Finite Group Databasebeta

← C53×#18265 #27225777

Subgroups — C53×#18265 #27225777

Order 13568.

1124 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
1C53×#3537C211yesmaximal, characteristic
2C53×#3092C211yesmaximal, characteristic, complemented
3C2×C53×#481C211yesmaximal, characteristic, complemented
4C53×#2501C211yesmaximal, characteristic, complemented
5C53×#1618C214yesmaximal, complemented
6C53×#1613C214yesmaximal, complemented
7C53×#1507C211yesmaximal, characteristic, complemented
8C53×#1501C211yesmaximal, characteristic, complemented
9C53×#1471C211yesmaximal, characteristic, complemented
10C53×#560—22no—
11C53×#553—21no—
12C53×#553—21no—
13D42×C53—22no—
14D42×C53—21no—
15D42×C53—21no—
16C53×#540—21no—
17C53×#538—21no—
18C53×#538C2212yes—
19C53×#535—21no—
20C53×#534—21no—
21C53×#534C2212yes—
22C53×#525C2211yescharacteristic
23C2×#123×C53—22no—
24C2×#121×C53C2211yescharacteristic
25C2×#121×C53C2211yescharacteristic
26C22×C4⋊C4×C53C2211yescharacteristic, complemented
27C53×#484C2212yescomplemented
28C53×#483—21no—
29C53×#481—22no—
30C53×#481C2214yescomplemented
31C53×#481C2212yescomplemented
32C53×#480—21no—
33C53×#428—21no—
34C2×#106×C53—21no—
35C53×#417—21no—
36C2×#102×C53—21no—
37C53×#407C2211yescharacteristic, complemented
38C2×C22⋊C8×C53C2211yescharacteristic, complemented
39C53×#399C2214yescomplemented
40C53×#390—21no—
41C53×#384—21no—
42C4×C22⋊C4×C53C2211yescharacteristic, complemented
43C53×#328C2212yescomplemented
44C53×#327C2212yescomplemented
45C53×#327C2214yescomplemented
46C53×#323C2214yescomplemented
47C2×#41×C53D411yescharacteristic
48C22×D4×C53—22no—
49C22×D4×C53—21no—
50C22×D4×C53—21no—
51C22×D4×C53—22no—
52C22×D4×C53D412yes—
53C22×D4×C53D411yescharacteristic
54C22×D4×C53D411yescharacteristic
55C23×C4×C53—21no—
56C23×C4×C53C2311yescharacteristic
57C2×M16×C53—21no—
58C22×C8×C53—21no—
59#127×C53—22no—
60#127×C53—21no—
61#126×C53—22no—
62#126×C53—22no—
63#125×C53—21no—
64#124×C53—22no—
65#123×C53—22no—
66#123×C53—22no—
67#123×C53—22no—
68#123×C53—22no—
69#123×C53—21no—
70#123×C53—48no—
71C22⋊Q8×C53D411yescharacteristic
72C22⋊Q8×C53C2311yescharacteristic
73#121×C53—22no—
74#121×C53—21no—
75#121×C53—21no—
76#121×C53—22no—
77#121×C53—22no—
78#121×C53—22no—
79#121×C53—22no—
80#121×C53—21no—
81#121×C53—21no—
82#121×C53—21no—
83#121×C53—21no—
84#121×C53—21no—
85#121×C53—21no—
86#121×C53—22no—
87#121×C53D412yes—
88#121×C53D411yescharacteristic
89#121×C53C2312yes—
90#121×C53C2311yescharacteristic
91#120×C53—22no—
92#120×C53—22no—
93#120×C53—22no—
94#120×C53—21no—
95#120×C53—22no—
96#120×C53—22no—
97#120×C53—22no—
98C4×D4×C53—22no—
99C4×D4×C53—21no—
100C4×D4×C53—21no—
101C4×D4×C53—21no—
102C4×D4×C53—21no—
103C4×D4×C53—22no—
104#117×C53—21no—
105#117×C53D411yescharacteristic
106C2×C4⋊C4×C53—21no—
107C2×C4⋊C4×C53—24no—
108C2×C4⋊C4×C53—21no—
109C2×C4⋊C4×C53—21no—
110C2×C4⋊C4×C53—21no—
111C2×C4⋊C4×C53—21no—
112C2×C4⋊C4×C53D411yescharacteristic
113C2×C4⋊C4×C53D411yescharacteristic
114C2×C4⋊C4×C53C2312yes—
115C2×C22⋊C4×C53—21no—
116C2×C22⋊C4×C53—24no—
117C2×C22⋊C4×C53—21no—
118C2×C22⋊C4×C53C2312yes—
119C2×C42×C53—21no—
120#107×C53—22no—
121#106×C53—24no—
122#106×C53—22no—
123#103×C53—22no—
124#102×C53—24no—
125#102×C53—22no—
126C22⋊C8×C53—21no—
127C22⋊C8×C53—21no—
128C22⋊C8×C53C2312yes—
129C22⋊C8×C53C2312yes—
130#95×C53C2312yes—
131C24×C53—22no—
132C24×C53—21no—
133C24×C53C2×D411yescharacteristic
134C24×C53C2×D412yes—
135#41×C53—42no—
136C2×Q8×C53C2×D411yescharacteristic
137C2×D4×C53—22no—
138C2×D4×C53—21no—
139C2×D4×C53—21no—
140C2×D4×C53—22no—
141C2×D4×C53—22no—
142C2×D4×C53—22no—
143C2×D4×C53—22no—
144C2×D4×C53—22no—
145C2×D4×C53—22no—
146C2×D4×C53—21no—
147C2×D4×C53—22no—
148C2×D4×C53—22no—
149C2×D4×C53—21no—
150C2×D4×C53—21no—
151C2×D4×C53—21no—
152C2×D4×C53—42no—
153C2×D4×C53—42no—
154C2×D4×C53—44no—
155C2×D4×C53—82no—
156C2×D4×C53—82no—
157C2×D4×C53—84no—
158C2×D4×C53—84no—
159C2×D4×C53C2×D411yescharacteristic
160C2×D4×C53C2×D412yes—
161C22×C4×C53—22no—
162C22×C4×C53—22no—
163C22×C4×C53—22no—
164C22×C4×C53—21no—
165C22×C4×C53—21no—
166C22×C4×C53—22no—
167C22×C4×C53—42no—
168C22×C4×C53—44no—
169C22×C4×C53—41no—
170C22×C4×C53—41no—
171C22×C4×C53C2×D411yescharacteristic
172C22×C4×C53C2×D411yescharacteristic
173C22×C4×C53C2×D411yescharacteristic
174C22×C4×C53C2×D412yes—
175C22×C4×C53C2×D411yescharacteristic
176C22×C4×C53C2×D411yescharacteristic
177C22×C4×C53C2411yescharacteristic
178M16×C53—42no—
179C2×C8×C53—22no—
180C2×C8×C53—22no—
181C2×C8×C53—42no—
182C4⋊C4×C53—22no—
183C4⋊C4×C53—22no—
184C4⋊C4×C53—21no—
185C4⋊C4×C53—21no—
186C4⋊C4×C53—21no—
187C4⋊C4×C53—21no—
188C4⋊C4×C53—21no—
189C4⋊C4×C53—22no—
190C4⋊C4×C53—22no—
191C4⋊C4×C53—48no—
192C4⋊C4×C53—41no—
193C4⋊C4×C53—41no—
194C4⋊C4×C53C2×D412yes—
195C4⋊C4×C53C2×D411yescharacteristic
196C4⋊C4×C53C2×D411yescharacteristic
197C22⋊C4×C53—24no—
198C22⋊C4×C53—22no—
199C22⋊C4×C53—22no—
200C22⋊C4×C53—22no—
201C22⋊C4×C53—22no—
202C22⋊C4×C53—22no—
203C22⋊C4×C53—21no—
204C22⋊C4×C53—21no—
205C22⋊C4×C53—21no—
206C22⋊C4×C53—22no—
207C22⋊C4×C53—21no—
208C22⋊C4×C53—22no—
209C22⋊C4×C53—22no—
210C22⋊C4×C53—22no—
211C22⋊C4×C53—21no—
212C22⋊C4×C53—48no—
213C42×C53—22no—
214C42×C53—21no—
215C42×C53—42no—
216C23×C53—22no—
217C23×C53—22no—
218C23×C53—22no—
219C23×C53—22no—
220C23×C53—21no—
221C23×C53—24no—
222C23×C53—41no—
223C23×C53—42no—
224C23×C53—42no—
225C23×C53—42no—
226C23×C53—81no—
227C23×C53—81no—
228C23×C53—82no—
229C23×C53—82no—
230C23×C53—82no—
231C23×C53#12011yescharacteristic
232C23×C53#12011yescharacteristic
233C23×C53C22×D411yescharacteristic
234Q8×C53—41no—
235D4×C53—42no—
236D4×C53—41no—
237D4×C53—44no—
238D4×C53—42no—
239D4×C53—44no—
240D4×C53—42no—
241D4×C53—42no—
242D4×C53—42no—
243D4×C53—81no—
244D4×C53—82no—
245D4×C53—82no—
246D4×C53—81no—
247D4×C53—82no—
248D4×C53—82no—
249D4×C53—82no—
250D4×C53—82no—
251D4×C53—82no—
252C2×C4×C53—22no—
253C2×C4×C53—22no—
254C2×C4×C53—22no—
255C2×C4×C53—24no—
256C2×C4×C53—21no—
257C2×C4×C53—22no—
258C2×C4×C53—21no—
259C2×C4×C53—22no—
260C2×C4×C53—21no—
261C2×C4×C53—41no—
262C2×C4×C53—41no—
263C2×C4×C53—41no—
264C2×C4×C53—42no—
265C2×C4×C53—42no—
266C2×C4×C53—42no—
267C2×C4×C53—42no—
268C2×C4×C53—44no—
269C2×C4×C53—42no—
270C2×C4×C53—81no—
271C2×C4×C53—82no—
272C2×C4×C53—82no—
273C2×C4×C53—82no—
274C2×C4×C53—82no—
275C2×C4×C53#12011yescharacteristic
276C2×C4×C53#12011yescharacteristic
277C2×C4×C53C22×D411yescharacteristic
278C2×C4×C53C22×D411yescharacteristic
279C8×C53—42no—
280C8×C53—42no—
281#18265C5311yesmaximal, characteristic, direct product factor, Sylow
282C22×C53—21no—
283C22×C53—21no—
284C22×C53—21no—
285C22×C53—21no—
286C22×C53—21no—
287C22×C53—21no—
288C22×C53—21no—
289C22×C53—44no—
290C22×C53—42no—
291C22×C53—42no—
292C22×C53—41no—
293C22×C53—42no—
294C22×C53—42no—
295C22×C53—41no—
296C22×C53—41no—
297C22×C53—42no—
298C22×C53—42no—
299C22×C53—84no—
300C22×C53—162no—
301C22×C53—164no—
302C22×C53#45312yes—
303C22×C53C2×#12011yescharacteristic
304C22×C53#57712yes—
305C4×C53—21no—
306C4×C53—21no—
307C4×C53—21no—
308C4×C53—44no—
309C4×C53—42no—
310C4×C53—41no—
311C4×C53—42no—
312C4×C53—41no—
313C4×C53—41no—
314C4×C53—82no—
315C4×C53#45812yes—
316#3537C2×C5311yescharacteristic
317#3092C2×C5311yescharacteristic, complemented
318C2×#481C2×C5311yescharacteristic, complemented
319#2501C2×C5311yescharacteristic, complemented
320#1618C2×C5314yescomplemented
321#1613C2×C5314yescomplemented
322#1507C2×C5311yescharacteristic, complemented
323#1501C2×C5311yescharacteristic, complemented
324#1471C2×C5311yescharacteristic, complemented
325C2×C53—22no—
326C2×C53—22no—
327C2×C53—41no—
328C2×C53—82no—
329C2×C53—82no—
330C2×C53—81no—
331C2×C53#300911yescharacteristic
332C2×C53C2×#45311yescharacteristic
333C2×C53C2×#45811yescharacteristic
334#560—22no—
335#553—21no—
336#553—21no—
337D42—22no—
338D42—21no—
339D42—21no—
340#540—21no—
341#538—21no—
342#538C22×C5312yes—
343#535—21no—
344#534—21no—
345#534C22×C5312yes—
346#525C22×C5311yescharacteristic
347C2×#123—22no—
348C2×#121C22×C5311yescharacteristic
349C2×#121C22×C5311yescharacteristic
350C22×C4⋊C4C22×C5311yescharacteristic, complemented
351#484C22×C5312yescomplemented
352#483—21no—
353#481—22no—
354#481C22×C5314yescomplemented
355#481C22×C5312yescomplemented
356#480—21no—
357#428—21no—
358C2×#106—21no—
359#417—21no—
360C2×#102—21no—
361#407C22×C5311yescharacteristic, complemented
362C2×C22⋊C8C22×C5311yescharacteristic, complemented
363#399C22×C5314yescomplemented
364#390—21no—
365#384—21no—
366C4×C22⋊C4C22×C5311yescharacteristic, complemented
367#328C22×C5312yescomplemented
368#327C22×C5312yescomplemented
369#327C22×C5314yescomplemented
370#323C22×C5314yescomplemented
371C53#1826511yescharacteristic, direct product factor, Sylow
372C2×#41D4×C5311yescharacteristic
373C22×D4—22no—
374C22×D4—21no—
375C22×D4—21no—
376C22×D4—22no—
377C22×D4D4×C5312yes—
378C22×D4D4×C5311yescharacteristic
379C22×D4D4×C5311yescharacteristic
380C23×C4—21no—
381C23×C4C23×C5311yescharacteristic
382C2×M16—21no—
383C22×C8—21no—
384#127—22no—
385#127—21no—
386#126—22no—
387#126—22no—
388#125—21no—
389#124—22no—
390#123—22no—
391#123—21no—
392#123—22no—
393#123—22no—
394#123—22no—
395#123—48no—
396C22⋊Q8D4×C5311yescharacteristic
397C22⋊Q8C23×C5311yescharacteristic
398#121—22no—
399#121—21no—
400#121—21no—
401#121—22no—
402#121—22no—
403#121—22no—
404#121—22no—
405#121—21no—
406#121—21no—
407#121—21no—
408#121—21no—
409#121—21no—
410#121—21no—
411#121—22no—
412#121D4×C5312yes—
413#121D4×C5311yescharacteristic
414#121C23×C5312yes—
415#121C23×C5311yescharacteristic
416#120—22no—
417#120—22no—
418#120—22no—
419#120—21no—
420#120—22no—
421#120—22no—
422#120—22no—
423C4×D4—22no—
424C4×D4—21no—
425C4×D4—21no—
426C4×D4—21no—
427C4×D4—21no—
428C4×D4—22no—
429#117—21no—
430#117D4×C5311yescharacteristic
431C2×C4⋊C4—21no—
432C2×C4⋊C4—21no—
433C2×C4⋊C4—21no—
434C2×C4⋊C4—21no—
435C2×C4⋊C4—21no—
436C2×C4⋊C4—24no—
437C2×C4⋊C4D4×C5311yescharacteristic
438C2×C4⋊C4D4×C5311yescharacteristic
439C2×C4⋊C4C23×C5312yes—
440C2×C22⋊C4—21no—
441C2×C22⋊C4—21no—
442C2×C22⋊C4—24no—
443C2×C22⋊C4C23×C5312yes—
444C2×C42—21no—
445#107—22no—
446#106—24no—
447#106—22no—
448#103—22no—
449#102—24no—
450#102—22no—
451C22⋊C8—21no—
452C22⋊C8—21no—
453C22⋊C8C23×C5312yes—
454C22⋊C8C23×C5312yes—
455#95C23×C5312yes—
456C24—22no—
457C24—21no—
458C24C2×D4×C5311yescharacteristic
459C24C2×D4×C5312yes—
460#41—42no—
461C2×Q8C2×D4×C5311yescharacteristic
462C2×D4—22no—
463C2×D4—21no—
464C2×D4—21no—
465C2×D4—22no—
466C2×D4—22no—
467C2×D4—22no—
468C2×D4—22no—
469C2×D4—22no—
470C2×D4—22no—
471C2×D4—21no—
472C2×D4—22no—
473C2×D4—22no—
474C2×D4—21no—
475C2×D4—21no—
476C2×D4—21no—
477C2×D4—42no—
478C2×D4—42no—
479C2×D4—44no—
480C2×D4—82no—
481C2×D4—82no—
482C2×D4—84no—
483C2×D4—84no—
484C2×D4C2×D4×C5311yescharacteristic
485C2×D4C2×D4×C5312yes—
486C22×C4—22no—
487C22×C4—22no—
488C22×C4—22no—
489C22×C4—21no—
490C22×C4—21no—
491C22×C4—22no—
492C22×C4—42no—
493C22×C4—44no—
494C22×C4—41no—
495C22×C4—41no—
496C22×C4C2×D4×C5311yescharacteristic
497C22×C4C2×D4×C5311yescharacteristic
498C22×C4C2×D4×C5311yescharacteristic
499C22×C4C2×D4×C5312yes—
500C22×C4C2×D4×C5311yescharacteristic
501C22×C4C2×D4×C5311yescharacteristic
502C22×C4C24×C5311yescharacteristic
503M16—42no—
504C2×C8—22no—
505C2×C8—22no—
506C2×C8—42no—
507C4⋊C4—22no—
508C4⋊C4—22no—
509C4⋊C4—21no—
510C4⋊C4—21no—
511C4⋊C4—21no—
512C4⋊C4—21no—
513C4⋊C4—21no—
514C4⋊C4—22no—
515C4⋊C4—22no—
516C4⋊C4—48no—
517C4⋊C4—41no—
518C4⋊C4—41no—
519C4⋊C4C2×D4×C5312yes—
520C4⋊C4C2×D4×C5311yescharacteristic
521C4⋊C4C2×D4×C5311yescharacteristic
522C22⋊C4—24no—
523C22⋊C4—22no—
524C22⋊C4—22no—
525C22⋊C4—22no—
526C22⋊C4—22no—
527C22⋊C4—22no—
528C22⋊C4—21no—
529C22⋊C4—21no—
530C22⋊C4—21no—
531C22⋊C4—22no—
532C22⋊C4—21no—
533C22⋊C4—22no—
534C22⋊C4—22no—
535C22⋊C4—22no—
536C22⋊C4—21no—
537C22⋊C4—48no—
538C42—22no—
539C42—21no—
540C42—42no—
541C23—22no—
542C23—22no—
543C23—22no—
544C23—22no—
545C23—21no—
546C23—24no—
547C23—41no—
548C23—42no—
549C23—42no—
550C23—42no—
551C23—81no—
552C23—81no—
553C23—82no—
554C23—82no—
555C23—82no—
556C23#120×C5311yescharacteristic
557C23#120×C5311yescharacteristic
558C23C22×D4×C5311yescharacteristic
559Q8—41no—
560D4—42no—
561D4—41no—
562D4—44no—
563D4—42no—
564D4—44no—
565D4—42no—
566D4—42no—
567D4—42no—
568D4—81no—
569D4—82no—
570D4—82no—
571D4—81no—
572D4—82no—
573D4—82no—
574D4—82no—
575D4—82no—
576D4—82no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—24no—
581C2×C4—21no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—44no—
594C2×C4—42no—
595C2×C4—81no—
596C2×C4—82no—
597C2×C4—82no—
598C2×C4—82no—
599C2×C4—82no—
600C2×C4#120×C5311yescharacteristic
601C2×C4#120×C5311yescharacteristic
602C2×C4C22×D4×C5311yescharacteristic
603C2×C4C22×D4×C5311yescharacteristic
604C8—42no—
605C8—42no—
606C22—21no—
607C22—21no—
608C22—21no—
609C22—21no—
610C22—21no—
611C22—21no—
612C22—21no—
613C22—44no—
614C22—42no—
615C22—42no—
616C22—41no—
617C22—42no—
618C22—42no—
619C22—41no—
620C22—41no—
621C22—42no—
622C22—42no—
623C22—84no—
624C22—162no—
625C22—164no—
626C22C53×#45312yes—
627C22C2×#120×C5311yescharacteristic
628C22C53×#57712yes—
629C4—21no—
630C4—21no—
631C4—21no—
632C4—44no—
633C4—42no—
634C4—41no—
635C4—42no—
636C4—41no—
637C4—41no—
638C4—82no—
639C4C53×#45812yes—
640C2—22no—
641C2—22no—
642C2—41no—
643C2—82no—
644C2—82no—
645C2—81no—
646C2C53×#300911yescharacteristic
647C2C2×C53×#45311yescharacteristic
648C2C2×C53×#45811yescharacteristic
subgroupG/H
center, socleC22×C53C2×#120
commutator, Frattini, cosocleC22×C4C24×C53
prime radicalC53×#3537C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C53 ◁ C23×C4×C53 ◁ G

Complements

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

NH ≅ G/Ntype
2. C53×#3092C2semidirect
3. C2×C53×#481C2semidirect
4. C53×#2501C2semidirect
5. C53×#1618C2semidirect
6. C53×#1613C2semidirect
7. C53×#1507C2semidirect
8. C53×#1501C2semidirect
9. C53×#1471C2semidirect
26. C22×C4⋊C4×C53C22semidirect
27. C53×#484C22semidirect
30. C53×#481C22semidirect
31. C53×#481C22semidirect
37. C53×#407C22semidirect
38. C2×C22⋊C8×C53C22semidirect
39. C53×#399C22semidirect
42. C4×C22⋊C4×C53C22semidirect
43. C53×#328C22semidirect
44. C53×#327C22semidirect
45. C53×#327C22semidirect
46. C53×#323C22semidirect
281. #18265C53direct (N × H)
317. #3092C2×C53semidirect
318. C2×#481C2×C53semidirect
319. #2501C2×C53semidirect
320. #1618C2×C53semidirect
321. #1613C2×C53semidirect
322. #1507C2×C53semidirect
323. #1501C2×C53semidirect
324. #1471C2×C53semidirect
350. C22×C4⋊C4C22×C53semidirect
351. #484C22×C53semidirect
354. #481C22×C53semidirect
355. #481C22×C53semidirect
361. #407C22×C53semidirect
362. C2×C22⋊C8C22×C53semidirect
363. #399C22×C53semidirect
366. C4×C22⋊C4C22×C53semidirect
367. #328C22×C53semidirect
368. #327C22×C53semidirect
369. #327C22×C53semidirect
370. #323C22×C53semidirect
371. C53#18265direct (N × H)

Maximal subgroups

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

subgroupindex
C53×#35372
C53×#30922
C2×C53×#4812
C53×#25012
C53×#16182
C53×#16132
C53×#15072
C53×#15012
C53×#14712
#1826553