Finite Group Databasebeta

← #27414010

Subgroups — #27414010

Order 13568.

810 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#20277196C211yesmaximal, characteristic
2#20277121C211yesmaximal, characteristic
3#20276284C211yesmaximal, characteristic, complemented
4#20276283C211yesmaximal, characteristic, complemented
5#20276282C211yesmaximal, characteristic, complemented
6#20276281C211yesmaximal, characteristic, complemented
7#20276280C211yesmaximal, characteristic, complemented
8#20276276C211yesmaximal, characteristic, complemented
9#20276275C211yesmaximal, characteristic, complemented
10#20274430C211yesmaximal, characteristic
11#20274430C211yesmaximal, characteristic
12#20274430C211yesmaximal, characteristic
13#20274430C211yesmaximal, characteristic
14#336×D53C211yesmaximal, characteristic
15C53×#2101C211yesmaximal, characteristic, complemented
16C22×C8×D53C2211yescharacteristic
17C22×C8×D53C2211yescharacteristic
18C2×#13133175C2211yescharacteristic
19C2×#13133175C2211yescharacteristic
20C2×#13133175C2211yescharacteristic
21C2×#13133175C2211yescharacteristic
22C2×C42×D53C2211yescharacteristic
23#16810522C2211yescharacteristic
24#16810521C2211yescharacteristic
25#16810519C2211yescharacteristic
26#16810497—21no—
27#16810494—21no—
28#16810492C2211yescharacteristic
29#16810492C2211yescharacteristic
30#16810489C2211yescharacteristic
31#16810482—21no—
32#16810482—21no—
33C4⋊C4×Dic53—21no—
34#16810474—21no—
35#16810474—21no—
36#16810474—21no—
37#16810386C2211yescharacteristic
38#16810386C2211yescharacteristic
39#16810386C2211yescharacteristic
40#16810386C2211yescharacteristic
41#16810385C2211yescharacteristic
42#16810385C2211yescharacteristic
43#16810385C2211yescharacteristic
44#16810385C2211yescharacteristic
45#16810385C2211yescharacteristic
46#16810385C2211yescharacteristic
47#16810385C2211yescharacteristic
48#16810385C2211yescharacteristic
49#16810378C2211yescharacteristic
50#16810377C2211yescharacteristic
51#16810377C2211yescharacteristic
52C53×#389C2211yescharacteristic
53C53×#382C2211yescharacteristic
54C53×#340C2211yescharacteristic
55C53×#340C2211yescharacteristic
56C53×#340C2211yescharacteristic
57C53×#340C2211yescharacteristic
58C53×#336C2211yescharacteristic
59C22×C4×D53D411yescharacteristic
60C22×C4×D53D411yescharacteristic
61C22×C4×D53C2311yescharacteristic
62C22×#88322C2311yescharacteristic
63C22×#88322C2311yescharacteristic
64C2×C8×D53—22no—
65C2×C8×D53—22no—
66C2×C8×D53—22no—
67C2×C8×D53—22no—
68C2×C8×D53—22no—
69C2×C8×D53—22no—
70#13133175—22no—
71#13133175—22no—
72#13133175—22no—
73#13133175—22no—
74#13133175—22no—
75#13133175—22no—
76#13133175—22no—
77#13133175Q814yes—
78C2×C4⋊Dic53—21no—
79C2×C4⋊Dic53C2311yescharacteristic
80C2×C4⋊Dic53C2311yescharacteristic
81C2×C4⋊Dic53C2311yescharacteristic
82C2×C4⋊Dic53C2311yescharacteristic
83C2×#11719919—21no—
84C2×#11719919—21no—
85C2×#11719919—21no—
86C2×#11719919—21no—
87C2×#11719919—21no—
88C2×#11719919—21no—
89C2×#11719921—21no—
90C2×#11719921—21no—
91C2×#11719921—21no—
92C2×#11719921—21no—
93C2×C4×Dic53—21no—
94C2×C4×Dic53—21no—
95C2×C4×Dic53—21no—
96C2×C4×Dic53—21no—
97C2×C4×Dic53D411yescharacteristic
98C2×C4×Dic53D411yescharacteristic
99C2×C4×Dic53C2311yescharacteristic
100C42×D53—22no—
101C42×D53—22no—
102#13133124—21no—
103#13133124—21no—
104#13133124—21no—
105#13133124—21no—
106#13133124—21no—
107C22×C8×C53C2311yescharacteristic
108C22×C8×C53C2311yescharacteristic
109C2×C4⋊C4×C53—21no—
110C2×C4⋊C4×C53C2311yescharacteristic
111C2×C4⋊C4×C53C2311yescharacteristic
112C2×C4⋊C4×C53C2311yescharacteristic
113C2×C4⋊C4×C53C2311yescharacteristic
114C2×C42×C53C2311yescharacteristic
115#95×C53—21no—
116#95×C53—21no—
117#95×C53—21no—
118C23×D53C2×D411yescharacteristic
119C22×C4×C53—21no—
120C22×C4×C53—21no—
121C22×C4×C53—21no—
122C22×C4×C53—21no—
123C22×C4×C53C2×D411yescharacteristic
124C22×C4×C53C2×D411yescharacteristic
125C22×C4×C53C2411yescharacteristic
126C22×Dic53—21no—
127C22×Dic53—21no—
128C22×Dic53—21no—
129C22×Dic53—21no—
130C22×Dic53C2×D411yescharacteristic
131C22×Dic53C2×D411yescharacteristic
132C22×Dic53C2×D411yescharacteristic
133C2×C4×D53—22no—
134C2×C4×D53—22no—
135C2×C4×D53—22no—
136C2×C4×D53—22no—
137C2×C4×D53—41no—
138C2×C4×D53—41no—
139C2×C4×D53C2×Q812yes—
140C2×C4×D53#4112yes—
141C2×C4×D53#4112yes—
142C2×C4×D53#4112yes—
143C2×C4×D53#4112yes—
144C2×C4×D53#4112yes—
145C2×C8×C53—21no—
146C2×C8×C53—21no—
147C2×C8×C53—21no—
148C2×C8×C53—21no—
149C2×C8×C53—21no—
150C2×C8×C53—21no—
151C4⋊C4×C53—21no—
152C4⋊C4×C53—21no—
153C4⋊C4×C53—21no—
154C4⋊C4×C53—21no—
155C4⋊C4×C53—21no—
156C4⋊C4×C53—21no—
157C4⋊C4×C53—21no—
158C4⋊C4×C53—42no—
159C4⋊C4×C53C2×Q812yes—
160C42×C53—21no—
161C42×C53—21no—
162#11719921—24no—
163#11719921—42no—
164#11719921—42no—
165#11719921—42no—
166C4⋊Dic53—21no—
167C4⋊Dic53—21no—
168C4⋊Dic53—21no—
169C4⋊Dic53—21no—
170C4⋊Dic53—21no—
171C4⋊Dic53—21no—
172C4⋊Dic53—21no—
173C4⋊Dic53—42no—
174C4⋊Dic53C2×Q812yes—
175#11719919—42no—
176#11719919—42no—
177#11719919—42no—
178#11719919—42no—
179#11719919—42no—
180#11719919—42no—
181C4×Dic53—21no—
182C4×Dic53—21no—
183C4×Dic53—24no—
184C4×Dic53—22no—
185C4×Dic53—22no—
186C4×Dic53—42no—
187C4×Dic53—42no—
188C4×Dic53—42no—
189C2×#88322—21no—
190C2×#88322—21no—
191C2×#88322—21no—
192C2×#88322—21no—
193C2×#88322—21no—
194C2×#88322—21no—
195C8×D53—44no—
196C8×D53—44no—
197C23×C53C22×D411yescharacteristic
198C22×D53—21no—
199C22×D53—21no—
200C22×D53—21no—
201C22×D53—21no—
202C22×D53C22⋊Q812yes—
203C22×D53#12312yes—
204C22×D53#12412yes—
205C2×C4×C53—22no—
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—41no—
213C2×C4×C53—41no—
214C2×C4×C53—41no—
215C2×C4×C53—41no—
216C2×C4×C53—41no—
217C2×C4×C53—41no—
218C2×C4×C53—41no—
219C2×C4×C53—41no—
220C2×C4×C53—41no—
221C2×C4×C53—41no—
222C2×C4×C53—41no—
223C2×C4×C53C22×Q811yescharacteristic
224C2×C4×C53C2×#4111yescharacteristic
225C2×C4×C53C2×#4111yescharacteristic
226C2×C4×C53C2×#4111yescharacteristic
227C2×C4×C53C2×#4111yescharacteristic
228C2×C4×C53C2×#4111yescharacteristic
229C2×Dic53—21no—
230C2×Dic53—21no—
231C2×Dic53—21no—
232C2×Dic53—21no—
233C2×Dic53—22no—
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×Dic53C22⋊Q812yes—
246C2×Dic53#12312yes—
247C2×Dic53#12412yes—
248C2×Dic53#13512yes—
249C2×Dic53#13512yes—
250C4×D53—44no—
251C4×D53—44no—
252C4×D53#12514yes—
253C4×D53#12514yes—
254C4×D53#12614yes—
255C4×D53#12614yes—
256C8×C53—41no—
257C8×C53—41no—
258#88322—41no—
259#88322—41no—
260#23744—531nomaximal, Sylow
261C22×C53C2×C22⋊Q811yescharacteristic
262C22×C53C2×#12311yescharacteristic
263C22×C53C2×#12411yescharacteristic
264C22×C53C2×#13511yescharacteristic
265C22×C53C2×#13511yescharacteristic
266C22×C53#57511yescharacteristic
267C22×C53#57511yescharacteristic
268C2×D53—22no—
269C2×D53—22no—
270C2×D53—22no—
271C2×D53—22no—
272C2×D53—22no—
273C2×D53—22no—
274C2×D53#40014yes—
275C4×C53—42no—
276C4×C53—41no—
277C4×C53—41no—
278C4×C53—81no—
279C4×C53—81no—
280C4×C53—81no—
281C4×C53C2×#12511yescharacteristic
282C4×C53C2×#12511yescharacteristic
283C4×C53C2×#12611yescharacteristic
284C4×C53C2×#12611yescharacteristic
285Dic53—22no—
286Dic53—22no—
287Dic53—42no—
288Dic53—81no—
289Dic53—81no—
290Dic53—81no—
291Dic53#40014yes—
292#2296—531no—
293#2285—531no—
294#2101—531no—
295#2101—531no—
296#2101—531no—
297#2101D5311yescharacteristic, complemented
298#2100—531no—
299#2100—531no—
300#2099—531no—
301#2099—531no—
302#1735—531no—
303#1735—531no—
304#1735—531no—
305#1735—531no—
306C2×#336—531no—
307C2×C53C2×#40011yescharacteristic
308C2×C53#307611yescharacteristic
309C2×C53#307611yescharacteristic
310C2×C53#309111yescharacteristic
311C2×C53#309111yescharacteristic
312C2×C53#313111yescharacteristic
313C2×C53#313511yescharacteristic
314D53—24no—
315C23×C8—531no—
316C23×C8—531no—
317C2×#117—531no—
318C2×#117—531no—
319C2×#117—531no—
320C2×#117—531no—
321C22×C42—531no—
322#389—531no—
323#389—531no—
324#389—531no—
325#389C2×D5311yescharacteristic
326#385—1061no—
327#385—1061no—
328#384—531no—
329#384—531no—
330#382—531no—
331#382C2×D5311yescharacteristic
332#380—1061no—
333#380—1061no—
334C4×C4⋊C4—1061no—
335C4×C4⋊C4—1061no—
336C4×C4⋊C4—1061no—
337C4×C4⋊C4—1061no—
338#340—531no—
339#340—531no—
340#340—531no—
341#340—531no—
342#340—531no—
343#340—531no—
344#340—531no—
345#340—531no—
346#340—531no—
347#340—531no—
348#340—531no—
349#340—531no—
350#340C2×D5311yescharacteristic
351#340C2×D5311yescharacteristic
352#340C2×D5311yescharacteristic
353#340C2×D5311yescharacteristic
354#336—531no—
355#336—531no—
356#336—531no—
357#336C2×D5311yescharacteristic
358C53#2374411yescharacteristic, complemented, Sylow
359C23×C4—531no—
360C23×C4—531no—
361C23×C4—531no—
362C22×C8—531no—
363C22×C8—531no—
364C22×C8—1062no—
365C22×C8—1062no—
366C22×C8—1062no—
367C22×C8—1062no—
368C22×C8—1062no—
369C22×C8—1062no—
370C22×C8C22×D5311yescharacteristic
371C22×C8C22×D5311yescharacteristic
372#117—534no—
373#117—1062no—
374#117—1062no—
375#117—1062no—
376#117—1062no—
377#117—1062no—
378#117—1062no—
379#117—1062no—
380C2×C4⋊C4—21no—
381C2×C4⋊C4—531no—
382C2×C4⋊C4—531no—
383C2×C4⋊C4—531no—
384C2×C4⋊C4—531no—
385C2×C4⋊C4—1061no—
386C2×C4⋊C4—1061no—
387C2×C4⋊C4—1061no—
388C2×C4⋊C4—1061no—
389C2×C4⋊C4—1061no—
390C2×C4⋊C4—1061no—
391C2×C4⋊C4—1061no—
392C2×C4⋊C4C22×D5311yescharacteristic
393C2×C4⋊C4C22×D5311yescharacteristic
394C2×C4⋊C4C22×D5311yescharacteristic
395C2×C4⋊C4C22×D5311yescharacteristic
396C2×C22⋊C4—1061no—
397C2×C22⋊C4—1061no—
398C2×C22⋊C4—1061no—
399C2×C22⋊C4—1061no—
400C2×C42—531no—
401C2×C42—531no—
402C2×C42—531no—
403C2×C42—1062no—
404C2×C42—1062no—
405C2×C42—1061no—
406C2×C42—1061no—
407C2×C42—1061no—
408C2×C42—1061no—
409C2×C42C22×D5311yescharacteristic
410#95—21no—
411#95—21no—
412#95—21no—
413#95—1061no—
414#95—1061no—
415#95—1061no—
416#95—1061no—
417#95—1061no—
418C24—531no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—531no—
424C22×C4—532no—
425C22×C4—532no—
426C22×C4—532no—
427C22×C4—532no—
428C22×C4—532no—
429C22×C4—532no—
430C22×C4—531no—
431C22×C4—531no—
432C22×C4—1062no—
433C22×C4—1062no—
434C22×C4—1062no—
435C22×C4—1062no—
436C22×C4—1061no—
437C22×C4—1061no—
438C22×C4—1061no—
439C22×C4—1061no—
440C22×C4—2121no—
441C22×C4—2121no—
442C22×C4D4×D5311yescharacteristic
443C22×C4D4×D5311yescharacteristic
444C22×C4C23×D5311yescharacteristic
445C2×C8—21no—
446C2×C8—21no—
447C2×C8—21no—
448C2×C8—21no—
449C2×C8—21no—
450C2×C8—21no—
451C2×C8—1061no—
452C2×C8—1061no—
453C2×C8—1061no—
454C2×C8—1061no—
455C2×C8—1061no—
456C2×C8—1061no—
457C2×C8—2124no—
458C2×C8—2124no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—42no—
467C4⋊C4—532no—
468C4⋊C4—1061no—
469C4⋊C4—1061no—
470C4⋊C4—1061no—
471C4⋊C4—1061no—
472C4⋊C4—1061no—
473C4⋊C4—1061no—
474C4⋊C4—1061no—
475C4⋊C4—2122no—
476C4⋊C4—2122no—
477C4⋊C4—2122no—
478C4⋊C4—2122no—
479C4⋊C4—2122no—
480C4⋊C4—2122no—
481C4⋊C4—2122no—
482C4⋊C4Q8×D5312yes—
483C22⋊C4—1064no—
484C22⋊C4—2122no—
485C22⋊C4—2122no—
486C22⋊C4—2122no—
487C42—21no—
488C42—21no—
489C42—1062no—
490C42—1062no—
491C42—1064no—
492C42—1061no—
493C42—1061no—
494C42—2122no—
495C42—2122no—
496C42—2122no—
497C23—532no—
498C23—532no—
499C23—532no—
500C23—1061no—
501C23—1061no—
502C23—1061no—
503C23—1061no—
504C23C2×D4×D5311yescharacteristic
505C2×C4—22no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—532no—
524C2×C4—532no—
525C2×C4—532no—
526C2×C4—534no—
527C2×C4—534no—
528C2×C4—534no—
529C2×C4—534no—
530C2×C4—532no—
531C2×C4—532no—
532C2×C4—1061no—
533C2×C4—1061no—
534C2×C4—1061no—
535C2×C4—1061no—
536C2×C4—1062no—
537C2×C4—2124no—
538C2×C4—2124no—
539C2×C4—2121no—
540C2×C4—2121no—
541C2×C4—2121no—
542C2×C4—2121no—
543C2×C4—2121no—
544C2×C4—2121no—
545C2×C4—2121no—
546C2×C4—2121no—
547C2×C4—2121no—
548C2×C4—2121no—
549C2×C4—2121no—
550C2×C4C2×Q8×D5311yescharacteristic
551C2×C4#41×D5311yescharacteristic
552C2×C4#41×D5311yescharacteristic
553C2×C4#41×D5311yescharacteristic
554C2×C4#41×D5311yescharacteristic
555C2×C4#41×D5311yescharacteristic
556C8—41no—
557C8—41no—
558C8—2121no—
559C8—2121no—
560C22—534no—
561C22—1062no—
562C22—1062no—
563C22—1062no—
564C22—1062no—
565C22—1062no—
566C22—1062no—
567C22C22⋊Q8×D5311yescharacteristic
568C22#123×D5311yescharacteristic
569C22#124×D5311yescharacteristic
570C22#135×D5311yescharacteristic
571C22#135×D5311yescharacteristic
572C22#1681141911yescharacteristic
573C22#1681142211yescharacteristic
574C4—42no—
575C4—41no—
576C4—41no—
577C4—81no—
578C4—81no—
579C4—81no—
580C4—534no—
581C4—1062no—
582C4—1062no—
583C4—4241no—
584C4—4241no—
585C4—4241no—
586C4—2122no—
587C4#125×D5311yescharacteristic
588C4#125×D5311yescharacteristic
589C4#126×D5311yescharacteristic
590C4#126×D5311yescharacteristic
591C2—1064no—
592C2#400×D5311yescharacteristic
593C2#2028450011yescharacteristic
594C2#2028450211yescharacteristic
595C2#2028462711yescharacteristic
596C2#2028463311yescharacteristic
597C2#2028503611yescharacteristic
598C2#2028505711yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2101C2
socleC23×C53C22×D4
prime radicalC23×D53C2×D4
hypercenter#2101D53
hypocenterC53#23744
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2101

Complements

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

NH ≅ G/Ntype
3. #20276284C2semidirect
4. #20276283C2semidirect
5. #20276282C2semidirect
6. #20276281C2semidirect
7. #20276280C2semidirect
8. #20276276C2semidirect
9. #20276275C2semidirect
15. C53×#2101C2semidirect
297. #2101D53semidirect
358. C53#23744semidirect

Maximal subgroups

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

subgroupindex
#202771962
#202771212
#202762842
#202762832
#202762822
#202762812
#202762802
#202762762
#202762752
#202744302
#336×D532
C53×#21012
#2374453