Finite Group Databasebeta

← C53×#13082 #27220594

Subgroups — C53×#13082 #27220594

Order 13568.

1088 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#100×C53C211yesmaximal, characteristic, complemented
2C53×#2403C211yesmaximal, characteristic
3C2×C53×AΣL(1,𝔽16)C212yesmaximal, complemented
4C53×#1781C211yesmaximal, characteristic, complemented
5C53×#1390C212yesmaximal, complemented
6C23×D4×C53C411yescharacteristic, complemented
7C23×D4×C53C2211yescharacteristic
8C22×M16×C53—21no—
9C2×#120×C53—21no—
10C2×#120×C53C411yescharacteristic, complemented
11C2×#120×C53C2211yescharacteristic
12C2×#100×C53—21no—
13C2×#100×C53—24no—
14C2×#100×C53C2212yes—
15C2×#99×C53—21no—
16C2×#99×C53C2212yes—
17C53×#392—21no—
18C53×#392—21no—
19C53×#386—21no—
20C53×#386C411yescharacteristic, complemented
21C53×#386C2211yescharacteristic
22C53×#379—21no—
23C53×#379C411yescharacteristic, complemented
24C53×AΣL(1,𝔽16)—24no—
25C25×C53C2×C411yescharacteristic
26C25×C53C2×C411yescharacteristic
27C25×C53D411yescharacteristic
28C22×D4×C53—21no—
29C22×D4×C53—22no—
30C22×D4×C53—22no—
31C22×D4×C53—21no—
32C22×D4×C53—21no—
33C22×D4×C53—21no—
34C22×D4×C53—42no—
35C22×D4×C53—41no—
36C22×D4×C53—41no—
37C22×D4×C53—44no—
38C22×D4×C53—41no—
39C22×D4×C53—41no—
40C22×D4×C53C2×C411yescharacteristic
41C22×D4×C53D411yescharacteristic
42C22×D4×C53D411yescharacteristic
43C22×D4×C53C2311yescharacteristic
44C23×C4×C53C2×C411yescharacteristic
45C23×C4×C53D411yescharacteristic
46C2×M16×C53—24no—
47C2×M16×C53—22no—
48C2×M16×C53—41no—
49C2×M16×C53—42no—
50C22×C8×C53—41no—
51#120×C53—22no—
52#120×C53—22no—
53#120×C53—42no—
54#120×C53—44no—
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—22no—
62C2×C22⋊C4×C53—41no—
63C2×C22⋊C4×C53—41no—
64C2×C22⋊C4×C53C2×C411yescharacteristic
65C2×C22⋊C4×C53C2×C411yescharacteristic
66#100×C53—24no—
67#100×C53—22no—
68#100×C53D414yescomplemented
69#99×C53—22no—
70#99×C53—22no—
71#99×C53—42no—
72#95×C53—21no—
73#95×C53—21no—
74C24×C53—21no—
75C24×C53—21no—
76C24×C53—21no—
77C24×C53—21no—
78C24×C53—21no—
79C24×C53—22no—
80C24×C53—22no—
81C24×C53—22no—
82C24×C53—41no—
83C24×C53—41no—
84C24×C53—42no—
85C24×C53—41no—
86C24×C53—42no—
87C24×C53—42no—
88C24×C53—41no—
89C24×C53—81no—
90C24×C53—81no—
91C24×C53—81no—
92C24×C53C22⋊C411yescharacteristic
93C24×C53C22⋊C411yescharacteristic
94C24×C53C22⋊C411yescharacteristic
95C24×C53C22×C411yescharacteristic
96C24×C53C2×D411yescharacteristic
97C2×D4×C53—24no—
98C2×D4×C53—22no—
99C2×D4×C53—22no—
100C2×D4×C53—22no—
101C2×D4×C53—22no—
102C2×D4×C53—21no—
103C2×D4×C53—21no—
104C2×D4×C53—41no—
105C2×D4×C53—41no—
106C2×D4×C53—41no—
107C2×D4×C53—41no—
108C2×D4×C53—41no—
109C2×D4×C53—41no—
110C2×D4×C53—44no—
111C2×D4×C53—44no—
112C2×D4×C53—44no—
113C2×D4×C53—44no—
114C2×D4×C53—42no—
115C2×D4×C53—42no—
116C2×D4×C53—42no—
117C2×D4×C53—42no—
118C2×D4×C53—44no—
119C2×D4×C53—44no—
120C2×D4×C53—44no—
121C2×D4×C53—41no—
122C2×D4×C53—41no—
123C2×D4×C53—44no—
124C2×D4×C53—42no—
125C2×D4×C53—42no—
126C2×D4×C53—42no—
127C2×D4×C53—42no—
128C2×D4×C53—81no—
129C2×D4×C53—81no—
130C2×D4×C53C2×D412yes—
131C2×D4×C53#4112yes—
132C22×C4×C53—21no—
133C22×C4×C53—21no—
134C22×C4×C53—22no—
135C22×C4×C53—21no—
136C22×C4×C53—21no—
137C22×C4×C53—21no—
138C22×C4×C53—21no—
139C22×C4×C53—21no—
140C22×C4×C53—42no—
141C22×C4×C53—41no—
142C22×C4×C53—42no—
143C22×C4×C53C22⋊C411yescharacteristic
144C22×C4×C53C2×D411yescharacteristic
145M16×C53—24no—
146M16×C53—44no—
147M16×C53—42no—
148C2×C8×C53—44no—
149C2×C8×C53—42no—
150C22⋊C4×C53—22no—
151C22⋊C4×C53—42no—
152C22⋊C4×C53—42no—
153C22⋊C4×C53—42no—
154C22⋊C4×C53—42no—
155C22⋊C4×C53—41no—
156C22⋊C4×C53—41no—
157C22⋊C4×C53—41no—
158C22⋊C4×C53—42no—
159C22⋊C4×C53—42no—
160C22⋊C4×C53—81no—
161C22⋊C4×C53—82no—
162C22⋊C4×C53—82no—
163C23×C53—22no—
164C23×C53—21no—
165C23×C53—21no—
166C23×C53—21no—
167C23×C53—21no—
168C23×C53—21no—
169C23×C53—21no—
170C23×C53—22no—
171C23×C53—21no—
172C23×C53—21no—
173C23×C53—21no—
174C23×C53—21no—
175C23×C53—21no—
176C23×C53—21no—
177C23×C53—41no—
178C23×C53—42no—
179C23×C53—42no—
180C23×C53—42no—
181C23×C53—42no—
182C23×C53—41no—
183C23×C53—41no—
184C23×C53—41no—
185C23×C53—41no—
186C23×C53—41no—
187C23×C53—41no—
188C23×C53—41no—
189C23×C53—42no—
190C23×C53—41no—
191C23×C53—41no—
192C23×C53—42no—
193C23×C53—41no—
194C23×C53—41no—
195C23×C53—42no—
196C23×C53—42no—
197C23×C53—42no—
198C23×C53—42no—
199C23×C53—41no—
200C23×C53—41no—
201C23×C53—41no—
202C23×C53—81no—
203C23×C53—81no—
204C23×C53—81no—
205C23×C53—81no—
206C23×C53—82no—
207C23×C53—84no—
208C23×C53—84no—
209C23×C53—84no—
210C23×C53—84no—
211C23×C53—84no—
212C23×C53—84no—
213C23×C53—82no—
214C23×C53#9911yescharacteristic
215C23×C53#9911yescharacteristic
216C23×C53C2×C22⋊C411yescharacteristic
217C23×C53C4×D412yes—
218C23×C53#12112yes—
219D4×C53—42no—
220D4×C53—42no—
221D4×C53—44no—
222D4×C53—44no—
223D4×C53—42no—
224D4×C53—42no—
225D4×C53—44no—
226D4×C53—48no—
227D4×C53—44no—
228D4×C53—84no—
229D4×C53—84no—
230C2×C4×C53—22no—
231C2×C4×C53—22no—
232C2×C4×C53—22no—
233C2×C4×C53—22no—
234C2×C4×C53—21no—
235C2×C4×C53—21no—
236C2×C4×C53—21no—
237C2×C4×C53—21no—
238C2×C4×C53—22no—
239C2×C4×C53—41no—
240C2×C4×C53—42no—
241C2×C4×C53—41no—
242C2×C4×C53—42no—
243C2×C4×C53—41no—
244C2×C4×C53—41no—
245C2×C4×C53—41no—
246C2×C4×C53—41no—
247C2×C4×C53—41no—
248C2×C4×C53—41no—
249C2×C4×C53—82no—
250C2×C4×C53—82no—
251C2×C4×C53—82no—
252C2×C4×C53#9912yes—
253C2×C4×C53#12011yescharacteristic
254C2×C4×C53#12311yescharacteristic
255C8×C53—44no—
256#13082C5311yesmaximal, 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—22no—
265C22×C53—22no—
266C22×C53—42no—
267C22×C53—42no—
268C22×C53—42no—
269C22×C53—42no—
270C22×C53—42no—
271C22×C53—42no—
272C22×C53—42no—
273C22×C53—42no—
274C22×C53—41no—
275C22×C53—41no—
276C22×C53—41no—
277C22×C53—41no—
278C22×C53—42no—
279C22×C53—41no—
280C22×C53—41no—
281C22×C53—41no—
282C22×C53—41no—
283C22×C53—41no—
284C22×C53—41no—
285C22×C53—81no—
286C22×C53—84no—
287C22×C53—84no—
288C22×C53—81no—
289C22×C53—81no—
290C22×C53—81no—
291C22×C53—81no—
292C22×C53—81no—
293C22×C53—84no—
294C22×C53—84no—
295C22×C53—84no—
296C22×C53—84no—
297C22×C53—81no—
298C22×C53—81no—
299C22×C53AΣL(1,𝔽16)12yescomplemented
300C22×C53#38611yescharacteristic
301C22×C53C2×#9911yescharacteristic
302C22×C53C2×#9911yescharacteristic
303C4×C53—21no—
304C4×C53—22no—
305C4×C53—21no—
306C4×C53—42no—
307C4×C53—81no—
308C4×C53—81no—
309C4×C53—162no—
310C22×#100C2×C5311yescharacteristic, complemented
311#2403C2×C5311yescharacteristic
312C2×AΣL(1,𝔽16)C2×C5312yescomplemented
313#1781C2×C5311yescharacteristic, complemented
314#1390C2×C5312yescomplemented
315C2×C53—21no—
316C2×C53—21no—
317C2×C53—22no—
318C2×C53—42no—
319C2×C53—41no—
320C2×C53—42no—
321C2×C53—81no—
322C2×C53—82no—
323C2×C53—81no—
324C2×C53#189811yescharacteristic
325C2×C53C2×AΣL(1,𝔽16)11yescharacteristic
326C2×C53#212111yescharacteristic
327C23×D4C4×C5311yescharacteristic, complemented
328C23×D4C22×C5311yescharacteristic
329C22×M16—21no—
330C2×#120—21no—
331C2×#120C4×C5311yescharacteristic, complemented
332C2×#120C22×C5311yescharacteristic
333C2×#100—21no—
334C2×#100—24no—
335C2×#100C22×C5312yes—
336C2×#99—21no—
337C2×#99C22×C5312yes—
338#392—21no—
339#392—21no—
340#386—21no—
341#386C4×C5311yescharacteristic, complemented
342#386C22×C5311yescharacteristic
343#379—21no—
344#379C4×C5311yescharacteristic, complemented
345AΣL(1,𝔽16)—24no—
346C53#1308211yescharacteristic, direct product factor, Sylow
347C25C2×C4×C5311yescharacteristic
348C25C2×C4×C5311yescharacteristic
349C25D4×C5311yescharacteristic
350C22×D4—21no—
351C22×D4—22no—
352C22×D4—22no—
353C22×D4—21no—
354C22×D4—21no—
355C22×D4—21no—
356C22×D4—42no—
357C22×D4—41no—
358C22×D4—41no—
359C22×D4—44no—
360C22×D4—41no—
361C22×D4—41no—
362C22×D4C2×C4×C5311yescharacteristic
363C22×D4D4×C5311yescharacteristic
364C22×D4D4×C5311yescharacteristic
365C22×D4C23×C5311yescharacteristic
366C23×C4C2×C4×C5311yescharacteristic
367C23×C4D4×C5311yescharacteristic
368C2×M16—24no—
369C2×M16—22no—
370C2×M16—41no—
371C2×M16—42no—
372C22×C8—41no—
373#120—22no—
374#120—22no—
375#120—44no—
376#120—42no—
377C2×C22⋊C4—21no—
378C2×C22⋊C4—21no—
379C2×C22⋊C4—21no—
380C2×C22⋊C4—21no—
381C2×C22⋊C4—21no—
382C2×C22⋊C4—22no—
383C2×C22⋊C4—21no—
384C2×C22⋊C4—41no—
385C2×C22⋊C4—41no—
386C2×C22⋊C4C2×C4×C5311yescharacteristic
387C2×C22⋊C4C2×C4×C5311yescharacteristic
388#100—24no—
389#100—22no—
390#100D4×C5314yescomplemented
391#99—22no—
392#99—22no—
393#99—42no—
394#95—21no—
395#95—21no—
396C24—21no—
397C24—21no—
398C24—21no—
399C24—21no—
400C24—21no—
401C24—22no—
402C24—22no—
403C24—22no—
404C24—41no—
405C24—41no—
406C24—42no—
407C24—41no—
408C24—42no—
409C24—42no—
410C24—41no—
411C24—81no—
412C24—81no—
413C24—81no—
414C24C22⋊C4×C5311yescharacteristic
415C24C22⋊C4×C5311yescharacteristic
416C24C22⋊C4×C5311yescharacteristic
417C24C22×C4×C5311yescharacteristic
418C24C2×D4×C5311yescharacteristic
419C2×D4—24no—
420C2×D4—22no—
421C2×D4—22no—
422C2×D4—22no—
423C2×D4—22no—
424C2×D4—21no—
425C2×D4—21no—
426C2×D4—41no—
427C2×D4—41no—
428C2×D4—41no—
429C2×D4—41no—
430C2×D4—41no—
431C2×D4—41no—
432C2×D4—44no—
433C2×D4—44no—
434C2×D4—44no—
435C2×D4—44no—
436C2×D4—42no—
437C2×D4—42no—
438C2×D4—42no—
439C2×D4—42no—
440C2×D4—44no—
441C2×D4—44no—
442C2×D4—44no—
443C2×D4—41no—
444C2×D4—41no—
445C2×D4—44no—
446C2×D4—42no—
447C2×D4—42no—
448C2×D4—42no—
449C2×D4—42no—
450C2×D4—81no—
451C2×D4—81no—
452C2×D4C2×D4×C5312yes—
453C2×D4#41×C5312yes—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—22no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—42no—
463C22×C4—41no—
464C22×C4—42no—
465C22×C4C22⋊C4×C5311yescharacteristic
466C22×C4C2×D4×C5311yescharacteristic
467M16—24no—
468M16—44no—
469M16—42no—
470C2×C8—44no—
471C2×C8—42no—
472C22⋊C4—22no—
473C22⋊C4—42no—
474C22⋊C4—42no—
475C22⋊C4—42no—
476C22⋊C4—42no—
477C22⋊C4—41no—
478C22⋊C4—41no—
479C22⋊C4—41no—
480C22⋊C4—42no—
481C22⋊C4—42no—
482C22⋊C4—81no—
483C22⋊C4—82no—
484C22⋊C4—82no—
485C23—22no—
486C23—21no—
487C23—21no—
488C23—21no—
489C23—21no—
490C23—21no—
491C23—21no—
492C23—22no—
493C23—21no—
494C23—21no—
495C23—21no—
496C23—21no—
497C23—21no—
498C23—21no—
499C23—41no—
500C23—42no—
501C23—42no—
502C23—42no—
503C23—42no—
504C23—41no—
505C23—41no—
506C23—41no—
507C23—41no—
508C23—41no—
509C23—41no—
510C23—42no—
511C23—41no—
512C23—42no—
513C23—41no—
514C23—41no—
515C23—41no—
516C23—41no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23—41no—
522C23—41no—
523C23—41no—
524C23—81no—
525C23—81no—
526C23—81no—
527C23—81no—
528C23—82no—
529C23—84no—
530C23—84no—
531C23—84no—
532C23—84no—
533C23—84no—
534C23—84no—
535C23—82no—
536C23#99×C5311yescharacteristic
537C23#99×C5311yescharacteristic
538C23C2×C22⋊C4×C5311yescharacteristic
539C23C4×D4×C5312yes—
540C23#121×C5312yes—
541D4—42no—
542D4—42no—
543D4—44no—
544D4—44no—
545D4—42no—
546D4—42no—
547D4—44no—
548D4—48no—
549D4—44no—
550D4—84no—
551D4—84no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—22no—
561C2×C4—41no—
562C2×C4—42no—
563C2×C4—41no—
564C2×C4—42no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—82no—
572C2×C4—82no—
573C2×C4—82no—
574C2×C4#99×C5312yes—
575C2×C4#120×C5311yescharacteristic
576C2×C4#123×C5311yescharacteristic
577C8—44no—
578C22—21no—
579C22—21no—
580C22—21no—
581C22—21no—
582C22—21no—
583C22—21no—
584C22—21no—
585C22—22no—
586C22—22no—
587C22—42no—
588C22—42no—
589C22—42no—
590C22—42no—
591C22—42no—
592C22—42no—
593C22—42no—
594C22—42no—
595C22—41no—
596C22—41no—
597C22—41no—
598C22—41no—
599C22—42no—
600C22—41no—
601C22—41no—
602C22—41no—
603C22—41no—
604C22—41no—
605C22—41no—
606C22—81no—
607C22—84no—
608C22—84no—
609C22—81no—
610C22—81no—
611C22—81no—
612C22—81no—
613C22—81no—
614C22—84no—
615C22—84no—
616C22—84no—
617C22—84no—
618C22—81no—
619C22—81no—
620C22C53×AΣL(1,𝔽16)12yescomplemented
621C22C53×#38611yescharacteristic
622C22C2×#99×C5311yescharacteristic
623C22C2×#99×C5311yescharacteristic
624C4—21no—
625C4—22no—
626C4—21no—
627C4—42no—
628C4—81no—
629C4—81no—
630C4—162no—
631C2—21no—
632C2—21no—
633C2—22no—
634C2—42no—
635C2—41no—
636C2—42no—
637C2—81no—
638C2—82no—
639C2—81no—
640C2C53×#189811yescharacteristic
641C2C2×C53×AΣL(1,𝔽16)11yescharacteristic
642C2C53×#212111yescharacteristic
subgroupG/H
center, socleC22×C53C2×#99
commutatorC24C22×C4×C53
Frattini, cosocleC22×D4C23×C53
prime radicalC53×#2403C2
LCSG ▷ C24 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C53 ◁ C24×C53 ◁ C23×D4×C53 ◁ G

Complements

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

NH ≅ G/Ntype
1. C22×#100×C53C2semidirect
3. C2×C53×AΣL(1,𝔽16)C2semidirect
4. C53×#1781C2semidirect
5. C53×#1390C2semidirect
6. C23×D4×C53C4semidirect
10. C2×#120×C53C4semidirect
20. C53×#386C4semidirect
23. C53×#379C4semidirect
68. #100×C53D4semidirect
256. #13082C53direct (N × H)
299. C22×C53AΣL(1,𝔽16)semidirect
310. C22×#100C2×C53semidirect
312. C2×AΣL(1,𝔽16)C2×C53semidirect
313. #1781C2×C53semidirect
314. #1390C2×C53semidirect
327. C23×D4C4×C53semidirect
331. C2×#120C4×C53semidirect
341. #386C4×C53semidirect
344. #379C4×C53semidirect
346. C53#13082direct (N × H)
390. #100D4×C53semidirect
620. C22C53×AΣL(1,𝔽16)semidirect

Maximal subgroups

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

subgroupindex
C22×#100×C532
C53×#24032
C2×C53×AΣL(1,𝔽16)2
C53×#17812
C53×#13902
#1308253