Finite Group Databasebeta

← C53×#18289 #27225801

Subgroups — C53×#18289 #27225801

Order 13568.

972 conjugacy classes of subgroups.

The subgroup lattice has 574 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C53×#539C211yesmaximal, characteristic
2C53×#3189C211yesmaximal, characteristic, complemented
3C53×#3187C211yesmaximal, characteristic
4C53×#2627C211yesmaximal, characteristic, complemented
5C53×#1615C214yesmaximal, complemented
6C53×#1603C214yesmaximal, complemented
7C53×#1520C211yesmaximal, characteristic, complemented
8C53×#1519C211yesmaximal, characteristic
9C2×C53×#324C211yesmaximal, characteristic, complemented
10C53×#539—22no—
11C53×#539—22no—
12C53×#539—22no—
13C53×#539C2214yes—
14C2×#126×C53—21no—
15C2×#125×C53—21no—
16C2×#123×C53—21no—
17C2×#123×C53—21no—
18C2×C22⋊Q8×C53—21no—
19C2×C22⋊Q8×C53C2211yescharacteristic
20C2×C22⋊Q8×C53C2211yescharacteristic
21C2×#121×C53C2211yescharacteristic
22C2×C4×D4×C53—21no—
23C22×C4⋊C4×C53C2211yescharacteristic
24C53×#484—21no—
25C53×#484C2212yescomplemented
26C53×#483—21no—
27C53×#481C2212yes—
28C53×#467—22no—
29C53×#465—22no—
30C53×#451—21no—
31C53×#451C2212yescomplemented
32C53×#450—21no—
33C53×#448C2212yes—
34C2×C22⋊C8×C53C2211yescharacteristic
35C2×C22⋊C8×C53C2211yescharacteristic
36C53×#400C2212yescomplemented
37C53×#398—22no—
38C53×#395C2212yescomplemented
39C53×#385—21no—
40C53×#381—21no—
41C53×#380C2211yescharacteristic
42C53×#328C2214yescomplemented
43C53×#327C2214yes—
44C53×#324C2214yescomplemented
45C22×Q8×C53D411yescharacteristic
46C22×Q8×C53C2311yescharacteristic
47C22×D4×C53—21no—
48C22×D4×C53D411yescharacteristic
49C23×C4×C53—21no—
50C23×C4×C53D411yescharacteristic
51C23×C4×C53D411yescharacteristic
52C23×C4×C53C2311yescharacteristic
53C2×Q16×C53—22no—
54C2×SD16×C53—22no—
55C22×C8×C53—21no—
56C22×C8×C53—21no—
57#126×C53—24no—
58#126×C53—24no—
59#125×C53—24no—
60#125×C53—24no—
61#123×C53—24no—
62#123×C53—24no—
63#123×C53—24no—
64#123×C53—24no—
65C22⋊Q8×C53—24no—
66C22⋊Q8×C53—24no—
67C22⋊Q8×C53—21no—
68C22⋊Q8×C53—21no—
69C22⋊Q8×C53—21no—
70C22⋊Q8×C53—21no—
71C22⋊Q8×C53—21no—
72C22⋊Q8×C53—21no—
73C22⋊Q8×C53D412yes—
74C22⋊Q8×C53C2312yes—
75#121×C53—21no—
76#121×C53—21no—
77#121×C53D412yes—
78#121×C53C2312yes—
79C4×D4×C53—24no—
80C4×D4×C53—24no—
81C2×C4⋊C4×C53—21no—
82C2×C4⋊C4×C53—21no—
83C2×C4⋊C4×C53—21no—
84C2×C4⋊C4×C53—21no—
85C2×C4⋊C4×C53—21no—
86C2×C4⋊C4×C53—21no—
87C2×C4⋊C4×C53—21no—
88C2×C4⋊C4×C53—21no—
89C2×C4⋊C4×C53—21no—
90C2×C4⋊C4×C53—21no—
91C2×C4⋊C4×C53—22no—
92C2×C4⋊C4×C53D411yescharacteristic
93C2×C4⋊C4×C53D411yescharacteristic
94C2×C4⋊C4×C53D412yes—
95C2×C4⋊C4×C53C2311yescharacteristic
96C2×C4⋊C4×C53C2312yes—
97C2×C22⋊C4×C53—21no—
98C2×C22⋊C4×C53—21no—
99C2×C22⋊C4×C53—21no—
100C2×C22⋊C4×C53—21no—
101C2×C22⋊C4×C53—21no—
102C2×C22⋊C4×C53—21no—
103C2×C22⋊C4×C53—21no—
104C2×C42×C53—21no—
105#107×C53—22no—
106#106×C53—22no—
107#103×C53—22no—
108#103×C53—22no—
109#103×C53—22no—
110#102×C53—22no—
111C22⋊C8×C53—21no—
112C22⋊C8×C53—21no—
113C22⋊C8×C53C2312yes—
114C22⋊C8×C53C2312yes—
115#95×C53—22no—
116#95×C53—22no—
117#95×C53—22no—
118#95×C53C2312yes—
119C24×C53—21no—
120C24×C53C2×D411yescharacteristic
121C2×Q8×C53—21no—
122C2×Q8×C53—21no—
123C2×Q8×C53—21no—
124C2×Q8×C53—81no—
125C2×Q8×C53—81no—
126C2×Q8×C53C2×D412yes—
127C2×D4×C53—22no—
128C2×D4×C53—22no—
129C2×D4×C53—21no—
130C2×D4×C53—42no—
131C2×D4×C53—42no—
132C2×D4×C53—81no—
133C2×D4×C53C2×D412yes—
134C22×C4×C53—21no—
135C22×C4×C53—21no—
136C22×C4×C53—21no—
137C22×C4×C53—21no—
138C22×C4×C53—21no—
139C22×C4×C53—21no—
140C22×C4×C53—22no—
141C22×C4×C53—22no—
142C22×C4×C53—24no—
143C22×C4×C53—21no—
144C22×C4×C53—21no—
145C22×C4×C53—22no—
146C22×C4×C53—21no—
147C22×C4×C53—81no—
148C22×C4×C53—81no—
149C22×C4×C53—82no—
150C22×C4×C53C2×D412yes—
151C22×C4×C53C2×D412yes—
152C22×C4×C53C2×D411yescharacteristic
153C22×C4×C53C2×D411yescharacteristic
154C22×C4×C53C2×D411yescharacteristic
155C22×C4×C53C2×D411yescharacteristic
156C22×C4×C53C2×D411yescharacteristic
157C22×C4×C53C2411yescharacteristic
158Q16×C53—82no—
159SD16×C53—82no—
160C2×C8×C53—22no—
161C2×C8×C53—22no—
162C2×C8×C53—81no—
163C2×C8×C53—81no—
164C4⋊C4×C53—24no—
165C4⋊C4×C53—24no—
166C4⋊C4×C53—24no—
167C4⋊C4×C53—22no—
168C4⋊C4×C53—22no—
169C4⋊C4×C53—22no—
170C4⋊C4×C53—21no—
171C4⋊C4×C53—21no—
172C4⋊C4×C53—22no—
173C4⋊C4×C53—22no—
174C4⋊C4×C53—22no—
175C4⋊C4×C53—21no—
176C4⋊C4×C53—21no—
177C4⋊C4×C53—41no—
178C4⋊C4×C53—41no—
179C4⋊C4×C53—81no—
180C4⋊C4×C53—82no—
181C4⋊C4×C53C2×D412yes—
182C4⋊C4×C53C2×D412yes—
183C22⋊C4×C53—24no—
184C22⋊C4×C53—24no—
185C22⋊C4×C53—24no—
186C22⋊C4×C53—22no—
187C22⋊C4×C53—22no—
188C22⋊C4×C53—22no—
189C22⋊C4×C53—22no—
190C22⋊C4×C53—41no—
191C22⋊C4×C53—41no—
192C22⋊C4×C53—42no—
193C42×C53—24no—
194C23×C53—22no—
195C23×C53—41no—
196C23×C53—41no—
197C23×C53—42no—
198C23×C53—42no—
199C23×C53—42no—
200C23×C53#12011yescharacteristic
201C23×C53#12011yescharacteristic
202C23×C53#13512yes—
203C23×C53#13712yes—
204C23×C53C22×D411yescharacteristic
205Q8×C53—42no—
206Q8×C53—42no—
207Q8×C53—42no—
208Q8×C53—81no—
209D4×C53—42no—
210D4×C53—42no—
211D4×C53—42no—
212D4×C53—44no—
213D4×C53—81no—
214C2×C4×C53—22no—
215C2×C4×C53—22no—
216C2×C4×C53—22no—
217C2×C4×C53—22no—
218C2×C4×C53—22no—
219C2×C4×C53—22no—
220C2×C4×C53—22no—
221C2×C4×C53—41no—
222C2×C4×C53—41no—
223C2×C4×C53—41no—
224C2×C4×C53—41no—
225C2×C4×C53—41no—
226C2×C4×C53—41no—
227C2×C4×C53—42no—
228C2×C4×C53—42no—
229C2×C4×C53—42no—
230C2×C4×C53—42no—
231C2×C4×C53—42no—
232C2×C4×C53—42no—
233C2×C4×C53—42no—
234C2×C4×C53—42no—
235C2×C4×C53—42no—
236C2×C4×C53—42no—
237C2×C4×C53—42no—
238C2×C4×C53—41no—
239C2×C4×C53—41no—
240C2×C4×C53—42no—
241C2×C4×C53—81no—
242C2×C4×C53—84no—
243C2×C4×C53#12011yescharacteristic
244C2×C4×C53#12011yescharacteristic
245C2×C4×C53C22×D411yescharacteristic
246C2×C4×C53C22×D411yescharacteristic
247C8×C53—42no—
248C8×C53—42no—
249#18289C5311yesmaximal, characteristic, direct product factor, Sylow
250C22×C53—21no—
251C22×C53—21no—
252C22×C53—21no—
253C22×C53—21no—
254C22×C53—21no—
255C22×C53—21no—
256C22×C53—42no—
257C22×C53—42no—
258C22×C53—42no—
259C22×C53—42no—
260C22×C53—42no—
261C22×C53—84no—
262C22×C53#45212yes—
263C22×C53#45212yes—
264C22×C53C2×#12011yescharacteristic
265C22×C53C2×#13511yescharacteristic
266C22×C53C2×#13711yescharacteristic
267C4×C53—21no—
268C4×C53—21no—
269C4×C53—21no—
270C4×C53—21no—
271C4×C53—42no—
272C4×C53—42no—
273C4×C53—42no—
274C4×C53—42no—
275C4×C53—42no—
276C4×C53—82no—
277C4×C53—81no—
278C4×C53—81no—
279C2×#539C2×C5311yescharacteristic
280#3189C2×C5311yescharacteristic, complemented
281#3187C2×C5311yescharacteristic
282#2627C2×C5311yescharacteristic, complemented
283#1615C2×C5314yescomplemented
284#1603C2×C5314yescomplemented
285#1520C2×C5311yescharacteristic, complemented
286#1519C2×C5311yescharacteristic
287C2×#324C2×C5311yescharacteristic, complemented
288C2×C53—21no—
289C2×C53—22no—
290C2×C53—21no—
291C2×C53—22no—
292C2×C53—82no—
293C2×C53C2×#45211yescharacteristic
294C2×C53#302011yescharacteristic
295C2×C53#302611yescharacteristic
296#539—22no—
297#539—22no—
298#539—22no—
299#539C22×C5314yes—
300C2×#126—21no—
301C2×#125—21no—
302C2×#123—21no—
303C2×#123—21no—
304C2×C22⋊Q8—21no—
305C2×C22⋊Q8C22×C5311yescharacteristic
306C2×C22⋊Q8C22×C5311yescharacteristic
307C2×#121C22×C5311yescharacteristic
308C2×C4×D4—21no—
309C22×C4⋊C4C22×C5311yescharacteristic
310#484—21no—
311#484C22×C5312yescomplemented
312#483—21no—
313#481C22×C5312yes—
314#467—22no—
315#465—22no—
316#451—21no—
317#451C22×C5312yescomplemented
318#450—21no—
319#448C22×C5312yes—
320C2×C22⋊C8C22×C5311yescharacteristic
321C2×C22⋊C8C22×C5311yescharacteristic
322#400C22×C5312yescomplemented
323#398—22no—
324#395C22×C5312yescomplemented
325#385—21no—
326#381—21no—
327#380C22×C5311yescharacteristic
328#328C22×C5314yescomplemented
329#327C22×C5314yes—
330#324C22×C5314yescomplemented
331C53#1828911yescharacteristic, direct product factor, Sylow
332C22×Q8D4×C5311yescharacteristic
333C22×Q8C23×C5311yescharacteristic
334C22×D4—21no—
335C22×D4D4×C5311yescharacteristic
336C23×C4—21no—
337C23×C4D4×C5311yescharacteristic
338C23×C4D4×C5311yescharacteristic
339C23×C4C23×C5311yescharacteristic
340C2×Q16—22no—
341C2×SD16—22no—
342C22×C8—21no—
343C22×C8—21no—
344#126—24no—
345#126—24no—
346#125—24no—
347#125—24no—
348#123—24no—
349#123—24no—
350#123—24no—
351#123—24no—
352C22⋊Q8—24no—
353C22⋊Q8—24no—
354C22⋊Q8—21no—
355C22⋊Q8—21no—
356C22⋊Q8—21no—
357C22⋊Q8—21no—
358C22⋊Q8—21no—
359C22⋊Q8—21no—
360C22⋊Q8D4×C5312yes—
361C22⋊Q8C23×C5312yes—
362#121—21no—
363#121—21no—
364#121D4×C5312yes—
365#121C23×C5312yes—
366C4×D4—24no—
367C4×D4—24no—
368C2×C4⋊C4—21no—
369C2×C4⋊C4—21no—
370C2×C4⋊C4—21no—
371C2×C4⋊C4—21no—
372C2×C4⋊C4—21no—
373C2×C4⋊C4—21no—
374C2×C4⋊C4—21no—
375C2×C4⋊C4—21no—
376C2×C4⋊C4—21no—
377C2×C4⋊C4—21no—
378C2×C4⋊C4—22no—
379C2×C4⋊C4D4×C5312yes—
380C2×C4⋊C4D4×C5311yescharacteristic
381C2×C4⋊C4D4×C5311yescharacteristic
382C2×C4⋊C4C23×C5311yescharacteristic
383C2×C4⋊C4C23×C5312yes—
384C2×C22⋊C4—21no—
385C2×C22⋊C4—21no—
386C2×C22⋊C4—21no—
387C2×C22⋊C4—21no—
388C2×C22⋊C4—21no—
389C2×C22⋊C4—21no—
390C2×C22⋊C4—21no—
391C2×C42—21no—
392#107—22no—
393#106—22no—
394#103—22no—
395#103—22no—
396#103—22no—
397#102—22no—
398C22⋊C8—21no—
399C22⋊C8—21no—
400C22⋊C8C23×C5312yes—
401C22⋊C8C23×C5312yes—
402#95—22no—
403#95—22no—
404#95—22no—
405#95C23×C5312yes—
406C24—21no—
407C24C2×D4×C5311yescharacteristic
408C2×Q8—21no—
409C2×Q8—21no—
410C2×Q8—21no—
411C2×Q8—81no—
412C2×Q8—81no—
413C2×Q8C2×D4×C5312yes—
414C2×D4—22no—
415C2×D4—22no—
416C2×D4—21no—
417C2×D4—42no—
418C2×D4—42no—
419C2×D4—81no—
420C2×D4C2×D4×C5312yes—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—24no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—22no—
433C22×C4—21no—
434C22×C4—81no—
435C22×C4—81no—
436C22×C4—82no—
437C22×C4C2×D4×C5312yes—
438C22×C4C2×D4×C5312yes—
439C22×C4C2×D4×C5311yescharacteristic
440C22×C4C2×D4×C5311yescharacteristic
441C22×C4C2×D4×C5311yescharacteristic
442C22×C4C2×D4×C5311yescharacteristic
443C22×C4C2×D4×C5311yescharacteristic
444C22×C4C24×C5311yescharacteristic
445Q16—82no—
446SD16—82no—
447C2×C8—22no—
448C2×C8—22no—
449C2×C8—81no—
450C2×C8—81no—
451C4⋊C4—24no—
452C4⋊C4—24no—
453C4⋊C4—24no—
454C4⋊C4—22no—
455C4⋊C4—22no—
456C4⋊C4—22no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—22no—
460C4⋊C4—22no—
461C4⋊C4—22no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—41no—
465C4⋊C4—41no—
466C4⋊C4—81no—
467C4⋊C4—82no—
468C4⋊C4C2×D4×C5312yes—
469C4⋊C4C2×D4×C5312yes—
470C22⋊C4—24no—
471C22⋊C4—24no—
472C22⋊C4—24no—
473C22⋊C4—22no—
474C22⋊C4—22no—
475C22⋊C4—22no—
476C22⋊C4—22no—
477C22⋊C4—41no—
478C22⋊C4—41no—
479C22⋊C4—42no—
480C42—24no—
481C23—22no—
482C23—41no—
483C23—41no—
484C23—42no—
485C23—42no—
486C23—42no—
487C23#120×C5311yescharacteristic
488C23#120×C5311yescharacteristic
489C23#135×C5312yes—
490C23#137×C5312yes—
491C23C22×D4×C5311yescharacteristic
492Q8—42no—
493Q8—42no—
494Q8—42no—
495Q8—81no—
496D4—42no—
497D4—42no—
498D4—42no—
499D4—44no—
500D4—81no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—42no—
528C2×C4—81no—
529C2×C4—84no—
530C2×C4#120×C5311yescharacteristic
531C2×C4#120×C5311yescharacteristic
532C2×C4C22×D4×C5311yescharacteristic
533C2×C4C22×D4×C5311yescharacteristic
534C8—42no—
535C8—42no—
536C22—21no—
537C22—21no—
538C22—21no—
539C22—21no—
540C22—21no—
541C22—21no—
542C22—42no—
543C22—42no—
544C22—42no—
545C22—42no—
546C22—42no—
547C22—84no—
548C22C53×#45212yes—
549C22C53×#45212yes—
550C22C2×#120×C5311yescharacteristic
551C22C2×#135×C5311yescharacteristic
552C22C2×#137×C5311yescharacteristic
553C4—21no—
554C4—21no—
555C4—21no—
556C4—21no—
557C4—42no—
558C4—42no—
559C4—42no—
560C4—42no—
561C4—42no—
562C4—82no—
563C4—81no—
564C4—81no—
565C2—21no—
566C2—22no—
567C2—21no—
568C2—22no—
569C2—82no—
570C2C2×C53×#45211yescharacteristic
571C2C53×#302011yescharacteristic
572C2C53×#302611yescharacteristic
subgroupG/H
center, socleC22×C53C2×#120
commutator, Frattini, cosocleC22×C4C24×C53
prime radicalC2×#121×C53C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C53 ◁ C23×C4×C53 ◁ G

Complements

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

NH ≅ G/Ntype
2. C53×#3189C2semidirect
4. C53×#2627C2semidirect
5. C53×#1615C2semidirect
6. C53×#1603C2semidirect
7. C53×#1520C2semidirect
9. C2×C53×#324C2semidirect
25. C53×#484C22semidirect
31. C53×#451C22semidirect
36. C53×#400C22semidirect
38. C53×#395C22semidirect
42. C53×#328C22semidirect
44. C53×#324C22semidirect
249. #18289C53direct (N × H)
280. #3189C2×C53semidirect
282. #2627C2×C53semidirect
283. #1615C2×C53semidirect
284. #1603C2×C53semidirect
285. #1520C2×C53semidirect
287. C2×#324C2×C53semidirect
311. #484C22×C53semidirect
317. #451C22×C53semidirect
322. #400C22×C53semidirect
324. #395C22×C53semidirect
328. #328C22×C53semidirect
330. #324C22×C53semidirect
331. C53#18289direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C53×#5392
C53×#31892
C53×#31872
C53×#26272
C53×#16152
C53×#16032
C53×#15202
C53×#15192
C2×C53×#3242
#1828953