Finite Group Databasebeta

← C53×#10745 #27218257

Subgroups — C53×#10745 #27218257

Order 13568.

1794 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C53×#518C211yesmaximal, characteristic
2C22×#99×C53C211yesmaximal, characteristic
3C2×C53×#386C211yesmaximal, characteristic
4C2×C53×#342C212yesmaximal, complemented
5C53×C22⋊#95C212yesmaximal, complemented
6C23×D4×C53C2211yescharacteristic
7C24×C4×C53C2211yescharacteristic
8C53×#518—22no—
9C53×#518—24no—
10C53×#518C414yescomplemented
11C2×C4×D4×C53—24no—
12C2×#117×C53—21no—
13C22×C22⋊C4×C53—21no—
14C22×C22⋊C4×C53—21no—
15C22×C22⋊C4×C53C2212yes—
16C22×C22⋊C4×C53C2211yescharacteristic
17C22×C22⋊C4×C53C2212yes—
18C2×#99×C53—22no—
19C2×#99×C53—22no—
20C2×#99×C53C414yescomplemented
21C53×#386—22no—
22C53×#386C414yescomplemented
23C2×#95×C53—21no—
24C2×#95×C53—24no—
25C53×#342—24no—
26C25×C53D411yescharacteristic
27C25×C53C2311yescharacteristic
28C22×D4×C53—22no—
29C22×D4×C53—22no—
30C22×D4×C53—22no—
31C22×D4×C53—22no—
32C22×D4×C53C2×C412yes—
33C22×D4×C53C2×C412yes—
34C22×D4×C53C2×C412yes—
35C22×D4×C53D412yes—
36C22×D4×C53D412yes—
37C22×D4×C53Q812yes—
38C23×C4×C53—21no—
39C23×C4×C53—22no—
40C23×C4×C53—22no—
41C23×C4×C53—21no—
42C23×C4×C53—22no—
43C23×C4×C53—21no—
44C23×C4×C53—22no—
45C23×C4×C53—44no—
46C23×C4×C53C2×C412yescomplemented
47C23×C4×C53D411yescharacteristic
48C23×C4×C53D414yes—
49C4×D4×C53—216no—
50C4×D4×C53—216no—
51#117×C53—24no—
52#117×C53—24no—
53C2×C4⋊C4×C53—22no—
54C2×C22⋊C4×C53—22no—
55C2×C22⋊C4×C53—24no—
56C2×C22⋊C4×C53—24no—
57C2×C22⋊C4×C53—24no—
58C2×C22⋊C4×C53—22no—
59C2×C22⋊C4×C53—22no—
60C2×C22⋊C4×C53—22no—
61C2×C22⋊C4×C53—22no—
62C2×C22⋊C4×C53—21no—
63C2×C22⋊C4×C53—22no—
64C2×C22⋊C4×C53—22no—
65C2×C22⋊C4×C53—22no—
66C2×C22⋊C4×C53—22no—
67C2×C22⋊C4×C53—22no—
68C2×C22⋊C4×C53—22no—
69C2×C22⋊C4×C53—22no—
70C2×C22⋊C4×C53—44no—
71C2×C22⋊C4×C53C2×C414yes—
72C2×C22⋊C4×C53C2×C412yescomplemented
73C2×C22⋊C4×C53C2×C414yescomplemented
74C2×C22⋊C4×C53D414yes—
75C2×C42×C53—22no—
76#99×C53—28no—
77#95×C53—24no—
78#95×C53—48no—
79C24×C53—21no—
80C24×C53—21no—
81C24×C53—21no—
82C24×C53—21no—
83C24×C53—42no—
84C24×C53—42no—
85C24×C53—42no—
86C24×C53—42no—
87C24×C53C22⋊C412yes—
88C24×C53C22⋊C412yes—
89C24×C53C4⋊C412yes—
90C24×C53C22×C411yescharacteristic
91C24×C53C22×C411yescharacteristic
92C24×C53C22×C411yescharacteristic
93C24×C53C2×D411yescharacteristic
94C24×C53C2×D412yes—
95C24×C53C2×D411yescharacteristic
96C24×C53C2×D411yescharacteristic
97C24×C53C2×D412yes—
98C24×C53C2×Q811yescharacteristic
99C24×C53#4112yes—
100C24×C53#4112yes—
101C2×D4×C53—24no—
102C2×D4×C53—24no—
103C2×D4×C53—24no—
104C2×D4×C53—24no—
105C2×D4×C53—24no—
106C2×D4×C53—24no—
107C2×D4×C53—24no—
108C2×D4×C53—24no—
109C2×D4×C53—24no—
110C2×D4×C53—24no—
111C2×D4×C53—24no—
112C2×D4×C53—24no—
113C2×D4×C53C4214yes—
114C2×D4×C53C22⋊C414yes—
115C2×D4×C53C4⋊C414yes—
116C2×D4×C53C4⋊C414yes—
117C22×C4×C53—21no—
118C22×C4×C53—21no—
119C22×C4×C53—21no—
120C22×C4×C53—21no—
121C22×C4×C53—28no—
122C22×C4×C53—24no—
123C22×C4×C53—22no—
124C22×C4×C53—24no—
125C22×C4×C53—24no—
126C22×C4×C53—28no—
127C22×C4×C53—24no—
128C22×C4×C53—22no—
129C22×C4×C53—24no—
130C22×C4×C53—44no—
131C22×C4×C53—48no—
132C22×C4×C53—44no—
133C22×C4×C53—44no—
134C22×C4×C53—42no—
135C22×C4×C53—41no—
136C22×C4×C53—42no—
137C22×C4×C53—41no—
138C22×C4×C53—44no—
139C22×C4×C53—44no—
140C22×C4×C53—44no—
141C22×C4×C53—42no—
142C22×C4×C53—44no—
143C22×C4×C53—42no—
144C22×C4×C53—42no—
145C22×C4×C53—41no—
146C22×C4×C53—42no—
147C22×C4×C53—41no—
148C22×C4×C53—82no—
149C22×C4×C53—82no—
150C22×C4×C53C22⋊C412yes—
151C22×C4×C53C22⋊C412yes—
152C22×C4×C53C22⋊C418yescomplemented
153C22×C4×C53C4⋊C412yes—
154C4⋊C4×C53—28no—
155C22⋊C4×C53—28no—
156C22⋊C4×C53—24no—
157C22⋊C4×C53—24no—
158C22⋊C4×C53—24no—
159C22⋊C4×C53—24no—
160C22⋊C4×C53—24no—
161C22⋊C4×C53—24no—
162C22⋊C4×C53—24no—
163C22⋊C4×C53—48no—
164C22⋊C4×C53—42no—
165C22⋊C4×C53—42no—
166C22⋊C4×C53—42no—
167C42×C53—28no—
168C23×C53—24no—
169C23×C53—22no—
170C23×C53—24no—
171C23×C53—22no—
172C23×C53—22no—
173C23×C53—22no—
174C23×C53—22no—
175C23×C53—22no—
176C23×C53—22no—
177C23×C53—22no—
178C23×C53—41no—
179C23×C53—41no—
180C23×C53—42no—
181C23×C53—42no—
182C23×C53—41no—
183C23×C53—42no—
184C23×C53—41no—
185C23×C53—44no—
186C23×C53—42no—
187C23×C53—42no—
188C23×C53—44no—
189C23×C53—42no—
190C23×C53—42no—
191C23×C53—44no—
192C23×C53—42no—
193C23×C53—44no—
194C23×C53—42no—
195C23×C53—42no—
196C23×C53—82no—
197C23×C53—81no—
198C23×C53—81no—
199C23×C53—82no—
200C23×C53#9514yescomplemented
201C23×C53C2×C4211yescharacteristic
202C23×C53C2×C22⋊C411yescharacteristic
203C23×C53C2×C22⋊C411yescharacteristic
204C23×C53C2×C22⋊C412yes—
205C23×C53C2×C22⋊C411yescharacteristic
206C23×C53C2×C4⋊C411yescharacteristic
207C23×C53C2×C4⋊C411yescharacteristic
208C23×C53C2×C4⋊C411yescharacteristic
209C23×C53#11712yes—
210C23×C53C4×D412yes—
211C23×C53C4×D412yes—
212C23×C53C4×D412yes—
213C23×C53C4×D412yes—
214C23×C53#12012yes—
215C23×C53#12011yescharacteristic
216C23×C53#12011yescharacteristic
217C23×C53#12112yes—
218C23×C53#12112yes—
219C23×C53C22⋊Q812yes—
220C23×C53C22⋊Q812yes—
221C23×C53#12311yescharacteristic
222C23×C53#12311yescharacteristic
223C23×C53#12312yes—
224D4×C53—28no—
225D4×C53—28no—
226D4×C53—28no—
227D4×C53—28no—
228C2×C4×C53—22no—
229C2×C4×C53—22no—
230C2×C4×C53—22no—
231C2×C4×C53—22no—
232C2×C4×C53—22no—
233C2×C4×C53—22no—
234C2×C4×C53—28no—
235C2×C4×C53—24no—
236C2×C4×C53—28no—
237C2×C4×C53—42no—
238C2×C4×C53—44no—
239C2×C4×C53—48no—
240C2×C4×C53—44no—
241C2×C4×C53—42no—
242C2×C4×C53—44no—
243C2×C4×C53—44no—
244C2×C4×C53—44no—
245C2×C4×C53—48no—
246C2×C4×C53—48no—
247C2×C4×C53—42no—
248C2×C4×C53—42no—
249C2×C4×C53—42no—
250C2×C4×C53—44no—
251C2×C4×C53—84no—
252C2×C4×C53—88no—
253C2×C4×C53—82no—
254C2×C4×C53—82no—
255C2×C4×C53—82no—
256C2×C4×C53—84no—
257C2×C4×C53—82no—
258C2×C4×C53#9514yescomplemented
259#10745C5311yesmaximal, characteristic, direct product factor, Sylow
260C22×C53—24no—
261C22×C53—22no—
262C22×C53—22no—
263C22×C53—24no—
264C22×C53—21no—
265C22×C53—22no—
266C22×C53—22no—
267C22×C53—21no—
268C22×C53—21no—
269C22×C53—22no—
270C22×C53—21no—
271C22×C53—22no—
272C22×C53—24no—
273C22×C53—24no—
274C22×C53—22no—
275C22×C53—22no—
276C22×C53—24no—
277C22×C53—41no—
278C22×C53—41no—
279C22×C53—41no—
280C22×C53—41no—
281C22×C53—42no—
282C22×C53—42no—
283C22×C53—42no—
284C22×C53—42no—
285C22×C53—42no—
286C22×C53—42no—
287C22×C53—44no—
288C22×C53—44no—
289C22×C53—42no—
290C22×C53—42no—
291C22×C53—44no—
292C22×C53—82no—
293C22×C53—82no—
294C22×C53—82no—
295C22×C53C2×#9511yescharacteristic
296C22×C53C4×C22⋊C412yes—
297C22×C53#37911yescharacteristic
298C22×C53#38012yes—
299C22×C53#38111yescharacteristic
300C22×C53#38511yescharacteristic
301C22×C53#38512yes—
302C22×C53#38511yescharacteristic
303C22×C53#38611yescharacteristic
304C22×C53#38612yes—
305C22×C53#38611yescharacteristic
306C22×C53B(4,𝔽2)12yes—
307C22×C53#45812yes—
308C4×C53—24no—
309C4×C53—44no—
310C4×C53—48no—
311C4×C53—82no—
312C4×C53—84no—
313C2×#518C2×C5311yescharacteristic
314C22×#99C2×C5311yescharacteristic
315C2×#386C2×C5311yescharacteristic
316C2×#342C2×C5312yescomplemented
317C22⋊#95C2×C5312yescomplemented
318C2×C53—21no—
319C2×C53—22no—
320C2×C53—21no—
321C2×C53—21no—
322C2×C53—21no—
323C2×C53—22no—
324C2×C53—22no—
325C2×C53—22no—
326C2×C53—44no—
327C2×C53C22⋊#9511yescharacteristic
328C2×C53#179012yes—
329C2×C53#186712yes—
330C2×C53#188911yescharacteristic
331C2×C53#189011yescharacteristic
332C23×D4C22×C5311yescharacteristic
333C24×C4C22×C5311yescharacteristic
334#518—22no—
335#518—24no—
336#518C4×C5314yescomplemented
337C2×C4×D4—24no—
338C2×#117—21no—
339C22×C22⋊C4—21no—
340C22×C22⋊C4—21no—
341C22×C22⋊C4C22×C5312yes—
342C22×C22⋊C4C22×C5311yescharacteristic
343C22×C22⋊C4C22×C5312yes—
344C2×#99—22no—
345C2×#99—22no—
346C2×#99C4×C5314yescomplemented
347#386—22no—
348#386C4×C5314yescomplemented
349C2×#95—21no—
350C2×#95—24no—
351#342—24no—
352C53#1074511yescharacteristic, direct product factor, Sylow
353C25D4×C5311yescharacteristic
354C25C23×C5311yescharacteristic
355C22×D4—22no—
356C22×D4—22no—
357C22×D4—22no—
358C22×D4—22no—
359C22×D4C2×C4×C5312yes—
360C22×D4C2×C4×C5312yes—
361C22×D4C2×C4×C5312yes—
362C22×D4D4×C5312yes—
363C22×D4D4×C5312yes—
364C22×D4Q8×C5312yes—
365C23×C4—21no—
366C23×C4—22no—
367C23×C4—22no—
368C23×C4—21no—
369C23×C4—22no—
370C23×C4—21no—
371C23×C4—22no—
372C23×C4—44no—
373C23×C4C2×C4×C5312yescomplemented
374C23×C4D4×C5311yescharacteristic
375C23×C4D4×C5314yes—
376C4×D4—216no—
377C4×D4—216no—
378#117—24no—
379#117—24no—
380C2×C4⋊C4—22no—
381C2×C22⋊C4—22no—
382C2×C22⋊C4—24no—
383C2×C22⋊C4—22no—
384C2×C22⋊C4—22no—
385C2×C22⋊C4—22no—
386C2×C22⋊C4—22no—
387C2×C22⋊C4—24no—
388C2×C22⋊C4—22no—
389C2×C22⋊C4—22no—
390C2×C22⋊C4—22no—
391C2×C22⋊C4—22no—
392C2×C22⋊C4—22no—
393C2×C22⋊C4—22no—
394C2×C22⋊C4—21no—
395C2×C22⋊C4—22no—
396C2×C22⋊C4—24no—
397C2×C22⋊C4—44no—
398C2×C22⋊C4C2×C4×C5314yes—
399C2×C22⋊C4C2×C4×C5312yescomplemented
400C2×C22⋊C4C2×C4×C5314yescomplemented
401C2×C22⋊C4D4×C5314yes—
402C2×C42—22no—
403#99—28no—
404#95—24no—
405#95—48no—
406C24—21no—
407C24—21no—
408C24—21no—
409C24—21no—
410C24—42no—
411C24—42no—
412C24—42no—
413C24—42no—
414C24C22⋊C4×C5312yes—
415C24C22⋊C4×C5312yes—
416C24C4⋊C4×C5312yes—
417C24C22×C4×C5311yescharacteristic
418C24C22×C4×C5311yescharacteristic
419C24C22×C4×C5311yescharacteristic
420C24C2×D4×C5311yescharacteristic
421C24C2×D4×C5312yes—
422C24C2×D4×C5311yescharacteristic
423C24C2×D4×C5311yescharacteristic
424C24C2×D4×C5312yes—
425C24C2×Q8×C5311yescharacteristic
426C24#41×C5312yes—
427C24#41×C5312yes—
428C2×D4—24no—
429C2×D4—24no—
430C2×D4—24no—
431C2×D4—24no—
432C2×D4—24no—
433C2×D4—24no—
434C2×D4—24no—
435C2×D4—24no—
436C2×D4—24no—
437C2×D4—24no—
438C2×D4—24no—
439C2×D4—24no—
440C2×D4C42×C5314yes—
441C2×D4C22⋊C4×C5314yes—
442C2×D4C4⋊C4×C5314yes—
443C2×D4C4⋊C4×C5314yes—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—28no—
449C22×C4—24no—
450C22×C4—22no—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4—28no—
454C22×C4—24no—
455C22×C4—22no—
456C22×C4—24no—
457C22×C4—44no—
458C22×C4—44no—
459C22×C4—48no—
460C22×C4—44no—
461C22×C4—42no—
462C22×C4—41no—
463C22×C4—42no—
464C22×C4—41no—
465C22×C4—44no—
466C22×C4—44no—
467C22×C4—44no—
468C22×C4—42no—
469C22×C4—44no—
470C22×C4—42no—
471C22×C4—41no—
472C22×C4—42no—
473C22×C4—42no—
474C22×C4—41no—
475C22×C4—82no—
476C22×C4—82no—
477C22×C4C22⋊C4×C5312yes—
478C22×C4C22⋊C4×C5312yes—
479C22×C4C22⋊C4×C5318yescomplemented
480C22×C4C4⋊C4×C5312yes—
481C4⋊C4—28no—
482C22⋊C4—28no—
483C22⋊C4—24no—
484C22⋊C4—24no—
485C22⋊C4—24no—
486C22⋊C4—24no—
487C22⋊C4—24no—
488C22⋊C4—24no—
489C22⋊C4—24no—
490C22⋊C4—48no—
491C22⋊C4—42no—
492C22⋊C4—42no—
493C22⋊C4—42no—
494C42—28no—
495C23—24no—
496C23—22no—
497C23—24no—
498C23—22no—
499C23—22no—
500C23—22no—
501C23—22no—
502C23—22no—
503C23—22no—
504C23—22no—
505C23—41no—
506C23—41no—
507C23—42no—
508C23—42no—
509C23—42no—
510C23—41no—
511C23—44no—
512C23—42no—
513C23—41no—
514C23—42no—
515C23—44no—
516C23—42no—
517C23—42no—
518C23—44no—
519C23—42no—
520C23—44no—
521C23—42no—
522C23—42no—
523C23—82no—
524C23—82no—
525C23—81no—
526C23—81no—
527C23#95×C5314yescomplemented
528C23C2×C42×C5311yescharacteristic
529C23C2×C22⋊C4×C5311yescharacteristic
530C23C2×C22⋊C4×C5311yescharacteristic
531C23C2×C22⋊C4×C5312yes—
532C23C2×C22⋊C4×C5311yescharacteristic
533C23C2×C4⋊C4×C5311yescharacteristic
534C23C2×C4⋊C4×C5311yescharacteristic
535C23C2×C4⋊C4×C5311yescharacteristic
536C23#117×C5312yes—
537C23C4×D4×C5312yes—
538C23C4×D4×C5312yes—
539C23C4×D4×C5312yes—
540C23C4×D4×C5312yes—
541C23#120×C5311yescharacteristic
542C23#120×C5312yes—
543C23#120×C5311yescharacteristic
544C23#121×C5312yes—
545C23#121×C5312yes—
546C23C22⋊Q8×C5312yes—
547C23C22⋊Q8×C5312yes—
548C23#123×C5311yescharacteristic
549C23#123×C5311yescharacteristic
550C23#123×C5312yes—
551D4—28no—
552D4—28no—
553D4—28no—
554D4—28no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—24no—
562C2×C4—28no—
563C2×C4—28no—
564C2×C4—42no—
565C2×C4—44no—
566C2×C4—48no—
567C2×C4—44no—
568C2×C4—42no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—48no—
573C2×C4—42no—
574C2×C4—48no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—44no—
578C2×C4—84no—
579C2×C4—88no—
580C2×C4—84no—
581C2×C4—82no—
582C2×C4—82no—
583C2×C4—82no—
584C2×C4—82no—
585C2×C4#95×C5314yescomplemented
586C22—24no—
587C22—24no—
588C22—22no—
589C22—22no—
590C22—24no—
591C22—22no—
592C22—22no—
593C22—24no—
594C22—21no—
595C22—22no—
596C22—22no—
597C22—21no—
598C22—21no—
599C22—22no—
600C22—21no—
601C22—22no—
602C22—24no—
603C22—41no—
604C22—41no—
605C22—41no—
606C22—41no—
607C22—42no—
608C22—42no—
609C22—42no—
610C22—42no—
611C22—42no—
612C22—42no—
613C22—44no—
614C22—44no—
615C22—44no—
616C22—42no—
617C22—42no—
618C22—82no—
619C22—82no—
620C22—82no—
621C22C2×#95×C5311yescharacteristic
622C22C4×C22⋊C4×C5312yes—
623C22C53×#37911yescharacteristic
624C22C53×#38012yes—
625C22C53×#38111yescharacteristic
626C22C53×#38511yescharacteristic
627C22C53×#38512yes—
628C22C53×#38511yescharacteristic
629C22C53×#38611yescharacteristic
630C22C53×#38612yes—
631C22C53×#38611yescharacteristic
632C22C53×B(4,𝔽2)12yes—
633C22C53×#45812yes—
634C4—24no—
635C4—48no—
636C4—44no—
637C4—82no—
638C4—84no—
639C2—21no—
640C2—22no—
641C2—21no—
642C2—21no—
643C2—21no—
644C2—22no—
645C2—22no—
646C2—22no—
647C2—44no—
648C2C53×C22⋊#9511yescharacteristic
649C2C53×#179012yes—
650C2C53×#186712yes—
651C2C53×#188911yescharacteristic
652C2C53×#189011yescharacteristic
subgroupG/H
center, socleC23×C53#120
commutatorC23C2×C42×C53
Frattini, cosocleC25C23×C53
prime radicalC23×D4×C53C22
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23×C53 ◁ C25×C53 ◁ G

Complements

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

NH ≅ G/Ntype
4. C2×C53×#342C2semidirect
5. C53×C22⋊#95C2semidirect
10. C53×#518C4semidirect
20. C2×#99×C53C4semidirect
22. C53×#386C4semidirect
46. C23×C4×C53C2×C4semidirect
72. C2×C22⋊C4×C53C2×C4semidirect
73. C2×C22⋊C4×C53C2×C4semidirect
152. C22×C4×C53C22⋊C4semidirect
200. C23×C53#95semidirect
258. C2×C4×C53#95semidirect
259. #10745C53direct (N × H)
316. C2×#342C2×C53semidirect
317. C22⋊#95C2×C53semidirect
336. #518C4×C53semidirect
346. C2×#99C4×C53semidirect
348. #386C4×C53semidirect
352. C53#10745direct (N × H)
373. C23×C4C2×C4×C53semidirect
399. C2×C22⋊C4C2×C4×C53semidirect
400. C2×C22⋊C4C2×C4×C53semidirect
479. C22×C4C22⋊C4×C53semidirect
527. C23#95×C53semidirect
585. C2×C4#95×C53semidirect

Maximal subgroups

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

subgroupindex
C2×C53×#5182
C22×#99×C532
C2×C53×#3862
C2×C53×#3422
C53×C22⋊#952
#1074553