Finite Group Databasebeta

← #27368416

Subgroups — #27368416

Order 13568.

654 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285453C211yesmaximal, characteristic
2#20284837C211yesmaximal, characteristic, complemented
3#20284187C211yesmaximal, characteristic, complemented
4#20274283C211yesmaximal, characteristic, complemented
5#20274255C211yesmaximal, characteristic, complemented
6#20274221C211yesmaximal, characteristic, complemented
7#20274199C211yesmaximal, characteristic, complemented
8#20274159C211yesmaximal, characteristic, complemented
9#20274131C211yesmaximal, characteristic, complemented
10#20274097C211yesmaximal, characteristic, complemented
11#20274075C211yesmaximal, characteristic, complemented
12#20273482C211yesmaximal, characteristic, complemented
13#20273448C211yesmaximal, characteristic, complemented
14#20273435C211yesmaximal, characteristic, complemented
15Q8×M16×C53C211yesmaximal, characteristic, complemented
16C2×#13133248—21no—
17#16811173C2211yescharacteristic
18#16811173C2211yescharacteristic
19#16811021C2211yescharacteristic
20#16811000C2211yescharacteristic
21#16810970C2211yescharacteristic
22#16810970C2211yescharacteristic
23#16810936—21no—
24#16810918C2211yescharacteristic
25#16810918C2211yescharacteristic
26#16810876—21no—
27#16810876—21no—
28#16810845—21no—
29#16810845—21no—
30#16810833C2211yescharacteristic
31#16810833C2211yescharacteristic
32#16810823C2211yescharacteristic
33#16810823C2211yescharacteristic
34#16810823C2211yescharacteristic
35#16810823C2211yescharacteristic
36#16810762—21no—
37#16810736—21no—
38#16810736—21no—
39#16810721—21no—
40#16810721—21no—
41#16810680C2211yescharacteristic
42#16810647C2211yescharacteristic
43#16810622—21no—
44#16810376C2211yescharacteristic
45#16810376C2211yescharacteristic
46#16810374C2211yescharacteristic
47#16810374C2211yescharacteristic
48#16810371C2211yescharacteristic
49#16810371C2211yescharacteristic
50#16810368C2211yescharacteristic
51#16810368C2211yescharacteristic
52#16810366C2211yescharacteristic
53#16810366C2211yescharacteristic
54#16810366C2211yescharacteristic
55#16810366C2211yescharacteristic
56C2×C4×Q8×C53C2211yescharacteristic
57C53×#446—21no—
58C53×#446—21no—
59C53×#446C2211yescharacteristic
60C53×#446C2211yescharacteristic
61C8×Q8×C53C2211yescharacteristic
62C8×Q8×C53C2211yescharacteristic
63C53×#423—21no—
64C53×#423C2211yescharacteristic
65C4×M16×C53—21no—
66C4×M16×C53C2211yescharacteristic
67C2×#11719945—21no—
68C2×#11719945—21no—
69#13133248—44no—
70C2×#11719925—21no—
71C2×#11719925—21no—
72C2×#11719924—21no—
73C2×#11719924—21no—
74C2×#11719924—21no—
75C2×#11719924—21no—
76C2×D424—21no—
77C2×D424—21no—
78C2×#11719917—21no—
79#13133217C2311yescharacteristic
80#13133217C2311yescharacteristic
81#13133214C2311yescharacteristic
82#13133214C2311yescharacteristic
83#13133205—21no—
84#13133205—21no—
85#13133191—21no—
86#13133191—21no—
87#13133191—21no—
88#13133191—21no—
89C4×D212—21no—
90C4×D212—21no—
91C4×D212—21no—
92C4×D212—21no—
93#13133150—21no—
94#13133150—21no—
95#13133149—21no—
96#13133149—21no—
97C4⋊#88322—21no—
98C4⋊#88322C2311yescharacteristic
99C4⋊#88322C2311yescharacteristic
100C4⋊#88322C2311yescharacteristic
101C4⋊#88322C2311yescharacteristic
102#13133141—21no—
103#13133141—21no—
104#13133140—21no—
105#13133140—21no—
106#13133139—21no—
107#13133139—21no—
108#13133139—21no—
109#13133139—21no—
110#13133137—21no—
111#13133137—21no—
112#13133137—21no—
113#13133137—21no—
114C22⋊#88322—21no—
115#13133128—21no—
116C22×Q8×C53D411yescharacteristic
117C2×M16×C53—21no—
118C2×M16×C53D411yescharacteristic
119C2×M16×C53D411yescharacteristic
120C4×Q8×C53—21no—
121C4×Q8×C53—21no—
122C4×Q8×C53D411yescharacteristic
123C4×Q8×C53D411yescharacteristic
124C4×Q8×C53C2311yescharacteristic
125C4×Q8×C53C2311yescharacteristic
126C2×C4⋊C4×C53—21no—
127C2×C4⋊C4×C53D411yescharacteristic
128C2×C42×C53—21no—
129C2×C42×C53C2311yescharacteristic
130C4⋊C8×C53—21no—
131C4⋊C8×C53—21no—
132C4⋊C8×C53—21no—
133C4⋊C8×C53—21no—
134C4⋊C8×C53—21no—
135C4⋊C8×C53C2311yescharacteristic
136C4⋊C8×C53C2311yescharacteristic
137#97×C53—21no—
138#97×C53—21no—
139#97×C53D411yescharacteristic
140#97×C53D411yescharacteristic
141C4×C8×C53—21no—
142C4×C8×C53—21no—
143C4×C8×C53C2311yescharacteristic
144C4×C8×C53C2311yescharacteristic
145C2×Q8×C53—21no—
146C2×Q8×C53—22no—
147C2×Q8×C53—41no—
148C2×Q8×C53C2×D411yescharacteristic
149C2×Q8×C53C2×D411yescharacteristic
150C22×C4×C53—21no—
151C22×C4×C53—21no—
152C22×C4×C53C2×D411yescharacteristic
153C22×C4×C53C2×D411yescharacteristic
154C22×C4×C53#4111yescharacteristic
155C2×C22⋊D53—41no—
156C2×C22⋊D53—41no—
157#11719945—42no—
158#11719945—42no—
159#11719945—82no—
160C2×D212—21no—
161C2×D212—21no—
162C2×D212—21no—
163C2×D212—21no—
164C2×D212—21no—
165C2×D212—21no—
166C2×C4×D53—41no—
167C2×C4×D53—41no—
168C2×Dic106—21no—
169C2×Dic106—21no—
170M16×C53—22no—
171M16×C53—41no—
172M16×C53—42no—
173C2×C8×C53—21no—
174C2×C8×C53—21no—
175C2×C8×C53C2×D411yescharacteristic
176C2×C8×C53C2×D411yescharacteristic
177C2×C8×C53C2×D411yescharacteristic
178C2×C8×C53C2×D411yescharacteristic
179C4⋊C4×C53—21no—
180C4⋊C4×C53—21no—
181C4⋊C4×C53—21no—
182C4⋊C4×C53—21no—
183C4⋊C4×C53—21no—
184C4⋊C4×C53C2×D411yescharacteristic
185C4⋊C4×C53C2×D411yescharacteristic
186C42×C53—21no—
187C42×C53—21no—
188C42×C53—21no—
189C42×C53—21no—
190C42×C53C2×D411yescharacteristic
191C42×C53C2×D411yescharacteristic
192C42×C53#4111yescharacteristic
193C42×C53C2411yescharacteristic
194#11719925—42no—
195#11719925—42no—
196#11719924—42no—
197#11719924—42no—
198#11719924—42no—
199#11719924—42no—
200#11719921—41no—
201#11719921—41no—
202#11719921—41no—
203#11719921—41no—
204C4⋊Dic53—21no—
205C4⋊Dic53—21no—
206C4⋊Dic53—21no—
207C4⋊Dic53—21no—
208#11719917—42no—
209C2×#88322—21no—
210C2×#88322—21no—
211C2×#88322—21no—
212C2×#88322—21no—
213D424—42no—
214D424—42no—
215C23×C53#12111yescharacteristic
216C22×D53—41no—
217C22×D53—41no—
218Q8×C53—21no—
219Q8×C53—21no—
220Q8×C53—21no—
221Q8×C53—42no—
222Q8×C53AGL(1,ℤ/8)12yes—
223C2×C4×C53—21no—
224C2×C4×C53—21no—
225C2×C4×C53—21no—
226C2×C4×C53—21no—
227C2×C4×C53—21no—
228C2×C4×C53—21no—
229C2×C4×C53—21no—
230C2×C4×C53—21no—
231C2×C4×C53—41no—
232C2×C4×C53—42no—
233C2×C4×C53—81no—
234C2×C4×C53#12111yescharacteristic
235C2×C4×C53#12111yescharacteristic
236C2×C4×C53#12111yescharacteristic
237C2×C4×C53C22×D411yescharacteristic
238C2×C4×C53C22×D411yescharacteristic
239C2×C4×C53C2×#4111yescharacteristic
240C22⋊D53—82no—
241C22⋊D53—82no—
242C2×Dic53—41no—
243C2×Dic53—41no—
244D212—41no—
245D212—41no—
246D212—41no—
247D212—41no—
248D212—41no—
249D212—41no—
250D212—41no—
251D212—41no—
252D212—81no—
253D212—81no—
254C4×D53—82no—
255C4×D53—82no—
256Dic106—41no—
257Dic106—41no—
258Dic106—81no—
259C8×C53—21no—
260C8×C53—21no—
261C8×C53—21no—
262C8×C53—41no—
263C8×C53—41no—
264C8×C53#13712yescomplemented
265#88322—41no—
266#88322—41no—
267#88322—41no—
268#88322—41no—
269#19310—531nomaximal, Sylow
270C22×C53—21no—
271C22×C53—21no—
272C22×C53—21no—
273C22×C53C2×#12111yescharacteristic
274C2×D53—81no—
275C2×D53—81no—
276C2×D53—81no—
277C2×D53—81no—
278C2×D53—81no—
279C2×D53—81no—
280C4×C53—21no—
281C4×C53—21no—
282C4×C53—22no—
283C4×C53—21no—
284C4×C53—21no—
285C4×C53—41no—
286C4×C53—41no—
287C4×C53—41no—
288C4×C53C2×AGL(1,ℤ/8)11yescharacteristic
289C4×C53C2×#13711yescharacteristic
290C4×C53#57511yescharacteristic
291C4×C53#57511yescharacteristic
292Dic53—81no—
293Dic53—81no—
294#3184—531no—
295#3112—531no—
296#3037—531no—
297Q8×M16D5311yescharacteristic, complemented
298#1702—531no—
299#1698—531no—
300#1692—531no—
301#1688—531no—
302#1682—531no—
303#1678—531no—
304#1672—531no—
305#1668—531no—
306#1569—531no—
307#1560—531no—
308#1557—531no—
309C2×C53—41no—
310C2×C53#303711yescharacteristic
311C2×C53#305511yescharacteristic
312C2×C53#306811yescharacteristic
313D53—161no—
314D53—161no—
315C2×#137—1061no—
316#532—531no—
317#532—531no—
318C2×C4×Q8C2×D5311yescharacteristic
319#498—531no—
320#493—531no—
321#486—531no—
322#486—531no—
323#480—1061no—
324#477—531no—
325#477—531no—
326#469—1061no—
327#469—1061no—
328#464—1061no—
329#464—1061no—
330#462—531no—
331#462—531no—
332#460—531no—
333#460—531no—
334#460—531no—
335#460—531no—
336#448—1061no—
337#446—21no—
338#446—21no—
339#446C2×D5311yescharacteristic
340#446C2×D5311yescharacteristic
341C8×Q8C2×D5311yescharacteristic
342C8×Q8C2×D5311yescharacteristic
343#442—1061no—
344#442—1061no—
345#440—1061no—
346#440—1061no—
347#432—531no—
348#423—21no—
349#423—531no—
350#423C2×D5311yescharacteristic
351#417—1061no—
352C4×M16—21no—
353C4×M16C2×D5311yescharacteristic
354#335—531no—
355#335—531no—
356#334—531no—
357#334—531no—
358#333—531no—
359#333—531no—
360#332—531no—
361#332—531no—
362#331—531no—
363#331—531no—
364#331—531no—
365#331—531no—
366C53#1931011yescharacteristic, complemented, Sylow
367C2×#41—1061no—
368C2×#41—1061no—
369C22×Q8D21211yescharacteristic
370#137—2124no—
371C2×Q16—1061no—
372C2×Q16—1061no—
373C2×SD16—1061no—
374C2×SD16—1061no—
375C2×SD16—1061no—
376C2×SD16—1061no—
377C2×D8—1061no—
378C2×D8—1061no—
379C2×M16—21no—
380C2×M16—1061no—
381C2×M16C22⋊D5311yescharacteristic
382C2×M16C22⋊D5311yescharacteristic
383C4⋊Q8—531no—
384C4⋊Q8—531no—
385#127—531no—
386#127—531no—
387#124—1061no—
388#124—1061no—
389#121—1061no—
390#121—1061no—
391#121—1061no—
392#121—1061no—
393C4×Q8—21no—
394C4×Q8—21no—
395C4×Q8D21211yescharacteristic
396C4×Q8D21211yescharacteristic
397C4×Q8C22×D5311yescharacteristic
398C4×Q8C22×D5311yescharacteristic
399C4×D4—1061no—
400C4×D4—1061no—
401C4×D4—1061no—
402C4×D4—1061no—
403C2×C4⋊C4—21no—
404C2×C4⋊C4D21211yescharacteristic
405C2×C42—21no—
406C2×C42C22×D5311yescharacteristic
407#107—1061no—
408#107—1061no—
409#106—1061no—
410#106—1061no—
411C4⋊C8—21no—
412C4⋊C8—21no—
413C4⋊C8—21no—
414C4⋊C8—21no—
415C4⋊C8—21no—
416C4⋊C8—531no—
417C4⋊C8—531no—
418C4⋊C8—531no—
419C4⋊C8—531no—
420C4⋊C8—1061no—
421C4⋊C8C22×D5311yescharacteristic
422C4⋊C8C22×D5311yescharacteristic
423#103—1061no—
424#103—1061no—
425#103—1061no—
426#103—1061no—
427#102—1061no—
428#102—1061no—
429#102—1061no—
430#102—1061no—
431#102—1061no—
432#102—1061no—
433#102—1061no—
434#102—1061no—
435C22⋊C8—1061no—
436#97—21no—
437#97—21no—
438#97—1061no—
439#97C22⋊D5311yescharacteristic
440#97C22⋊D5311yescharacteristic
441C4×C8—21no—
442C4×C8—21no—
443C4×C8C22×D5311yescharacteristic
444C4×C8C22×D5311yescharacteristic
445#41—4242no—
446#41—2122no—
447#41—2122no—
448C2×Q8—21no—
449C2×Q8—22no—
450C2×Q8—41no—
451C2×Q8—1061no—
452C2×Q8—1061no—
453C2×Q8C2×D21211yescharacteristic
454C2×Q8C2×D21211yescharacteristic
455C2×D4—1061no—
456C2×D4—1061no—
457C2×D4—1061no—
458C2×D4—1061no—
459C2×D4—1061no—
460C2×D4—1061no—
461C2×D4—2121no—
462C2×D4—2121no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—2121no—
466C22×C4—2121no—
467C22×C4C2×D21211yescharacteristic
468C22×C4#1171994511yescharacteristic
469C22×C4C2×C22⋊D5311yescharacteristic
470Q16—2122no—
471Q16—2122no—
472SD16—2122no—
473SD16—2122no—
474SD16—2122no—
475SD16—2122no—
476D8—2122no—
477D8—2122no—
478M16—22no—
479M16—41no—
480M16—42no—
481M16—2122no—
482C2×C8—21no—
483C2×C8—21no—
484C2×C8—1061no—
485C2×C8—1061no—
486C2×C8—1061no—
487C2×C8—1061no—
488C2×C8C2×C22⋊D5311yescharacteristic
489C2×C8C2×C22⋊D5311yescharacteristic
490C2×C8C2×C22⋊D5311yescharacteristic
491C2×C8C2×C22⋊D5311yescharacteristic
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—1061no—
498C4⋊C4—1061no—
499C4⋊C4—1061no—
500C4⋊C4—1061no—
501C4⋊C4C2×D21211yescharacteristic
502C4⋊C4C2×D21211yescharacteristic
503C22⋊C4—2121no—
504C22⋊C4—2121no—
505C22⋊C4—2121no—
506C22⋊C4—2121no—
507C42—21no—
508C42—21no—
509C42—21no—
510C42—21no—
511C42C2×D21211yescharacteristic
512C42#1171994511yescharacteristic
513C42C2×C22⋊D5311yescharacteristic
514C42C23×D5311yescharacteristic
515C23—2121no—
516C23—2121no—
517C23#1313319111yescharacteristic
518Q8—21no—
519Q8—21no—
520Q8—21no—
521Q8—42no—
522Q8—4241no—
523Q8—2121no—
524Q8—2121no—
525Q8#1313324512yes—
526D4—4241no—
527D4—4241no—
528D4—4242no—
529D4—4242no—
530D4—2121no—
531D4—2121no—
532D4—2121no—
533D4—2121no—
534D4—2121no—
535D4—2121no—
536D4—2121no—
537D4—2121no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—41no—
547C2×C4—42no—
548C2×C4—81no—
549C2×C4—4242no—
550C2×C4—4242no—
551C2×C4—2121no—
552C2×C4—2121no—
553C2×C4#1313319111yescharacteristic
554C2×C4#1313319111yescharacteristic
555C2×C4#1313319111yescharacteristic
556C2×C4C22×C22⋊D5311yescharacteristic
557C2×C4C22×D21211yescharacteristic
558C2×C4C2×#1171994511yescharacteristic
559C8—21no—
560C8—21no—
561C8—21no—
562C8—41no—
563C8—41no—
564C8—2121no—
565C8—2121no—
566C8—2121no—
567C8—2121no—
568C8#1313324812yescomplemented
569C22—21no—
570C22—21no—
571C22—21no—
572C22—4241no—
573C22—4241no—
574C22—4241no—
575C22—4241no—
576C22—4241no—
577C22—4241no—
578C22C2×#1313319111yescharacteristic
579C4—21no—
580C4—21no—
581C4—21no—
582C4—21no—
583C4—22no—
584C4—41no—
585C4—41no—
586C4—41no—
587C4—4241no—
588C4—4241no—
589C4C2×#1313324511yescharacteristic
590C4C2×#1313324811yescharacteristic
591C4#1681142111yescharacteristic
592C4#1681142311yescharacteristic
593C2—41no—
594C2—8481no—
595C2—8481no—
596C2#2028418711yescharacteristic
597C2#2028433411yescharacteristic
598C2#2028444311yescharacteristic
subgroupG/H
centerC22C2×#13133191
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingQ8×M16×C53C2
socleC22×C53C2×#121
prime radical#20285453C2
hypercenterQ8×M16D53
hypocenterC53#19310
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ Q8×M16

Complements

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

NH ≅ G/Ntype
2. #20284837C2semidirect
3. #20284187C2semidirect
4. #20274283C2semidirect
5. #20274255C2semidirect
6. #20274221C2semidirect
7. #20274199C2semidirect
8. #20274159C2semidirect
9. #20274131C2semidirect
10. #20274097C2semidirect
11. #20274075C2semidirect
12. #20273482C2semidirect
13. #20273448C2semidirect
14. #20273435C2semidirect
15. Q8×M16×C53C2semidirect
264. C8×C53#137semidirect
297. Q8×M16D53semidirect
366. C53#19310semidirect
568. C8#13133248semidirect

Maximal subgroups

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

subgroupindex
#202854532
#202848372
#202841872
#202742832
#202742552
#202742212
#202741992
#202741592
#202741312
#202740972
#202740752
#202734822
#202734482
#202734352
Q8×M16×C532
#1931053