Finite Group Databasebeta

← C53×#10395 #27217907

Subgroups — C53×#10395 #27217907

Order 13568.

772 conjugacy classes of subgroups.

The subgroup lattice has 642 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C53×#2617C211yesmaximal, characteristic
2C53×#1874C211yesmaximal, characteristic, complemented
3C53×#1853C211yesmaximal, characteristic, complemented
4C2×C53×#327C211yesmaximal, characteristic, complemented
5C2×C53×#323C211yesmaximal, characteristic, complemented
6C53×#1439C211yesmaximal, characteristic, complemented
7C53×#1294C211yesmaximal, characteristic, complemented
8C2×#121×C53C2211yescharacteristic
9C2×#120×C53—21no—
10C22×C4⋊C4×C53C2211yescharacteristic
11C22×C22⋊C4×C53—21no—
12C22×C22⋊C4×C53C2211yescharacteristic
13C2×#106×C53—21no—
14C2×#102×C53—21no—
15C2×C22⋊C8×C53C2211yescharacteristic
16C2×C22⋊C8×C53C2211yescharacteristic
17C53×#397—21no—
18C53×#394—21no—
19C53×#394—21no—
20C53×#392—21no—
21C53×#386—21no—
22C53×#380C2211yescharacteristic
23C2×#95×C53—21no—
24C2×#95×C53C2211yescharacteristic
25C53×#340—21no—
26C53×#340—21no—
27C53×#327—22no—
28C53×#323—22no—
29C25×C53D411yescharacteristic
30C22×D4×C53—21no—
31C22×D4×C53—21no—
32C22×D4×C53—21no—
33C22×D4×C53D411yescharacteristic
34C23×C4×C53—21no—
35C23×C4×C53—21no—
36C23×C4×C53D411yescharacteristic
37C23×C4×C53C2311yescharacteristic
38C22×C8×C53—21no—
39C22×C8×C53—21no—
40#121×C53—21no—
41#121×C53—21no—
42#121×C53—21no—
43#121×C53—21no—
44#120×C53—42no—
45#120×C53—42no—
46C2×C4⋊C4×C53—21no—
47C2×C4⋊C4×C53—21no—
48C2×C4⋊C4×C53—42no—
49C2×C4⋊C4×C53D411yescharacteristic
50C2×C4⋊C4×C53D411yescharacteristic
51C2×C4⋊C4×C53D411yescharacteristic
52C2×C22⋊C4×C53—21no—
53C2×C22⋊C4×C53—21no—
54C2×C22⋊C4×C53—21no—
55C2×C22⋊C4×C53—21no—
56C2×C22⋊C4×C53—21no—
57C2×C22⋊C4×C53—21no—
58C2×C22⋊C4×C53—21no—
59C2×C22⋊C4×C53—21no—
60C2×C22⋊C4×C53—21no—
61C2×C22⋊C4×C53—21no—
62C2×C22⋊C4×C53—21no—
63C2×C22⋊C4×C53—21no—
64C2×C22⋊C4×C53—42no—
65C2×C22⋊C4×C53—42no—
66C2×C22⋊C4×C53—42no—
67#106×C53—42no—
68#102×C53—42no—
69C22⋊C8×C53—21no—
70C22⋊C8×C53—21no—
71C22⋊C8×C53—21no—
72C22⋊C8×C53D412yescomplemented
73#95×C53—21no—
74#95×C53—21no—
75#95×C53—21no—
76#95×C53—21no—
77#95×C53—21no—
78#95×C53—42no—
79C24×C53—21no—
80C24×C53—21no—
81C24×C53—21no—
82C24×C53—21no—
83C24×C53—21no—
84C24×C53—21no—
85C24×C53—21no—
86C24×C53—41no—
87C24×C53—41no—
88C24×C53—41no—
89C24×C53—41no—
90C24×C53SD1612yes—
91C24×C53C2×D411yescharacteristic
92C2×D4×C53—21no—
93C2×D4×C53—21no—
94C2×D4×C53—41no—
95C2×D4×C53—41no—
96C2×D4×C53—42no—
97C2×D4×C53—42no—
98C2×D4×C53—41no—
99C2×D4×C53—41no—
100C2×D4×C53—41no—
101C2×D4×C53—41no—
102C2×D4×C53—41no—
103C2×D4×C53—41no—
104C2×D4×C53—81no—
105C2×D4×C53—81no—
106C2×D4×C53—81no—
107C2×D4×C53—81no—
108C22×C4×C53—22no—
109C22×C4×C53—22no—
110C22×C4×C53—21no—
111C22×C4×C53—21no—
112C22×C4×C53—21no—
113C22×C4×C53—21no—
114C22×C4×C53—21no—
115C22×C4×C53—21no—
116C22×C4×C53—41no—
117C22×C4×C53—41no—
118C22×C4×C53—41no—
119C22×C4×C53—41no—
120C22×C4×C53—41no—
121C22×C4×C53—41no—
122C22×C4×C53—41no—
123C22×C4×C53—41no—
124C22×C4×C53—41no—
125C22×C4×C53—41no—
126C22×C4×C53—41no—
127C22×C4×C53—41no—
128C22×C4×C53—41no—
129C22×C4×C53—41no—
130C22×C4×C53—41no—
131C22×C4×C53—41no—
132C22×C4×C53C2×D411yescharacteristic
133C22×C4×C53C2×D411yescharacteristic
134C22×C4×C53C2×D411yescharacteristic
135C22×C4×C53#4111yescharacteristic
136C22×C4×C53#4111yescharacteristic
137C22×C4×C53#4111yescharacteristic
138C2×C8×C53—22no—
139C2×C8×C53—41no—
140C2×C8×C53—41no—
141C2×C8×C53—41no—
142C2×C8×C53—41no—
143C2×C8×C53—41no—
144C4⋊C4×C53—22no—
145C4⋊C4×C53—22no—
146C4⋊C4×C53—21no—
147C4⋊C4×C53—21no—
148C4⋊C4×C53—41no—
149C4⋊C4×C53—41no—
150C4⋊C4×C53—41no—
151C4⋊C4×C53—41no—
152C22⋊C4×C53—42no—
153C22⋊C4×C53—42no—
154C22⋊C4×C53—48no—
155C22⋊C4×C53—41no—
156C22⋊C4×C53—41no—
157C22⋊C4×C53—41no—
158C22⋊C4×C53—41no—
159C22⋊C4×C53—41no—
160C22⋊C4×C53—41no—
161C22⋊C4×C53—41no—
162C22⋊C4×C53—41no—
163C22⋊C4×C53—81no—
164C22⋊C4×C53—81no—
165C23×C53—21no—
166C23×C53—21no—
167C23×C53—21no—
168C23×C53—21no—
169C23×C53—22no—
170C23×C53—22no—
171C23×C53—22no—
172C23×C53—42no—
173C23×C53—42no—
174C23×C53—42no—
175C23×C53—42no—
176C23×C53—42no—
177C23×C53—42no—
178C23×C53—41no—
179C23×C53—41no—
180C23×C53—41no—
181C23×C53—41no—
182C23×C53—41no—
183C23×C53—41no—
184C23×C53—41no—
185C23×C53—41no—
186C23×C53—41no—
187C23×C53—41no—
188C23×C53—41no—
189C23×C53—41no—
190C23×C53—41no—
191C23×C53—41no—
192C23×C53—41no—
193C23×C53—81no—
194C23×C53—81no—
195C23×C53—81no—
196C23×C53—81no—
197C23×C53—81no—
198C23×C53—81no—
199C23×C53#12011yescharacteristic
200C23×C53#12111yescharacteristic
201C23×C53#12311yescharacteristic
202C23×C53C2×SD1611yescharacteristic
203C23×C53#13511yescharacteristic
204C23×C53AGL(1,ℤ/8)11yescharacteristic
205C23×C53AGL(1,ℤ/8)11yescharacteristic
206D4×C53—42no—
207D4×C53—81no—
208D4×C53—84no—
209D4×C53—84no—
210D4×C53—81no—
211D4×C53—81no—
212D4×C53—81no—
213D4×C53—81no—
214C2×C4×C53—21no—
215C2×C4×C53—21no—
216C2×C4×C53—21no—
217C2×C4×C53—21no—
218C2×C4×C53—41no—
219C2×C4×C53—41no—
220C2×C4×C53—41no—
221C2×C4×C53—41no—
222C2×C4×C53—42no—
223C2×C4×C53—42no—
224C2×C4×C53—42no—
225C2×C4×C53—42no—
226C2×C4×C53—42no—
227C2×C4×C53—42no—
228C2×C4×C53—41no—
229C2×C4×C53—41no—
230C2×C4×C53—41no—
231C2×C4×C53—41no—
232C2×C4×C53—41no—
233C2×C4×C53—41no—
234C2×C4×C53—41no—
235C2×C4×C53—41no—
236C2×C4×C53—41no—
237C2×C4×C53—41no—
238C2×C4×C53—42no—
239C2×C4×C53—41no—
240C2×C4×C53—41no—
241C2×C4×C53—81no—
242C2×C4×C53—81no—
243C2×C4×C53—81no—
244C2×C4×C53—81no—
245C2×C4×C53—81no—
246C2×C4×C53—81no—
247C2×C4×C53—81no—
248C2×C4×C53—81no—
249C2×C4×C53—81no—
250C2×C4×C53#12111yescharacteristic
251C2×C4×C53#12111yescharacteristic
252C2×C4×C53#12311yescharacteristic
253C2×C4×C53#12411yescharacteristic
254C8×C53—42no—
255C8×C53—81no—
256#10395C5311yesmaximal, characteristic, direct product factor, Sylow
257C22×C53—21no—
258C22×C53—21no—
259C22×C53—21no—
260C22×C53—21no—
261C22×C53—21no—
262C22×C53—21no—
263C22×C53—21no—
264C22×C53—21no—
265C22×C53—21no—
266C22×C53—21no—
267C22×C53—21no—
268C22×C53—21no—
269C22×C53—44no—
270C22×C53—42no—
271C22×C53—42no—
272C22×C53—42no—
273C22×C53—82no—
274C22×C53—82no—
275C22×C53—82no—
276C22×C53—81no—
277C22×C53—81no—
278C22×C53—81no—
279C22×C53—81no—
280C22×C53—81no—
281C22×C53—81no—
282C22×C53—81no—
283C22×C53—81no—
284C22×C53—81no—
285C22×C53—81no—
286C22×C53—81no—
287C22×C53#39411yescharacteristic
288C22×C53#44911yescharacteristic
289C22×C53#45011yescharacteristic
290C22×C53#45311yescharacteristic
291C22×C53#45511yescharacteristic
292C22×C53B(4,𝔽2)11yescharacteristic
293C22×C53#45811yescharacteristic
294C22×C53#46511yescharacteristic
295C22×C53#46911yescharacteristic
296C22×C53#48111yescharacteristic
297C22×C53#48211yescharacteristic
298C4×C53—21no—
299C4×C53—21no—
300C4×C53—21no—
301C4×C53—21no—
302C4×C53—82no—
303C4×C53—82no—
304C4×C53—81no—
305C4×C53—81no—
306C4×C53—81no—
307C4×C53—81no—
308#2617C2×C5311yescharacteristic
309#1874C2×C5311yescharacteristic, complemented
310#1853C2×C5311yescharacteristic, complemented
311C2×#327C2×C5311yescharacteristic, complemented
312C2×#323C2×C5311yescharacteristic, complemented
313#1439C2×C5311yescharacteristic, complemented
314#1294C2×C5311yescharacteristic, complemented
315C2×C53—21no—
316C2×C53—21no—
317C2×C53—21no—
318C2×C53—21no—
319C2×C53—82no—
320C2×C53—161no—
321C2×C53#160011yescharacteristic
322C2×C53#160111yescharacteristic
323C2×C53#161211yescharacteristic
324C2×C53#161311yescharacteristic
325C2×C53#203411yescharacteristic
326C2×C53#203711yescharacteristic
327C2×C53#204411yescharacteristic
328C2×#121C22×C5311yescharacteristic
329C2×#120—21no—
330C22×C4⋊C4C22×C5311yescharacteristic
331C22×C22⋊C4—21no—
332C22×C22⋊C4C22×C5311yescharacteristic
333C2×#106—21no—
334C2×#102—21no—
335C2×C22⋊C8C22×C5311yescharacteristic
336C2×C22⋊C8C22×C5311yescharacteristic
337#397—21no—
338#394—21no—
339#394—21no—
340#392—21no—
341#386—21no—
342#380C22×C5311yescharacteristic
343C2×#95—21no—
344C2×#95C22×C5311yescharacteristic
345#340—21no—
346#340—21no—
347#327—22no—
348#323—22no—
349C53#1039511yescharacteristic, direct product factor, Sylow
350C25D4×C5311yescharacteristic
351C22×D4—21no—
352C22×D4—21no—
353C22×D4—21no—
354C22×D4D4×C5311yescharacteristic
355C23×C4—21no—
356C23×C4—21no—
357C23×C4D4×C5311yescharacteristic
358C23×C4C23×C5311yescharacteristic
359C22×C8—21no—
360C22×C8—21no—
361#121—21no—
362#121—21no—
363#121—21no—
364#121—21no—
365#120—42no—
366#120—42no—
367C2×C4⋊C4—21no—
368C2×C4⋊C4—21no—
369C2×C4⋊C4—42no—
370C2×C4⋊C4D4×C5311yescharacteristic
371C2×C4⋊C4D4×C5311yescharacteristic
372C2×C4⋊C4D4×C5311yescharacteristic
373C2×C22⋊C4—21no—
374C2×C22⋊C4—21no—
375C2×C22⋊C4—21no—
376C2×C22⋊C4—21no—
377C2×C22⋊C4—21no—
378C2×C22⋊C4—21no—
379C2×C22⋊C4—21no—
380C2×C22⋊C4—21no—
381C2×C22⋊C4—21no—
382C2×C22⋊C4—21no—
383C2×C22⋊C4—21no—
384C2×C22⋊C4—21no—
385C2×C22⋊C4—42no—
386C2×C22⋊C4—42no—
387C2×C22⋊C4—42no—
388#106—42no—
389#102—42no—
390C22⋊C8—21no—
391C22⋊C8—21no—
392C22⋊C8—21no—
393C22⋊C8D4×C5312yescomplemented
394#95—21no—
395#95—21no—
396#95—21no—
397#95—21no—
398#95—21no—
399#95—42no—
400C24—21no—
401C24—21no—
402C24—21no—
403C24—21no—
404C24—21no—
405C24—21no—
406C24—21no—
407C24—41no—
408C24—41no—
409C24—41no—
410C24—41no—
411C24SD16×C5312yes—
412C24C2×D4×C5311yescharacteristic
413C2×D4—21no—
414C2×D4—21no—
415C2×D4—41no—
416C2×D4—41no—
417C2×D4—42no—
418C2×D4—42no—
419C2×D4—41no—
420C2×D4—41no—
421C2×D4—41no—
422C2×D4—41no—
423C2×D4—41no—
424C2×D4—41no—
425C2×D4—81no—
426C2×D4—81no—
427C2×D4—81no—
428C2×D4—81no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—41no—
438C22×C4—41no—
439C22×C4—41no—
440C22×C4—41no—
441C22×C4—41no—
442C22×C4—41no—
443C22×C4—41no—
444C22×C4—41no—
445C22×C4—41no—
446C22×C4—41no—
447C22×C4—41no—
448C22×C4—41no—
449C22×C4—41no—
450C22×C4—41no—
451C22×C4—41no—
452C22×C4—41no—
453C22×C4C2×D4×C5311yescharacteristic
454C22×C4C2×D4×C5311yescharacteristic
455C22×C4C2×D4×C5311yescharacteristic
456C22×C4#41×C5311yescharacteristic
457C22×C4#41×C5311yescharacteristic
458C22×C4#41×C5311yescharacteristic
459C2×C8—22no—
460C2×C8—41no—
461C2×C8—41no—
462C2×C8—41no—
463C2×C8—41no—
464C2×C8—41no—
465C4⋊C4—22no—
466C4⋊C4—22no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—41no—
470C4⋊C4—41no—
471C4⋊C4—41no—
472C4⋊C4—41no—
473C22⋊C4—42no—
474C22⋊C4—42no—
475C22⋊C4—48no—
476C22⋊C4—41no—
477C22⋊C4—41no—
478C22⋊C4—41no—
479C22⋊C4—41no—
480C22⋊C4—41no—
481C22⋊C4—41no—
482C22⋊C4—41no—
483C22⋊C4—41no—
484C22⋊C4—81no—
485C22⋊C4—81no—
486C23—21no—
487C23—21no—
488C23—21no—
489C23—21no—
490C23—22no—
491C23—22no—
492C23—22no—
493C23—42no—
494C23—42no—
495C23—42no—
496C23—42no—
497C23—42no—
498C23—42no—
499C23—41no—
500C23—41no—
501C23—41no—
502C23—41no—
503C23—41no—
504C23—41no—
505C23—41no—
506C23—41no—
507C23—41no—
508C23—41no—
509C23—41no—
510C23—41no—
511C23—41no—
512C23—41no—
513C23—41no—
514C23—81no—
515C23—81no—
516C23—81no—
517C23—81no—
518C23—81no—
519C23—81no—
520C23#120×C5311yescharacteristic
521C23#121×C5311yescharacteristic
522C23#123×C5311yescharacteristic
523C23C2×SD16×C5311yescharacteristic
524C23#135×C5311yescharacteristic
525C23AGL(1,ℤ/8)×C5311yescharacteristic
526C23AGL(1,ℤ/8)×C5311yescharacteristic
527D4—42no—
528D4—81no—
529D4—84no—
530D4—84no—
531D4—81no—
532D4—81no—
533D4—81no—
534D4—81no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—42no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—81no—
563C2×C4—81no—
564C2×C4—81no—
565C2×C4—81no—
566C2×C4—81no—
567C2×C4—81no—
568C2×C4—81no—
569C2×C4—81no—
570C2×C4—81no—
571C2×C4#121×C5311yescharacteristic
572C2×C4#121×C5311yescharacteristic
573C2×C4#123×C5311yescharacteristic
574C2×C4#124×C5311yescharacteristic
575C8—42no—
576C8—81no—
577C22—21no—
578C22—21no—
579C22—21no—
580C22—21no—
581C22—21no—
582C22—21no—
583C22—21no—
584C22—21no—
585C22—21no—
586C22—21no—
587C22—21no—
588C22—21no—
589C22—44no—
590C22—42no—
591C22—42no—
592C22—42no—
593C22—82no—
594C22—82no—
595C22—82no—
596C22—81no—
597C22—81no—
598C22—81no—
599C22—81no—
600C22—81no—
601C22—81no—
602C22—81no—
603C22—81no—
604C22—81no—
605C22—81no—
606C22—81no—
607C22C53×#39411yescharacteristic
608C22C53×#44911yescharacteristic
609C22C53×#45011yescharacteristic
610C22C53×#45311yescharacteristic
611C22C53×#45511yescharacteristic
612C22C53×B(4,𝔽2)11yescharacteristic
613C22C53×#45811yescharacteristic
614C22C53×#46511yescharacteristic
615C22C53×#46911yescharacteristic
616C22C53×#48111yescharacteristic
617C22C53×#48211yescharacteristic
618C4—21no—
619C4—21no—
620C4—21no—
621C4—21no—
622C4—82no—
623C4—81no—
624C4—81no—
625C4—82no—
626C4—81no—
627C4—81no—
628C2—21no—
629C2—21no—
630C2—21no—
631C2—21no—
632C2—82no—
633C2—161no—
634C2C53×#160011yescharacteristic
635C2C53×#160111yescharacteristic
636C2C53×#161211yescharacteristic
637C2C53×#161311yescharacteristic
638C2C53×#203411yescharacteristic
639C2C53×#203711yescharacteristic
640C2C53×#204411yescharacteristic
subgroupG/H
center, socleC23×C53#120
commutator, Frattini, cosocleC23×C4C23×C53
prime radicalC53×#2617C2
LCSG ▷ C23×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×#1874C2semidirect
3. C53×#1853C2semidirect
4. C2×C53×#327C2semidirect
5. C2×C53×#323C2semidirect
6. C53×#1439C2semidirect
7. C53×#1294C2semidirect
72. C22⋊C8×C53D4semidirect
256. #10395C53direct (N × H)
309. #1874C2×C53semidirect
310. #1853C2×C53semidirect
311. C2×#327C2×C53semidirect
312. C2×#323C2×C53semidirect
313. #1439C2×C53semidirect
314. #1294C2×C53semidirect
349. C53#10395direct (N × H)
393. C22⋊C8D4×C53semidirect

Maximal subgroups

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

subgroupindex
C53×#26172
C53×#18742
C53×#18532
C2×C53×#3272
C2×C53×#3232
C53×#14392
C53×#12942
#1039553