Finite Group Databasebeta

← C53×#18468 #27225980

Subgroups — C53×#18468 #27225980

Order 13568.

870 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C53×#3559C211yesmaximal, characteristic
2C53×#3234C211yesmaximal, characteristic, complemented
3C53×#3232C211yesmaximal, characteristic, complemented
4C53×#2822C211yesmaximal, characteristic, complemented
5C53×#1616C212yesmaximal, complemented
6C53×#1615C212yesmaximal, complemented
7C53×#1610C212yesmaximal, complemented
8C53×#1609C212yesmaximal, 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×#125×C53C2211yescharacteristic
28C53×#501—21no—
29C53×#499—21no—
30C53×#490—21no—
31C53×#488—21no—
32C53×#484C2211yescharacteristic
33C53×#483C2211yescharacteristic
34C53×#481C2211yescharacteristic
35C53×#481C2212yes—
36C53×#480C2211yescharacteristic
37C53×#480C2212yes—
38C53×#479—21no—
39C53×#478—21no—
40C53×#478—22no—
41C53×#432C2211yescharacteristic
42C53×#431C2211yescharacteristic
43C53×#400—21no—
44C53×#400C2212yes—
45C53×#398—21no—
46C53×#398—22no—
47C53×#398C2212yes—
48C53×#382C2211yescharacteristic
49C53×#328C2212yes—
50C53×#327C2212yes—
51C53×#327C2212yes—
52C53×#327C2212yes—
53C53×#324C2212yes—
54C53×#324C2212yes—
55C2×#41×C53—21no—
56C2×#41×C53—22no—
57C4⋊Q8×C53—21no—
58C4⋊Q8×C53—22no—
59C4⋊Q8×C53D411yescharacteristic
60#127×C53D411yescharacteristic
61#126×C53—22no—
62#126×C53—22no—
63#126×C53—21no—
64#125×C53—21no—
65#125×C53—22no—
66#125×C53—21no—
67#125×C53—21no—
68#125×C53—21no—
69#125×C53—21no—
70#125×C53—21no—
71#125×C53—21no—
72#125×C53—21no—
73#125×C53—21no—
74#125×C53D411yescharacteristic
75#124×C53—21no—
76#124×C53D411yescharacteristic
77#123×C53—22no—
78#123×C53—22no—
79#123×C53—22no—
80#123×C53—22no—
81#123×C53—22no—
82#123×C53—21no—
83#123×C53—22no—
84C22⋊Q8×C53—22no—
85C22⋊Q8×C53—22no—
86C22⋊Q8×C53—22no—
87C22⋊Q8×C53—22no—
88C22⋊Q8×C53—22no—
89C22⋊Q8×C53—22no—
90C22⋊Q8×C53C2311yescharacteristic
91#121×C53—21no—
92#121×C53—22no—
93#121×C53—21no—
94#121×C53—22no—
95#121×C53—22no—
96#121×C53—21no—
97#121×C53—22no—
98#121×C53C2311yescharacteristic
99#121×C53C2312yes—
100C4×Q8×C53D411yescharacteristic
101C4×D4×C53—22no—
102C4×D4×C53—21no—
103C4×D4×C53—21no—
104C4×D4×C53—21no—
105C4×D4×C53—21no—
106C4×D4×C53—22no—
107C4×D4×C53D411yescharacteristic
108C4×D4×C53D412yes—
109#117×C53—21no—
110#117×C53—21no—
111C2×C4⋊C4×C53—22no—
112C2×C4⋊C4×C53—21no—
113C2×C4⋊C4×C53—21no—
114C2×C4⋊C4×C53—21no—
115C2×C4⋊C4×C53—21no—
116C2×C4⋊C4×C53—22no—
117C2×C4⋊C4×C53D411yescharacteristic
118C2×C4⋊C4×C53D412yes—
119C2×C4⋊C4×C53D411yescharacteristic
120C2×C4⋊C4×C53C2311yescharacteristic
121C2×C4⋊C4×C53C2311yescharacteristic
122C2×C4⋊C4×C53C2312yes—
123C2×C42×C53C2311yescharacteristic
124#107×C53—21no—
125#107×C53—21no—
126#107×C53—22no—
127#106×C53—21no—
128#106×C53—21no—
129#106×C53—22no—
130C4⋊C8×C53—21no—
131C4⋊C8×C53—21no—
132#103×C53—21no—
133#103×C53—21no—
134#102×C53—21no—
135#102×C53—21no—
136#102×C53—22no—
137#102×C53—22no—
138C22⋊C8×C53C2311yescharacteristic
139C22⋊C8×C53C2311yescharacteristic
140C22⋊C8×C53C2312yes—
141#97×C53—21no—
142C4×C8×C53—21no—
143#95×C53—21no—
144#95×C53—22no—
145#95×C53—22no—
146#95×C53—22no—
147#95×C53C2312yes—
148#41×C53—81no—
149#41×C53—81no—
150#41×C53—82no—
151#41×C53—82no—
152C2×Q8×C53—22no—
153C2×Q8×C53—21no—
154C2×Q8×C53C2×D411yescharacteristic
155C2×D4×C53—21no—
156C2×D4×C53—22no—
157C2×D4×C53—22no—
158C2×D4×C53—22no—
159C2×D4×C53—21no—
160C2×D4×C53—21no—
161C2×D4×C53C2×D411yescharacteristic
162C2×D4×C53C2×D412yes—
163C22×C4×C53—22no—
164C22×C4×C53—22no—
165C22×C4×C53—22no—
166C22×C4×C53—21no—
167C22×C4×C53—22no—
168C22×C4×C53—22no—
169C22×C4×C53—21no—
170C22×C4×C53—21no—
171C22×C4×C53C2×D412yes—
172C22×C4×C53C2×D411yescharacteristic
173C22×C4×C53C2×D412yes—
174C22×C4×C53C2×D411yescharacteristic
175C22×C4×C53C2411yescharacteristic
176C2×C8×C53—21no—
177C2×C8×C53—21no—
178C2×C8×C53—22no—
179C4⋊C4×C53—22no—
180C4⋊C4×C53—22no—
181C4⋊C4×C53—22no—
182C4⋊C4×C53—22no—
183C4⋊C4×C53—21no—
184C4⋊C4×C53—21no—
185C4⋊C4×C53—22no—
186C4⋊C4×C53—21no—
187C4⋊C4×C53—21no—
188C4⋊C4×C53—22no—
189C4⋊C4×C53—21no—
190C4⋊C4×C53—21no—
191C4⋊C4×C53—21no—
192C4⋊C4×C53—21no—
193C4⋊C4×C53—22no—
194C4⋊C4×C53—22no—
195C4⋊C4×C53—21no—
196C4⋊C4×C53—21no—
197C4⋊C4×C53—21no—
198C4⋊C4×C53—21no—
199C4⋊C4×C53—21no—
200C4⋊C4×C53—41no—
201C4⋊C4×C53—41no—
202C4⋊C4×C53—41no—
203C4⋊C4×C53—41no—
204C4⋊C4×C53—81no—
205C4⋊C4×C53—82no—
206C4⋊C4×C53C2×D412yes—
207C4⋊C4×C53C2×D411yescharacteristic
208C4⋊C4×C53C2×D411yescharacteristic
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—21no—
217C22⋊C4×C53—21no—
218C22⋊C4×C53—22no—
219C22⋊C4×C53—21no—
220C22⋊C4×C53—22no—
221C42×C53—22no—
222C42×C53—21no—
223C42×C53C2×D411yescharacteristic
224C42×C53C2×D411yescharacteristic
225C42×C53C2×D411yescharacteristic
226C42×C53C2×D411yescharacteristic
227C23×C53—21no—
228C23×C53—22no—
229C23×C53C22×D411yescharacteristic
230Q8×C53—41no—
231Q8×C53—81no—
232Q8×C53—82no—
233D4×C53—42no—
234D4×C53—41no—
235D4×C53—42no—
236D4×C53—42no—
237D4×C53—41no—
238D4×C53—42no—
239D4×C53—41no—
240D4×C53—81no—
241D4×C53—82no—
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—21no—
251C2×C4×C53—22no—
252C2×C4×C53—22no—
253C2×C4×C53—41no—
254C2×C4×C53—41no—
255C2×C4×C53—42no—
256C2×C4×C53—42no—
257C2×C4×C53—42no—
258C2×C4×C53—42no—
259C2×C4×C53—42no—
260C2×C4×C53—41no—
261C2×C4×C53—41no—
262C2×C4×C53—42no—
263C2×C4×C53—41no—
264C2×C4×C53—41no—
265C2×C4×C53—82no—
266C2×C4×C53—82no—
267C2×C4×C53—82no—
268C2×C4×C53—82no—
269C2×C4×C53#12011yescharacteristic
270C2×C4×C53#12011yescharacteristic
271C2×C4×C53#12011yescharacteristic
272C2×C4×C53#12011yescharacteristic
273C2×C4×C53C22×D411yescharacteristic
274C2×C4×C53C22×D411yescharacteristic
275C8×C53—42no—
276C8×C53—41no—
277C8×C53—41no—
278#18468C5311yesmaximal, characteristic, direct product factor, Sylow
279C22×C53—21no—
280C22×C53—21no—
281C22×C53—42no—
282C22×C53—41no—
283C22×C53—41no—
284C22×C53—42no—
285C22×C53—42no—
286C22×C53—41no—
287C22×C53C2×#12011yescharacteristic
288C22×C53#57611yescharacteristic
289C22×C53#57711yescharacteristic
290C4×C53—21no—
291C4×C53—21no—
292C4×C53—21no—
293C4×C53—21no—
294C4×C53—21no—
295C4×C53—21no—
296C4×C53—42no—
297C4×C53—42no—
298C4×C53—41no—
299C4×C53—41no—
300C4×C53—42no—
301C4×C53—41no—
302C4×C53—82no—
303C4×C53—81no—
304C4×C53—81no—
305#3559C2×C5311yescharacteristic
306#3234C2×C5311yescharacteristic, complemented
307#3232C2×C5311yescharacteristic, complemented
308#2822C2×C5311yescharacteristic, complemented
309#1616C2×C5312yescomplemented
310#1615C2×C5312yescomplemented
311#1610C2×C5312yescomplemented
312#1609C2×C5312yescomplemented
313#1517C2×C5311yescharacteristic, complemented
314#1512C2×C5311yescharacteristic, complemented
315#1470C2×C5311yescharacteristic, complemented
316C2×C53—21no—
317C2×C53—21no—
318C2×C53—81no—
319C2×C53—82no—
320C2×C53#301211yescharacteristic
321C2×C53#302011yescharacteristic
322C2×C53#302811yescharacteristic
323#563—21no—
324#562—21no—
325#554—21no—
326#553—21no—
327#553—21no—
328#547—22no—
329#547—21no—
330#544—21no—
331#540—21no—
332#539—22no—
333#539C22×C5312yes—
334#537C22×C5312yes—
335#532C22×C5311yescharacteristic
336#529C22×C5311yescharacteristic
337C2×#125C22×C5311yescharacteristic
338C2×#125C22×C5311yescharacteristic
339#501—21no—
340#499—21no—
341#490—21no—
342#488—21no—
343#484C22×C5311yescharacteristic
344#483C22×C5311yescharacteristic
345#481C22×C5312yes—
346#481C22×C5311yescharacteristic
347#480C22×C5312yes—
348#480C22×C5311yescharacteristic
349#479—21no—
350#478—21no—
351#478—22no—
352#432C22×C5311yescharacteristic
353#431C22×C5311yescharacteristic
354#400—21no—
355#400C22×C5312yes—
356#398—21no—
357#398—22no—
358#398C22×C5312yes—
359#382C22×C5311yescharacteristic
360#328C22×C5312yes—
361#327C22×C5312yes—
362#327C22×C5312yes—
363#327C22×C5312yes—
364#324C22×C5312yes—
365#324C22×C5312yes—
366C53#1846811yescharacteristic, direct product factor, Sylow
367C2×#41—21no—
368C2×#41—22no—
369C4⋊Q8—21no—
370C4⋊Q8—22no—
371C4⋊Q8D4×C5311yescharacteristic
372#127D4×C5311yescharacteristic
373#126—22no—
374#126—22no—
375#126—21no—
376#125—21no—
377#125—22no—
378#125—21no—
379#125—21no—
380#125—21no—
381#125—21no—
382#125—21no—
383#125—21no—
384#125—21no—
385#125—21no—
386#125D4×C5311yescharacteristic
387#124—21no—
388#124D4×C5311yescharacteristic
389#123—22no—
390#123—22no—
391#123—22no—
392#123—22no—
393#123—22no—
394#123—21no—
395#123—22no—
396C22⋊Q8—22no—
397C22⋊Q8—22no—
398C22⋊Q8—22no—
399C22⋊Q8—22no—
400C22⋊Q8—22no—
401C22⋊Q8—22no—
402C22⋊Q8C23×C5311yescharacteristic
403#121—21no—
404#121—22no—
405#121—21no—
406#121—22no—
407#121—22no—
408#121—21no—
409#121—22no—
410#121C23×C5311yescharacteristic
411#121C23×C5312yes—
412C4×Q8D4×C5311yescharacteristic
413C4×D4—22no—
414C4×D4—21no—
415C4×D4—21no—
416C4×D4—21no—
417C4×D4—22no—
418C4×D4—21no—
419C4×D4D4×C5311yescharacteristic
420C4×D4D4×C5312yes—
421#117—21no—
422#117—21no—
423C2×C4⋊C4—22no—
424C2×C4⋊C4—21no—
425C2×C4⋊C4—21no—
426C2×C4⋊C4—21no—
427C2×C4⋊C4—21no—
428C2×C4⋊C4—22no—
429C2×C4⋊C4D4×C5311yescharacteristic
430C2×C4⋊C4D4×C5312yes—
431C2×C4⋊C4D4×C5311yescharacteristic
432C2×C4⋊C4C23×C5311yescharacteristic
433C2×C4⋊C4C23×C5311yescharacteristic
434C2×C4⋊C4C23×C5312yes—
435C2×C42C23×C5311yescharacteristic
436#107—22no—
437#107—21no—
438#107—21no—
439#106—22no—
440#106—21no—
441#106—21no—
442C4⋊C8—21no—
443C4⋊C8—21no—
444#103—21no—
445#103—21no—
446#102—21no—
447#102—22no—
448#102—21no—
449#102—22no—
450C22⋊C8C23×C5311yescharacteristic
451C22⋊C8C23×C5311yescharacteristic
452C22⋊C8C23×C5312yes—
453#97—21no—
454C4×C8—21no—
455#95—21no—
456#95—22no—
457#95—22no—
458#95—22no—
459#95C23×C5312yes—
460#41—81no—
461#41—81no—
462#41—82no—
463#41—82no—
464C2×Q8—22no—
465C2×Q8—21no—
466C2×Q8C2×D4×C5311yescharacteristic
467C2×D4—21no—
468C2×D4—22no—
469C2×D4—22no—
470C2×D4—22no—
471C2×D4—21no—
472C2×D4—21no—
473C2×D4C2×D4×C5311yescharacteristic
474C2×D4C2×D4×C5312yes—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—21no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—21no—
482C22×C4—21no—
483C22×C4C2×D4×C5312yes—
484C22×C4C2×D4×C5311yescharacteristic
485C22×C4C2×D4×C5312yes—
486C22×C4C2×D4×C5311yescharacteristic
487C22×C4C24×C5311yescharacteristic
488C2×C8—21no—
489C2×C8—21no—
490C2×C8—22no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—22no—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—22no—
501C4⋊C4—21no—
502C4⋊C4—21no—
503C4⋊C4—21no—
504C4⋊C4—21no—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—21no—
510C4⋊C4—21no—
511C4⋊C4—21no—
512C4⋊C4—41no—
513C4⋊C4—41no—
514C4⋊C4—41no—
515C4⋊C4—41no—
516C4⋊C4—81no—
517C4⋊C4—82no—
518C4⋊C4C2×D4×C5312yes—
519C4⋊C4C2×D4×C5311yescharacteristic
520C4⋊C4C2×D4×C5311yescharacteristic
521C22⋊C4—22no—
522C22⋊C4—22no—
523C22⋊C4—22no—
524C22⋊C4—22no—
525C22⋊C4—22no—
526C22⋊C4—21no—
527C22⋊C4—21no—
528C22⋊C4—21no—
529C22⋊C4—21no—
530C22⋊C4—22no—
531C22⋊C4—21no—
532C22⋊C4—22no—
533C42—22no—
534C42—21no—
535C42C2×D4×C5311yescharacteristic
536C42C2×D4×C5311yescharacteristic
537C42C2×D4×C5311yescharacteristic
538C42C2×D4×C5311yescharacteristic
539C23—21no—
540C23—22no—
541C23C22×D4×C5311yescharacteristic
542Q8—41no—
543Q8—81no—
544Q8—82no—
545D4—42no—
546D4—41no—
547D4—42no—
548D4—42no—
549D4—41no—
550D4—42no—
551D4—41no—
552D4—81no—
553D4—82no—
554D4—82no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—21no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—42no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—82no—
578C2×C4—82no—
579C2×C4—82no—
580C2×C4—82no—
581C2×C4#120×C5311yescharacteristic
582C2×C4#120×C5311yescharacteristic
583C2×C4#120×C5311yescharacteristic
584C2×C4#120×C5311yescharacteristic
585C2×C4C22×D4×C5311yescharacteristic
586C2×C4C22×D4×C5311yescharacteristic
587C8—42no—
588C8—41no—
589C8—41no—
590C22—21no—
591C22—21no—
592C22—42no—
593C22—41no—
594C22—41no—
595C22—42no—
596C22—42no—
597C22—41no—
598C22C2×#120×C5311yescharacteristic
599C22C53×#57611yescharacteristic
600C22C53×#57711yescharacteristic
601C4—21no—
602C4—21no—
603C4—21no—
604C4—21no—
605C4—21no—
606C4—21no—
607C4—42no—
608C4—42no—
609C4—41no—
610C4—41no—
611C4—42no—
612C4—41no—
613C4—82no—
614C4—81no—
615C4—81no—
616C2—21no—
617C2—21no—
618C2—81no—
619C2—82no—
620C2C53×#301211yescharacteristic
621C2C53×#302011yescharacteristic
622C2C53×#302811yescharacteristic
subgroupG/H
center, socleC22×C53C2×#120
commutator, Frattini, cosocleC22×C4C24×C53
prime radicalC53×#3559C2
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×#3234C2semidirect
3. C53×#3232C2semidirect
4. C53×#2822C2semidirect
5. C53×#1616C2semidirect
6. C53×#1615C2semidirect
7. C53×#1610C2semidirect
8. C53×#1609C2semidirect
9. C53×#1517C2semidirect
10. C53×#1512C2semidirect
11. C53×#1470C2semidirect
278. #18468C53direct (N × H)
306. #3234C2×C53semidirect
307. #3232C2×C53semidirect
308. #2822C2×C53semidirect
309. #1616C2×C53semidirect
310. #1615C2×C53semidirect
311. #1610C2×C53semidirect
312. #1609C2×C53semidirect
313. #1517C2×C53semidirect
314. #1512C2×C53semidirect
315. #1470C2×C53semidirect
366. C53#18468direct (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×#32342
C53×#32322
C53×#28222
C53×#16162
C53×#16152
C53×#16102
C53×#16092
C53×#15172
C53×#15122
C53×#14702
#1846853