Finite Group Databasebeta

← #27465824

Subgroups — #27465824

Order 13568.

828 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810939C211yesmaximal, characteristic
2#20277410C211yesmaximal, characteristic
3#20277335C211yesmaximal, characteristic, complemented
4#20276199C211yesmaximal, characteristic, complemented
5#20276194C211yesmaximal, characteristic, complemented
6#20276131C211yesmaximal, characteristic, complemented
7#20276121C211yesmaximal, characteristic, complemented
8#20275480C211yesmaximal, characteristic, complemented
9#20275475C211yesmaximal, characteristic, complemented
10#20274810C211yesmaximal, characteristic, complemented
11#20274802C211yesmaximal, characteristic, complemented
12#20274723C211yesmaximal, characteristic, complemented
13#20274713C211yesmaximal, characteristic, complemented
14#20274697C211yesmaximal, characteristic, complemented
15C2×C53×#482C211yesmaximal, characteristic
16C2×D4×Dic53C2211yescharacteristic
17C2×#13133176C2211yescharacteristic
18C22×C4⋊Dic53C2211yescharacteristic
19#16810939—24no—
20C2×#13133148C2211yescharacteristic
21C2×#13133148C2211yescharacteristic
22C2×#13133146C2211yescharacteristic
23C2×#13133146C2211yescharacteristic
24C2×#13133138C2211yescharacteristic
25C2×#13133138C2211yescharacteristic
26C2×C22⋊#88322C2211yescharacteristic
27#16810526—21no—
28#16810526C2211yescharacteristic
29#16810522C2211yescharacteristic
30#16810521—21no—
31#16810521—21no—
32#16810516—21no—
33#16810503—21no—
34#16810498—21no—
35#16810495—21no—
36#16810495—21no—
37#16810492C2211yescharacteristic
38#16810492C2211yescharacteristic
39#16810488C2211yescharacteristic
40#16810488C2211yescharacteristic
41#16810483C2211yescharacteristic
42C4⋊C4×Dic53C2211yescharacteristic
43#16810386C2211yescharacteristic
44#16810386C2211yescharacteristic
45#16810386C2211yescharacteristic
46#16810386C2211yescharacteristic
47#16810385C2211yescharacteristic
48#16810385C2211yescharacteristic
49#16810385C2211yescharacteristic
50#16810385C2211yescharacteristic
51#16810377C2211yescharacteristic
52#16810377C2211yescharacteristic
53C2×#121×C53C2211yescharacteristic
54C2×#117×C53C2211yescharacteristic
55C53×#482—24no—
56C2×#107×C53C2211yescharacteristic
57C2×#106×C53C2211yescharacteristic
58C2×#102×C53C2211yescharacteristic
59C2×#102×C53C2211yescharacteristic
60C2×C22⋊C8×C53C2211yescharacteristic
61C23×Dic53—21no—
62C23×Dic53—21no—
63C22×#88322C2311yescharacteristic
64C22×#88322C2311yescharacteristic
65D4×Dic53—22no—
66D4×Dic53—22no—
67#13133176—22no—
68#13133176—22no—
69C2×C4⋊Dic53—21no—
70C2×C4⋊Dic53—21no—
71C2×C4⋊Dic53—22no—
72C2×C4⋊Dic53—22no—
73C2×C4⋊Dic53D411yescharacteristic
74C2×C4⋊Dic53D411yescharacteristic
75C2×C4⋊Dic53C2311yescharacteristic
76C2×C4⋊Dic53C2311yescharacteristic
77C2×C4⋊Dic53C2311yescharacteristic
78C2×C4⋊Dic53C2311yescharacteristic
79C2×#11719919—21no—
80C2×#11719919—21no—
81C2×#11719919—21no—
82C2×#11719919—21no—
83C2×C22⋊Dic53—21no—
84C2×C22⋊Dic53—21no—
85C2×C22⋊Dic53—21no—
86C2×C22⋊Dic53—21no—
87C2×C22⋊Dic53D411yescharacteristic
88C2×C22⋊Dic53D411yescharacteristic
89C2×C4×Dic53—21no—
90C2×C4×Dic53C2311yescharacteristic
91C2×C4×Dic53C2311yescharacteristic
92#13133148—22no—
93#13133148—22no—
94#13133146—22no—
95#13133146—22no—
96#13133138—22no—
97#13133138—22no—
98C22⋊#88322—22no—
99#13133124—21no—
100#13133124—21no—
101#13133124—21no—
102#13133124—21no—
103#13133124—21no—
104#13133124—21no—
105#13133124—21no—
106C22×D4×C53—21no—
107C22×D4×C53C2311yescharacteristic
108C23×C4×C53C2311yescharacteristic
109C22×C8×C53C2311yescharacteristic
110C22×C8×C53C2311yescharacteristic
111#121×C53—22no—
112#121×C53D414yes—
113#117×C53—22no—
114#117×C53—22no—
115C2×C4⋊C4×C53C2311yescharacteristic
116C2×C4⋊C4×C53C2311yescharacteristic
117C2×C4⋊C4×C53C2311yescharacteristic
118C2×C22⋊C4×C53—21no—
119C2×C22⋊C4×C53—21no—
120C2×C42×C53—21no—
121#107×C53—22no—
122#106×C53—22no—
123#102×C53—22no—
124#102×C53—22no—
125C22⋊C8×C53—22no—
126C24×C53—21no—
127C24×C53C2×D411yescharacteristic
128C2×D4×C53—24no—
129C2×D4×C53—21no—
130C2×D4×C53—21no—
131C2×D4×C53—21no—
132C2×D4×C53—21no—
133C2×D4×C53—21no—
134C2×D4×C53—81no—
135C2×D4×C53—81no—
136C2×D4×C53C2×D412yes—
137C22×C4×C53—21no—
138C22×C4×C53—21no—
139C22×C4×C53—21no—
140C22×C4×C53—21no—
141C22×C4×C53—21no—
142C22×C4×C53—21no—
143C22×C4×C53—21no—
144C22×C4×C53C2×D411yescharacteristic
145C22×C4×C53C2×D412yes—
146C22×C4×C53#4112yes—
147C22×C4×C53C2411yescharacteristic
148C22×Dic53—21no—
149C22×Dic53—21no—
150C22×Dic53—21no—
151C22×Dic53—21no—
152C22×Dic53—42no—
153C22×Dic53—42no—
154C22×Dic53—81no—
155C22×Dic53—81no—
156C22×Dic53—81no—
157C22×Dic53—81no—
158C22×Dic53C2×D411yescharacteristic
159C22×Dic53C2×D411yescharacteristic
160C22×Dic53C2×D411yescharacteristic
161C22×Dic53C2×D411yescharacteristic
162C2×C8×C53—21no—
163C2×C8×C53—21no—
164C2×C8×C53—21no—
165C2×C8×C53—21no—
166C2×C8×C53—21no—
167C2×C8×C53—21no—
168C4⋊C4×C53—21no—
169C4⋊C4×C53—21no—
170C4⋊C4×C53—21no—
171C4⋊C4×C53—21no—
172C4⋊C4×C53—21no—
173C4⋊C4×C53C2×D412yes—
174C22⋊C4×C53—24no—
175C22⋊C4×C53—42no—
176C42×C53—42no—
177C22⋊Dic53—22no—
178C22⋊Dic53—22no—
179C22⋊Dic53—42no—
180C22⋊Dic53—81no—
181C22⋊Dic53—81no—
182C22⋊Dic53—81no—
183C4⋊Dic53—22no—
184C4⋊Dic53—22no—
185C4⋊Dic53—21no—
186C4⋊Dic53—21no—
187C4⋊Dic53—21no—
188C4⋊Dic53—21no—
189C4⋊Dic53—21no—
190C4⋊Dic53—21no—
191C4⋊Dic53—21no—
192C4⋊Dic53—21no—
193C4⋊Dic53—42no—
194C4⋊Dic53—81no—
195#11719919—81no—
196#11719919—81no—
197#11719919—81no—
198#11719919—81no—
199C4×Dic53—21no—
200C4×Dic53—21no—
201C4×Dic53—81no—
202C4×Dic53#4112yes—
203C4×Dic53#4112yes—
204C2×#88322—21no—
205C2×#88322—21no—
206C2×#88322—21no—
207C2×#88322—21no—
208C2×#88322—21no—
209C2×#88322—21no—
210C23×C53—21no—
211C23×C53—21no—
212C23×C53—21no—
213C23×C53—21no—
214C23×C53—21no—
215C23×C53—21no—
216C23×C53—22no—
217C23×C53—41no—
218C23×C53—41no—
219C23×C53—41no—
220C23×C53—41no—
221C23×C53—41no—
222C23×C53—41no—
223C23×C53#12112yes—
224C23×C53C22×D411yescharacteristic
225D4×C53—22no—
226D4×C53—22no—
227D4×C53—41no—
228D4×C53—41no—
229D4×C53—84no—
230C2×C4×C53—21no—
231C2×C4×C53—21no—
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#12112yes—
246C2×C4×C53C22×D411yescharacteristic
247C2×C4×C53C2×#4111yescharacteristic
248C2×C4×C53C2×#4111yescharacteristic
249C2×C4×C53C2×#4111yescharacteristic
250C2×C4×C532−1+411yescharacteristic
251C2×C4×C532−1+411yescharacteristic
252C2×Dic53—21no—
253C2×Dic53—21no—
254C2×Dic53—21no—
255C2×Dic53—21no—
256C2×Dic53—21no—
257C2×Dic53—21no—
258C2×Dic53—21no—
259C2×Dic53—21no—
260C2×Dic53—21no—
261C2×Dic53—21no—
262C2×Dic53—41no—
263C2×Dic53—41no—
264C2×Dic53—41no—
265C2×Dic53—41no—
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—84no—
275C2×Dic53—84no—
276C2×Dic53#12312yes—
277C2×Dic53#12312yes—
278C8×C53—41no—
279C8×C53—41no—
280#88322—41no—
281#88322—41no—
282#27654—531nomaximal, Sylow
283C22×C53—22no—
284C22×C53—22no—
285C22×C53—22no—
286C22×C53—41no—
287C22×C53—41no—
288C22×C53—41no—
289C22×C53—41no—
290C22×C53—42no—
291C22×C53—42no—
292C22×C53—42no—
293C22×C53—81no—
294C22×C53—81no—
295C22×C53—81no—
296C22×C53—81no—
297C22×C53C2×#12111yescharacteristic
298C22×C53C2×#12311yescharacteristic
299C22×C53#53611yescharacteristic
300C22×C53#57611yescharacteristic
301C22×C53#57711yescharacteristic
302C22×C53#57711yescharacteristic
303C22×C53#57711yescharacteristic
304C4×C53—22no—
305C4×C53—42no—
306C4×C53—81no—
307C4×C53—81no—
308C4×C53#54711yescharacteristic
309C4×C53#54711yescharacteristic
310C4×C53#55211yescharacteristic
311C4×C53#55211yescharacteristic
312Dic53—22no—
313Dic53—22no—
314Dic53—41no—
315Dic53—41no—
316Dic53—81no—
317Dic53—81no—
318Dic53—81no—
319Dic53—81no—
320C2×#481—531no—
321C2×#482D5311yescharacteristic
322#2324—531no—
323#2317—531no—
324#2085—531no—
325#2084—531no—
326#2073—531no—
327#2071—531no—
328#1949—531no—
329#1948—531no—
330#1825—531no—
331#1823—531no—
332#1804—531no—
333#1802—531no—
334#1798—531no—
335C2×C53—42no—
336C2×C53—82no—
337C2×C53#248611yescharacteristic
338C2×C53#304211yescharacteristic
339C2×C53#304511yescharacteristic
340C2×C53#309611yescharacteristic
341C2×C53#309911yescharacteristic
342C2×C53#324111yescharacteristic
343C2×C53#324411yescharacteristic
344C2×#121C2×D5311yescharacteristic
345C2×C4×D4—531no—
346C2×#117—531no—
347C2×#117C2×D5311yescharacteristic
348C22×C4⋊C4—531no—
349#482—24no—
350#481—1064no—
351C2×#107C2×D5311yescharacteristic
352C2×#106—531no—
353C2×#106—531no—
354C2×#106C2×D5311yescharacteristic
355C2×C4⋊C8—531no—
356C2×C4⋊C8—531no—
357C2×#102—531no—
358C2×#102—531no—
359C2×#102C2×D5311yescharacteristic
360C2×#102C2×D5311yescharacteristic
361C2×C22⋊C8—531no—
362C2×C22⋊C8C2×D5311yescharacteristic
363#390—531no—
364#390—1061no—
365#389—531no—
366#389—1061no—
367#389—1061no—
368#388—1061no—
369#386—1061no—
370#385—1061no—
371#385—1061no—
372#385—1061no—
373#384—531no—
374#384—531no—
375#382—531no—
376#382—531no—
377#380—531no—
378C4×C4⋊C4—531no—
379#340—531no—
380#340—531no—
381#340—531no—
382#340—531no—
383#340—531no—
384#340—531no—
385#340—531no—
386#340—531no—
387#336—531no—
388#336—531no—
389C53#2765411yescharacteristic, complemented, Sylow
390C22×D4—21no—
391C22×D4C22×D5311yescharacteristic
392C23×C4—1061no—
393C23×C4—1061no—
394C23×C4C22×D5311yescharacteristic
395C22×C8—531no—
396C22×C8—531no—
397C22×C8C22×D5311yescharacteristic
398C22×C8C22×D5311yescharacteristic
399#121—22no—
400#121C22⋊D5314yes—
401C4×D4—1062no—
402C4×D4—1062no—
403#117—22no—
404#117—22no—
405#117—1062no—
406#117—1062no—
407C2×C4⋊C4—531no—
408C2×C4⋊C4—531no—
409C2×C4⋊C4—531no—
410C2×C4⋊C4—531no—
411C2×C4⋊C4—531no—
412C2×C4⋊C4—531no—
413C2×C4⋊C4—1062no—
414C2×C4⋊C4—1062no—
415C2×C4⋊C4—1061no—
416C2×C4⋊C4—1061no—
417C2×C4⋊C4—1061no—
418C2×C4⋊C4—1061no—
419C2×C4⋊C4—1061no—
420C2×C4⋊C4—1061no—
421C2×C4⋊C4C22×D5311yescharacteristic
422C2×C4⋊C4C22×D5311yescharacteristic
423C2×C4⋊C4C22×D5311yescharacteristic
424C2×C22⋊C4—21no—
425C2×C22⋊C4—21no—
426C2×C22⋊C4—531no—
427C2×C22⋊C4—531no—
428C2×C22⋊C4—1061no—
429C2×C22⋊C4—1061no—
430C2×C22⋊C4—1061no—
431C2×C22⋊C4—1061no—
432C2×C42—21no—
433C2×C42—531no—
434C2×C42—531no—
435C2×C42—1061no—
436#107—22no—
437#106—22no—
438#106—1062no—
439#106—1062no—
440C4⋊C8—1062no—
441C4⋊C8—1062no—
442#102—22no—
443#102—22no—
444#102—1062no—
445#102—1062no—
446C22⋊C8—22no—
447C22⋊C8—1062no—
448#95—1061no—
449#95—1061no—
450#95—1061no—
451#95—1061no—
452#95—1061no—
453#95—1061no—
454#95—1061no—
455C24—21no—
456C24D4×D5311yescharacteristic
457C2×D4—24no—
458C2×D4—21no—
459C2×D4—21no—
460C2×D4—21no—
461C2×D4—21no—
462C2×D4—21no—
463C2×D4—81no—
464C2×D4—81no—
465C2×D4C2×C22⋊D5312yes—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—531no—
474C22×C4—531no—
475C22×C4—531no—
476C22×C4—531no—
477C22×C4—1061no—
478C22×C4—1061no—
479C22×C4—1061no—
480C22×C4—1061no—
481C22×C4—4241no—
482C22×C4—4241no—
483C22×C4—4241no—
484C22×C4—4241no—
485C22×C4—2122no—
486C22×C4—2122no—
487C22×C4D4×D5311yescharacteristic
488C22×C4#1171994712yes—
489C22×C4C2×C22⋊D5312yes—
490C22×C4C23×D5311yescharacteristic
491C2×C8—21no—
492C2×C8—21no—
493C2×C8—21no—
494C2×C8—21no—
495C2×C8—21no—
496C2×C8—21no—
497C2×C8—1061no—
498C2×C8—1061no—
499C2×C8—1061no—
500C2×C8—1061no—
501C2×C8—1061no—
502C2×C8—1061no—
503C4⋊C4—21no—
504C4⋊C4—21no—
505C4⋊C4—21no—
506C4⋊C4—21no—
507C4⋊C4—21no—
508C4⋊C4—1062no—
509C4⋊C4—1062no—
510C4⋊C4—1061no—
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—4241no—
520C4⋊C4—4241no—
521C4⋊C4—4241no—
522C4⋊C4—4241no—
523C4⋊C4—2122no—
524C4⋊C4C2×C22⋊D5312yes—
525C22⋊C4—24no—
526C22⋊C4—42no—
527C22⋊C4—1062no—
528C22⋊C4—1062no—
529C22⋊C4—4241no—
530C22⋊C4—4241no—
531C22⋊C4—4241no—
532C22⋊C4—2122no—
533C42—42no—
534C42—532no—
535C42—532no—
536C42—1061no—
537C42—1061no—
538C42—4241no—
539C23—21no—
540C23—21no—
541C23—21no—
542C23—21no—
543C23—21no—
544C23—21no—
545C23—22no—
546C23—41no—
547C23—41no—
548C23—41no—
549C23—41no—
550C23—41no—
551C23—41no—
552C23#1313318912yes—
553C23C2×D4×D5311yescharacteristic
554D4—22no—
555D4—22no—
556D4—41no—
557D4—41no—
558D4—84no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
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—4244no—
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—2121no—
600C2×C4#1313318912yes—
601C2×C4C22×C22⋊D5311yescharacteristic
602C2×C4#41×D5311yescharacteristic
603C2×C4#41×D5311yescharacteristic
604C2×C4C2×#1171994711yescharacteristic
605C2×C4#1313326411yescharacteristic
606C2×C4#1313326411yescharacteristic
607C8—41no—
608C8—41no—
609C8—2121no—
610C8—2121no—
611C22—22no—
612C22—22no—
613C22—22no—
614C22—41no—
615C22—41no—
616C22—41no—
617C22—41no—
618C22—42no—
619C22—42no—
620C22—42no—
621C22—81no—
622C22—81no—
623C22—81no—
624C22—81no—
625C22C2×#1313318911yescharacteristic
626C22#123×D5311yescharacteristic
627C22#1681119511yescharacteristic
628C22#1681142511yescharacteristic
629C22#1681143011yescharacteristic
630C22#1681143311yescharacteristic
631C22#1681143311yescharacteristic
632C4—22no—
633C4—42no—
634C4—81no—
635C4—81no—
636C4—1062no—
637C4—1062no—
638C4—4241no—
639C4—4241no—
640C4—4241no—
641C4—4241no—
642C4—2121no—
643C4—2121no—
644C4#1681127211yescharacteristic
645C4#1681127211yescharacteristic
646C4#1681131011yescharacteristic
647C4#1681131011yescharacteristic
648C2—42no—
649C2—82no—
650C2#2027902011yescharacteristic
651C2#2028422911yescharacteristic
652C2#2028425211yescharacteristic
653C2#2028467011yescharacteristic
654C2#2028470711yescharacteristic
655C2#2028597411yescharacteristic
656C2#2028601111yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC2×C53×#482C2
socleC23×C53C22×D4
prime radicalC2×#121×C53C22
hypercenterC2×#482D53
hypocenterC53#27654
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#482

Complements

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

NH ≅ G/Ntype
3. #20277335C2semidirect
4. #20276199C2semidirect
5. #20276194C2semidirect
6. #20276131C2semidirect
7. #20276121C2semidirect
8. #20275480C2semidirect
9. #20275475C2semidirect
10. #20274810C2semidirect
11. #20274802C2semidirect
12. #20274723C2semidirect
13. #20274713C2semidirect
14. #20274697C2semidirect
389. C53#27654semidirect

Maximal subgroups

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

subgroupindex
C2×#168109392
#202774102
#202773352
#202761992
#202761942
#202761312
#202761212
#202754802
#202754752
#202748102
#202748022
#202747232
#202747132
#202746972
C2×C53×#4822
#2765453