Finite Group Databasebeta

← #27432536

Subgroups — #27432536

Order 13568.

824 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810951C211yesmaximal, characteristic
2#20279194C211yesmaximal, characteristic
3#20277746C211yesmaximal, characteristic
4#20276194C211yesmaximal, characteristic, complemented
5#20276126C211yesmaximal, characteristic, complemented
6#20276092C211yesmaximal, characteristic, complemented
7#20276035C211yesmaximal, characteristic, complemented
8#20275475C211yesmaximal, characteristic, complemented
9#20275352C211yesmaximal, characteristic, complemented
10#20275058C211yesmaximal, characteristic, complemented
11#20275028C211yesmaximal, characteristic
12#20274898C211yesmaximal, characteristic
13#20274718C211yesmaximal, characteristic, complemented
14#20274435C211yesmaximal, characteristic
15C2×C53×#470C211yesmaximal, characteristic
16C2×Q8×Dic53C2211yescharacteristic
17C2×#13133176C2211yescharacteristic
18C22×C4⋊Dic53C2211yescharacteristic
19#16810951—24no—
20C2×#13133148C2211yescharacteristic
21C2×#13133148C2211yescharacteristic
22C2×#13133146C2211yescharacteristic
23C2×#13133146C2211yescharacteristic
24C2×#13133141C2211yescharacteristic
25C2×#13133141C2211yescharacteristic
26C2×#13133141C2211yescharacteristic
27C2×#13133141C2211yescharacteristic
28C2×C22⋊#88322C2211yescharacteristic
29#16810553—21no—
30#16810553—21no—
31#16810530C2211yescharacteristic
32#16810530C2211yescharacteristic
33#16810530C2211yescharacteristic
34#16810522C2211yescharacteristic
35#16810516—21no—
36#16810516—21no—
37#16810509—21no—
38#16810495—21no—
39#16810495—21no—
40#16810492C2211yescharacteristic
41#16810492C2211yescharacteristic
42C4⋊C4×Dic53—21no—
43C4⋊C4×Dic53C2211yescharacteristic
44C4×C22⋊Dic53C2211yescharacteristic
45#16810386C2211yescharacteristic
46#16810386C2211yescharacteristic
47#16810386C2211yescharacteristic
48#16810386C2211yescharacteristic
49#16810385C2211yescharacteristic
50#16810385C2211yescharacteristic
51#16810377C2211yescharacteristic
52#16810377C2211yescharacteristic
53C22×Q16×C53C2211yescharacteristic
54C22×M16×C53C2211yescharacteristic
55C2×C22⋊Q8×C53C2211yescharacteristic
56C2×C22⋊Q8×C53C2211yescharacteristic
57C53×#470—24no—
58C2×#106×C53C2211yescharacteristic
59C2×#103×C53C2211yescharacteristic
60C2×#103×C53C2211yescharacteristic
61C23×Dic53—21no—
62C22×#88322C2311yescharacteristic
63C22×#88322C2311yescharacteristic
64Q8×Dic53—22no—
65Q8×Dic53—41no—
66#13133176—22no—
67#13133176—41no—
68C2×C4⋊Dic53—21no—
69C2×C4⋊Dic53—21no—
70C2×C4⋊Dic53—21no—
71C2×C4⋊Dic53—22no—
72C2×C4⋊Dic53—22no—
73C2×C4⋊Dic53C2311yescharacteristic
74C2×C4⋊Dic53C2311yescharacteristic
75C2×C4⋊Dic53C2311yescharacteristic
76C2×C4⋊Dic53C2311yescharacteristic
77C2×#11719919—21no—
78C2×#11719919—21no—
79C2×#11719919—21no—
80C2×#11719919—21no—
81C2×C22⋊Dic53—21no—
82C2×C22⋊Dic53D411yescharacteristic
83C2×C22⋊Dic53D411yescharacteristic
84C2×C4×Dic53—21no—
85C2×C4×Dic53—21no—
86C2×C4×Dic53—21no—
87C2×C4×Dic53C2311yescharacteristic
88C2×C4×Dic53C2311yescharacteristic
89#13133148—22no—
90#13133148—22no—
91#13133146—22no—
92#13133146—41no—
93#13133141—22no—
94#13133141—22no—
95#13133141—22no—
96#13133141—41no—
97C22⋊#88322—22no—
98#13133124—21no—
99#13133124—21no—
100#13133124—21no—
101#13133124—21no—
102#13133124—21no—
103#13133124—21no—
104#13133124—21no—
105#13133124—21no—
106#13133124D411yescharacteristic
107#13133124D411yescharacteristic
108C22×Q8×C53C2311yescharacteristic
109C22×Q8×C53C2311yescharacteristic
110C23×C4×C53C2311yescharacteristic
111C2×Q16×C53—22no—
112C2×Q16×C53—22no—
113C2×Q16×C53—41no—
114C2×M16×C53—22no—
115C2×M16×C53—22no—
116C2×M16×C53—41no—
117C22×C8×C53C2311yescharacteristic
118C22×C8×C53C2311yescharacteristic
119C22⋊Q8×C53—22no—
120C22⋊Q8×C53—22no—
121C22⋊Q8×C53—22no—
122C22⋊Q8×C53D414yes—
123C2×C4⋊C4×C53—21no—
124C2×C4⋊C4×C53—21no—
125C2×C4⋊C4×C53C2311yescharacteristic
126C2×C4⋊C4×C53C2311yescharacteristic
127C2×C22⋊C4×C53—21no—
128C2×C22⋊C4×C53—21no—
129#106×C53—22no—
130#103×C53—22no—
131#103×C53—22no—
132C24×C53C2×D411yescharacteristic
133C2×Q8×C53—21no—
134C2×Q8×C53—21no—
135C2×Q8×C53—21no—
136C2×Q8×C53—21no—
137C2×Q8×C53—21no—
138C2×Q8×C53—21no—
139C2×Q8×C53—21no—
140C2×Q8×C53—21no—
141C2×Q8×C53—21no—
142C2×Q8×C53—21no—
143C2×Q8×C53—21no—
144C2×Q8×C53C2×D412yes—
145C22×C4×C53—21no—
146C22×C4×C53—21no—
147C22×C4×C53—21no—
148C22×C4×C53—21no—
149C22×C4×C53—21no—
150C22×C4×C53—21no—
151C22×C4×C53—21no—
152C22×C4×C53—21no—
153C22×C4×C53C2×D411yescharacteristic
154C22×C4×C53C2×D412yes—
155C22×C4×C53#4112yes—
156C22×C4×C53C2411yescharacteristic
157C22×Dic53—21no—
158C22×Dic53—21no—
159C22×Dic53—21no—
160C22×Dic53—21no—
161C22×Dic53—42no—
162C22×Dic53—81no—
163C22×Dic53—81no—
164C22×Dic53C2×D411yescharacteristic
165C22×Dic53C2×D411yescharacteristic
166C22×Dic53C2×D411yescharacteristic
167C22×Dic53C2×D411yescharacteristic
168Q16×C53—44no—
169M16×C53—44no—
170C2×C8×C53—21no—
171C2×C8×C53—21no—
172C2×C8×C53—21no—
173C2×C8×C53—21no—
174C2×C8×C53—21no—
175C2×C8×C53#4112yes—
176C4⋊C4×C53—24no—
177C4⋊C4×C53—21no—
178C4⋊C4×C53—21no—
179C4⋊C4×C53—21no—
180C4⋊C4×C53—42no—
181C4⋊C4×C53C2×D412yes—
182C22⋊C4×C53—24no—
183C22⋊C4×C53—42no—
184C22⋊Dic53—22no—
185C22⋊Dic53—22no—
186C22⋊Dic53—42no—
187C4⋊Dic53—21no—
188C4⋊Dic53—21no—
189C4⋊Dic53—21no—
190C4⋊Dic53—21no—
191C4⋊Dic53—21no—
192C4⋊Dic53—21no—
193C4⋊Dic53—21no—
194C4⋊Dic53—42no—
195C4⋊Dic53—42no—
196C4⋊Dic53—42no—
197C4⋊Dic53#4112yes—
198#11719919—81no—
199#11719919—81no—
200#11719919—81no—
201#11719919—81no—
202C4×Dic53—21no—
203C4×Dic53—21no—
204C4×Dic53—21no—
205C4×Dic53—21no—
206C4×Dic53—42no—
207C4×Dic53—42no—
208C4×Dic53—42no—
209C2×#88322—21no—
210C2×#88322—21no—
211C2×#88322—21no—
212C2×#88322—21no—
213C2×#88322—21no—
214C2×#88322—21no—
215C23×C53—21no—
216C23×C53—21no—
217C23×C53—21no—
218C23×C53—21no—
219C23×C53—21no—
220C23×C53—21no—
221C23×C53#12112yes—
222C23×C53C22×D411yescharacteristic
223Q8×C53—22no—
224Q8×C53—22no—
225Q8×C53—41no—
226Q8×C53—41no—
227Q8×C53—41no—
228Q8×C53—41no—
229Q8×C53—41no—
230Q8×C53—41no—
231C2×C4×C53—21no—
232C2×C4×C53—21no—
233C2×C4×C53—22no—
234C2×C4×C53—22no—
235C2×C4×C53—22no—
236C2×C4×C53—22no—
237C2×C4×C53—22no—
238C2×C4×C53—22no—
239C2×C4×C53—41no—
240C2×C4×C53—41no—
241C2×C4×C53—41no—
242C2×C4×C53—41no—
243C2×C4×C53—41no—
244C2×C4×C53—41no—
245C2×C4×C53—41no—
246C2×C4×C53—41no—
247C2×C4×C53—41no—
248C2×C4×C53—41no—
249C2×C4×C53#12112yes—
250C2×C4×C53C22×D411yescharacteristic
251C2×C4×C53C2×#4111yescharacteristic
252C2×C4×C53C2×#4111yescharacteristic
253C2×C4×C53C2×#4111yescharacteristic
254C2×C4×C53H(5,𝔽2)11yescharacteristic
255C2×C4×C532−1+411yescharacteristic
256C2×Dic53—21no—
257C2×Dic53—21no—
258C2×Dic53—21no—
259C2×Dic53—21no—
260C2×Dic53—22no—
261C2×Dic53—21no—
262C2×Dic53—21no—
263C2×Dic53—21no—
264C2×Dic53—21no—
265C2×Dic53—21no—
266C2×Dic53—21no—
267C2×Dic53—41no—
268C2×Dic53—41no—
269C2×Dic53—41no—
270C2×Dic53—41no—
271C2×Dic53—41no—
272C2×Dic53—41no—
273C2×Dic53—41no—
274C2×Dic53—41no—
275C2×Dic53—41no—
276C2×Dic53—41no—
277C2×Dic53—41no—
278C2×Dic53—84no—
279C2×Dic53#13712yes—
280C2×Dic53#13712yes—
281C8×C53—22no—
282C8×C53—41no—
283#88322—41no—
284#88322—41no—
285#25219—531nomaximal, Sylow
286C22×C53—22no—
287C22×C53—22no—
288C22×C53—22no—
289C22×C53—41no—
290C22×C53—41no—
291C22×C53—41no—
292C22×C53—41no—
293C22×C53C2×#12111yescharacteristic
294C22×C53#54611yescharacteristic
295C22×C53#54711yescharacteristic
296C22×C53C2×#13711yescharacteristic
297C22×C53C2×#13711yescharacteristic
298C22×C53#57711yescharacteristic
299C22×C53#57711yescharacteristic
300C4×C53—22no—
301C4×C53—42no—
302C4×C53—42no—
303C4×C53—81no—
304C4×C53—81no—
305C4×C53#52911yescharacteristic
306C4×C53#53911yescharacteristic
307C4×C53#54611yescharacteristic
308C4×C53#54711yescharacteristic
309Dic53—22no—
310Dic53—22no—
311Dic53—41no—
312Dic53—42no—
313Dic53—41no—
314Dic53—81no—
315Dic53—81no—
316Dic53—81no—
317C2×#483—531no—
318C2×#470D5311yescharacteristic
319#2502—531no—
320#2362—531no—
321#2084—531no—
322#2072—531no—
323#2066—531no—
324#2056—531no—
325#1948—531no—
326#1927—531no—
327#1878—531no—
328#1873—531no—
329#1851—531no—
330#1803—531no—
331#1736—531no—
332C2×C53—42no—
333C2×C53#242511yescharacteristic
334C2×C53#303911yescharacteristic
335C2×C53#304311yescharacteristic
336C2×C53#331611yescharacteristic
337C2×C53#333111yescharacteristic
338C2×C53#334411yescharacteristic
339C2×C53#335611yescharacteristic
340C22×Q16C2×D5311yescharacteristic
341C22×M16C2×D5311yescharacteristic
342C2×C22⋊Q8C2×D5311yescharacteristic
343C2×C22⋊Q8C2×D5311yescharacteristic
344C2×C4×Q8—531no—
345C2×#117—531no—
346C22×C4⋊C4—531no—
347#483—1064no—
348#470—24no—
349C2×#106—531no—
350C2×#106—531no—
351C2×#106C2×D5311yescharacteristic
352C2×C4⋊C8—531no—
353C2×C4⋊C8—531no—
354C2×#103—531no—
355C2×#103—531no—
356C2×#103—531no—
357C2×#103—531no—
358C2×#103C2×D5311yescharacteristic
359C2×#103C2×D5311yescharacteristic
360C2×C22⋊C8—531no—
361#398—1061no—
362#398—1061no—
363#391—531no—
364#391—531no—
365#391—531no—
366#389—531no—
367#388—1061no—
368#388—1061no—
369#387—1061no—
370#385—1061no—
371#385—1061no—
372#384—531no—
373#384—531no—
374C4×C4⋊C4—531no—
375C4×C4⋊C4—1061no—
376C4×C22⋊C4—531no—
377#340—531no—
378#340—531no—
379#340—531no—
380#340—531no—
381#340—531no—
382#340—531no—
383#336—531no—
384#336—531no—
385C53#2521911yescharacteristic, complemented, Sylow
386C22×Q8C22×D5311yescharacteristic
387C22×Q8C22×D5311yescharacteristic
388C23×C4—1061no—
389C23×C4C22×D5311yescharacteristic
390C2×Q16—22no—
391C2×Q16—22no—
392C2×Q16—41no—
393C2×M16—22no—
394C2×M16—22no—
395C2×M16—41no—
396C22×C8—531no—
397C22×C8—531no—
398C22×C8C22×D5311yescharacteristic
399C22×C8C22×D5311yescharacteristic
400C22⋊Q8—22no—
401C22⋊Q8—22no—
402C22⋊Q8—22no—
403C22⋊Q8C22⋊D5314yes—
404C4×Q8—1062no—
405C4×Q8—2121no—
406#117—1062no—
407#117—2121no—
408C2×C4⋊C4—21no—
409C2×C4⋊C4—21no—
410C2×C4⋊C4—531no—
411C2×C4⋊C4—531no—
412C2×C4⋊C4—531no—
413C2×C4⋊C4—531no—
414C2×C4⋊C4—1062no—
415C2×C4⋊C4—1062no—
416C2×C4⋊C4—1061no—
417C2×C4⋊C4—1061no—
418C2×C4⋊C4—1061no—
419C2×C4⋊C4—1061no—
420C2×C4⋊C4—1061no—
421C2×C4⋊C4—1061no—
422C2×C4⋊C4—1061no—
423C2×C4⋊C4C22×D5311yescharacteristic
424C2×C4⋊C4C22×D5311yescharacteristic
425C2×C22⋊C4—21no—
426C2×C22⋊C4—21no—
427C2×C22⋊C4—531no—
428C2×C22⋊C4—531no—
429C2×C22⋊C4—1061no—
430C2×C42—531no—
431C2×C42—531no—
432C2×C42—1061no—
433C2×C42—1061no—
434C2×C42—1061no—
435#106—22no—
436#106—1062no—
437#106—1062no—
438C4⋊C8—1062no—
439C4⋊C8—2121no—
440#103—22no—
441#103—22no—
442#103—1062no—
443#103—1062no—
444#103—1062no—
445#103—2121no—
446C22⋊C8—1062no—
447#95—531no—
448#95—531no—
449#95—1061no—
450#95—1061no—
451#95—1061no—
452#95—1061no—
453#95—1061no—
454#95—1061no—
455#95—1061no—
456#95—1061no—
457C24D4×D5311yescharacteristic
458C2×Q8—21no—
459C2×Q8—21no—
460C2×Q8—21no—
461C2×Q8—21no—
462C2×Q8—21no—
463C2×Q8—21no—
464C2×Q8—21no—
465C2×Q8—21no—
466C2×Q8—21no—
467C2×Q8—21no—
468C2×Q8—21no—
469C2×Q8C2×C22⋊D5312yes—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—21no—
478C22×C4—531no—
479C22×C4—531no—
480C22×C4—531no—
481C22×C4—531no—
482C22×C4—1061no—
483C22×C4—1061no—
484C22×C4—1061no—
485C22×C4—1061no—
486C22×C4—4241no—
487C22×C4—4241no—
488C22×C4—2122no—
489C22×C4D4×D5311yescharacteristic
490C22×C4#1171994712yes—
491C22×C4C2×C22⋊D5312yes—
492C22×C4C23×D5311yescharacteristic
493Q16—44no—
494M16—44no—
495C2×C8—21no—
496C2×C8—21no—
497C2×C8—21no—
498C2×C8—21no—
499C2×C8—21no—
500C2×C8—1061no—
501C2×C8—1061no—
502C2×C8—1061no—
503C2×C8—1061no—
504C2×C8—1061no—
505C2×C8—1061no—
506C2×C8#1171994712yes—
507C4⋊C4—24no—
508C4⋊C4—21no—
509C4⋊C4—21no—
510C4⋊C4—21no—
511C4⋊C4—42no—
512C4⋊C4—532no—
513C4⋊C4—1061no—
514C4⋊C4—1061no—
515C4⋊C4—1061no—
516C4⋊C4—1061no—
517C4⋊C4—1061no—
518C4⋊C4—1061no—
519C4⋊C4—1061no—
520C4⋊C4—4241no—
521C4⋊C4—4241no—
522C4⋊C4—4241no—
523C4⋊C4—4241no—
524C4⋊C4—2122no—
525C4⋊C4—2122no—
526C4⋊C4—2122no—
527C4⋊C4C2×C22⋊D5312yes—
528C22⋊C4—24no—
529C22⋊C4—42no—
530C22⋊C4—1062no—
531C22⋊C4—1062no—
532C22⋊C4—2122no—
533C42—1061no—
534C42—1061no—
535C42—1061no—
536C42—1061no—
537C42—2122no—
538C42—2122no—
539C42—2122no—
540C23—21no—
541C23—21no—
542C23—21no—
543C23—21no—
544C23—21no—
545C23—21no—
546C23#1313318912yes—
547C23C2×D4×D5311yescharacteristic
548Q8—22no—
549Q8—22no—
550Q8—41no—
551Q8—41no—
552Q8—41no—
553Q8—41no—
554Q8—41no—
555Q8—41no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—532no—
575C2×C4—532no—
576C2×C4—1061no—
577C2×C4—1061no—
578C2×C4—1061no—
579C2×C4—1061no—
580C2×C4—1061no—
581C2×C4—1061no—
582C2×C4—1061no—
583C2×C4—1061no—
584C2×C4—1061no—
585C2×C4—1061no—
586C2×C4—1062no—
587C2×C4—4244no—
588C2×C4—2121no—
589C2×C4—2121no—
590C2×C4—2121no—
591C2×C4—2121no—
592C2×C4—2121no—
593C2×C4—2121no—
594C2×C4—2121no—
595C2×C4—2121no—
596C2×C4—2121no—
597C2×C4—2121no—
598C2×C4—2121no—
599C2×C4#1313318912yes—
600C2×C4C22×C22⋊D5311yescharacteristic
601C2×C4#41×D5311yescharacteristic
602C2×C4C2×#1171994711yescharacteristic
603C2×C4C2×#1171994711yescharacteristic
604C2×C4#1313326211yescharacteristic
605C2×C4#1313326411yescharacteristic
606C8—22no—
607C8—41no—
608C8—2121no—
609C8—2121no—
610C22—22no—
611C22—22no—
612C22—22no—
613C22—41no—
614C22—41no—
615C22—41no—
616C22—41no—
617C22C2×#1313318911yescharacteristic
618C22#1681126211yescharacteristic
619C22#1681127311yescharacteristic
620C22#137×D5311yescharacteristic
621C22#137×D5311yescharacteristic
622C22#1681143211yescharacteristic
623C22#1681143211yescharacteristic
624C4—22no—
625C4—42no—
626C4—42no—
627C4—81no—
628C4—81no—
629C4—1062no—
630C4—1062no—
631C4—4241no—
632C4—4241no—
633C4—4241no—
634C4—2121no—
635C4—2122no—
636C4—2121no—
637C4#1681115411yescharacteristic
638C4#1681121811yescharacteristic
639C4#1681126711yescharacteristic
640C4#1681127211yescharacteristic
641C2—42no—
642C2#2027827211yescharacteristic
643C2#2028420011yescharacteristic
644C2#2028423811yescharacteristic
645C2#2028682111yescharacteristic
646C2#2028702611yescharacteristic
647C2#2028716511yescharacteristic
648C2#2028732511yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC2×C53×#470C2
socleC23×C53C22×D4
prime radicalC24×C53C2×D4
hypercenterC2×#470D53
hypocenterC53#25219
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#470

Complements

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

NH ≅ G/Ntype
4. #20276194C2semidirect
5. #20276126C2semidirect
6. #20276092C2semidirect
7. #20276035C2semidirect
8. #20275475C2semidirect
9. #20275352C2semidirect
10. #20275058C2semidirect
13. #20274718C2semidirect
385. C53#25219semidirect

Maximal subgroups

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

subgroupindex
C2×#168109512
#202791942
#202777462
#202761942
#202761262
#202760922
#202760352
#202754752
#202753522
#202750582
#202750282
#202748982
#202747182
#202744352
C2×C53×#4702
#2521953