Finite Group Databasebeta

← C53×#9041 #27216553

Subgroups — C53×#9041 #27216553

Order 13568.

1050 conjugacy classes of subgroups.

The subgroup lattice has 720 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C53×#447C211yesmaximal, characteristic
2C53×#2325C211yesmaximal, characteristic, complemented
3C53×#1786C211yesmaximal, characteristic, complemented
4C2×C53×#327C211yesmaximal, characteristic, complemented
5C53×#1438C211yesmaximal, characteristic, complemented
6C53×#1294C211yesmaximal, characteristic, complemented
7C53×#1280C211yesmaximal, characteristic, complemented
8C23×D4×C53C2211yescharacteristic
9C22×D8×C53—21no—
10C2×#121×C53C2211yescharacteristic
11C2×C4×D4×C53—21no—
12C22×C22⋊C4×C53—21no—
13C22×C22⋊C4×C53—21no—
14C53×#447—22no—
15C53×#447C414yescomplemented
16C2×#102×C53—21no—
17C2×#102×C53—21no—
18C2×#102×C53—21no—
19C2×C22⋊C8×C53C2211yescharacteristic
20C2×C22⋊C8×C53C2211yescharacteristic
21C53×#388—21no—
22C53×#387—21no—
23C53×#386—21no—
24C53×#386—21no—
25C53×#385—21no—
26C53×#380C2211yescharacteristic
27C2×#95×C53C2211yescharacteristic
28C2×#95×C53C2211yescharacteristic
29C53×#327—22no—
30C25×C53—21no—
31C22×D4×C53—21no—
32C22×D4×C53—21no—
33C22×D4×C53—21no—
34C22×D4×C53—21no—
35C22×D4×C53—21no—
36C22×D4×C53—21no—
37C22×D4×C53—42no—
38C22×D4×C53—42no—
39C22×D4×C53C2×C412yescomplemented
40C22×D4×C53D411yescharacteristic
41C23×C4×C53—21no—
42C23×C4×C53—21no—
43C23×C4×C53D411yescharacteristic
44C23×C4×C53D411yescharacteristic
45C23×C4×C53C2311yescharacteristic
46C2×D8×C53—24no—
47C2×D8×C53—42no—
48C2×D8×C53—42no—
49C22×C8×C53—21no—
50C22×C8×C53—21no—
51#121×C53—21no—
52#121×C53—21no—
53#121×C53C2×C412yes—
54#121×C53D412yes—
55C4×D4×C53—42no—
56C4×D4×C53—42no—
57C2×C4⋊C4×C53—21no—
58C2×C4⋊C4×C53D411yescharacteristic
59C2×C4⋊C4×C53D411yescharacteristic
60C2×C4⋊C4×C53D411yescharacteristic
61C2×C22⋊C4×C53—21no—
62C2×C22⋊C4×C53—21no—
63C2×C22⋊C4×C53—21no—
64C2×C22⋊C4×C53—21no—
65C2×C22⋊C4×C53—21no—
66C2×C22⋊C4×C53—42no—
67C2×C22⋊C4×C53—42no—
68C2×C22⋊C4×C53—42no—
69C2×C22⋊C4×C53—42no—
70C2×C22⋊C4×C53—42no—
71C2×C22⋊C4×C53—42no—
72C2×C42×C53—21no—
73#102×C53—24no—
74#102×C53—42no—
75#102×C53—42no—
76C22⋊C8×C53—21no—
77C22⋊C8×C53—21no—
78C22⋊C8×C53—21no—
79C22⋊C8×C53C2×C412yescomplemented
80#95×C53—21no—
81#95×C53—21no—
82#95×C53—21no—
83#95×C53—21no—
84#95×C53—21no—
85#95×C53—21no—
86#95×C53—21no—
87C24×C53—22no—
88C24×C53—22no—
89C24×C53—22no—
90C24×C53—21no—
91C24×C53—41no—
92C24×C53—41no—
93C24×C53—41no—
94C24×C53—41no—
95C24×C53—41no—
96C24×C53—41no—
97C24×C53—41no—
98C24×C53—41no—
99C24×C53—41no—
100C24×C53—41no—
101C24×C53—41no—
102C24×C53—41no—
103C24×C53C2×D411yescharacteristic
104C2×D4×C53—22no—
105C2×D4×C53—22no—
106C2×D4×C53—22no—
107C2×D4×C53—22no—
108C2×D4×C53—22no—
109C2×D4×C53—22no—
110C2×D4×C53—22no—
111C2×D4×C53—22no—
112C2×D4×C53—21no—
113C2×D4×C53—41no—
114C2×D4×C53—41no—
115C2×D4×C53—41no—
116C2×D4×C53—41no—
117C2×D4×C53—41no—
118C2×D4×C53—41no—
119C2×D4×C53—44no—
120C2×D4×C53—44no—
121C2×D4×C53—44no—
122C2×D4×C53—44no—
123C2×D4×C53—41no—
124C2×D4×C53—41no—
125C2×D4×C53—41no—
126C2×D4×C53—41no—
127C2×D4×C53—41no—
128C2×D4×C53—41no—
129C2×D4×C53C22⋊C412yescomplemented
130C22×C4×C53—21no—
131C22×C4×C53—21no—
132C22×C4×C53—21no—
133C22×C4×C53—22no—
134C22×C4×C53—21no—
135C22×C4×C53—21no—
136C22×C4×C53—21no—
137C22×C4×C53—21no—
138C22×C4×C53—21no—
139C22×C4×C53—41no—
140C22×C4×C53—41no—
141C22×C4×C53—42no—
142C22×C4×C53—41no—
143C22×C4×C53—41no—
144C22×C4×C53—41no—
145C22×C4×C53—41no—
146C22×C4×C53—41no—
147C22×C4×C53—41no—
148C22×C4×C53—41no—
149C22×C4×C53—41no—
150C22×C4×C53—41no—
151C22×C4×C53—41no—
152C22×C4×C53—81no—
153C22×C4×C53—81no—
154C22×C4×C53D812yescomplemented
155C22×C4×C53C22×C411yescharacteristic
156C22×C4×C53C2×D411yescharacteristic
157C22×C4×C53C2×D411yescharacteristic
158C22×C4×C53C2×D411yescharacteristic
159C22×C4×C53#4111yescharacteristic
160C22×C4×C53#4111yescharacteristic
161D8×C53—48no—
162C2×C8×C53—22no—
163C2×C8×C53—41no—
164C2×C8×C53—41no—
165C2×C8×C53—41no—
166C2×C8×C53—41no—
167C2×C8×C53—41no—
168C4⋊C4×C53—21no—
169C4⋊C4×C53—42no—
170C4⋊C4×C53—41no—
171C4⋊C4×C53—41no—
172C4⋊C4×C53C22⋊C412yescomplemented
173C22⋊C4×C53—22no—
174C22⋊C4×C53—22no—
175C22⋊C4×C53—42no—
176C22⋊C4×C53—48no—
177C22⋊C4×C53—48no—
178C22⋊C4×C53—41no—
179C22⋊C4×C53—41no—
180C22⋊C4×C53—81no—
181C22⋊C4×C53—81no—
182C42×C53—42no—
183C23×C53—21no—
184C23×C53—21no—
185C23×C53—21no—
186C23×C53—21no—
187C23×C53—24no—
188C23×C53—22no—
189C23×C53—42no—
190C23×C53—42no—
191C23×C53—42no—
192C23×C53—42no—
193C23×C53—42no—
194C23×C53—42no—
195C23×C53—42no—
196C23×C53—42no—
197C23×C53—42no—
198C23×C53—42no—
199C23×C53—42no—
200C23×C53—42no—
201C23×C53—42no—
202C23×C53—42no—
203C23×C53—42no—
204C23×C53—42no—
205C23×C53—41no—
206C23×C53—41no—
207C23×C53—41no—
208C23×C53—41no—
209C23×C53—41no—
210C23×C53—42no—
211C23×C53—42no—
212C23×C53—41no—
213C23×C53—81no—
214C23×C53—81no—
215C23×C53—81no—
216C23×C53—81no—
217C23×C53—81no—
218C23×C53—81no—
219C23×C53—81no—
220C23×C53—81no—
221C23×C53—81no—
222C23×C53—81no—
223C23×C53—81no—
224C23×C53—81no—
225C23×C53—81no—
226C23×C53—81no—
227C23×C53—81no—
228C23×C53—81no—
229C23×C53C4×D411yescharacteristic
230C23×C53#12011yescharacteristic
231C23×C53#12111yescharacteristic
232C23×C53C2×D811yescharacteristic
233C23×C53#13511yescharacteristic
234C23×C53AGL(1,ℤ/8)11yescharacteristic
235C23×C53AGL(1,ℤ/8)11yescharacteristic
236D4×C53—24no—
237D4×C53—24no—
238D4×C53—42no—
239D4×C53—42no—
240D4×C53—42no—
241D4×C53—42no—
242D4×C53—42no—
243D4×C53—42no—
244D4×C53—42no—
245D4×C53—42no—
246D4×C53—42no—
247D4×C53—42no—
248D4×C53—42no—
249D4×C53—81no—
250C2×C4×C53—21no—
251C2×C4×C53—21no—
252C2×C4×C53—21no—
253C2×C4×C53—21no—
254C2×C4×C53—22no—
255C2×C4×C53—22no—
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—42no—
265C2×C4×C53—42no—
266C2×C4×C53—42no—
267C2×C4×C53—42no—
268C2×C4×C53—42no—
269C2×C4×C53—41no—
270C2×C4×C53—41no—
271C2×C4×C53—41no—
272C2×C4×C53—41no—
273C2×C4×C53—41no—
274C2×C4×C53—41no—
275C2×C4×C53—41no—
276C2×C4×C53—41no—
277C2×C4×C53—41no—
278C2×C4×C53—81no—
279C2×C4×C53—84no—
280C2×C4×C53—81no—
281C2×C4×C53—81no—
282C2×C4×C53—81no—
283C2×C4×C53—81no—
284C2×C4×C53—81no—
285C2×C4×C53C2×C22⋊C411yescharacteristic
286C2×C4×C53C4×D411yescharacteristic
287C2×C4×C53#12111yescharacteristic
288C2×C4×C53#12311yescharacteristic
289C8×C53—42no—
290C8×C53—81no—
291#9041C5311yesmaximal, characteristic, direct product factor, Sylow
292C22×C53—21no—
293C22×C53—21no—
294C22×C53—21no—
295C22×C53—21no—
296C22×C53—21no—
297C22×C53—21no—
298C22×C53—21no—
299C22×C53—21no—
300C22×C53—21no—
301C22×C53—21no—
302C22×C53—21no—
303C22×C53—21no—
304C22×C53—44no—
305C22×C53—44no—
306C22×C53—42no—
307C22×C53—42no—
308C22×C53—44no—
309C22×C53—42no—
310C22×C53—82no—
311C22×C53—82no—
312C22×C53—82no—
313C22×C53—82no—
314C22×C53—82no—
315C22×C53—82no—
316C22×C53—81no—
317C22×C53—82no—
318C22×C53—81no—
319C22×C53—82no—
320C22×C53—81no—
321C22×C53—82no—
322C22×C53—81no—
323C22×C53—82no—
324C22×C53—82no—
325C22×C53—82no—
326C22×C53#38611yescharacteristic
327C22×C53C4×D811yescharacteristic
328C22×C53#44211yescharacteristic
329C22×C53#44711yescharacteristic
330C22×C53#44911yescharacteristic
331C22×C53#45311yescharacteristic
332C22×C53#45411yescharacteristic
333C22×C53B(4,𝔽2)11yescharacteristic
334C22×C53B(4,𝔽2)11yescharacteristic
335C22×C53#45911yescharacteristic
336C22×C53#46311yescharacteristic
337C4×C53—21no—
338C4×C53—21no—
339C4×C53—21no—
340C4×C53—21no—
341C4×C53—42no—
342C4×C53—42no—
343C4×C53—82no—
344C4×C53—81no—
345C4×C53—81no—
346C4×C53—81no—
347C2×#447C2×C5311yescharacteristic
348#2325C2×C5311yescharacteristic, complemented
349#1786C2×C5311yescharacteristic, complemented
350C2×#327C2×C5311yescharacteristic, complemented
351#1438C2×C5311yescharacteristic, complemented
352#1294C2×C5311yescharacteristic, complemented
353#1280C2×C5311yescharacteristic, complemented
354C2×C53—21no—
355C2×C53—21no—
356C2×C53—21no—
357C2×C53—21no—
358C2×C53—82no—
359C2×C53—84no—
360C2×C53#159511yescharacteristic
361C2×C53#159711yescharacteristic
362C2×C53#159811yescharacteristic
363C2×C53#160011yescharacteristic
364C2×C53#187911yescharacteristic
365C2×C53#188311yescharacteristic
366C2×C53#188911yescharacteristic
367C23×D4C22×C5311yescharacteristic
368C22×D8—21no—
369C2×#121C22×C5311yescharacteristic
370C2×C4×D4—21no—
371C22×C22⋊C4—21no—
372C22×C22⋊C4—21no—
373#447—22no—
374#447C4×C5314yescomplemented
375C2×#102—21no—
376C2×#102—21no—
377C2×#102—21no—
378C2×C22⋊C8C22×C5311yescharacteristic
379C2×C22⋊C8C22×C5311yescharacteristic
380#388—21no—
381#387—21no—
382#386—21no—
383#386—21no—
384#385—21no—
385#380C22×C5311yescharacteristic
386C2×#95C22×C5311yescharacteristic
387C2×#95C22×C5311yescharacteristic
388#327—22no—
389C53#904111yescharacteristic, direct product factor, Sylow
390C25—21no—
391C22×D4—21no—
392C22×D4—21no—
393C22×D4—21no—
394C22×D4—21no—
395C22×D4—21no—
396C22×D4—21no—
397C22×D4—42no—
398C22×D4—42no—
399C22×D4C2×C4×C5312yescomplemented
400C22×D4D4×C5311yescharacteristic
401C23×C4—21no—
402C23×C4—21no—
403C23×C4D4×C5311yescharacteristic
404C23×C4D4×C5311yescharacteristic
405C23×C4C23×C5311yescharacteristic
406C2×D8—24no—
407C2×D8—42no—
408C2×D8—42no—
409C22×C8—21no—
410C22×C8—21no—
411#121—21no—
412#121—21no—
413#121C2×C4×C5312yes—
414#121D4×C5312yes—
415C4×D4—42no—
416C4×D4—42no—
417C2×C4⋊C4—21no—
418C2×C4⋊C4D4×C5311yescharacteristic
419C2×C4⋊C4D4×C5311yescharacteristic
420C2×C4⋊C4D4×C5311yescharacteristic
421C2×C22⋊C4—21no—
422C2×C22⋊C4—21no—
423C2×C22⋊C4—21no—
424C2×C22⋊C4—21no—
425C2×C22⋊C4—21no—
426C2×C22⋊C4—42no—
427C2×C22⋊C4—42no—
428C2×C22⋊C4—42no—
429C2×C22⋊C4—42no—
430C2×C22⋊C4—42no—
431C2×C22⋊C4—42no—
432C2×C42—21no—
433#102—24no—
434#102—42no—
435#102—42no—
436C22⋊C8—21no—
437C22⋊C8—21no—
438C22⋊C8—21no—
439C22⋊C8C2×C4×C5312yescomplemented
440#95—21no—
441#95—21no—
442#95—21no—
443#95—21no—
444#95—21no—
445#95—21no—
446#95—21no—
447C24—22no—
448C24—22no—
449C24—22no—
450C24—21no—
451C24—41no—
452C24—41no—
453C24—41no—
454C24—41no—
455C24—41no—
456C24—41no—
457C24—41no—
458C24—41no—
459C24—41no—
460C24—41no—
461C24—41no—
462C24—41no—
463C24C2×D4×C5311yescharacteristic
464C2×D4—22no—
465C2×D4—22no—
466C2×D4—22no—
467C2×D4—22no—
468C2×D4—22no—
469C2×D4—22no—
470C2×D4—22no—
471C2×D4—22no—
472C2×D4—21no—
473C2×D4—41no—
474C2×D4—41no—
475C2×D4—41no—
476C2×D4—41no—
477C2×D4—41no—
478C2×D4—41no—
479C2×D4—44no—
480C2×D4—44no—
481C2×D4—44no—
482C2×D4—44no—
483C2×D4—41no—
484C2×D4—41no—
485C2×D4—41no—
486C2×D4—41no—
487C2×D4—41no—
488C2×D4—41no—
489C2×D4C22⋊C4×C5312yescomplemented
490C22×C4—21no—
491C22×C4—21no—
492C22×C4—21no—
493C22×C4—22no—
494C22×C4—21no—
495C22×C4—21no—
496C22×C4—21no—
497C22×C4—21no—
498C22×C4—21no—
499C22×C4—41no—
500C22×C4—41no—
501C22×C4—42no—
502C22×C4—41no—
503C22×C4—41no—
504C22×C4—41no—
505C22×C4—41no—
506C22×C4—41no—
507C22×C4—41no—
508C22×C4—41no—
509C22×C4—41no—
510C22×C4—41no—
511C22×C4—41no—
512C22×C4—81no—
513C22×C4—81no—
514C22×C4D8×C5312yescomplemented
515C22×C4C22×C4×C5311yescharacteristic
516C22×C4C2×D4×C5311yescharacteristic
517C22×C4C2×D4×C5311yescharacteristic
518C22×C4C2×D4×C5311yescharacteristic
519C22×C4#41×C5311yescharacteristic
520C22×C4#41×C5311yescharacteristic
521D8—48no—
522C2×C8—22no—
523C2×C8—41no—
524C2×C8—41no—
525C2×C8—41no—
526C2×C8—41no—
527C2×C8—41no—
528C4⋊C4—21no—
529C4⋊C4—41no—
530C4⋊C4—41no—
531C4⋊C4—42no—
532C4⋊C4C22⋊C4×C5312yescomplemented
533C22⋊C4—22no—
534C22⋊C4—22no—
535C22⋊C4—42no—
536C22⋊C4—48no—
537C22⋊C4—48no—
538C22⋊C4—41no—
539C22⋊C4—41no—
540C22⋊C4—81no—
541C22⋊C4—81no—
542C42—42no—
543C23—21no—
544C23—21no—
545C23—21no—
546C23—21no—
547C23—24no—
548C23—22no—
549C23—42no—
550C23—42no—
551C23—42no—
552C23—42no—
553C23—42no—
554C23—42no—
555C23—42no—
556C23—42no—
557C23—42no—
558C23—42no—
559C23—42no—
560C23—42no—
561C23—42no—
562C23—42no—
563C23—42no—
564C23—42no—
565C23—41no—
566C23—41no—
567C23—41no—
568C23—41no—
569C23—41no—
570C23—41no—
571C23—42no—
572C23—42no—
573C23—81no—
574C23—81no—
575C23—81no—
576C23—81no—
577C23—81no—
578C23—81no—
579C23—81no—
580C23—81no—
581C23—81no—
582C23—81no—
583C23—81no—
584C23—81no—
585C23—81no—
586C23—81no—
587C23—81no—
588C23—81no—
589C23C4×D4×C5311yescharacteristic
590C23#120×C5311yescharacteristic
591C23#121×C5311yescharacteristic
592C23C2×D8×C5311yescharacteristic
593C23#135×C5311yescharacteristic
594C23AGL(1,ℤ/8)×C5311yescharacteristic
595C23AGL(1,ℤ/8)×C5311yescharacteristic
596D4—24no—
597D4—24no—
598D4—42no—
599D4—42no—
600D4—42no—
601D4—42no—
602D4—42no—
603D4—42no—
604D4—42no—
605D4—42no—
606D4—42no—
607D4—42no—
608D4—42no—
609D4—81no—
610C2×C4—21no—
611C2×C4—21no—
612C2×C4—21no—
613C2×C4—21no—
614C2×C4—22no—
615C2×C4—22no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—41no—
623C2×C4—41no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—41no—
630C2×C4—41no—
631C2×C4—41no—
632C2×C4—41no—
633C2×C4—41no—
634C2×C4—41no—
635C2×C4—41no—
636C2×C4—41no—
637C2×C4—41no—
638C2×C4—81no—
639C2×C4—84no—
640C2×C4—81no—
641C2×C4—81no—
642C2×C4—81no—
643C2×C4—81no—
644C2×C4—81no—
645C2×C4C2×C22⋊C4×C5311yescharacteristic
646C2×C4C4×D4×C5311yescharacteristic
647C2×C4#121×C5311yescharacteristic
648C2×C4#123×C5311yescharacteristic
649C8—42no—
650C8—81no—
651C22—21no—
652C22—21no—
653C22—21no—
654C22—21no—
655C22—21no—
656C22—21no—
657C22—21no—
658C22—21no—
659C22—21no—
660C22—21no—
661C22—21no—
662C22—21no—
663C22—44no—
664C22—44no—
665C22—42no—
666C22—42no—
667C22—44no—
668C22—42no—
669C22—82no—
670C22—82no—
671C22—82no—
672C22—82no—
673C22—82no—
674C22—82no—
675C22—81no—
676C22—82no—
677C22—82no—
678C22—81no—
679C22—82no—
680C22—81no—
681C22—82no—
682C22—81no—
683C22—82no—
684C22—82no—
685C22C53×#38611yescharacteristic
686C22C4×D8×C5311yescharacteristic
687C22C53×#44211yescharacteristic
688C22C53×#44711yescharacteristic
689C22C53×#44911yescharacteristic
690C22C53×#45311yescharacteristic
691C22C53×#45411yescharacteristic
692C22C53×B(4,𝔽2)11yescharacteristic
693C22C53×B(4,𝔽2)11yescharacteristic
694C22C53×#45911yescharacteristic
695C22C53×#46311yescharacteristic
696C4—21no—
697C4—21no—
698C4—21no—
699C4—21no—
700C4—42no—
701C4—42no—
702C4—82no—
703C4—81no—
704C4—81no—
705C4—81no—
706C2—21no—
707C2—21no—
708C2—21no—
709C2—21no—
710C2—84no—
711C2—82no—
712C2C53×#159511yescharacteristic
713C2C53×#159711yescharacteristic
714C2C53×#159811yescharacteristic
715C2C53×#160011yescharacteristic
716C2C53×#187911yescharacteristic
717C2C53×#188311yescharacteristic
718C2C53×#188911yescharacteristic
subgroupG/H
center, socleC23×C53#120
commutatorC22×C4C22×C4×C53
Frattini, cosocleC23×C4C23×C53
prime radicalC2×C53×#447C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23×C53 ◁ C23×C4×C53 ◁ G

Complements

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

NH ≅ G/Ntype
2. C53×#2325C2semidirect
3. C53×#1786C2semidirect
4. C2×C53×#327C2semidirect
5. C53×#1438C2semidirect
6. C53×#1294C2semidirect
7. C53×#1280C2semidirect
15. C53×#447C4semidirect
39. C22×D4×C53C2×C4semidirect
79. C22⋊C8×C53C2×C4semidirect
129. C2×D4×C53C22⋊C4semidirect
154. C22×C4×C53D8semidirect
172. C4⋊C4×C53C22⋊C4semidirect
291. #9041C53direct (N × H)
348. #2325C2×C53semidirect
349. #1786C2×C53semidirect
350. C2×#327C2×C53semidirect
351. #1438C2×C53semidirect
352. #1294C2×C53semidirect
353. #1280C2×C53semidirect
374. #447C4×C53semidirect
389. C53#9041direct (N × H)
399. C22×D4C2×C4×C53semidirect
439. C22⋊C8C2×C4×C53semidirect
489. C2×D4C22⋊C4×C53semidirect
514. C22×C4D8×C53semidirect
532. C4⋊C4C22⋊C4×C53semidirect

Maximal subgroups

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

subgroupindex
C2×C53×#4472
C53×#23252
C53×#17862
C2×C53×#3272
C53×#14382
C53×#12942
C53×#12802
#904153