Finite Group Databasebeta

← #27430278

Subgroups — #27430278

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#20280073C211yesmaximal, characteristic
2#20277672C211yesmaximal, characteristic, complemented
3#20276235C211yesmaximal, characteristic
4#20276235C211yesmaximal, characteristic
5#20276198C211yesmaximal, characteristic, complemented
6#20276198C211yesmaximal, characteristic, complemented
7#20275485C211yesmaximal, characteristic, complemented
8#20275343C211yesmaximal, characteristic
9#20275336C211yesmaximal, characteristic
10#20275294C211yesmaximal, characteristic, complemented
11#20275287C211yesmaximal, characteristic, complemented
12#20274883C211yesmaximal, characteristic, complemented
13#20274716C211yesmaximal, characteristic
14#20274541C211yesmaximal, characteristic
15C53×#1985C211yesmaximal, characteristic, complemented
16C2×C4×D212C2211yescharacteristic
17C2×#13133151C2211yescharacteristic
18C2×#13133151C2211yescharacteristic
19C2×#13133151C2211yescharacteristic
20C2×#13133149C2211yescharacteristic
21C2×#13133149C2211yescharacteristic
22C2×#13133149C2211yescharacteristic
23C2×#13133139C2211yescharacteristic
24C2×#13133139C2211yescharacteristic
25C2×#13133139C2211yescharacteristic
26C2×#13133139C2211yescharacteristic
27#16810558—21no—
28#16810534—21no—
29#16810530—21no—
30#16810525C2211yescharacteristic
31#16810525C2211yescharacteristic
32#16810521C2211yescharacteristic
33#16810521C2211yescharacteristic
34#16810509—21no—
35#16810509—21no—
36#16810494—21no—
37#16810494—21no—
38#16810493C2211yescharacteristic
39#16810489C2211yescharacteristic
40#16810489C2211yescharacteristic
41#16810489C2211yescharacteristic
42C4⋊C4×Dic53—21no—
43C4×C4⋊Dic53C2211yescharacteristic
44C4×C4⋊Dic53C2211yescharacteristic
45C4×C4⋊Dic53C2211yescharacteristic
46#16810386C2211yescharacteristic
47#16810386C2211yescharacteristic
48#16810386C2211yescharacteristic
49#16810386C2211yescharacteristic
50#16810386C2211yescharacteristic
51#16810386C2211yescharacteristic
52C2×C4×Q8×C53C2211yescharacteristic
53C2×C4⋊C8×C53C2211yescharacteristic
54C2×#97×C53C2211yescharacteristic
55C53×#336C2211yescharacteristic
56C53×#336C2211yescharacteristic
57C53×#336C2211yescharacteristic
58C53×#336C2211yescharacteristic
59C22×D212C2311yescharacteristic
60C22×C4×D53—21no—
61C4×D212—22no—
62C4×D212—22no—
63C2×C4⋊Dic53—21no—
64C2×C4⋊Dic53—21no—
65C2×C4⋊Dic53C2311yescharacteristic
66C2×C4⋊Dic53C2311yescharacteristic
67C2×C4⋊Dic53C2311yescharacteristic
68C2×C4⋊Dic53C2311yescharacteristic
69C2×C4⋊Dic53C2311yescharacteristic
70C2×C4⋊Dic53C2311yescharacteristic
71C2×C4⋊Dic53C2311yescharacteristic
72C2×#11719919—21no—
73C2×#11719919—21no—
74C2×#11719921—21no—
75C2×#11719921—21no—
76C2×#11719921—21no—
77C2×#11719921—21no—
78C2×#11719921—21no—
79C2×C4×Dic53—21no—
80C2×C4×Dic53—21no—
81C2×C4×Dic53—21no—
82#13133151—22no—
83#13133151—41no—
84#13133151—41no—
85#13133149—22no—
86#13133149—41no—
87#13133149—41no—
88#13133139—22no—
89#13133139—41no—
90#13133139—41no—
91#13133139Q814yescomplemented
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—
102#13133124—21no—
103C22×Q8×C53D411yescharacteristic
104C22×C8×C53C2311yescharacteristic
105C22×C8×C53C2311yescharacteristic
106C22×C8×C53C2311yescharacteristic
107C22×C8×C53C2311yescharacteristic
108C4×Q8×C53—22no—
109C4×Q8×C53—22no—
110C2×C4⋊C4×C53D411yescharacteristic
111C2×C4⋊C4×C53D411yescharacteristic
112C2×C4⋊C4×C53D411yescharacteristic
113C2×C42×C53C2311yescharacteristic
114C2×C42×C53C2311yescharacteristic
115C2×C42×C53C2311yescharacteristic
116C4⋊C8×C53—22no—
117#97×C53—22no—
118C23×D53—21no—
119C2×Q8×C53—22no—
120C2×Q8×C53—41no—
121C2×Q8×C53—41no—
122C22×C4×C53C2×D411yescharacteristic
123C22×C4×C53C2×D411yescharacteristic
124C22×C4×C53C2×D411yescharacteristic
125C22×C4×C53C2×D411yescharacteristic
126C22×C4×C53C2×D411yescharacteristic
127C22×C4×C53C2×D411yescharacteristic
128C22×C4×C53C2411yescharacteristic
129C22×Dic53—21no—
130C22×Dic53—21no—
131C22×Dic53—21no—
132C22×Dic53—21no—
133C22×Dic53—21no—
134C22×Dic53—21no—
135C22×Dic53—21no—
136C2×D212—21no—
137C2×D212—21no—
138C2×D212—21no—
139C2×D212—21no—
140C2×D212—21no—
141C2×D212C2×Q812yes—
142C2×C4×D53—42no—
143C2×C4×D53—81no—
144C2×C4×D53—81no—
145C2×C8×C53—21no—
146C2×C8×C53—21no—
147C2×C8×C53—21no—
148C2×C8×C53—21no—
149C2×C8×C53—21no—
150C2×C8×C53—21no—
151C2×C8×C53—21no—
152C2×C8×C53—21no—
153C2×C8×C53—21no—
154C2×C8×C53—21no—
155C2×C8×C53C2×Q812yes—
156C2×C8×C53#4112yes—
157C4⋊C4×C53—22no—
158C4⋊C4×C53—22no—
159C4⋊C4×C53—22no—
160C42×C53—21no—
161C42×C53—21no—
162C42×C53—21no—
163C42×C53—21no—
164C42×C53#4112yes—
165C42×C53#4112yes—
166#11719921—42no—
167#11719921—42no—
168#11719921—42no—
169#11719921—81no—
170#11719921—81no—
171C4⋊Dic53—21no—
172C4⋊Dic53—21no—
173C4⋊Dic53—21no—
174C4⋊Dic53—21no—
175C4⋊Dic53—21no—
176C4⋊Dic53—21no—
177C4⋊Dic53—21no—
178C4⋊Dic53—21no—
179C4⋊Dic53—21no—
180C4⋊Dic53—21no—
181C4⋊Dic53—21no—
182C4⋊Dic53—21no—
183C4⋊Dic53—21no—
184C4⋊Dic53—42no—
185C4⋊Dic53—81no—
186C4⋊Dic53C2×Q812yes—
187#11719919—81no—
188#11719919—81no—
189C4×Dic53—42no—
190C4×Dic53—81no—
191C4×Dic53—81no—
192C23×C53C22×D411yescharacteristic
193C22×D53—22no—
194C22×D53—41no—
195C22×D53—41no—
196C22×D53—41no—
197C22×D53—41no—
198C22×D53—41no—
199C22×D53—41no—
200Q8×C53—44no—
201C2×C4×C53—21no—
202C2×C4×C53—21no—
203C2×C4×C53—21no—
204C2×C4×C53—21no—
205C2×C4×C53—21no—
206C2×C4×C53—21no—
207C2×C4×C53—21no—
208C2×C4×C53—21no—
209C2×C4×C53—21no—
210C2×C4×C53—21no—
211C2×C4×C53—21no—
212C2×C4×C53—21no—
213C2×C4×C53—21no—
214C2×C4×C53—21no—
215C2×C4×C53—21no—
216C2×C4×C53—21no—
217C2×C4×C53#12312yes—
218C2×C4×C53#12312yes—
219C2×C4×C53C22×Q811yescharacteristic
220C2×C4×C53C2×#4111yescharacteristic
221C2×C4×C53C2×#4111yescharacteristic
222C2×C4×C53C2×#4111yescharacteristic
223C2×C4×C53H(5,𝔽2)11yescharacteristic
224C2×C4×C532−1+411yescharacteristic
225C2×Dic53—22no—
226C2×Dic53—41no—
227C2×Dic53—41no—
228C2×Dic53—41no—
229C2×Dic53—41no—
230C2×Dic53—41no—
231C2×Dic53—41no—
232C2×Dic53—41no—
233C2×Dic53—41no—
234C2×Dic53—41no—
235C2×Dic53—41no—
236C2×Dic53—41no—
237C2×Dic53—41no—
238C2×Dic53—41no—
239C2×Dic53—41no—
240C2×Dic53—41no—
241C2×Dic53—41no—
242C2×Dic53—41no—
243C2×Dic53—41no—
244C2×Dic53—41no—
245C2×Dic53—41no—
246D212—22no—
247D212—22no—
248D212—41no—
249D212—41no—
250C4×D53—84no—
251C8×C53—22no—
252C8×C53—22no—
253C8×C53—41no—
254C8×C53—41no—
255#25069—531nomaximal, Sylow
256C22×C53C2×#12311yescharacteristic
257C22×C53D4×Q811yescharacteristic
258C22×C53#55011yescharacteristic
259C22×C53#57511yescharacteristic
260C22×C53#57511yescharacteristic
261C22×C53#57611yescharacteristic
262C22×C53#57711yescharacteristic
263C2×D53—42no—
264C2×D53—42no—
265C2×D53—42no—
266C2×D53—81no—
267C2×D53—81no—
268C2×D53—81no—
269C2×D53—81no—
270C4×C53—22no—
271C4×C53—22no—
272C4×C53—41no—
273C4×C53—41no—
274C4×C53—41no—
275C4×C53—41no—
276C4×C53#53311yescharacteristic
277C4×C53#54211yescharacteristic
278C4×C53#55211yescharacteristic
279C4×C53#55411yescharacteristic
280Dic53—42no—
281Dic53—81no—
282Dic53—81no—
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#1985D5311yescharacteristic, complemented
294#1949—531no—
295#1926—531no—
296#1925—531no—
297#1919—531no—
298#1918—531no—
299#1847—531no—
300#1802—531no—
301#1762—531no—
302C2×C53#242311yescharacteristic
303C2×C53#309311yescharacteristic
304C2×C53#309611yescharacteristic
305C2×C53#338811yescharacteristic
306C2×C53#338911yescharacteristic
307C2×C53#340111yescharacteristic
308C2×C53#340311yescharacteristic
309D53—82no—
310C2×C4×Q8C2×D5311yescharacteristic
311C2×C4×D4—531no—
312C2×#107—531no—
313C2×#107—531no—
314C2×#107—531no—
315C2×#106—531no—
316C2×#106—531no—
317C2×#106—531no—
318C2×C4⋊C8C2×D5311yescharacteristic
319C2×#102—531no—
320C2×#102—531no—
321C2×#102—531no—
322C2×#102—531no—
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—531no—
342C4×C4⋊C4—1061no—
343#340—531no—
344#340—531no—
345#340—531no—
346#340—531no—
347#340—531no—
348#340—531no—
349#336C2×D5311yescharacteristic
350#336C2×D5311yescharacteristic
351#336C2×D5311yescharacteristic
352#336C2×D5311yescharacteristic
353C53#2506911yescharacteristic, complemented, Sylow
354C22×Q8D21211yescharacteristic
355C22×D4—531no—
356C23×C4—1061no—
357C22×C8C22×D5311yescharacteristic
358C22×C8C22×D5311yescharacteristic
359C22×C8C22×D5311yescharacteristic
360C22×C8C22×D5311yescharacteristic
361C4×Q8—22no—
362C4×Q8—22no—
363C4×D4—1062no—
364C4×D4—1062no—
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⋊C4D21211yescharacteristic
377C2×C4⋊C4D21211yescharacteristic
378C2×C4⋊C4D21211yescharacteristic
379C2×C22⋊C4—1061no—
380C2×C22⋊C4—1061no—
381C2×C22⋊C4—1061no—
382C2×C22⋊C4—1061no—
383C2×C22⋊C4—1061no—
384C2×C42—1061no—
385C2×C42—1061no—
386C2×C42—1061no—
387C2×C42C22×D5311yescharacteristic
388C2×C42C22×D5311yescharacteristic
389C2×C42C22×D5311yescharacteristic
390#107—1062no—
391#107—2121no—
392#107—2121no—
393#106—1062no—
394#106—2121no—
395#106—2121no—
396C4⋊C8—22no—
397#102—534no—
398#102—1062no—
399#102—2121no—
400#102—2121no—
401#97—22no—
402#95—1061no—
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—1061no—
414C2×Q8—22no—
415C2×Q8—41no—
416C2×Q8—41no—
417C2×D4—532no—
418C2×D4—1061no—
419C2×D4—1061no—
420C2×D4—1061no—
421C2×D4—1061no—
422C2×D4—1061no—
423C22×C4—1061no—
424C22×C4—1061no—
425C22×C4—1061no—
426C22×C4—1061no—
427C22×C4—1061no—
428C22×C4—1061no—
429C22×C4—1061no—
430C22×C4—4241no—
431C22×C4—4241no—
432C22×C4—2122no—
433C22×C4C2×D21211yescharacteristic
434C22×C4C2×D21211yescharacteristic
435C22×C4C2×D21211yescharacteristic
436C22×C4C2×D21211yescharacteristic
437C22×C4C2×D21211yescharacteristic
438C22×C4C2×D21211yescharacteristic
439C22×C4C23×D5311yescharacteristic
440C2×C8—21no—
441C2×C8—21no—
442C2×C8—21no—
443C2×C8—21no—
444C2×C8—21no—
445C2×C8—21no—
446C2×C8—21no—
447C2×C8—21no—
448C2×C8—21no—
449C2×C8—21no—
450C2×C8Q8×D5312yes—
451C2×C8#1171994912yes—
452C4⋊C4—22no—
453C4⋊C4—22no—
454C4⋊C4—22no—
455C4⋊C4—532no—
456C4⋊C4—1061no—
457C4⋊C4—1061no—
458C4⋊C4—1061no—
459C4⋊C4—1061no—
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—1061no—
469C4⋊C4—4241no—
470C4⋊C4—4241no—
471C4⋊C4—4241no—
472C4⋊C4—2122no—
473C22⋊C4—4241no—
474C22⋊C4—4241no—
475C22⋊C4—2122no—
476C22⋊C4—2122no—
477C22⋊C4—2122no—
478C42—21no—
479C42—21no—
480C42—21no—
481C42—21no—
482C42—4241no—
483C42—4241no—
484C42—2122no—
485C42#1171994712yes—
486C42#1171994712yes—
487C23—1062no—
488C23—2121no—
489C23—2121no—
490C23—2121no—
491C23—2121no—
492C23—2121no—
493C23—2121no—
494C23C22×D21211yescharacteristic
495Q8—44no—
496D4—1062no—
497D4—1062no—
498D4—2121no—
499D4—2121no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—1062no—
517C2×C4—4244no—
518C2×C4—2121no—
519C2×C4—2121no—
520C2×C4—2121no—
521C2×C4—2121no—
522C2×C4—2121no—
523C2×C4—2121no—
524C2×C4—2121no—
525C2×C4—2121no—
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#1313320112yes—
539C2×C4#1313320112yes—
540C2×C4C2×Q8×D5311yescharacteristic
541C2×C4C2×#1171994711yescharacteristic
542C2×C4C2×#1171994711yescharacteristic
543C2×C4C2×#1171994911yescharacteristic
544C2×C4#1313326311yescharacteristic
545C2×C4#1313326411yescharacteristic
546C8—22no—
547C8—22no—
548C8—41no—
549C8—41no—
550C22—4241no—
551C22—4241no—
552C22—4241no—
553C22—4241no—
554C22—2122no—
555C22—2122no—
556C22—2122no—
557C22C2×#1313320111yescharacteristic
558C22Q8×D21211yescharacteristic
559C22#1681129511yescharacteristic
560C22#1681142311yescharacteristic
561C22#1681142311yescharacteristic
562C22#1681142811yescharacteristic
563C22#1681143511yescharacteristic
564C4—22no—
565C4—22no—
566C4—41no—
567C4—41no—
568C4—41no—
569C4—41no—
570C4—4241no—
571C4—4241no—
572C4—4241no—
573C4—4241no—
574C4—4241no—
575C4—4241no—
576C4—2122no—
577C4#1681117811yescharacteristic
578C4#1681124211yescharacteristic
579C4#1681131511yescharacteristic
580C4#1681133111yescharacteristic
581C2—4242no—
582C2#2027823811yescharacteristic
583C2#2028464811yescharacteristic
584C2#2028467411yescharacteristic
585C2#2028766211yescharacteristic
586C2#2028767511yescharacteristic
587C2#2028778311yescharacteristic
588C2#2028780511yescharacteristic
subgroupG/H
centerC23C22×D212
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#1985C2
socleC23×C53C22×D4
prime radicalC22×D212C23
hypercenter#1985D53
hypocenterC53#25069
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ #1985

Complements

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

NH ≅ G/Ntype
2. #20277672C2semidirect
5. #20276198C2semidirect
6. #20276198C2semidirect
7. #20275485C2semidirect
10. #20275294C2semidirect
11. #20275287C2semidirect
12. #20274883C2semidirect
15. C53×#1985C2semidirect
91. #13133139Q8semidirect
293. #1985D53semidirect
353. C53#25069semidirect

Maximal subgroups

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

subgroupindex
#202800732
#202776722
#202762352
#202761982
#202754852
#202753432
#202753362
#202752942
#202752872
#202748832
#202747162
#202745412
C53×#19852
#2506953