Finite Group Databasebeta

← #27417238

Subgroups — #27417238

Order 13568.

1472 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810934C211yesmaximal, characteristic, complemented
2C2×#16810624C211yesmaximal, characteristic, complemented
3#20279032C211yesmaximal, characteristic, complemented
4#20275951C212yesmaximal, complemented
5#20275915C212yesmaximal, complemented
6#20275154C211yesmaximal, characteristic, complemented
7#20275013C211yesmaximal, characteristic, complemented
8#20274962C212yesmaximal, complemented
9#20274655C211yesmaximal, characteristic, complemented
10#20274646C212yesmaximal, complemented
11C2×C53×#546C211yesmaximal, characteristic
12C22×#11719917C2211yescharacteristic, complemented
13C22×C4⋊Dic53C2211yescharacteristic, complemented
14#16810934—24no—
15C2×#13133150C2212yescomplemented
16#16810624—24no—
17C2×#13133141C2212yescomplemented
18C2×#13133141C2212yescomplemented
19C2×#13133138C2212yes—
20C2×#13133138C2212yes—
21C2×#13133138C2212yes—
22C2×#13133138C2212yes—
23C2×C22⋊#88322C2212yes—
24C2×C22⋊#88322C2211yescharacteristic, complemented
25#16810562—21no—
26#16810559—21no—
27#16810521C2212yescomplemented
28#16810503—21no—
29#16810495—21no—
30#16810483C2212yes—
31#16810483C2211yescharacteristic, complemented
32#16810385C2212yescomplemented
33#16810378C2212yescomplemented
34C22×#41×C53C2211yescharacteristic
35C23×D4×C53C2211yescharacteristic
36C53×#546—24no—
37C53×#546—24no—
38C2×#124×C53C2211yescharacteristic
39C2×#123×C53—21no—
40C2×C22⋊Q8×C53C2211yescharacteristic
41C2×#121×C53—21no—
42C2×#121×C53C2211yescharacteristic
43C2×#120×C53—21no—
44C2×C4×D4×C53C2211yescharacteristic
45C2×C4×D4×C53C2211yescharacteristic
46C22×C22⋊C4×C53—21no—
47C23×Dic53—21no—
48C2×#11719917—22no—
49C2×#11719917—42no—
50C22×#88322C2312yes—
51C22×#88322C2312yes—
52C2×C4⋊Dic53—21no—
53C2×C4⋊Dic53—22no—
54C2×C4⋊Dic53—22no—
55C2×C4⋊Dic53C2312yes—
56C2×C4⋊Dic53C2312yes—
57C2×#11719919—22no—
58C2×C22⋊Dic53—21no—
59C2×C22⋊Dic53—22no—
60#13133150—24no—
61#13133141—24no—
62#13133141—42no—
63#13133138—24no—
64#13133138—24no—
65#13133138—24no—
66#13133138—42no—
67C22⋊#88322—22no—
68C22⋊#88322—42no—
69#13133124—22no—
70#13133124—21no—
71#13133124—22no—
72C25×C53—21no—
73C2×#41×C53—22no—
74C2×#41×C53—24no—
75C2×#41×C53—42no—
76C2×#41×C53D414yes—
77C22×Q8×C53C2311yescharacteristic
78C22×D4×C53—21no—
79C22×D4×C53—21no—
80C22×D4×C53—22no—
81C22×D4×C53—22no—
82C22×D4×C53—24no—
83C22×D4×C53—21no—
84C22×D4×C53—42no—
85C22×D4×C53D411yescharacteristic
86C22×D4×C53C2311yescharacteristic
87C22×D4×C53C2311yescharacteristic
88C23×C4×C53—21no—
89C23×C4×C53—21no—
90C23×C4×C53C2311yescharacteristic
91C23×C4×C53C2311yescharacteristic
92#124×C53—22no—
93#124×C53—22no—
94#123×C53—28no—
95C22⋊Q8×C53—22no—
96C22⋊Q8×C53—22no—
97#121×C53—28no—
98#121×C53—22no—
99#121×C53D414yes—
100#120×C53—28no—
101C4×D4×C53—22no—
102C4×D4×C53—22no—
103C4×D4×C53—22no—
104C4×D4×C53—22no—
105C2×C4⋊C4×C53—21no—
106C2×C4⋊C4×C53D411yescharacteristic
107C2×C4⋊C4×C53C2311yescharacteristic
108C2×C22⋊C4×C53—28no—
109C2×C22⋊C4×C53—21no—
110C2×C22⋊C4×C53—21no—
111C2×C22⋊C4×C53—21no—
112C2×C22⋊C4×C53—21no—
113C2×C22⋊C4×C53—21no—
114C2×C22⋊C4×C53D411yescharacteristic
115C2×C22⋊C4×C53D411yescharacteristic
116C2×C42×C53C2311yescharacteristic
117C24×C53—21no—
118C24×C53—24no—
119C24×C53—21no—
120C24×C53—44no—
121C24×C53—42no—
122C24×C53—42no—
123C24×C53—82no—
124C24×C53C2×D411yescharacteristic
125C24×C53C2×D411yescharacteristic
126#41×C53—28no—
127#41×C53—44no—
128#41×C53—48no—
129C2×Q8×C53—21no—
130C2×Q8×C53—22no—
131C2×Q8×C53—22no—
132C2×Q8×C53C2×D412yes—
133C2×D4×C53—24no—
134C2×D4×C53—22no—
135C2×D4×C53—24no—
136C2×D4×C53—22no—
137C2×D4×C53—24no—
138C2×D4×C53—21no—
139C2×D4×C53—21no—
140C2×D4×C53—22no—
141C2×D4×C53—22no—
142C2×D4×C53—22no—
143C2×D4×C53—41no—
144C2×D4×C53—44no—
145C2×D4×C53—48no—
146C2×D4×C53—44no—
147C2×D4×C53—44no—
148C2×D4×C53—44no—
149C2×D4×C53—82no—
150C2×D4×C53C2×D412yes—
151C2×D4×C53C2×D412yes—
152C2×D4×C53#4114yes—
153C22×C4×C53—22no—
154C22×C4×C53—22no—
155C22×C4×C53—21no—
156C22×C4×C53—21no—
157C22×C4×C53—24no—
158C22×C4×C53—24no—
159C22×C4×C53—21no—
160C22×C4×C53—21no—
161C22×C4×C53—22no—
162C22×C4×C53—22no—
163C22×C4×C53—44no—
164C22×C4×C53—82no—
165C22×C4×C53C2×D411yescharacteristic
166C22×C4×C53C2×D411yescharacteristic
167C22×C4×C53C2×D411yescharacteristic
168C22×C4×C53C2×D411yescharacteristic
169C22×C4×C53C2×D412yes—
170C22×C4×C53C2×D412yes—
171C22×C4×C53C2411yescharacteristic
172C22×Dic53—22no—
173C22×Dic53—22no—
174C22×Dic53—42no—
175C22×Dic53—82no—
176C4⋊C4×C53—24no—
177C4⋊C4×C53—22no—
178C4⋊C4×C53—21no—
179C4⋊C4×C53C2×D412yes—
180C22⋊C4×C53—24no—
181C22⋊C4×C53—24no—
182C22⋊C4×C53—24no—
183C22⋊C4×C53—24no—
184C22⋊C4×C53—22no—
185C22⋊C4×C53—22no—
186C22⋊C4×C53—24no—
187C42×C53—21no—
188C42×C53—21no—
189C22⋊Dic53—42no—
190C22⋊Dic53—82no—
191C4⋊Dic53—22no—
192C4⋊Dic53—22no—
193C4⋊Dic53—22no—
194C4⋊Dic53—22no—
195C4⋊Dic53—42no—
196#11719919—82no—
197#11719917—44no—
198C2×#88322—22no—
199C2×#88322—22no—
200C2×#88322—22no—
201C2×#88322—22no—
202C2×#88322—22no—
203C2×#88322—22no—
204C23×C53—22no—
205C23×C53—21no—
206C23×C53—22no—
207C23×C53—22no—
208C23×C53—24no—
209C23×C53—21no—
210C23×C53—21no—
211C23×C53—21no—
212C23×C53—22no—
213C23×C53—22no—
214C23×C53—22no—
215C23×C53—48no—
216C23×C53—48no—
217C23×C53—41no—
218C23×C53—41no—
219C23×C53—42no—
220C23×C53—42no—
221C23×C53—41no—
222C23×C53—41no—
223C23×C53—42no—
224C23×C53—88no—
225C23×C53—84no—
226C23×C53—88no—
227C23×C53#12012yes—
228C23×C53#13712yes—
229C23×C53C22×D411yescharacteristic
230Q8×C53—22no—
231Q8×C53—22no—
232Q8×C53—42no—
233D4×C53—22no—
234D4×C53—24no—
235D4×C53—24no—
236D4×C53—44no—
237D4×C53—48no—
238D4×C53—48no—
239D4×C53—41no—
240D4×C53—41no—
241D4×C53—84no—
242D4×C53#12314yes—
243C2×C4×C53—22no—
244C2×C4×C53—22no—
245C2×C4×C53—22no—
246C2×C4×C53—24no—
247C2×C4×C53—21no—
248C2×C4×C53—22no—
249C2×C4×C53—22no—
250C2×C4×C53—21no—
251C2×C4×C53—21no—
252C2×C4×C53—22no—
253C2×C4×C53—21no—
254C2×C4×C53—21no—
255C2×C4×C53—22no—
256C2×C4×C53—48no—
257C2×C4×C53—42no—
258C2×C4×C53—42no—
259C2×C4×C53—41no—
260C2×C4×C53—41no—
261C2×C4×C53—42no—
262C2×C4×C53—84no—
263C2×C4×C53#12012yes—
264C2×C4×C53C22×D411yescharacteristic
265C2×C4×C53C22×D411yescharacteristic
266C2×C4×C53C2×#4112yes—
267C2×C4×C53H(5,𝔽2)12yes—
268C2×Dic53—42no—
269C2×Dic53—42no—
270C2×Dic53—42no—
271C2×Dic53—42no—
272C2×Dic53—42no—
273C2×Dic53—42no—
274C2×Dic53—84no—
275#88322—42no—
276#88322—42no—
277#24049—531nomaximal, Sylow
278C22×C53—22no—
279C22×C53—24no—
280C22×C53—22no—
281C22×C53—22no—
282C22×C53—22no—
283C22×C53—24no—
284C22×C53—41no—
285C22×C53—42no—
286C22×C53—42no—
287C22×C53—41no—
288C22×C53—42no—
289C22×C53—44no—
290C22×C53—42no—
291C22×C53—42no—
292C22×C53—44no—
293C22×C53—42no—
294C22×C53—42no—
295C22×C53—42no—
296C22×C53—816no—
297C22×C53—82no—
298C22×C53—82no—
299C22×C53C2×#12011yescharacteristic
300C22×C53#54612yes—
301C22×C53C2×#13711yescharacteristic
302C22×C53#57511yescharacteristic
303C22×C53#57612yes—
304C4×C53—22no—
305C4×C53—41no—
306C4×C53—42no—
307C4×C53—41no—
308C4×C53—41no—
309C4×C53—42no—
310C4×C53C2×#12311yescharacteristic
311C4×C53#53411yescharacteristic
312C4×C53#54612yes—
313Dic53—82no—
314Dic53—82no—
315C2×#546D5311yescharacteristic
316C2×#480—531no—
317C2×#417—531no—
318#2487—531no—
319#2042—532no—
320#2036—532no—
321#1895—531no—
322#1870—531no—
323#1862—532no—
324#1788—531no—
325#1786—532no—
326C2×C53—42no—
327C2×C53—42no—
328C2×C53—44no—
329C2×C53—82no—
330C2×C53#240511yescharacteristic
331C2×C53#300711yescharacteristic
332C2×C53#300811yescharacteristic
333C2×C53#331412yes—
334C2×C53#332212yes—
335C22×#41C2×D5311yescharacteristic
336C23×D4C2×D5311yescharacteristic
337C22×M16—531no—
338#546—24no—
339#546—24no—
340C2×#124C2×D5311yescharacteristic
341C2×#123—21no—
342C2×C22⋊Q8C2×D5311yescharacteristic
343C2×#121—21no—
344C2×#121C2×D5311yescharacteristic
345C2×#120—21no—
346C2×C4×D4C2×D5311yescharacteristic
347C2×C4×D4C2×D5311yescharacteristic
348C22×C4⋊C4—531no—
349C22×C22⋊C4—21no—
350#480—1064no—
351C2×#107—532no—
352#417—1064no—
353C2×#103—532no—
354C2×#103—532no—
355C2×#102—532no—
356C2×#102—532no—
357C2×#102—532no—
358C2×#102—532no—
359C2×C22⋊C8—531no—
360C2×C22⋊C8—532no—
361#400—1061no—
362#399—1061no—
363#389—532no—
364#386—1061no—
365#385—1061no—
366#380—531no—
367#380—532no—
368#340—532no—
369#336—532no—
370C53#2404911yescharacteristic, complemented, Sylow
371C25—21no—
372C2×#41—22no—
373C2×#41—24no—
374C2×#41—42no—
375C2×#41C22⋊D5314yes—
376C22×Q8C22×D5311yescharacteristic
377C22×D4—21no—
378C22×D4—21no—
379C22×D4—22no—
380C22×D4—22no—
381C22×D4—24no—
382C22×D4—21no—
383C22×D4—42no—
384C22×D4C22⋊D5311yescharacteristic
385C22×D4C22×D5311yescharacteristic
386C22×D4C22×D5311yescharacteristic
387C23×C4—21no—
388C23×C4—21no—
389C23×C4—1061no—
390C23×C4C22×D5311yescharacteristic
391C23×C4C22×D5311yescharacteristic
392C2×M16—1062no—
393C2×M16—2122no—
394C22×C8—532no—
395C22×C8—532no—
396#124—22no—
397#124—22no—
398#123—28no—
399C22⋊Q8—22no—
400C22⋊Q8—22no—
401#121—28no—
402#121—22no—
403#121C22⋊D5314yes—
404#120—28no—
405C4×D4—22no—
406C4×D4—22no—
407C4×D4—22no—
408C4×D4—22no—
409C2×C4⋊C4—21no—
410C2×C4⋊C4—532no—
411C2×C4⋊C4—532no—
412C2×C4⋊C4—1062no—
413C2×C4⋊C4—1062no—
414C2×C4⋊C4—1062no—
415C2×C4⋊C4—1061no—
416C2×C4⋊C4C22⋊D5311yescharacteristic
417C2×C4⋊C4C22×D5311yescharacteristic
418C2×C22⋊C4—28no—
419C2×C22⋊C4—21no—
420C2×C22⋊C4—21no—
421C2×C22⋊C4—21no—
422C2×C22⋊C4—21no—
423C2×C22⋊C4—21no—
424C2×C22⋊C4—1062no—
425C2×C22⋊C4—1061no—
426C2×C22⋊C4C22⋊D5311yescharacteristic
427C2×C22⋊C4C22⋊D5311yescharacteristic
428C2×C42C22×D5311yescharacteristic
429#107—1064no—
430#103—1064no—
431#103—2122no—
432#102—1064no—
433#102—1064no—
434#102—1064no—
435#102—2122no—
436C22⋊C8—1062no—
437C22⋊C8—2122no—
438#95—1062no—
439#95—1061no—
440#95—1062no—
441C24—21no—
442C24—24no—
443C24—21no—
444C24—44no—
445C24—42no—
446C24—42no—
447C24—82no—
448C24C2×C22⋊D5311yescharacteristic
449C24C2×C22⋊D5311yescharacteristic
450#41—28no—
451#41—44no—
452#41—48no—
453C2×Q8—21no—
454C2×Q8—22no—
455C2×Q8—22no—
456C2×Q8C2×C22⋊D5312yes—
457C2×D4—24no—
458C2×D4—22no—
459C2×D4—24no—
460C2×D4—22no—
461C2×D4—24no—
462C2×D4—21no—
463C2×D4—21no—
464C2×D4—22no—
465C2×D4—22no—
466C2×D4—22no—
467C2×D4—41no—
468C2×D4—44no—
469C2×D4—48no—
470C2×D4—44no—
471C2×D4—44no—
472C2×D4—44no—
473C2×D4—82no—
474C2×D4#1171994714yes—
475C2×D4C2×C22⋊D5312yes—
476C2×D4C2×C22⋊D5312yes—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—24no—
482C22×C4—24no—
483C22×C4—21no—
484C22×C4—21no—
485C22×C4—22no—
486C22×C4—22no—
487C22×C4—44no—
488C22×C4—82no—
489C22×C4—1062no—
490C22×C4—1062no—
491C22×C4—4242no—
492C22×C4—2122no—
493C22×C4C2×C22⋊D5311yescharacteristic
494C22×C4C2×C22⋊D5311yescharacteristic
495C22×C4C2×C22⋊D5311yescharacteristic
496C22×C4C2×C22⋊D5311yescharacteristic
497C22×C4C2×C22⋊D5312yes—
498C22×C4C2×C22⋊D5312yes—
499C22×C4C23×D5311yescharacteristic
500M16—2124no—
501C2×C8—1062no—
502C2×C8—1062no—
503C2×C8—1062no—
504C2×C8—1062no—
505C2×C8—1062no—
506C2×C8—1062no—
507C4⋊C4—24no—
508C4⋊C4—22no—
509C4⋊C4—21no—
510C4⋊C4—1062no—
511C4⋊C4—1062no—
512C4⋊C4—1062no—
513C4⋊C4—1062no—
514C4⋊C4—4242no—
515C4⋊C4—2122no—
516C4⋊C4C2×C22⋊D5312yes—
517C22⋊C4—24no—
518C22⋊C4—24no—
519C22⋊C4—24no—
520C22⋊C4—24no—
521C22⋊C4—22no—
522C22⋊C4—22no—
523C22⋊C4—24no—
524C22⋊C4—4242no—
525C22⋊C4—2122no—
526C42—21no—
527C42—21no—
528C23—22no—
529C23—22no—
530C23—24no—
531C23—21no—
532C23—22no—
533C23—21no—
534C23—21no—
535C23—21no—
536C23—22no—
537C23—22no—
538C23—22no—
539C23—48no—
540C23—48no—
541C23—41no—
542C23—41no—
543C23—42no—
544C23—42no—
545C23—41no—
546C23—41no—
547C23—42no—
548C23—88no—
549C23—84no—
550C23—88no—
551C23#1313318612yes—
552C23#1313324712yes—
553C23C22×C22⋊D5311yescharacteristic
554Q8—22no—
555Q8—22no—
556Q8—42no—
557D4—24no—
558D4—24no—
559D4—22no—
560D4—44no—
561D4—48no—
562D4—48no—
563D4—41no—
564D4—41no—
565D4—84no—
566D4#1313319814yes—
567C2×C4—22no—
568C2×C4—24no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—22no—
580C2×C4—48no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—84no—
587C2×C4—4244no—
588C2×C4—2122no—
589C2×C4—2122no—
590C2×C4—2122no—
591C2×C4—2122no—
592C2×C4—2122no—
593C2×C4—2122no—
594C2×C4#1313318612yes—
595C2×C4C22×C22⋊D5311yescharacteristic
596C2×C4C22×C22⋊D5311yescharacteristic
597C2×C4C2×#1171994712yes—
598C2×C4#1313326212yes—
599C8—2122no—
600C8—2122no—
601C22—22no—
602C22—24no—
603C22—22no—
604C22—22no—
605C22—24no—
606C22—22no—
607C22—41no—
608C22—42no—
609C22—42no—
610C22—44no—
611C22—42no—
612C22—41no—
613C22—42no—
614C22—42no—
615C22—44no—
616C22—42no—
617C22—42no—
618C22—42no—
619C22—816no—
620C22—82no—
621C22—82no—
622C22C2×#1313318611yescharacteristic
623C22#1681126712yes—
624C22C2×#1313324711yescharacteristic
625C22#1681142111yescharacteristic
626C22#1681142612yes—
627C4—22no—
628C4—41no—
629C4—42no—
630C4—41no—
631C4—41no—
632C4—42no—
633C4—4242no—
634C4—4242no—
635C4C2×#1313319811yescharacteristic
636C4#1681118211yescharacteristic
637C4#1681126712yes—
638C2—42no—
639C2—42no—
640C2—44no—
641C2—82no—
642C2#2027803811yescharacteristic
643C2#2028397511yescharacteristic
644C2#2028398311yescharacteristic
645C2#2028679612yes—
646C2#2028691612yes—
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
Fitting, prime radicalC2×C53×#546C2
socleC23×C53C22×D4
hypercenterC2×#546D53
hypocenterC53#24049
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×#546

Complements

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

NH ≅ G/Ntype
1. C2×#16810934C2semidirect
2. C2×#16810624C2semidirect
3. #20279032C2semidirect
4. #20275951C2semidirect
5. #20275915C2semidirect
6. #20275154C2semidirect
7. #20275013C2semidirect
8. #20274962C2semidirect
9. #20274655C2semidirect
10. #20274646C2semidirect
12. C22×#11719917C22semidirect
13. C22×C4⋊Dic53C22semidirect
15. C2×#13133150C22semidirect
17. C2×#13133141C22semidirect
18. C2×#13133141C22semidirect
24. C2×C22⋊#88322C22semidirect
27. #16810521C22semidirect
31. #16810483C22semidirect
32. #16810385C22semidirect
33. #16810378C22semidirect
370. C53#24049semidirect

Maximal subgroups

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

subgroupindex
C2×#168109342
C2×#168106242
#202790322
#202759512
#202759152
#202751542
#202750132
#202749622
#202746552
#202746462
C2×C53×#5462
#2404953