Finite Group Databasebeta

← C53×#18460 #27225972

Subgroups — C53×#18460 #27225972

Order 13568.

936 conjugacy classes of subgroups.

The subgroup lattice has 670 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C53×#3559C211yesmaximal, characteristic, complemented
2C53×#3213C211yesmaximal, characteristic, complemented
3C53×#3211C211yesmaximal, characteristic, complemented
4C53×#2820C211yesmaximal, characteristic, complemented
5C53×#1604C212yesmaximal, complemented
6C53×#1603C212yesmaximal, complemented
7C53×#1598C212yesmaximal, complemented
8C53×#1597C212yesmaximal, complemented
9C53×#1517C211yesmaximal, characteristic, complemented
10C53×#1512C211yesmaximal, characteristic, complemented
11C53×#1470C211yesmaximal, characteristic, complemented
12C53×#563—21no—
13C53×#562—21no—
14C53×#554—21no—
15C53×#553—21no—
16C53×#553—21no—
17C53×#547—22no—
18C53×#547—21no—
19C53×#544—21no—
20C53×#540—21no—
21C53×#539—22no—
22C53×#539C2212yes—
23C53×#537C2212yes—
24C53×#532C2211yescharacteristic
25C53×#529C2211yescharacteristic
26C2×#125×C53C2211yescharacteristic
27C2×#124×C53C2211yescharacteristic
28C53×#496—21no—
29C53×#495—21no—
30C53×#489—21no—
31C53×#488—21no—
32C53×#464—21no—
33C53×#463—21no—
34C53×#463—22no—
35C53×#451C2211yescharacteristic
36C53×#450C2211yescharacteristic
37C53×#448C2211yescharacteristic
38C53×#448C2212yes—
39C53×#447C2211yescharacteristic
40C53×#447C2212yes—
41C53×#432C2211yescharacteristic
42C53×#431C2211yescharacteristic
43C53×#399—21no—
44C53×#397—21no—
45C53×#396—22no—
46C53×#395C2212yes—
47C53×#394C2212yes—
48C53×#382C2211yescharacteristic
49C53×#328C2212yes—
50C53×#327C2212yes—
51C53×#327C2212yes—
52C53×#327C2212yes—
53C53×#324C2212yes—
54C53×#324C2212yes—
55C2×#41×C53—21no—
56C2×#41×C53—22no—
57C22×Q8×C53C2311yescharacteristic
58C22×D4×C53C2311yescharacteristic
59C2×Q16×C53—21no—
60C2×SD16×C53—21no—
61C2×SD16×C53—21no—
62C2×SD16×C53—22no—
63C2×D8×C53—21no—
64C2×D8×C53—22no—
65C4⋊Q8×C53—21no—
66C4⋊Q8×C53—22no—
67C4⋊Q8×C53D411yescharacteristic
68#127×C53D411yescharacteristic
69#126×C53—22no—
70#126×C53—22no—
71#126×C53—21no—
72#125×C53—21no—
73#125×C53—22no—
74#125×C53—21no—
75#125×C53—21no—
76#125×C53—21no—
77#125×C53—21no—
78#125×C53D411yescharacteristic
79#124×C53—21no—
80#124×C53—21no—
81#124×C53—21no—
82#124×C53—21no—
83#124×C53—21no—
84#124×C53D411yescharacteristic
85#123×C53—22no—
86#123×C53—22no—
87#123×C53—22no—
88#123×C53—22no—
89#123×C53—22no—
90#123×C53—21no—
91#123×C53—22no—
92C22⋊Q8×C53—22no—
93C22⋊Q8×C53—22no—
94C22⋊Q8×C53—22no—
95C22⋊Q8×C53—22no—
96C22⋊Q8×C53—22no—
97C22⋊Q8×C53—22no—
98C22⋊Q8×C53C2311yescharacteristic
99#121×C53—21no—
100#121×C53—22no—
101#121×C53—21no—
102#121×C53—22no—
103#121×C53—22no—
104#121×C53—21no—
105#121×C53—22no—
106#121×C53C2311yescharacteristic
107#121×C53C2312yes—
108C4×Q8×C53D411yescharacteristic
109C4×D4×C53—22no—
110C4×D4×C53—21no—
111C4×D4×C53—21no—
112C4×D4×C53—21no—
113C4×D4×C53—21no—
114C4×D4×C53—22no—
115C4×D4×C53D411yescharacteristic
116C4×D4×C53D412yes—
117#117×C53—21no—
118#117×C53—21no—
119C2×C4⋊C4×C53—22no—
120C2×C4⋊C4×C53—21no—
121C2×C4⋊C4×C53—21no—
122C2×C4⋊C4×C53—22no—
123C2×C4⋊C4×C53D412yes—
124C2×C4⋊C4×C53D411yescharacteristic
125C2×C4⋊C4×C53D411yescharacteristic
126C2×C4⋊C4×C53C2312yes—
127C2×C22⋊C4×C53—21no—
128C2×C22⋊C4×C53—21no—
129C2×C22⋊C4×C53—22no—
130C2×C22⋊C4×C53—22no—
131C2×C42×C53C2311yescharacteristic
132C4⋊C8×C53—21no—
133C4⋊C8×C53—21no—
134#103×C53—21no—
135#103×C53—21no—
136#103×C53—22no—
137#102×C53—21no—
138#102×C53—21no—
139#102×C53—22no—
140C22⋊C8×C53C2311yescharacteristic
141C22⋊C8×C53C2311yescharacteristic
142C22⋊C8×C53C2312yes—
143#97×C53—21no—
144C4×C8×C53—21no—
145#95×C53—21no—
146#95×C53—22no—
147#95×C53C2312yes—
148C24×C53—21no—
149#41×C53—81no—
150#41×C53—81no—
151#41×C53—82no—
152#41×C53—82no—
153C2×Q8×C53—22no—
154C2×Q8×C53—21no—
155C2×Q8×C53—21no—
156C2×Q8×C53—21no—
157C2×Q8×C53—81no—
158C2×Q8×C53C2×D411yescharacteristic
159C2×D4×C53—21no—
160C2×D4×C53—22no—
161C2×D4×C53—22no—
162C2×D4×C53—22no—
163C2×D4×C53—21no—
164C2×D4×C53—21no—
165C2×D4×C53—21no—
166C2×D4×C53—21no—
167C2×D4×C53—81no—
168C2×D4×C53C2×D411yescharacteristic
169C2×D4×C53C2×D412yes—
170C22×C4×C53—22no—
171C22×C4×C53—22no—
172C22×C4×C53—22no—
173C22×C4×C53—21no—
174C22×C4×C53—22no—
175C22×C4×C53—22no—
176C22×C4×C53—21no—
177C22×C4×C53C2×D412yes—
178C22×C4×C53C2×D411yescharacteristic
179C22×C4×C53C2×D412yes—
180C22×C4×C53C2×D411yescharacteristic
181C22×C4×C53C2411yescharacteristic
182Q16×C53—81no—
183SD16×C53—82no—
184SD16×C53—81no—
185SD16×C53—81no—
186D8×C53—82no—
187D8×C53—81no—
188C2×C8×C53—21no—
189C2×C8×C53—21no—
190C2×C8×C53—22no—
191C4⋊C4×C53—22no—
192C4⋊C4×C53—22no—
193C4⋊C4×C53—22no—
194C4⋊C4×C53—22no—
195C4⋊C4×C53—21no—
196C4⋊C4×C53—21no—
197C4⋊C4×C53—22no—
198C4⋊C4×C53—21no—
199C4⋊C4×C53—21no—
200C4⋊C4×C53—22no—
201C4⋊C4×C53—21no—
202C4⋊C4×C53—21no—
203C4⋊C4×C53—21no—
204C4⋊C4×C53—21no—
205C4⋊C4×C53—22no—
206C4⋊C4×C53—22no—
207C4⋊C4×C53—21no—
208C4⋊C4×C53—81no—
209C4⋊C4×C53—82no—
210C4⋊C4×C53C2×D412yes—
211C4⋊C4×C53C2×D411yescharacteristic
212C4⋊C4×C53C2×D411yescharacteristic
213C22⋊C4×C53—22no—
214C22⋊C4×C53—22no—
215C22⋊C4×C53—22no—
216C22⋊C4×C53—22no—
217C22⋊C4×C53—22no—
218C22⋊C4×C53—21no—
219C22⋊C4×C53—21no—
220C22⋊C4×C53—21no—
221C22⋊C4×C53—21no—
222C22⋊C4×C53—22no—
223C22⋊C4×C53—21no—
224C22⋊C4×C53—22no—
225C22⋊C4×C53—41no—
226C22⋊C4×C53—41no—
227C22⋊C4×C53—41no—
228C22⋊C4×C53—41no—
229C22⋊C4×C53—82no—
230C22⋊C4×C53—82no—
231C42×C53—22no—
232C42×C53—21no—
233C42×C53C2×D411yescharacteristic
234C42×C53C2×D411yescharacteristic
235C42×C53C2×D411yescharacteristic
236C42×C53C2×D411yescharacteristic
237C23×C53—21no—
238C23×C53—22no—
239C23×C53—42no—
240C23×C53—42no—
241C23×C53—41no—
242C23×C53—41no—
243C23×C53—41no—
244C23×C53C22×D411yescharacteristic
245Q8×C53—41no—
246Q8×C53—42no—
247Q8×C53—41no—
248Q8×C53—41no—
249Q8×C53—81no—
250Q8×C53—82no—
251D4×C53—42no—
252D4×C53—41no—
253D4×C53—42no—
254D4×C53—42no—
255D4×C53—41no—
256D4×C53—41no—
257D4×C53—42no—
258D4×C53—41no—
259D4×C53—42no—
260D4×C53—41no—
261D4×C53—81no—
262D4×C53—82no—
263D4×C53—82no—
264C2×C4×C53—21no—
265C2×C4×C53—21no—
266C2×C4×C53—22no—
267C2×C4×C53—21no—
268C2×C4×C53—22no—
269C2×C4×C53—21no—
270C2×C4×C53—22no—
271C2×C4×C53—21no—
272C2×C4×C53—22no—
273C2×C4×C53—22no—
274C2×C4×C53—41no—
275C2×C4×C53—41no—
276C2×C4×C53—42no—
277C2×C4×C53—42no—
278C2×C4×C53—42no—
279C2×C4×C53—42no—
280C2×C4×C53—42no—
281C2×C4×C53—41no—
282C2×C4×C53—42no—
283C2×C4×C53—82no—
284C2×C4×C53—82no—
285C2×C4×C53—82no—
286C2×C4×C53—82no—
287C2×C4×C53#12011yescharacteristic
288C2×C4×C53#12011yescharacteristic
289C2×C4×C53#12011yescharacteristic
290C2×C4×C53#12011yescharacteristic
291C2×C4×C53C22×D411yescharacteristic
292C2×C4×C53C22×D411yescharacteristic
293C8×C53—42no—
294C8×C53—41no—
295C8×C53—41no—
296#18460C5311yesmaximal, characteristic, direct product factor, Sylow
297C22×C53—21no—
298C22×C53—21no—
299C22×C53—42no—
300C22×C53—41no—
301C22×C53—41no—
302C22×C53—42no—
303C22×C53—42no—
304C22×C53—41no—
305C22×C53—82no—
306C22×C53—82no—
307C22×C53—81no—
308C22×C53—81no—
309C22×C53—81no—
310C22×C53C2×#12011yescharacteristic
311C22×C53#57611yescharacteristic
312C22×C53#57711yescharacteristic
313C4×C53—21no—
314C4×C53—21no—
315C4×C53—21no—
316C4×C53—21no—
317C4×C53—21no—
318C4×C53—21no—
319C4×C53—42no—
320C4×C53—42no—
321C4×C53—41no—
322C4×C53—41no—
323C4×C53—42no—
324C4×C53—41no—
325C4×C53—82no—
326C4×C53—81no—
327#3559C2×C5311yescharacteristic, complemented
328#3213C2×C5311yescharacteristic, complemented
329#3211C2×C5311yescharacteristic, complemented
330#2820C2×C5311yescharacteristic, complemented
331#1604C2×C5312yescomplemented
332#1603C2×C5312yescomplemented
333#1598C2×C5312yescomplemented
334#1597C2×C5312yescomplemented
335#1517C2×C5311yescharacteristic, complemented
336#1512C2×C5311yescharacteristic, complemented
337#1470C2×C5311yescharacteristic, complemented
338C2×C53—21no—
339C2×C53—21no—
340C2×C53—81no—
341C2×C53—82no—
342C2×C53—161no—
343C2×C53#301211yescharacteristic
344C2×C53#302011yescharacteristic
345C2×C53#302711yescharacteristic
346#563—21no—
347#562—21no—
348#554—21no—
349#553—21no—
350#553—21no—
351#547—22no—
352#547—21no—
353#544—21no—
354#540—21no—
355#539—22no—
356#539C22×C5312yes—
357#537C22×C5312yes—
358#532C22×C5311yescharacteristic
359#529C22×C5311yescharacteristic
360C2×#125C22×C5311yescharacteristic
361C2×#124C22×C5311yescharacteristic
362#496—21no—
363#495—21no—
364#489—21no—
365#488—21no—
366#464—21no—
367#463—21no—
368#463—22no—
369#451C22×C5311yescharacteristic
370#450C22×C5311yescharacteristic
371#448C22×C5312yes—
372#448C22×C5311yescharacteristic
373#447C22×C5312yes—
374#447C22×C5311yescharacteristic
375#432C22×C5311yescharacteristic
376#431C22×C5311yescharacteristic
377#399—21no—
378#397—21no—
379#396—22no—
380#395C22×C5312yes—
381#394C22×C5312yes—
382#382C22×C5311yescharacteristic
383#328C22×C5312yes—
384#327C22×C5312yes—
385#327C22×C5312yes—
386#327C22×C5312yes—
387#324C22×C5312yes—
388#324C22×C5312yes—
389C53#1846011yescharacteristic, direct product factor, Sylow
390C2×#41—21no—
391C2×#41—22no—
392C22×Q8C23×C5311yescharacteristic
393C22×D4C23×C5311yescharacteristic
394C2×Q16—21no—
395C2×SD16—21no—
396C2×SD16—21no—
397C2×SD16—22no—
398C2×D8—21no—
399C2×D8—22no—
400C4⋊Q8—21no—
401C4⋊Q8—22no—
402C4⋊Q8D4×C5311yescharacteristic
403#127D4×C5311yescharacteristic
404#126—22no—
405#126—22no—
406#126—21no—
407#125—21no—
408#125—22no—
409#125—21no—
410#125—21no—
411#125—21no—
412#125—21no—
413#125D4×C5311yescharacteristic
414#124—21no—
415#124—21no—
416#124—21no—
417#124—21no—
418#124—21no—
419#124D4×C5311yescharacteristic
420#123—22no—
421#123—22no—
422#123—22no—
423#123—22no—
424#123—22no—
425#123—21no—
426#123—22no—
427C22⋊Q8—22no—
428C22⋊Q8—22no—
429C22⋊Q8—22no—
430C22⋊Q8—22no—
431C22⋊Q8—22no—
432C22⋊Q8—22no—
433C22⋊Q8C23×C5311yescharacteristic
434#121—21no—
435#121—22no—
436#121—21no—
437#121—22no—
438#121—22no—
439#121—21no—
440#121—22no—
441#121C23×C5311yescharacteristic
442#121C23×C5312yes—
443C4×Q8D4×C5311yescharacteristic
444C4×D4—22no—
445C4×D4—21no—
446C4×D4—21no—
447C4×D4—21no—
448C4×D4—22no—
449C4×D4—21no—
450C4×D4D4×C5311yescharacteristic
451C4×D4D4×C5312yes—
452#117—21no—
453#117—21no—
454C2×C4⋊C4—22no—
455C2×C4⋊C4—21no—
456C2×C4⋊C4—21no—
457C2×C4⋊C4—22no—
458C2×C4⋊C4D4×C5312yes—
459C2×C4⋊C4D4×C5311yescharacteristic
460C2×C4⋊C4D4×C5311yescharacteristic
461C2×C4⋊C4C23×C5312yes—
462C2×C22⋊C4—21no—
463C2×C22⋊C4—21no—
464C2×C22⋊C4—22no—
465C2×C22⋊C4—22no—
466C2×C42C23×C5311yescharacteristic
467C4⋊C8—21no—
468C4⋊C8—21no—
469#103—22no—
470#103—21no—
471#103—21no—
472#102—22no—
473#102—21no—
474#102—21no—
475C22⋊C8C23×C5311yescharacteristic
476C22⋊C8C23×C5311yescharacteristic
477C22⋊C8C23×C5312yes—
478#97—21no—
479C4×C8—21no—
480#95—21no—
481#95—22no—
482#95C23×C5312yes—
483C24—21no—
484#41—81no—
485#41—81no—
486#41—82no—
487#41—82no—
488C2×Q8—22no—
489C2×Q8—21no—
490C2×Q8—21no—
491C2×Q8—21no—
492C2×Q8—81no—
493C2×Q8C2×D4×C5311yescharacteristic
494C2×D4—21no—
495C2×D4—22no—
496C2×D4—22no—
497C2×D4—22no—
498C2×D4—21no—
499C2×D4—21no—
500C2×D4—21no—
501C2×D4—21no—
502C2×D4—81no—
503C2×D4C2×D4×C5311yescharacteristic
504C2×D4C2×D4×C5312yes—
505C22×C4—22no—
506C22×C4—22no—
507C22×C4—22no—
508C22×C4—21no—
509C22×C4—22no—
510C22×C4—22no—
511C22×C4—21no—
512C22×C4C2×D4×C5312yes—
513C22×C4C2×D4×C5311yescharacteristic
514C22×C4C2×D4×C5312yes—
515C22×C4C2×D4×C5311yescharacteristic
516C22×C4C24×C5311yescharacteristic
517Q16—81no—
518SD16—82no—
519SD16—81no—
520SD16—81no—
521D8—82no—
522D8—81no—
523C2×C8—21no—
524C2×C8—21no—
525C2×C8—22no—
526C4⋊C4—22no—
527C4⋊C4—22no—
528C4⋊C4—22no—
529C4⋊C4—22no—
530C4⋊C4—21no—
531C4⋊C4—21no—
532C4⋊C4—22no—
533C4⋊C4—21no—
534C4⋊C4—21no—
535C4⋊C4—22no—
536C4⋊C4—21no—
537C4⋊C4—21no—
538C4⋊C4—21no—
539C4⋊C4—21no—
540C4⋊C4—22no—
541C4⋊C4—22no—
542C4⋊C4—21no—
543C4⋊C4—81no—
544C4⋊C4—82no—
545C4⋊C4C2×D4×C5312yes—
546C4⋊C4C2×D4×C5311yescharacteristic
547C4⋊C4C2×D4×C5311yescharacteristic
548C22⋊C4—22no—
549C22⋊C4—22no—
550C22⋊C4—22no—
551C22⋊C4—22no—
552C22⋊C4—22no—
553C22⋊C4—21no—
554C22⋊C4—21no—
555C22⋊C4—21no—
556C22⋊C4—21no—
557C22⋊C4—22no—
558C22⋊C4—21no—
559C22⋊C4—22no—
560C22⋊C4—41no—
561C22⋊C4—41no—
562C22⋊C4—41no—
563C22⋊C4—41no—
564C22⋊C4—82no—
565C22⋊C4—82no—
566C42—22no—
567C42—21no—
568C42C2×D4×C5311yescharacteristic
569C42C2×D4×C5311yescharacteristic
570C42C2×D4×C5311yescharacteristic
571C42C2×D4×C5311yescharacteristic
572C23—21no—
573C23—22no—
574C23—42no—
575C23—42no—
576C23—41no—
577C23—41no—
578C23—41no—
579C23C22×D4×C5311yescharacteristic
580Q8—41no—
581Q8—42no—
582Q8—41no—
583Q8—41no—
584Q8—81no—
585Q8—82no—
586D4—42no—
587D4—41no—
588D4—42no—
589D4—42no—
590D4—41no—
591D4—41no—
592D4—42no—
593D4—41no—
594D4—42no—
595D4—41no—
596D4—81no—
597D4—82no—
598D4—82no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—21no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—41no—
617C2×C4—42no—
618C2×C4—82no—
619C2×C4—82no—
620C2×C4—82no—
621C2×C4—82no—
622C2×C4#120×C5311yescharacteristic
623C2×C4#120×C5311yescharacteristic
624C2×C4#120×C5311yescharacteristic
625C2×C4#120×C5311yescharacteristic
626C2×C4C22×D4×C5311yescharacteristic
627C2×C4C22×D4×C5311yescharacteristic
628C8—42no—
629C8—41no—
630C8—41no—
631C22—21no—
632C22—21no—
633C22—42no—
634C22—41no—
635C22—41no—
636C22—42no—
637C22—42no—
638C22—41no—
639C22—82no—
640C22—82no—
641C22—81no—
642C22—81no—
643C22—81no—
644C22C2×#120×C5311yescharacteristic
645C22C53×#57611yescharacteristic
646C22C53×#57711yescharacteristic
647C4—21no—
648C4—21no—
649C4—21no—
650C4—21no—
651C4—21no—
652C4—21no—
653C4—42no—
654C4—42no—
655C4—41no—
656C4—41no—
657C4—42no—
658C4—41no—
659C4—82no—
660C4—81no—
661C2—21no—
662C2—21no—
663C2—81no—
664C2—82no—
665C2—161no—
666C2C53×#301211yescharacteristic
667C2C53×#302011yescharacteristic
668C2C53×#302711yescharacteristic
subgroupG/H
center, socleC22×C53C2×#120
commutator, Frattini, cosocleC22×C4C24×C53
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C53 ◁ C2×C42×C53 ◁ G

Complements

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

NH ≅ G/Ntype
1. C53×#3559C2semidirect
2. C53×#3213C2semidirect
3. C53×#3211C2semidirect
4. C53×#2820C2semidirect
5. C53×#1604C2semidirect
6. C53×#1603C2semidirect
7. C53×#1598C2semidirect
8. C53×#1597C2semidirect
9. C53×#1517C2semidirect
10. C53×#1512C2semidirect
11. C53×#1470C2semidirect
296. #18460C53direct (N × H)
327. #3559C2×C53semidirect
328. #3213C2×C53semidirect
329. #3211C2×C53semidirect
330. #2820C2×C53semidirect
331. #1604C2×C53semidirect
332. #1603C2×C53semidirect
333. #1598C2×C53semidirect
334. #1597C2×C53semidirect
335. #1517C2×C53semidirect
336. #1512C2×C53semidirect
337. #1470C2×C53semidirect
389. C53#18460direct (N × H)

Maximal subgroups

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

subgroupindex
C53×#35592
C53×#32132
C53×#32112
C53×#28202
C53×#16042
C53×#16032
C53×#15982
C53×#15972
C53×#15172
C53×#15122
C53×#14702
#1846053