Finite Group Databasebeta

← C53×#23901 #27231413

Subgroups — C53×#23901 #27231413

Order 13568.

2056 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C53×#3484C211yesmaximal, characteristic
2C22×#123×C53C211yesmaximal, characteristic, complemented
3C2×C53×#458C212yesmaximal, complemented
4C53×#2849C212yesmaximal, complemented
5C53×#2022C214yesmaximal, complemented
6C53×#1898C212yesmaximal, complemented
7C53×#1890C212yesmaximal, complemented
8C53×#1781C211yesmaximal, characteristic, complemented
9C2×H(5,𝔽2)×C53C2211yescharacteristic
10C23×D4×C53C2211yescharacteristic
11C24×C4×C53C2211yescharacteristic, complemented
12C53×#546—24no—
13D42×C53—24no—
14C53×#535—21no—
15C53×#534—21no—
16C53×#534C2212yes—
17C53×#525—21no—
18C2×#123×C53—21no—
19C2×#123×C53—21no—
20C2×#123×C53—28no—
21C2×#123×C53C2212yes—
22C2×#123×C53C2212yescomplemented
23C2×#123×C53C2214yescomplemented
24C2×#120×C53—21no—
25C2×#120×C53C2212yes—
26C53×#518C2211yescharacteristic
27C22×C4⋊C4×C53—21no—
28C22×C22⋊C4×C53—21no—
29C22×C22⋊C4×C53C2211yescharacteristic, complemented
30C53×#458—28no—
31C2×#99×C53C2212yes—
32C2×#99×C53C2214yescomplemented
33C2×#99×C53C2212yescomplemented
34C53×#386—24no—
35C53×#385—24no—
36C53×#381C2214yescomplemented
37C53×#379C2212yescomplemented
38C53×#342C2214yescomplemented
39C25×C53D411yescharacteristic
40C25×C53C2311yescharacteristic
41H(5,𝔽2)×C53—24no—
42H(5,𝔽2)×C53—42no—
43C2×#41×C53—21no—
44C2×#41×C53—22no—
45C22×D4×C53—21no—
46C22×D4×C53—22no—
47C22×D4×C53—24no—
48C22×D4×C53—22no—
49C22×D4×C53—24no—
50C22×D4×C53—42no—
51C22×D4×C53D411yescharacteristic
52C22×D4×C53D411yescharacteristic
53C22×D4×C53D414yes—
54C22×D4×C53C2312yes—
55C22×D4×C53C2311yescharacteristic
56C22×D4×C53C2311yescharacteristic
57C23×C4×C53—21no—
58C23×C4×C53—21no—
59C23×C4×C53—21no—
60C23×C4×C53—24no—
61C23×C4×C53—24no—
62C23×C4×C53—42no—
63C23×C4×C53D411yescharacteristic
64C23×C4×C53C2312yes—
65#127×C53—22no—
66#124×C53—22no—
67#123×C53—28no—
68#123×C53—28no—
69#123×C53—22no—
70#123×C53—24no—
71#123×C53—24no—
72#123×C53—22no—
73#123×C53—22no—
74#123×C53—24no—
75#123×C53—28no—
76#123×C53—44no—
77#123×C53D418yescomplemented
78C22⋊Q8×C53—22no—
79#121×C53—22no—
80#121×C53—24no—
81#121×C53—24no—
82#121×C53—24no—
83#120×C53—24no—
84#120×C53—22no—
85#120×C53—22no—
86#120×C53—22no—
87#120×C53—22no—
88#120×C53—24no—
89#120×C53—22no—
90C4×D4×C53—24no—
91C4×D4×C53—24no—
92#117×C53—21no—
93C2×C4⋊C4×C53—28no—
94C2×C4⋊C4×C53—22no—
95C2×C4⋊C4×C53—22no—
96C2×C4⋊C4×C53—24no—
97C2×C22⋊C4×C53—28no—
98C2×C22⋊C4×C53—22no—
99C2×C22⋊C4×C53—24no—
100C2×C22⋊C4×C53—22no—
101C2×C22⋊C4×C53—22no—
102C2×C22⋊C4×C53—21no—
103C2×C22⋊C4×C53—21no—
104C2×C22⋊C4×C53—24no—
105C2×C22⋊C4×C53—24no—
106C2×C22⋊C4×C53D414yes—
107C2×C22⋊C4×C53C2314yes—
108C2×C22⋊C4×C53C2312yes—
109C2×C22⋊C4×C53C2312yes—
110#99×C53—28no—
111#99×C53—24no—
112#99×C53—42no—
113#95×C53—28no—
114C24×C53—22no—
115C24×C53—22no—
116C24×C53—21no—
117C24×C53—22no—
118C24×C53—22no—
119C24×C53—44no—
120C24×C53—41no—
121C24×C53—44no—
122C24×C53C2×D411yescharacteristic
123C24×C53C2×D414yes—
124C24×C53C2×D411yescharacteristic
125C24×C53C2×D412yes—
126C24×C53C2×D411yescharacteristic
127C24×C53C2×D411yescharacteristic
128C24×C53C2411yescharacteristic
129#41×C53—48no—
130#41×C53—42no—
131#41×C53—42no—
132C2×Q8×C53—21no—
133C2×D4×C53—22no—
134C2×D4×C53—22no—
135C2×D4×C53—22no—
136C2×D4×C53—21no—
137C2×D4×C53—24no—
138C2×D4×C53—22no—
139C2×D4×C53—22no—
140C2×D4×C53—22no—
141C2×D4×C53—24no—
142C2×D4×C53—22no—
143C2×D4×C53—22no—
144C2×D4×C53—22no—
145C2×D4×C53—22no—
146C2×D4×C53—22no—
147C2×D4×C53—24no—
148C2×D4×C53—24no—
149C2×D4×C53—24no—
150C2×D4×C53—22no—
151C2×D4×C53—41no—
152C2×D4×C53—41no—
153C2×D4×C53—416no—
154C2×D4×C53—48no—
155C2×D4×C53—42no—
156C2×D4×C53—42no—
157C2×D4×C53—48no—
158C2×D4×C53—84no—
159C2×D4×C53C2×D414yes—
160C22×C4×C53—24no—
161C22×C4×C53—24no—
162C22×C4×C53—21no—
163C22×C4×C53—22no—
164C22×C4×C53—24no—
165C22×C4×C53—22no—
166C22×C4×C53—24no—
167C22×C4×C53—22no—
168C22×C4×C53—22no—
169C22×C4×C53—22no—
170C22×C4×C53—24no—
171C22×C4×C53—22no—
172C22×C4×C53—22no—
173C22×C4×C53—24no—
174C22×C4×C53—41no—
175C22×C4×C53—416no—
176C22×C4×C53—44no—
177C22×C4×C53—42no—
178C22×C4×C53—42no—
179C22×C4×C53—42no—
180C22×C4×C53—42no—
181C22×C4×C53C2×D412yes—
182C22×C4×C53C2×D411yescharacteristic
183C22×C4×C53C2×D411yescharacteristic
184C22×C4×C53C2×D414yes—
185C22×C4×C53C2×D414yescomplemented
186C4⋊C4×C53—28no—
187C4⋊C4×C53—22no—
188C4⋊C4×C53—24no—
189C4⋊C4×C53—28no—
190C4⋊C4×C53—48no—
191C22⋊C4×C53—28no—
192C22⋊C4×C53—24no—
193C22⋊C4×C53—22no—
194C22⋊C4×C53—24no—
195C22⋊C4×C53—24no—
196C22⋊C4×C53—28no—
197C22⋊C4×C53—24no—
198C22⋊C4×C53—24no—
199C22⋊C4×C53—22no—
200C22⋊C4×C53—22no—
201C22⋊C4×C53—22no—
202C22⋊C4×C53—24no—
203C22⋊C4×C53—24no—
204C22⋊C4×C53—48no—
205C22⋊C4×C53—48no—
206C22⋊C4×C53C2×D414yescomplemented
207C42×C53—22no—
208C23×C53—21no—
209C23×C53—21no—
210C23×C53—24no—
211C23×C53—21no—
212C23×C53—21no—
213C23×C53—21no—
214C23×C53—28no—
215C23×C53—28no—
216C23×C53—21no—
217C23×C53—21no—
218C23×C53—22no—
219C23×C53—21no—
220C23×C53—21no—
221C23×C53—24no—
222C23×C53—44no—
223C23×C53—42no—
224C23×C53—42no—
225C23×C53—44no—
226C23×C53—48no—
227C23×C53—48no—
228C23×C53—42no—
229C23×C53—42no—
230C23×C53—48no—
231C23×C53—42no—
232C23×C53—44no—
233C23×C53—44no—
234C23×C53—82no—
235C23×C53—84no—
236C23×C53—82no—
237C23×C53#12012yes—
238C23×C53#12012yes—
239C23×C53#12012yes—
240C23×C53C22×D411yescharacteristic
241C23×C53C22×D412yes—
242C23×C53C22×D412yes—
243C23×C53H(5,𝔽2)12yes—
244Q8×C53—42no—
245D4×C53—28no—
246D4×C53—44no—
247D4×C53—42no—
248D4×C53—44no—
249D4×C53—44no—
250D4×C53—42no—
251D4×C53—42no—
252D4×C53—44no—
253D4×C53—44no—
254D4×C53—48no—
255D4×C53—84no—
256D4×C53—84no—
257C2×C4×C53—21no—
258C2×C4×C53—21no—
259C2×C4×C53—24no—
260C2×C4×C53—24no—
261C2×C4×C53—21no—
262C2×C4×C53—21no—
263C2×C4×C53—22no—
264C2×C4×C53—24no—
265C2×C4×C53—21no—
266C2×C4×C53—28no—
267C2×C4×C53—24no—
268C2×C4×C53—44no—
269C2×C4×C53—44no—
270C2×C4×C53—48no—
271C2×C4×C53—44no—
272C2×C4×C53—48no—
273C2×C4×C53—48no—
274C2×C4×C53—42no—
275C2×C4×C53—42no—
276C2×C4×C53—42no—
277C2×C4×C53—44no—
278C2×C4×C53—44no—
279C2×C4×C53—48no—
280C2×C4×C53—44no—
281C2×C4×C53—82no—
282C2×C4×C53#12012yes—
283#23901C5311yesmaximal, characteristic, direct product factor, Sylow
284C22×C53—24no—
285C22×C53—24no—
286C22×C53—22no—
287C22×C53—24no—
288C22×C53—22no—
289C22×C53—22no—
290C22×C53—22no—
291C22×C53—22no—
292C22×C53—24no—
293C22×C53—42no—
294C22×C53—42no—
295C22×C53—416no—
296C22×C53—42no—
297C22×C53—42no—
298C22×C53—42no—
299C22×C53—42no—
300C22×C53—42no—
301C22×C53—42no—
302C22×C53—41no—
303C22×C53—44no—
304C22×C53—42no—
305C22×C53—81no—
306C22×C53—88no—
307C22×C53—84no—
308C22×C53—81no—
309C22×C53—162no—
310C22×C53#45814yescomplemented
311C22×C53C2×#12011yescharacteristic
312C22×C53C2×#12011yescharacteristic
313C22×C53#53411yescharacteristic
314C22×C53D4214yes—
315C4×C53—22no—
316C4×C53—44no—
317C4×C53—44no—
318C4×C53—41no—
319C4×C53—42no—
320C4×C53—44no—
321C4×C53—84no—
322#3484C2×C5311yescharacteristic
323C22×#123C2×C5311yescharacteristic, complemented
324C2×#458C2×C5312yescomplemented
325#2849C2×C5312yescomplemented
326#2022C2×C5314yescomplemented
327#1898C2×C5312yescomplemented
328#1890C2×C5312yescomplemented
329#1781C2×C5311yescharacteristic, complemented
330C2×C53—21no—
331C2×C53—24no—
332C2×C53—21no—
333C2×C53—24no—
334C2×C53—42no—
335C2×C53—42no—
336C2×C53—81no—
337C2×C53—82no—
338C2×C53#240311yescharacteristic
339C2×C53C2×#45812yes—
340C2×H(5,𝔽2)C22×C5311yescharacteristic
341C23×D4C22×C5311yescharacteristic
342C24×C4C22×C5311yescharacteristic, complemented
343#546—24no—
344D42—24no—
345#535—21no—
346#534—21no—
347#534C22×C5312yes—
348#525—21no—
349C2×#123—21no—
350C2×#123—21no—
351C2×#123—28no—
352C2×#123C22×C5312yes—
353C2×#123C22×C5312yescomplemented
354C2×#123C22×C5314yescomplemented
355C2×#120—21no—
356C2×#120C22×C5312yes—
357#518C22×C5311yescharacteristic
358C22×C4⋊C4—21no—
359C22×C22⋊C4—21no—
360C22×C22⋊C4C22×C5311yescharacteristic, complemented
361#458—28no—
362C2×#99C22×C5312yes—
363C2×#99C22×C5314yescomplemented
364C2×#99C22×C5312yescomplemented
365#386—24no—
366#385—24no—
367#381C22×C5314yescomplemented
368#379C22×C5312yescomplemented
369#342C22×C5314yescomplemented
370C53#2390111yescharacteristic, direct product factor, Sylow
371C25D4×C5311yescharacteristic
372C25C23×C5311yescharacteristic
373H(5,𝔽2)—24no—
374H(5,𝔽2)—42no—
375C2×#41—21no—
376C2×#41—22no—
377C22×D4—21no—
378C22×D4—22no—
379C22×D4—24no—
380C22×D4—22no—
381C22×D4—24no—
382C22×D4—42no—
383C22×D4D4×C5311yescharacteristic
384C22×D4D4×C5311yescharacteristic
385C22×D4D4×C5314yes—
386C22×D4C23×C5312yes—
387C22×D4C23×C5311yescharacteristic
388C22×D4C23×C5311yescharacteristic
389C23×C4—21no—
390C23×C4—21no—
391C23×C4—21no—
392C23×C4—24no—
393C23×C4—24no—
394C23×C4—42no—
395C23×C4D4×C5311yescharacteristic
396C23×C4C23×C5312yes—
397#127—22no—
398#124—22no—
399#123—28no—
400#123—28no—
401#123—22no—
402#123—24no—
403#123—22no—
404#123—22no—
405#123—24no—
406#123—24no—
407#123—28no—
408#123—44no—
409#123D4×C5318yescomplemented
410C22⋊Q8—22no—
411#121—22no—
412#121—24no—
413#121—24no—
414#121—24no—
415#120—24no—
416#120—22no—
417#120—22no—
418#120—22no—
419#120—22no—
420#120—24no—
421#120—22no—
422C4×D4—24no—
423C4×D4—24no—
424#117—21no—
425C2×C4⋊C4—28no—
426C2×C4⋊C4—22no—
427C2×C4⋊C4—22no—
428C2×C4⋊C4—24no—
429C2×C22⋊C4—28no—
430C2×C22⋊C4—22no—
431C2×C22⋊C4—22no—
432C2×C22⋊C4—22no—
433C2×C22⋊C4—21no—
434C2×C22⋊C4—21no—
435C2×C22⋊C4—24no—
436C2×C22⋊C4—24no—
437C2×C22⋊C4—24no—
438C2×C22⋊C4D4×C5314yes—
439C2×C22⋊C4C23×C5314yes—
440C2×C22⋊C4C23×C5312yes—
441C2×C22⋊C4C23×C5312yes—
442#99—24no—
443#99—28no—
444#99—42no—
445#95—28no—
446C24—22no—
447C24—22no—
448C24—21no—
449C24—22no—
450C24—22no—
451C24—44no—
452C24—41no—
453C24—44no—
454C24C2×D4×C5311yescharacteristic
455C24C2×D4×C5314yes—
456C24C2×D4×C5311yescharacteristic
457C24C2×D4×C5312yes—
458C24C2×D4×C5311yescharacteristic
459C24C2×D4×C5311yescharacteristic
460C24C24×C5311yescharacteristic
461#41—48no—
462#41—42no—
463#41—42no—
464C2×Q8—21no—
465C2×D4—22no—
466C2×D4—22no—
467C2×D4—22no—
468C2×D4—21no—
469C2×D4—24no—
470C2×D4—22no—
471C2×D4—22no—
472C2×D4—22no—
473C2×D4—24no—
474C2×D4—22no—
475C2×D4—22no—
476C2×D4—22no—
477C2×D4—22no—
478C2×D4—22no—
479C2×D4—24no—
480C2×D4—24no—
481C2×D4—24no—
482C2×D4—22no—
483C2×D4—41no—
484C2×D4—41no—
485C2×D4—416no—
486C2×D4—48no—
487C2×D4—42no—
488C2×D4—42no—
489C2×D4—48no—
490C2×D4—84no—
491C2×D4C2×D4×C5314yes—
492C22×C4—24no—
493C22×C4—24no—
494C22×C4—21no—
495C22×C4—22no—
496C22×C4—24no—
497C22×C4—22no—
498C22×C4—24no—
499C22×C4—22no—
500C22×C4—22no—
501C22×C4—22no—
502C22×C4—24no—
503C22×C4—22no—
504C22×C4—22no—
505C22×C4—24no—
506C22×C4—41no—
507C22×C4—416no—
508C22×C4—44no—
509C22×C4—42no—
510C22×C4—42no—
511C22×C4—42no—
512C22×C4—42no—
513C22×C4C2×D4×C5312yes—
514C22×C4C2×D4×C5311yescharacteristic
515C22×C4C2×D4×C5311yescharacteristic
516C22×C4C2×D4×C5314yes—
517C22×C4C2×D4×C5314yescomplemented
518C4⋊C4—28no—
519C4⋊C4—22no—
520C4⋊C4—24no—
521C4⋊C4—28no—
522C4⋊C4—48no—
523C22⋊C4—24no—
524C22⋊C4—28no—
525C22⋊C4—22no—
526C22⋊C4—24no—
527C22⋊C4—24no—
528C22⋊C4—28no—
529C22⋊C4—24no—
530C22⋊C4—24no—
531C22⋊C4—22no—
532C22⋊C4—22no—
533C22⋊C4—22no—
534C22⋊C4—24no—
535C22⋊C4—24no—
536C22⋊C4—48no—
537C22⋊C4—48no—
538C22⋊C4C2×D4×C5314yescomplemented
539C42—22no—
540C23—21no—
541C23—21no—
542C23—24no—
543C23—21no—
544C23—21no—
545C23—21no—
546C23—28no—
547C23—28no—
548C23—21no—
549C23—21no—
550C23—21no—
551C23—22no—
552C23—21no—
553C23—24no—
554C23—44no—
555C23—42no—
556C23—42no—
557C23—44no—
558C23—48no—
559C23—48no—
560C23—42no—
561C23—42no—
562C23—48no—
563C23—42no—
564C23—44no—
565C23—44no—
566C23—82no—
567C23—84no—
568C23—82no—
569C23#120×C5312yes—
570C23#120×C5312yes—
571C23#120×C5312yes—
572C23C22×D4×C5311yescharacteristic
573C23C22×D4×C5312yes—
574C23C22×D4×C5312yes—
575C23H(5,𝔽2)×C5312yes—
576Q8—42no—
577D4—28no—
578D4—44no—
579D4—42no—
580D4—44no—
581D4—44no—
582D4—42no—
583D4—42no—
584D4—44no—
585D4—44no—
586D4—48no—
587D4—84no—
588D4—84no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—22no—
594C2×C4—24no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—28no—
599C2×C4—24no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—48no—
603C2×C4—44no—
604C2×C4—48no—
605C2×C4—48no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—48no—
612C2×C4—44no—
613C2×C4—82no—
614C2×C4#120×C5312yes—
615C22—24no—
616C22—24no—
617C22—24no—
618C22—22no—
619C22—22no—
620C22—22no—
621C22—24no—
622C22—22no—
623C22—22no—
624C22—42no—
625C22—42no—
626C22—416no—
627C22—42no—
628C22—42no—
629C22—42no—
630C22—42no—
631C22—42no—
632C22—44no—
633C22—42no—
634C22—42no—
635C22—41no—
636C22—81no—
637C22—88no—
638C22—81no—
639C22—84no—
640C22—162no—
641C22C53×#45814yescomplemented
642C22C2×#120×C5311yescharacteristic
643C22C2×#120×C5311yescharacteristic
644C22C53×#53411yescharacteristic
645C22D42×C5314yes—
646C4—22no—
647C4—44no—
648C4—44no—
649C4—44no—
650C4—41no—
651C4—42no—
652C4—84no—
653C2—21no—
654C2—24no—
655C2—21no—
656C2—24no—
657C2—42no—
658C2—42no—
659C2—81no—
660C2—82no—
661C2C53×#240311yescharacteristic
662C2C2×C53×#45812yes—
subgroupG/H
center, socleC22×C53C2×#120
commutator, Frattini, cosocleC24C24×C53
prime radicalC53×#3484C2
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C22×C53 ◁ C25×C53 ◁ G

Complements

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

NH ≅ G/Ntype
2. C22×#123×C53C2semidirect
3. C2×C53×#458C2semidirect
4. C53×#2849C2semidirect
5. C53×#2022C2semidirect
6. C53×#1898C2semidirect
7. C53×#1890C2semidirect
8. C53×#1781C2semidirect
11. C24×C4×C53C22semidirect
22. C2×#123×C53C22semidirect
23. C2×#123×C53C22semidirect
29. C22×C22⋊C4×C53C22semidirect
32. C2×#99×C53C22semidirect
33. C2×#99×C53C22semidirect
36. C53×#381C22semidirect
37. C53×#379C22semidirect
38. C53×#342C22semidirect
77. #123×C53D4semidirect
185. C22×C4×C53C2×D4semidirect
206. C22⋊C4×C53C2×D4semidirect
283. #23901C53direct (N × H)
310. C22×C53#458semidirect
323. C22×#123C2×C53semidirect
324. C2×#458C2×C53semidirect
325. #2849C2×C53semidirect
326. #2022C2×C53semidirect
327. #1898C2×C53semidirect
328. #1890C2×C53semidirect
329. #1781C2×C53semidirect
342. C24×C4C22×C53semidirect
353. C2×#123C22×C53semidirect
354. C2×#123C22×C53semidirect
360. C22×C22⋊C4C22×C53semidirect
363. C2×#99C22×C53semidirect
364. C2×#99C22×C53semidirect
367. #381C22×C53semidirect
368. #379C22×C53semidirect
369. #342C22×C53semidirect
370. C53#23901direct (N × H)
409. #123D4×C53semidirect
517. C22×C4C2×D4×C53semidirect
538. C22⋊C4C2×D4×C53semidirect
641. C22C53×#458semidirect

Maximal subgroups

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

subgroupindex
C53×#34842
C22×#123×C532
C2×C53×#4582
C53×#28492
C53×#20222
C53×#18982
C53×#18902
C53×#17812
#2390153