Finite Group Databasebeta

← #27447741

Subgroups — #27447741

Order 13568.

746 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20278402C211yesmaximal, characteristic
2#20277747C211yesmaximal, characteristic
3#20276287C211yesmaximal, characteristic, complemented
4#20276219C211yesmaximal, characteristic, complemented
5#20276096C211yesmaximal, characteristic, complemented
6#20275664C211yesmaximal, characteristic, complemented
7#20275287C211yesmaximal, characteristic, complemented
8#20275135C211yesmaximal, characteristic
9#20275024C211yesmaximal, characteristic
10#20274971C211yesmaximal, characteristic, complemented
11#20274917C211yesmaximal, characteristic
12#20274894C211yesmaximal, characteristic
13#20274728C211yesmaximal, characteristic, complemented
14#20274430C211yesmaximal, characteristic
15C53×#2024C211yesmaximal, characteristic, complemented
16C22×C8×D53C2211yescharacteristic
17C2×#13133193C2211yescharacteristic
18C2×Q8×Dic53C2211yescharacteristic
19C2×#13133175C2211yescharacteristic
20C2×#13133175C2211yescharacteristic
21C2×#13133151C2211yescharacteristic
22C2×#13133148C2211yescharacteristic
23C2×C4⋊#88322C2211yescharacteristic
24C2×#13133146C2211yescharacteristic
25C2×#13133141C2211yescharacteristic
26C2×#13133141C2211yescharacteristic
27C2×#13133129C2211yescharacteristic
28#16810545—21no—
29#16810534—21no—
30#16810530—21no—
31#16810522C2211yescharacteristic
32#16810521C2211yescharacteristic
33#16810519C2211yescharacteristic
34#16810515—21no—
35#16810512—21no—
36#16810510—21no—
37#16810504—21no—
38#16810488C2211yescharacteristic
39#16810480C2211yescharacteristic
40C4×C4⋊Dic53C2211yescharacteristic
41#16810474—21no—
42C4×#11719921C2211yescharacteristic
43#16810386C2211yescharacteristic
44#16810386C2211yescharacteristic
45#16810386C2211yescharacteristic
46#16810385C2211yescharacteristic
47#16810385C2211yescharacteristic
48#16810385C2211yescharacteristic
49#16810385C2211yescharacteristic
50#16810385C2211yescharacteristic
51#16810377C2211yescharacteristic
52C2×C4⋊Q8×C53C2211yescharacteristic
53C2×#103×C53C2211yescharacteristic
54C2×#103×C53C2211yescharacteristic
55C53×#391C2211yescharacteristic
56C53×#340C2211yescharacteristic
57C53×#340C2211yescharacteristic
58C53×#336C2211yescharacteristic
59C22×C4×D53C2311yescharacteristic
60C22×#88322C2311yescharacteristic
61C22×#88322C2311yescharacteristic
62C2×C8×D53—22no—
63C2×C8×D53—41no—
64C2×C8×D53—41no—
65#13133193—22no—
66#13133193Q814yescomplemented
67Q8×Dic53—22no—
68Q8×Dic53—41no—
69#13133175—22no—
70#13133175—22no—
71#13133175—41no—
72#13133175—41no—
73C2×C4⋊Dic53—21no—
74C2×C4⋊Dic53—21no—
75C2×C4⋊Dic53C2311yescharacteristic
76C2×C4⋊Dic53C2311yescharacteristic
77C2×C4⋊Dic53C2311yescharacteristic
78C2×C4⋊Dic53C2311yescharacteristic
79C2×#11719919—21no—
80C2×#11719919—21no—
81C2×#11719919—21no—
82C2×#11719921—21no—
83C2×#11719921—21no—
84C2×#11719921—21no—
85C2×#11719921—21no—
86C2×#11719921D411yescharacteristic
87C2×#11719921D411yescharacteristic
88C2×C4×Dic53—21no—
89C2×C4×Dic53—21no—
90C2×C4×Dic53—21no—
91C2×C4×Dic53—21no—
92C2×C4×Dic53C2311yescharacteristic
93#13133151—22no—
94#13133148—41no—
95C4⋊#88322—41no—
96#13133146—41no—
97#13133141—41no—
98#13133141—41no—
99#13133129—22no—
100#13133124—21no—
101#13133124—21no—
102#13133124—21no—
103#13133124—21no—
104#13133124—21no—
105#13133124—21no—
106#13133124—21no—
107#13133124D411yescharacteristic
108#13133124D411yescharacteristic
109C22×Q8×C53—21no—
110C22×Q8×C53C2311yescharacteristic
111C22×C8×C53C2311yescharacteristic
112C22×C8×C53C2311yescharacteristic
113C4⋊Q8×C53—22no—
114C4⋊Q8×C53—41no—
115C2×C4⋊C4×C53—21no—
116C2×C4⋊C4×C53C2311yescharacteristic
117C2×C4⋊C4×C53C2311yescharacteristic
118C2×C4⋊C4×C53C2311yescharacteristic
119C2×C42×C53C2311yescharacteristic
120#103×C53—22no—
121#103×C53—41no—
122#95×C53—21no—
123#95×C53—21no—
124C23×D53C2×D411yescharacteristic
125C2×Q8×C53—21no—
126C2×Q8×C53—21no—
127C2×Q8×C53—21no—
128C2×Q8×C53—21no—
129C2×Q8×C53—21no—
130C2×Q8×C53—42no—
131C2×Q8×C53—42no—
132C2×Q8×C53—81no—
133C2×Q8×C53C2×Q812yes—
134C22×C4×C53—21no—
135C22×C4×C53—21no—
136C22×C4×C53—21no—
137C22×C4×C53—21no—
138C22×C4×C53C2×D411yescharacteristic
139C22×C4×C53C2×D411yescharacteristic
140C22×C4×C53C2411yescharacteristic
141C22×Dic53—21no—
142C22×Dic53—21no—
143C22×Dic53—21no—
144C22×Dic53—21no—
145C22×Dic53C2×D411yescharacteristic
146C22×Dic53C2×D411yescharacteristic
147C22×Dic53C2×D411yescharacteristic
148C2×C4×D53—21no—
149C2×C4×D53—21no—
150C2×C4×D53—21no—
151C2×C4×D53—21no—
152C2×C4×D53C2×Q812yes—
153C2×C4×D53#4112yes—
154C2×C8×C53—21no—
155C2×C8×C53—21no—
156C2×C8×C53—21no—
157C2×C8×C53—21no—
158C2×C8×C53—21no—
159C2×C8×C53—21no—
160C4⋊C4×C53—21no—
161C4⋊C4×C53—21no—
162C4⋊C4×C53—21no—
163C4⋊C4×C53—21no—
164C4⋊C4×C53—21no—
165C4⋊C4×C53—42no—
166C4⋊C4×C53#4112yes—
167C42×C53—21no—
168C42×C53—21no—
169#11719921—24no—
170#11719921—22no—
171#11719921—41no—
172#11719921—42no—
173#11719921—42no—
174#11719921—81no—
175C4⋊Dic53—21no—
176C4⋊Dic53—21no—
177C4⋊Dic53—21no—
178C4⋊Dic53—21no—
179C4⋊Dic53—21no—
180C4⋊Dic53—21no—
181C4⋊Dic53—42no—
182C4⋊Dic53—81no—
183C4⋊Dic53C2×Q812yes—
184C4⋊Dic53#4112yes—
185#11719919—24no—
186#11719919—42no—
187#11719919—42no—
188C4×Dic53—21no—
189C4×Dic53—21no—
190C4×Dic53—42no—
191C4×Dic53—42no—
192C4×Dic53—42no—
193C4×Dic53—42no—
194C2×#88322—21no—
195C2×#88322—21no—
196C2×#88322—21no—
197C2×#88322—21no—
198C2×#88322—21no—
199C2×#88322—21no—
200C8×D53—44no—
201C23×C53C22×D411yescharacteristic
202C22×D53—21no—
203C22×D53—21no—
204C22×D53—21no—
205C22×D53—21no—
206C22×D53—21no—
207C22×D53—21no—
208C22×D53C22⋊Q812yes—
209Q8×C53—22no—
210Q8×C53—22no—
211Q8×C53—41no—
212Q8×C53—41no—
213Q8×C53—84no—
214C2×C4×C53—21no—
215C2×C4×C53—21no—
216C2×C4×C53—22no—
217C2×C4×C53—22no—
218C2×C4×C53—21no—
219C2×C4×C53—21no—
220C2×C4×C53—21no—
221C2×C4×C53—41no—
222C2×C4×C53—41no—
223C2×C4×C53—41no—
224C2×C4×C53—41no—
225C2×C4×C53—41no—
226C2×C4×C53—41no—
227C2×C4×C53—41no—
228C2×C4×C53—41no—
229C2×C4×C53—41no—
230C2×C4×C53—41no—
231C2×C4×C53#13512yes—
232C2×C4×C53C22×Q811yescharacteristic
233C2×C4×C53C2×#4111yescharacteristic
234C2×C4×C53C2×#4111yescharacteristic
235C2×C4×C53C2×#4111yescharacteristic
236C2×C4×C53H(5,𝔽2)11yescharacteristic
237C2×C4×C53H(5,𝔽2)11yescharacteristic
238C2×Dic53—21no—
239C2×Dic53—21no—
240C2×Dic53—22no—
241C2×Dic53—22no—
242C2×Dic53—21no—
243C2×Dic53—21no—
244C2×Dic53—21no—
245C2×Dic53—21no—
246C2×Dic53—21no—
247C2×Dic53—21no—
248C2×Dic53—41no—
249C2×Dic53—41no—
250C2×Dic53—41no—
251C2×Dic53—41no—
252C2×Dic53—41no—
253C2×Dic53—41no—
254C2×Dic53—41no—
255C2×Dic53—41no—
256C2×Dic53—41no—
257C2×Dic53—41no—
258C2×Dic53C22⋊Q812yes—
259C4×D53—22no—
260C4×D53—22no—
261C4×D53—22no—
262C4×D53—22no—
263C8×C53—41no—
264C8×C53—41no—
265#88322—41no—
266#88322—41no—
267#26415—531nomaximal, Sylow
268C22×C53C2×C22⋊Q811yescharacteristic
269C22×C53#54611yescharacteristic
270C22×C53#54611yescharacteristic
271C22×C53C2×#13511yescharacteristic
272C22×C53#57511yescharacteristic
273C22×C53#57711yescharacteristic
274C22×C53#57811yescharacteristic
275C2×D53—22no—
276C2×D53—22no—
277C2×D53—22no—
278C2×D53—41no—
279C2×D53—41no—
280C2×D53—41no—
281C2×D53—41no—
282C4×C53—22no—
283C4×C53—42no—
284C4×C53—41no—
285C4×C53—42no—
286C4×C53—81no—
287C4×C53—81no—
288C4×C53#52911yescharacteristic
289C4×C53#53811yescharacteristic
290C4×C53#55411yescharacteristic
291C4×C53#55411yescharacteristic
292Dic53—22no—
293Dic53—42no—
294Dic53—41no—
295Dic53—42no—
296Dic53—41no—
297Dic53—81no—
298Dic53—81no—
299#2439—531no—
300#2362—531no—
301#2101—531no—
302#2088—531no—
303#2066—531no—
304#2024D5311yescharacteristic, complemented
305#1987—531no—
306#1918—531no—
307#1892—531no—
308#1873—531no—
309#1863—531no—
310#1854—531no—
311#1850—531no—
312#1805—531no—
313#1735—531no—
314C2×C53#245011yescharacteristic
315C2×C53#307611yescharacteristic
316C2×C53#308211yescharacteristic
317C2×C53#330011yescharacteristic
318C2×C53#330311yescharacteristic
319C2×C53#331111yescharacteristic
320C2×C53#331311yescharacteristic
321D53—42no—
322C23×C8—531no—
323C2×C4⋊Q8C2×D5311yescharacteristic
324C2×C22⋊Q8—531no—
325C2×C4×Q8—531no—
326C2×#117—531no—
327C2×#117—531no—
328C2×#107—531no—
329C2×#106—531no—
330C2×C4⋊C8—531no—
331C2×C4⋊C8—531no—
332C2×#103—531no—
333C2×#103—531no—
334C2×#103C2×D5311yescharacteristic
335C2×#103C2×D5311yescharacteristic
336C2×C22⋊C8—531no—
337#396—1061no—
338#393—1061no—
339#391—1061no—
340#391C2×D5311yescharacteristic
341#389—531no—
342#389—531no—
343#389—531no—
344#388—1061no—
345#388—1061no—
346#387—1061no—
347#387—1061no—
348#382—531no—
349#380—531no—
350C4×C4⋊C4—531no—
351C4×C4⋊C4—1061no—
352C4×C22⋊C4—531no—
353#340—531no—
354#340—531no—
355#340—531no—
356#340—531no—
357#340—531no—
358#340—531no—
359#340—531no—
360#340—531no—
361#340C2×D5311yescharacteristic
362#340C2×D5311yescharacteristic
363#336—531no—
364#336C2×D5311yescharacteristic
365C53#2641511yescharacteristic, complemented, Sylow
366C22×Q8—21no—
367C22×Q8C22×D5311yescharacteristic
368C23×C4—531no—
369C22×C8—531no—
370C22×C8—531no—
371C22×C8—1062no—
372C22×C8—2121no—
373C22×C8—2121no—
374C22×C8C22×D5311yescharacteristic
375C22×C8C22×D5311yescharacteristic
376C4⋊Q8—22no—
377C4⋊Q8—41no—
378C22⋊Q8—534no—
379C22⋊Q8—1062no—
380C4×Q8—1062no—
381C4×Q8—2121no—
382#117—1062no—
383#117—1062no—
384#117—2121no—
385#117—2121no—
386C2×C4⋊C4—21no—
387C2×C4⋊C4—531no—
388C2×C4⋊C4—531no—
389C2×C4⋊C4—531no—
390C2×C4⋊C4—531no—
391C2×C4⋊C4—1061no—
392C2×C4⋊C4—1061no—
393C2×C4⋊C4—1061no—
394C2×C4⋊C4—1061no—
395C2×C4⋊C4—1061no—
396C2×C4⋊C4C22×D5311yescharacteristic
397C2×C4⋊C4C22×D5311yescharacteristic
398C2×C4⋊C4C22×D5311yescharacteristic
399C2×C22⋊C4—531no—
400C2×C22⋊C4—531no—
401C2×C22⋊C4—1061no—
402C2×C22⋊C4—1061no—
403C2×C22⋊C4—1061no—
404C2×C22⋊C4—1061no—
405C2×C42—531no—
406C2×C42—1061no—
407C2×C42—1061no—
408C2×C42—1061no—
409C2×C42—1061no—
410C2×C42C22×D5311yescharacteristic
411#107—1062no—
412#106—2121no—
413C4⋊C8—2121no—
414C4⋊C8—2121no—
415#103—22no—
416#103—41no—
417#103—2121no—
418#103—2121no—
419C22⋊C8—1062no—
420#95—21no—
421#95—21no—
422#95—531no—
423#95—531no—
424#95—1061no—
425#95—1061no—
426#95—1061no—
427#95—1061no—
428#95—1061no—
429#95—1061no—
430#95—1061no—
431C24—531no—
432C2×Q8—21no—
433C2×Q8—21no—
434C2×Q8—21no—
435C2×Q8—21no—
436C2×Q8—21no—
437C2×Q8—42no—
438C2×Q8—42no—
439C2×Q8—81no—
440C2×Q8Q8×D5312yes—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—531no—
446C22×C4—532no—
447C22×C4—532no—
448C22×C4—531no—
449C22×C4—531no—
450C22×C4—1061no—
451C22×C4—1061no—
452C22×C4—1061no—
453C22×C4—1061no—
454C22×C4—1061no—
455C22×C4—1061no—
456C22×C4—1061no—
457C22×C4—1061no—
458C22×C4D4×D5311yescharacteristic
459C22×C4D4×D5311yescharacteristic
460C22×C4C23×D5311yescharacteristic
461C2×C8—21no—
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—21no—
467C2×C8—1061no—
468C2×C8—1061no—
469C2×C8—1061no—
470C2×C8—1061no—
471C2×C8—1061no—
472C2×C8—1061no—
473C2×C8—2124no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—42no—
480C4⋊C4—532no—
481C4⋊C4—532no—
482C4⋊C4—1064no—
483C4⋊C4—1061no—
484C4⋊C4—1061no—
485C4⋊C4—1061no—
486C4⋊C4—1061no—
487C4⋊C4—1061no—
488C4⋊C4—1061no—
489C4⋊C4—4241no—
490C4⋊C4—2122no—
491C4⋊C4—2122no—
492C4⋊C4—2122no—
493C4⋊C4#1171994712yes—
494C22⋊C4—1064no—
495C22⋊C4—1062no—
496C22⋊C4—4241no—
497C22⋊C4—2121no—
498C22⋊C4—2122no—
499C22⋊C4—2122no—
500C42—21no—
501C42—21no—
502C42—1061no—
503C42—1061no—
504C42—2122no—
505C42—2122no—
506C42—2122no—
507C42—2122no—
508C23—532no—
509C23—1061no—
510C23—1061no—
511C23—1061no—
512C23—1061no—
513C23—1061no—
514C23—1061no—
515C23C2×D4×D5311yescharacteristic
516Q8—22no—
517Q8—22no—
518Q8—41no—
519Q8—41no—
520Q8—84no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—22no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—22no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—532no—
539C2×C4—1061no—
540C2×C4—1061no—
541C2×C4—1062no—
542C2×C4—1062no—
543C2×C4—1062no—
544C2×C4—1062no—
545C2×C4—1061no—
546C2×C4—1061no—
547C2×C4—1062no—
548C2×C4—1062no—
549C2×C4—1061no—
550C2×C4—1061no—
551C2×C4—1061no—
552C2×C4—1061no—
553C2×C4—2121no—
554C2×C4—2121no—
555C2×C4—2121no—
556C2×C4—2121no—
557C2×C4—2121no—
558C2×C4—2121no—
559C2×C4—2121no—
560C2×C4—2121no—
561C2×C4—2121no—
562C2×C4—2121no—
563C2×C4#1313324012yes—
564C2×C4C2×Q8×D5311yescharacteristic
565C2×C4#41×D5311yescharacteristic
566C2×C4#41×D5311yescharacteristic
567C2×C4C2×#1171994711yescharacteristic
568C2×C4#1313326211yescharacteristic
569C2×C4#1313326311yescharacteristic
570C8—41no—
571C8—41no—
572C8—2121no—
573C8—2121no—
574C22—1062no—
575C22—1062no—
576C22—1062no—
577C22—2121no—
578C22—2121no—
579C22—2121no—
580C22—2121no—
581C22C22⋊Q8×D5311yescharacteristic
582C22#1681126011yescharacteristic
583C22#1681126611yescharacteristic
584C22C2×#1313324011yescharacteristic
585C22#1681141911yescharacteristic
586C22#1681142911yescharacteristic
587C22#1681143911yescharacteristic
588C4—22no—
589C4—42no—
590C4—41no—
591C4—42no—
592C4—81no—
593C4—81no—
594C4—1062no—
595C4—4241no—
596C4—4241no—
597C4—2121no—
598C4—2122no—
599C4—2121no—
600C4—2122no—
601C4#1681115211yescharacteristic
602C4#1681120711yescharacteristic
603C4#1681132711yescharacteristic
604C4#1681133111yescharacteristic
605C2—2122no—
606C2#2027853211yescharacteristic
607C2#2028450611yescharacteristic
608C2#2028455011yescharacteristic
609C2#2028660611yescharacteristic
610C2#2028664211yescharacteristic
611C2#2028674411yescharacteristic
612C2#2028678011yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2024C2
socleC23×C53C22×D4
prime radicalC23×D53C2×D4
hypercenter#2024D53
hypocenterC53#26415
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2024

Complements

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

NH ≅ G/Ntype
3. #20276287C2semidirect
4. #20276219C2semidirect
5. #20276096C2semidirect
6. #20275664C2semidirect
7. #20275287C2semidirect
10. #20274971C2semidirect
13. #20274728C2semidirect
15. C53×#2024C2semidirect
66. #13133193Q8semidirect
304. #2024D53semidirect
365. C53#26415semidirect

Maximal subgroups

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

subgroupindex
#202784022
#202777472
#202762872
#202762192
#202760962
#202756642
#202752872
#202751352
#202750242
#202749712
#202749172
#202748942
#202747282
#202744302
C53×#20242
#2641553