Finite Group Databasebeta

← #27534049

Subgroups — #27534049

Order 13568.

864 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×#16810930C211yesmaximal, characteristic, complemented
2#20283085C211yesmaximal, characteristic
3C4×Q8×Dic53C211yesmaximal, characteristic, complemented
4#20275670C211yesmaximal, characteristic, complemented
5#20275648C211yesmaximal, characteristic
6#20275359C211yesmaximal, characteristic, complemented
7#20275352C211yesmaximal, characteristic, complemented
8#20275345C211yesmaximal, characteristic
9#20275338C211yesmaximal, characteristic
10#20275072C211yesmaximal, characteristic
11#20275072C211yesmaximal, characteristic
12#20275058C211yesmaximal, characteristic, complemented
13#20275058C211yesmaximal, characteristic, complemented
14#20274831C211yesmaximal, characteristic, complemented
15C2×C53×#464C211yesmaximal, characteristic
16C2×#13133209C2211yescharacteristic
17C2×Q8×Dic53C2211yescharacteristic
18C2×Q8×Dic53C2211yescharacteristic
19#16810930—24no—
20C2×#13133150C2211yescharacteristic
21C2×#13133148C2211yescharacteristic
22C2×C4⋊#88322C2211yescharacteristic
23C2×#13133141C2211yescharacteristic
24C2×#13133141C2211yescharacteristic
25C2×#13133141C2211yescharacteristic
26C2×#13133141C2211yescharacteristic
27C2×#13133138C2211yescharacteristic
28C2×#13133138C2211yescharacteristic
29#16810530C2211yescharacteristic
30#16810530C2211yescharacteristic
31#16810526C2211yescharacteristic
32#16810526C2211yescharacteristic
33#16810517—21no—
34#16810517—21no—
35#16810517—21no—
36#16810517—21no—
37C4⋊C4×Dic53—21no—
38C4⋊C4×Dic53—21no—
39C4⋊C4×Dic53C2211yescharacteristic
40C4⋊C4×Dic53C2211yescharacteristic
41C4×C4⋊Dic53—21no—
42C4×C4⋊Dic53C2211yescharacteristic
43C42×Dic53—21no—
44C42×Dic53C2211yescharacteristic
45#16810386C2211yescharacteristic
46#16810386C2211yescharacteristic
47#16810386C2211yescharacteristic
48#16810386C2211yescharacteristic
49#16810377C2211yescharacteristic
50#16810377C2211yescharacteristic
51#16810377C2211yescharacteristic
52#16810377C2211yescharacteristic
53C22×Q16×C53C2211yescharacteristic
54C22×SD16×C53C2211yescharacteristic
55C2×#124×C53C2211yescharacteristic
56C2×C4×Q8×C53C2211yescharacteristic
57C53×#464—24no—
58C2×C4⋊C8×C53C2211yescharacteristic
59C2×#103×C53C2211yescharacteristic
60C2×#102×C53C2211yescharacteristic
61C22×#88322C2311yescharacteristic
62C22×#88322C2311yescharacteristic
63#13133209—22no—
64#13133209—22no—
65Q8×Dic53—22no—
66Q8×Dic53—22no—
67Q8×Dic53—22no—
68Q8×Dic53—22no—
69C2×C4⋊Dic53—21no—
70C2×C4⋊Dic53—21no—
71C2×C4⋊Dic53C2311yescharacteristic
72C2×C4⋊Dic53C2311yescharacteristic
73C2×C4⋊Dic53C2311yescharacteristic
74C2×C4⋊Dic53C2311yescharacteristic
75C2×#11719919—21no—
76C2×#11719919—21no—
77C2×C22⋊Dic53—21no—
78C2×C22⋊Dic53—21no—
79C2×C22⋊Dic53—21no—
80C2×C22⋊Dic53—21no—
81C2×C4×Dic53—21no—
82C2×C4×Dic53—21no—
83C2×C4×Dic53—21no—
84C2×C4×Dic53—21no—
85C2×C4×Dic53—21no—
86C2×C4×Dic53D411yescharacteristic
87C2×C4×Dic53D411yescharacteristic
88C2×C4×Dic53D411yescharacteristic
89C2×C4×Dic53D411yescharacteristic
90C2×C4×Dic53C2311yescharacteristic
91C2×C4×Dic53C2311yescharacteristic
92#13133150—22no—
93#13133148—22no—
94C4⋊#88322—22no—
95#13133141—22no—
96#13133141—22no—
97#13133141—22no—
98#13133141—22no—
99#13133138—22no—
100#13133138—22no—
101#13133124—21no—
102#13133124—21no—
103#13133124—21no—
104#13133124—21no—
105#13133124—21no—
106#13133124—21no—
107#13133124—21no—
108C22×Q8×C53C2311yescharacteristic
109C22×Q8×C53C2311yescharacteristic
110C22×D4×C53C2311yescharacteristic
111C2×Q16×C53—22no—
112C2×Q16×C53—22no—
113C2×Q16×C53—22no—
114C2×SD16×C53—22no—
115C2×SD16×C53—22no—
116C2×SD16×C53—22no—
117C22×C8×C53C2311yescharacteristic
118C22×C8×C53C2311yescharacteristic
119#124×C53—22no—
120#124×C53—22no—
121C4×Q8×C53—22no—
122C4×Q8×C53D414yes—
123C2×C4⋊C4×C53—21no—
124C2×C4⋊C4×C53C2311yescharacteristic
125C2×C22⋊C4×C53—21no—
126C2×C22⋊C4×C53—21no—
127C2×C42×C53—21no—
128C2×C42×C53C2311yescharacteristic
129C4⋊C8×C53—22no—
130#103×C53—22no—
131#102×C53—22no—
132C24×C53—21no—
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×C53C2×D412yes—
143C2×Q8×C53#4112yes—
144C2×Q8×C53#4112yes—
145C2×D4×C53—21no—
146C2×D4×C53—21no—
147C2×D4×C53—21no—
148C2×D4×C53—21no—
149C2×D4×C53—21no—
150C2×D4×C53—21no—
151C22×C4×C53—21no—
152C22×C4×C53—21no—
153C22×C4×C53—21no—
154C22×C4×C53C2×D411yescharacteristic
155C22×C4×C53C2×D411yescharacteristic
156C22×C4×C53C2411yescharacteristic
157C22×Dic53—21no—
158C22×Dic53—21no—
159C22×Dic53—21no—
160C22×Dic53—21no—
161C22×Dic53C2×D411yescharacteristic
162C22×Dic53C2×D411yescharacteristic
163C22×Dic53C2×D411yescharacteristic
164C22×Dic53C2×D411yescharacteristic
165Q16×C53—44no—
166SD16×C53—44no—
167C2×C8×C53—21no—
168C2×C8×C53—21no—
169C2×C8×C53—21no—
170C2×C8×C53—21no—
171C2×C8×C53—21no—
172C2×C8×C53—21no—
173C4⋊C4×C53—24no—
174C4⋊C4×C53—21no—
175C4⋊C4×C53C2×D412yes—
176C22⋊C4×C53—42no—
177C22⋊C4×C53—42no—
178C42×C53—24no—
179C42×C53C2×D412yes—
180C42×C53#4112yes—
181C22⋊Dic53—42no—
182C22⋊Dic53—42no—
183C22⋊Dic53—42no—
184C22⋊Dic53—42no—
185C4⋊Dic53—24no—
186C4⋊Dic53—24no—
187C4⋊Dic53—21no—
188C4⋊Dic53—21no—
189C4⋊Dic53—21no—
190C4⋊Dic53—21no—
191C4⋊Dic53—21no—
192C4⋊Dic53—21no—
193C4⋊Dic53#4112yes—
194C4⋊Dic53#4112yes—
195#11719919—42no—
196#11719919—42no—
197C4×Dic53—24no—
198C4×Dic53—24no—
199C4×Dic53—21no—
200C4×Dic53—21no—
201C4×Dic53—21no—
202C4×Dic53—21no—
203C4×Dic53—22no—
204C4×Dic53—22no—
205C4×Dic53—41no—
206C4×Dic53—41no—
207C4×Dic53—42no—
208C4×Dic53—42no—
209C4×Dic53—42no—
210C2×#88322—21no—
211C2×#88322—21no—
212C2×#88322—21no—
213C2×#88322—21no—
214C2×#88322—21no—
215C2×#88322—21no—
216C23×C53—41no—
217C23×C53—41no—
218C23×C53—41no—
219C23×C53—41no—
220C23×C53—41no—
221C23×C53—41no—
222C23×C53—41no—
223C23×C53C22×D411yescharacteristic
224Q8×C53—22no—
225Q8×C53—22no—
226Q8×C53—22no—
227Q8×C53—41no—
228Q8×C53—41no—
229Q8×C53—41no—
230Q8×C53—41no—
231Q8×C53#12414yes—
232D4×C53—41no—
233D4×C53—41no—
234D4×C53—41no—
235D4×C53—41no—
236C2×C4×C53—22no—
237C2×C4×C53—22no—
238C2×C4×C53—22no—
239C2×C4×C53—21no—
240C2×C4×C53—21no—
241C2×C4×C53—21no—
242C2×C4×C53—21no—
243C2×C4×C53—22no—
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×C53#12112yes—
251C2×C4×C53C22×D411yescharacteristic
252C2×C4×C53C2×#4111yescharacteristic
253C2×C4×C53C2×#4111yescharacteristic
254C2×C4×C53C2×#4111yescharacteristic
255C2×C4×C53C2×#4111yescharacteristic
256C2×C4×C53C2×#4111yescharacteristic
257C2×Dic53—21no—
258C2×Dic53—21no—
259C2×Dic53—21no—
260C2×Dic53—21no—
261C2×Dic53—22no—
262C2×Dic53—22no—
263C2×Dic53—21no—
264C2×Dic53—21no—
265C2×Dic53—21no—
266C2×Dic53—21no—
267C2×Dic53—21no—
268C2×Dic53—21no—
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—41no—
279C2×Dic53#13512yes—
280C2×Dic53#13712yes—
281C8×C53—41no—
282C8×C53—41no—
283#88322—41no—
284#88322—41no—
285#33029—531nomaximal, Sylow
286C22×C53—81no—
287C22×C53—81no—
288C22×C53—81no—
289C22×C53—81no—
290C22×C53—81no—
291C22×C53—81no—
292C22×C53—81no—
293C22×C53C2×#12111yescharacteristic
294C22×C53#52511yescharacteristic
295C22×C53#52511yescharacteristic
296C22×C53C2×#13511yescharacteristic
297C22×C53C2×#13711yescharacteristic
298C22×C53#57511yescharacteristic
299C22×C53#57511yescharacteristic
300C4×C53—24no—
301C4×C53—22no—
302C4×C53—22no—
303C4×C53—42no—
304C4×C53—81no—
305C4×C53C2×#12411yescharacteristic
306C4×C53#52911yescharacteristic
307C4×C53#52911yescharacteristic
308C4×C53#53211yescharacteristic
309Dic53—22no—
310Dic53—22no—
311Dic53—41no—
312Dic53—42no—
313Dic53—41no—
314Dic53—42no—
315Dic53—81no—
316Dic53—81no—
317C2×#479—531no—
318C2×#464D5311yescharacteristic
319#2861—531no—
320C42×Q8—531no—
321#1989—531no—
322#1983—531no—
323#1928—531no—
324#1927—531no—
325#1926—531no—
326#1925—531no—
327#1880—531no—
328#1880—531no—
329#1878—531no—
330#1878—531no—
331#1831—531no—
332C2×C53—161no—
333C2×C53#285011yescharacteristic
334C2×C53#303511yescharacteristic
335C2×C53#303711yescharacteristic
336C2×C53#310211yescharacteristic
337C2×C53#310811yescharacteristic
338C2×C53#311111yescharacteristic
339C2×C53#311711yescharacteristic
340C22×Q16C2×D5311yescharacteristic
341C22×SD16C2×D5311yescharacteristic
342C2×#125—531no—
343C2×#124C2×D5311yescharacteristic
344C2×C4×Q8—531no—
345C2×C4×Q8—531no—
346C2×C4×Q8C2×D5311yescharacteristic
347#479—1064no—
348#464—24no—
349C2×#107—531no—
350C2×#106—531no—
351C2×C4⋊C8—531no—
352C2×C4⋊C8C2×D5311yescharacteristic
353C2×#103—531no—
354C2×#103—531no—
355C2×#103—531no—
356C2×#103—531no—
357C2×#103C2×D5311yescharacteristic
358C2×#102—531no—
359C2×#102—531no—
360C2×#102C2×D5311yescharacteristic
361#391—531no—
362#391—531no—
363#390—531no—
364#390—531no—
365#388—1061no—
366#388—1061no—
367#388—1061no—
368#388—1061no—
369C4×C4⋊C4—531no—
370C4×C4⋊C4—531no—
371C4×C4⋊C4—531no—
372C4×C4⋊C4—1061no—
373C4×C4⋊C4—1061no—
374C4×C4⋊C4—1061no—
375C43—531no—
376C43—1061no—
377#340—531no—
378#340—531no—
379#340—531no—
380#340—531no—
381#336—531no—
382#336—531no—
383#336—531no—
384#336—531no—
385C53#3302911yescharacteristic, complemented, Sylow
386C22×Q8C22×D5311yescharacteristic
387C22×Q8C22×D5311yescharacteristic
388C22×D4C22×D5311yescharacteristic
389C2×Q16—22no—
390C2×Q16—22no—
391C2×Q16—22no—
392C2×SD16—22no—
393C2×SD16—22no—
394C2×SD16—22no—
395C22×C8—531no—
396C22×C8—531no—
397C22×C8C22×D5311yescharacteristic
398C22×C8C22×D5311yescharacteristic
399#125—1062no—
400#125—1062no—
401#124—22no—
402#124—22no—
403C4×Q8—22no—
404C4×Q8—1062no—
405C4×Q8—1062no—
406C4×Q8—1062no—
407C4×Q8—1062no—
408C4×Q8C22⋊D5314yes—
409C2×C4⋊C4—21no—
410C2×C4⋊C4—531no—
411C2×C4⋊C4—531no—
412C2×C4⋊C4—531no—
413C2×C4⋊C4—531no—
414C2×C4⋊C4—1061no—
415C2×C4⋊C4—1061no—
416C2×C4⋊C4—1061no—
417C2×C4⋊C4—1061no—
418C2×C4⋊C4C22×D5311yescharacteristic
419C2×C22⋊C4—21no—
420C2×C22⋊C4—21no—
421C2×C22⋊C4—1061no—
422C2×C22⋊C4—1061no—
423C2×C22⋊C4—1061no—
424C2×C22⋊C4—1061no—
425C2×C42—21no—
426C2×C42—531no—
427C2×C42—531no—
428C2×C42—531no—
429C2×C42—531no—
430C2×C42—531no—
431C2×C42—531no—
432C2×C42—1061no—
433C2×C42—1061no—
434C2×C42—1061no—
435C2×C42—1061no—
436C2×C42—1061no—
437C2×C42C22×D5311yescharacteristic
438#107—1062no—
439#106—1062no—
440C4⋊C8—22no—
441C4⋊C8—1062no—
442#103—22no—
443#103—1062no—
444#103—1062no—
445#103—1062no—
446#103—1062no—
447#102—22no—
448#102—1062no—
449#102—1062no—
450#95—1061no—
451#95—1061no—
452#95—1061no—
453#95—1061no—
454#95—1061no—
455#95—1061no—
456#95—1061no—
457C24—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×Q8#1171994712yes—
468C2×Q8#1171994712yes—
469C2×Q8C2×C22⋊D5312yes—
470C2×D4—21no—
471C2×D4—21no—
472C2×D4—21no—
473C2×D4—21no—
474C2×D4—21no—
475C2×D4—21no—
476C22×C4—21no—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—531no—
480C22×C4—531no—
481C22×C4—531no—
482C22×C4—531no—
483C22×C4—1061no—
484C22×C4—1061no—
485C22×C4—1061no—
486C22×C4—1061no—
487C22×C4D4×D5311yescharacteristic
488C22×C4D4×D5311yescharacteristic
489C22×C4C23×D5311yescharacteristic
490Q16—44no—
491SD16—44no—
492C2×C8—21no—
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—
504C4⋊C4—24no—
505C4⋊C4—21no—
506C4⋊C4—532no—
507C4⋊C4—532no—
508C4⋊C4—1064no—
509C4⋊C4—1064no—
510C4⋊C4—1061no—
511C4⋊C4—1061no—
512C4⋊C4—1061no—
513C4⋊C4—1061no—
514C4⋊C4—1061no—
515C4⋊C4—1061no—
516C4⋊C4—2122no—
517C4⋊C4—2122no—
518C4⋊C4C2×C22⋊D5312yes—
519C22⋊C4—42no—
520C22⋊C4—42no—
521C22⋊C4—2122no—
522C22⋊C4—2122no—
523C22⋊C4—2122no—
524C22⋊C4—2122no—
525C42—24no—
526C42—1062no—
527C42—1062no—
528C42—1064no—
529C42—1064no—
530C42—1061no—
531C42—1061no—
532C42—1061no—
533C42—1061no—
534C42—2121no—
535C42—2121no—
536C42—2122no—
537C42—2122no—
538C42—2122no—
539C42#1171994712yes—
540C42C2×C22⋊D5312yes—
541C23—41no—
542C23—41no—
543C23—41no—
544C23—41no—
545C23—41no—
546C23—41no—
547C23—41no—
548C23C2×D4×D5311yescharacteristic
549Q8—22no—
550Q8—22no—
551Q8—22no—
552Q8—41no—
553Q8—41no—
554Q8—41no—
555Q8—41no—
556Q8#1313320414yes—
557D4—41no—
558D4—41no—
559D4—41no—
560D4—41no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—22no—
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—1062no—
581C2×C4—1062no—
582C2×C4—1061no—
583C2×C4—1061no—
584C2×C4—1061no—
585C2×C4—1061no—
586C2×C4—1061no—
587C2×C4—1061no—
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#1313318912yes—
599C2×C4#1313318912yes—
600C2×C4C22×C22⋊D5311yescharacteristic
601C2×C4#41×D5311yescharacteristic
602C2×C4#41×D5311yescharacteristic
603C2×C4C2×#1171994711yescharacteristic
604C2×C4C2×#1171994711yescharacteristic
605C2×C4C2×#1171994711yescharacteristic
606C8—41no—
607C8—41no—
608C8—2121no—
609C8—2121no—
610C22—81no—
611C22—81no—
612C22—81no—
613C22—81no—
614C22—81no—
615C22—81no—
616C22—81no—
617C22C2×#1313318911yescharacteristic
618C22#1681113311yescharacteristic
619C22#1681113311yescharacteristic
620C22#135×D5311yescharacteristic
621C22#137×D5311yescharacteristic
622C22#1681142011yescharacteristic
623C22#1681142211yescharacteristic
624C4—22no—
625C4—22no—
626C4—24no—
627C4—42no—
628C4—81no—
629C4—1062no—
630C4—1062no—
631C4—4241no—
632C4—4241no—
633C4—2121no—
634C4—2122no—
635C4—2121no—
636C4—2122no—
637C4C2×#1313320411yescharacteristic
638C4#1681115411yescharacteristic
639C4#1681115411yescharacteristic
640C4#1681116911yescharacteristic
641C2—161no—
642C2#2028301411yescharacteristic
643C2#2028416611yescharacteristic
644C2#2028418411yescharacteristic
645C2#2028473211yescharacteristic
646C2#2028479411yescharacteristic
647C2#2028482311yescharacteristic
648C2#2028488511yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC2×C53×#464C2
socleC23×C53C22×D4
prime radicalC22×D4×C53C23
hypercenterC2×#464D53
hypocenterC53#33029
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#464

Complements

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

NH ≅ G/Ntype
1. C2×#16810930C2semidirect
3. C4×Q8×Dic53C2semidirect
4. #20275670C2semidirect
6. #20275359C2semidirect
7. #20275352C2semidirect
12. #20275058C2semidirect
13. #20275058C2semidirect
14. #20274831C2semidirect
385. C53#33029semidirect

Maximal subgroups

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

subgroupindex
C2×#168109302
#202830852
C4×Q8×Dic532
#202756702
#202756482
#202753592
#202753522
#202753452
#202753382
#202750722
#202750582
#202748312
C2×C53×#4642
#3302953