Finite Group Databasebeta

← #27432551

Subgroups — #27432551

Order 13568.

816 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810961C211yesmaximal, characteristic
2#20279169C211yesmaximal, characteristic
3#20277724C211yesmaximal, characteristic
4#20276194C211yesmaximal, characteristic, complemented
5#20276136C211yesmaximal, characteristic, complemented
6#20276106C211yesmaximal, characteristic, complemented
7#20276035C211yesmaximal, characteristic, complemented
8#20275484C211yesmaximal, characteristic, complemented
9#20275352C211yesmaximal, characteristic, complemented
10#20275058C211yesmaximal, characteristic, complemented
11#20275023C211yesmaximal, characteristic
12#20274898C211yesmaximal, characteristic
13#20274718C211yesmaximal, characteristic, complemented
14#20274435C211yesmaximal, characteristic
15C2×C53×#470C211yesmaximal, characteristic
16C2×Q8×Dic53C2211yescharacteristic
17C2×#13133176C2211yescharacteristic
18C22×C4⋊Dic53C2211yescharacteristic
19#16810961—24no—
20C2×#13133150C2211yescharacteristic
21C2×#13133148C2211yescharacteristic
22C2×#13133146C2211yescharacteristic
23C2×#13133146C2211yescharacteristic
24C2×#13133141C2211yescharacteristic
25C2×#13133141C2211yescharacteristic
26C2×#13133141C2211yescharacteristic
27C2×#13133141C2211yescharacteristic
28C2×C22⋊#88322C2211yescharacteristic
29#16810551—21no—
30#16810551—21no—
31#16810530C2211yescharacteristic
32#16810530C2211yescharacteristic
33#16810530C2211yescharacteristic
34#16810522—21no—
35#16810522C2211yescharacteristic
36#16810509—21no—
37#16810504—21no—
38#16810504—21no—
39#16810498—21no—
40#16810492C2211yescharacteristic
41#16810488C2211yescharacteristic
42C4⋊C4×Dic53C2211yescharacteristic
43C22⋊C4×Dic53—21no—
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—22no—
68C2×C4⋊Dic53—21no—
69C2×C4⋊Dic53—21no—
70C2×C4⋊Dic53—21no—
71C2×C4⋊Dic53—22no—
72C2×C4⋊Dic53—41no—
73C2×C4⋊Dic53C2311yescharacteristic
74C2×C4⋊Dic53C2311yescharacteristic
75C2×C4⋊Dic53C2311yescharacteristic
76C2×C4⋊Dic53C2311yescharacteristic
77C2×#11719919—21no—
78C2×#11719919—21no—
79C2×C22⋊Dic53—21no—
80C2×C22⋊Dic53D411yescharacteristic
81C2×C22⋊Dic53D411yescharacteristic
82C2×C4×Dic53—21no—
83C2×C4×Dic53—21no—
84C2×C4×Dic53—21no—
85C2×C4×Dic53C2311yescharacteristic
86C2×C4×Dic53C2311yescharacteristic
87#13133150—22no—
88#13133148—22no—
89#13133146—22no—
90#13133146—41no—
91#13133141—22no—
92#13133141—22no—
93#13133141—22no—
94#13133141—41no—
95C22⋊#88322—22no—
96#13133124—21no—
97#13133124—21no—
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—42no—
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—41no—
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—81no—
197C4⋊Dic53#4112yes—
198#11719919—42no—
199#11719919—42no—
200C4×Dic53—21no—
201C4×Dic53—21no—
202C4×Dic53—21no—
203C4×Dic53—21no—
204C4×Dic53—42no—
205C4×Dic53—42no—
206C4×Dic53—81no—
207C2×#88322—21no—
208C2×#88322—21no—
209C2×#88322—21no—
210C2×#88322—21no—
211C2×#88322—21no—
212C2×#88322—21no—
213C23×C53—21no—
214C23×C53—21no—
215C23×C53—21no—
216C23×C53—21no—
217C23×C53—21no—
218C23×C53—21no—
219C23×C53#12112yes—
220C23×C53C22×D411yescharacteristic
221Q8×C53—22no—
222Q8×C53—22no—
223Q8×C53—41no—
224Q8×C53—41no—
225Q8×C53—41no—
226Q8×C53—41no—
227Q8×C53—41no—
228Q8×C53—41no—
229C2×C4×C53—21no—
230C2×C4×C53—21no—
231C2×C4×C53—22no—
232C2×C4×C53—22no—
233C2×C4×C53—22no—
234C2×C4×C53—22no—
235C2×C4×C53—22no—
236C2×C4×C53—22no—
237C2×C4×C53—41no—
238C2×C4×C53—41no—
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#12112yes—
248C2×C4×C53C22×D411yescharacteristic
249C2×C4×C53C2×#4111yescharacteristic
250C2×C4×C53C2×#4111yescharacteristic
251C2×C4×C53C2×#4111yescharacteristic
252C2×C4×C53H(5,𝔽2)11yescharacteristic
253C2×C4×C532−1+411yescharacteristic
254C2×Dic53—21no—
255C2×Dic53—21no—
256C2×Dic53—22no—
257C2×Dic53—21no—
258C2×Dic53—21no—
259C2×Dic53—21no—
260C2×Dic53—21no—
261C2×Dic53—21no—
262C2×Dic53—21no—
263C2×Dic53—21no—
264C2×Dic53—21no—
265C2×Dic53—21no—
266C2×Dic53—41no—
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—84no—
278C2×Dic53#13712yes—
279C8×C53—22no—
280C8×C53—41no—
281#88322—41no—
282#88322—41no—
283#25220—531nomaximal, Sylow
284C22×C53—22no—
285C22×C53—22no—
286C22×C53—22no—
287C22×C53—41no—
288C22×C53—41no—
289C22×C53—41no—
290C22×C53—41no—
291C22×C53C2×#12111yescharacteristic
292C22×C53#54611yescharacteristic
293C22×C53#54711yescharacteristic
294C22×C53C2×#13711yescharacteristic
295C22×C53#57511yescharacteristic
296C22×C53#57711yescharacteristic
297C22×C53#57811yescharacteristic
298C4×C53—22no—
299C4×C53—42no—
300C4×C53—42no—
301C4×C53—81no—
302C4×C53—81no—
303C4×C53#52911yescharacteristic
304C4×C53#53911yescharacteristic
305C4×C53#54611yescharacteristic
306C4×C53#54711yescharacteristic
307Dic53—22no—
308Dic53—41no—
309Dic53—41no—
310Dic53—41no—
311Dic53—42no—
312Dic53—81no—
313Dic53—81no—
314Dic53—81no—
315C2×#485—531no—
316C2×#470D5311yescharacteristic
317#2499—531no—
318#2360—531no—
319#2084—531no—
320#2074—531no—
321#2068—531no—
322#2056—531no—
323#1949—531no—
324#1927—531no—
325#1878—531no—
326#1872—531no—
327#1851—531no—
328#1803—531no—
329#1736—531no—
330C2×C53—42no—
331C2×C53#242511yescharacteristic
332C2×C53#303711yescharacteristic
333C2×C53#304611yescharacteristic
334C2×C53#331211yescharacteristic
335C2×C53#332311yescharacteristic
336C2×C53#334911yescharacteristic
337C2×C53#335611yescharacteristic
338C22×Q16C2×D5311yescharacteristic
339C22×M16C2×D5311yescharacteristic
340C2×C22⋊Q8C2×D5311yescharacteristic
341C2×C22⋊Q8C2×D5311yescharacteristic
342C2×C4×Q8—531no—
343C2×#117—531no—
344C22×C4⋊C4—531no—
345#485—1064no—
346#470—24no—
347C2×#107—531no—
348C2×#106—531no—
349C2×#106C2×D5311yescharacteristic
350C2×C4⋊C8—531no—
351C2×C4⋊C8—531no—
352C2×#103—531no—
353C2×#103—531no—
354C2×#103—531no—
355C2×#103—531no—
356C2×#103C2×D5311yescharacteristic
357C2×#103C2×D5311yescharacteristic
358C2×C22⋊C8—531no—
359#397—1061no—
360#397—1061no—
361#391—531no—
362#391—531no—
363#391—531no—
364#389—531no—
365#389—1061no—
366#387—1061no—
367#387—1061no—
368#387—1061no—
369#385—1061no—
370#384—531no—
371#382—531no—
372C4×C4⋊C4—531no—
373C4×C22⋊C4—531no—
374C4×C22⋊C4—1061no—
375#340—531no—
376#340—531no—
377#340—531no—
378#340—531no—
379#340—531no—
380#340—531no—
381#336—531no—
382#336—531no—
383C53#2522011yescharacteristic, complemented, Sylow
384C22×Q8C22×D5311yescharacteristic
385C22×Q8C22×D5311yescharacteristic
386C23×C4—1061no—
387C23×C4C22×D5311yescharacteristic
388C2×Q16—22no—
389C2×Q16—22no—
390C2×Q16—41no—
391C2×M16—22no—
392C2×M16—22no—
393C2×M16—41no—
394C22×C8—531no—
395C22×C8—531no—
396C22×C8C22×D5311yescharacteristic
397C22×C8C22×D5311yescharacteristic
398C22⋊Q8—22no—
399C22⋊Q8—22no—
400C22⋊Q8—22no—
401C22⋊Q8C22⋊D5314yes—
402C4×Q8—1062no—
403C4×Q8—2121no—
404#117—1062no—
405#117—1062no—
406C2×C4⋊C4—21no—
407C2×C4⋊C4—21no—
408C2×C4⋊C4—531no—
409C2×C4⋊C4—531no—
410C2×C4⋊C4—531no—
411C2×C4⋊C4—531no—
412C2×C4⋊C4—1062no—
413C2×C4⋊C4—1061no—
414C2×C4⋊C4—1061no—
415C2×C4⋊C4—1061no—
416C2×C4⋊C4—1061no—
417C2×C4⋊C4—1061no—
418C2×C4⋊C4—2121no—
419C2×C4⋊C4C22×D5311yescharacteristic
420C2×C4⋊C4C22×D5311yescharacteristic
421C2×C22⋊C4—21no—
422C2×C22⋊C4—21no—
423C2×C22⋊C4—531no—
424C2×C22⋊C4—531no—
425C2×C22⋊C4—1061no—
426C2×C42—531no—
427C2×C42—531no—
428C2×C42—1061no—
429C2×C42—1061no—
430C2×C42—1061no—
431#107—1062no—
432#106—22no—
433#106—1062no—
434C4⋊C8—1062no—
435C4⋊C8—2121no—
436#103—22no—
437#103—22no—
438#103—1062no—
439#103—1062no—
440#103—1062no—
441#103—2121no—
442C22⋊C8—1062no—
443#95—531no—
444#95—531no—
445#95—1061no—
446#95—1061no—
447#95—1061no—
448#95—1061no—
449#95—1061no—
450#95—1061no—
451#95—1061no—
452#95—1061no—
453#95—1061no—
454#95—1061no—
455C24D4×D5311yescharacteristic
456C2×Q8—21no—
457C2×Q8—21no—
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×Q8C2×C22⋊D5312yes—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—531no—
477C22×C4—531no—
478C22×C4—531no—
479C22×C4—531no—
480C22×C4—1061no—
481C22×C4—1061no—
482C22×C4—1061no—
483C22×C4—1061no—
484C22×C4—4241no—
485C22×C4—2122no—
486C22×C4—2122no—
487C22×C4D4×D5311yescharacteristic
488C22×C4#1171994712yes—
489C22×C4C2×C22⋊D5312yes—
490C22×C4C23×D5311yescharacteristic
491Q16—44no—
492M16—44no—
493C2×C8—21no—
494C2×C8—21no—
495C2×C8—21no—
496C2×C8—21no—
497C2×C8—21no—
498C2×C8—1061no—
499C2×C8—1061no—
500C2×C8—1061no—
501C2×C8—1061no—
502C2×C8—1061no—
503C2×C8—1061no—
504C2×C8#1171994712yes—
505C4⋊C4—24no—
506C4⋊C4—21no—
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—42no—
510C4⋊C4—532no—
511C4⋊C4—1061no—
512C4⋊C4—1061no—
513C4⋊C4—1061no—
514C4⋊C4—1061no—
515C4⋊C4—1061no—
516C4⋊C4—1061no—
517C4⋊C4—1061no—
518C4⋊C4—4241no—
519C4⋊C4—2122no—
520C4⋊C4—2122no—
521C4⋊C4—2122no—
522C4⋊C4—2122no—
523C4⋊C4C2×C22⋊D5312yes—
524C22⋊C4—24no—
525C22⋊C4—42no—
526C22⋊C4—1062no—
527C22⋊C4—2121no—
528C22⋊C4—2122no—
529C42—1061no—
530C42—1061no—
531C42—1061no—
532C42—1061no—
533C42—4241no—
534C42—2122no—
535C42—2122no—
536C23—21no—
537C23—21no—
538C23—21no—
539C23—21no—
540C23—21no—
541C23—21no—
542C23#1313318912yes—
543C23C2×D4×D5311yescharacteristic
544Q8—22no—
545Q8—22no—
546Q8—41no—
547Q8—41no—
548Q8—41no—
549Q8—41no—
550Q8—41no—
551Q8—41no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—532no—
571C2×C4—1061no—
572C2×C4—1061no—
573C2×C4—1061no—
574C2×C4—1061no—
575C2×C4—1061no—
576C2×C4—1061no—
577C2×C4—1061no—
578C2×C4—1061no—
579C2×C4—1061no—
580C2×C4—1061no—
581C2×C4—1061no—
582C2×C4—1062no—
583C2×C4—4244no—
584C2×C4—2121no—
585C2×C4—2121no—
586C2×C4—2121no—
587C2×C4—2121no—
588C2×C4—2121no—
589C2×C4—2121no—
590C2×C4—2121no—
591C2×C4—2121no—
592C2×C4—2121no—
593C2×C4—2121no—
594C2×C4—2121no—
595C2×C4#1313318912yes—
596C2×C4C22×C22⋊D5311yescharacteristic
597C2×C4#41×D5311yescharacteristic
598C2×C4C2×#1171994711yescharacteristic
599C2×C4C2×#1171994711yescharacteristic
600C2×C4#1313326211yescharacteristic
601C2×C4#1313326411yescharacteristic
602C8—22no—
603C8—41no—
604C8—2121no—
605C8—2121no—
606C22—22no—
607C22—22no—
608C22—22no—
609C22—41no—
610C22—41no—
611C22—41no—
612C22—41no—
613C22C2×#1313318911yescharacteristic
614C22#1681126211yescharacteristic
615C22#1681127311yescharacteristic
616C22#137×D5311yescharacteristic
617C22#1681142211yescharacteristic
618C22#1681143211yescharacteristic
619C22#1681143911yescharacteristic
620C4—22no—
621C4—42no—
622C4—42no—
623C4—81no—
624C4—81no—
625C4—1062no—
626C4—4241no—
627C4—4241no—
628C4—4241no—
629C4—2121no—
630C4—2121no—
631C4—2121no—
632C4—2122no—
633C4#1681115411yescharacteristic
634C4#1681121811yescharacteristic
635C4#1681126711yescharacteristic
636C4#1681127211yescharacteristic
637C2—42no—
638C2#2027827211yescharacteristic
639C2#2028418411yescharacteristic
640C2#2028426911yescharacteristic
641C2#2028676111yescharacteristic
642C2#2028693711yescharacteristic
643C2#2028724011yescharacteristic
644C2#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#25220
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. #20276136C2semidirect
6. #20276106C2semidirect
7. #20276035C2semidirect
8. #20275484C2semidirect
9. #20275352C2semidirect
10. #20275058C2semidirect
13. #20274718C2semidirect
383. C53#25220semidirect

Maximal subgroups

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

subgroupindex
C2×#168109612
#202791692
#202777242
#202761942
#202761362
#202761062
#202760352
#202754842
#202753522
#202750582
#202750232
#202748982
#202747182
#202744352
C2×C53×#4702
#2522053