Finite Group Databasebeta

← #27355245

Subgroups — #27355245

Order 13568.

650 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20284760C211yesmaximal, characteristic
2#20283870C211yesmaximal, characteristic, complemented
3#20283747C211yesmaximal, characteristic, complemented
4#20273603C211yesmaximal, characteristic, complemented
5#20273600C211yesmaximal, characteristic, complemented
6#20273596C211yesmaximal, characteristic, complemented
7#20273582C211yesmaximal, characteristic, complemented
8#20273574C211yesmaximal, characteristic, complemented
9#20273514C211yesmaximal, characteristic, complemented
10C8×#11719922C211yesmaximal, characteristic, complemented
11#20273497C211yesmaximal, characteristic, complemented
12#20273172C211yesmaximal, characteristic, complemented
13D4⋊C8×D53C211yesmaximal, characteristic, complemented
14#20273000C211yesmaximal, characteristic, complemented
15C53×#1586C211yesmaximal, characteristic, complemented
16#16811155C2211yescharacteristic
17C4×D4×D53C2211yescharacteristic
18#16810960C411yescharacteristic
19#16810940C411yescharacteristic
20#16810926C411yescharacteristic
21#16810925C411yescharacteristic
22#16810837C411yescharacteristic
23#16810836C411yescharacteristic
24#16810775C411yescharacteristic
25#16810755C411yescharacteristic
26#16810753C2211yescharacteristic, complemented
27#16810747C2211yescharacteristic, complemented
28C4×#11719913C2211yescharacteristic
29C4×#11719923C2211yescharacteristic
30C4×#11719922C2211yescharacteristic
31#16810701C2211yescharacteristic
32#16810701C2211yescharacteristic, complemented
33#16810697—21no—
34#16810697—21no—
35#16810692—21no—
36C8×D212C2211yescharacteristic
37#16810688C2211yescharacteristic
38C8×C22⋊D53—21no—
39#16810681—21no—
40#16810679C2211yescharacteristic
41#16810674C2211yescharacteristic
42#16810674C2211yescharacteristic
43#16810649—21no—
44C4⋊C8×D53C2211yescharacteristic
45#16810586—21no—
46C22⋊C8×D53—21no—
47C4×C8×D53C2211yescharacteristic
48#16810376C2211yescharacteristic, complemented
49#16810375C2211yescharacteristic, complemented
50#16810373C2211yescharacteristic, complemented
51#16810355C2211yescharacteristic, complemented
52#16810353C2211yescharacteristic, complemented
53#16810352C2211yescharacteristic
54D4⋊#88322C2211yescharacteristic
55D4⋊#88322C2211yescharacteristic, complemented
56#16810350C2211yescharacteristic
57#16810344C2211yescharacteristic, complemented
58C8×#88322C2211yescharacteristic, complemented
59C8×#88322C2211yescharacteristic, complemented
60C4×D8×C53C2211yescharacteristic
61C53×#436C2211yescharacteristic
62C8×D4×C53C2211yescharacteristic, complemented
63C53×#334C2211yescharacteristic, complemented
64C53×D4⋊C8C2211yescharacteristic
65C53×D4⋊C8C2211yescharacteristic, complemented
66C53×#322C2211yescharacteristic, complemented
67C2×D4×D53C2×C411yescharacteristic
68C22×C4×D53—21no—
69C2×#11719913C2×C411yescharacteristic
70C2×#11719923C2×C411yescharacteristic
71C2×#11719922C2×C411yescharacteristic
72C2×#11719917—21no—
73C2×#11719917—21no—
74C2×C8×D53—21no—
75C2×C8×D53—21no—
76C2×C8×D53—21no—
77C2×C8×D53—21no—
78#13133210—21no—
79#13133208C2×C411yescharacteristic
80#13133204C2×C411yescharacteristic
81#13133199—21no—
82#13133197C2×C411yescharacteristic
83#13133188C2×C411yescharacteristic
84C4×Dic106C2311yescharacteristic
85C4×D212C2311yescharacteristic
86D4×Dic53C2×C411yescharacteristic
87#13133179C2×C411yescharacteristic
88C4×C22⋊D53—21no—
89C4×C22⋊D53—21no—
90#13133177—21no—
91#13133176—21no—
92C4⋊C4×D53C2×C411yescharacteristic
93C22⋊C4×D53—21no—
94C42×D53C2311yescharacteristic
95#13133150C2×C411yescharacteristic
96#13133149C2×C411yescharacteristic
97#13133148C2×C411yescharacteristic
98C4⋊#88322C2311yescharacteristic
99C4⋊#88322C2311yescharacteristic
100#13133146—21no—
101#13133146—21no—
102#13133146—21no—
103#13133146C2×C411yescharacteristic
104#13133146C2×C411yescharacteristic
105#13133146D411yescharacteristic
106#13133146D411yescharacteristic
107#13133142C2×C411yescharacteristic
108#13133140C2×C411yescharacteristic
109#13133139C2×C411yescharacteristic
110#13133138C2×C411yescharacteristic
111#13133138C2×C411yescharacteristic
112#13133137C2×C411yescharacteristic
113C22⋊#88322—21no—
114C22⋊#88322—21no—
115#13133129—21no—
116#13133129—21no—
117#13133129—21no—
118#13133129C2×C411yescharacteristic
119#13133129C2×C411yescharacteristic
120#13133129D411yescharacteristic
121#13133129D411yescharacteristic
122#13133127—21no—
123#13133127—21no—
124C4×#88322C2311yescharacteristic
125C4×#88322C2311yescharacteristic
126C4×#88322C2311yescharacteristic
127C8×Dic53—21no—
128C8×Dic53—21no—
129C2×D8×C53C2×C411yescharacteristic
130C2×M16×C53—21no—
131C22×C8×C53—21no—
132C4×D4×C53C2311yescharacteristic
133C4×D4×C53C2311yescharacteristic
134#107×C53C2×C411yescharacteristic
135C4⋊C8×C53C2311yescharacteristic
136C4⋊C8×C53C2311yescharacteristic
137#102×C53C2×C411yescharacteristic
138#102×C53C2×C411yescharacteristic
139C22⋊C8×C53—21no—
140C22⋊C8×C53—21no—
141C4×C8×C53C2311yescharacteristic
142C4×C8×C53C2311yescharacteristic
143C4×C8×C53C2311yescharacteristic
144C23×D53—21no—
145C2×D4×C53C22×C411yescharacteristic
146C2×D4×C53C22×C411yescharacteristic
147C22×C4×C53—21no—
148C22×C4×C53—21no—
149C2×C22⋊D53—21no—
150C2×C22⋊D53—21no—
151C22×Dic53—21no—
152D4×D53—22no—
153D4×D53—41no—
154C2×D212C22×C411yescharacteristic
155C2×C4×D53—21no—
156C2×C4×D53—21no—
157C2×C4×D53—21no—
158C2×C4×D53—81no—
159C2×C4×D53—81no—
160C2×C4×D53C22×C411yescharacteristic
161C2×C4×D53C2×D411yescharacteristic
162C2×C4×D53#4111yescharacteristic
163C2×Dic106C22×C411yescharacteristic
164D8×C53—41no—
165M16×C53—81no—
166C2×C8×C53—21no—
167C2×C8×C53—21no—
168C2×C8×C53—21no—
169C2×C8×C53—81no—
170C2×C8×C53C22×C411yescharacteristic
171C2×C8×C53C22×C411yescharacteristic
172C2×C8×C53C2×D411yescharacteristic
173C2×C8×C53C2×D411yescharacteristic
174C4⋊C4×C53C22×C411yescharacteristic
175C4⋊C4×C53C22×C411yescharacteristic
176C22⋊C4×C53—21no—
177C22⋊C4×C53—21no—
178C42×C53C2411yescharacteristic
179C22⋊Dic53—21no—
180C22⋊Dic53—21no—
181C22⋊Dic53—21no—
182C22⋊Dic53—21no—
183#11719923—41no—
184#11719922—22no—
185#11719921—21no—
186#11719921—21no—
187#11719921—21no—
188C4⋊Dic53—21no—
189C4⋊Dic53—21no—
190C4⋊Dic53C22×C411yescharacteristic
191C4⋊Dic53C22×C411yescharacteristic
192#11719919—21no—
193#11719919—21no—
194#11719919—21no—
195C4×Dic53—21no—
196C4×Dic53C22×C411yescharacteristic
197C4×Dic53C2×D411yescharacteristic
198C4×Dic53#4111yescharacteristic
199#11719917—81no—
200#11719917—81no—
201C2×#88322—21no—
202C2×#88322—21no—
203C2×#88322—21no—
204C2×#88322C22×C411yescharacteristic
205C2×#88322C22×C411yescharacteristic
206C2×#88322C2×D411yescharacteristic
207C2×#88322C2×D411yescharacteristic
208#11719913—22no—
209C8×D53—24no—
210C8×D53—81no—
211C8×D53—81no—
212C8×D53—81no—
213C23×C53—21no—
214C23×C53—21no—
215C22×D53—21no—
216C22×D53—42no—
217C22×D53—42no—
218C22×D53—81no—
219C22×D53C4×D411yescharacteristic
220D4×C53—21no—
221D4×C53—21no—
222D4×C53—21no—
223D4×C53—21no—
224C2×C4×C53—21no—
225C2×C4×C53—21no—
226C2×C4×C53—41no—
227C2×C4×C53—41no—
228C2×C4×C53—41no—
229C2×C4×C53—41no—
230C2×C4×C53C23×C411yescharacteristic
231C2×C4×C53C22×D411yescharacteristic
232C2×C4×C53C2×#4111yescharacteristic
233C22⋊D53—42no—
234C22⋊D53—42no—
235C2×Dic53—21no—
236C2×Dic53—21no—
237C2×Dic53—21no—
238C2×Dic53—81no—
239C2×Dic53C4×D411yescharacteristic
240C2×Dic53C4×D411yescharacteristic
241C2×Dic53C4×D411yescharacteristic
242D212—21no—
243D212#13112yes—
244C4×D53—22no—
245C4×D53—21no—
246C4×D53—21no—
247C4×D53—21no—
248C4×D53—21no—
249C4×D53—41no—
250C4×D53—41no—
251C4×D53—81no—
252C4×D53—81no—
253Dic106—21no—
254Dic106—21no—
255C8×C53—22no—
256C8×C53—21no—
257C8×C53—21no—
258C8×C53—21no—
259C8×C53—21no—
260C8×C53—41no—
261C8×C53—41no—
262#88322—22no—
263#88322—21no—
264#88322—21no—
265#88322—21no—
266#88322—21no—
267#88322—41no—
268#88322—41no—
269#18014—531nomaximal, Sylow
270C22×C53—41no—
271C22×C53—41no—
272C22×C53—41no—
273C22×C53—41no—
274C22×C53—41no—
275C22×C53—41no—
276C22×C53C2×C4×D411yescharacteristic
277C2×D53—21no—
278C2×D53—21no—
279C2×D53—22no—
280C2×D53—41no—
281C2×D53—41no—
282C2×D53—84no—
283C2×D53#44312yes—
284C4×C53—21no—
285C4×C53—41no—
286C4×C53—41no—
287C4×C53C2×#13111yescharacteristic
288C4×C53#56811yescharacteristic
289C4×C53#57611yescharacteristic
290C4×C53#57711yescharacteristic
291Dic53—21no—
292Dic53—21no—
293Dic53—21no—
294Dic53—41no—
295Dic53—41no—
296Dic53—41no—
297#3105—531no—
298#2994—531no—
299#2974—531no—
300#1590—531no—
301#1589—531no—
302#1588—531no—
303#1586—531no—
304#1586D5311yescharacteristic, complemented
305#1585—531no—
306C8×SD16—531no—
307C8×D8—531no—
308#1573—531no—
309#1494—531no—
310C2×D4⋊C8—531no—
311#1450—531no—
312C2×C53—81no—
313C2×C53—81no—
314C2×C53#293311yescharacteristic
315C2×C53#294611yescharacteristic
316C2×C53C2×#44311yescharacteristic
317D53—22no—
318D53—42no—
319#529—531no—
320C2×C4×D4—531no—
321#484—531no—
322#481—531no—
323#478—531no—
324#478—531no—
325#463—531no—
326#463—531no—
327#450—531no—
328#447—531no—
329#446—531no—
330C8×Q8—531no—
331C4×SD16—531no—
332C4×SD16—531no—
333C4×D8—531no—
334C4×D8C2×D5311yescharacteristic
335#436—531no—
336#436—531no—
337#436C2×D5311yescharacteristic
338#435—1061no—
339#435—1061no—
340#435—1061no—
341C8×D4—531no—
342C8×D4—531no—
343C8×D4—1061no—
344C8×D4C2×D5311yescharacteristic, complemented
345#433—1061no—
346#432—531no—
347#431—531no—
348#431—531no—
349#424—1061no—
350C2×C4⋊C8—531no—
351#407—1061no—
352C2×C22⋊C8—1061no—
353C2×C4×C8—531no—
354#335—531no—
355#335—531no—
356#334—531no—
357#334C2×D5311yescharacteristic, complemented
358Q8⋊C8—531no—
359Q8⋊C8—531no—
360D4⋊C8—531no—
361D4⋊C8—531no—
362D4⋊C8—531no—
363D4⋊C8—531no—
364D4⋊C8C2×D5311yescharacteristic
365D4⋊C8C2×D5311yescharacteristic, complemented
366#322—531no—
367#322C2×D5311yescharacteristic, complemented
368C82—531no—
369C82—531no—
370C53#1801411yescharacteristic, complemented, Sylow
371C22×D4—531no—
372C23×C4—1061no—
373C2×SD16—531no—
374C2×SD16—531no—
375C2×D8—531no—
376C2×D8C4×D5311yescharacteristic
377C2×M16—21no—
378C2×M16—1061no—
379C2×M16—1061no—
380C22×C8—21no—
381C22×C8—1061no—
382C22×C8—1061no—
383C22×C8—1061no—
384C22×C8—1061no—
385#126—1061no—
386#125—531no—
387#124—531no—
388#123—1061no—
389C22⋊Q8—531no—
390#121—531no—
391C4×Q8—531no—
392C4×D4—531no—
393C4×D4—531no—
394C4×D4—531no—
395C4×D4—1061no—
396C4×D4—1061no—
397C4×D4—1061no—
398C4×D4C22×D5311yescharacteristic
399C4×D4C22×D5311yescharacteristic
400#117—1061no—
401C2×C4⋊C4—531no—
402C2×C22⋊C4—1061no—
403C2×C42—531no—
404#107—531no—
405#107C4×D5311yescharacteristic
406#106—531no—
407#106—531no—
408C4⋊C8—531no—
409C4⋊C8—531no—
410C4⋊C8—531no—
411C4⋊C8—531no—
412C4⋊C8—531no—
413C4⋊C8—531no—
414C4⋊C8—1061no—
415C4⋊C8—1061no—
416C4⋊C8—1061no—
417C4⋊C8C22×D5311yescharacteristic
418C4⋊C8C22×D5311yescharacteristic
419#103—531no—
420#103—531no—
421#102—531no—
422#102—531no—
423#102—531no—
424#102—531no—
425#102C4×D5311yescharacteristic
426#102C4×D5311yescharacteristic
427C22⋊C8—21no—
428C22⋊C8—21no—
429C22⋊C8—531no—
430C22⋊C8—531no—
431C22⋊C8—531no—
432C22⋊C8—531no—
433C22⋊C8—1061no—
434C22⋊C8—1061no—
435C22⋊C8—1061no—
436C22⋊C8—1061no—
437C22⋊C8—1061no—
438#97—1061no—
439#97—1061no—
440C4×C8—531no—
441C4×C8—531no—
442C4×C8—531no—
443C4×C8—1061no—
444C4×C8—1061no—
445C4×C8C22×D5311yescharacteristic
446C4×C8C22×D5311yescharacteristic
447C4×C8C22×D5311yescharacteristic
448C24—1061no—
449C2×Q8—531no—
450C2×D4—531no—
451C2×D4—1061no—
452C2×D4—1062no—
453C2×D4—1061no—
454C2×D4—2121no—
455C2×D4C2×C4×D5311yescharacteristic
456C2×D4C2×C4×D5311yescharacteristic
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—531no—
460C22×C4—531no—
461C22×C4—531no—
462C22×C4—1061no—
463C22×C4—1061no—
464C22×C4—1061no—
465C22×C4—1061no—
466C22×C4—4241no—
467C22×C4—4241no—
468SD16—1062no—
469SD16—2121no—
470D8—41no—
471D8—1062no—
472M16—81no—
473M16—4241no—
474M16—4241no—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—21no—
478C2×C8—81no—
479C2×C8—531no—
480C2×C8—531no—
481C2×C8—531no—
482C2×C8—531no—
483C2×C8—1064no—
484C2×C8—1061no—
485C2×C8—1061no—
486C2×C8—1061no—
487C2×C8—4241no—
488C2×C8—4241no—
489C2×C8—4241no—
490C2×C8C2×C4×D5311yescharacteristic
491C2×C8C2×C4×D5311yescharacteristic
492C2×C8D4×D5311yescharacteristic
493C2×C8D4×D5311yescharacteristic
494C4⋊C4—531no—
495C4⋊C4—531no—
496C4⋊C4—1061no—
497C4⋊C4—1061no—
498C4⋊C4—1061no—
499C4⋊C4—1061no—
500C4⋊C4—1061no—
501C4⋊C4C2×C4×D5311yescharacteristic
502C4⋊C4C2×C4×D5311yescharacteristic
503C22⋊C4—21no—
504C22⋊C4—21no—
505C22⋊C4—1061no—
506C22⋊C4—1061no—
507C22⋊C4—1061no—
508C22⋊C4—1061no—
509C22⋊C4—1061no—
510C22⋊C4—1061no—
511C22⋊C4—1061no—
512C42—531no—
513C42—531no—
514C42—531no—
515C42—1061no—
516C42C23×D5311yescharacteristic
517C23—21no—
518C23—21no—
519C23—531no—
520C23—1061no—
521C23—4241no—
522C23—2122no—
523C23—2122no—
524Q8—1061no—
525Q8—1061no—
526D4—21no—
527D4—21no—
528D4—21no—
529D4—21no—
530D4—532no—
531D4—1061no—
532D4—2122no—
533D4—2122no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—531no—
541C2×C4—531no—
542C2×C4—531no—
543C2×C4—1062no—
544C2×C4—1061no—
545C2×C4—1061no—
546C2×C4—1061no—
547C2×C4—1061no—
548C2×C4—1061no—
549C2×C4—1061no—
550C2×C4—1061no—
551C2×C4—4241no—
552C2×C4—4241no—
553C2×C4—4241no—
554C2×C4—2121no—
555C2×C4—2121no—
556C2×C4C22×C4×D5311yescharacteristic
557C2×C4C2×D4×D5311yescharacteristic
558C2×C4#41×D5311yescharacteristic
559C8—22no—
560C8—21no—
561C8—21no—
562C8—21no—
563C8—21no—
564C8—41no—
565C8—41no—
566C8—1062no—
567C8—1061no—
568C8—1061no—
569C8—1061no—
570C8—1061no—
571C8—2121no—
572C8—2121no—
573C22—41no—
574C22—41no—
575C22—41no—
576C22—41no—
577C22—41no—
578C22—41no—
579C22—532no—
580C22—1061no—
581C22—1061no—
582C22—1062no—
583C22—4244no—
584C22—2121no—
585C22—2121no—
586C22C4×D4×D5311yescharacteristic
587C4—21no—
588C4—41no—
589C4—41no—
590C4—1061no—
591C4—1061no—
592C4—1061no—
593C4—2121no—
594C4—2121no—
595C4—2121no—
596C4#131×D5311yescharacteristic
597C4#1681138911yescharacteristic
598C4#1681142711yescharacteristic
599C4#1681143311yescharacteristic
600C2—81no—
601C2—81no—
602C2—1062no—
603C2—2122no—
604C2#2028343811yescharacteristic
605C2#2028354411yescharacteristic
606C2#443×D5311yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC42C23×D53
FittingC53×#1586C2
socleC22×C53C2×C4×D4
prime radical#16810755C4
cosocleC42×C53C24
hypercenter#1586D53
hypocenterC53#18014
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ #1586

Complements

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

NH ≅ G/Ntype
2. #20283870C2semidirect
3. #20283747C2semidirect
4. #20273603C2semidirect
5. #20273600C2semidirect
6. #20273596C2semidirect
7. #20273582C2semidirect
8. #20273574C2semidirect
9. #20273514C2semidirect
10. C8×#11719922C2semidirect
11. #20273497C2semidirect
12. #20273172C2semidirect
13. D4⋊C8×D53C2semidirect
14. #20273000C2semidirect
15. C53×#1586C2semidirect
26. #16810753C22semidirect
27. #16810747C22semidirect
32. #16810701C22semidirect
48. #16810376C22semidirect
49. #16810375C22semidirect
50. #16810373C22semidirect
51. #16810355C22semidirect
52. #16810353C22semidirect
55. D4⋊#88322C22semidirect
57. #16810344C22semidirect
58. C8×#88322C22semidirect
59. C8×#88322C22semidirect
62. C8×D4×C53C22semidirect
63. C53×#334C22semidirect
65. C53×D4⋊C8C22semidirect
66. C53×#322C22semidirect
304. #1586D53semidirect
344. C8×D4C2×D53semidirect
357. #334C2×D53semidirect
365. D4⋊C8C2×D53semidirect
367. #322C2×D53semidirect
370. C53#18014semidirect

Maximal subgroups

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

subgroupindex
#202847602
#202838702
#202837472
#202736032
#202736002
#202735962
#202735822
#202735742
#202735142
C8×#117199222
#202734972
#202731722
D4⋊C8×D532
#202730002
C53×#15862
#1801453