Finite Group Databasebeta

← C53×#9914 #27217426

Subgroups — C53×#9914 #27217426

Order 13568.

1356 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C53×#447C211yesmaximal, characteristic, complemented
2C53×#2617C211yesmaximal, characteristic, complemented
3C2×C53×#392C211yesmaximal, characteristic, complemented
4C53×#1786C211yesmaximal, characteristic, complemented
5C2×C53×#327C211yesmaximal, characteristic, complemented
6C2×C53×#323C211yesmaximal, characteristic, complemented
7C53×#1294C211yesmaximal, characteristic, complemented
8C23×D4×C53—21no—
9C23×D4×C53C2211yescharacteristic
10C22×D8×C53—21no—
11C2×#121×C53C2211yescharacteristic
12C2×#120×C53—21no—
13C22×C22⋊C4×C53—21no—
14C22×C22⋊C4×C53C2211yescharacteristic
15C53×#447—22no—
16C53×#447—22no—
17C2×#102×C53—21no—
18C2×#102×C53—22no—
19C2×C22⋊C8×C53C2211yescharacteristic
20C2×C22⋊C8×C53C2211yescharacteristic
21C53×#397—21no—
22C53×#394—22no—
23C53×#392—21no—
24C53×#392—22no—
25C53×#392—22no—
26C53×#386—21no—
27C53×#380C2211yescharacteristic
28C2×#95×C53C2211yescharacteristic
29C53×#327—22no—
30C53×#323—22no—
31C25×C53—21no—
32C25×C53D411yescharacteristic
33C22×D4×C53—24no—
34C22×D4×C53—21no—
35C22×D4×C53—22no—
36C22×D4×C53—21no—
37C22×D4×C53—21no—
38C22×D4×C53—21no—
39C22×D4×C53—21no—
40C22×D4×C53—21no—
41C22×D4×C53—44no—
42C22×D4×C53—44no—
43C22×D4×C53—44no—
44C22×D4×C53—42no—
45C22×D4×C53—42no—
46C22×D4×C53D411yescharacteristic, complemented
47C22×D4×C53D412yescomplemented
48C23×C4×C53—21no—
49C23×C4×C53D411yescharacteristic
50C23×C4×C53C2311yescharacteristic
51C2×D8×C53—24no—
52C2×D8×C53—44no—
53C22×C8×C53—21no—
54C22×C8×C53—21no—
55#121×C53—21no—
56#121×C53—21no—
57#121×C53—21no—
58#121×C53—21no—
59#120×C53—42no—
60#120×C53—42no—
61C2×C4⋊C4×C53D411yescharacteristic, complemented
62C2×C4⋊C4×C53D412yescomplemented
63C2×C22⋊C4×C53—24no—
64C2×C22⋊C4×C53—22no—
65C2×C22⋊C4×C53—22no—
66C2×C22⋊C4×C53—21no—
67C2×C22⋊C4×C53—21no—
68C2×C22⋊C4×C53—21no—
69C2×C22⋊C4×C53—21no—
70C2×C22⋊C4×C53—21no—
71C2×C22⋊C4×C53—21no—
72C2×C22⋊C4×C53—21no—
73C2×C22⋊C4×C53—21no—
74C2×C22⋊C4×C53—44no—
75#102×C53—24no—
76#102×C53—44no—
77C22⋊C8×C53—21no—
78C22⋊C8×C53—21no—
79C22⋊C8×C53—21no—
80C22⋊C8×C53D412yescomplemented
81#95×C53—22no—
82#95×C53—21no—
83#95×C53—21no—
84#95×C53D412yescomplemented
85C24×C53—24no—
86C24×C53—21no—
87C24×C53—22no—
88C24×C53—21no—
89C24×C53—21no—
90C24×C53—21no—
91C24×C53—21no—
92C24×C53—21no—
93C24×C53—21no—
94C24×C53—42no—
95C24×C53—42no—
96C24×C53—42no—
97C24×C53—42no—
98C24×C53—42no—
99C24×C53—41no—
100C24×C53—41no—
101C24×C53—41no—
102C24×C53—41no—
103C24×C53—41no—
104C24×C53—41no—
105C24×C53D812yescomplemented
106C24×C53C2×D411yescharacteristic
107C2×D4×C53—24no—
108C2×D4×C53—24no—
109C2×D4×C53—22no—
110C2×D4×C53—21no—
111C2×D4×C53—22no—
112C2×D4×C53—21no—
113C2×D4×C53—42no—
114C2×D4×C53—42no—
115C2×D4×C53—48no—
116C2×D4×C53—42no—
117C2×D4×C53—44no—
118C2×D4×C53—44no—
119C2×D4×C53—44no—
120C2×D4×C53—44no—
121C2×D4×C53—48no—
122C2×D4×C53—48no—
123C2×D4×C53—416no—
124C2×D4×C53—416no—
125C2×D4×C53—44no—
126C2×D4×C53—41no—
127C2×D4×C53—41no—
128C2×D4×C53—41no—
129C2×D4×C53—41no—
130C2×D4×C53—41no—
131C2×D4×C53—41no—
132C2×D4×C53—41no—
133C2×D4×C53—41no—
134C2×D4×C53—82no—
135C2×D4×C53—82no—
136C22×C4×C53—24no—
137C22×C4×C53—22no—
138C22×C4×C53—21no—
139C22×C4×C53—21no—
140C22×C4×C53—21no—
141C22×C4×C53—22no—
142C22×C4×C53—21no—
143C22×C4×C53—42no—
144C22×C4×C53—42no—
145C22×C4×C53—41no—
146C22×C4×C53—41no—
147C22×C4×C53—42no—
148C22×C4×C53C2×D411yescharacteristic
149C22×C4×C53C2×D411yescharacteristic
150C22×C4×C53C2×D411yescharacteristic
151C22×C4×C53C2×D411yescharacteristic
152C22×C4×C53C2×D411yescharacteristic
153C22×C4×C53#4111yescharacteristic
154D8×C53—48no—
155C2×C8×C53—22no—
156C2×C8×C53—42no—
157C2×C8×C53—42no—
158C2×C8×C53—41no—
159C4⋊C4×C53—21no—
160C4⋊C4×C53—21no—
161C4⋊C4×C53—42no—
162C22⋊C4×C53—48no—
163C22⋊C4×C53—44no—
164C22⋊C4×C53—41no—
165C22⋊C4×C53—41no—
166C22⋊C4×C53—41no—
167C22⋊C4×C53—41no—
168C22⋊C4×C53—41no—
169C22⋊C4×C53—41no—
170C22⋊C4×C53—41no—
171C22⋊C4×C53—41no—
172C22⋊C4×C53—82no—
173C23×C53—22no—
174C23×C53—22no—
175C23×C53—24no—
176C23×C53—22no—
177C23×C53—22no—
178C23×C53—22no—
179C23×C53—44no—
180C23×C53—44no—
181C23×C53—44no—
182C23×C53—44no—
183C23×C53—42no—
184C23×C53—42no—
185C23×C53—42no—
186C23×C53—42no—
187C23×C53—44no—
188C23×C53—44no—
189C23×C53—44no—
190C23×C53—42no—
191C23×C53—42no—
192C23×C53—41no—
193C23×C53—42no—
194C23×C53—41no—
195C23×C53—41no—
196C23×C53—42no—
197C23×C53—42no—
198C23×C53—44no—
199C23×C53—42no—
200C23×C53—41no—
201C23×C53—41no—
202C23×C53—41no—
203C23×C53—41no—
204C23×C53—42no—
205C23×C53—42no—
206C23×C53—82no—
207C23×C53—82no—
208C23×C53—81no—
209C23×C53—81no—
210C23×C53—82no—
211C23×C53—82no—
212C23×C53—82no—
213C23×C53—82no—
214C23×C53—82no—
215C23×C53—81no—
216C23×C53—81no—
217C23×C53—82no—
218C23×C53—81no—
219C23×C53—81no—
220C23×C53#12011yescharacteristic
221C23×C53#12011yescharacteristic
222C23×C53#12111yescharacteristic
223C23×C53C2×D811yescharacteristic
224C23×C53#13511yescharacteristic
225C23×C53AGL(1,ℤ/8)11yescharacteristic
226C23×C53AGL(1,ℤ/8)11yescharacteristic
227D4×C53—24no—
228D4×C53—24no—
229D4×C53—42no—
230D4×C53—44no—
231D4×C53—42no—
232D4×C53—44no—
233D4×C53—42no—
234D4×C53—432no—
235D4×C53—81no—
236D4×C53—88no—
237D4×C53—81no—
238D4×C53—81no—
239D4×C53—81no—
240D4×C53—81no—
241C2×C4×C53—22no—
242C2×C4×C53—22no—
243C2×C4×C53—24no—
244C2×C4×C53—41no—
245C2×C4×C53—41no—
246C2×C4×C53—41no—
247C2×C4×C53—41no—
248C2×C4×C53—44no—
249C2×C4×C53—44no—
250C2×C4×C53—42no—
251C2×C4×C53—42no—
252C2×C4×C53—42no—
253C2×C4×C53—41no—
254C2×C4×C53—41no—
255C2×C4×C53—42no—
256C2×C4×C53—41no—
257C2×C4×C53—41no—
258C2×C4×C53—42no—
259C2×C4×C53—42no—
260C2×C4×C53—42no—
261C2×C4×C53—81no—
262C2×C4×C53#12011yescharacteristic
263C2×C4×C53#12111yescharacteristic
264C2×C4×C53#12111yescharacteristic
265C2×C4×C53#12711yescharacteristic
266C8×C53—42no—
267C8×C53—81no—
268#9914C5311yesmaximal, characteristic, direct product factor, Sylow
269C22×C53—22no—
270C22×C53—22no—
271C22×C53—22no—
272C22×C53—22no—
273C22×C53—21no—
274C22×C53—21no—
275C22×C53—21no—
276C22×C53—21no—
277C22×C53—44no—
278C22×C53—44no—
279C22×C53—44no—
280C22×C53—44no—
281C22×C53—42no—
282C22×C53—44no—
283C22×C53—82no—
284C22×C53—82no—
285C22×C53—82no—
286C22×C53—82no—
287C22×C53—84no—
288C22×C53—82no—
289C22×C53—84no—
290C22×C53—84no—
291C22×C53—82no—
292C22×C53—84no—
293C22×C53—82no—
294C22×C53—81no—
295C22×C53—82no—
296C22×C53—81no—
297C22×C53—82no—
298C22×C53—82no—
299C22×C53—81no—
300C22×C53—81no—
301C22×C53—81no—
302C22×C53#39211yescharacteristic
303C22×C53#44711yescharacteristic
304C22×C53#44711yescharacteristic
305C22×C53#44911yescharacteristic
306C22×C53#44911yescharacteristic
307C22×C53#45312yes—
308C22×C53B(4,𝔽2)12yes—
309C22×C53#46611yescharacteristic
310C22×C53#46911yescharacteristic
311C4×C53—21no—
312C4×C53—21no—
313C4×C53—21no—
314C4×C53—21no—
315C4×C53—44no—
316C4×C53—82no—
317C4×C53—81no—
318C4×C53—81no—
319C2×#447C2×C5311yescharacteristic, complemented
320#2617C2×C5311yescharacteristic, complemented
321C2×#392C2×C5311yescharacteristic, complemented
322#1786C2×C5311yescharacteristic, complemented
323C2×#327C2×C5311yescharacteristic, complemented
324C2×#323C2×C5311yescharacteristic, complemented
325#1294C2×C5311yescharacteristic, complemented
326C2×C53—21no—
327C2×C53—21no—
328C2×C53—21no—
329C2×C53—21no—
330C2×C53—82no—
331C2×C53—84no—
332C2×C53—161no—
333C2×C53#159512yes—
334C2×C53#160012yes—
335C2×C53#199911yescharacteristic
336C2×C53#200211yescharacteristic
337C2×C53#202111yescharacteristic
338C23×D4—21no—
339C23×D4C22×C5311yescharacteristic
340C22×D8—21no—
341C2×#121C22×C5311yescharacteristic
342C2×#120—21no—
343C22×C22⋊C4—21no—
344C22×C22⋊C4C22×C5311yescharacteristic
345#447—22no—
346#447—22no—
347C2×#102—21no—
348C2×#102—22no—
349C2×C22⋊C8C22×C5311yescharacteristic
350C2×C22⋊C8C22×C5311yescharacteristic
351#397—21no—
352#394—22no—
353#392—21no—
354#392—22no—
355#392—22no—
356#386—21no—
357#380C22×C5311yescharacteristic
358C2×#95C22×C5311yescharacteristic
359#327—22no—
360#323—22no—
361C53#991411yescharacteristic, direct product factor, Sylow
362C25—21no—
363C25D4×C5311yescharacteristic
364C22×D4—24no—
365C22×D4—21no—
366C22×D4—22no—
367C22×D4—21no—
368C22×D4—21no—
369C22×D4—21no—
370C22×D4—21no—
371C22×D4—21no—
372C22×D4—44no—
373C22×D4—44no—
374C22×D4—44no—
375C22×D4—42no—
376C22×D4—42no—
377C22×D4D4×C5311yescharacteristic, complemented
378C22×D4D4×C5312yescomplemented
379C23×C4—21no—
380C23×C4D4×C5311yescharacteristic
381C23×C4C23×C5311yescharacteristic
382C2×D8—24no—
383C2×D8—44no—
384C22×C8—21no—
385C22×C8—21no—
386#121—21no—
387#121—21no—
388#121—21no—
389#121—21no—
390#120—42no—
391#120—42no—
392C2×C4⋊C4D4×C5311yescharacteristic, complemented
393C2×C4⋊C4D4×C5312yescomplemented
394C2×C22⋊C4—24no—
395C2×C22⋊C4—22no—
396C2×C22⋊C4—21no—
397C2×C22⋊C4—21no—
398C2×C22⋊C4—21no—
399C2×C22⋊C4—21no—
400C2×C22⋊C4—21no—
401C2×C22⋊C4—21no—
402C2×C22⋊C4—21no—
403C2×C22⋊C4—21no—
404C2×C22⋊C4—22no—
405C2×C22⋊C4—44no—
406#102—24no—
407#102—44no—
408C22⋊C8—21no—
409C22⋊C8—21no—
410C22⋊C8—21no—
411C22⋊C8D4×C5312yescomplemented
412#95—22no—
413#95—21no—
414#95—21no—
415#95D4×C5312yescomplemented
416C24—24no—
417C24—21no—
418C24—21no—
419C24—22no—
420C24—21no—
421C24—21no—
422C24—21no—
423C24—21no—
424C24—21no—
425C24—42no—
426C24—42no—
427C24—42no—
428C24—42no—
429C24—42no—
430C24—41no—
431C24—41no—
432C24—41no—
433C24—41no—
434C24—41no—
435C24—41no—
436C24D8×C5312yescomplemented
437C24C2×D4×C5311yescharacteristic
438C2×D4—24no—
439C2×D4—24no—
440C2×D4—22no—
441C2×D4—21no—
442C2×D4—22no—
443C2×D4—21no—
444C2×D4—42no—
445C2×D4—42no—
446C2×D4—48no—
447C2×D4—42no—
448C2×D4—44no—
449C2×D4—44no—
450C2×D4—44no—
451C2×D4—44no—
452C2×D4—48no—
453C2×D4—48no—
454C2×D4—416no—
455C2×D4—416no—
456C2×D4—44no—
457C2×D4—41no—
458C2×D4—41no—
459C2×D4—41no—
460C2×D4—41no—
461C2×D4—41no—
462C2×D4—41no—
463C2×D4—41no—
464C2×D4—41no—
465C2×D4—82no—
466C2×D4—82no—
467C22×C4—24no—
468C22×C4—22no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—22no—
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—42no—
475C22×C4—42no—
476C22×C4—41no—
477C22×C4—41no—
478C22×C4—42no—
479C22×C4C2×D4×C5311yescharacteristic
480C22×C4C2×D4×C5311yescharacteristic
481C22×C4C2×D4×C5311yescharacteristic
482C22×C4C2×D4×C5311yescharacteristic
483C22×C4C2×D4×C5311yescharacteristic
484C22×C4#41×C5311yescharacteristic
485D8—48no—
486C2×C8—22no—
487C2×C8—42no—
488C2×C8—42no—
489C2×C8—41no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—42no—
493C22⋊C4—48no—
494C22⋊C4—44no—
495C22⋊C4—41no—
496C22⋊C4—41no—
497C22⋊C4—41no—
498C22⋊C4—41no—
499C22⋊C4—41no—
500C22⋊C4—41no—
501C22⋊C4—41no—
502C22⋊C4—41no—
503C22⋊C4—82no—
504C23—22no—
505C23—22no—
506C23—24no—
507C23—22no—
508C23—22no—
509C23—22no—
510C23—44no—
511C23—42no—
512C23—42no—
513C23—42no—
514C23—42no—
515C23—42no—
516C23—44no—
517C23—44no—
518C23—44no—
519C23—44no—
520C23—44no—
521C23—44no—
522C23—41no—
523C23—42no—
524C23—42no—
525C23—41no—
526C23—41no—
527C23—42no—
528C23—42no—
529C23—44no—
530C23—42no—
531C23—41no—
532C23—41no—
533C23—41no—
534C23—41no—
535C23—42no—
536C23—42no—
537C23—82no—
538C23—81no—
539C23—81no—
540C23—82no—
541C23—82no—
542C23—82no—
543C23—82no—
544C23—82no—
545C23—82no—
546C23—81no—
547C23—81no—
548C23—82no—
549C23—81no—
550C23—81no—
551C23#120×C5311yescharacteristic
552C23#120×C5311yescharacteristic
553C23#121×C5311yescharacteristic
554C23C2×D8×C5311yescharacteristic
555C23#135×C5311yescharacteristic
556C23AGL(1,ℤ/8)×C5311yescharacteristic
557C23AGL(1,ℤ/8)×C5311yescharacteristic
558D4—24no—
559D4—24no—
560D4—42no—
561D4—44no—
562D4—42no—
563D4—44no—
564D4—42no—
565D4—432no—
566D4—81no—
567D4—88no—
568D4—81no—
569D4—81no—
570D4—81no—
571D4—81no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—24no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—42no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—42no—
588C2×C4—41no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—81no—
593C2×C4#120×C5311yescharacteristic
594C2×C4#121×C5311yescharacteristic
595C2×C4#121×C5311yescharacteristic
596C2×C4#127×C5311yescharacteristic
597C8—42no—
598C8—81no—
599C22—21no—
600C22—21no—
601C22—21no—
602C22—21no—
603C22—22no—
604C22—22no—
605C22—22no—
606C22—22no—
607C22—44no—
608C22—44no—
609C22—44no—
610C22—42no—
611C22—44no—
612C22—44no—
613C22—82no—
614C22—82no—
615C22—82no—
616C22—82no—
617C22—84no—
618C22—82no—
619C22—84no—
620C22—82no—
621C22—81no—
622C22—82no—
623C22—81no—
624C22—84no—
625C22—82no—
626C22—84no—
627C22—81no—
628C22—81no—
629C22—81no—
630C22—82no—
631C22—82no—
632C22C53×#39211yescharacteristic
633C22C53×#44711yescharacteristic
634C22C53×#44711yescharacteristic
635C22C53×#44911yescharacteristic
636C22C53×#44911yescharacteristic
637C22C53×#45312yes—
638C22C53×B(4,𝔽2)12yes—
639C22C53×#46611yescharacteristic
640C22C53×#46911yescharacteristic
641C4—21no—
642C4—21no—
643C4—21no—
644C4—21no—
645C4—44no—
646C4—81no—
647C4—81no—
648C4—82no—
649C2—21no—
650C2—21no—
651C2—21no—
652C2—21no—
653C2—82no—
654C2—84no—
655C2—161no—
656C2C53×#159512yes—
657C2C53×#160012yes—
658C2C53×#199911yescharacteristic
659C2C53×#200211yescharacteristic
660C2C53×#202111yescharacteristic
subgroupG/H
center, socleC23×C53#120
commutator, Frattini, cosocleC23×C4C23×C53
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
1. C2×C53×#447C2semidirect
2. C53×#2617C2semidirect
3. C2×C53×#392C2semidirect
4. C53×#1786C2semidirect
5. C2×C53×#327C2semidirect
6. C2×C53×#323C2semidirect
7. C53×#1294C2semidirect
46. C22×D4×C53D4semidirect
47. C22×D4×C53D4semidirect
61. C2×C4⋊C4×C53D4semidirect
62. C2×C4⋊C4×C53D4semidirect
80. C22⋊C8×C53D4semidirect
84. #95×C53D4semidirect
105. C24×C53D8semidirect
268. #9914C53direct (N × H)
319. C2×#447C2×C53semidirect
320. #2617C2×C53semidirect
321. C2×#392C2×C53semidirect
322. #1786C2×C53semidirect
323. C2×#327C2×C53semidirect
324. C2×#323C2×C53semidirect
325. #1294C2×C53semidirect
361. C53#9914direct (N × H)
377. C22×D4D4×C53semidirect
378. C22×D4D4×C53semidirect
392. C2×C4⋊C4D4×C53semidirect
393. C2×C4⋊C4D4×C53semidirect
411. C22⋊C8D4×C53semidirect
415. #95D4×C53semidirect
436. C24D8×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×#26172
C2×C53×#3922
C53×#17862
C2×C53×#3272
C2×C53×#3232
C53×#12942
#991453