2122no—
480C2×C8C2×C22⋊D5311yescharacteristic
481C2×C8C2×C22⋊D5311yescharacteristic
482C2×C8C2×C22⋊D5311yescharacteristic
483C2×C8C2×C22⋊D5311yescharacteristic
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—41no—
491C4⋊C4—41no—
492C4⋊C4—531no—
493C4⋊C4—531no—
494C4⋊C4—1061no—
495C4⋊C4—1061no—
496C22⋊C4—41no—
497C22⋊C4—41no—
498C22⋊C4—41no—
499C22⋊C4—1061no—
500C22⋊C4—1061no—
501C22⋊C4—1061no—
502C42—41no—
503C42—1061no—
504C42—1061no—
505C42—1061no—
506C42D4×D5311yescharacteristic
507C42#1171994711yescharacteristic
508C42C2×C22⋊D5311yescharacteristic
509C42C23×D5311yescharacteristic
510C23—41no—
511C23—1061no—
512C23#1313318811yescharacteristic
513Q8—41no—
514Q8—41no—
515Q8—41no—
516Q8—41no—
517Q8—81no—
518Q8—532no—
519Q8—1061no—
520D4—41no—
521D4—41no—
522D4—41no—
523D4—41no—
524D4—81no—
525D4—82no—
526D4—532no—
527D4—1064no—
528D4—1061no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—82no—
536C2×C4—1061no—
537C2×C4—1064no—
538C2×C4—1061no—
539C2×C4—1061no—
540C2×C4—2122no—
541C2×C4—2121no—
542C2×C4—2121no—
543C2×C4#1313318811yescharacteristic
544C2×C4#1313318811yescharacteristic
545C2×C4#1313318811yescharacteristic
546C2×C4SD16×D5312yes—
547C2×C4#1313323812yes—
548C2×C4C2×D4×D5311yescharacteristic
549C2×C4C22×C22⋊D5311yescharacteristic
550C2×C4C2×#1171994711yescharacteristic
551C8—21no—
552C8—21no—
553C8—21no—
554C8—2121no—
555C8—2121no—
556C8—2121no—
557C8—2121no—
558C8—2121no—
559C8—2121no—
560C8#1313323712yescomplemented
561C22—21no—
562C22—21no—
563C22—81no—
564C22—81no—
565C22—81no—
566C22—1062no—
567C22—2121no—
568C22—2121no—
569C22#1681084912yes—
570C22C2×#1313318811yescharacteristic
571C4—21no—
572C4—21no—
573C4—21no—
574C4—81no—
575C4—81no—
576C4—81no—
577C4—1062no—
578C4—2121no—
579C4—2121no—
580C4#1681084912yes—
581C4C2×SD16×D5311yescharacteristic
582C4C2×#1313323711yescharacteristic
583C4C2×#1313323811yescharacteristic
584C4#1681142111yescharacteristic
585C2—22no—
586C2—161no—
587C2—2122no—
588C2#2028416411yescharacteristic
589C2C2×#1681084911yescharacteristic
590C2#2028443111yescharacteristic
subgroupG/H
centerC2×C4#13133188
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#3167C2
socleC22×C53C2×#121
prime radical#20284164C2
hypercenter#3167D53
hypocenterC53#19124
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3167

Complements

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

NH ≅ G/Ntype
1. #20284497C2semidirect
3. C8×#11719945C2semidirect
4. #20274142C2semidirect
5. #20274135C2semidirect
6. #20274128C2semidirect
7. #20274121C2semidirect
8. #20274094C2semidirect
9. #20274087C2semidirect
10. #20274060C2semidirect
11. #20274053C2semidirect
12. C2×#16810374C2semidirect
13. #20273372C2semidirect
14. #20273364C2semidirect
15. C53×#3167C2semidirect
160. #11719945SD16semidirect
258. C8×C53#135semidirect
290. #3167D53semidirect
362. C53#19124semidirect
560. C8#13133237semidirect

Maximal subgroups

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

subgroupindex
#202844972
#202841642
C8×#117199452
#202741422
#202741352
#202741282
#202741212
#202740942
#202740872
#202740602
#202740532
C2×#168103742
#202733722
#202733642
C53×#31672
#1912453