Finite Group Databasebeta

← C53×#18437 #27225949

Subgroups — C53×#18437 #27225949

Order 13568.

1028 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C53×#3558C211yesmaximal, characteristic
2C53×#3239C211yesmaximal, characteristic, complemented
3C53×#3227C211yesmaximal, characteristic, complemented
4C53×#2658C211yesmaximal, characteristic, complemented
5C53×#1618C214yesmaximal, complemented
6C53×#1613C214yesmaximal, complemented
7C53×#1505C211yesmaximal, characteristic, complemented
8C53×#1500C211yesmaximal, characteristic, complemented
9C53×#1469C211yesmaximal, characteristic, complemented
10C53×#562—21no—
11C53×H(3,𝔽4)—21no—
12C53×#559—21no—
13C53×#548—21no—
14C53×#546—21no—
15C53×#546—22no—
16D42×C53—21no—
17C53×#542—21no—
18C53×#538—22no—
19C53×#538C2212yes—
20C53×#535—21no—
21C53×#534C2212yes—
22C53×#532C2211yescharacteristic
23C2×C4⋊Q8×C53—21no—
24C2×C4⋊Q8×C53C2211yescharacteristic, complemented
25C53×#529C2211yescharacteristic
26C2×#124×C53—21no—
27C2×#124×C53C2211yescharacteristic
28C53×#501—21no—
29C53×#498—21no—
30C53×#489—21no—
31C53×#486—21no—
32C53×#484C2212yescomplemented
33C53×#481C2212yes—
34C53×#481C2214yescomplemented
35C53×#476—22no—
36C53×#475—21no—
37C53×#474—21no—
38C53×#432C2211yescharacteristic, complemented
39C53×#431C2211yescharacteristic, complemented
40C53×#399—22no—
41C53×#399C2214yescomplemented
42C53×#376C2211yescharacteristic, complemented
43C53×#328C2212yescomplemented
44C53×#327C2212yes—
45C53×#327C2214yescomplemented
46C53×#323C2214yescomplemented
47C2×#41×C53—21no—
48C2×#41×C53—21no—
49C22×Q8×C53—21no—
50C22×Q8×C53D411yescharacteristic
51C22×D4×C53—21no—
52C22×D4×C53—22no—
53C22×D4×C53D411yescharacteristic
54C4⋊Q8×C53—21no—
55C4⋊Q8×C53—21no—
56C4⋊Q8×C53—21no—
57C4⋊Q8×C53—21no—
58C4⋊Q8×C53—44no—
59C4⋊Q8×C53D411yescharacteristic
60#127×C53—21no—
61#127×C53D411yescharacteristic
62#126×C53—22no—
63#126×C53—21no—
64#125×C53D411yescharacteristic
65#124×C53—21no—
66#124×C53—22no—
67#124×C53—22no—
68#124×C53—21no—
69#124×C53—21no—
70#124×C53—21no—
71#124×C53—21no—
72#124×C53—21no—
73#124×C53—22no—
74#124×C53—44no—
75#124×C53D411yescharacteristic
76#123×C53—22no—
77#123×C53—22no—
78#123×C53—22no—
79#123×C53—22no—
80#123×C53—21no—
81C22⋊Q8×C53—22no—
82C22⋊Q8×C53—21no—
83C22⋊Q8×C53—22no—
84C22⋊Q8×C53C2311yescharacteristic
85#121×C53—22no—
86#121×C53—22no—
87#121×C53—22no—
88#121×C53—21no—
89#121×C53—22no—
90#121×C53—21no—
91#121×C53—22no—
92#121×C53C2311yescharacteristic
93#121×C53C2312yes—
94#120×C53—22no—
95#120×C53—22no—
96#120×C53—22no—
97#120×C53—22no—
98#120×C53—22no—
99C4×Q8×C53—21no—
100C4×Q8×C53D411yescharacteristic
101C4×D4×C53—22no—
102C4×D4×C53—21no—
103C4×D4×C53—21no—
104C4×D4×C53—21no—
105C4×D4×C53—22no—
106C4×D4×C53D411yescharacteristic
107C4×D4×C53D412yes—
108#117×C53—21no—
109C2×C4⋊C4×C53—24no—
110C2×C4⋊C4×C53—21no—
111C2×C4⋊C4×C53C2312yes—
112C2×C22⋊C4×C53—22no—
113C2×C22⋊C4×C53—24no—
114C2×C22⋊C4×C53D412yes—
115C2×C22⋊C4×C53C2312yes—
116C2×C42×C53—21no—
117C2×C42×C53C2311yescharacteristic
118#107×C53—22no—
119#106×C53—24no—
120#106×C53—22no—
121C4⋊C8×C53—21no—
122C4⋊C8×C53—21no—
123#103×C53—22no—
124#102×C53—24no—
125#102×C53—22no—
126C22⋊C8×C53C2312yes—
127C22⋊C8×C53C2312yes—
128#97×C53—21no—
129C4×C8×C53—21no—
130#95×C53—21no—
131#95×C53C2312yes—
132C24×C53—22no—
133C24×C53C2×D412yes—
134#41×C53—82no—
135#41×C53—82no—
136C2×Q8×C53—21no—
137C2×Q8×C53—21no—
138C2×Q8×C53—21no—
139C2×Q8×C53—42no—
140C2×Q8×C53—42no—
141C2×Q8×C53—42no—
142C2×Q8×C53—41no—
143C2×Q8×C53—41no—
144C2×Q8×C53C2×D411yescharacteristic
145C2×D4×C53—22no—
146C2×D4×C53—21no—
147C2×D4×C53—22no—
148C2×D4×C53—22no—
149C2×D4×C53—22no—
150C2×D4×C53—22no—
151C2×D4×C53—22no—
152C2×D4×C53—21no—
153C2×D4×C53—21no—
154C2×D4×C53—21no—
155C2×D4×C53—21no—
156C2×D4×C53—42no—
157C2×D4×C53—82no—
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—22no—
166C22×C4×C53—22no—
167C22×C4×C53C2×D411yescharacteristic
168C22×C4×C53C2×D411yescharacteristic
169C22×C4×C53C2×D412yes—
170C22×C4×C53C2411yescharacteristic
171C2×C8×C53—22no—
172C2×C8×C53—22no—
173C4⋊C4×C53—22no—
174C4⋊C4×C53—22no—
175C4⋊C4×C53—21no—
176C4⋊C4×C53—21no—
177C4⋊C4×C53—21no—
178C4⋊C4×C53—21no—
179C4⋊C4×C53—21no—
180C4⋊C4×C53—22no—
181C4⋊C4×C53—22no—
182C4⋊C4×C53—48no—
183C4⋊C4×C53—41no—
184C4⋊C4×C53—41no—
185C4⋊C4×C53C2×D412yes—
186C4⋊C4×C53C2×D411yescharacteristic
187C4⋊C4×C53C2×D411yescharacteristic
188C22⋊C4×C53—22no—
189C22⋊C4×C53—22no—
190C22⋊C4×C53—22no—
191C22⋊C4×C53—22no—
192C22⋊C4×C53—22no—
193C22⋊C4×C53—21no—
194C22⋊C4×C53—21no—
195C22⋊C4×C53—22no—
196C22⋊C4×C53—22no—
197C22⋊C4×C53—21no—
198C22⋊C4×C53—21no—
199C22⋊C4×C53—22no—
200C22⋊C4×C53—22no—
201C22⋊C4×C53—22no—
202C22⋊C4×C53—21no—
203C22⋊C4×C53—22no—
204C22⋊C4×C53—48no—
205C42×C53—24no—
206C42×C53—22no—
207C42×C53—21no—
208C42×C53C2×D411yescharacteristic
209C42×C53C2×D411yescharacteristic
210C42×C53C2×D411yescharacteristic
211C42×C53C2×D411yescharacteristic
212C23×C53—22no—
213C23×C53—22no—
214C23×C53—21no—
215C23×C53—24no—
216C23×C53—42no—
217C23×C53—42no—
218C23×C53—82no—
219C23×C53—82no—
220C23×C53—82no—
221C23×C53C22×D411yescharacteristic
222Q8×C53—41no—
223Q8×C53—42no—
224Q8×C53—81no—
225Q8×C53—81no—
226Q8×C53—82no—
227Q8×C53—82no—
228D4×C53—42no—
229D4×C53—41no—
230D4×C53—42no—
231D4×C53—44no—
232D4×C53—42no—
233D4×C53—42no—
234D4×C53—81no—
235D4×C53—81no—
236D4×C53—82no—
237D4×C53—82no—
238D4×C53—82no—
239D4×C53—82no—
240D4×C53—82no—
241C2×C4×C53—22no—
242C2×C4×C53—22no—
243C2×C4×C53—24no—
244C2×C4×C53—21no—
245C2×C4×C53—21no—
246C2×C4×C53—22no—
247C2×C4×C53—22no—
248C2×C4×C53—21no—
249C2×C4×C53—41no—
250C2×C4×C53—42no—
251C2×C4×C53—42no—
252C2×C4×C53—42no—
253C2×C4×C53—42no—
254C2×C4×C53—42no—
255C2×C4×C53—42no—
256C2×C4×C53—42no—
257C2×C4×C53—82no—
258C2×C4×C53—82no—
259C2×C4×C53—82no—
260C2×C4×C53#12011yescharacteristic
261C2×C4×C53#12011yescharacteristic
262C2×C4×C53#12011yescharacteristic
263C2×C4×C53#12011yescharacteristic
264C2×C4×C53C22×D411yescharacteristic
265C2×C4×C53C22×D411yescharacteristic
266C8×C53—42no—
267C8×C53—42no—
268#18437C5311yesmaximal, characteristic, direct product factor, Sylow
269C22×C53—21no—
270C22×C53—21no—
271C22×C53—42no—
272C22×C53—42no—
273C22×C53—41no—
274C22×C53—41no—
275C22×C53—42no—
276C22×C53—42no—
277C22×C53—42no—
278C22×C53—41no—
279C22×C53—42no—
280C22×C53—84no—
281C22×C53—164no—
282C22×C53C2×#12011yescharacteristic
283C22×C53#57712yes—
284C4×C53—21no—
285C4×C53—21no—
286C4×C53—21no—
287C4×C53—21no—
288C4×C53—21no—
289C4×C53—44no—
290C4×C53—41no—
291C4×C53—42no—
292C4×C53—41no—
293C4×C53—42no—
294C4×C53—41no—
295C4×C53—82no—
296C4×C53#45312yes—
297#3558C2×C5311yescharacteristic
298#3239C2×C5311yescharacteristic, complemented
299#3227C2×C5311yescharacteristic, complemented
300#2658C2×C5311yescharacteristic, complemented
301#1618C2×C5314yescomplemented
302#1613C2×C5314yescomplemented
303#1505C2×C5311yescharacteristic, complemented
304#1500C2×C5311yescharacteristic, complemented
305#1469C2×C5311yescharacteristic, complemented
306C2×C53—22no—
307C2×C53—82no—
308C2×C53—82no—
309C2×C53—81no—
310C2×C53#300911yescharacteristic
311C2×C53C2×#45311yescharacteristic
312C2×C53#302811yescharacteristic
313#562—21no—
314H(3,𝔽4)—21no—
315#559—21no—
316#548—21no—
317#546—21no—
318#546—22no—
319D42—21no—
320#542—21no—
321#538—22no—
322#538C22×C5312yes—
323#535—21no—
324#534C22×C5312yes—
325#532C22×C5311yescharacteristic
326C2×C4⋊Q8—21no—
327C2×C4⋊Q8C22×C5311yescharacteristic, complemented
328#529C22×C5311yescharacteristic
329C2×#124—21no—
330C2×#124C22×C5311yescharacteristic
331#501—21no—
332#498—21no—
333#489—21no—
334#486—21no—
335#484C22×C5312yescomplemented
336#481C22×C5312yes—
337#481C22×C5314yescomplemented
338#476—22no—
339#475—21no—
340#474—21no—
341#432C22×C5311yescharacteristic, complemented
342#431C22×C5311yescharacteristic, complemented
343#399—22no—
344#399C22×C5314yescomplemented
345#376C22×C5311yescharacteristic, complemented
346#328C22×C5312yescomplemented
347#327C22×C5312yes—
348#327C22×C5314yescomplemented
349#323C22×C5314yescomplemented
350C53#1843711yescharacteristic, direct product factor, Sylow
351C2×#41—21no—
352C2×#41—21no—
353C22×Q8—21no—
354C22×Q8D4×C5311yescharacteristic
355C22×D4—21no—
356C22×D4—22no—
357C22×D4D4×C5311yescharacteristic
358C4⋊Q8—21no—
359C4⋊Q8—21no—
360C4⋊Q8—21no—
361C4⋊Q8—21no—
362C4⋊Q8—44no—
363C4⋊Q8D4×C5311yescharacteristic
364#127—21no—
365#127D4×C5311yescharacteristic
366#126—22no—
367#126—21no—
368#125D4×C5311yescharacteristic
369#124—21no—
370#124—22no—
371#124—22no—
372#124—22no—
373#124—21no—
374#124—21no—
375#124—21no—
376#124—21no—
377#124—21no—
378#124—44no—
379#124D4×C5311yescharacteristic
380#123—22no—
381#123—21no—
382#123—22no—
383#123—22no—
384#123—22no—
385C22⋊Q8—22no—
386C22⋊Q8—21no—
387C22⋊Q8—22no—
388C22⋊Q8C23×C5311yescharacteristic
389#121—22no—
390#121—22no—
391#121—22no—
392#121—21no—
393#121—22no—
394#121—21no—
395#121—22no—
396#121C23×C5311yescharacteristic
397#121C23×C5312yes—
398#120—22no—
399#120—22no—
400#120—22no—
401#120—22no—
402#120—22no—
403C4×Q8—21no—
404C4×Q8D4×C5311yescharacteristic
405C4×D4—22no—
406C4×D4—21no—
407C4×D4—21no—
408C4×D4—22no—
409C4×D4—21no—
410C4×D4D4×C5311yescharacteristic
411C4×D4D4×C5312yes—
412#117—21no—
413C2×C4⋊C4—24no—
414C2×C4⋊C4—21no—
415C2×C4⋊C4C23×C5312yes—
416C2×C22⋊C4—22no—
417C2×C22⋊C4—24no—
418C2×C22⋊C4D4×C5312yes—
419C2×C22⋊C4C23×C5312yes—
420C2×C42—21no—
421C2×C42C23×C5311yescharacteristic
422#107—22no—
423#106—22no—
424#106—24no—
425C4⋊C8—21no—
426C4⋊C8—21no—
427#103—22no—
428#102—22no—
429#102—24no—
430C22⋊C8C23×C5312yes—
431C22⋊C8C23×C5312yes—
432#97—21no—
433C4×C8—21no—
434#95—21no—
435#95C23×C5312yes—
436C24—22no—
437C24C2×D4×C5312yes—
438#41—82no—
439#41—82no—
440C2×Q8—21no—
441C2×Q8—21no—
442C2×Q8—21no—
443C2×Q8—42no—
444C2×Q8—42no—
445C2×Q8—42no—
446C2×Q8—41no—
447C2×Q8—41no—
448C2×Q8C2×D4×C5311yescharacteristic
449C2×D4—22no—
450C2×D4—21no—
451C2×D4—22no—
452C2×D4—22no—
453C2×D4—22no—
454C2×D4—22no—
455C2×D4—22no—
456C2×D4—21no—
457C2×D4—21no—
458C2×D4—21no—
459C2×D4—21no—
460C2×D4—42no—
461C2×D4—82no—
462C2×D4—84no—
463C2×D4C2×D4×C5311yescharacteristic
464C2×D4C2×D4×C5312yes—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—21no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4C2×D4×C5311yescharacteristic
472C22×C4C2×D4×C5311yescharacteristic
473C22×C4C2×D4×C5312yes—
474C22×C4C24×C5311yescharacteristic
475C2×C8—22no—
476C2×C8—22no—
477C4⋊C4—22no—
478C4⋊C4—22no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—48no—
487C4⋊C4—41no—
488C4⋊C4—41no—
489C4⋊C4C2×D4×C5312yes—
490C4⋊C4C2×D4×C5311yescharacteristic
491C4⋊C4C2×D4×C5311yescharacteristic
492C22⋊C4—22no—
493C22⋊C4—22no—
494C22⋊C4—22no—
495C22⋊C4—22no—
496C22⋊C4—22no—
497C22⋊C4—21no—
498C22⋊C4—21no—
499C22⋊C4—22no—
500C22⋊C4—22no—
501C22⋊C4—21no—
502C22⋊C4—21no—
503C22⋊C4—22no—
504C22⋊C4—22no—
505C22⋊C4—22no—
506C22⋊C4—21no—
507C22⋊C4—22no—
508C22⋊C4—48no—
509C42—24no—
510C42—22no—
511C42—21no—
512C42C2×D4×C5311yescharacteristic
513C42C2×D4×C5311yescharacteristic
514C42C2×D4×C5311yescharacteristic
515C42C2×D4×C5311yescharacteristic
516C23—22no—
517C23—21no—
518C23—22no—
519C23—24no—
520C23—42no—
521C23—42no—
522C23—82no—
523C23—82no—
524C23—82no—
525C23C22×D4×C5311yescharacteristic
526Q8—41no—
527Q8—42no—
528Q8—81no—
529Q8—81no—
530Q8—82no—
531Q8—82no—
532D4—42no—
533D4—41no—
534D4—42no—
535D4—44no—
536D4—42no—
537D4—42no—
538D4—81no—
539D4—81no—
540D4—82no—
541D4—82no—
542D4—82no—
543D4—82no—
544D4—82no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—24no—
548C2×C4—21no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—41no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—82no—
562C2×C4—82no—
563C2×C4—82no—
564C2×C4#120×C5311yescharacteristic
565C2×C4#120×C5311yescharacteristic
566C2×C4#120×C5311yescharacteristic
567C2×C4#120×C5311yescharacteristic
568C2×C4C22×D4×C5311yescharacteristic
569C2×C4C22×D4×C5311yescharacteristic
570C8—42no—
571C8—42no—
572C22—21no—
573C22—21no—
574C22—42no—
575C22—42no—
576C22—41no—
577C22—41no—
578C22—42no—
579C22—42no—
580C22—42no—
581C22—41no—
582C22—42no—
583C22—84no—
584C22—164no—
585C22C2×#120×C5311yescharacteristic
586C22C53×#57712yes—
587C4—21no—
588C4—21no—
589C4—21no—
590C4—21no—
591C4—21no—
592C4—44no—
593C4—41no—
594C4—42no—
595C4—41no—
596C4—42no—
597C4—41no—
598C4—82no—
599C4C53×#45312yes—
600C2—22no—
601C2—82no—
602C2—81no—
603C2—82no—
604C2C53×#300911yescharacteristic
605C2C2×C53×#45311yescharacteristic
606C2C53×#302811yescharacteristic
subgroupG/H
center, socleC22×C53C2×#120
commutator, Frattini, cosocleC22×C4C24×C53
prime radicalC53×#3558C2
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×#3239C2semidirect
3. C53×#3227C2semidirect
4. C53×#2658C2semidirect
5. C53×#1618C2semidirect
6. C53×#1613C2semidirect
7. C53×#1505C2semidirect
8. C53×#1500C2semidirect
9. C53×#1469C2semidirect
24. C2×C4⋊Q8×C53C22semidirect
32. C53×#484C22semidirect
34. C53×#481C22semidirect
38. C53×#432C22semidirect
39. C53×#431C22semidirect
41. C53×#399C22semidirect
42. C53×#376C22semidirect
43. C53×#328C22semidirect
45. C53×#327C22semidirect
46. C53×#323C22semidirect
268. #18437C53direct (N × H)
298. #3239C2×C53semidirect
299. #3227C2×C53semidirect
300. #2658C2×C53semidirect
301. #1618C2×C53semidirect
302. #1613C2×C53semidirect
303. #1505C2×C53semidirect
304. #1500C2×C53semidirect
305. #1469C2×C53semidirect
327. C2×C4⋊Q8C22×C53semidirect
335. #484C22×C53semidirect
337. #481C22×C53semidirect
341. #432C22×C53semidirect
342. #431C22×C53semidirect
344. #399C22×C53semidirect
345. #376C22×C53semidirect
346. #328C22×C53semidirect
348. #327C22×C53semidirect
349. #323C22×C53semidirect
350. C53#18437direct (N × H)

Maximal subgroups

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

subgroupindex
C53×#35582
C53×#32392
C53×#32272
C53×#26582
C53×#16182
C53×#16132
C53×#15052
C53×#15002
C53×#14692
#1843753