Finite Group Databasebeta

← C53×#13877 #27221389

Subgroups — C53×#13877 #27221389

Order 13568.

4046 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C22⋊C4×C53C211yesmaximal, characteristic
2C23×C22⋊C4×C53C211yesmaximal, characteristic
3C2×C53×#385C212yesmaximal, complemented
4C2×C53×#380C212yesmaximal, complemented
5C2×C53×#379C211yesmaximal, characteristic
6C53×#1439C214yesmaximal, complemented
7C53×C22⋊#95C212yesmaximal, complemented
8C53×C22⋊#95C212yesmaximal, complemented
9C26×C53C2211yescharacteristic
10C24×C4×C53C2212yes—
11C24×C4×C53C2212yes—
12C22×C4⋊C4×C53C2214yescomplemented
13C22×C22⋊C4×C53—24no—
14C22×C22⋊C4×C53—28no—
15C22×C22⋊C4×C53—28no—
16C22×C22⋊C4×C53C418yescomplemented
17C22×C22⋊C4×C53C2211yescharacteristic
18C22×C22⋊C4×C53C2212yes—
19C22×C22⋊C4×C53C2212yes—
20C22×C22⋊C4×C53C2211yescharacteristic
21C22×C22⋊C4×C53C2211yescharacteristic
22C22×C22⋊C4×C53C2212yes—
23C22×C22⋊C4×C53C2212yes—
24C22×C22⋊C4×C53C2212yes—
25C22×C22⋊C4×C53C2211yescharacteristic
26C53×#385—28no—
27C53×#380—28no—
28C53×#379—24no—
29C2×#95×C53C2214yescomplemented
30C2×#95×C53C2214yescomplemented
31C2×#95×C53C2214yescomplemented
32C25×C53—24no—
33C25×C53—24no—
34C25×C53—24no—
35C25×C53—24no—
36C25×C53—44no—
37C25×C53C2×C414yes—
38C25×C53D414yes—
39C25×C53Q814yes—
40C25×C53C2311yescharacteristic
41C25×C53C2311yescharacteristic
42C25×C53C2311yescharacteristic
43C23×C4×C53—24no—
44C23×C4×C53—24no—
45C23×C4×C53—24no—
46C23×C4×C53—28no—
47C23×C4×C53—28no—
48C23×C4×C53—24no—
49C23×C4×C53—24no—
50C23×C4×C53—44no—
51C23×C4×C53—44no—
52C23×C4×C53C2×C418yescomplemented
53C23×C4×C53C2312yes—
54C23×C4×C53C2312yes—
55C23×C4×C53C2312yes—
56C23×C4×C53C2314yes—
57C23×C4×C53C2312yes—
58C2×C4⋊C4×C53—28no—
59C2×C4⋊C4×C53—28no—
60C2×C4⋊C4×C53—44no—
61C2×C22⋊C4×C53—232no—
62C2×C22⋊C4×C53—232no—
63C2×C22⋊C4×C53—24no—
64C2×C22⋊C4×C53—24no—
65C2×C22⋊C4×C53—24no—
66C2×C22⋊C4×C53—24no—
67C2×C22⋊C4×C53—24no—
68C2×C22⋊C4×C53—22no—
69C2×C22⋊C4×C53—22no—
70C2×C22⋊C4×C53—24no—
71C2×C22⋊C4×C53—24no—
72C2×C22⋊C4×C53—24no—
73C2×C22⋊C4×C53—24no—
74C2×C22⋊C4×C53—24no—
75C2×C22⋊C4×C53—28no—
76C2×C22⋊C4×C53—28no—
77C2×C22⋊C4×C53—44no—
78C2×C22⋊C4×C53C2×C414yes—
79C2×C22⋊C4×C53C2×C414yes—
80C2×C22⋊C4×C53C2×C418yescomplemented
81C2×C22⋊C4×C53D418yes—
82C2×C22⋊C4×C53D418yes—
83#95×C53—28no—
84#95×C53—28no—
85#95×C53—44no—
86C24×C53—22no—
87C24×C53—22no—
88C24×C53—22no—
89C24×C53—22no—
90C24×C53—24no—
91C24×C53—24no—
92C24×C53—28no—
93C24×C53—28no—
94C24×C53—216no—
95C24×C53—28no—
96C24×C53—216no—
97C24×C53—216no—
98C24×C53—216no—
99C24×C53—28no—
100C24×C53—22no—
101C24×C53—22no—
102C24×C53—22no—
103C24×C53—22no—
104C24×C53—24no—
105C24×C53—48no—
106C24×C53—48no—
107C24×C53—48no—
108C24×C53—416no—
109C24×C53—416no—
110C24×C53—416no—
111C24×C53—416no—
112C24×C53C4⋊C4116yescomplemented
113C24×C53C22×C412yes—
114C24×C53C22×C412yes—
115C24×C53C22×C412yes—
116C24×C53C2×D412yes—
117C24×C53C2×D414yes—
118C24×C53C2×D412yes—
119C24×C53C2×D412yes—
120C24×C53C2×D414yes—
121C24×C53C2×Q812yes—
122C24×C53C2×Q812yes—
123C24×C53C2×Q812yes—
124C24×C53#4114yes—
125C24×C53#4114yes—
126C24×C53C2411yescharacteristic
127C22×C4×C53—22no—
128C22×C4×C53—22no—
129C22×C4×C53—24no—
130C22×C4×C53—216no—
131C22×C4×C53—216no—
132C22×C4×C53—22no—
133C22×C4×C53—24no—
134C22×C4×C53—24no—
135C22×C4×C53—24no—
136C22×C4×C53—24no—
137C22×C4×C53—22no—
138C22×C4×C53—24no—
139C22×C4×C53—24no—
140C22×C4×C53—24no—
141C22×C4×C53—24no—
142C22×C4×C53—216no—
143C22×C4×C53—22no—
144C22×C4×C53—24no—
145C22×C4×C53—22no—
146C22×C4×C53—22no—
147C22×C4×C53—22no—
148C22×C4×C53—22no—
149C22×C4×C53—22no—
150C22×C4×C53—24no—
151C22×C4×C53—24no—
152C22×C4×C53—24no—
153C22×C4×C53—48no—
154C22×C4×C53—416no—
155C22×C4×C53—416no—
156C22×C4×C53—416no—
157C22×C4×C53—416no—
158C22×C4×C53—48no—
159C22×C4×C53—48no—
160C22×C4×C53C22×C414yes—
161C22×C4×C53C22×C414yes—
162C22×C4×C53C2×D418yes—
163C22×C4×C53C2×D418yes—
164C4⋊C4×C53—416no—
165C22⋊C4×C53—216no—
166C22⋊C4×C53—28no—
167C22⋊C4×C53—216no—
168C22⋊C4×C53—216no—
169C22⋊C4×C53—48no—
170C22⋊C4×C53—44no—
171C22⋊C4×C53—48no—
172C22⋊C4×C53—44no—
173C22⋊C4×C53C22⋊C4116yescomplemented
174C23×C53—22no—
175C23×C53—22no—
176C23×C53—24no—
177C23×C53—24no—
178C23×C53—24no—
179C23×C53—22no—
180C23×C53—24no—
181C23×C53—24no—
182C23×C53—24no—
183C23×C53—22no—
184C23×C53—24no—
185C23×C53—24no—
186C23×C53—24no—
187C23×C53—24no—
188C23×C53—28no—
189C23×C53—28no—
190C23×C53—28no—
191C23×C53—28no—
192C23×C53—28no—
193C23×C53—28no—
194C23×C53—28no—
195C23×C53—28no—
196C23×C53—22no—
197C23×C53—22no—
198C23×C53—24no—
199C23×C53—24no—
200C23×C53—22no—
201C23×C53—24no—
202C23×C53—24no—
203C23×C53—24no—
204C23×C53—22no—
205C23×C53—24no—
206C23×C53—24no—
207C23×C53—232no—
208C23×C53—232no—
209C23×C53—416no—
210C23×C53—432no—
211C23×C53—432no—
212C23×C53—432no—
213C23×C53—432no—
214C23×C53—464no—
215C23×C53—44no—
216C23×C53—44no—
217C23×C53—44no—
218C23×C53—44no—
219C23×C53C2×C22⋊C414yes—
220C23×C53C2×C4⋊C414yes—
221C23×C53C2×C4⋊C414yes—
222C23×C53C2×C4⋊C414yes—
223C23×C53#11714yes—
224C23×C53C4×D418yes—
225C23×C53#12014yes—
226C23×C53#12118yes—
227C23×C53C22⋊Q818yes—
228C23×C53C22⋊Q818yes—
229C23×C53#12314yes—
230C23×C53C23×C411yescharacteristic
231C23×C53C22×D412yes—
232C23×C53C22×D411yescharacteristic
233C23×C53C22×D412yes—
234C23×C53C22×Q811yescharacteristic
235C23×C53C2×#4112yes—
236C23×C53C2×#4112yes—
237C23×C53H(5,𝔽2)14yes—
238C2×C4×C53—24no—
239C2×C4×C53—24no—
240C2×C4×C53—28no—
241C2×C4×C53—28no—
242C2×C4×C53—24no—
243C2×C4×C53—24no—
244C2×C4×C53—24no—
245C2×C4×C53—28no—
246C2×C4×C53—28no—
247C2×C4×C53—28no—
248C2×C4×C53—432no—
249C2×C4×C53—432no—
250C2×C4×C53—44no—
251C2×C4×C53—44no—
252C2×C4×C53—44no—
253C2×C4×C53—44no—
254C2×C4×C53—44no—
255C2×C4×C53—44no—
256C2×C4×C53—42no—
257C2×C4×C53—42no—
258C2×C4×C53—42no—
259C2×C4×C53—42no—
260C2×C4×C53—42no—
261C2×C4×C53—42no—
262C2×C4×C53—44no—
263C2×C4×C53—44no—
264C2×C4×C53—44no—
265C2×C4×C53—44no—
266C2×C4×C53C2×C22⋊C418yescomplemented
267#13877C5311yesmaximal, characteristic, direct product factor, Sylow
268C22×C53—24no—
269C22×C53—24no—
270C22×C53—28no—
271C22×C53—28no—
272C22×C53—28no—
273C22×C53—28no—
274C22×C53—28no—
275C22×C53—28no—
276C22×C53—28no—
277C22×C53—28no—
278C22×C53—216no—
279C22×C53—216no—
280C22×C53—24no—
281C22×C53—24no—
282C22×C53—24no—
283C22×C53—24no—
284C22×C53—24no—
285C22×C53—464no—
286C22×C53—44no—
287C22×C53—44no—
288C22×C53—44no—
289C22×C53—44no—
290C22×C53—48no—
291C22×C53#38018yescomplemented
292C22×C53#38518yescomplemented
293C22×C53C22×C22⋊C411yescharacteristic
294C22×C53C22×C4⋊C411yescharacteristic
295C22×C53C2×#11711yescharacteristic
296C22×C53C2×C4×D412yes—
297C22×C53#51812yes—
298C22×C53C2×#12011yescharacteristic
299C22×C53C2×#12112yes—
300C22×C53C2×C22⋊Q812yes—
301C22×C53C2×C22⋊Q812yes—
302C22×C53C2×#12311yescharacteristic
303C22×C53#53412yes—
304C22×C53#54312yes—
305C22×C53D4214yes—
306C22×C53#54614yes—
307C22×C53#54614yes—
308C22×C53#55114yes—
309C4×C53—28no—
310C4×C53—44no—
311C4×C53—48no—
312C4×C53—82no—
313C4×C53—82no—
314C23×C22⋊C4C2×C5311yescharacteristic
315C23×C22⋊C4C2×C5311yescharacteristic
316C2×#385C2×C5312yescomplemented
317C2×#380C2×C5312yescomplemented
318C2×#379C2×C5311yescharacteristic
319#1439C2×C5314yescomplemented
320C22⋊#95C2×C5312yescomplemented
321C22⋊#95C2×C5312yescomplemented
322C2×C53—28no—
323C2×C53—28no—
324C2×C53—44no—
325C2×C53C2×#38011yescharacteristic
326C2×C53C2×#38511yescharacteristic
327C2×C53#231511yescharacteristic
328C2×C53D4×C22⋊C412yes—
329C2×C53#234212yes—
330C2×C53#241012yes—
331C2×C53#241212yes—
332C2×C53#243714yes—
333C26C22×C5311yescharacteristic
334C24×C4C22×C5312yes—
335C24×C4C22×C5312yes—
336C22×C4⋊C4C22×C5314yescomplemented
337C22×C22⋊C4—24no—
338C22×C22⋊C4—28no—
339C22×C22⋊C4—28no—
340C22×C22⋊C4C4×C5318yescomplemented
341C22×C22⋊C4C22×C5312yes—
342C22×C22⋊C4C22×C5311yescharacteristic
343C22×C22⋊C4C22×C5312yes—
344C22×C22⋊C4C22×C5311yescharacteristic
345C22×C22⋊C4C22×C5311yescharacteristic
346C22×C22⋊C4C22×C5312yes—
347C22×C22⋊C4C22×C5312yes—
348C22×C22⋊C4C22×C5312yes—
349C22×C22⋊C4C22×C5311yescharacteristic
350#385—28no—
351#380—28no—
352#379—24no—
353C2×#95C22×C5314yescomplemented
354C2×#95C22×C5314yescomplemented
355C2×#95C22×C5314yescomplemented
356C53#1387711yescharacteristic, direct product factor, Sylow
357C25—24no—
358C25—24no—
359C25—24no—
360C25—24no—
361C25—44no—
362C25C2×C4×C5314yes—
363C25D4×C5314yes—
364C25Q8×C5314yes—
365C25C23×C5311yescharacteristic
366C25C23×C5311yescharacteristic
367C25C23×C5311yescharacteristic
368C23×C4—24no—
369C23×C4—24no—
370C23×C4—24no—
371C23×C4—28no—
372C23×C4—28no—
373C23×C4—24no—
374C23×C4—24no—
375C23×C4—44no—
376C23×C4—44no—
377C23×C4C2×C4×C5318yescomplemented
378C23×C4C23×C5312yes—
379C23×C4C23×C5312yes—
380C23×C4C23×C5312yes—
381C23×C4C23×C5314yes—
382C23×C4C23×C5312yes—
383C2×C4⋊C4—28no—
384C2×C4⋊C4—28no—
385C2×C4⋊C4—44no—
386C2×C22⋊C4—232no—
387C2×C22⋊C4—24no—
388C2×C22⋊C4—24no—
389C2×C22⋊C4—24no—
390C2×C22⋊C4—24no—
391C2×C22⋊C4—28no—
392C2×C22⋊C4—28no—
393C2×C22⋊C4—232no—
394C2×C22⋊C4—24no—
395C2×C22⋊C4—22no—
396C2×C22⋊C4—24no—
397C2×C22⋊C4—22no—
398C2×C22⋊C4—24no—
399C2×C22⋊C4—24no—
400C2×C22⋊C4—24no—
401C2×C22⋊C4—24no—
402C2×C22⋊C4—44no—
403C2×C22⋊C4C2×C4×C5314yes—
404C2×C22⋊C4C2×C4×C5314yes—
405C2×C22⋊C4C2×C4×C5318yescomplemented
406C2×C22⋊C4D4×C5318yes—
407C2×C22⋊C4D4×C5318yes—
408#95—28no—
409#95—28no—
410#95—44no—
411C24—22no—
412C24—22no—
413C24—22no—
414C24—22no—
415C24—24no—
416C24—24no—
417C24—28no—
418C24—28no—
419C24—216no—
420C24—28no—
421C24—216no—
422C24—216no—
423C24—216no—
424C24—28no—
425C24—22no—
426C24—22no—
427C24—22no—
428C24—22no—
429C24—24no—
430C24—48no—
431C24—48no—
432C24—48no—
433C24—416no—
434C24—416no—
435C24—416no—
436C24—416no—
437C24C4⋊C4×C53116yescomplemented
438C24C22×C4×C5312yes—
439C24C22×C4×C5312yes—
440C24C22×C4×C5312yes—
441C24C2×D4×C5312yes—
442C24C2×D4×C5314yes—
443C24C2×D4×C5312yes—
444C24C2×D4×C5312yes—
445C24C2×D4×C5314yes—
446C24C2×Q8×C5312yes—
447C24C2×Q8×C5312yes—
448C24C2×Q8×C5312yes—
449C24#41×C5314yes—
450C24#41×C5314yes—
451C24C24×C5311yescharacteristic
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—24no—
455C22×C4—216no—
456C22×C4—216no—
457C22×C4—22no—
458C22×C4—24no—
459C22×C4—24no—
460C22×C4—24no—
461C22×C4—24no—
462C22×C4—22no—
463C22×C4—24no—
464C22×C4—24no—
465C22×C4—24no—
466C22×C4—24no—
467C22×C4—216no—
468C22×C4—22no—
469C22×C4—24no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—24no—
476C22×C4—24no—
477C22×C4—24no—
478C22×C4—48no—
479C22×C4—416no—
480C22×C4—416no—
481C22×C4—416no—
482C22×C4—416no—
483C22×C4—48no—
484C22×C4—48no—
485C22×C4C22×C4×C5314yes—
486C22×C4C22×C4×C5314yes—
487C22×C4C2×D4×C5318yes—
488C22×C4C2×D4×C5318yes—
489C4⋊C4—416no—
490C22⋊C4—216no—
491C22⋊C4—28no—
492C22⋊C4—216no—
493C22⋊C4—216no—
494C22⋊C4—48no—
495C22⋊C4—44no—
496C22⋊C4—48no—
497C22⋊C4—44no—
498C22⋊C4C22⋊C4×C53116yescomplemented
499C23—22no—
500C23—22no—
501C23—24no—
502C23—24no—
503C23—24no—
504C23—22no—
505C23—24no—
506C23—24no—
507C23—24no—
508C23—22no—
509C23—24no—
510C23—24no—
511C23—24no—
512C23—24no—
513C23—28no—
514C23—28no—
515C23—28no—
516C23—28no—
517C23—28no—
518C23—28no—
519C23—28no—
520C23—28no—
521C23—22no—
522C23—22no—
523C23—24no—
524C23—24no—
525C23—22no—
526C23—24no—
527C23—24no—
528C23—24no—
529C23—22no—
530C23—24no—
531C23—24no—
532C23—232no—
533C23—232no—
534C23—416no—
535C23—464no—
536C23—432no—
537C23—432no—
538C23—432no—
539C23—432no—
540C23—44no—
541C23—44no—
542C23—44no—
543C23—44no—
544C23C2×C22⋊C4×C5314yes—
545C23C2×C4⋊C4×C5314yes—
546C23C2×C4⋊C4×C5314yes—
547C23C2×C4⋊C4×C5314yes—
548C23#117×C5314yes—
549C23C4×D4×C5318yes—
550C23#120×C5314yes—
551C23#121×C5318yes—
552C23C22⋊Q8×C5318yes—
553C23C22⋊Q8×C5318yes—
554C23#123×C5314yes—
555C23C23×C4×C5311yescharacteristic
556C23C22×D4×C5312yes—
557C23C22×D4×C5311yescharacteristic
558C23C22×D4×C5312yes—
559C23C22×Q8×C5311yescharacteristic
560C23C2×#41×C5312yes—
561C23C2×#41×C5312yes—
562C23H(5,𝔽2)×C5314yes—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—28no—
566C2×C4—28no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—28no—
571C2×C4—28no—
572C2×C4—28no—
573C2×C4—432no—
574C2×C4—432no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4C2×C22⋊C4×C5318yescomplemented
592C22—216no—
593C22—216no—
594C22—24no—
595C22—24no—
596C22—24no—
597C22—24no—
598C22—28no—
599C22—28no—
600C22—28no—
601C22—28no—
602C22—28no—
603C22—28no—
604C22—28no—
605C22—28no—
606C22—24no—
607C22—24no—
608C22—24no—
609C22—464no—
610C22—48no—
611C22—44no—
612C22—44no—
613C22—44no—
614C22—44no—
615C22C53×#38018yescomplemented
616C22C53×#38518yescomplemented
617C22C22×C22⋊C4×C5311yescharacteristic
618C22C22×C4⋊C4×C5311yescharacteristic
619C22C2×#117×C5311yescharacteristic
620C22C2×C4×D4×C5312yes—
621C22C53×#51812yes—
622C22C2×#120×C5311yescharacteristic
623C22C2×#121×C5312yes—
624C22C2×C22⋊Q8×C5312yes—
625C22C2×C22⋊Q8×C5312yes—
626C22C2×#123×C5311yescharacteristic
627C22C53×#53412yes—
628C22C53×#54312yes—
629C22D42×C5314yes—
630C22C53×#54614yes—
631C22C53×#54614yes—
632C22C53×#55114yes—
633C4—28no—
634C4—44no—
635C4—48no—
636C4—82no—
637C4—82no—
638C2—28no—
639C2—28no—
640C2—44no—
641C2C2×C53×#38011yescharacteristic
642C2C2×C53×#38511yescharacteristic
643C2C53×#231511yescharacteristic
644C2D4×C22⋊C4×C5312yes—
645C2C53×#234212yes—
646C2C53×#241012yes—
647C2C53×#241212yes—
648C2C53×#243714yes—
subgroupG/H
center, socleC24×C53C24
commutatorC23C23×C4×C53
Frattini, cosocleC24C24×C53
prime radicalC26×C53C22
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24×C53 ◁ G

Complements

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

NH ≅ G/Ntype
3. C2×C53×#385C2semidirect
4. C2×C53×#380C2semidirect
6. C53×#1439C2semidirect
7. C53×C22⋊#95C2semidirect
8. C53×C22⋊#95C2semidirect
12. C22×C4⋊C4×C53C22semidirect
16. C22×C22⋊C4×C53C4semidirect
29. C2×#95×C53C22semidirect
30. C2×#95×C53C22semidirect
31. C2×#95×C53C22semidirect
52. C23×C4×C53C2×C4semidirect
80. C2×C22⋊C4×C53C2×C4semidirect
112. C24×C53C4⋊C4semidirect
173. C22⋊C4×C53C22⋊C4semidirect
266. C2×C4×C53C2×C22⋊C4semidirect
267. #13877C53direct (N × H)
291. C22×C53#380semidirect
292. C22×C53#385semidirect
316. C2×#385C2×C53semidirect
317. C2×#380C2×C53semidirect
319. #1439C2×C53semidirect
320. C22⋊#95C2×C53semidirect
321. C22⋊#95C2×C53semidirect
336. C22×C4⋊C4C22×C53semidirect
340. C22×C22⋊C4C4×C53semidirect
353. C2×#95C22×C53semidirect
354. C2×#95C22×C53semidirect
355. C2×#95C22×C53semidirect
356. C53#13877direct (N × H)
377. C23×C4C2×C4×C53semidirect
405. C2×C22⋊C4C2×C4×C53semidirect
437. C24C4⋊C4×C53semidirect
498. C22⋊C4C22⋊C4×C53semidirect
591. C2×C4C2×C22⋊C4×C53semidirect
615. C22C53×#380semidirect
616. C22C53×#385semidirect

Maximal subgroups

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

subgroupindex
C23×C22⋊C4×C532
C2×C53×#3852
C2×C53×#3802
C2×C53×#3792
C53×#14392
C53×C22⋊#952
#1387753