Finite Group Databasebeta

← C53×#18127 #27225639

Subgroups — C53×#18127 #27225639

Order 13568.

950 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C53×#3455C211yesmaximal, characteristic
2C53×#3104C211yesmaximal, characteristic, complemented
3C53×#3103C211yesmaximal, characteristic, complemented
4C53×#2396C211yesmaximal, characteristic, complemented
5C53×#1618C212yesmaximal, complemented
6C53×#1615C212yesmaximal, complemented
7C53×#1606C212yesmaximal, complemented
8C53×#1603C212yesmaximal, complemented
9C53×#1503C211yesmaximal, characteristic, complemented
10C53×#1502C211yesmaximal, characteristic, complemented
11C53×#1458C211yesmaximal, characteristic, complemented
12C53×#542—21no—
13C53×#542—21no—
14C53×#541—21no—
15C53×#540—21no—
16C53×#539—21no—
17C53×#539C2212yes—
18C53×#538—21no—
19C53×#538C2212yes—
20C53×#533C2211yescharacteristic
21C53×#532C2211yescharacteristic
22C53×#529—22no—
23C53×#529—22no—
24C53×#525—21no—
25C4×#41×C53—21no—
26C2×C4×Q8×C53C2211yescharacteristic
27C2×#117×C53C2211yescharacteristic
28C53×#484C2212yes—
29C53×#481C2212yes—
30C53×#477—21no—
31C53×#475—21no—
32C53×#462—21no—
33C53×#460—21no—
34C53×#451C2212yes—
35C53×#448C2212yes—
36C4×Q16×C53—22no—
37C4×SD16×C53—22no—
38C53×#431C2211yescharacteristic
39C53×#431C2211yescharacteristic
40C53×#400C2211yescharacteristic
41C53×#399C2211yescharacteristic
42C53×#398—21no—
43C53×#396—21no—
44C53×#395C2211yescharacteristic
45C53×#393C2211yescharacteristic
46C4×C4⋊C4×C53—21no—
47C4×C4⋊C4×C53C2211yescharacteristic
48C4×C22⋊C4×C53—21no—
49C53×#328C2212yes—
50C53×#328C2212yes—
51C53×#327C2212yes—
52C53×#327C2212yes—
53C53×#324C2212yes—
54C53×#323C2212yes—
55C2×#41×C53—21no—
56C22×Q8×C53C2311yescharacteristic
57C23×C4×C53D411yescharacteristic
58C2×Q16×C53—22no—
59C2×SD16×C53—22no—
60C4⋊Q8×C53D411yescharacteristic
61C4⋊Q8×C53D411yescharacteristic
62C4⋊Q8×C53D411yescharacteristic
63#127×C53D411yescharacteristic
64#126×C53—22no—
65#126×C53—22no—
66#126×C53—22no—
67#126×C53—22no—
68#125×C53—22no—
69#125×C53—22no—
70#125×C53—21no—
71#124×C53—22no—
72#124×C53—21no—
73#124×C53—22no—
74#123×C53—22no—
75#123×C53—22no—
76#123×C53—22no—
77#123×C53—22no—
78#123×C53—22no—
79C22⋊Q8×C53—21no—
80C22⋊Q8×C53—21no—
81C22⋊Q8×C53—22no—
82C22⋊Q8×C53—22no—
83C22⋊Q8×C53—21no—
84C22⋊Q8×C53C2312yes—
85#121×C53—21no—
86#121×C53—21no—
87#121×C53—22no—
88#121×C53—22no—
89#121×C53—21no—
90#121×C53C2312yes—
91#120×C53—22no—
92C4×Q8×C53—21no—
93C4×Q8×C53—21no—
94C4×Q8×C53—21no—
95C4×Q8×C53—21no—
96C4×Q8×C53—21no—
97C4×Q8×C53—21no—
98C4×Q8×C53—21no—
99C4×Q8×C53D412yes—
100C4×D4×C53—21no—
101C4×D4×C53—21no—
102C4×D4×C53—22no—
103C4×D4×C53—22no—
104C4×D4×C53—21no—
105C4×D4×C53—22no—
106C4×D4×C53—22no—
107C4×D4×C53D412yes—
108#117×C53—22no—
109#117×C53—21no—
110#117×C53—21no—
111#117×C53—21no—
112#117×C53—21no—
113#117×C53—21no—
114C2×C4⋊C4×C53—21no—
115C2×C4⋊C4×C53—21no—
116C2×C4⋊C4×C53—21no—
117C2×C4⋊C4×C53—21no—
118C2×C4⋊C4×C53D411yescharacteristic
119C2×C4⋊C4×C53C2311yescharacteristic
120C2×C4⋊C4×C53C2311yescharacteristic
121C2×C22⋊C4×C53—21no—
122C2×C22⋊C4×C53—21no—
123C2×C22⋊C4×C53D411yescharacteristic
124C2×C22⋊C4×C53C2311yescharacteristic
125C2×C42×C53—22no—
126C2×C42×C53—21no—
127C2×C42×C53—21no—
128C2×C42×C53D411yescharacteristic
129C2×C42×C53C2311yescharacteristic
130#107×C53—22no—
131#106×C53—22no—
132C4⋊C8×C53—21no—
133C4⋊C8×C53—21no—
134#103×C53—22no—
135#103×C53—22no—
136#103×C53—22no—
137#102×C53—22no—
138C22⋊C8×C53C2312yes—
139C22⋊C8×C53C2312yes—
140C4×C8×C53—21no—
141C4×C8×C53—21no—
142#95×C53—21no—
143#95×C53—21no—
144#95×C53—21no—
145#95×C53—21no—
146#95×C53C2311yescharacteristic
147#95×C53C2311yescharacteristic
148C24×C53C2×D411yescharacteristic
149#41×C53—42no—
150#41×C53—42no—
151C2×Q8×C53—21no—
152C2×Q8×C53—21no—
153C2×Q8×C53—21no—
154C2×Q8×C53—21no—
155C2×Q8×C53—42no—
156C2×Q8×C53C2×D412yes—
157C2×D4×C53—22no—
158C2×D4×C53—22no—
159C2×D4×C53—22no—
160C2×D4×C53—21no—
161C2×D4×C53—21no—
162C2×D4×C53C2×D412yes—
163C22×C4×C53—21no—
164C22×C4×C53—21no—
165C22×C4×C53—22no—
166C22×C4×C53—22no—
167C22×C4×C53—21no—
168C22×C4×C53—22no—
169C22×C4×C53—21no—
170C22×C4×C53—22no—
171C22×C4×C53—22no—
172C22×C4×C53—21no—
173C22×C4×C53—21no—
174C22×C4×C53C2×D411yescharacteristic
175C22×C4×C53C2×D411yescharacteristic
176C22×C4×C53C2×D411yescharacteristic
177C22×C4×C53C2×D411yescharacteristic
178C22×C4×C53C2×D411yescharacteristic
179C22×C4×C53C2411yescharacteristic
180Q16×C53—44no—
181SD16×C53—44no—
182C2×C8×C53—22no—
183C2×C8×C53—22no—
184C4⋊C4×C53—22no—
185C4⋊C4×C53—22no—
186C4⋊C4×C53—22no—
187C4⋊C4×C53—22no—
188C4⋊C4×C53—21no—
189C4⋊C4×C53—21no—
190C4⋊C4×C53—21no—
191C4⋊C4×C53—21no—
192C4⋊C4×C53—21no—
193C4⋊C4×C53—21no—
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—42no—
201C4⋊C4×C53—42no—
202C4⋊C4×C53—42no—
203C4⋊C4×C53—41no—
204C4⋊C4×C53—41no—
205C4⋊C4×C53C2×D412yes—
206C4⋊C4×C53C2×D412yes—
207C22⋊C4×C53—22no—
208C22⋊C4×C53—22no—
209C22⋊C4×C53—22no—
210C22⋊C4×C53—22no—
211C22⋊C4×C53—22no—
212C22⋊C4×C53—22no—
213C22⋊C4×C53—22no—
214C22⋊C4×C53—21no—
215C22⋊C4×C53—21no—
216C22⋊C4×C53—22no—
217C22⋊C4×C53—21no—
218C22⋊C4×C53—21no—
219C22⋊C4×C53—22no—
220C22⋊C4×C53—42no—
221C22⋊C4×C53—42no—
222C42×C53—22no—
223C42×C53—22no—
224C42×C53—21no—
225C42×C53—21no—
226C42×C53—22no—
227C42×C53—41no—
228C42×C53—41no—
229C42×C53—42no—
230C42×C53C2×D411yescharacteristic
231C42×C53C2×D411yescharacteristic
232C42×C53C2×D411yescharacteristic
233C42×C53C2×D411yescharacteristic
234C23×C53—22no—
235C23×C53—21no—
236C23×C53—21no—
237C23×C53—21no—
238C23×C53—41no—
239C23×C53—41no—
240C23×C53C22×D411yescharacteristic
241Q8×C53—22no—
242Q8×C53—22no—
243Q8×C53—41no—
244Q8×C53—41no—
245Q8×C53—42no—
246Q8×C53—42no—
247Q8×C53—81no—
248D4×C53—42no—
249D4×C53—42no—
250D4×C53—42no—
251D4×C53—42no—
252D4×C53—81no—
253D4×C53—82no—
254D4×C53—82no—
255C2×C4×C53—22no—
256C2×C4×C53—22no—
257C2×C4×C53—21no—
258C2×C4×C53—21no—
259C2×C4×C53—21no—
260C2×C4×C53—22no—
261C2×C4×C53—22no—
262C2×C4×C53—22no—
263C2×C4×C53—21no—
264C2×C4×C53—21no—
265C2×C4×C53—41no—
266C2×C4×C53—41no—
267C2×C4×C53—41no—
268C2×C4×C53—41no—
269C2×C4×C53—41no—
270C2×C4×C53—41no—
271C2×C4×C53—42no—
272C2×C4×C53—42no—
273C2×C4×C53—44no—
274C2×C4×C53—41no—
275C2×C4×C53—41no—
276C2×C4×C53—41no—
277C2×C4×C53—41no—
278C2×C4×C53—41no—
279C2×C4×C53—41no—
280C2×C4×C53—41no—
281C2×C4×C53—41no—
282C2×C4×C53—82no—
283C2×C4×C53—82no—
284C2×C4×C53#12011yescharacteristic
285C2×C4×C53#12011yescharacteristic
286C2×C4×C53#12011yescharacteristic
287C2×C4×C53#12011yescharacteristic
288C2×C4×C53#13512yes—
289C2×C4×C53#13712yes—
290C2×C4×C53C22×D411yescharacteristic
291C2×C4×C53C22×D411yescharacteristic
292C8×C53—42no—
293C8×C53—42no—
294#18127C5311yesmaximal, characteristic, direct product factor, Sylow
295C22×C53—21no—
296C22×C53—21no—
297C22×C53—42no—
298C22×C53—41no—
299C22×C53—42no—
300C22×C53—42no—
301C22×C53—41no—
302C22×C53—42no—
303C22×C53—41no—
304C22×C53—81no—
305C22×C53C2×#12011yescharacteristic
306C22×C53C2×#13511yescharacteristic
307C22×C53C2×#13711yescharacteristic
308C4×C53—21no—
309C4×C53—21no—
310C4×C53—21no—
311C4×C53—21no—
312C4×C53—42no—
313C4×C53—41no—
314C4×C53—42no—
315C4×C53—41no—
316C4×C53—42no—
317C4×C53—42no—
318C4×C53—41no—
319C4×C53—42no—
320C4×C53—81no—
321C4×C53—81no—
322C4×C53#45212yes—
323C4×C53#45212yes—
324#3455C2×C5311yescharacteristic
325#3104C2×C5311yescharacteristic, complemented
326#3103C2×C5311yescharacteristic, complemented
327#2396C2×C5311yescharacteristic, complemented
328#1618C2×C5312yescomplemented
329#1615C2×C5312yescomplemented
330#1606C2×C5312yescomplemented
331#1603C2×C5312yescomplemented
332#1503C2×C5311yescharacteristic, complemented
333#1502C2×C5311yescharacteristic, complemented
334#1458C2×C5311yescharacteristic, complemented
335C2×C53—21no—
336C2×C53—21no—
337C2×C53—82no—
338C2×C53—81no—
339C2×C53C2×#45211yescharacteristic
340C2×C53#301811yescharacteristic
341C2×C53#302511yescharacteristic
342#542—21no—
343#542—21no—
344#541—21no—
345#540—21no—
346#539—21no—
347#539C22×C5312yes—
348#538—21no—
349#538C22×C5312yes—
350#533C22×C5311yescharacteristic
351#532C22×C5311yescharacteristic
352#529—22no—
353#529—22no—
354#525—21no—
355C4×#41—21no—
356C2×C4×Q8C22×C5311yescharacteristic
357C2×#117C22×C5311yescharacteristic
358#484C22×C5312yes—
359#481C22×C5312yes—
360#477—21no—
361#475—21no—
362#462—21no—
363#460—21no—
364#451C22×C5312yes—
365#448C22×C5312yes—
366C4×Q16—22no—
367C4×SD16—22no—
368#431C22×C5311yescharacteristic
369#431C22×C5311yescharacteristic
370#400C22×C5311yescharacteristic
371#399C22×C5311yescharacteristic
372#398—21no—
373#396—21no—
374#395C22×C5311yescharacteristic
375#393C22×C5311yescharacteristic
376C4×C4⋊C4—21no—
377C4×C4⋊C4C22×C5311yescharacteristic
378C4×C22⋊C4—21no—
379#328C22×C5312yes—
380#328C22×C5312yes—
381#327C22×C5312yes—
382#327C22×C5312yes—
383#324C22×C5312yes—
384#323C22×C5312yes—
385C53#1812711yescharacteristic, direct product factor, Sylow
386C2×#41—21no—
387C22×Q8C23×C5311yescharacteristic
388C23×C4D4×C5311yescharacteristic
389C2×Q16—22no—
390C2×SD16—22no—
391C4⋊Q8D4×C5311yescharacteristic
392C4⋊Q8D4×C5311yescharacteristic
393C4⋊Q8D4×C5311yescharacteristic
394#127D4×C5311yescharacteristic
395#126—22no—
396#126—22no—
397#126—22no—
398#126—22no—
399#125—22no—
400#125—22no—
401#125—21no—
402#124—22no—
403#124—22no—
404#124—21no—
405#123—22no—
406#123—22no—
407#123—22no—
408#123—22no—
409#123—22no—
410C22⋊Q8—21no—
411C22⋊Q8—21no—
412C22⋊Q8—22no—
413C22⋊Q8—22no—
414C22⋊Q8—21no—
415C22⋊Q8C23×C5312yes—
416#121—21no—
417#121—21no—
418#121—22no—
419#121—22no—
420#121—21no—
421#121C23×C5312yes—
422#120—22no—
423C4×Q8—21no—
424C4×Q8—21no—
425C4×Q8—21no—
426C4×Q8—21no—
427C4×Q8—21no—
428C4×Q8—21no—
429C4×Q8—21no—
430C4×Q8D4×C5312yes—
431C4×D4—21no—
432C4×D4—21no—
433C4×D4—22no—
434C4×D4—22no—
435C4×D4—22no—
436C4×D4—22no—
437C4×D4—21no—
438C4×D4D4×C5312yes—
439#117—22no—
440#117—21no—
441#117—21no—
442#117—21no—
443#117—21no—
444#117—21no—
445C2×C4⋊C4—21no—
446C2×C4⋊C4—21no—
447C2×C4⋊C4—21no—
448C2×C4⋊C4—21no—
449C2×C4⋊C4D4×C5311yescharacteristic
450C2×C4⋊C4C23×C5311yescharacteristic
451C2×C4⋊C4C23×C5311yescharacteristic
452C2×C22⋊C4—21no—
453C2×C22⋊C4—21no—
454C2×C22⋊C4D4×C5311yescharacteristic
455C2×C22⋊C4C23×C5311yescharacteristic
456C2×C42—22no—
457C2×C42—21no—
458C2×C42—21no—
459C2×C42D4×C5311yescharacteristic
460C2×C42C23×C5311yescharacteristic
461#107—22no—
462#106—22no—
463C4⋊C8—21no—
464C4⋊C8—21no—
465#103—22no—
466#103—22no—
467#103—22no—
468#102—22no—
469C22⋊C8C23×C5312yes—
470C22⋊C8C23×C5312yes—
471C4×C8—21no—
472C4×C8—21no—
473#95—21no—
474#95—21no—
475#95—21no—
476#95—21no—
477#95C23×C5311yescharacteristic
478#95C23×C5311yescharacteristic
479C24C2×D4×C5311yescharacteristic
480#41—42no—
481#41—42no—
482C2×Q8—21no—
483C2×Q8—21no—
484C2×Q8—21no—
485C2×Q8—21no—
486C2×Q8—42no—
487C2×Q8C2×D4×C5312yes—
488C2×D4—22no—
489C2×D4—22no—
490C2×D4—22no—
491C2×D4—21no—
492C2×D4—21no—
493C2×D4C2×D4×C5312yes—
494C22×C4—21no—
495C22×C4—21no—
496C22×C4—22no—
497C22×C4—22no—
498C22×C4—22no—
499C22×C4—21no—
500C22×C4—21no—
501C22×C4—22no—
502C22×C4—22no—
503C22×C4—21no—
504C22×C4—21no—
505C22×C4C2×D4×C5311yescharacteristic
506C22×C4C2×D4×C5311yescharacteristic
507C22×C4C2×D4×C5311yescharacteristic
508C22×C4C2×D4×C5311yescharacteristic
509C22×C4C2×D4×C5311yescharacteristic
510C22×C4C24×C5311yescharacteristic
511Q16—44no—
512SD16—44no—
513C2×C8—22no—
514C2×C8—22no—
515C4⋊C4—22no—
516C4⋊C4—22no—
517C4⋊C4—22no—
518C4⋊C4—22no—
519C4⋊C4—21no—
520C4⋊C4—21no—
521C4⋊C4—21no—
522C4⋊C4—21no—
523C4⋊C4—21no—
524C4⋊C4—21no—
525C4⋊C4—22no—
526C4⋊C4—21no—
527C4⋊C4—21no—
528C4⋊C4—22no—
529C4⋊C4—21no—
530C4⋊C4—21no—
531C4⋊C4—42no—
532C4⋊C4—42no—
533C4⋊C4—42no—
534C4⋊C4—41no—
535C4⋊C4—41no—
536C4⋊C4C2×D4×C5312yes—
537C4⋊C4C2×D4×C5312yes—
538C22⋊C4—22no—
539C22⋊C4—22no—
540C22⋊C4—22no—
541C22⋊C4—22no—
542C22⋊C4—22no—
543C22⋊C4—22no—
544C22⋊C4—22no—
545C22⋊C4—21no—
546C22⋊C4—21no—
547C22⋊C4—22no—
548C22⋊C4—21no—
549C22⋊C4—21no—
550C22⋊C4—22no—
551C22⋊C4—42no—
552C22⋊C4—42no—
553C42—22no—
554C42—22no—
555C42—21no—
556C42—22no—
557C42—21no—
558C42—41no—
559C42—41no—
560C42—42no—
561C42C2×D4×C5311yescharacteristic
562C42C2×D4×C5311yescharacteristic
563C42C2×D4×C5311yescharacteristic
564C42C2×D4×C5311yescharacteristic
565C23—21no—
566C23—22no—
567C23—21no—
568C23—21no—
569C23—41no—
570C23—41no—
571C23C22×D4×C5311yescharacteristic
572Q8—22no—
573Q8—22no—
574Q8—42no—
575Q8—41no—
576Q8—41no—
577Q8—42no—
578Q8—81no—
579D4—42no—
580D4—42no—
581D4—42no—
582D4—42no—
583D4—81no—
584D4—82no—
585D4—82no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—44no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—82no—
614C2×C4—82no—
615C2×C4#120×C5311yescharacteristic
616C2×C4#120×C5311yescharacteristic
617C2×C4#120×C5311yescharacteristic
618C2×C4#120×C5311yescharacteristic
619C2×C4#135×C5312yes—
620C2×C4#137×C5312yes—
621C2×C4C22×D4×C5311yescharacteristic
622C2×C4C22×D4×C5311yescharacteristic
623C8—42no—
624C8—42no—
625C22—21no—
626C22—21no—
627C22—42no—
628C22—41no—
629C22—42no—
630C22—42no—
631C22—41no—
632C22—42no—
633C22—41no—
634C22—81no—
635C22C2×#120×C5311yescharacteristic
636C22C2×#135×C5311yescharacteristic
637C22C2×#137×C5311yescharacteristic
638C4—21no—
639C4—21no—
640C4—21no—
641C4—21no—
642C4—42no—
643C4—41no—
644C4—42no—
645C4—41no—
646C4—42no—
647C4—42no—
648C4—41no—
649C4—42no—
650C4—81no—
651C4—81no—
652C4C53×#45212yes—
653C4C53×#45212yes—
654C2—21no—
655C2—21no—
656C2—81no—
657C2—82no—
658C2C2×C53×#45211yescharacteristic
659C2C53×#301811yescharacteristic
660C2C53×#302511yescharacteristic
subgroupG/H
centerC2×C4×C53#120
commutator, Frattini, cosocleC22×C4C24×C53
socleC22×C53C2×#120
prime radicalC53×#3455C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4×C53 ◁ C2×C42×C53 ◁ G

Complements

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

NH ≅ G/Ntype
2. C53×#3104C2semidirect
3. C53×#3103C2semidirect
4. C53×#2396C2semidirect
5. C53×#1618C2semidirect
6. C53×#1615C2semidirect
7. C53×#1606C2semidirect
8. C53×#1603C2semidirect
9. C53×#1503C2semidirect
10. C53×#1502C2semidirect
11. C53×#1458C2semidirect
294. #18127C53direct (N × H)
325. #3104C2×C53semidirect
326. #3103C2×C53semidirect
327. #2396C2×C53semidirect
328. #1618C2×C53semidirect
329. #1615C2×C53semidirect
330. #1606C2×C53semidirect
331. #1603C2×C53semidirect
332. #1503C2×C53semidirect
333. #1502C2×C53semidirect
334. #1458C2×C53semidirect
385. C53#18127direct (N × H)

Maximal subgroups

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

subgroupindex
C53×#34552
C53×#31042
C53×#31032
C53×#23962
C53×#16182
C53×#16152
C53×#16062
C53×#16032
C53×#15032
C53×#15022
C53×#14582
#1812753