Finite Group Databasebeta

← C53×#8482 #27215994

Subgroups — C53×#8482 #27215994

Order 13568.

768 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
1C53×#3538C211yesmaximal, characteristic
2C53×#1996C211yesmaximal, characteristic, complemented
3C53×#1901C211yesmaximal, characteristic, complemented
4C53×#1894C211yesmaximal, characteristic, complemented
5C53×#1871C211yesmaximal, characteristic
6C53×#1513C211yesmaximal, characteristic
7C53×#1508C211yesmaximal, characteristic, complemented
8C22×M16×C53C2211yescharacteristic
9C53×#563—22no—
10C53×#552—21no—
11C53×#552—21no—
12C53×#551—22no—
13D4×Q8×C53—21no—
14D4×Q8×C53—21no—
15C53×#544—21no—
16C53×#544C412yescomplemented
17C53×#541—21no—
18C53×#539—21no—
19C53×#539C412yescomplemented
20C53×#536—21no—
21C53×#525C2211yescharacteristic
22C2×C22⋊Q8×C53C2211yescharacteristic
23C2×C22⋊Q8×C53C2211yescharacteristic
24C2×#104×C53—21no—
25C2×#103×C53—21no—
26C2×#101×C53—21no—
27C53×#407C2211yescharacteristic
28C53×#380C2211yescharacteristic
29C4×C22⋊C4×C53C2211yescharacteristic
30C53×#343—21no—
31C53×#342—22no—
32C53×#340—21no—
33C53×#339—21no—
34C53×#328—22no—
35C53×#328—21no—
36C53×#327—21no—
37C2×#41×C53D411yescharacteristic
38C22×Q8×C53—21no—
39C22×Q8×C53D411yescharacteristic
40C22×Q8×C53D411yescharacteristic
41C23×C4×C53C2311yescharacteristic
42C2×M16×C53—21no—
43C2×M16×C53—21no—
44C2×M16×C53—21no—
45C2×M16×C53—42no—
46C22×C8×C53—21no—
47C4⋊Q8×C53—21no—
48C4⋊Q8×C53—22no—
49C4⋊Q8×C53—22no—
50C4⋊Q8×C53—22no—
51#126×C53—22no—
52#126×C53—22no—
53#126×C53—22no—
54#126×C53—22no—
55#125×C53—21no—
56#125×C53—22no—
57#125×C53—22no—
58#125×C53—22no—
59#125×C53—22no—
60#124×C53—22no—
61#123×C53—22no—
62#123×C53—22no—
63#123×C53—22no—
64#123×C53—22no—
65#123×C53—21no—
66C22⋊Q8×C53—21no—
67C22⋊Q8×C53—21no—
68C22⋊Q8×C53—22no—
69C22⋊Q8×C53—22no—
70C22⋊Q8×C53—22no—
71C22⋊Q8×C53—22no—
72C22⋊Q8×C53—21no—
73C22⋊Q8×C53—21no—
74C22⋊Q8×C53—21no—
75C22⋊Q8×C53—21no—
76C22⋊Q8×C53—21no—
77C22⋊Q8×C53—21no—
78C22⋊Q8×C53C2×C412yes—
79C22⋊Q8×C53C2×C411yescharacteristic
80C22⋊Q8×C53D412yes—
81C22⋊Q8×C53D411yescharacteristic
82#121×C53C2×C411yescharacteristic
83#121×C53D411yescharacteristic
84#120×C53—21no—
85C4×Q8×C53—21no—
86C4×Q8×C53—21no—
87C4×D4×C53—22no—
88C4×D4×C53—22no—
89C4×D4×C53—21no—
90C4×D4×C53—21no—
91#117×C53—21no—
92#117×C53—22no—
93#117×C53—22no—
94#117×C53D411yescharacteristic
95C2×C4⋊C4×C53—21no—
96C2×C4⋊C4×C53—21no—
97C2×C4⋊C4×C53C2×C412yes—
98C2×C4⋊C4×C53D411yescharacteristic
99C2×C4⋊C4×C53D411yescharacteristic
100C2×C22⋊C4×C53—21no—
101C2×C22⋊C4×C53—21no—
102C2×C22⋊C4×C53—21no—
103C2×C22⋊C4×C53D412yes—
104C2×C42×C53—21no—
105#104×C53—42no—
106#103×C53—42no—
107#101×C53—42no—
108C22⋊C8×C53—21no—
109C22⋊C8×C53—21no—
110#95×C53—21no—
111#95×C53—21no—
112C24×C53C2×D411yescharacteristic
113#41×C53—41no—
114#41×C53—41no—
115C2×Q8×C53—22no—
116C2×Q8×C53—21no—
117C2×Q8×C53—21no—
118C2×Q8×C53—21no—
119C2×Q8×C53—41no—
120C2×Q8×C53—41no—
121C2×Q8×C53—41no—
122C2×Q8×C53—41no—
123C2×Q8×C53—81no—
124C2×Q8×C53—81no—
125C2×Q8×C53C22⋊C412yes—
126C2×Q8×C53C22⋊C411yescharacteristic
127C2×D4×C53—21no—
128C2×D4×C53—21no—
129C2×D4×C53—21no—
130C2×D4×C53C22⋊C411yescharacteristic
131C22×C4×C53—21no—
132C22×C4×C53—22no—
133C22×C4×C53—21no—
134C22×C4×C53—21no—
135C22×C4×C53—41no—
136C22×C4×C53—41no—
137C22×C4×C53—41no—
138C22×C4×C53—41no—
139C22×C4×C53C22⋊C412yes—
140C22×C4×C53C22⋊C412yes—
141C22×C4×C53C22×C411yescharacteristic
142C22×C4×C53C2×D411yescharacteristic
143C22×C4×C53C2×D411yescharacteristic
144C22×C4×C53C2×D411yescharacteristic
145C22×C4×C53C2×D411yescharacteristic
146C22×C4×C53C2×D411yescharacteristic
147M16×C53—22no—
148M16×C53—22no—
149M16×C53—41no—
150M16×C53—41no—
151M16×C53—41no—
152M16×C53—41no—
153C2×C8×C53—22no—
154C2×C8×C53—41no—
155C2×C8×C53—41no—
156C2×C8×C53—41no—
157C4⋊C4×C53—22no—
158C4⋊C4×C53—22no—
159C4⋊C4×C53—22no—
160C4⋊C4×C53—22no—
161C4⋊C4×C53—22no—
162C4⋊C4×C53—22no—
163C4⋊C4×C53—22no—
164C4⋊C4×C53—22no—
165C4⋊C4×C53—22no—
166C4⋊C4×C53—22no—
167C4⋊C4×C53—22no—
168C4⋊C4×C53—21no—
169C4⋊C4×C53—22no—
170C4⋊C4×C53—21no—
171C4⋊C4×C53—21no—
172C4⋊C4×C53—21no—
173C4⋊C4×C53—21no—
174C4⋊C4×C53—41no—
175C4⋊C4×C53—41no—
176C4⋊C4×C53C22⋊C412yes—
177C4⋊C4×C53C22⋊C411yescharacteristic
178C4⋊C4×C53C22⋊C411yescharacteristic
179C22⋊C4×C53—22no—
180C22⋊C4×C53—22no—
181C22⋊C4×C53—22no—
182C22⋊C4×C53—22no—
183C22⋊C4×C53—21no—
184C22⋊C4×C53—22no—
185C22⋊C4×C53—22no—
186C22⋊C4×C53—22no—
187C22⋊C4×C53—21no—
188C22⋊C4×C53—21no—
189C22⋊C4×C53—21no—
190C22⋊C4×C53—21no—
191C22⋊C4×C53—22no—
192C22⋊C4×C53—41no—
193C22⋊C4×C53—42no—
194C42×C53—22no—
195C42×C53—22no—
196C42×C53—22no—
197C42×C53—21no—
198C42×C53—41no—
199C42×C53—41no—
200C23×C53—21no—
201C23×C53—21no—
202C23×C53—21no—
203C23×C53—41no—
204C23×C53—41no—
205C23×C53C2×C22⋊C411yescharacteristic
206C23×C53#12011yescharacteristic
207C23×C53#12011yescharacteristic
208Q8×C53—42no—
209Q8×C53—41no—
210Q8×C53—41no—
211Q8×C53—41no—
212Q8×C53—41no—
213Q8×C53—41no—
214Q8×C53—81no—
215Q8×C53—82no—
216D4×C53—41no—
217D4×C53—42no—
218D4×C53—41no—
219D4×C53—41no—
220D4×C53—81no—
221C2×C4×C53—21no—
222C2×C4×C53—21no—
223C2×C4×C53—21no—
224C2×C4×C53—21no—
225C2×C4×C53—21no—
226C2×C4×C53—21no—
227C2×C4×C53—22no—
228C2×C4×C53—22no—
229C2×C4×C53—21no—
230C2×C4×C53—22no—
231C2×C4×C53—22no—
232C2×C4×C53—21no—
233C2×C4×C53—21no—
234C2×C4×C53—41no—
235C2×C4×C53—41no—
236C2×C4×C53—41no—
237C2×C4×C53—41no—
238C2×C4×C53—41no—
239C2×C4×C53—41no—
240C2×C4×C53—41no—
241C2×C4×C53—42no—
242C2×C4×C53—42no—
243C2×C4×C53—42no—
244C2×C4×C53—81no—
245C2×C4×C53—82no—
246C2×C4×C53—81no—
247C2×C4×C53—81no—
248C2×C4×C53—81no—
249C2×C4×C53—81no—
250C2×C4×C53—81no—
251C2×C4×C53C2×C22⋊C411yescharacteristic
252C2×C4×C53C2×C22⋊C411yescharacteristic
253C2×C4×C53#12011yescharacteristic
254C2×C4×C53#12011yescharacteristic
255C8×C53—42no—
256C8×C53—81no—
257#8482C5311yesmaximal, characteristic, direct product factor, Sylow
258C22×C53—21no—
259C22×C53—21no—
260C22×C53—21no—
261C22×C53—21no—
262C22×C53—21no—
263C22×C53—21no—
264C22×C53—21no—
265C22×C53—21no—
266C22×C53—41no—
267C22×C53—41no—
268C22×C53—41no—
269C22×C53—81no—
270C22×C53—81no—
271C22×C53#37911yescharacteristic
272C22×C53#45411yescharacteristic
273C22×C53#45611yescharacteristic
274C4×C53—21no—
275C4×C53—21no—
276C4×C53—21no—
277C4×C53—42no—
278C4×C53—41no—
279C4×C53—42no—
280C4×C53—41no—
281C4×C53—41no—
282C4×C53—42no—
283C4×C53—42no—
284C4×C53—82no—
285C4×C53—161no—
286C4×C53B(4,𝔽2)11yescharacteristic
287C4×C53#45811yescharacteristic
288#3538C2×C5311yescharacteristic
289#1996C2×C5311yescharacteristic, complemented
290#1901C2×C5311yescharacteristic, complemented
291#1894C2×C5311yescharacteristic, complemented
292#1871C2×C5311yescharacteristic
293#1513C2×C5311yescharacteristic
294#1508C2×C5311yescharacteristic, complemented
295C2×C53—21no—
296C2×C53—21no—
297C2×C53—41no—
298C2×C53—41no—
299C2×C53—81no—
300C2×C53#179011yescharacteristic
301C2×C53#179311yescharacteristic
302C2×C53#179611yescharacteristic
303C22×M16C22×C5311yescharacteristic
304#563—22no—
305#552—21no—
306#552—21no—
307#551—22no—
308D4×Q8—21no—
309D4×Q8—21no—
310#544—21no—
311#544C4×C5312yescomplemented
312#541—21no—
313#539—21no—
314#539C4×C5312yescomplemented
315#536—21no—
316#525C22×C5311yescharacteristic
317C2×C22⋊Q8C22×C5311yescharacteristic
318C2×C22⋊Q8C22×C5311yescharacteristic
319C2×#104—21no—
320C2×#103—21no—
321C2×#101—21no—
322#407C22×C5311yescharacteristic
323#380C22×C5311yescharacteristic
324C4×C22⋊C4C22×C5311yescharacteristic
325#343—21no—
326#342—22no—
327#340—21no—
328#339—21no—
329#328—22no—
330#328—21no—
331#327—21no—
332C53#848211yescharacteristic, direct product factor, Sylow
333C2×#41D4×C5311yescharacteristic
334C22×Q8—21no—
335C22×Q8D4×C5311yescharacteristic
336C22×Q8D4×C5311yescharacteristic
337C23×C4C23×C5311yescharacteristic
338C2×M16—21no—
339C2×M16—21no—
340C2×M16—21no—
341C2×M16—42no—
342C22×C8—21no—
343C4⋊Q8—21no—
344C4⋊Q8—22no—
345C4⋊Q8—22no—
346C4⋊Q8—22no—
347#126—22no—
348#126—22no—
349#126—22no—
350#126—22no—
351#125—21no—
352#125—22no—
353#125—22no—
354#125—22no—
355#125—22no—
356#124—22no—
357#123—22no—
358#123—22no—
359#123—22no—
360#123—21no—
361#123—22no—
362C22⋊Q8—21no—
363C22⋊Q8—21no—
364C22⋊Q8—22no—
365C22⋊Q8—22no—
366C22⋊Q8—22no—
367C22⋊Q8—21no—
368C22⋊Q8—21no—
369C22⋊Q8—21no—
370C22⋊Q8—21no—
371C22⋊Q8—21no—
372C22⋊Q8—21no—
373C22⋊Q8—22no—
374C22⋊Q8C2×C4×C5312yes—
375C22⋊Q8C2×C4×C5311yescharacteristic
376C22⋊Q8D4×C5312yes—
377C22⋊Q8D4×C5311yescharacteristic
378#121C2×C4×C5311yescharacteristic
379#121D4×C5311yescharacteristic
380#120—21no—
381C4×Q8—21no—
382C4×Q8—21no—
383C4×D4—22no—
384C4×D4—21no—
385C4×D4—21no—
386C4×D4—22no—
387#117—21no—
388#117—22no—
389#117—22no—
390#117D4×C5311yescharacteristic
391C2×C4⋊C4—21no—
392C2×C4⋊C4—21no—
393C2×C4⋊C4C2×C4×C5312yes—
394C2×C4⋊C4D4×C5311yescharacteristic
395C2×C4⋊C4D4×C5311yescharacteristic
396C2×C22⋊C4—21no—
397C2×C22⋊C4—21no—
398C2×C22⋊C4—21no—
399C2×C22⋊C4D4×C5312yes—
400C2×C42—21no—
401#104—42no—
402#103—42no—
403#101—42no—
404C22⋊C8—21no—
405C22⋊C8—21no—
406#95—21no—
407#95—21no—
408C24C2×D4×C5311yescharacteristic
409#41—41no—
410#41—41no—
411C2×Q8—22no—
412C2×Q8—21no—
413C2×Q8—21no—
414C2×Q8—21no—
415C2×Q8—41no—
416C2×Q8—41no—
417C2×Q8—41no—
418C2×Q8—41no—
419C2×Q8—81no—
420C2×Q8—81no—
421C2×Q8C22⋊C4×C5312yes—
422C2×Q8C22⋊C4×C5311yescharacteristic
423C2×D4—21no—
424C2×D4—21no—
425C2×D4—21no—
426C2×D4C22⋊C4×C5311yescharacteristic
427C22×C4—21no—
428C22×C4—22no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—41no—
432C22×C4—41no—
433C22×C4—41no—
434C22×C4—41no—
435C22×C4C22⋊C4×C5312yes—
436C22×C4C22⋊C4×C5312yes—
437C22×C4C22×C4×C5311yescharacteristic
438C22×C4C2×D4×C5311yescharacteristic
439C22×C4C2×D4×C5311yescharacteristic
440C22×C4C2×D4×C5311yescharacteristic
441C22×C4C2×D4×C5311yescharacteristic
442C22×C4C2×D4×C5311yescharacteristic
443M16—22no—
444M16—22no—
445M16—41no—
446M16—41no—
447M16—41no—
448M16—41no—
449C2×C8—22no—
450C2×C8—41no—
451C2×C8—41no—
452C2×C8—41no—
453C4⋊C4—22no—
454C4⋊C4—22no—
455C4⋊C4—22no—
456C4⋊C4—22no—
457C4⋊C4—22no—
458C4⋊C4—22no—
459C4⋊C4—22no—
460C4⋊C4—22no—
461C4⋊C4—22no—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—21no—
465C4⋊C4—22no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—41no—
471C4⋊C4—41no—
472C4⋊C4C22⋊C4×C5312yes—
473C4⋊C4C22⋊C4×C5311yescharacteristic
474C4⋊C4C22⋊C4×C5311yescharacteristic
475C22⋊C4—22no—
476C22⋊C4—22no—
477C22⋊C4—22no—
478C22⋊C4—22no—
479C22⋊C4—21no—
480C22⋊C4—22no—
481C22⋊C4—22no—
482C22⋊C4—22no—
483C22⋊C4—21no—
484C22⋊C4—21no—
485C22⋊C4—21no—
486C22⋊C4—21no—
487C22⋊C4—22no—
488C22⋊C4—41no—
489C22⋊C4—42no—
490C42—22no—
491C42—22no—
492C42—22no—
493C42—21no—
494C42—41no—
495C42—41no—
496C23—21no—
497C23—21no—
498C23—21no—
499C23—41no—
500C23—41no—
501C23C2×C22⋊C4×C5311yescharacteristic
502C23#120×C5311yescharacteristic
503C23#120×C5311yescharacteristic
504Q8—42no—
505Q8—41no—
506Q8—41no—
507Q8—41no—
508Q8—41no—
509Q8—41no—
510Q8—81no—
511Q8—82no—
512D4—41no—
513D4—42no—
514D4—41no—
515D4—41no—
516D4—81no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—21no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—81no—
541C2×C4—82no—
542C2×C4—81no—
543C2×C4—81no—
544C2×C4—81no—
545C2×C4—81no—
546C2×C4—81no—
547C2×C4C2×C22⋊C4×C5311yescharacteristic
548C2×C4C2×C22⋊C4×C5311yescharacteristic
549C2×C4#120×C5311yescharacteristic
550C2×C4#120×C5311yescharacteristic
551C8—42no—
552C8—81no—
553C22—21no—
554C22—21no—
555C22—21no—
556C22—21no—
557C22—21no—
558C22—21no—
559C22—21no—
560C22—21no—
561C22—41no—
562C22—41no—
563C22—41no—
564C22—81no—
565C22—81no—
566C22C53×#37911yescharacteristic
567C22C53×#45411yescharacteristic
568C22C53×#45611yescharacteristic
569C4—21no—
570C4—21no—
571C4—21no—
572C4—41no—
573C4—42no—
574C4—42no—
575C4—42no—
576C4—41no—
577C4—42no—
578C4—41no—
579C4—82no—
580C4—161no—
581C4C53×B(4,𝔽2)11yescharacteristic
582C4C53×#45811yescharacteristic
583C2—21no—
584C2—21no—
585C2—41no—
586C2—41no—
587C2—81no—
588C2C53×#179011yescharacteristic
589C2C53×#179311yescharacteristic
590C2C53×#179611yescharacteristic
subgroupG/H
center, socleC22×C53#379
commutatorC22×C4C22×C4×C53
Frattini, cosocleC23×C4C23×C53
prime radicalC53×#525C22
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×#1996C2semidirect
3. C53×#1901C2semidirect
4. C53×#1894C2semidirect
7. C53×#1508C2semidirect
16. C53×#544C4semidirect
19. C53×#539C4semidirect
257. #8482C53direct (N × H)
289. #1996C2×C53semidirect
290. #1901C2×C53semidirect
291. #1894C2×C53semidirect
294. #1508C2×C53semidirect
311. #544C4×C53semidirect
314. #539C4×C53semidirect
332. C53#8482direct (N × H)

Maximal subgroups

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

subgroupindex
C53×#35382
C53×#19962
C53×#19012
C53×#18942
C53×#18712
C53×#15132
C53×#15082
#848253