Finite Group Databasebeta

← C53×#14506 #27222018

Subgroups — C53×#14506 #27222018

Order 13568.

1962 conjugacy classes of subgroups.

The subgroup lattice has 668 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#123×C53C211yesmaximal, characteristic
2C2×C53×#388C212yesmaximal, complemented
3C53×#1445C212yesmaximal, complemented
4C53×#1439C212yesmaximal, complemented
5C53×#1438C211yesmaximal, characteristic
6C53×#1438C211yesmaximal, characteristic
7C53×#1437C212yesmaximal, complemented
8C53×C4⋊#95C212yesmaximal, complemented
9C53×C22⋊#95C212yesmaximal, complemented
10C23×D4×C53C2211yescharacteristic
11C24×C4×C53C2211yescharacteristic
12C2×#123×C53—28no—
13C2×#123×C53C418yescomplemented
14C22×C4⋊C4×C53C2212yes—
15C22×C4⋊C4×C53C2212yes—
16C22×C22⋊C4×C53C2212yes—
17C22×C22⋊C4×C53C2212yes—
18C22×C22⋊C4×C53C2211yescharacteristic
19C22×C22⋊C4×C53C2212yes—
20C22×C22⋊C4×C53C2212yes—
21C22×C22⋊C4×C53C2211yescharacteristic
22C22×C22⋊C4×C53C2211yescharacteristic
23C22×C42×C53C2212yes—
24C53×#388—28no—
25C2×#95×C53C2212yes—
26C2×#95×C53C2212yes—
27C2×#95×C53C2212yes—
28C2×#95×C53C2212yes—
29C2×#95×C53C2212yes—
30C2×#95×C53C2212yes—
31C2×#95×C53C2211yescharacteristic
32C2×#95×C53C2212yes—
33C2×#95×C53C2211yescharacteristic
34C25×C53C2311yescharacteristic
35C25×C53C2311yescharacteristic
36C22×D4×C53—24no—
37C22×D4×C53—42no—
38C22×D4×C53—41no—
39C22×D4×C53—41no—
40C22×D4×C53C2×C414yes—
41C23×C4×C53—24no—
42C23×C4×C53—42no—
43C23×C4×C53—41no—
44C23×C4×C53—41no—
45C23×C4×C53C2×C414yes—
46C23×C4×C53C2312yes—
47C23×C4×C53C2312yes—
48C23×C4×C53C2312yes—
49C23×C4×C53C2311yescharacteristic
50C23×C4×C53C2312yes—
51C23×C4×C53C2312yes—
52C23×C4×C53C2312yes—
53#123×C53—232no—
54C2×C4⋊C4×C53—24no—
55C2×C4⋊C4×C53—24no—
56C2×C4⋊C4×C53—24no—
57C2×C4⋊C4×C53—24no—
58C2×C4⋊C4×C53—42no—
59C2×C4⋊C4×C53C2×C418yes—
60C2×C22⋊C4×C53—24no—
61C2×C22⋊C4×C53—24no—
62C2×C22⋊C4×C53—24no—
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—41no—
69C2×C22⋊C4×C53—42no—
70C2×C22⋊C4×C53—42no—
71C2×C22⋊C4×C53—42no—
72C2×C22⋊C4×C53—42no—
73C2×C22⋊C4×C53—41no—
74C2×C22⋊C4×C53—42no—
75C2×C22⋊C4×C53C2×C418yes—
76C2×C22⋊C4×C53C2×C414yes—
77C2×C22⋊C4×C53D418yes—
78C2×C42×C53—24no—
79C2×C42×C53—24no—
80C2×C42×C53—42no—
81#95×C53—24no—
82#95×C53—24no—
83#95×C53—41no—
84#95×C53—42no—
85#95×C53—42no—
86#95×C53—42no—
87#95×C53—42no—
88#95×C53—42no—
89#95×C53—41no—
90C24×C53—22no—
91C24×C53—22no—
92C24×C53—22no—
93C24×C53—22no—
94C24×C53—21no—
95C24×C53—21no—
96C24×C53—22no—
97C24×C53—22no—
98C24×C53—22no—
99C24×C53—22no—
100C24×C53—22no—
101C24×C53—22no—
102C24×C53—21no—
103C24×C53—21no—
104C24×C53—22no—
105C24×C53C22×C412yes—
106C24×C53C22×C412yes—
107C24×C53C2×D414yes—
108C24×C53C2411yescharacteristic
109C2×D4×C53—28no—
110C2×D4×C53—44no—
111C2×D4×C53—44no—
112C2×D4×C53—48no—
113C2×D4×C53—48no—
114C22×C4×C53—22no—
115C22×C4×C53—22no—
116C22×C4×C53—22no—
117C22×C4×C53—22no—
118C22×C4×C53—28no—
119C22×C4×C53—22no—
120C22×C4×C53—22no—
121C22×C4×C53—22no—
122C22×C4×C53—22no—
123C22×C4×C53—22no—
124C22×C4×C53—22no—
125C22×C4×C53—22no—
126C22×C4×C53—22no—
127C22×C4×C53—22no—
128C22×C4×C53—22no—
129C22×C4×C53—22no—
130C22×C4×C53—22no—
131C22×C4×C53—22no—
132C22×C4×C53—22no—
133C22×C4×C53—22no—
134C22×C4×C53—22no—
135C22×C4×C53—21no—
136C22×C4×C53—21no—
137C22×C4×C53—22no—
138C22×C4×C53—22no—
139C22×C4×C53—22no—
140C22×C4×C53—22no—
141C22×C4×C53—22no—
142C22×C4×C53—22no—
143C22×C4×C53—22no—
144C22×C4×C53—22no—
145C22×C4×C53—22no—
146C22×C4×C53—22no—
147C22×C4×C53—22no—
148C22×C4×C53—22no—
149C22×C4×C53—22no—
150C22×C4×C53—22no—
151C22×C4×C53—22no—
152C22×C4×C53—22no—
153C22×C4×C53—22no—
154C22×C4×C53—22no—
155C22×C4×C53—44no—
156C22×C4×C53—44no—
157C22×C4×C53—48no—
158C22×C4×C53—48no—
159C22×C4×C53C22×C414yes—
160C22×C4×C53C22×C412yes—
161C22×C4×C53C22×C414yes—
162C22×C4×C53C2×D414yes—
163C22×C4×C53C2×D414yes—
164C22×C4×C53#4114yes—
165C22×C4×C53#4114yes—
166C4⋊C4×C53—216no—
167C4⋊C4×C53—48no—
168C22⋊C4×C53—216no—
169C22⋊C4×C53—44no—
170C22⋊C4×C53—48no—
171C22⋊C4×C53—48no—
172C22⋊C4×C53—48no—
173C22⋊C4×C53—44no—
174C22⋊C4×C53C22⋊C4116yescomplemented
175C42×C53—48no—
176C23×C53—24no—
177C23×C53—22no—
178C23×C53—22no—
179C23×C53—24no—
180C23×C53—24no—
181C23×C53—24no—
182C23×C53—24no—
183C23×C53—24no—
184C23×C53—24no—
185C23×C53—24no—
186C23×C53—22no—
187C23×C53—24no—
188C23×C53—22no—
189C23×C53—22no—
190C23×C53—24no—
191C23×C53—42no—
192C23×C53—42no—
193C23×C53—42no—
194C23×C53—42no—
195C23×C53—41no—
196C23×C53—42no—
197C23×C53—41no—
198C23×C53—41no—
199C23×C53—42no—
200C23×C53—41no—
201C23×C53—42no—
202C23×C53—42no—
203C23×C53—42no—
204C23×C53—42no—
205C23×C53—42no—
206C23×C53—42no—
207C23×C53—42no—
208C23×C53—42no—
209C23×C53—42no—
210C23×C53—42no—
211C23×C53—41no—
212C23×C53—42no—
213C23×C53—41no—
214C23×C53—41no—
215C23×C53—42no—
216C23×C53—41no—
217C23×C53C2×C22⋊C414yes—
218C23×C53C23×C411yescharacteristic
219C23×C53C22×D412yes—
220C23×C53C2×#4112yes—
221C23×C53C2×#4112yes—
222C23×C53H(5,𝔽2)12yes—
223C23×C53H(5,𝔽2)11yescharacteristic
224C23×C53H(5,𝔽2)11yescharacteristic
225C23×C53H(5,𝔽2)12yes—
226C23×C532−1+412yes—
227D4×C53—416no—
228C2×C4×C53—24no—
229C2×C4×C53—24no—
230C2×C4×C53—24no—
231C2×C4×C53—24no—
232C2×C4×C53—24no—
233C2×C4×C53—24no—
234C2×C4×C53—24no—
235C2×C4×C53—22no—
236C2×C4×C53—24no—
237C2×C4×C53—24no—
238C2×C4×C53—22no—
239C2×C4×C53—22no—
240C2×C4×C53—24no—
241C2×C4×C53—24no—
242C2×C4×C53—24no—
243C2×C4×C53—24no—
244C2×C4×C53—24no—
245C2×C4×C53—24no—
246C2×C4×C53—416no—
247C2×C4×C53—42no—
248C2×C4×C53—42no—
249C2×C4×C53—42no—
250C2×C4×C53—42no—
251C2×C4×C53—42no—
252C2×C4×C53—42no—
253C2×C4×C53—42no—
254C2×C4×C53—42no—
255C2×C4×C53—42no—
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—42no—
263C2×C4×C53—42no—
264C2×C4×C53—42no—
265C2×C4×C53—42no—
266C2×C4×C53—42no—
267C2×C4×C53—42no—
268C2×C4×C53—42no—
269C2×C4×C53—42no—
270C2×C4×C53—42no—
271C2×C4×C53—42no—
272C2×C4×C53—42no—
273C2×C4×C53—42no—
274C2×C4×C53—42no—
275C2×C4×C53C2×C22⋊C418yes—
276#14506C5311yesmaximal, characteristic, direct product factor, Sylow
277C22×C53—24no—
278C22×C53—28no—
279C22×C53—44no—
280C22×C53—42no—
281C22×C53—42no—
282C22×C53—44no—
283C22×C53—44no—
284C22×C53—44no—
285C22×C53—44no—
286C22×C53—42no—
287C22×C53—44no—
288C22×C53—44no—
289C22×C53—42no—
290C22×C53—42no—
291C22×C53—82no—
292C22×C53—82no—
293C22×C53—82no—
294C22×C53—82no—
295C22×C53C22×C22⋊C411yescharacteristic
296C22×C53C4×#4112yes—
297C22×C53#51811yescharacteristic
298C22×C53#51811yescharacteristic
299C22×C53#52012yes—
300C22×C53#53512yes—
301C22×C53#53512yes—
302C22×C53#53812yes—
303C22×C53#53912yes—
304C22×C53#54012yes—
305C22×C53#54212yes—
306C22×C53#54612yes—
307C22×C53#54612yes—
308C22×C53#54712yes—
309C22×C53#54712yes—
310C22×C53#55111yescharacteristic
311C22×C53#55312yes—
312C22×C53#55911yescharacteristic
313C22×C53#56011yescharacteristic
314C22×C53#56212yes—
315C22×C53#56311yescharacteristic
316C4×C53—28no—
317C4×C53—44no—
318C4×C53—42no—
319C4×C53—44no—
320C4×C53—44no—
321C4×C53—82no—
322C4×C53—82no—
323C22×#123C2×C5311yescharacteristic
324C2×#388C2×C5312yescomplemented
325#1445C2×C5312yescomplemented
326#1439C2×C5312yescomplemented
327#1438C2×C5311yescharacteristic
328#1438C2×C5311yescharacteristic
329#1437C2×C5312yescomplemented
330C4⋊#95C2×C5312yescomplemented
331C22⋊#95C2×C5312yescomplemented
332C2×C53—44no—
333C2×C53—82no—
334C2×C53#230911yescharacteristic
335C2×C53#233512yes—
336C2×C53#233912yes—
337C2×C53#237411yescharacteristic
338C2×C53#237611yescharacteristic
339C2×C53#270312yes—
340C2×C53#270612yes—
341C2×C53#271012yes—
342C2×C53#271812yes—
343C23×D4C22×C5311yescharacteristic
344C24×C4C22×C5311yescharacteristic
345C2×#123—28no—
346C2×#123C4×C5318yescomplemented
347C22×C4⋊C4C22×C5312yes—
348C22×C4⋊C4C22×C5312yes—
349C22×C22⋊C4C22×C5312yes—
350C22×C22⋊C4C22×C5312yes—
351C22×C22⋊C4C22×C5311yescharacteristic
352C22×C22⋊C4C22×C5312yes—
353C22×C22⋊C4C22×C5312yes—
354C22×C22⋊C4C22×C5311yescharacteristic
355C22×C22⋊C4C22×C5311yescharacteristic
356C22×C42C22×C5312yes—
357#388—28no—
358C2×#95C22×C5312yes—
359C2×#95C22×C5312yes—
360C2×#95C22×C5312yes—
361C2×#95C22×C5312yes—
362C2×#95C22×C5312yes—
363C2×#95C22×C5312yes—
364C2×#95C22×C5311yescharacteristic
365C2×#95C22×C5312yes—
366C2×#95C22×C5311yescharacteristic
367C53#1450611yescharacteristic, direct product factor, Sylow
368C25C23×C5311yescharacteristic
369C25C23×C5311yescharacteristic
370C22×D4—24no—
371C22×D4—42no—
372C22×D4—41no—
373C22×D4—41no—
374C22×D4C2×C4×C5314yes—
375C23×C4—24no—
376C23×C4—42no—
377C23×C4—41no—
378C23×C4—41no—
379C23×C4C2×C4×C5314yes—
380C23×C4C23×C5312yes—
381C23×C4C23×C5312yes—
382C23×C4C23×C5312yes—
383C23×C4C23×C5311yescharacteristic
384C23×C4C23×C5312yes—
385C23×C4C23×C5312yes—
386C23×C4C23×C5312yes—
387#123—232no—
388C2×C4⋊C4—24no—
389C2×C4⋊C4—24no—
390C2×C4⋊C4—24no—
391C2×C4⋊C4—24no—
392C2×C4⋊C4—42no—
393C2×C4⋊C4C2×C4×C5318yes—
394C2×C22⋊C4—24no—
395C2×C22⋊C4—24no—
396C2×C22⋊C4—24no—
397C2×C22⋊C4—24no—
398C2×C22⋊C4—24no—
399C2×C22⋊C4—24no—
400C2×C22⋊C4—24no—
401C2×C22⋊C4—24no—
402C2×C22⋊C4—41no—
403C2×C22⋊C4—42no—
404C2×C22⋊C4—42no—
405C2×C22⋊C4—42no—
406C2×C22⋊C4—42no—
407C2×C22⋊C4—41no—
408C2×C22⋊C4—42no—
409C2×C22⋊C4C2×C4×C5318yes—
410C2×C22⋊C4C2×C4×C5314yes—
411C2×C22⋊C4D4×C5318yes—
412C2×C42—24no—
413C2×C42—24no—
414C2×C42—42no—
415#95—24no—
416#95—24no—
417#95—41no—
418#95—42no—
419#95—42no—
420#95—42no—
421#95—42no—
422#95—42no—
423#95—41no—
424C24—22no—
425C24—22no—
426C24—22no—
427C24—22no—
428C24—21no—
429C24—21no—
430C24—22no—
431C24—22no—
432C24—22no—
433C24—22no—
434C24—22no—
435C24—22no—
436C24—21no—
437C24—21no—
438C24—22no—
439C24C22×C4×C5312yes—
440C24C22×C4×C5312yes—
441C24C2×D4×C5314yes—
442C24C24×C5311yescharacteristic
443C2×D4—28no—
444C2×D4—44no—
445C2×D4—44no—
446C2×D4—48no—
447C2×D4—48no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—28no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—22no—
485C22×C4—22no—
486C22×C4—22no—
487C22×C4—22no—
488C22×C4—22no—
489C22×C4—44no—
490C22×C4—44no—
491C22×C4—48no—
492C22×C4—48no—
493C22×C4C22×C4×C5314yes—
494C22×C4C22×C4×C5312yes—
495C22×C4C22×C4×C5314yes—
496C22×C4C2×D4×C5314yes—
497C22×C4C2×D4×C5314yes—
498C22×C4#41×C5314yes—
499C22×C4#41×C5314yes—
500C4⋊C4—216no—
501C4⋊C4—48no—
502C22⋊C4—216no—
503C22⋊C4—44no—
504C22⋊C4—48no—
505C22⋊C4—48no—
506C22⋊C4—48no—
507C22⋊C4—44no—
508C22⋊C4C22⋊C4×C53116yescomplemented
509C42—48no—
510C23—24no—
511C23—22no—
512C23—22no—
513C23—24no—
514C23—24no—
515C23—24no—
516C23—24no—
517C23—24no—
518C23—24no—
519C23—24no—
520C23—22no—
521C23—24no—
522C23—22no—
523C23—22no—
524C23—24no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—41no—
530C23—42no—
531C23—41no—
532C23—41no—
533C23—42no—
534C23—41no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23—42no—
539C23—42no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23—42no—
544C23—42no—
545C23—41no—
546C23—42no—
547C23—41no—
548C23—41no—
549C23—42no—
550C23—41no—
551C23C2×C22⋊C4×C5314yes—
552C23C23×C4×C5311yescharacteristic
553C23C22×D4×C5312yes—
554C23C2×#41×C5312yes—
555C23C2×#41×C5312yes—
556C23H(5,𝔽2)×C5312yes—
557C23H(5,𝔽2)×C5311yescharacteristic
558C23H(5,𝔽2)×C5311yescharacteristic
559C23H(5,𝔽2)×C5312yes—
560C232−1+4×C5312yes—
561D4—416no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—22no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—416no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4C2×C22⋊C4×C5318yes—
610C22—24no—
611C22—28no—
612C22—44no—
613C22—44no—
614C22—44no—
615C22—44no—
616C22—44no—
617C22—42no—
618C22—42no—
619C22—42no—
620C22—44no—
621C22—44no—
622C22—42no—
623C22—42no—
624C22—82no—
625C22—82no—
626C22—82no—
627C22—82no—
628C22C22×C22⋊C4×C5311yescharacteristic
629C22C4×#41×C5312yes—
630C22C53×#51811yescharacteristic
631C22C53×#51811yescharacteristic
632C22C53×#52012yes—
633C22C53×#53512yes—
634C22C53×#53512yes—
635C22C53×#53812yes—
636C22C53×#53912yes—
637C22C53×#54012yes—
638C22C53×#54212yes—
639C22C53×#54612yes—
640C22C53×#54612yes—
641C22C53×#54712yes—
642C22C53×#54712yes—
643C22C53×#55111yescharacteristic
644C22C53×#55312yes—
645C22C53×#55911yescharacteristic
646C22C53×#56011yescharacteristic
647C22C53×#56212yes—
648C22C53×#56311yescharacteristic
649C4—28no—
650C4—44no—
651C4—44no—
652C4—44no—
653C4—42no—
654C4—82no—
655C4—82no—
656C2—44no—
657C2—82no—
658C2C53×#230911yescharacteristic
659C2C53×#233512yes—
660C2C53×#233912yes—
661C2C53×#237411yescharacteristic
662C2C53×#237611yescharacteristic
663C2C53×#270312yes—
664C2C53×#270612yes—
665C2C53×#271012yes—
666C2C53×#271812yes—
subgroupG/H
center, socleC24×C53C24
commutatorC23C23×C4×C53
Frattini, cosocleC24C24×C53
prime radicalC23×D4×C53C22
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24×C53 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C53×#388C2semidirect
3. C53×#1445C2semidirect
4. C53×#1439C2semidirect
7. C53×#1437C2semidirect
8. C53×C4⋊#95C2semidirect
9. C53×C22⋊#95C2semidirect
13. C2×#123×C53C4semidirect
174. C22⋊C4×C53C22⋊C4semidirect
276. #14506C53direct (N × H)
324. C2×#388C2×C53semidirect
325. #1445C2×C53semidirect
326. #1439C2×C53semidirect
329. #1437C2×C53semidirect
330. C4⋊#95C2×C53semidirect
331. C22⋊#95C2×C53semidirect
346. C2×#123C4×C53semidirect
367. C53#14506direct (N × H)
508. C22⋊C4C22⋊C4×C53semidirect

Maximal subgroups

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

subgroupindex
C22×#123×C532
C2×C53×#3882
C53×#14452
C53×#14392
C53×#14382
C53×#14372
C53×C4⋊#952
C53×C22⋊#952
#1450653