Finite Group Databasebeta

← #27351879

Subgroups — #27351879

Order 13568.

1014 conjugacy classes of subgroups.

The subgroup lattice has 674 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20288554C211yesmaximal, characteristic
2#20283756C211yesmaximal, characteristic, complemented
3#20273290C212yesmaximal, complemented
4#20273256C212yesmaximal, complemented
5#20273247C211yesmaximal, characteristic, complemented
6#20273243C211yesmaximal, characteristic, complemented
7#20273234C211yesmaximal, characteristic, complemented
8#20273230C211yesmaximal, characteristic, complemented
9#20273217C212yesmaximal, complemented
10#20273199C212yesmaximal, complemented
11C53×#2353C211yesmaximal, characteristic, complemented
12#16811322—22no—
13#16811319C412yescomplemented
14#16811313C412yescomplemented
15#16811302—22no—
16#16811282C412yescomplemented
17#16811269C412yescomplemented
18#16811220—21no—
19#16811211—21no—
20#16811182—21no—
21#16811157C2212yes—
22#16811134C2212yes—
23C2×#13133196C2211yescharacteristic
24C2×#13133191C2211yescharacteristic
25#16811096—21no—
26#16810701—22no—
27#16810701—22no—
28#16810676C2212yescomplemented
29#16810587C2211yescharacteristic
30#16810587C2211yescharacteristic
31#16810587C2212yescomplemented
32#16810359C2212yescomplemented
33#16810359C2212yescomplemented
34#16810359C2212yescomplemented
35#16810359C2212yescomplemented
36#16810356C2212yes—
37#16810356C2212yes—
38#16810356C2212yes—
39#16810356C2212yes—
40C2×C4×D4×C53C2211yescharacteristic
41C53×#388—21no—
42C53×#387—21no—
43C53×#386—21no—
44C53×#383C2211yescharacteristic, complemented
45C53×#381C2211yescharacteristic
46C53×#380C2211yescharacteristic, complemented
47C4×C22⋊C4×C53—21no—
48C4×C22⋊C4×C53C2211yescharacteristic
49C53×#376C2211yescharacteristic, complemented
50C2×#95×C53C2211yescharacteristic, complemented
51C2×#11719945C2×C412yes—
52C22×Dic106C2×C411yescharacteristic, complemented
53C22×D212C2×C411yescharacteristic, complemented
54C22×C22⋊D53—21no—
55C2×#11719917—21no—
56C2×#11719917—21no—
57C2×#11719917—22no—
58#13133212—22no—
59#13133211—22no—
60#13133210—22no—
61#13133209C2×C412yescomplemented
62#13133207—22no—
63#13133205C2×C412yescomplemented
64#13133203—22no—
65#13133202—22no—
66#13133200—22no—
67#13133200—22no—
68#13133199—22no—
69#13133198—22no—
70#13133196—21no—
71#13133196—21no—
72#13133196C2×C412yes—
73#13133196C2×C412yescomplemented
74#13133196C2312yes—
75#13133195—22no—
76#13133191—21no—
77#13133191—21no—
78#13133191C2×C412yes—
79#13133191C2×C412yescomplemented
80#13133191C2312yes—
81#13133190—22no—
82#13133189—22no—
83#13133189—22no—
84#13133186—22no—
85#13133185—22no—
86C4×Dic106C2×C412yescomplemented
87C4×D212C2×C412yescomplemented
88C4×C22⋊D53—22no—
89C4×C22⋊D53—22no—
90C4×C22⋊D53—22no—
91#13133177—22no—
92#13133177—22no—
93#13133176C2×C412yes—
94#13133174—22no—
95#13133173—22no—
96C2×C4⋊Dic53C2×C411yescharacteristic, complemented
97C2×C4⋊Dic53C2×C411yescharacteristic, complemented
98C2×#11719919—21no—
99C2×C22⋊Dic53—21no—
100C2×#11719921—21no—
101C22⋊C4×D53—22no—
102C4⋊#88322—22no—
103C22⋊#88322—21no—
104C22⋊#88322—21no—
105C22⋊#88322—22no—
106C22⋊#88322C2312yes—
107C22⋊#88322C2312yes—
108C4×#88322—22no—
109C22×D4×C53D411yescharacteristic
110C23×C4×C53—21no—
111C23×C4×C53C2311yescharacteristic
112C23×C4×C53C2311yescharacteristic
113C4×D4×C53C2×C412yes—
114C4×D4×C53C2×C412yes—
115C4×D4×C53D412yes—
116C4×D4×C53D412yes—
117C2×C4⋊C4×C53—21no—
118C2×C4⋊C4×C53D411yescharacteristic
119C2×C22⋊C4×C53—21no—
120C2×C22⋊C4×C53—21no—
121C2×C22⋊C4×C53—21no—
122C2×C22⋊C4×C53D411yescharacteristic
123C2×C22⋊C4×C53D411yescharacteristic
124C2×C42×C53—21no—
125C2×C42×C53—21no—
126C2×C42×C53C2311yescharacteristic
127#95×C53—21no—
128#95×C53—21no—
129#95×C53—21no—
130#95×C53—21no—
131#95×C53—21no—
132#95×C53C2311yescharacteristic
133#95×C53C2311yescharacteristic
134#95×C53C2311yescharacteristic
135#95×C53C2311yescharacteristic
136C24×C53C2×D411yescharacteristic
137C24×C53C2×D411yescharacteristic
138C23×D53—21no—
139C2×D4×C53—41no—
140C2×D4×C53—41no—
141C2×D4×C53C22⋊C412yes—
142C2×D4×C53C22⋊C412yes—
143C22×C4×C53—21no—
144C22×C4×C53—21no—
145C22×C4×C53—21no—
146C22×C4×C53—21no—
147C22×C4×C53—42no—
148C22×C4×C53—42no—
149C22×C4×C53—81no—
150C22×C4×C53—81no—
151C22×C4×C53—81no—
152C22×C4×C53C22×C411yescharacteristic
153C22×C4×C53C22×C411yescharacteristic
154C22×C4×C53C22×C412yes—
155C22×C4×C53C2×D411yescharacteristic
156C22×C4×C53C2×D411yescharacteristic
157C22×C4×C53C2×D411yescharacteristic
158C22×C4×C53C2×D411yescharacteristic
159C22×C4×C53C2×D411yescharacteristic
160C22×C4×C53C2×D411yescharacteristic
161C22×C4×C53C2×D412yes—
162C22×C4×C53C2411yescharacteristic
163C2×C22⋊D53—22no—
164C2×C22⋊D53—22no—
165C2×C22⋊D53—22no—
166C2×C22⋊D53—22no—
167C2×C22⋊D53—42no—
168C2×C22⋊D53—42no—
169C22×Dic53—21no—
170C22×Dic53—21no—
171C22×Dic53—21no—
172#11719945—82no—
173C2×D212—21no—
174C2×D212—81no—
175C2×D212C22×C412yes—
176C2×C4×D53—22no—
177C2×C4×D53—22no—
178C2×Dic106—21no—
179C2×Dic106—81no—
180C2×Dic106C22×C412yes—
181C4⋊C4×C53—81no—
182C4⋊C4×C53C22⋊C412yes—
183C4⋊C4×C53C22⋊C412yes—
184C22⋊C4×C53—42no—
185C22⋊C4×C53—42no—
186C22⋊C4×C53—81no—
187C22⋊C4×C53C22⋊C412yes—
188C22⋊C4×C53C22⋊C412yes—
189C22⋊C4×C53C22⋊C412yes—
190C22⋊C4×C53C22⋊C412yes—
191C42×C53—42no—
192C42×C53—81no—
193C42×C53C22×C412yes—
194C42×C53C2×D412yes—
195C22⋊Dic53—22no—
196C22⋊Dic53—22no—
197C22⋊Dic53—22no—
198C22⋊Dic53—22no—
199#11719921—22no—
200#11719921—22no—
201#11719921—22no—
202#11719921—22no—
203#11719921—22no—
204#11719921—22no—
205C4⋊Dic53—21no—
206C4⋊Dic53—21no—
207C4⋊Dic53C22×C412yes—
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—
217#11719917—81no—
218#11719917—81no—
219#11719917—82no—
220C2×#88322—22no—
221C2×#88322—22no—
222C23×C53—21no—
223C23×C53—21no—
224C23×C53—21no—
225C23×C53—21no—
226C23×C53—41no—
227C23×C53—41no—
228C23×C53—41no—
229C23×C53—41no—
230C23×C53C2×C22⋊C411yescharacteristic
231C23×C53C2×C22⋊C411yescharacteristic
232C23×C53C2×C22⋊C412yes—
233C23×C53C22×D411yescharacteristic
234C22×D53—22no—
235C22×D53—42no—
236C22×D53—81no—
237C22×D53—81no—
238D4×C53—42no—
239D4×C53—42no—
240C2×C4×C53—21no—
241C2×C4×C53—21no—
242C2×C4×C53—22no—
243C2×C4×C53—41no—
244C2×C4×C53—41no—
245C2×C4×C53—42no—
246C2×C4×C53—41no—
247C2×C4×C53—41no—
248C2×C4×C53—42no—
249C2×C4×C53—41no—
250C2×C4×C53—41no—
251C2×C4×C53—41no—
252C2×C4×C53—41no—
253C2×C4×C53—41no—
254C2×C4×C53—41no—
255C2×C4×C53—41no—
256C2×C4×C53—41no—
257C2×C4×C53—41no—
258C2×C4×C53—41no—
259C2×C4×C53—41no—
260C2×C4×C53—41no—
261C2×C4×C53—41no—
262C2×C4×C53—41no—
263C2×C4×C53—41no—
264C2×C4×C53—84no—
265C2×C4×C53C2×C22⋊C411yescharacteristic
266C2×C4×C53C2×C22⋊C412yes—
267C2×C4×C53C2×C22⋊C412yes—
268C2×C4×C53C2×C22⋊C411yescharacteristic
269C2×C4×C53C2×C22⋊C412yes—
270C2×C4×C53C23×C411yescharacteristic
271C2×C4×C53C22×D411yescharacteristic
272C22⋊D53—44no—
273C22⋊D53—42no—
274C22⋊D53—42no—
275C22⋊D53—82no—
276C22⋊D53—82no—
277C2×Dic53—22no—
278C2×Dic53—22no—
279C2×Dic53—22no—
280C2×Dic53—42no—
281C2×Dic53—81no—
282C2×Dic53—81no—
283D212—42no—
284D212—81no—
285C4×D53—82no—
286C4×D53—82no—
287Dic106—42no—
288Dic106—81no—
289#88322—42no—
290#88322—42no—
291#17623—531nomaximal, Sylow
292C22×C53—21no—
293C22×C53—21no—
294C22×C53—21no—
295C22×C53—22no—
296C22×C53—21no—
297C22×C53—21no—
298C22×C53—21no—
299C22×C53—22no—
300C22×C53—21no—
301C22×C53—22no—
302C22×C53—21no—
303C22×C53—42no—
304C22×C53—42no—
305C22×C53—81no—
306C22×C53—81no—
307C22×C53C22×C22⋊C411yescharacteristic
308C22×C53#57511yescharacteristic
309C22×C53#57511yescharacteristic
310C2×D53—42no—
311C2×D53—42no—
312C2×D53—42no—
313C2×D53—162no—
314C4×C53—21no—
315C4×C53—21no—
316C4×C53—21no—
317C4×C53—22no—
318C4×C53—21no—
319C4×C53—22no—
320C4×C53—22no—
321C4×C53—42no—
322C4×C53—81no—
323C4×C53—81no—
324C4×C53—81no—
325Dic53—42no—
326Dic53—42no—
327Dic53—42no—
328#3505—531no—
329#2975—531no—
330#2353D5311yescharacteristic, complemented
331#1521—532no—
332#1513—532no—
333#1511—531no—
334#1510—531no—
335#1508—531no—
336#1507—531no—
337#1504—532no—
338#1500—532no—
339C2×C53—21no—
340C2×C53—21no—
341C2×C53—41no—
342C2×C53—42no—
343C2×C53—41no—
344C2×C53#288311yescharacteristic
345C2×C53#289611yescharacteristic
346C2×C53#290111yescharacteristic
347D53—82no—
348#553—532no—
349#553—1062no—
350#552—532no—
351#551—1062no—
352#548—532no—
353#546—532no—
354#539—1061no—
355#538—1061no—
356#534—1061no—
357#529—532no—
358#525—532no—
359C2×C22⋊Q8—531no—
360C2×#121—531no—
361#518—1061no—
362C2×C4×D4C2×D5311yescharacteristic
363#436—1062no—
364#436—1062no—
365#431—532no—
366#407—531no—
367#407—531no—
368#407—532no—
369#388—21no—
370#387—21no—
371#386—21no—
372#383C2×D5311yescharacteristic, complemented
373#381C2×D5311yescharacteristic
374#380C2×D5311yescharacteristic, complemented
375C4×C22⋊C4—21no—
376C4×C22⋊C4C2×D5311yescharacteristic
377#376C2×D5311yescharacteristic, complemented
378C2×#95C2×D5311yescharacteristic, complemented
379#328—532no—
380#328—532no—
381#328—532no—
382#328—532no—
383#327—532no—
384#327—532no—
385#327—532no—
386#327—532no—
387C53#1762311yescharacteristic, complemented, Sylow
388C2×#41—532no—
389C22×Q8—531no—
390C22×D4—531no—
391C22×D4—1061no—
392C22×D4C22⋊D5311yescharacteristic
393C23×C4—21no—
394C23×C4C22×D5311yescharacteristic
395C23×C4C22×D5311yescharacteristic
396C2×M16—1061no—
397C2×M16—1061no—
398C2×M16—1062no—
399#126—1062no—
400#126—1062no—
401#126—1062no—
402#125—532no—
403#125—1062no—
404#124—532no—
405#124—1062no—
406#123—1062no—
407#123—1062no—
408#123—1062no—
409#123—1062no—
410#123—1062no—
411C22⋊Q8—532no—
412C22⋊Q8—532no—
413C22⋊Q8—532no—
414C22⋊Q8—1061no—
415C22⋊Q8—1061no—
416C22⋊Q8—1062no—
417#121—532no—
418#121—532no—
419#121—532no—
420#121—1062no—
421#121—1062no—
422#121—1061no—
423#121—1061no—
424#121—1062no—
425#120—1062no—
426#120—1062no—
427C4×Q8—532no—
428C4×D4—532no—
429C4×D4—1062no—
430C4×D4—1062no—
431C4×D4—1062no—
432C4×D4—1062no—
433C4×D4—1062no—
434C4×D4C4×D5312yes—
435C4×D4C4×D5312yes—
436C4×D4D21212yes—
437C4×D4D21212yes—
438#117—532no—
439#117—1062no—
440#117—1062no—
441C2×C4⋊C4—21no—
442C2×C4⋊C4—531no—
443C2×C4⋊C4—531no—
444C2×C4⋊C4—1061no—
445C2×C4⋊C4C22⋊D5311yescharacteristic
446C2×C22⋊C4—21no—
447C2×C22⋊C4—21no—
448C2×C22⋊C4—21no—
449C2×C22⋊C4—1061no—
450C2×C22⋊C4—1061no—
451C2×C22⋊C4—1062no—
452C2×C22⋊C4C22⋊D5311yescharacteristic
453C2×C22⋊C4C22⋊D5311yescharacteristic
454C2×C42—21no—
455C2×C42—21no—
456C2×C42C22×D5311yescharacteristic
457C4⋊C8—1062no—
458C22⋊C8—532no—
459C22⋊C8—532no—
460C22⋊C8—1061no—
461C22⋊C8—1061no—
462C22⋊C8—1062no—
463C4×C8—1062no—
464#95—21no—
465#95—21no—
466#95—21no—
467#95—21no—
468#95—21no—
469#95C22×D5311yescharacteristic
470#95C22×D5311yescharacteristic
471#95C22×D5311yescharacteristic
472#95C22×D5311yescharacteristic
473C24—1061no—
474C24C2×C22⋊D5311yescharacteristic
475C24C2×C22⋊D5311yescharacteristic
476#41—4242no—
477C2×Q8—532no—
478C2×Q8—1061no—
479C2×Q8—4241no—
480C2×D4—41no—
481C2×D4—41no—
482C2×D4—532no—
483C2×D4—1062no—
484C2×D4—1062no—
485C2×D4—1062no—
486C2×D4—1062no—
487C2×D4—1061no—
488C2×D4—4241no—
489C2×D4—2122no—
490C2×D4—2122no—
491C2×D4#1171992112yes—
492C2×D4#1171992112yes—
493C22×C4—21no—
494C22×C4—21no—
495C22×C4—21no—
496C22×C4—21no—
497C22×C4—42no—
498C22×C4—42no—
499C22×C4—81no—
500C22×C4—81no—
501C22×C4—81no—
502C22×C4—1061no—
503C22×C4—1061no—
504C22×C4—1061no—
505C22×C4—1062no—
506C22×C4—1062no—
507C22×C4C2×C4×D5311yescharacteristic
508C22×C4C2×C4×D5311yescharacteristic
509C22×C4C2×C4×D5312yes—
510C22×C4C2×D21211yescharacteristic
511C22×C4C2×D21211yescharacteristic
512C22×C4C2×D21212yes—
513C22×C4C2×C22⋊D5311yescharacteristic
514C22×C4C2×C22⋊D5311yescharacteristic
515C22×C4C2×C22⋊D5311yescharacteristic
516C22×C4C2×C22⋊D5311yescharacteristic
517C22×C4C23×D5311yescharacteristic
518M16—4242no—
519M16—4241no—
520M16—4241no—
521C2×C8—1062no—
522C2×C8—1062no—
523C4⋊C4—81no—
524C4⋊C4—532no—
525C4⋊C4—532no—
526C4⋊C4—1061no—
527C4⋊C4—1061no—
528C4⋊C4—1062no—
529C4⋊C4—1062no—
530C4⋊C4—1062no—
531C4⋊C4—1062no—
532C4⋊C4—1062no—
533C4⋊C4—1062no—
534C4⋊C4#1171992112yes—
535C4⋊C4#1171992112yes—
536C22⋊C4—42no—
537C22⋊C4—42no—
538C22⋊C4—81no—
539C22⋊C4—1062no—
540C22⋊C4—1062no—
541C22⋊C4—1062no—
542C22⋊C4—1062no—
543C22⋊C4—1062no—
544C22⋊C4—1062no—
545C22⋊C4—1062no—
546C22⋊C4—1062no—
547C22⋊C4—1062no—
548C22⋊C4—1062no—
549C22⋊C4#1171992112yes—
550C22⋊C4#1171992112yes—
551C22⋊C4#1171992112yes—
552C22⋊C4#1171992112yes—
553C42—42no—
554C42—81no—
555C42—1062no—
556C42—1062no—
557C42C2×C4×D5312yes—
558C42C2×D21212yes—
559C23—21no—
560C23—21no—
561C23—21no—
562C23—21no—
563C23—41no—
564C23—41no—
565C23—41no—
566C23—41no—
567C23—1062no—
568C23—4241no—
569C23—4241no—
570C23—2122no—
571C23C2×#1171992111yescharacteristic
572C23C2×#1171992112yes—
573C23C2×#1171992111yescharacteristic
574C23C22×C22⋊D5311yescharacteristic
575Q8—4241no—
576Q8—2122no—
577D4—42no—
578D4—42no—
579D4—4241no—
580D4—4242no—
581D4—4242no—
582D4—2122no—
583D4—2124no—
584D4—2122no—
585D4—2122no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—22no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—42no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—42no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—84no—
611C2×C4—1062no—
612C2×C4—1062no—
613C2×C4—1062no—
614C2×C4—4242no—
615C2×C4—4242no—
616C2×C4—4241no—
617C2×C4—4241no—
618C2×C4—2122no—
619C2×C4C2×#1171992112yes—
620C2×C4C2×#1171992112yes—
621C2×C4C2×#1171992112yes—
622C2×C4C2×#1171992111yescharacteristic
623C2×C4C2×#1171992111yescharacteristic
624C2×C4C22×C4×D5311yescharacteristic
625C2×C4C22×D21211yescharacteristic
626C8—2122no—
627C8—2122no—
628C22—21no—
629C22—21no—
630C22—21no—
631C22—22no—
632C22—21no—
633C22—21no—
634C22—21no—
635C22—22no—
636C22—21no—
637C22—22no—
638C22—21no—
639C22—42no—
640C22—42no—
641C22—81no—
642C22—81no—
643C22—8482no—
644C22—2122no—
645C22—2122no—
646C22—2122no—
647C22C22×#1171992111yescharacteristic
648C22#1681142111yescharacteristic
649C22#1681142111yescharacteristic
650C4—21no—
651C4—21no—
652C4—22no—
653C4—21no—
654C4—21no—
655C4—22no—
656C4—22no—
657C4—42no—
658C4—81no—
659C4—81no—
660C4—81no—
661C4—2122no—
662C4—2122no—
663C4—2122no—
664C2—21no—
665C2—21no—
666C2—42no—
667C2—41no—
668C2—41no—
669C2—4242no—
670C2#2028317511yescharacteristic
671C2#2028323211yescharacteristic
672C2#2028326311yescharacteristic
subgroupG/H
centerC22C22×#11719921
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#2353C2
socleC22×C53C22×C22⋊C4
prime radical#20288554C2
cosocleC22×C4×C53C24
hypercenter#2353D53
hypocenterC53#17623
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C4×D4 ◁ #2353

Complements

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

NH ≅ G/Ntype
2. #20283756C2semidirect
3. #20273290C2semidirect
4. #20273256C2semidirect
5. #20273247C2semidirect
6. #20273243C2semidirect
7. #20273234C2semidirect
8. #20273230C2semidirect
9. #20273217C2semidirect
10. #20273199C2semidirect
11. C53×#2353C2semidirect
13. #16811319C4semidirect
14. #16811313C4semidirect
16. #16811282C4semidirect
17. #16811269C4semidirect
28. #16810676C22semidirect
31. #16810587C22semidirect
32. #16810359C22semidirect
33. #16810359C22semidirect
34. #16810359C22semidirect
35. #16810359C22semidirect
44. C53×#383C22semidirect
46. C53×#380C22semidirect
49. C53×#376C22semidirect
50. C2×#95×C53C22semidirect
52. C22×Dic106C2×C4semidirect
53. C22×D212C2×C4semidirect
61. #13133209C2×C4semidirect
63. #13133205C2×C4semidirect
73. #13133196C2×C4semidirect
79. #13133191C2×C4semidirect
86. C4×Dic106C2×C4semidirect
87. C4×D212C2×C4semidirect
96. C2×C4⋊Dic53C2×C4semidirect
97. C2×C4⋊Dic53C2×C4semidirect
330. #2353D53semidirect
372. #383C2×D53semidirect
374. #380C2×D53semidirect
377. #376C2×D53semidirect
378. C2×#95C2×D53semidirect
387. C53#17623semidirect

Maximal subgroups

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

subgroupindex
#202885542
#202837562
#202732902
#202732562
#202732472
#202732432
#202732342
#202732302
#202732172
#202731992
C53×#23532
#1762353