Finite Group Databasebeta

← #27355019

Subgroups — #27355019

Order 13568.

670 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#20284744C211yesmaximal, characteristic
2#20283667C211yesmaximal, characteristic, complemented
3#20283460C211yesmaximal, characteristic
4#20273602C211yesmaximal, characteristic, complemented
5#20273557C211yesmaximal, characteristic, complemented
6#20273556C211yesmaximal, characteristic, complemented
7#20273552C211yesmaximal, characteristic, complemented
8#20273549C211yesmaximal, characteristic, complemented
9#20273518C211yesmaximal, characteristic, complemented
10C8×#11719913C211yesmaximal, characteristic, complemented
11#20273484C211yesmaximal, characteristic, complemented
12#20273164C211yesmaximal, characteristic
13Q8⋊C8×D53C211yesmaximal, characteristic, complemented
14C8×#11719912C211yesmaximal, characteristic, complemented
15C53×#1590C211yesmaximal, characteristic, complemented
16C2×#13133225—21no—
17#16811170C2211yescharacteristic
18C4×Q8×D53C2211yescharacteristic
19#16810943C411yescharacteristic
20#16810940C411yescharacteristic
21#16810907C411yescharacteristic
22#16810906C411yescharacteristic
23#16810822C411yescharacteristic
24#16810821C411yescharacteristic
25#16810763C411yescharacteristic
26#16810760C411yescharacteristic
27#16810754C2211yescharacteristic
28#16810752—21no—
29#16810751—21no—
30C8×Dic106C2211yescharacteristic
31#16810747C2211yescharacteristic
32C4×#11719913C2211yescharacteristic
33C4×#11719924C2211yescharacteristic
34C4×#11719923C2211yescharacteristic
35#16810700C2211yescharacteristic
36#16810692—21no—
37#16810691—21no—
38C8×D212C2211yescharacteristic
39D4×#88322C2211yescharacteristic
40#16810674C2211yescharacteristic
41#16810645—21no—
42#16810645C2211yescharacteristic
43C4⋊C8×D53C2211yescharacteristic
44#16810588—21no—
45#16810575—21no—
46C4×#11719912C2211yescharacteristic
47C4×C8×D53C2211yescharacteristic
48#16810374C2211yescharacteristic
49#16810373C2211yescharacteristic
50#16810373C2211yescharacteristic
51#16810355C2211yescharacteristic
52Q8⋊#88322C2211yescharacteristic
53#16810353C2211yescharacteristic
54#16810352C2211yescharacteristic
55D4⋊#88322C2211yescharacteristic
56#16810350C2211yescharacteristic
57#16810345C2211yescharacteristic
58#16810344C2211yescharacteristic
59C8×#88322C2211yescharacteristic
60C53×#446C2211yescharacteristic
61C4×SD16×C53C2211yescharacteristic
62C53×#436C2211yescharacteristic
63C53×#334C2211yescharacteristic
64C53×Q8⋊C8C2211yescharacteristic
65C53×D4⋊C8C2211yescharacteristic
66C82×C53C2211yescharacteristic
67C2×#11719945—21no—
68C2×Q8×D53C2×C411yescharacteristic
69C2×#11719913C2×C411yescharacteristic
70C2×#11719923C2×C411yescharacteristic
71C2×#11719924C2×C411yescharacteristic
72#13133225—44no—
73C2×#11719912—21no—
74C2×#11719912—21no—
75C2×#11719912D411yescharacteristic
76C2×#11719912D411yescharacteristic
77C22×#88322—21no—
78C2×C8×D53—21no—
79C2×C8×D53—21no—
80C2×C8×D53—21no—
81#13133216C2×C411yescharacteristic
82#13133213C2×C411yescharacteristic
83#13133203—21no—
84#13133192C2×C411yescharacteristic
85#13133190—21no—
86#13133188C2×C411yescharacteristic
87Q8×Dic53C2×C411yescharacteristic
88C4×Dic106—21no—
89C4×Dic106C2311yescharacteristic
90#13133182—21no—
91#13133182C2×C411yescharacteristic
92C4×D212C2311yescharacteristic
93C4×C22⋊D53—21no—
94C4⋊C4×D53—21no—
95C4⋊C4×D53C2×C411yescharacteristic
96C42×D53—21no—
97C42×D53C2311yescharacteristic
98#13133149C2×C411yescharacteristic
99#13133148C2×C411yescharacteristic
100#13133148C2×C411yescharacteristic
101C4⋊#88322—21no—
102C4⋊#88322C2311yescharacteristic
103C4⋊#88322C2311yescharacteristic
104#13133146—21no—
105#13133146—21no—
106#13133146—21no—
107#13133146—21no—
108#13133146C2×C411yescharacteristic
109#13133146C2×C411yescharacteristic
110#13133142C2×C411yescharacteristic
111#13133141C2×C411yescharacteristic
112#13133140C2×C411yescharacteristic
113#13133139C2×C411yescharacteristic
114#13133138C2×C411yescharacteristic
115#13133137C2×C411yescharacteristic
116C22⋊#88322—21no—
117#13133129—21no—
118#13133129—21no—
119#13133129C2×C411yescharacteristic
120#13133129C2×C411yescharacteristic
121#13133128—21no—
122#13133127D411yescharacteristic
123#13133127D411yescharacteristic
124C4×#88322C2311yescharacteristic
125C4×#88322C2311yescharacteristic
126C4×#88322C2311yescharacteristic
127C8×Dic53—21no—
128C8×Dic53—21no—
129C8×Dic53—21no—
130C2×SD16×C53C2×C411yescharacteristic
131C2×M16×C53—21no—
132C4×Q8×C53C2311yescharacteristic
133C4×D4×C53C2311yescharacteristic
134#106×C53C2×C411yescharacteristic
135C4⋊C8×C53—21no—
136C4⋊C8×C53C2311yescharacteristic
137C4⋊C8×C53C2311yescharacteristic
138#103×C53C2×C411yescharacteristic
139#102×C53C2×C411yescharacteristic
140C22⋊C8×C53—21no—
141#97×C53—21no—
142C4×C8×C53C2311yescharacteristic
143C4×C8×C53C2311yescharacteristic
144C4×C8×C53C2311yescharacteristic
145C2×Q8×C53C22×C411yescharacteristic
146C2×D4×C53C22×C411yescharacteristic
147C22×C4×C53—21no—
148C2×C22⋊D53—21no—
149C2×C22⋊D53—21no—
150C2×C22⋊D53—21no—
151Q8×D53—22no—
152Q8×D53—41no—
153#11719945—42no—
154#11719945—42no—
155C2×D212—21no—
156C2×D212C22×C411yescharacteristic
157C2×C4×D53—21no—
158C2×C4×D53—21no—
159C2×C4×D53—21no—
160C2×C4×D53C22×C411yescharacteristic
161C2×C4×D53C2×D411yescharacteristic
162C2×C4×D53#4111yescharacteristic
163C2×Dic106—21no—
164C2×Dic106—21no—
165C2×Dic106C22×C411yescharacteristic
166SD16×C53—41no—
167M16×C53—42no—
168C2×C8×C53—21no—
169C2×C8×C53—21no—
170C2×C8×C53—21no—
171C2×C8×C53C22×C411yescharacteristic
172C2×C8×C53C22×C411yescharacteristic
173C2×C8×C53C2×D411yescharacteristic
174C2×C8×C53C2×D411yescharacteristic
175C4⋊C4×C53—21no—
176C4⋊C4×C53C22×C411yescharacteristic
177C4⋊C4×C53C22×C411yescharacteristic
178C22⋊C4×C53—21no—
179C42×C53—21no—
180C42×C53C2411yescharacteristic
181C22⋊Dic53—21no—
182#11719924—41no—
183#11719923—22no—
184#11719921—21no—
185#11719921—21no—
186C4⋊Dic53—21no—
187C4⋊Dic53C22×C411yescharacteristic
188C4⋊Dic53C22×C411yescharacteristic
189#11719919—21no—
190#11719919—21no—
191#11719919—21no—
192#11719919—21no—
193C4×Dic53—21no—
194C4×Dic53—21no—
195C4×Dic53—21no—
196C4×Dic53C22×C411yescharacteristic
197C4×Dic53C2×D411yescharacteristic
198C4×Dic53#4111yescharacteristic
199C2×#88322—21no—
200C2×#88322—21no—
201C2×#88322—21no—
202C2×#88322—42no—
203C2×#88322C22×C411yescharacteristic
204C2×#88322C22×C411yescharacteristic
205C2×#88322C2×D411yescharacteristic
206C2×#88322C2×D411yescharacteristic
207#11719913—22no—
208#11719912—22no—
209#11719912—42no—
210#11719912—42no—
211#11719912SD1614yescomplemented
212C8×D53—42no—
213C8×D53—42no—
214C8×D53—42no—
215C23×C53—21no—
216C22×D53—21no—
217C22×D53C4×D411yescharacteristic
218Q8×C53—21no—
219Q8×C53—21no—
220D4×C53—21no—
221D4×C53—21no—
222C2×C4×C53—21no—
223C2×C4×C53—21no—
224C2×C4×C53—21no—
225C2×C4×C53—41no—
226C2×C4×C53—41no—
227C2×C4×C53C23×C411yescharacteristic
228C2×C4×C53C22×D411yescharacteristic
229C2×C4×C53C2×#4111yescharacteristic
230C22⋊D53—42no—
231C22⋊D53—42no—
232C22⋊D53—81no—
233C2×Dic53—21no—
234C2×Dic53—21no—
235C2×Dic53—21no—
236C2×Dic53C4×D411yescharacteristic
237C2×Dic53C4×D411yescharacteristic
238C2×Dic53C4×D411yescharacteristic
239D212—21no—
240D212—21no—
241D212—42no—
242C4×D53—21no—
243C4×D53—21no—
244C4×D53—41no—
245C4×D53—42no—
246C4×D53—41no—
247C4×D53—41no—
248C4×D53—81no—
249C4×D53C2×SD1612yes—
250C4×D53#13512yes—
251Dic106—21no—
252Dic106—42no—
253Dic106—42no—
254Dic106#13112yescomplemented
255C8×C53—22no—
256C8×C53—21no—
257C8×C53—21no—
258C8×C53—21no—
259C8×C53—41no—
260C8×C53—41no—
261C8×C53C2×SD1612yes—
262#88322—21no—
263#88322—21no—
264#88322—21no—
265#88322—41no—
266#88322—41no—
267#88322—41no—
268#88322C2×SD1612yes—
269#17997—531nomaximal, Sylow
270C22×C53—41no—
271C22×C53—41no—
272C22×C53—41no—
273C22×C53C2×C4×D411yescharacteristic
274C2×D53—21no—
275C2×D53—21no—
276C2×D53—41no—
277C2×D53—41no—
278C2×D53—41no—
279C2×D53C4×SD1612yes—
280C4×C53—21no—
281C4×C53—41no—
282C4×C53—41no—
283C4×C53—41no—
284C4×C53C2×#13111yescharacteristic
285C4×C53#56811yescharacteristic
286C4×C53C22×SD1611yescharacteristic
287C4×C53C2×#13511yescharacteristic
288Dic53—22no—
289Dic53—21no—
290Dic53—21no—
291Dic53—41no—
292Dic53—41no—
293Dic53C4×SD1612yes—
294#3103—531no—
295Q8×M16—531no—
296#2935—531no—
297#1590—531no—
298#1590D5311yescharacteristic, complemented
299#1583—531no—
300#1583—531no—
301#1582—531no—
302#1582—531no—
303C8×SD16—531no—
304C8×SD16—531no—
305#1570—531no—
306#1492—531no—
307C2×Q8⋊C8—531no—
308C8×M16—531no—
309C2×C53—81no—
310C2×C53#293311yescharacteristic
311C2×C53C2×C4×SD1611yescharacteristic
312C2×C53#295611yescharacteristic
313D53—22no—
314D53—81no—
315C2×#131—1061no—
316#532—531no—
317C2×C4×Q8—531no—
318#481—531no—
319#481—531no—
320#475—531no—
321#475—531no—
322#460—531no—
323#460—531no—
324#448—531no—
325#448—531no—
326#446—531no—
327#446—1061no—
328#446—1061no—
329#446C2×D5311yescharacteristic
330C8×Q8—531no—
331C8×Q8—531no—
332C4×SD16—531no—
333C4×SD16—531no—
334C4×SD16—531no—
335C4×SD16C2×D5311yescharacteristic
336#436—531no—
337#436C2×D5311yescharacteristic
338#435—1061no—
339#435—1061no—
340C8×D4—531no—
341C8×D4—531no—
342#431—531no—
343#423—531no—
344#423—1061no—
345C2×C4⋊C8—531no—
346#408—1061no—
347C4×M16—531no—
348C4×M16—1061no—
349C2×C4×C8—531no—
350#334—531no—
351#334—531no—
352#334—531no—
353#334C2×D5311yescharacteristic
354Q8⋊C8—531no—
355Q8⋊C8—531no—
356Q8⋊C8—531no—
357Q8⋊C8C2×D5311yescharacteristic
358D4⋊C8—531no—
359D4⋊C8—531no—
360D4⋊C8—531no—
361D4⋊C8C2×D5311yescharacteristic
362#322—531no—
363#322—531no—
364C82—531no—
365C82C2×D5311yescharacteristic
366C53#1799711yescharacteristic, complemented, Sylow
367C2×#41—1061no—
368C22×Q8—531no—
369C2×SD16—531no—
370C2×SD16—531no—
371C2×SD16—531no—
372C2×SD16C4×D5311yescharacteristic
373#131—2124no—
374C2×M16—21no—
375C2×M16—531no—
376C2×M16—531no—
377C2×M16—1061no—
378C2×M16—1061no—
379C22×C8—1061no—
380C22×C8—1061no—
381C22×C8—1061no—
382C22×C8—1061no—
383C4⋊Q8—531no—
384#127—531no—
385#124—1061no—
386#121—531no—
387#121—531no—
388#121—1061no—
389C4×Q8—531no—
390C4×Q8—531no—
391C4×Q8—531no—
392C4×Q8—1061no—
393C4×Q8—1061no—
394C4×Q8C22×D5311yescharacteristic
395C4×D4—531no—
396C4×D4—1061no—
397C4×D4C22×D5311yescharacteristic
398C2×C4⋊C4—531no—
399C2×C4⋊C4—1061no—
400C2×C42—531no—
401C2×C42—1061no—
402#106—531no—
403#106—531no—
404#106—531no—
405#106C4×D5311yescharacteristic
406C4⋊C8—21no—
407C4⋊C8—531no—
408C4⋊C8—531no—
409C4⋊C8—531no—
410C4⋊C8—531no—
411C4⋊C8—1061no—
412C4⋊C8—1061no—
413C4⋊C8—1061no—
414C4⋊C8—1061no—
415C4⋊C8—1061no—
416C4⋊C8C22×D5311yescharacteristic
417C4⋊C8C22×D5311yescharacteristic
418#103—531no—
419#103—531no—
420#103—531no—
421#103C4×D5311yescharacteristic
422#102—531no—
423#102—531no—
424#102—531no—
425#102C4×D5311yescharacteristic
426C22⋊C8—21no—
427C22⋊C8—531no—
428C22⋊C8—531no—
429C22⋊C8—1061no—
430C22⋊C8—1061no—
431C22⋊C8—1061no—
432#97—21no—
433#97—531no—
434#97—531no—
435#97—1061no—
436C4×C8—531no—
437C4×C8—531no—
438C4×C8—531no—
439C4×C8—1061no—
440C4×C8—1061no—
441C4×C8—1061no—
442C4×C8C22×D5311yescharacteristic
443C4×C8C22×D5311yescharacteristic
444C4×C8C22×D5311yescharacteristic
445#41—2122no—
446#41—2122no—
447C2×Q8—531no—
448C2×Q8—1061no—
449C2×Q8—1061no—
450C2×Q8—1062no—
451C2×Q8—2121no—
452C2×Q8C2×C4×D5311yescharacteristic
453C2×D4—531no—
454C2×D4—1061no—
455C2×D4—1061no—
456C2×D4—1061no—
457C2×D4—1061no—
458C2×D4C2×C4×D5311yescharacteristic
459C22×C4—21no—
460C22×C4—531no—
461C22×C4—531no—
462C22×C4—531no—
463C22×C4—1061no—
464C22×C4—1061no—
465C22×C4—1061no—
466SD16—41no—
467SD16—1062no—
468SD16—1062no—
469SD16—2121no—
470M16—42no—
471M16—534no—
472M16—1062no—
473M16—2122no—
474M16—2122no—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—21no—
478C2×C8—531no—
479C2×C8—531no—
480C2×C8—531no—
481C2×C8—531no—
482C2×C8—1061no—
483C2×C8—1061no—
484C2×C8—1061no—
485C2×C8—2122no—
486C2×C8—2122no—
487C2×C8—2122no—
488C2×C8—2122no—
489C2×C8C2×C4×D5311yescharacteristic
490C2×C8C2×C4×D5311yescharacteristic
491C2×C8D4×D5311yescharacteristic
492C2×C8D4×D5311yescharacteristic
493C4⋊C4—21no—
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—1061no—
505C22⋊C4—1061no—
506C22⋊C4—1061no—
507C42—21no—
508C42—531no—
509C42—531no—
510C42—531no—
511C42—1061no—
512C42—1061no—
513C42—1061no—
514C42C23×D5311yescharacteristic
515C23—21no—
516C23—531no—
517C23—1061no—
518Q8—21no—
519Q8—21no—
520Q8—532no—
521Q8—1061no—
522Q8—2122no—
523Q8—2122no—
524D4—21no—
525D4—21no—
526D4—1061no—
527D4—1061no—
528D4—4241no—
529D4—2122no—
530D4—2122no—
531D4—2122no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—531no—
538C2×C4—532no—
539C2×C4—532no—
540C2×C4—531no—
541C2×C4—531no—
542C2×C4—1061no—
543C2×C4—1061no—
544C2×C4—1061no—
545C2×C4—1061no—
546C2×C4—1061no—
547C2×C4—4241no—
548C2×C4—2121no—
549C2×C4—2122no—
550C2×C4—2121no—
551C2×C4—2121no—
552C2×C4C22×C4×D5311yescharacteristic
553C2×C4C2×D4×D5311yescharacteristic
554C2×C4#41×D5311yescharacteristic
555C8—22no—
556C8—21no—
557C8—21no—
558C8—21no—
559C8—41no—
560C8—41no—
561C8—532no—
562C8—1061no—
563C8—1061no—
564C8—1061no—
565C8—2121no—
566C8—2121no—
567C8—2121no—
568C8SD16×D5312yes—
569C22—41no—
570C22—41no—
571C22—41no—
572C22—532no—
573C22—1061no—
574C22—1061no—
575C22—2121no—
576C22—2121no—
577C22—2121no—
578C22C4×D4×D5311yescharacteristic
579C4—21no—
580C4—41no—
581C4—41no—
582C4—41no—
583C4—532no—
584C4—1061no—
585C4—1062no—
586C4—1061no—
587C4—2121no—
588C4—2121no—
589C4#131×D5311yescharacteristic
590C4#1681138911yescharacteristic
591C4C2×SD16×D5311yescharacteristic
592C4#135×D5311yescharacteristic
593C2—81no—
594C2—1062no—
595C2—4241no—
596C2#2028344411yescharacteristic
597C2C4×SD16×D5311yescharacteristic
598C2#2028362011yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC42C23×D53
FittingC53×#1590C2
socleC22×C53C2×C4×D4
prime radical#16811170C22
cosocleC42×C53C24
hypercenter#1590D53
hypocenterC53#17997
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ #1590

Complements

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

NH ≅ G/Ntype
2. #20283667C2semidirect
4. #20273602C2semidirect
5. #20273557C2semidirect
6. #20273556C2semidirect
7. #20273552C2semidirect
8. #20273549C2semidirect
9. #20273518C2semidirect
10. C8×#11719913C2semidirect
11. #20273484C2semidirect
13. Q8⋊C8×D53C2semidirect
14. C8×#11719912C2semidirect
15. C53×#1590C2semidirect
211. #11719912SD16semidirect
254. Dic106#131semidirect
298. #1590D53semidirect
366. C53#17997semidirect

Maximal subgroups

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

subgroupindex
#202847442
#202836672
#202834602
#202736022
#202735572
#202735562
#202735522
#202735492
#202735182
C8×#117199132
#202734842
#202731642
Q8⋊C8×D532
C8×#117199122
C53×#15902
#1799753