Finite Group Databasebeta

← #27350957

Subgroups — #27350957

Order 13568.

980 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20288279C211yesmaximal, characteristic
2#41×#88322C211yesmaximal, characteristic, complemented
3#20273173C212yesmaximal, complemented
4#20273169C211yesmaximal, characteristic, complemented
5#20273147C212yesmaximal, complemented
6#20273139C211yesmaximal, characteristic, complemented
7#20273121C212yesmaximal, complemented
8#20273117C211yesmaximal, characteristic, complemented
9#20273095C212yesmaximal, complemented
10C2×#16810350C211yesmaximal, characteristic, complemented
11C53×#2925C211yesmaximal, characteristic, complemented
12#16811346C411yescharacteristic
13#16811338C411yescharacteristic
14#16811333C411yescharacteristic
15#16811323—21no—
16#16811314—21no—
17#16811313C412yes—
18#16811302—22no—
19#16811278C411yescharacteristic
20#16811269C412yes—
21#16811180C2211yescharacteristic
22#16811157C2212yes—
23#16811151—22no—
24#16811130—21no—
25C4×#11719945C2212yes—
26C2×C4×D212C2211yescharacteristic
27#117×D53—21no—
28Q8×#88322—21no—
29D4×#88322—22no—
30D4×#88322—21no—
31#16810676C2211yescharacteristic
32#16810676C2212yescomplemented
33C2×C4×#88322C2211yescharacteristic
34C2×C4×#88322C2212yescomplemented
35#16810353C2212yescomplemented
36#16810353C2212yescomplemented
37#16810353C2212yescomplemented
38#16810353C2212yescomplemented
39#16810350C2211yescharacteristic
40#16810350C2212yes—
41#16810350C2211yescharacteristic
42#16810350C2211yescharacteristic
43#16810350C2211yescharacteristic
44#16810350C2212yes—
45C4×#41×C53C2211yescharacteristic
46C53×#433—22no—
47C53×#432C2212yescomplemented
48C53×#431C2212yescomplemented
49C53×#423C2211yescharacteristic
50C2×C4⋊C8×C53C2211yescharacteristic
51C53×#408—21no—
52C53×#405—21no—
53C2×#11719945C2×C412yes—
54C22×D212C2×C411yescharacteristic
55C22×C4×D53—21no—
56C22×#88322—21no—
57C22×#88322—22no—
58#13133217C2×C411yescharacteristic
59#13133217C2×C411yescharacteristic
60#13133212—22no—
61#13133211—22no—
62#13133210—22no—
63#13133209—21no—
64#13133209C2×C412yes—
65#13133207—22no—
66#13133205C2×C412yes—
67#13133203—22no—
68#13133202—22no—
69#13133200—22no—
70#13133199—22no—
71#13133198—22no—
72#13133196—21no—
73#13133196C2×C412yes—
74#13133196C2×C412yes—
75#13133195—22no—
76#13133193—21no—
77#13133193—21no—
78#13133191C2×C412yes—
79#13133190—22no—
80#13133188—21no—
81#13133188—21no—
82#13133185—22no—
83C4×Dic106—21no—
84C4×Dic106C2×C412yes—
85C4×Dic106C2312yes—
86C4×D212—21no—
87C4×D212—21no—
88C4×D212C2×C412yes—
89C4×D212C2×C411yescharacteristic
90C4×D212C2×C411yescharacteristic
91C4×D212C2311yescharacteristic
92C4×D212C2311yescharacteristic
93C4×C22⋊D53—22no—
94C4×C22⋊D53—22no—
95C4×C22⋊D53—22no—
96#13133176—21no—
97#13133176C2×C412yes—
98#13133175—21no—
99#13133175—21no—
100#13133174—22no—
101#13133174—22no—
102#13133173—22no—
103C2×C4⋊Dic53C2×C411yescharacteristic
104C4⋊C4×D53—21no—
105C4⋊C4×D53—21no—
106C2×#11719921—21no—
107C22⋊C4×D53—22no—
108C42×D53—22no—
109C4⋊#88322—21no—
110C4⋊#88322—22no—
111C22⋊#88322—21no—
112C22⋊#88322—22no—
113C4×#88322—21no—
114C4×#88322—22no—
115C4×#88322C2311yescharacteristic
116C4×#88322C2311yescharacteristic
117C4×#88322C2312yes—
118C2×#41×C53D411yescharacteristic
119C2×M16×C53—21no—
120C22×C8×C53—21no—
121C4×Q8×C53C2×C411yescharacteristic
122C4×Q8×C53D411yescharacteristic
123C4×D4×C53C2×C412yes—
124C4×D4×C53C2×C411yescharacteristic
125C4×D4×C53D412yes—
126C4×D4×C53D411yescharacteristic
127#117×C53D411yescharacteristic
128#117×C53D412yes—
129C2×C42×C53C2311yescharacteristic
130C2×C42×C53C2312yes—
131C4⋊C8×C53—21no—
132C4⋊C8×C53—21no—
133C4⋊C8×C53C2311yescharacteristic
134C4⋊C8×C53C2311yescharacteristic
135C4⋊C8×C53C2312yes—
136C22⋊C8×C53—22no—
137C22⋊C8×C53—22no—
138#97×C53—22no—
139C4×C8×C53—22no—
140C23×D53—21no—
141#41×C53—22no—
142#41×C53—22no—
143C2×Q8×C53C22⋊C411yescharacteristic
144C2×D4×C53C22⋊C412yes—
145C2×D4×C53C22⋊C411yescharacteristic
146C22×C4×C53C22×C411yescharacteristic
147C22×C4×C53C22×C412yes—
148C22×C4×C53C2×D412yes—
149C22×C4×C53C2×D411yescharacteristic
150C22×C4×C53C2×D412yes—
151C22×C4×C53C2×D411yescharacteristic
152C2×C22⋊D53—22no—
153C2×C22⋊D53—22no—
154C22×Dic53—21no—
155#11719945—24no—
156#11719945—42no—
157C2×D212—22no—
158C2×D212—21no—
159C2×D212—41no—
160C2×D212C22×C411yescharacteristic
161C2×D212C22×C411yescharacteristic
162C2×C4×D53—22no—
163C2×C4×D53—21no—
164C2×C4×D53—21no—
165C2×C4×D53—21no—
166C2×C4×D53—21no—
167C2×C4×D53—22no—
168C2×C4×D53—42no—
169C2×C4×D53—42no—
170C2×Dic106—21no—
171C2×Dic106C22×C412yes—
172M16×C53—42no—
173C2×C8×C53—21no—
174C2×C8×C53—22no—
175C2×C8×C53—21no—
176C2×C8×C53—42no—
177C4⋊C4×C53C22⋊C411yescharacteristic
178C4⋊C4×C53C22⋊C412yes—
179C4⋊C4×C53C22⋊C411yescharacteristic
180C4⋊C4×C53C22⋊C412yes—
181C22⋊C4×C53C22⋊C412yes—
182C22⋊C4×C53C22⋊C412yes—
183C22⋊C4×C53C22⋊C412yes—
184C42×C53C22×C411yescharacteristic
185C42×C53C22×C412yes—
186C42×C53C2×D412yes—
187C42×C53C2×D411yescharacteristic
188C42×C53C2×D412yes—
189C42×C53C2×D411yescharacteristic
190C42×C53C2411yescharacteristic
191C22⋊Dic53—22no—
192C22⋊Dic53—22no—
193C22⋊Dic53—22no—
194#11719921—22no—
195#11719921—22no—
196#11719921—21no—
197#11719921—21no—
198#11719921—22no—
199#11719921—22no—
200#11719921—21no—
201#11719921—21no—
202C4⋊Dic53—21no—
203C4⋊Dic53—21no—
204C4⋊Dic53—21no—
205C4⋊Dic53—21no—
206C4⋊Dic53C22×C411yescharacteristic
207C4⋊Dic53C22×C411yescharacteristic
208C4⋊Dic53C22×C412yes—
209#11719919—22no—
210#11719919—22no—
211#11719919—22no—
212#11719919—22no—
213#11719919—22no—
214#11719919—22no—
215C4×Dic53—22no—
216C4×Dic53—22no—
217C4×Dic53—22no—
218C2×#88322—21no—
219C2×#88322—21no—
220C2×#88322—22no—
221C2×#88322—42no—
222C2×#88322—44no—
223C23×C53C2×C22⋊C412yes—
224C23×C53C2×C22⋊C411yescharacteristic
225C22×D53—21no—
226C22×D53—21no—
227C22×D53—42no—
228C22×D53—42no—
229C22×D53—81no—
230Q8×C53—22no—
231D4×C53—24no—
232D4×C53—22no—
233C2×C4×C53—21no—
234C2×C4×C53—22no—
235C2×C4×C53—21no—
236C2×C4×C53—22no—
237C2×C4×C53—22no—
238C2×C4×C53—22no—
239C2×C4×C53—41no—
240C2×C4×C53—42no—
241C2×C4×C53C2×C22⋊C411yescharacteristic
242C2×C4×C53C2×C22⋊C412yes—
243C2×C4×C53C2×C22⋊C412yes—
244C2×C4×C53C2×C22⋊C411yescharacteristic
245C2×C4×C53C2×C22⋊C412yes—
246C2×C4×C53C2×C22⋊C411yescharacteristic
247C2×C4×C53#13512yes—
248C2×C4×C53#13512yes—
249C2×C4×C53C23×C411yescharacteristic
250C2×C4×C53C22×D411yescharacteristic
251C2×C4×C53C22×D411yescharacteristic
252C22⋊D53—44no—
253C22⋊D53—82no—
254C2×Dic53—22no—
255C2×Dic53—22no—
256C2×Dic53—21no—
257C2×Dic53—21no—
258C2×Dic53—42no—
259D212—21no—
260D212—21no—
261D212—21no—
262D212—42no—
263D212#13112yes—
264C4×D53—44no—
265C4×D53—41no—
266C4×D53—41no—
267C4×D53—41no—
268C4×D53—41no—
269C4×D53—42no—
270C4×D53—42no—
271C4×D53—82no—
272Dic106—22no—
273Dic106—22no—
274Dic106—81no—
275C8×C53—41no—
276C8×C53—41no—
277C8×C53—42no—
278#88322—22no—
279#88322—41no—
280#88322—42no—
281#17508—531nomaximal, Sylow
282C22×C53—21no—
283C22×C53—21no—
284C22×C53—22no—
285C22×C53—22no—
286C22×C53—22no—
287C22×C53#41612yes—
288C22×C53C22×C22⋊C411yescharacteristic
289C2×D53—22no—
290C2×D53—41no—
291C2×D53—41no—
292C2×D53—41no—
293C2×D53—41no—
294C2×D53—41no—
295C2×D53—84no—
296C4×C53—22no—
297C4×C53—21no—
298C4×C53—22no—
299C4×C53—21no—
300C4×C53—21no—
301C4×C53—22no—
302C4×C53#41612yes—
303C4×C53C2×#13111yescharacteristic
304C4×C53#56811yescharacteristic
305C4×C53C2×#13511yescharacteristic
306C4×C53C2×#13511yescharacteristic
307Dic53—42no—
308Dic53—41no—
309Dic53—41no—
310Dic53—42no—
311#3475—531no—
312C8×#41—531no—
313#2925D5311yescharacteristic, complemented
314#1494—532no—
315#1493—531no—
316#1488—532no—
317#1486—531no—
318#1482—532no—
319#1481—531no—
320#1476—532no—
321C2×D4⋊C8—531no—
322C2×C53—22no—
323C2×C53—42no—
324C2×C53#288011yescharacteristic
325C2×C53C2×#41611yescharacteristic
326C2×C53#290111yescharacteristic
327D53—42no—
328D53—81no—
329#556—531no—
330#555—531no—
331#554—531no—
332#553—1061no—
333#552—532no—
334#552—1061no—
335#551—1062no—
336#547—531no—
337#546—532no—
338#533—531no—
339#529—532no—
340#529—1062no—
341#525—1061no—
342C4×#41—532no—
343C4×#41C2×D5311yescharacteristic
344C2×C4×D4—531no—
345C2×#117—1061no—
346C8×Q8—1061no—
347C8×D4—1062no—
348C8×D4—1061no—
349#433—22no—
350#432C2×D5312yescomplemented
351#431—531no—
352#431—532no—
353#431C2×D5312yescomplemented
354#423C2×D5311yescharacteristic
355C2×C4⋊C8C2×D5311yescharacteristic
356#408—21no—
357#405—21no—
358C2×C4×C8—531no—
359C2×C4×C8—532no—
360Q8⋊C8—532no—
361Q8⋊C8—532no—
362Q8⋊C8—532no—
363Q8⋊C8—532no—
364D4⋊C8—531no—
365D4⋊C8—531no—
366D4⋊C8—531no—
367D4⋊C8—531no—
368D4⋊C8—532no—
369D4⋊C8—532no—
370C53#1750811yescharacteristic, complemented, Sylow
371C2×#41—532no—
372C2×#41C22⋊D5311yescharacteristic
373C22×D4—531no—
374C23×C4—1061no—
375C2×M16—21no—
376C22×C8—21no—
377C22×C8—1061no—
378C22×C8—1062no—
379C4⋊Q8—531no—
380C4⋊Q8—531no—
381#126—1062no—
382#126—1062no—
383#126—1062no—
384#125—532no—
385#125—1062no—
386#125—1061no—
387#124—532no—
388#124—1062no—
389#123—1062no—
390#123—1062no—
391#123—1062no—
392#123—1062no—
393C22⋊Q8—532no—
394C22⋊Q8—532no—
395C22⋊Q8—1061no—
396C22⋊Q8—1061no—
397C22⋊Q8—1061no—
398C22⋊Q8—1062no—
399#121—532no—
400#121—1061no—
401#121—1061no—
402#121—1062no—
403#120—1062no—
404C4×Q8—532no—
405C4×Q8—532no—
406C4×Q8—1061no—
407C4×Q8C4×D5311yescharacteristic
408C4×Q8D21211yescharacteristic
409C4×D4—531no—
410C4×D4—531no—
411C4×D4—532no—
412C4×D4—531no—
413C4×D4—531no—
414C4×D4—1062no—
415C4×D4—1062no—
416C4×D4—1062no—
417C4×D4—1061no—
418C4×D4—1061no—
419C4×D4C4×D5312yes—
420C4×D4C4×D5311yescharacteristic
421C4×D4D21212yes—
422C4×D4D21211yescharacteristic
423#117—532no—
424#117—1061no—
425#117—1061no—
426#117—1061no—
427#117—1062no—
428#117—1062no—
429#117—1062no—
430#117C22⋊D5311yescharacteristic
431#117C22⋊D5312yes—
432C2×C4⋊C4—531no—
433C2×C4⋊C4—1061no—
434C2×C4⋊C4—1061no—
435C2×C22⋊C4—1061no—
436C2×C22⋊C4—1062no—
437C2×C42—1062no—
438C2×C42C22×D5311yescharacteristic
439C2×C42C22×D5312yes—
440C4⋊C8—21no—
441C4⋊C8—21no—
442C4⋊C8—1062no—
443C4⋊C8—1061no—
444C4⋊C8C22×D5311yescharacteristic
445C4⋊C8C22×D5311yescharacteristic
446C4⋊C8C22×D5312yes—
447C22⋊C8—22no—
448C22⋊C8—22no—
449C22⋊C8—1061no—
450C22⋊C8—1062no—
451#97—22no—
452C4×C8—22no—
453C4×C8—531no—
454C4×C8—531no—
455C4×C8—532no—
456C4×C8—1062no—
457C4×C8—1061no—
458C24—1061no—
459#41—22no—
460#41—22no—
461#41—1064no—
462#41—2122no—
463C2×Q8—532no—
464C2×Q8—1061no—
465C2×Q8#1171992111yescharacteristic
466C2×D4—531no—
467C2×D4—531no—
468C2×D4—1062no—
469C2×D4—1062no—
470C2×D4—1062no—
471C2×D4—1061no—
472C2×D4—2121no—
473C2×D4#1171992112yes—
474C2×D4#1171992111yescharacteristic
475C22×C4—1061no—
476C22×C4—1062no—
477C22×C4—1062no—
478C22×C4—1061no—
479C22×C4—1061no—
480C22×C4—1061no—
481C22×C4—1061no—
482C22×C4—2122no—
483C22×C4—2122no—
484C22×C4C2×C4×D5311yescharacteristic
485C22×C4C2×C4×D5312yes—
486C22×C4C2×D21211yescharacteristic
487C22×C4C2×D21212yes—
488C22×C4C2×C22⋊D5312yes—
489C22×C4C2×C22⋊D5311yescharacteristic
490M16—42no—
491C2×C8—21no—
492C2×C8—22no—
493C2×C8—21no—
494C2×C8—42no—
495C2×C8—1061no—
496C2×C8—1061no—
497C2×C8—1062no—
498C2×C8—2124no—
499C2×C8—2122no—
500C4⋊C4—531no—
501C4⋊C4—531no—
502C4⋊C4—532no—
503C4⋊C4—1061no—
504C4⋊C4—1061no—
505C4⋊C4—1061no—
506C4⋊C4—1062no—
507C4⋊C4—1062no—
508C4⋊C4—1062no—
509C4⋊C4—1062no—
510C4⋊C4—1062no—
511C4⋊C4—1061no—
512C4⋊C4—1062no—
513C4⋊C4#1171992111yescharacteristic
514C4⋊C4#1171992111yescharacteristic
515C4⋊C4#1171992112yes—
516C4⋊C4#1171992112yes—
517C22⋊C4—1062no—
518C22⋊C4—1062no—
519C22⋊C4—1062no—
520C22⋊C4—1062no—
521C22⋊C4—1062no—
522C22⋊C4—1061no—
523C22⋊C4—1061no—
524C22⋊C4—1062no—
525C22⋊C4—1062no—
526C22⋊C4—1061no—
527C22⋊C4—1061no—
528C22⋊C4#1171992112yes—
529C22⋊C4#1171992112yes—
530C22⋊C4#1171992112yes—
531C42—1062no—
532C42—1062no—
533C42—1062no—
534C42C2×C4×D5311yescharacteristic
535C42C2×C4×D5312yes—
536C42C2×D21212yes—
537C42C2×D21211yescharacteristic
538C42C2×C22⋊D5312yes—
539C42C2×C22⋊D5311yescharacteristic
540C42C23×D5311yescharacteristic
541C23—1061no—
542C23—1061no—
543C23—4241no—
544C23—2122no—
545C23—2122no—
546C23C2×#1171992112yes—
547C23C2×#1171992111yescharacteristic
548Q8—22no—
549Q8—1062no—
550Q8—1062no—
551Q8—4241no—
552D4—24no—
553D4—22no—
554D4—532no—
555D4—1061no—
556D4—1061no—
557D4—1061no—
558D4—4242no—
559D4—2122no—
560D4—2124no—
561C2×C4—21no—
562C2×C4—22no—
563C2×C4—21no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—42no—
568C2×C4—41no—
569C2×C4—1061no—
570C2×C4—1062no—
571C2×C4—1062no—
572C2×C4—1061no—
573C2×C4—4242no—
574C2×C4—2124no—
575C2×C4—2122no—
576C2×C4—2122no—
577C2×C4—2122no—
578C2×C4—2121no—
579C2×C4—2121no—
580C2×C4—2121no—
581C2×C4—2121no—
582C2×C4C2×#1171992112yes—
583C2×C4C2×#1171992112yes—
584C2×C4C2×#1171992112yes—
585C2×C4C2×#1171992111yescharacteristic
586C2×C4C2×#1171992111yescharacteristic
587C2×C4C2×#1171992111yescharacteristic
588C2×C4#1313323712yes—
589C2×C4#1313323712yes—
590C2×C4C22×C4×D5311yescharacteristic
591C2×C4C22×C22⋊D5311yescharacteristic
592C2×C4C22×D21211yescharacteristic
593C8—41no—
594C8—41no—
595C8—42no—
596C8—1062no—
597C8—2121no—
598C8—2122no—
599C22—21no—
600C22—21no—
601C22—22no—
602C22—22no—
603C22—22no—
604C22—1062no—
605C22—4244no—
606C22—2121no—
607C22—2121no—
608C22—2121no—
609C22—2121no—
610C22—2121no—
611C22#1681061712yes—
612C22C22×#1171992111yescharacteristic
613C4—22no—
614C4—21no—
615C4—22no—
616C4—21no—
617C4—22no—
618C4—21no—
619C4—2122no—
620C4—2121no—
621C4—2121no—
622C4—2122no—
623C4#1681061712yes—
624C4#131×D5311yescharacteristic
625C4#1681138811yescharacteristic
626C4C2×#1313323711yescharacteristic
627C4C2×#1313323711yescharacteristic
628C2—22no—
629C2—42no—
630C2—4241no—
631C2—2122no—
632C2#2028316211yescharacteristic
633C2C2×#1681061711yescharacteristic
634C2#2028326011yescharacteristic
subgroupG/H
centerC2×C4C22×C22⋊D53
commutatorC2×C4×C53C23×C4
FrattiniC42C23×D53
FittingC53×#2925C2
socleC22×C53C22×C22⋊C4
prime radical#20288279C2
cosocleC42×C53C24
hypercenter#2925D53
hypocenterC53#17508
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C4×#41 ◁ #2925

Complements

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

NH ≅ G/Ntype
2. #41×#88322C2semidirect
3. #20273173C2semidirect
4. #20273169C2semidirect
5. #20273147C2semidirect
6. #20273139C2semidirect
7. #20273121C2semidirect
8. #20273117C2semidirect
9. #20273095C2semidirect
10. C2×#16810350C2semidirect
11. C53×#2925C2semidirect
32. #16810676C22semidirect
34. C2×C4×#88322C22semidirect
35. #16810353C22semidirect
36. #16810353C22semidirect
37. #16810353C22semidirect
38. #16810353C22semidirect
47. C53×#432C22semidirect
48. C53×#431C22semidirect
313. #2925D53semidirect
350. #432C2×D53semidirect
353. #431C2×D53semidirect
370. C53#17508semidirect

Maximal subgroups

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

subgroupindex
#202882792
#41×#883222
#202731732
#202731692
#202731472
#202731392
#202731212
#202731172
#202730952
C2×#168103502
C53×#29252
#1750853