Finite Group Databasebeta

← C53×#18159 #27225671

Subgroups — C53×#18159 #27225671

Order 13568.

998 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C53×#3459C211yesmaximal, characteristic
2C53×#3117C212yesmaximal, complemented
3C53×#2580C211yesmaximal, characteristic, complemented
4C53×#1618C212yesmaximal, complemented
5C53×#1614C211yesmaximal, characteristic, complemented
6C53×#1613C211yesmaximal, characteristic, complemented
7C53×#1606C212yesmaximal, complemented
8C53×#1602C211yesmaximal, characteristic, complemented
9C53×#1601C211yesmaximal, characteristic, complemented
10C53×#1512C212yesmaximal, complemented
11C53×#1466C211yesmaximal, characteristic, complemented
12C53×#543C2211yescharacteristic
13C53×#539—22no—
14C53×#539—21no—
15C53×#538—22no—
16C53×#538C2212yes—
17C53×#534—21no—
18C53×#534C2211yescharacteristic
19C53×#529—22no—
20C53×#529C2212yes—
21C53×#525—21no—
22C53×#525—22no—
23C2×C4×Q8×C53C2211yescharacteristic, complemented
24C2×C4×D4×C53—21no—
25C2×C4×D4×C53C2211yescharacteristic
26C53×#484C2212yescomplemented
27C53×#481C2212yes—
28C53×#479—22no—
29C53×#464—22no—
30C53×#451C2212yescomplemented
31C53×#448C2212yes—
32C53×#441—22no—
33C53×#440—22no—
34C53×#432C2212yescomplemented
35C53×#399C2211yescharacteristic
36C53×#399C2211yescharacteristic, complemented
37C53×#393C2211yescharacteristic
38C53×#393C2211yescharacteristic, complemented
39C53×#390—21no—
40C53×#389—21no—
41C53×#385—21no—
42C53×#385—21no—
43C53×#382C2211yescharacteristic, complemented
44C53×#328C2212yescomplemented
45C53×#328C2212yescomplemented
46C53×#327C2212yes—
47C53×#327C2212yes—
48C53×#323C2211yescharacteristic
49C53×#323C2211yescharacteristic
50C53×#323C2212yescomplemented
51C2×#41×C53—21no—
52C22×Q8×C53C2311yescharacteristic
53C22×D4×C53—21no—
54C22×D4×C53D411yescharacteristic
55C23×C4×C53—21no—
56C23×C4×C53D411yescharacteristic
57C23×C4×C53D411yescharacteristic
58C2×Q16×C53—22no—
59C2×SD16×C53—22no—
60#126×C53—22no—
61#126×C53—22no—
62#126×C53—22no—
63#125×C53—22no—
64#125×C53D412yes—
65#124×C53—22no—
66#124×C53D412yes—
67#123×C53—22no—
68#123×C53—22no—
69#123×C53—22no—
70#123×C53—22no—
71#123×C53—22no—
72#123×C53—22no—
73C22⋊Q8×C53—22no—
74C22⋊Q8×C53—21no—
75C22⋊Q8×C53—22no—
76C22⋊Q8×C53—22no—
77C22⋊Q8×C53—21no—
78C22⋊Q8×C53C2312yes—
79#121×C53—22no—
80#121×C53—21no—
81#121×C53—22no—
82#121×C53—22no—
83#121×C53—21no—
84#121×C53C2312yes—
85#120×C53—22no—
86#120×C53—22no—
87#120×C53—22no—
88C4×Q8×C53—22no—
89C4×Q8×C53—22no—
90C4×Q8×C53D412yes—
91C4×D4×C53—22no—
92C4×D4×C53—22no—
93C4×D4×C53—22no—
94C4×D4×C53—22no—
95C4×D4×C53—22no—
96C4×D4×C53—22no—
97C4×D4×C53—22no—
98C4×D4×C53D412yes—
99#117×C53—21no—
100#117×C53—22no—
101C2×C4⋊C4×C53—21no—
102C2×C4⋊C4×C53—21no—
103C2×C4⋊C4×C53—21no—
104C2×C4⋊C4×C53—21no—
105C2×C4⋊C4×C53—21no—
106C2×C4⋊C4×C53D411yescharacteristic
107C2×C4⋊C4×C53C2311yescharacteristic
108C2×C22⋊C4×C53—21no—
109C2×C22⋊C4×C53—21no—
110C2×C22⋊C4×C53—21no—
111C2×C22⋊C4×C53—21no—
112C2×C22⋊C4×C53—21no—
113C2×C22⋊C4×C53—21no—
114C2×C22⋊C4×C53C2311yescharacteristic
115C2×C22⋊C4×C53C2311yescharacteristic
116C2×C42×C53—21no—
117C2×C42×C53—21no—
118C2×C42×C53C2311yescharacteristic
119#107×C53—22no—
120#106×C53—22no—
121C4⋊C8×C53—22no—
122#103×C53—22no—
123#103×C53—22no—
124#103×C53—22no—
125#102×C53—22no—
126C22⋊C8×C53C2312yes—
127C22⋊C8×C53C2312yes—
128#97×C53—22no—
129#95×C53—21no—
130#95×C53—21no—
131#95×C53—21no—
132#95×C53C2311yescharacteristic
133#95×C53C2311yescharacteristic
134C24×C53—21no—
135C24×C53C2×D411yescharacteristic
136C24×C53C2×D411yescharacteristic
137#41×C53—82no—
138C2×Q8×C53—21no—
139C2×Q8×C53—22no—
140C2×Q8×C53—42no—
141C2×Q8×C53C2×D412yes—
142C2×D4×C53—22no—
143C2×D4×C53—22no—
144C2×D4×C53—22no—
145C2×D4×C53—22no—
146C2×D4×C53—22no—
147C2×D4×C53—21no—
148C2×D4×C53—22no—
149C2×D4×C53—41no—
150C2×D4×C53—42no—
151C2×D4×C53—42no—
152C2×D4×C53C2×D412yes—
153C22×C4×C53—21no—
154C22×C4×C53—21no—
155C22×C4×C53—21no—
156C22×C4×C53—21no—
157C22×C4×C53—22no—
158C22×C4×C53—21no—
159C22×C4×C53—21no—
160C22×C4×C53—22no—
161C22×C4×C53—21no—
162C22×C4×C53—21no—
163C22×C4×C53—21no—
164C22×C4×C53—21no—
165C22×C4×C53—21no—
166C22×C4×C53—81no—
167C22×C4×C53—81no—
168C22×C4×C53—82no—
169C22×C4×C53C2×D411yescharacteristic
170C22×C4×C53C2×D411yescharacteristic
171C22×C4×C53C2×D411yescharacteristic
172C22×C4×C53C2×D411yescharacteristic
173C22×C4×C53C2411yescharacteristic
174Q16×C53—44no—
175SD16×C53—44no—
176C2×C8×C53—22no—
177C2×C8×C53—22no—
178C4⋊C4×C53—22no—
179C4⋊C4×C53—21no—
180C4⋊C4×C53—21no—
181C4⋊C4×C53—22no—
182C4⋊C4×C53—22no—
183C4⋊C4×C53—22no—
184C4⋊C4×C53—22no—
185C4⋊C4×C53—22no—
186C4⋊C4×C53—22no—
187C4⋊C4×C53—22no—
188C4⋊C4×C53—42no—
189C4⋊C4×C53—41no—
190C4⋊C4×C53—41no—
191C4⋊C4×C53—81no—
192C4⋊C4×C53—81no—
193C4⋊C4×C53C2×D412yes—
194C4⋊C4×C53C2×D412yes—
195C22⋊C4×C53—22no—
196C22⋊C4×C53—22no—
197C22⋊C4×C53—22no—
198C22⋊C4×C53—22no—
199C22⋊C4×C53—22no—
200C22⋊C4×C53—22no—
201C22⋊C4×C53—22no—
202C22⋊C4×C53—22no—
203C22⋊C4×C53—22no—
204C22⋊C4×C53—22no—
205C22⋊C4×C53—22no—
206C22⋊C4×C53—22no—
207C22⋊C4×C53—42no—
208C22⋊C4×C53—42no—
209C22⋊C4×C53—81no—
210C22⋊C4×C53—81no—
211C42×C53—22no—
212C42×C53—22no—
213C42×C53—42no—
214C42×C53C2×D412yes—
215C42×C53C2×D412yes—
216C23×C53—21no—
217C23×C53—21no—
218C23×C53—21no—
219C23×C53—22no—
220C23×C53—21no—
221C23×C53—21no—
222C23×C53—41no—
223C23×C53—41no—
224C23×C53—41no—
225C23×C53—41no—
226C23×C53—42no—
227C23×C53—81no—
228C23×C53—81no—
229C23×C53#13712yes—
230C23×C53C22×D411yescharacteristic
231Q8×C53—24no—
232Q8×C53—42no—
233Q8×C53—42no—
234Q8×C53—81no—
235D4×C53—44no—
236D4×C53—42no—
237D4×C53—44no—
238D4×C53—42no—
239D4×C53—42no—
240D4×C53—82no—
241D4×C53—81no—
242D4×C53—82no—
243C2×C4×C53—21no—
244C2×C4×C53—21no—
245C2×C4×C53—22no—
246C2×C4×C53—21no—
247C2×C4×C53—22no—
248C2×C4×C53—21no—
249C2×C4×C53—22no—
250C2×C4×C53—22no—
251C2×C4×C53—41no—
252C2×C4×C53—41no—
253C2×C4×C53—42no—
254C2×C4×C53—42no—
255C2×C4×C53—44no—
256C2×C4×C53—42no—
257C2×C4×C53—41no—
258C2×C4×C53—41no—
259C2×C4×C53—41no—
260C2×C4×C53—41no—
261C2×C4×C53—41no—
262C2×C4×C53—41no—
263C2×C4×C53—41no—
264C2×C4×C53—41no—
265C2×C4×C53—41no—
266C2×C4×C53—41no—
267C2×C4×C53—41no—
268C2×C4×C53—41no—
269C2×C4×C53—41no—
270C2×C4×C53—81no—
271C2×C4×C53—81no—
272C2×C4×C53—81no—
273C2×C4×C53—82no—
274C2×C4×C53—81no—
275C2×C4×C53#12011yescharacteristic
276C2×C4×C53#12011yescharacteristic
277C2×C4×C53#12012yes—
278C2×C4×C53C22×D412yes—
279C8×C53—42no—
280C8×C53—42no—
281#18159C5311yesmaximal, characteristic, direct product factor, Sylow
282C22×C53—21no—
283C22×C53—21no—
284C22×C53—22no—
285C22×C53—42no—
286C22×C53—42no—
287C22×C53—41no—
288C22×C53—41no—
289C22×C53—41no—
290C22×C53—42no—
291C22×C53—41no—
292C22×C53—42no—
293C22×C53—42no—
294C22×C53—41no—
295C22×C53—81no—
296C22×C53—81no—
297C22×C53—84no—
298C22×C53C2×#12011yescharacteristic
299C22×C53C2×#13711yescharacteristic
300C22×C53#57511yescharacteristic
301C4×C53—22no—
302C4×C53—21no—
303C4×C53—21no—
304C4×C53—22no—
305C4×C53—42no—
306C4×C53—41no—
307C4×C53—41no—
308C4×C53—42no—
309C4×C53—42no—
310C4×C53—42no—
311C4×C53—81no—
312C4×C53—81no—
313C4×C53—81no—
314#3459C2×C5311yescharacteristic
315#3117C2×C5312yescomplemented
316#2580C2×C5311yescharacteristic, complemented
317#1618C2×C5312yescomplemented
318#1614C2×C5311yescharacteristic, complemented
319#1613C2×C5311yescharacteristic, complemented
320#1606C2×C5312yescomplemented
321#1602C2×C5311yescharacteristic, complemented
322#1601C2×C5311yescharacteristic, complemented
323#1512C2×C5312yescomplemented
324#1466C2×C5311yescharacteristic, complemented
325C2×C53—21no—
326C2×C53—21no—
327C2×C53—42no—
328C2×C53—81no—
329C2×C53—82no—
330C2×C53#300711yescharacteristic
331C2×C53#301911yescharacteristic
332C2×C53#302511yescharacteristic
333#543C22×C5311yescharacteristic
334#539—22no—
335#539—21no—
336#538—22no—
337#538C22×C5312yes—
338#534—21no—
339#534C22×C5311yescharacteristic
340#529—22no—
341#529C22×C5312yes—
342#525—21no—
343#525—22no—
344C2×C4×Q8C22×C5311yescharacteristic, complemented
345C2×C4×D4—21no—
346C2×C4×D4C22×C5311yescharacteristic
347#484C22×C5312yescomplemented
348#481C22×C5312yes—
349#479—22no—
350#464—22no—
351#451C22×C5312yescomplemented
352#448C22×C5312yes—
353#441—22no—
354#440—22no—
355#432C22×C5312yescomplemented
356#399C22×C5311yescharacteristic
357#399C22×C5311yescharacteristic, complemented
358#393C22×C5311yescharacteristic
359#393C22×C5311yescharacteristic, complemented
360#390—21no—
361#389—21no—
362#385—21no—
363#385—21no—
364#382C22×C5311yescharacteristic, complemented
365#328C22×C5312yescomplemented
366#328C22×C5312yescomplemented
367#327C22×C5312yes—
368#327C22×C5312yes—
369#323C22×C5311yescharacteristic
370#323C22×C5311yescharacteristic
371#323C22×C5312yescomplemented
372C53#1815911yescharacteristic, direct product factor, Sylow
373C2×#41—21no—
374C22×Q8C23×C5311yescharacteristic
375C22×D4—21no—
376C22×D4D4×C5311yescharacteristic
377C23×C4—21no—
378C23×C4D4×C5311yescharacteristic
379C23×C4D4×C5311yescharacteristic
380C2×Q16—22no—
381C2×SD16—22no—
382#126—22no—
383#126—22no—
384#126—22no—
385#125—22no—
386#125D4×C5312yes—
387#124—22no—
388#124D4×C5312yes—
389#123—22no—
390#123—22no—
391#123—22no—
392#123—22no—
393#123—22no—
394#123—22no—
395C22⋊Q8—21no—
396C22⋊Q8—22no—
397C22⋊Q8—22no—
398C22⋊Q8—21no—
399C22⋊Q8—22no—
400C22⋊Q8C23×C5312yes—
401#121—22no—
402#121—21no—
403#121—22no—
404#121—22no—
405#121—21no—
406#121C23×C5312yes—
407#120—22no—
408#120—22no—
409#120—22no—
410C4×Q8—22no—
411C4×Q8—22no—
412C4×Q8D4×C5312yes—
413C4×D4—22no—
414C4×D4—22no—
415C4×D4—22no—
416C4×D4—22no—
417C4×D4—22no—
418C4×D4—22no—
419C4×D4—22no—
420C4×D4D4×C5312yes—
421#117—21no—
422#117—22no—
423C2×C4⋊C4—21no—
424C2×C4⋊C4—21no—
425C2×C4⋊C4—21no—
426C2×C4⋊C4—21no—
427C2×C4⋊C4—21no—
428C2×C4⋊C4D4×C5311yescharacteristic
429C2×C4⋊C4C23×C5311yescharacteristic
430C2×C22⋊C4—21no—
431C2×C22⋊C4—21no—
432C2×C22⋊C4—21no—
433C2×C22⋊C4—21no—
434C2×C22⋊C4—21no—
435C2×C22⋊C4—21no—
436C2×C22⋊C4C23×C5311yescharacteristic
437C2×C22⋊C4C23×C5311yescharacteristic
438C2×C42—21no—
439C2×C42—21no—
440C2×C42C23×C5311yescharacteristic
441#107—22no—
442#106—22no—
443C4⋊C8—22no—
444#103—22no—
445#103—22no—
446#103—22no—
447#102—22no—
448C22⋊C8C23×C5312yes—
449C22⋊C8C23×C5312yes—
450#97—22no—
451#95—21no—
452#95—21no—
453#95—21no—
454#95C23×C5311yescharacteristic
455#95C23×C5311yescharacteristic
456C24—21no—
457C24C2×D4×C5311yescharacteristic
458C24C2×D4×C5311yescharacteristic
459#41—82no—
460C2×Q8—21no—
461C2×Q8—22no—
462C2×Q8—42no—
463C2×Q8C2×D4×C5312yes—
464C2×D4—22no—
465C2×D4—22no—
466C2×D4—22no—
467C2×D4—22no—
468C2×D4—21no—
469C2×D4—22no—
470C2×D4—22no—
471C2×D4—41no—
472C2×D4—42no—
473C2×D4—42no—
474C2×D4C2×D4×C5312yes—
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—22no—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—22no—
483C22×C4—21no—
484C22×C4—21no—
485C22×C4—21no—
486C22×C4—21no—
487C22×C4—21no—
488C22×C4—81no—
489C22×C4—81no—
490C22×C4—82no—
491C22×C4C2×D4×C5311yescharacteristic
492C22×C4C2×D4×C5311yescharacteristic
493C22×C4C2×D4×C5311yescharacteristic
494C22×C4C2×D4×C5311yescharacteristic
495C22×C4C24×C5311yescharacteristic
496Q16—44no—
497SD16—44no—
498C2×C8—22no—
499C2×C8—22no—
500C4⋊C4—22no—
501C4⋊C4—21no—
502C4⋊C4—21no—
503C4⋊C4—22no—
504C4⋊C4—22no—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4—22no—
508C4⋊C4—22no—
509C4⋊C4—22no—
510C4⋊C4—42no—
511C4⋊C4—41no—
512C4⋊C4—41no—
513C4⋊C4—81no—
514C4⋊C4—81no—
515C4⋊C4C2×D4×C5312yes—
516C4⋊C4C2×D4×C5312yes—
517C22⋊C4—22no—
518C22⋊C4—22no—
519C22⋊C4—22no—
520C22⋊C4—22no—
521C22⋊C4—22no—
522C22⋊C4—22no—
523C22⋊C4—22no—
524C22⋊C4—22no—
525C22⋊C4—22no—
526C22⋊C4—22no—
527C22⋊C4—22no—
528C22⋊C4—22no—
529C22⋊C4—42no—
530C22⋊C4—42no—
531C22⋊C4—81no—
532C22⋊C4—81no—
533C42—22no—
534C42—22no—
535C42—42no—
536C42C2×D4×C5312yes—
537C42C2×D4×C5312yes—
538C23—21no—
539C23—21no—
540C23—21no—
541C23—22no—
542C23—21no—
543C23—21no—
544C23—41no—
545C23—41no—
546C23—41no—
547C23—41no—
548C23—42no—
549C23—81no—
550C23—81no—
551C23#137×C5312yes—
552C23C22×D4×C5311yescharacteristic
553Q8—24no—
554Q8—42no—
555Q8—42no—
556Q8—81no—
557D4—44no—
558D4—42no—
559D4—44no—
560D4—42no—
561D4—42no—
562D4—82no—
563D4—81no—
564D4—82no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—22no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—44no—
578C2×C4—42no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—81no—
593C2×C4—81no—
594C2×C4—81no—
595C2×C4—82no—
596C2×C4—81no—
597C2×C4#120×C5311yescharacteristic
598C2×C4#120×C5312yes—
599C2×C4#120×C5311yescharacteristic
600C2×C4C22×D4×C5312yes—
601C8—42no—
602C8—42no—
603C22—21no—
604C22—21no—
605C22—22no—
606C22—42no—
607C22—42no—
608C22—41no—
609C22—41no—
610C22—41no—
611C22—42no—
612C22—41no—
613C22—42no—
614C22—42no—
615C22—41no—
616C22—81no—
617C22—81no—
618C22—84no—
619C22C2×#120×C5311yescharacteristic
620C22C2×#137×C5311yescharacteristic
621C22C53×#57511yescharacteristic
622C4—22no—
623C4—21no—
624C4—21no—
625C4—22no—
626C4—42no—
627C4—41no—
628C4—41no—
629C4—42no—
630C4—42no—
631C4—42no—
632C4—81no—
633C4—81no—
634C4—81no—
635C2—21no—
636C2—21no—
637C2—42no—
638C2—81no—
639C2—82no—
640C2C53×#300711yescharacteristic
641C2C53×#301911yescharacteristic
642C2C53×#302511yescharacteristic
subgroupG/H
center, socleC22×C53C2×#120
commutator, Frattini, cosocleC22×C4C24×C53
prime radicalC53×#3459C2
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
2. C53×#3117C2semidirect
3. C53×#2580C2semidirect
4. C53×#1618C2semidirect
5. C53×#1614C2semidirect
6. C53×#1613C2semidirect
7. C53×#1606C2semidirect
8. C53×#1602C2semidirect
9. C53×#1601C2semidirect
10. C53×#1512C2semidirect
11. C53×#1466C2semidirect
23. C2×C4×Q8×C53C22semidirect
26. C53×#484C22semidirect
30. C53×#451C22semidirect
34. C53×#432C22semidirect
36. C53×#399C22semidirect
38. C53×#393C22semidirect
43. C53×#382C22semidirect
44. C53×#328C22semidirect
45. C53×#328C22semidirect
50. C53×#323C22semidirect
281. #18159C53direct (N × H)
315. #3117C2×C53semidirect
316. #2580C2×C53semidirect
317. #1618C2×C53semidirect
318. #1614C2×C53semidirect
319. #1613C2×C53semidirect
320. #1606C2×C53semidirect
321. #1602C2×C53semidirect
322. #1601C2×C53semidirect
323. #1512C2×C53semidirect
324. #1466C2×C53semidirect
344. C2×C4×Q8C22×C53semidirect
347. #484C22×C53semidirect
351. #451C22×C53semidirect
355. #432C22×C53semidirect
357. #399C22×C53semidirect
359. #393C22×C53semidirect
364. #382C22×C53semidirect
365. #328C22×C53semidirect
366. #328C22×C53semidirect
371. #323C22×C53semidirect
372. C53#18159direct (N × H)

Maximal subgroups

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

subgroupindex
C53×#34592
C53×#31172
C53×#25802
C53×#16182
C53×#16142
C53×#16132
C53×#16062
C53×#16022
C53×#16012
C53×#15122
C53×#14662
#1815953