Finite Group Databasebeta

← C53×#9113 #27216625

Subgroups — C53×#9113 #27216625

Order 13568.

1062 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C22⋊C8×C53C211yesmaximal, characteristic, complemented
2C53×#2847C211yesmaximal, characteristic
3C2×C53×#327C212yesmaximal, complemented
4C53×C22⋊#95C211yesmaximal, characteristic, complemented
5C53×#1294C212yesmaximal, complemented
6C24×C4×C53C2211yescharacteristic
7C23×C8×C53—21no—
8C2×#121×C53C411yescharacteristic, complemented
9C2×#121×C53C411yescharacteristic, complemented
10C2×#121×C53C2211yescharacteristic
11C2×#120×C53—21no—
12C22×C22⋊C4×C53—21no—
13C2×C22⋊C8×C53—21no—
14C2×C22⋊C8×C53—24no—
15C2×C22⋊C8×C53C2212yes—
16C53×#386—21no—
17C53×#386—21no—
18C53×#386—21no—
19C53×#380C411yescharacteristic, complemented
20C53×#380C411yescharacteristic, complemented
21C53×#380C2211yescharacteristic
22C2×#95×C53—21no—
23C2×#95×C53C2212yes—
24C53×#327—24no—
25C25×C53D411yescharacteristic
26C22×D4×C53—21no—
27C22×D4×C53—21no—
28C22×D4×C53—21no—
29C22×D4×C53C2×C411yescharacteristic
30C23×C4×C53—22no—
31C23×C4×C53—22no—
32C23×C4×C53—22no—
33C23×C4×C53—42no—
34C23×C4×C53—42no—
35C23×C4×C53C2×C411yescharacteristic
36C23×C4×C53C2×C411yescharacteristic
37C23×C4×C53D411yescharacteristic
38C23×C4×C53D411yescharacteristic
39C23×C4×C53D411yescharacteristic
40C23×C4×C53C2311yescharacteristic
41C22×C8×C53—24no—
42C22×C8×C53—22no—
43C22×C8×C53—42no—
44C22×C8×C53—42no—
45#121×C53—22no—
46#121×C53—22no—
47#121×C53—42no—
48#121×C53—42no—
49#120×C53—42no—
50#120×C53—42no—
51C2×C4⋊C4×C53C2×C411yescharacteristic
52C2×C4⋊C4×C53C2×C411yescharacteristic
53C2×C4⋊C4×C53C2×C411yescharacteristic
54C2×C22⋊C4×C53—24no—
55C2×C22⋊C4×C53—21no—
56C2×C22⋊C4×C53—21no—
57C2×C22⋊C4×C53—21no—
58C2×C22⋊C4×C53—21no—
59C2×C22⋊C4×C53—21no—
60C2×C22⋊C4×C53—21no—
61C2×C22⋊C4×C53—42no—
62C2×C22⋊C4×C53—42no—
63C22⋊C8×C53—24no—
64C22⋊C8×C53—22no—
65C22⋊C8×C53D414yescomplemented
66#95×C53—21no—
67#95×C53—21no—
68#95×C53—21no—
69#95×C53—22no—
70#95×C53—22no—
71#95×C53—42no—
72C24×C53—22no—
73C24×C53—22no—
74C24×C53—22no—
75C24×C53—22no—
76C24×C53—21no—
77C24×C53—41no—
78C24×C53—41no—
79C24×C53C22⋊C411yescharacteristic
80C24×C53C22⋊C411yescharacteristic
81C24×C53D812yes—
82C24×C53SD1612yes—
83C24×C53C2×D411yescharacteristic
84C2×D4×C53—22no—
85C2×D4×C53—41no—
86C2×D4×C53—41no—
87C2×D4×C53—42no—
88C2×D4×C53—42no—
89C2×D4×C53—42no—
90C2×D4×C53—42no—
91C2×D4×C53—42no—
92C2×D4×C53—42no—
93C2×D4×C53—42no—
94C2×D4×C53—41no—
95C2×D4×C53—41no—
96C2×D4×C53—42no—
97C22×C4×C53—22no—
98C22×C4×C53—22no—
99C22×C4×C53—21no—
100C22×C4×C53—21no—
101C22×C4×C53—22no—
102C22×C4×C53—22no—
103C22×C4×C53—22no—
104C22×C4×C53—22no—
105C22×C4×C53—22no—
106C22×C4×C53—22no—
107C22×C4×C53—21no—
108C22×C4×C53—21no—
109C22×C4×C53—24no—
110C22×C4×C53—21no—
111C22×C4×C53—21no—
112C22×C4×C53—21no—
113C22×C4×C53—41no—
114C22×C4×C53—41no—
115C22×C4×C53—41no—
116C22×C4×C53—41no—
117C22×C4×C53—41no—
118C22×C4×C53—41no—
119C22×C4×C53—41no—
120C22×C4×C53—48no—
121C22×C4×C53—48no—
122C22×C4×C53—42no—
123C22×C4×C53—42no—
124C22×C4×C53—42no—
125C22×C4×C53—42no—
126C22×C4×C53—42no—
127C22×C4×C53—42no—
128C22×C4×C53C22⋊C411yescharacteristic
129C22×C4×C53C22⋊C411yescharacteristic
130C22×C4×C53C22×C411yescharacteristic
131C22×C4×C53C2×D411yescharacteristic
132C22×C4×C53C2×D412yes—
133C22×C4×C53#4112yes—
134C2×C8×C53—24no—
135C2×C8×C53—48no—
136C2×C8×C53—42no—
137C2×C8×C53—42no—
138C4⋊C4×C53—22no—
139C4⋊C4×C53—41no—
140C4⋊C4×C53—41no—
141C22⋊C4×C53—42no—
142C22⋊C4×C53—48no—
143C22⋊C4×C53—42no—
144C22⋊C4×C53—42no—
145C22⋊C4×C53—41no—
146C22⋊C4×C53—41no—
147C22⋊C4×C53—42no—
148C22⋊C4×C53—41no—
149C22⋊C4×C53—41no—
150C23×C53—21no—
151C23×C53—21no—
152C23×C53—21no—
153C23×C53—21no—
154C23×C53—21no—
155C23×C53—21no—
156C23×C53—21no—
157C23×C53—21no—
158C23×C53—24no—
159C23×C53—24no—
160C23×C53—21no—
161C23×C53—21no—
162C23×C53—21no—
163C23×C53—21no—
164C23×C53—21no—
165C23×C53—21no—
166C23×C53—24no—
167C23×C53—24no—
168C23×C53—42no—
169C23×C53—42no—
170C23×C53—44no—
171C23×C53—44no—
172C23×C53—44no—
173C23×C53—44no—
174C23×C53—41no—
175C23×C53—41no—
176C23×C53—41no—
177C23×C53—41no—
178C23×C53—41no—
179C23×C53—41no—
180C23×C53—41no—
181C23×C53#9912yes—
182C23×C53#10212yes—
183C23×C53#10212yes—
184C23×C53#10412yes—
185C23×C53C2×C22⋊C411yescharacteristic
186C23×C53#12112yes—
187C23×C53C2×D811yescharacteristic
188C23×C53C2×SD1611yescharacteristic
189C23×C53#13512yes—
190D4×C53—82no—
191D4×C53—84no—
192D4×C53—84no—
193D4×C53—82no—
194D4×C53—82no—
195C2×C4×C53—24no—
196C2×C4×C53—24no—
197C2×C4×C53—21no—
198C2×C4×C53—21no—
199C2×C4×C53—21no—
200C2×C4×C53—21no—
201C2×C4×C53—21no—
202C2×C4×C53—21no—
203C2×C4×C53—21no—
204C2×C4×C53—21no—
205C2×C4×C53—21no—
206C2×C4×C53—42no—
207C2×C4×C53—42no—
208C2×C4×C53—44no—
209C2×C4×C53—44no—
210C2×C4×C53—44no—
211C2×C4×C53—42no—
212C2×C4×C53—42no—
213C2×C4×C53—44no—
214C2×C4×C53—44no—
215C2×C4×C53—44no—
216C2×C4×C53—41no—
217C2×C4×C53—41no—
218C2×C4×C53—41no—
219C2×C4×C53—41no—
220C2×C4×C53—41no—
221C2×C4×C53—41no—
222C2×C4×C53—41no—
223C2×C4×C53—41no—
224C2×C4×C53—41no—
225C2×C4×C53—82no—
226C2×C4×C53—82no—
227C2×C4×C53—82no—
228C2×C4×C53—82no—
229C2×C4×C53#9912yes—
230C2×C4×C53C4×D412yes—
231C2×C4×C53#12011yescharacteristic
232C2×C4×C53#12311yescharacteristic
233C8×C53—44no—
234#9113C5311yesmaximal, characteristic, direct product factor, Sylow
235C22×C53—22no—
236C22×C53—22no—
237C22×C53—22no—
238C22×C53—22no—
239C22×C53—22no—
240C22×C53—21no—
241C22×C53—21no—
242C22×C53—22no—
243C22×C53—22no—
244C22×C53—22no—
245C22×C53—22no—
246C22×C53—21no—
247C22×C53—22no—
248C22×C53—21no—
249C22×C53—22no—
250C22×C53—22no—
251C22×C53—22no—
252C22×C53—48no—
253C22×C53—48no—
254C22×C53—41no—
255C22×C53—41no—
256C22×C53—41no—
257C22×C53—41no—
258C22×C53—81no—
259C22×C53—81no—
260C22×C53—81no—
261C22×C53—81no—
262C22×C53—81no—
263C22×C53—81no—
264C22×C53—81no—
265C22×C53#32714yescomplemented
266C22×C53#38611yescharacteristic
267C22×C53C2×#9911yescharacteristic
268C22×C53C2×#9911yescharacteristic
269C22×C53C2×#10211yescharacteristic
270C22×C53#41611yescharacteristic
271C22×C53C2×#10411yescharacteristic
272C22×C53#42111yescharacteristic
273C22×C53#46512yes—
274C22×C53#46612yes—
275C4×C53—21no—
276C4×C53—22no—
277C4×C53—21no—
278C4×C53—22no—
279C4×C53—41no—
280C4×C53—84no—
281C4×C53—81no—
282C4×C53—81no—
283C4×C53—81no—
284C22×C22⋊C8C2×C5311yescharacteristic, complemented
285#2847C2×C5311yescharacteristic
286C2×#327C2×C5312yescomplemented
287C22⋊#95C2×C5311yescharacteristic, complemented
288#1294C2×C5312yescomplemented
289C2×C53—22no—
290C2×C53—24no—
291C2×C53—22no—
292C2×C53—42no—
293C2×C53—161no—
294C2×C53C2×#32711yescharacteristic
295C2×C53#150111yescharacteristic
296C2×C53#151611yescharacteristic
297C2×C53#151911yescharacteristic
298C2×C53#187111yescharacteristic
299C2×C53#189311yescharacteristic
300C2×C53#189811yescharacteristic
301C24×C4C22×C5311yescharacteristic
302C23×C8—21no—
303C2×#121C4×C5311yescharacteristic, complemented
304C2×#121C4×C5311yescharacteristic, complemented
305C2×#121C22×C5311yescharacteristic
306C2×#120—21no—
307C22×C22⋊C4—21no—
308C2×C22⋊C8—21no—
309C2×C22⋊C8—24no—
310C2×C22⋊C8C22×C5312yes—
311#386—21no—
312#386—21no—
313#386—21no—
314#380C4×C5311yescharacteristic, complemented
315#380C4×C5311yescharacteristic, complemented
316#380C22×C5311yescharacteristic
317C2×#95—21no—
318C2×#95C22×C5312yes—
319#327—24no—
320C53#911311yescharacteristic, direct product factor, Sylow
321C25D4×C5311yescharacteristic
322C22×D4—21no—
323C22×D4—21no—
324C22×D4—21no—
325C22×D4C2×C4×C5311yescharacteristic
326C23×C4—22no—
327C23×C4—22no—
328C23×C4—22no—
329C23×C4—42no—
330C23×C4—42no—
331C23×C4C2×C4×C5311yescharacteristic
332C23×C4C2×C4×C5311yescharacteristic
333C23×C4D4×C5311yescharacteristic
334C23×C4D4×C5311yescharacteristic
335C23×C4D4×C5311yescharacteristic
336C23×C4C23×C5311yescharacteristic
337C22×C8—24no—
338C22×C8—22no—
339C22×C8—42no—
340C22×C8—42no—
341#121—22no—
342#121—22no—
343#121—42no—
344#121—42no—
345#120—42no—
346#120—42no—
347C2×C4⋊C4C2×C4×C5311yescharacteristic
348C2×C4⋊C4C2×C4×C5311yescharacteristic
349C2×C4⋊C4C2×C4×C5311yescharacteristic
350C2×C22⋊C4—24no—
351C2×C22⋊C4—21no—
352C2×C22⋊C4—21no—
353C2×C22⋊C4—21no—
354C2×C22⋊C4—21no—
355C2×C22⋊C4—21no—
356C2×C22⋊C4—21no—
357C2×C22⋊C4—42no—
358C2×C22⋊C4—42no—
359C22⋊C8—24no—
360C22⋊C8—22no—
361C22⋊C8D4×C5314yescomplemented
362#95—21no—
363#95—21no—
364#95—21no—
365#95—22no—
366#95—22no—
367#95—42no—
368C24—22no—
369C24—22no—
370C24—22no—
371C24—22no—
372C24—21no—
373C24—41no—
374C24—41no—
375C24C22⋊C4×C5311yescharacteristic
376C24C22⋊C4×C5311yescharacteristic
377C24D8×C5312yes—
378C24SD16×C5312yes—
379C24C2×D4×C5311yescharacteristic
380C2×D4—22no—
381C2×D4—41no—
382C2×D4—41no—
383C2×D4—42no—
384C2×D4—42no—
385C2×D4—42no—
386C2×D4—42no—
387C2×D4—42no—
388C2×D4—42no—
389C2×D4—42no—
390C2×D4—41no—
391C2×D4—41no—
392C2×D4—42no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—24no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—41no—
410C22×C4—41no—
411C22×C4—41no—
412C22×C4—41no—
413C22×C4—41no—
414C22×C4—41no—
415C22×C4—41no—
416C22×C4—48no—
417C22×C4—48no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4C22⋊C4×C5311yescharacteristic
425C22×C4C22⋊C4×C5311yescharacteristic
426C22×C4C22×C4×C5311yescharacteristic
427C22×C4C2×D4×C5311yescharacteristic
428C22×C4C2×D4×C5312yes—
429C22×C4#41×C5312yes—
430C2×C8—24no—
431C2×C8—48no—
432C2×C8—42no—
433C2×C8—42no—
434C4⋊C4—22no—
435C4⋊C4—41no—
436C4⋊C4—41no—
437C22⋊C4—42no—
438C22⋊C4—48no—
439C22⋊C4—42no—
440C22⋊C4—42no—
441C22⋊C4—41no—
442C22⋊C4—41no—
443C22⋊C4—42no—
444C22⋊C4—41no—
445C22⋊C4—41no—
446C23—21no—
447C23—21no—
448C23—21no—
449C23—21no—
450C23—21no—
451C23—21no—
452C23—21no—
453C23—21no—
454C23—24no—
455C23—24no—
456C23—21no—
457C23—21no—
458C23—21no—
459C23—21no—
460C23—21no—
461C23—21no—
462C23—24no—
463C23—24no—
464C23—42no—
465C23—42no—
466C23—44no—
467C23—44no—
468C23—44no—
469C23—44no—
470C23—41no—
471C23—41no—
472C23—41no—
473C23—41no—
474C23—41no—
475C23—41no—
476C23—41no—
477C23#99×C5312yes—
478C23#102×C5312yes—
479C23#102×C5312yes—
480C23#104×C5312yes—
481C23C2×C22⋊C4×C5311yescharacteristic
482C23#121×C5312yes—
483C23C2×D8×C5311yescharacteristic
484C23C2×SD16×C5311yescharacteristic
485C23#135×C5312yes—
486D4—82no—
487D4—84no—
488D4—84no—
489D4—82no—
490D4—82no—
491C2×C4—24no—
492C2×C4—24no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—44no—
505C2×C4—44no—
506C2×C4—44no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—44no—
510C2×C4—44no—
511C2×C4—44no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—82no—
522C2×C4—82no—
523C2×C4—82no—
524C2×C4—82no—
525C2×C4#99×C5312yes—
526C2×C4C4×D4×C5312yes—
527C2×C4#120×C5311yescharacteristic
528C2×C4#123×C5311yescharacteristic
529C8—44no—
530C22—21no—
531C22—21no—
532C22—22no—
533C22—22no—
534C22—22no—
535C22—22no—
536C22—21no—
537C22—22no—
538C22—21no—
539C22—22no—
540C22—22no—
541C22—22no—
542C22—22no—
543C22—22no—
544C22—22no—
545C22—22no—
546C22—22no—
547C22—48no—
548C22—48no—
549C22—41no—
550C22—41no—
551C22—41no—
552C22—41no—
553C22—81no—
554C22—81no—
555C22—81no—
556C22—81no—
557C22—81no—
558C22—81no—
559C22—81no—
560C22C53×#32714yescomplemented
561C22C53×#38611yescharacteristic
562C22C2×#99×C5311yescharacteristic
563C22C2×#99×C5311yescharacteristic
564C22C2×#102×C5311yescharacteristic
565C22C53×#41611yescharacteristic
566C22C2×#104×C5311yescharacteristic
567C22C53×#42111yescharacteristic
568C22C53×#46512yes—
569C22C53×#46612yes—
570C4—21no—
571C4—22no—
572C4—21no—
573C4—22no—
574C4—41no—
575C4—84no—
576C4—81no—
577C4—81no—
578C4—81no—
579C2—22no—
580C2—24no—
581C2—22no—
582C2—42no—
583C2—161no—
584C2C2×C53×#32711yescharacteristic
585C2C53×#150111yescharacteristic
586C2C53×#151611yescharacteristic
587C2C53×#151911yescharacteristic
588C2C53×#187111yescharacteristic
589C2C53×#189311yescharacteristic
590C2C53×#189811yescharacteristic
subgroupG/H
center, socleC23×C53C2×C22⋊C4
commutatorC22×C4C22×C4×C53
Frattini, cosocleC23×C4C23×C53
prime radicalC53×#2847C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23×C53 ◁ C24×C4×C53 ◁ G

Complements

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

NH ≅ G/Ntype
1. C22×C22⋊C8×C53C2semidirect
3. C2×C53×#327C2semidirect
4. C53×C22⋊#95C2semidirect
5. C53×#1294C2semidirect
8. C2×#121×C53C4semidirect
9. C2×#121×C53C4semidirect
19. C53×#380C4semidirect
20. C53×#380C4semidirect
65. C22⋊C8×C53D4semidirect
234. #9113C53direct (N × H)
265. C22×C53#327semidirect
284. C22×C22⋊C8C2×C53semidirect
286. C2×#327C2×C53semidirect
287. C22⋊#95C2×C53semidirect
288. #1294C2×C53semidirect
303. C2×#121C4×C53semidirect
304. C2×#121C4×C53semidirect
314. #380C4×C53semidirect
315. #380C4×C53semidirect
320. C53#9113direct (N × H)
361. C22⋊C8D4×C53semidirect
560. C22C53×#327semidirect

Maximal subgroups

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

subgroupindex
C22×C22⋊C8×C532
C53×#28472
C2×C53×#3272
C53×C22⋊#952
C53×#12942
#911353