Finite Group Databasebeta

← C53×#13043 #27220555

Subgroups — C53×#13043 #27220555

Order 13568.

844 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
1C2×C53×B(4,𝔽2)C211yesmaximal, characteristic
2C53×#2375C211yesmaximal, characteristic, complemented
3C2×C53×AΣL(1,𝔽16)C211yesmaximal, characteristic, complemented
4C53×#1792C211yesmaximal, characteristic, complemented
5C53×#1790C211yesmaximal, characteristic, complemented
6C53×#1393C211yesmaximal, characteristic, complemented
7C53×#1390C211yesmaximal, characteristic, complemented
8C2×H(5,𝔽2)×C53C2211yescharacteristic
9C2×#120×C53—21no—
10C2×#120×C53C2211yescharacteristic
11C53×B(4,𝔽2)—22no—
12C53×B(4,𝔽2)C414yescomplemented
13C2×#104×C53—21no—
14C53×#417—21no—
15C2×#100×C53C2211yescharacteristic
16C2×#99×C53—21no—
17C2×#99×C53—21no—
18C2×#99×C53C2211yescharacteristic
19C2×#99×C53C2211yescharacteristic
20C53×#390—21no—
21C53×#390C2211yescharacteristic
22C53×#388—21no—
23C53×#388—21no—
24C53×#388—21no—
25C53×#386—21no—
26C53×#386—21no—
27C53×#386C2211yescharacteristic
28C53×#379—21no—
29C53×AΣL(1,𝔽16)—22no—
30C25×C53D411yescharacteristic
31H(5,𝔽2)×C53—21no—
32H(5,𝔽2)×C53—21no—
33H(5,𝔽2)×C53—21no—
34H(5,𝔽2)×C53—42no—
35H(5,𝔽2)×C53C2×C412yes—
36C2×#41×C53—21no—
37C2×#41×C53—41no—
38C22×D4×C53—21no—
39C22×D4×C53—21no—
40C22×D4×C53—21no—
41C22×D4×C53—41no—
42C22×D4×C53—41no—
43C22×D4×C53C2311yescharacteristic
44C23×C4×C53D411yescharacteristic, complemented
45C2×M16×C53—21no—
46#120×C53—21no—
47#120×C53—21no—
48#120×C53—24no—
49#120×C53—42no—
50#120×C53C2×C412yes—
51#120×C53D412yes—
52C2×C4⋊C4×C53—21no—
53C2×C4⋊C4×C53D411yescharacteristic, complemented
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—41no—
65C2×C22⋊C4×C53—41no—
66C2×C22⋊C4×C53D411yescharacteristic
67C2×C22⋊C4×C53D411yescharacteristic, complemented
68C2×C42×C53—21no—
69C2×C42×C53D411yescharacteristic, complemented
70#104×C53—42no—
71#103×C53—41no—
72#102×C53—41no—
73#100×C53—21no—
74#100×C53—21no—
75#99×C53—24no—
76#99×C53—21no—
77#99×C53—21no—
78#99×C53—21no—
79#99×C53—42no—
80#99×C53C2×C412yes—
81#95×C53—21no—
82#95×C53—21no—
83#95×C53—21no—
84#95×C53—41no—
85C24×C53—21no—
86C24×C53—21no—
87C24×C53—21no—
88C24×C53—21no—
89C24×C53—41no—
90C24×C53—41no—
91C24×C53—81no—
92C24×C53—81no—
93C24×C53C22⋊C412yescomplemented
94C24×C53C2×D411yescharacteristic
95C24×C53C2×D411yescharacteristic
96#41×C53—22no—
97#41×C53—22no—
98#41×C53—42no—
99#41×C53—41no—
100#41×C53—42no—
101#41×C53—42no—
102#41×C53—42no—
103#41×C53—41no—
104C2×Q8×C53—21no—
105C2×D4×C53—21no—
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—41no—
113C2×D4×C53—41no—
114C2×D4×C53—42no—
115C2×D4×C53—42no—
116C2×D4×C53—41no—
117C2×D4×C53—41no—
118C2×D4×C53—41no—
119C2×D4×C53—41no—
120C2×D4×C53—42no—
121C2×D4×C53—42no—
122C2×D4×C53—41no—
123C2×D4×C53—41no—
124C2×D4×C53—41no—
125C2×D4×C53—44no—
126C2×D4×C53—81no—
127C2×D4×C53—81no—
128C2×D4×C53—82no—
129C2×D4×C53—82no—
130C2×D4×C53C22×C411yescharacteristic
131C2×D4×C53C2×D411yescharacteristic
132C2×D4×C53#4111yescharacteristic
133C2×D4×C53#4111yescharacteristic
134C22×C4×C53—21no—
135C22×C4×C53—21no—
136C22×C4×C53—21no—
137C22×C4×C53—21no—
138C22×C4×C53—21no—
139C22×C4×C53—21no—
140C22×C4×C53—21no—
141C22×C4×C53—41no—
142C22×C4×C53—41no—
143C22×C4×C53—41no—
144C22×C4×C53—41no—
145C22×C4×C53—81no—
146C22×C4×C53C2×D411yescharacteristic
147M16×C53—41no—
148M16×C53—41no—
149C2×C8×C53—41no—
150C4⋊C4×C53—41no—
151C4⋊C4×C53—81no—
152C22⋊C4×C53—21no—
153C22⋊C4×C53—22no—
154C22⋊C4×C53—22no—
155C22⋊C4×C53—22no—
156C22⋊C4×C53—44no—
157C22⋊C4×C53—41no—
158C22⋊C4×C53—41no—
159C22⋊C4×C53—41no—
160C22⋊C4×C53—41no—
161C22⋊C4×C53—81no—
162C22⋊C4×C53—81no—
163C22⋊C4×C53—81no—
164C22⋊C4×C53—81no—
165C22⋊C4×C53—81no—
166C22⋊C4×C53—81no—
167C22⋊C4×C53—81no—
168C22⋊C4×C53—82no—
169C22⋊C4×C53C22⋊C412yescomplemented
170C42×C53—41no—
171C42×C53—81no—
172C23×C53—21no—
173C23×C53—21no—
174C23×C53—21no—
175C23×C53—21no—
176C23×C53—22no—
177C23×C53—22no—
178C23×C53—22no—
179C23×C53—41no—
180C23×C53—42no—
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—42no—
191C23×C53—42no—
192C23×C53—41no—
193C23×C53—41no—
194C23×C53—41no—
195C23×C53—81no—
196C23×C53—82no—
197C23×C53—82no—
198C23×C53—81no—
199C23×C53—81no—
200C23×C53—84no—
201C23×C53—81no—
202C23×C53—81no—
203C23×C53—81no—
204C23×C53—81no—
205C23×C53C2×C22⋊C411yescharacteristic
206C23×C53C4×D411yescharacteristic
207C23×C53#12011yescharacteristic
208C23×C53#12111yescharacteristic
209C23×C53#12311yescharacteristic
210Q8×C53—22no—
211Q8×C53—41no—
212D4×C53—22no—
213D4×C53—41no—
214D4×C53—41no—
215D4×C53—41no—
216D4×C53—41no—
217D4×C53—41no—
218D4×C53—42no—
219D4×C53—42no—
220D4×C53—42no—
221D4×C53—42no—
222D4×C53—42no—
223D4×C53—42no—
224D4×C53—82no—
225D4×C53—82no—
226D4×C53—88no—
227C2×C4×C53—21no—
228C2×C4×C53—21no—
229C2×C4×C53—22no—
230C2×C4×C53—22no—
231C2×C4×C53—22no—
232C2×C4×C53—42no—
233C2×C4×C53—42no—
234C2×C4×C53—41no—
235C2×C4×C53—41no—
236C2×C4×C53—41no—
237C2×C4×C53—41no—
238C2×C4×C53—41no—
239C2×C4×C53—41no—
240C2×C4×C53—41no—
241C2×C4×C53—41no—
242C2×C4×C53—41no—
243C2×C4×C53—41no—
244C2×C4×C53—41no—
245C2×C4×C53—41no—
246C2×C4×C53—41no—
247C2×C4×C53—42no—
248C2×C4×C53—42no—
249C2×C4×C53—41no—
250C2×C4×C53—41no—
251C2×C4×C53—41no—
252C2×C4×C53—41no—
253C2×C4×C53—81no—
254C2×C4×C53—81no—
255C2×C4×C53—81no—
256C2×C4×C53—81no—
257C2×C4×C53—81no—
258C2×C4×C53—81no—
259C2×C4×C53C4×D411yescharacteristic
260C2×C4×C53#12111yescharacteristic
261C8×C53—81no—
262#13043C5311yesmaximal, characteristic, direct product factor, Sylow
263C22×C53—21no—
264C22×C53—21no—
265C22×C53—21no—
266C22×C53—21no—
267C22×C53—21no—
268C22×C53—21no—
269C22×C53—41no—
270C22×C53—41no—
271C22×C53—41no—
272C22×C53—41no—
273C22×C53—42no—
274C22×C53—42no—
275C22×C53—41no—
276C22×C53—41no—
277C22×C53—41no—
278C22×C53—41no—
279C22×C53—44no—
280C22×C53—42no—
281C22×C53—42no—
282C22×C53—81no—
283C22×C53—81no—
284C22×C53—81no—
285C22×C53—81no—
286C22×C53—82no—
287C22×C53—82no—
288C22×C53—82no—
289C22×C53—82no—
290C22×C53—82no—
291C22×C53—81no—
292C22×C53—81no—
293C22×C53—81no—
294C22×C53—81no—
295C22×C53—161no—
296C22×C53—161no—
297C22×C53#38611yescharacteristic
298C22×C53B(4,𝔽2)11yescharacteristic
299C22×C53B(4,𝔽2)11yescharacteristic
300C4×C53—21no—
301C4×C53—21no—
302C4×C53—42no—
303C4×C53—42no—
304C4×C53—81no—
305C4×C53—81no—
306C4×C53—82no—
307C4×C53—81no—
308C4×C53—81no—
309C4×C53—161no—
310C2×B(4,𝔽2)C2×C5311yescharacteristic
311#2375C2×C5311yescharacteristic, complemented
312C2×AΣL(1,𝔽16)C2×C5311yescharacteristic, complemented
313#1792C2×C5311yescharacteristic, complemented
314#1790C2×C5311yescharacteristic, complemented
315#1393C2×C5311yescharacteristic, complemented
316#1390C2×C5311yescharacteristic, complemented
317C2×C53—21no—
318C2×C53—21no—
319C2×C53—41no—
320C2×C53—41no—
321C2×C53—41no—
322C2×C53—41no—
323C2×C53—82no—
324C2×C53—82no—
325C2×C53#188911yescharacteristic
326C2×C53#219911yescharacteristic
327C2×C53#220011yescharacteristic
328C2×H(5,𝔽2)C22×C5311yescharacteristic
329C2×#120—21no—
330C2×#120C22×C5311yescharacteristic
331B(4,𝔽2)—22no—
332B(4,𝔽2)C4×C5314yescomplemented
333C2×#104—21no—
334#417—21no—
335C2×#100C22×C5311yescharacteristic
336C2×#99—21no—
337C2×#99—21no—
338C2×#99C22×C5311yescharacteristic
339C2×#99C22×C5311yescharacteristic
340#390—21no—
341#390C22×C5311yescharacteristic
342#388—21no—
343#388—21no—
344#388—21no—
345#386—21no—
346#386—21no—
347#386C22×C5311yescharacteristic
348#379—21no—
349AΣL(1,𝔽16)—22no—
350C53#1304311yescharacteristic, direct product factor, Sylow
351C25D4×C5311yescharacteristic
352H(5,𝔽2)—21no—
353H(5,𝔽2)—21no—
354H(5,𝔽2)—21no—
355H(5,𝔽2)—42no—
356H(5,𝔽2)C2×C4×C5312yes—
357C2×#41—21no—
358C2×#41—41no—
359C22×D4—21no—
360C22×D4—21no—
361C22×D4—21no—
362C22×D4—41no—
363C22×D4—41no—
364C22×D4C23×C5311yescharacteristic
365C23×C4D4×C5311yescharacteristic, complemented
366C2×M16—21no—
367#120—21no—
368#120—21no—
369#120—24no—
370#120—42no—
371#120C2×C4×C5312yes—
372#120D4×C5312yes—
373C2×C4⋊C4—21no—
374C2×C4⋊C4D4×C5311yescharacteristic, complemented
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—41no—
386C2×C22⋊C4—41no—
387C2×C22⋊C4D4×C5311yescharacteristic
388C2×C22⋊C4D4×C5311yescharacteristic, complemented
389C2×C42—21no—
390C2×C42D4×C5311yescharacteristic, complemented
391#104—42no—
392#103—41no—
393#102—41no—
394#100—21no—
395#100—21no—
396#99—21no—
397#99—24no—
398#99—21no—
399#99—21no—
400#99—42no—
401#99C2×C4×C5312yes—
402#95—21no—
403#95—21no—
404#95—21no—
405#95—41no—
406C24—21no—
407C24—21no—
408C24—21no—
409C24—21no—
410C24—41no—
411C24—41no—
412C24—81no—
413C24—81no—
414C24C22⋊C4×C5312yescomplemented
415C24C2×D4×C5311yescharacteristic
416C24C2×D4×C5311yescharacteristic
417#41—22no—
418#41—22no—
419#41—42no—
420#41—41no—
421#41—42no—
422#41—42no—
423#41—42no—
424#41—41no—
425C2×Q8—21no—
426C2×D4—21no—
427C2×D4—22no—
428C2×D4—22no—
429C2×D4—22no—
430C2×D4—22no—
431C2×D4—22no—
432C2×D4—22no—
433C2×D4—41no—
434C2×D4—41no—
435C2×D4—42no—
436C2×D4—42no—
437C2×D4—41no—
438C2×D4—41no—
439C2×D4—41no—
440C2×D4—41no—
441C2×D4—42no—
442C2×D4—42no—
443C2×D4—41no—
444C2×D4—41no—
445C2×D4—41no—
446C2×D4—44no—
447C2×D4—81no—
448C2×D4—81no—
449C2×D4—82no—
450C2×D4—82no—
451C2×D4C22×C4×C5311yescharacteristic
452C2×D4C2×D4×C5311yescharacteristic
453C2×D4#41×C5311yescharacteristic
454C2×D4#41×C5311yescharacteristic
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—41no—
463C22×C4—41no—
464C22×C4—41no—
465C22×C4—41no—
466C22×C4—81no—
467C22×C4C2×D4×C5311yescharacteristic
468M16—41no—
469M16—41no—
470C2×C8—41no—
471C4⋊C4—41no—
472C4⋊C4—81no—
473C22⋊C4—21no—
474C22⋊C4—22no—
475C22⋊C4—22no—
476C22⋊C4—22no—
477C22⋊C4—44no—
478C22⋊C4—41no—
479C22⋊C4—41no—
480C22⋊C4—41no—
481C22⋊C4—41no—
482C22⋊C4—81no—
483C22⋊C4—81no—
484C22⋊C4—81no—
485C22⋊C4—81no—
486C22⋊C4—81no—
487C22⋊C4—81no—
488C22⋊C4—81no—
489C22⋊C4—82no—
490C22⋊C4C22⋊C4×C5312yescomplemented
491C42—41no—
492C42—81no—
493C23—21no—
494C23—21no—
495C23—21no—
496C23—21no—
497C23—22no—
498C23—22no—
499C23—22no—
500C23—42no—
501C23—41no—
502C23—41no—
503C23—41no—
504C23—41no—
505C23—41no—
506C23—41no—
507C23—41no—
508C23—41no—
509C23—41no—
510C23—41no—
511C23—42no—
512C23—42no—
513C23—41no—
514C23—41no—
515C23—41no—
516C23—81no—
517C23—82no—
518C23—82no—
519C23—81no—
520C23—81no—
521C23—84no—
522C23—81no—
523C23—81no—
524C23—81no—
525C23—81no—
526C23C2×C22⋊C4×C5311yescharacteristic
527C23C4×D4×C5311yescharacteristic
528C23#120×C5311yescharacteristic
529C23#121×C5311yescharacteristic
530C23#123×C5311yescharacteristic
531Q8—22no—
532Q8—41no—
533D4—22no—
534D4—41no—
535D4—41no—
536D4—41no—
537D4—41no—
538D4—41no—
539D4—42no—
540D4—42no—
541D4—42no—
542D4—42no—
543D4—42no—
544D4—42no—
545D4—82no—
546D4—82no—
547D4—88no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—42no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—42no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—81no—
575C2×C4—81no—
576C2×C4—81no—
577C2×C4—81no—
578C2×C4—81no—
579C2×C4—81no—
580C2×C4C4×D4×C5311yescharacteristic
581C2×C4#121×C5311yescharacteristic
582C8—81no—
583C22—21no—
584C22—21no—
585C22—21no—
586C22—21no—
587C22—21no—
588C22—21no—
589C22—41no—
590C22—41no—
591C22—41no—
592C22—41no—
593C22—42no—
594C22—42no—
595C22—41no—
596C22—41no—
597C22—41no—
598C22—41no—
599C22—44no—
600C22—42no—
601C22—42no—
602C22—81no—
603C22—81no—
604C22—81no—
605C22—81no—
606C22—82no—
607C22—82no—
608C22—82no—
609C22—82no—
610C22—82no—
611C22—81no—
612C22—81no—
613C22—81no—
614C22—81no—
615C22—161no—
616C22—161no—
617C22C53×#38611yescharacteristic
618C22C53×B(4,𝔽2)11yescharacteristic
619C22C53×B(4,𝔽2)11yescharacteristic
620C4—21no—
621C4—21no—
622C4—42no—
623C4—42no—
624C4—81no—
625C4—81no—
626C4—82no—
627C4—81no—
628C4—81no—
629C4—161no—
630C2—21no—
631C2—21no—
632C2—41no—
633C2—41no—
634C2—41no—
635C2—41no—
636C2—82no—
637C2—82no—
638C2C53×#188911yescharacteristic
639C2C53×#219911yescharacteristic
640C2C53×#220011yescharacteristic
subgroupG/H
center, socleC22×C53B(4,𝔽2)
commutatorC2×D4C22×C4×C53
Frattini, cosocleC22×D4C23×C53
prime radicalC2×C53×B(4,𝔽2)C2
LCSG ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C53 ◁ C23×C53 ◁ C22×D4×C53 ◁ G

Complements

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

NH ≅ G/Ntype
2. C53×#2375C2semidirect
3. C2×C53×AΣL(1,𝔽16)C2semidirect
4. C53×#1792C2semidirect
5. C53×#1790C2semidirect
6. C53×#1393C2semidirect
7. C53×#1390C2semidirect
12. C53×B(4,𝔽2)C4semidirect
44. C23×C4×C53D4semidirect
53. C2×C4⋊C4×C53D4semidirect
67. C2×C22⋊C4×C53D4semidirect
69. C2×C42×C53D4semidirect
93. C24×C53C22⋊C4semidirect
169. C22⋊C4×C53C22⋊C4semidirect
262. #13043C53direct (N × H)
311. #2375C2×C53semidirect
312. C2×AΣL(1,𝔽16)C2×C53semidirect
313. #1792C2×C53semidirect
314. #1790C2×C53semidirect
315. #1393C2×C53semidirect
316. #1390C2×C53semidirect
332. B(4,𝔽2)C4×C53semidirect
350. C53#13043direct (N × H)
365. C23×C4D4×C53semidirect
374. C2×C4⋊C4D4×C53semidirect
388. C2×C22⋊C4D4×C53semidirect
390. C2×C42D4×C53semidirect
414. C24C22⋊C4×C53semidirect
490. C22⋊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×B(4,𝔽2)2
C53×#23752
C2×C53×AΣL(1,𝔽16)2
C53×#17922
C53×#17902
C53×#13932
C53×#13902
#1304353