Finite Group Databasebeta

← #27430280

Subgroups — #27430280

Order 13568.

688 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20280069C211yesmaximal, characteristic
2#20277670C211yesmaximal, characteristic, complemented
3#20276238C211yesmaximal, characteristic
4#20276234C211yesmaximal, characteristic
5#20276202C211yesmaximal, characteristic, complemented
6#20276201C211yesmaximal, characteristic, complemented
7#20275655C211yesmaximal, characteristic, complemented
8#20275483C211yesmaximal, characteristic, complemented
9#20275344C211yesmaximal, characteristic
10#20275337C211yesmaximal, characteristic
11#20275292C211yesmaximal, characteristic, complemented
12#20275285C211yesmaximal, characteristic, complemented
13#20274883C211yesmaximal, characteristic, complemented
14#20274715C211yesmaximal, characteristic
15C53×#1762C211yesmaximal, characteristic
16C2×C4×Dic106C2211yescharacteristic
17C2×#13133151C2211yescharacteristic
18C2×#13133151C2211yescharacteristic
19C2×#13133150C2211yescharacteristic
20C2×#13133149C2211yescharacteristic
21C2×#13133149C2211yescharacteristic
22C2×#13133148C2211yescharacteristic
23C2×C4⋊#88322C2211yescharacteristic
24C2×#13133138C2211yescharacteristic
25C2×#13133138C2211yescharacteristic
26#16810559—21no—
27#16810535—21no—
28#16810529—21no—
29#16810526C2211yescharacteristic
30#16810526C2211yescharacteristic
31#16810519C2211yescharacteristic
32#16810519C2211yescharacteristic
33#16810510—21no—
34#16810510—21no—
35#16810496—21no—
36#16810496—21no—
37#16810493C2211yescharacteristic
38#16810489C2211yescharacteristic
39#16810488C2211yescharacteristic
40#16810488C2211yescharacteristic
41C4⋊C4×Dic53C2211yescharacteristic
42C4⋊C4×Dic53C2211yescharacteristic
43C4×C4⋊Dic53—21no—
44#16810385C2211yescharacteristic
45#16810385C2211yescharacteristic
46#16810385C2211yescharacteristic
47#16810385C2211yescharacteristic
48#16810377C2211yescharacteristic
49#16810377C2211yescharacteristic
50#16810377C2211yescharacteristic
51#16810377C2211yescharacteristic
52C2×C4×D4×C53C2211yescharacteristic
53C2×#102×C53C2211yescharacteristic
54C2×#102×C53C2211yescharacteristic
55C2×#97×C53C2211yescharacteristic
56C4×C4⋊C4×C53C2211yescharacteristic
57C53×#340C2211yescharacteristic
58C53×#340C2211yescharacteristic
59C22×Dic106D411yescharacteristic
60C22×#88322C2311yescharacteristic
61C22×#88322C2311yescharacteristic
62C4×Dic106—22no—
63C4×Dic106—22no—
64C2×C4⋊Dic53—21no—
65C2×C4⋊Dic53—21no—
66C2×C4⋊Dic53D411yescharacteristic
67C2×C4⋊Dic53C2311yescharacteristic
68C2×C4⋊Dic53C2311yescharacteristic
69C2×C4⋊Dic53C2311yescharacteristic
70C2×C4⋊Dic53C2311yescharacteristic
71C2×#11719919—21no—
72C2×#11719919—21no—
73C2×#11719919D411yescharacteristic
74C2×#11719919D411yescharacteristic
75C2×C22⋊Dic53—21no—
76C2×C22⋊Dic53—21no—
77C2×C22⋊Dic53—21no—
78C2×C22⋊Dic53—21no—
79C2×C4×Dic53—21no—
80C2×C4×Dic53—21no—
81C2×C4×Dic53C2311yescharacteristic
82C2×C4×Dic53C2311yescharacteristic
83#13133151—22no—
84#13133151—41no—
85#13133150—41no—
86#13133149—22no—
87#13133149—41no—
88#13133148—41no—
89C4⋊#88322—22no—
90#13133138—41no—
91#13133138—41no—
92#13133124—21no—
93#13133124—21no—
94#13133124—21no—
95#13133124—21no—
96#13133124—21no—
97#13133124—21no—
98#13133124—21no—
99#13133124—21no—
100#13133124—21no—
101#13133124—21no—
102C22×D4×C53C2311yescharacteristic
103C23×C4×C53—21no—
104C22×C8×C53C2311yescharacteristic
105C22×C8×C53C2311yescharacteristic
106C4×D4×C53—22no—
107C4×D4×C53—22no—
108C2×C4⋊C4×C53C2311yescharacteristic
109C2×C4⋊C4×C53C2311yescharacteristic
110C2×C4⋊C4×C53C2311yescharacteristic
111C2×C22⋊C4×C53—21no—
112C2×C42×C53—21no—
113C2×C42×C53C2311yescharacteristic
114#102×C53—22no—
115#102×C53Q814yescomplemented
116#97×C53—22no—
117#95×C53—21no—
118C24×C53—21no—
119C2×D4×C53—21no—
120C2×D4×C53—21no—
121C2×D4×C53—21no—
122C2×D4×C53—21no—
123C2×D4×C53—21no—
124C2×D4×C53C2×Q812yes—
125C22×C4×C53—21no—
126C22×C4×C53—21no—
127C22×C4×C53—21no—
128C22×C4×C53—42no—
129C22×C4×C53—81no—
130C22×C4×C53—81no—
131C22×C4×C53C2×D411yescharacteristic
132C22×C4×C53C2×D411yescharacteristic
133C22×C4×C53C2411yescharacteristic
134C22×Dic53—21no—
135C22×Dic53—21no—
136C22×Dic53—21no—
137C22×Dic53—21no—
138C22×Dic53C2×D411yescharacteristic
139C22×Dic53C2×D411yescharacteristic
140C22×Dic53C2×D411yescharacteristic
141C22×Dic53C2×D411yescharacteristic
142C2×Dic106—22no—
143C2×Dic106—41no—
144C2×Dic106—41no—
145C2×C8×C53—21no—
146C2×C8×C53—21no—
147C2×C8×C53—21no—
148C2×C8×C53—21no—
149C2×C8×C53C2×Q812yes—
150C2×C8×C53#4112yes—
151C4⋊C4×C53—21no—
152C4⋊C4×C53—21no—
153C4⋊C4×C53—21no—
154C4⋊C4×C53—21no—
155C4⋊C4×C53—21no—
156C4⋊C4×C53C2×Q812yes—
157C22⋊C4×C53—42no—
158C42×C53—42no—
159C42×C53#4112yes—
160C42×C53#4112yes—
161C22⋊Dic53—42no—
162C22⋊Dic53—42no—
163C22⋊Dic53—81no—
164C22⋊Dic53—81no—
165C4⋊Dic53—22no—
166C4⋊Dic53—21no—
167C4⋊Dic53—21no—
168C4⋊Dic53—21no—
169C4⋊Dic53—21no—
170C4⋊Dic53—21no—
171C4⋊Dic53—21no—
172C4⋊Dic53—21no—
173C4⋊Dic53—21no—
174C4⋊Dic53—42no—
175C4⋊Dic53—81no—
176#11719919—22no—
177#11719919—22no—
178#11719919—81no—
179#11719919—81no—
180C4×Dic53—21no—
181C4×Dic53—21no—
182C4×Dic53—21no—
183C4×Dic53—21no—
184C4×Dic53—81no—
185C4×Dic53—81no—
186C2×#88322—21no—
187C2×#88322—21no—
188C2×#88322—21no—
189C2×#88322—21no—
190C2×#88322—21no—
191C2×#88322—21no—
192C23×C53—22no—
193C23×C53—41no—
194C23×C53—41no—
195C23×C53—41no—
196C23×C53—41no—
197C23×C53—41no—
198C23×C53—41no—
199C23×C53C22×D411yescharacteristic
200D4×C53—22no—
201D4×C53—22no—
202D4×C53—41no—
203D4×C53—41no—
204C2×C4×C53—21no—
205C2×C4×C53—21no—
206C2×C4×C53—21no—
207C2×C4×C53—21no—
208C2×C4×C53—22no—
209C2×C4×C53—41no—
210C2×C4×C53—41no—
211C2×C4×C53—41no—
212C2×C4×C53—41no—
213C2×C4×C53—41no—
214C2×C4×C53—41no—
215C2×C4×C53—41no—
216C2×C4×C53—41no—
217C2×C4×C53—84no—
218C2×C4×C53#12312yes—
219C2×C4×C53#12312yes—
220C2×C4×C53C22×Q811yescharacteristic
221C2×C4×C53C2×#4111yescharacteristic
222C2×C4×C53C2×#4111yescharacteristic
223C2×C4×C53C2×#4111yescharacteristic
224C2×C4×C53H(5,𝔽2)11yescharacteristic
225C2×C4×C532−1+411yescharacteristic
226C2×Dic53—21no—
227C2×Dic53—21no—
228C2×Dic53—21no—
229C2×Dic53—21no—
230C2×Dic53—21no—
231C2×Dic53—21no—
232C2×Dic53—21no—
233C2×Dic53—21no—
234C2×Dic53—21no—
235C2×Dic53—21no—
236C2×Dic53—21no—
237C2×Dic53—21no—
238C2×Dic53—41no—
239C2×Dic53—41no—
240C2×Dic53—41no—
241C2×Dic53—41no—
242C2×Dic53—41no—
243C2×Dic53—41no—
244C2×Dic53—41no—
245C2×Dic53—41no—
246C2×Dic53—41no—
247C2×Dic53—41no—
248C2×Dic53—41no—
249C2×Dic53—41no—
250Dic106—44no—
251C8×C53—22no—
252C8×C53—22no—
253#88322—41no—
254#88322—41no—
255#25069—531nomaximal, Sylow
256C22×C53—42no—
257C22×C53—42no—
258C22×C53—42no—
259C22×C53—81no—
260C22×C53—81no—
261C22×C53—81no—
262C22×C53—81no—
263C22×C53C2×#12311yescharacteristic
264C22×C53D4×Q811yescharacteristic
265C22×C53#55011yescharacteristic
266C22×C53#57511yescharacteristic
267C22×C53#57511yescharacteristic
268C22×C53#57611yescharacteristic
269C22×C53#57711yescharacteristic
270C4×C53—22no—
271C4×C53—22no—
272C4×C53—42no—
273C4×C53—81no—
274C4×C53—81no—
275C4×C53#53311yescharacteristic
276C4×C53#54211yescharacteristic
277C4×C53#55211yescharacteristic
278C4×C53#55411yescharacteristic
279Dic53—41no—
280Dic53—41no—
281Dic53—41no—
282Dic53—41no—
283Dic53—81no—
284Dic53—81no—
285Dic53—81no—
286Dic53—81no—
287#2580—531no—
288#2355—531no—
289#2092—531no—
290#2092—531no—
291#2085—531no—
292#2085—531no—
293#1985—531no—
294#1949—531no—
295#1926—531no—
296#1925—531no—
297#1919—531no—
298#1918—531no—
299#1847—531no—
300#1802—531no—
301#1762D5311yescharacteristic
302C2×C53—82no—
303C2×C53#242311yescharacteristic
304C2×C53#309311yescharacteristic
305C2×C53#309611yescharacteristic
306C2×C53#338811yescharacteristic
307C2×C53#338911yescharacteristic
308C2×C53#340111yescharacteristic
309C2×C53#340311yescharacteristic
310C2×C4×Q8—531no—
311C2×C4×D4C2×D5311yescharacteristic
312C2×#107—531no—
313C2×#107—531no—
314C2×#107—531no—
315C2×#106—531no—
316C2×#106—531no—
317C2×#106—531no—
318C2×C4⋊C8—531no—
319C2×#102—531no—
320C2×#102—531no—
321C2×#102C2×D5311yescharacteristic
322C2×#102C2×D5311yescharacteristic
323C2×#97C2×D5311yescharacteristic
324#399—1061no—
325#393—1061no—
326#391—1061no—
327#390—531no—
328#390—531no—
329#389—531no—
330#389—531no—
331#387—1061no—
332#387—1061no—
333#385—1061no—
334#385—1061no—
335#384—531no—
336#382—531no—
337#382—531no—
338#382—531no—
339C4×C4⋊C4—531no—
340C4×C4⋊C4—531no—
341C4×C4⋊C4—1061no—
342C4×C4⋊C4C2×D5311yescharacteristic
343#340—531no—
344#340—531no—
345#340—531no—
346#340—531no—
347#340C2×D5311yescharacteristic
348#340C2×D5311yescharacteristic
349#336—531no—
350#336—531no—
351#336—531no—
352#336—531no—
353C53#2506911yescharacteristic, complemented, Sylow
354C22×Q8—531no—
355C22×D4C22×D5311yescharacteristic
356C23×C4—21no—
357C22×C8—531no—
358C22×C8—531no—
359C22×C8C22×D5311yescharacteristic
360C22×C8C22×D5311yescharacteristic
361C4×Q8—1062no—
362C4×Q8—1062no—
363C4×D4—22no—
364C4×D4—22no—
365C2×C4⋊C4—531no—
366C2×C4⋊C4—531no—
367C2×C4⋊C4—531no—
368C2×C4⋊C4—531no—
369C2×C4⋊C4—531no—
370C2×C4⋊C4—531no—
371C2×C4⋊C4—531no—
372C2×C4⋊C4—1061no—
373C2×C4⋊C4—1061no—
374C2×C4⋊C4—1061no—
375C2×C4⋊C4—1061no—
376C2×C4⋊C4C22×D5311yescharacteristic
377C2×C4⋊C4C22×D5311yescharacteristic
378C2×C4⋊C4C22×D5311yescharacteristic
379C2×C22⋊C4—21no—
380C2×C22⋊C4—1061no—
381C2×C22⋊C4—1061no—
382C2×C22⋊C4—1061no—
383C2×C22⋊C4—1061no—
384C2×C42—21no—
385C2×C42—531no—
386C2×C42—531no—
387C2×C42—1061no—
388C2×C42—1061no—
389C2×C42C22×D5311yescharacteristic
390#107—1062no—
391#107—2121no—
392#107—2121no—
393#106—1062no—
394#106—2121no—
395#106—2121no—
396C4⋊C8—1062no—
397#102—22no—
398#102—2121no—
399#102—2121no—
400#102Dic10614yescomplemented
401#97—22no—
402#95—21no—
403#95—1061no—
404#95—1061no—
405#95—1061no—
406#95—1061no—
407#95—1061no—
408#95—1061no—
409#95—1061no—
410#95—1061no—
411#95—1061no—
412#95—1061no—
413C24—21no—
414C2×Q8—1062no—
415C2×Q8—2121no—
416C2×Q8—2121no—
417C2×D4—21no—
418C2×D4—21no—
419C2×D4—21no—
420C2×D4—21no—
421C2×D4—21no—
422C2×D4C2×Dic10612yes—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—42no—
427C22×C4—81no—
428C22×C4—81no—
429C22×C4—531no—
430C22×C4—531no—
431C22×C4—531no—
432C22×C4—531no—
433C22×C4—1061no—
434C22×C4—1061no—
435C22×C4—1061no—
436C22×C4—1061no—
437C22×C4D4×D5311yescharacteristic
438C22×C4D4×D5311yescharacteristic
439C22×C4C23×D5311yescharacteristic
440C2×C8—21no—
441C2×C8—21no—
442C2×C8—21no—
443C2×C8—21no—
444C2×C8—1061no—
445C2×C8—1061no—
446C2×C8—1061no—
447C2×C8—1061no—
448C2×C8—1061no—
449C2×C8—1061no—
450C2×C8C2×Dic10612yes—
451C2×C8#1171994512yes—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—1062no—
458C4⋊C4—1062no—
459C4⋊C4—1062no—
460C4⋊C4—1061no—
461C4⋊C4—1061no—
462C4⋊C4—1061no—
463C4⋊C4—1061no—
464C4⋊C4—1061no—
465C4⋊C4—1061no—
466C4⋊C4—1061no—
467C4⋊C4—1061no—
468C4⋊C4—4241no—
469C4⋊C4—4241no—
470C4⋊C4—4241no—
471C4⋊C4—2122no—
472C4⋊C4C2×Dic10612yes—
473C22⋊C4—42no—
474C22⋊C4—4241no—
475C22⋊C4—4241no—
476C22⋊C4—2122no—
477C22⋊C4—2122no—
478C42—42no—
479C42—1061no—
480C42—1061no—
481C42—1061no—
482C42—1061no—
483C42—4241no—
484C42—4241no—
485C42#1171994512yes—
486C42#1171994712yes—
487C23—22no—
488C23—41no—
489C23—41no—
490C23—41no—
491C23—41no—
492C23—41no—
493C23—41no—
494C23C2×D4×D5311yescharacteristic
495Q8—2124no—
496D4—22no—
497D4—22no—
498D4—41no—
499D4—41no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—22no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—84no—
514C2×C4—1061no—
515C2×C4—1061no—
516C2×C4—1061no—
517C2×C4—1061no—
518C2×C4—1061no—
519C2×C4—1061no—
520C2×C4—1061no—
521C2×C4—1061no—
522C2×C4—1061no—
523C2×C4—1061no—
524C2×C4—1061no—
525C2×C4—1061no—
526C2×C4—2121no—
527C2×C4—2121no—
528C2×C4—2121no—
529C2×C4—2121no—
530C2×C4—2121no—
531C2×C4—2121no—
532C2×C4—2121no—
533C2×C4—2121no—
534C2×C4—2121no—
535C2×C4—2121no—
536C2×C4—2121no—
537C2×C4—2121no—
538C2×C4#1313319912yes—
539C2×C4#1313319912yes—
540C2×C4C22×Dic10611yescharacteristic
541C2×C4C2×#1171994711yescharacteristic
542C2×C4C2×#1171994511yescharacteristic
543C2×C4C2×#1171994511yescharacteristic
544C2×C4#1313326311yescharacteristic
545C2×C4#1313326411yescharacteristic
546C8—22no—
547C8—22no—
548C8—2121no—
549C8—2121no—
550C22—42no—
551C22—42no—
552C22—42no—
553C22—81no—
554C22—81no—
555C22—81no—
556C22—81no—
557C22C2×#1313319911yescharacteristic
558C22D4×Dic10611yescharacteristic
559C22#1681129411yescharacteristic
560C22#1681141911yescharacteristic
561C22#1681142211yescharacteristic
562C22#1681142711yescharacteristic
563C22#1681143311yescharacteristic
564C4—22no—
565C4—22no—
566C4—42no—
567C4—81no—
568C4—81no—
569C4—4241no—
570C4—4241no—
571C4—4241no—
572C4—4241no—
573C4—2121no—
574C4—2121no—
575C4—2121no—
576C4—2121no—
577C4#1681118011yescharacteristic
578C4#1681124411yescharacteristic
579C4#1681131311yescharacteristic
580C4#1681133011yescharacteristic
581C2—82no—
582C2#2027824011yescharacteristic
583C2#2028464511yescharacteristic
584C2#2028467111yescharacteristic
585C2#2028766111yescharacteristic
586C2#2028767411yescharacteristic
587C2#2028778611yescharacteristic
588C2#2028780411yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#1762C2
socleC23×C53C22×D4
prime radicalC22×D4×C53C23
hypercenter#1762D53
hypocenterC53#25069
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1762

Complements

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

NH ≅ G/Ntype
2. #20277670C2semidirect
5. #20276202C2semidirect
6. #20276201C2semidirect
7. #20275655C2semidirect
8. #20275483C2semidirect
11. #20275292C2semidirect
12. #20275285C2semidirect
13. #20274883C2semidirect
115. #102×C53Q8semidirect
353. C53#25069semidirect
400. #102Dic106semidirect

Maximal subgroups

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

subgroupindex
#202800692
#202776702
#202762382
#202762342
#202762022
#202762012
#202756552
#202754832
#202753442
#202753372
#202752922
#202752852
#202748832
#202747152
C53×#17622
#2506953