Finite Group Databasebeta

← C53×#18129 #27225641

Subgroups — C53×#18129 #27225641

Order 13568.

1006 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C53×#3459C211yesmaximal, characteristic
2C53×#3108C212yesmaximal, complemented
3C53×#2398C211yesmaximal, characteristic, complemented
4C53×#1612C212yesmaximal, complemented
5C53×#1608C211yesmaximal, characteristic, complemented
6C53×#1607C211yesmaximal, characteristic, complemented
7C53×#1606C212yesmaximal, complemented
8C53×#1602C211yesmaximal, characteristic, complemented
9C53×#1601C211yesmaximal, characteristic, complemented
10C53×#1500C212yesmaximal, complemented
11C53×#1460C211yesmaximal, 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×#483C2212yescomplemented
27C53×#480C2212yes—
28C53×#479—22no—
29C53×#464—22no—
30C53×#451C2212yescomplemented
31C53×#448C2212yes—
32C4×Q16×C53—22no—
33C4×SD16×C53—22no—
34C53×#431C2212yescomplemented
35C53×#400—21no—
36C53×#397C2211yescharacteristic
37C53×#397C2211yescharacteristic, complemented
38C53×#394—21no—
39C53×#393C2211yescharacteristic
40C53×#393C2211yescharacteristic, complemented
41C4×C22⋊C4×C53—21no—
42C4×C22⋊C4×C53—21no—
43C53×#376C2211yescharacteristic, 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×C53D411yescharacteristic
104C2×C4⋊C4×C53C2311yescharacteristic
105C2×C22⋊C4×C53—21no—
106C2×C22⋊C4×C53—21no—
107C2×C22⋊C4×C53—21no—
108C2×C22⋊C4×C53—21no—
109C2×C22⋊C4×C53—21no—
110C2×C22⋊C4×C53—21no—
111C2×C22⋊C4×C53C2311yescharacteristic
112C2×C22⋊C4×C53C2311yescharacteristic
113C2×C42×C53—21no—
114C2×C42×C53—21no—
115C2×C42×C53—21no—
116C2×C42×C53C2311yescharacteristic
117#107×C53—22no—
118#106×C53—22no—
119C4⋊C8×C53—22no—
120#103×C53—22no—
121#103×C53—22no—
122#103×C53—22no—
123#102×C53—22no—
124C22⋊C8×C53C2312yes—
125C22⋊C8×C53C2312yes—
126C4×C8×C53—22no—
127#95×C53—21no—
128#95×C53—21no—
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—42no—
167C22×C4×C53—42no—
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×C53C2×D412yes—
190C4⋊C4×C53C2×D412yes—
191C22⋊C4×C53—22no—
192C22⋊C4×C53—22no—
193C22⋊C4×C53—22no—
194C22⋊C4×C53—22no—
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—42no—
204C22⋊C4×C53—42no—
205C22⋊C4×C53—42no—
206C22⋊C4×C53—42no—
207C42×C53—22no—
208C42×C53—22no—
209C42×C53—42no—
210C42×C53—41no—
211C42×C53—41no—
212C42×C53C2×D412yes—
213C42×C53C2×D412yes—
214C23×C53—21no—
215C23×C53—21no—
216C23×C53—21no—
217C23×C53—22no—
218C23×C53—21no—
219C23×C53—21no—
220C23×C53—41no—
221C23×C53—41no—
222C23×C53—41no—
223C23×C53—41no—
224C23×C53—42no—
225C23×C53—81no—
226C23×C53—81no—
227C23×C53#13512yes—
228C23×C53C22×D411yescharacteristic
229Q8×C53—24no—
230Q8×C53—42no—
231Q8×C53—42no—
232Q8×C53—81no—
233D4×C53—44no—
234D4×C53—42no—
235D4×C53—44no—
236D4×C53—42no—
237D4×C53—42no—
238D4×C53—82no—
239D4×C53—81no—
240D4×C53—82no—
241C2×C4×C53—21no—
242C2×C4×C53—21no—
243C2×C4×C53—22no—
244C2×C4×C53—22no—
245C2×C4×C53—21no—
246C2×C4×C53—22no—
247C2×C4×C53—21no—
248C2×C4×C53—22no—
249C2×C4×C53—22no—
250C2×C4×C53—41no—
251C2×C4×C53—42no—
252C2×C4×C53—41no—
253C2×C4×C53—41no—
254C2×C4×C53—42no—
255C2×C4×C53—42no—
256C2×C4×C53—44no—
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—81no—
269C2×C4×C53—81no—
270C2×C4×C53—81no—
271C2×C4×C53—82no—
272C2×C4×C53—81no—
273C2×C4×C53#12011yescharacteristic
274C2×C4×C53#12012yes—
275C2×C4×C53#12011yescharacteristic
276C2×C4×C53C22×D412yes—
277C8×C53—42no—
278C8×C53—42no—
279#18129C5311yesmaximal, characteristic, direct product factor, Sylow
280C22×C53—21no—
281C22×C53—21no—
282C22×C53—22no—
283C22×C53—42no—
284C22×C53—42no—
285C22×C53—41no—
286C22×C53—41no—
287C22×C53—41no—
288C22×C53—42no—
289C22×C53—41no—
290C22×C53—42no—
291C22×C53—42no—
292C22×C53—41no—
293C22×C53—81no—
294C22×C53—81no—
295C22×C53—84no—
296C22×C53C2×#12011yescharacteristic
297C22×C53C2×#13511yescharacteristic
298C22×C53#57511yescharacteristic
299C4×C53—22no—
300C4×C53—21no—
301C4×C53—22no—
302C4×C53—42no—
303C4×C53—41no—
304C4×C53—41no—
305C4×C53—42no—
306C4×C53—42no—
307C4×C53—42no—
308C4×C53—81no—
309C4×C53—81no—
310C4×C53—81no—
311C4×C53#45512yes—
312#3459C2×C5311yescharacteristic
313#3108C2×C5312yescomplemented
314#2398C2×C5311yescharacteristic, complemented
315#1612C2×C5312yescomplemented
316#1608C2×C5311yescharacteristic, complemented
317#1607C2×C5311yescharacteristic, complemented
318#1606C2×C5312yescomplemented
319#1602C2×C5311yescharacteristic, complemented
320#1601C2×C5311yescharacteristic, complemented
321#1500C2×C5312yescomplemented
322#1460C2×C5311yescharacteristic, complemented
323C2×C53—21no—
324C2×C53—21no—
325C2×C53—42no—
326C2×C53—81no—
327C2×C53—82no—
328C2×C53#300211yescharacteristic
329C2×C53C2×#45511yescharacteristic
330C2×C53#302511yescharacteristic
331#543C22×C5311yescharacteristic
332#539—22no—
333#539—21no—
334#538—22no—
335#538C22×C5312yes—
336#534—21no—
337#534C22×C5311yescharacteristic
338#529—22no—
339#529C22×C5312yes—
340#525—21no—
341#525—22no—
342C2×C4×Q8C22×C5311yescharacteristic, complemented
343C2×C4×D4—21no—
344C2×C4×D4C22×C5311yescharacteristic
345#483C22×C5312yescomplemented
346#480C22×C5312yes—
347#479—22no—
348#464—22no—
349#451C22×C5312yescomplemented
350#448C22×C5312yes—
351C4×Q16—22no—
352C4×SD16—22no—
353#431C22×C5312yescomplemented
354#400—21no—
355#397C22×C5311yescharacteristic
356#397C22×C5311yescharacteristic, complemented
357#394—21no—
358#393C22×C5311yescharacteristic
359#393C22×C5311yescharacteristic, complemented
360C4×C22⋊C4—21no—
361C4×C22⋊C4—21no—
362#376C22×C5311yescharacteristic, complemented
363#328C22×C5312yescomplemented
364#328C22×C5312yescomplemented
365#327C22×C5312yes—
366#327C22×C5312yes—
367#323C22×C5311yescharacteristic
368#323C22×C5311yescharacteristic
369#323C22×C5312yescomplemented
370C53#1812911yescharacteristic, direct product factor, Sylow
371C2×#41—21no—
372C22×Q8C23×C5311yescharacteristic
373C22×D4—21no—
374C22×D4D4×C5311yescharacteristic
375C23×C4—21no—
376C23×C4D4×C5311yescharacteristic
377C23×C4D4×C5311yescharacteristic
378C2×Q16—22no—
379C2×SD16—22no—
380#126—22no—
381#126—22no—
382#126—22no—
383#125—22no—
384#125D4×C5312yes—
385#124—22no—
386#124D4×C5312yes—
387#123—22no—
388#123—22no—
389#123—22no—
390#123—22no—
391#123—22no—
392#123—22no—
393C22⋊Q8—21no—
394C22⋊Q8—22no—
395C22⋊Q8—22no—
396C22⋊Q8—21no—
397C22⋊Q8—22no—
398C22⋊Q8C23×C5312yes—
399#121—22no—
400#121—21no—
401#121—22no—
402#121—22no—
403#121—21no—
404#121C23×C5312yes—
405#120—22no—
406#120—22no—
407#120—22no—
408C4×Q8—22no—
409C4×Q8—22no—
410C4×Q8D4×C5312yes—
411C4×D4—22no—
412C4×D4—22no—
413C4×D4—22no—
414C4×D4—22no—
415C4×D4—22no—
416C4×D4—22no—
417C4×D4—22no—
418C4×D4D4×C5312yes—
419#117—21no—
420#117—22no—
421C2×C4⋊C4—21no—
422C2×C4⋊C4—21no—
423C2×C4⋊C4D4×C5311yescharacteristic
424C2×C4⋊C4C23×C5311yescharacteristic
425C2×C22⋊C4—21no—
426C2×C22⋊C4—21no—
427C2×C22⋊C4—21no—
428C2×C22⋊C4—21no—
429C2×C22⋊C4—21no—
430C2×C22⋊C4—21no—
431C2×C22⋊C4C23×C5311yescharacteristic
432C2×C22⋊C4C23×C5311yescharacteristic
433C2×C42—21no—
434C2×C42—21no—
435C2×C42—21no—
436C2×C42C23×C5311yescharacteristic
437#107—22no—
438#106—22no—
439C4⋊C8—22no—
440#103—22no—
441#103—22no—
442#103—22no—
443#102—22no—
444C22⋊C8C23×C5312yes—
445C22⋊C8C23×C5312yes—
446C4×C8—22no—
447#95—21no—
448#95—21no—
449#95—21no—
450#95—21no—
451#95—21no—
452#95C23×C5311yescharacteristic
453#95C23×C5311yescharacteristic
454C24—21no—
455C24C2×D4×C5311yescharacteristic
456C24C2×D4×C5311yescharacteristic
457#41—82no—
458C2×Q8—21no—
459C2×Q8—22no—
460C2×Q8—42no—
461C2×Q8C2×D4×C5312yes—
462C2×D4—22no—
463C2×D4—22no—
464C2×D4—22no—
465C2×D4—22no—
466C2×D4—21no—
467C2×D4—22no—
468C2×D4—22no—
469C2×D4—41no—
470C2×D4—42no—
471C2×D4—42no—
472C2×D4C2×D4×C5312yes—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—22no—
478C22×C4—21no—
479C22×C4—21no—
480C22×C4—22no—
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—21no—
484C22×C4—21no—
485C22×C4—21no—
486C22×C4—42no—
487C22×C4—42no—
488C22×C4—82no—
489C22×C4C2×D4×C5311yescharacteristic
490C22×C4C2×D4×C5311yescharacteristic
491C22×C4C2×D4×C5311yescharacteristic
492C22×C4C2×D4×C5311yescharacteristic
493C22×C4C24×C5311yescharacteristic
494Q16—44no—
495SD16—44no—
496C2×C8—22no—
497C2×C8—22no—
498C4⋊C4—22no—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—22no—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4—22no—
508C4⋊C4—42no—
509C4⋊C4C2×D4×C5312yes—
510C4⋊C4C2×D4×C5312yes—
511C22⋊C4—22no—
512C22⋊C4—22no—
513C22⋊C4—22no—
514C22⋊C4—22no—
515C22⋊C4—22no—
516C22⋊C4—22no—
517C22⋊C4—22no—
518C22⋊C4—22no—
519C22⋊C4—22no—
520C22⋊C4—22no—
521C22⋊C4—22no—
522C22⋊C4—22no—
523C22⋊C4—42no—
524C22⋊C4—42no—
525C22⋊C4—42no—
526C22⋊C4—42no—
527C42—22no—
528C42—22no—
529C42—42no—
530C42—41no—
531C42—41no—
532C42C2×D4×C5312yes—
533C42C2×D4×C5312yes—
534C23—21no—
535C23—21no—
536C23—21no—
537C23—22no—
538C23—21no—
539C23—21no—
540C23—41no—
541C23—41no—
542C23—41no—
543C23—41no—
544C23—42no—
545C23—81no—
546C23—81no—
547C23#135×C5312yes—
548C23C22×D4×C5311yescharacteristic
549Q8—24no—
550Q8—42no—
551Q8—42no—
552Q8—81no—
553D4—44no—
554D4—42no—
555D4—44no—
556D4—42no—
557D4—42no—
558D4—82no—
559D4—81no—
560D4—82no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—22no—
570C2×C4—41no—
571C2×C4—42no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—44no—
577C2×C4—41no—
578C2×C4—41no—
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—81no—
589C2×C4—81no—
590C2×C4—81no—
591C2×C4—82no—
592C2×C4—81no—
593C2×C4#120×C5311yescharacteristic
594C2×C4#120×C5312yes—
595C2×C4#120×C5311yescharacteristic
596C2×C4C22×D4×C5312yes—
597C8—42no—
598C8—42no—
599C22—21no—
600C22—21no—
601C22—22no—
602C22—42no—
603C22—42no—
604C22—41no—
605C22—41no—
606C22—41no—
607C22—42no—
608C22—41no—
609C22—42no—
610C22—42no—
611C22—41no—
612C22—81no—
613C22—81no—
614C22—84no—
615C22C2×#120×C5311yescharacteristic
616C22C2×#135×C5311yescharacteristic
617C22C53×#57511yescharacteristic
618C4—22no—
619C4—21no—
620C4—22no—
621C4—42no—
622C4—41no—
623C4—41no—
624C4—42no—
625C4—42no—
626C4—42no—
627C4—81no—
628C4—81no—
629C4—81no—
630C4C53×#45512yes—
631C2—21no—
632C2—21no—
633C2—42no—
634C2—81no—
635C2—82no—
636C2C53×#300211yescharacteristic
637C2C2×C53×#45511yescharacteristic
638C2C53×#302511yescharacteristic
subgroupG/H
centerC2×C4×C53#120
commutator, Frattini, cosocleC22×C4C24×C53
socleC22×C53C2×#120
prime radicalC53×#3459C2
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×#3108C2semidirect
3. C53×#2398C2semidirect
4. C53×#1612C2semidirect
5. C53×#1608C2semidirect
6. C53×#1607C2semidirect
7. C53×#1606C2semidirect
8. C53×#1602C2semidirect
9. C53×#1601C2semidirect
10. C53×#1500C2semidirect
11. C53×#1460C2semidirect
23. C2×C4×Q8×C53C22semidirect
26. C53×#483C22semidirect
30. C53×#451C22semidirect
34. C53×#431C22semidirect
37. C53×#397C22semidirect
40. C53×#393C22semidirect
43. C53×#376C22semidirect
44. C53×#328C22semidirect
45. C53×#328C22semidirect
50. C53×#323C22semidirect
279. #18129C53direct (N × H)
313. #3108C2×C53semidirect
314. #2398C2×C53semidirect
315. #1612C2×C53semidirect
316. #1608C2×C53semidirect
317. #1607C2×C53semidirect
318. #1606C2×C53semidirect
319. #1602C2×C53semidirect
320. #1601C2×C53semidirect
321. #1500C2×C53semidirect
322. #1460C2×C53semidirect
342. C2×C4×Q8C22×C53semidirect
345. #483C22×C53semidirect
349. #451C22×C53semidirect
353. #431C22×C53semidirect
356. #397C22×C53semidirect
359. #393C22×C53semidirect
362. #376C22×C53semidirect
363. #328C22×C53semidirect
364. #328C22×C53semidirect
369. #323C22×C53semidirect
370. C53#18129direct (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×#31082
C53×#23982
C53×#16122
C53×#16082
C53×#16072
C53×#16062
C53×#16022
C53×#16012
C53×#15002
C53×#14602
#1812953