Finite Group Databasebeta

← C53×#8439 #27215951

Subgroups — C53×#8439 #27215951

Order 13568.

760 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Q8×#41×C53C211yesmaximal, characteristic
2C53×#1806C211yesmaximal, characteristic
3C4×#103×C53C211yesmaximal, characteristic, complemented
4C53×#1503C211yesmaximal, characteristic, complemented
5C53×#1493C211yesmaximal, characteristic, complemented
6C53×#1487C211yesmaximal, characteristic, complemented
7C53×#1476C211yesmaximal, characteristic
8Q82×C53C411yescharacteristic, complemented
9C53×#557—21no—
10C53×#557C411yescharacteristic, complemented
11C53×#554—22no—
12C53×#554—21no—
13C53×#554—21no—
14C53×#554C411yescharacteristic, complemented
15D4×Q8×C53—21no—
16D4×Q8×C53C411yescharacteristic, complemented
17C53×#533—22no—
18C53×#533—21no—
19C53×#533—21no—
20C53×#533C2211yescharacteristic
21C4×#41×C53—21no—
22C4×#41×C53C2211yescharacteristic
23C2×C4×Q8×C53—21no—
24C2×C4×Q8×C53C2211yescharacteristic
25C53×#431C2211yescharacteristic
26C53×#423C2211yescharacteristic
27C2×#104×C53—21no—
28C2×#103×C53—21no—
29C2×C4×C8×C53C2211yescharacteristic
30C4×C4⋊C4×C53C2211yescharacteristic
31C53×#340—21no—
32C53×#339—21no—
33C53×#328—22no—
34C53×Q8⋊C8—21no—
35C53×Q8⋊C8—21no—
36C53×Q8⋊C8—21no—
37C53×Q8⋊C8—22no—
38C53×D4⋊C8—21no—
39C2×#41×C53D411yescharacteristic
40C22×Q8×C53—21no—
41C22×Q8×C53D411yescharacteristic
42C2×M16×C53—21no—
43C22×C8×C53—21no—
44C4⋊Q8×C53—22no—
45C4⋊Q8×C53—21no—
46C4⋊Q8×C53—21no—
47C4⋊Q8×C53—21no—
48C4⋊Q8×C53—21no—
49C4⋊Q8×C53—22no—
50C4⋊Q8×C53C2×C411yescharacteristic
51C4⋊Q8×C53C2×C411yescharacteristic
52#125×C53—22no—
53#125×C53—22no—
54#125×C53—22no—
55#125×C53—21no—
56#125×C53—21no—
57#125×C53—21no—
58C22⋊Q8×C53—22no—
59C22⋊Q8×C53—22no—
60C22⋊Q8×C53—22no—
61C22⋊Q8×C53—22no—
62C22⋊Q8×C53—22no—
63C22⋊Q8×C53—22no—
64C22⋊Q8×C53—21no—
65C22⋊Q8×C53—21no—
66C22⋊Q8×C53—22no—
67C22⋊Q8×C53—21no—
68C22⋊Q8×C53D412yes—
69C4×Q8×C53—22no—
70C4×Q8×C53—22no—
71C4×Q8×C53—22no—
72C4×Q8×C53—21no—
73C4×Q8×C53—21no—
74C4×Q8×C53—21no—
75C4×Q8×C53—21no—
76C4×Q8×C53—21no—
77C4×Q8×C53—21no—
78C4×Q8×C53—21no—
79C4×Q8×C53C2×C411yescharacteristic
80C4×Q8×C53C2×C411yescharacteristic
81C4×Q8×C53C2×C411yescharacteristic
82C4×Q8×C53D411yescharacteristic
83C4×Q8×C53D411yescharacteristic
84C4×Q8×C53D411yescharacteristic
85C4×Q8×C53D412yes—
86C4×D4×C53—22no—
87C4×D4×C53—22no—
88C4×D4×C53—21no—
89C4×D4×C53—21no—
90C4×D4×C53—21no—
91C4×D4×C53—21no—
92C4×D4×C53C2×C411yescharacteristic
93C4×D4×C53D411yescharacteristic
94#117×C53—22no—
95#117×C53—21no—
96#117×C53—21no—
97#117×C53D411yescharacteristic
98C2×C4⋊C4×C53—22no—
99C2×C4⋊C4×C53—21no—
100C2×C4⋊C4×C53—21no—
101C2×C4⋊C4×C53—21no—
102C2×C4⋊C4×C53D411yescharacteristic
103C2×C42×C53—22no—
104C2×C42×C53—21no—
105C2×C42×C53—21no—
106C2×C42×C53—21no—
107C2×C42×C53C2311yescharacteristic
108C4⋊C8×C53—21no—
109C4⋊C8×C53—21no—
110C4⋊C8×C53—21no—
111#104×C53—42no—
112#103×C53—24no—
113C22⋊C8×C53—21no—
114C4×C8×C53—21no—
115C4×C8×C53—21no—
116C4×C8×C53—21no—
117#95×C53—21no—
118#41×C53—21no—
119#41×C53—21no—
120#41×C53SD1612yes—
121#41×C53Q1612yes—
122C2×Q8×C53—22no—
123C2×Q8×C53—22no—
124C2×Q8×C53—21no—
125C2×Q8×C53—21no—
126C2×Q8×C53—22no—
127C2×Q8×C53—41no—
128C2×Q8×C53—42no—
129C2×Q8×C53—42no—
130C2×Q8×C53C22⋊C411yescharacteristic
131C2×Q8×C53C22⋊C411yescharacteristic
132C2×Q8×C53C22⋊C411yescharacteristic
133C2×Q8×C53C22⋊C412yes—
134C2×D4×C53—21no—
135C2×D4×C53C22⋊C411yescharacteristic
136C22×C4×C53—21no—
137C22×C4×C53—22no—
138C22×C4×C53—21no—
139C22×C4×C53—22no—
140C22×C4×C53—21no—
141C22×C4×C53—21no—
142C22×C4×C53—21no—
143C22×C4×C53—41no—
144C22×C4×C53C2×D411yescharacteristic
145C22×C4×C53C2×D411yescharacteristic
146C22×C4×C53C2×D411yescharacteristic
147M16×C53—41no—
148M16×C53—41no—
149C2×C8×C53—22no—
150C2×C8×C53—41no—
151C2×C8×C53—41no—
152C2×C8×C53—41no—
153C2×C8×C53—41no—
154C4⋊C4×C53—22no—
155C4⋊C4×C53—22no—
156C4⋊C4×C53—22no—
157C4⋊C4×C53—22no—
158C4⋊C4×C53—22no—
159C4⋊C4×C53—22no—
160C4⋊C4×C53—22no—
161C4⋊C4×C53—22no—
162C4⋊C4×C53—21no—
163C4⋊C4×C53—21no—
164C4⋊C4×C53—21no—
165C4⋊C4×C53—21no—
166C4⋊C4×C53—21no—
167C4⋊C4×C53—21no—
168C4⋊C4×C53—21no—
169C4⋊C4×C53—22no—
170C4⋊C4×C53—21no—
171C4⋊C4×C53—21no—
172C4⋊C4×C53—21no—
173C4⋊C4×C53—22no—
174C4⋊C4×C53—22no—
175C4⋊C4×C53—41no—
176C4⋊C4×C53C22⋊C411yescharacteristic
177C4⋊C4×C53C22⋊C411yescharacteristic
178C4⋊C4×C53C22⋊C411yescharacteristic
179C4⋊C4×C53C22⋊C411yescharacteristic
180C4⋊C4×C53C22⋊C412yes—
181C22⋊C4×C53—22no—
182C22⋊C4×C53—22no—
183C22⋊C4×C53—22no—
184C22⋊C4×C53—21no—
185C22⋊C4×C53—21no—
186C22⋊C4×C53—21no—
187C42×C53—22no—
188C42×C53—22no—
189C42×C53—21no—
190C42×C53—21no—
191C42×C53—21no—
192C42×C53—21no—
193C42×C53—21no—
194C42×C53—22no—
195C42×C53—21no—
196C42×C53—21no—
197C42×C53—41no—
198C42×C53—41no—
199C42×C53C22×C411yescharacteristic
200C42×C53C2×D411yescharacteristic
201C42×C53C2×D411yescharacteristic
202C42×C53C2×D411yescharacteristic
203C23×C53—21no—
204C23×C53#12011yescharacteristic
205Q8×C53—21no—
206Q8×C53—21no—
207Q8×C53—21no—
208Q8×C53—21no—
209Q8×C53—42no—
210Q8×C53—44no—
211Q8×C53—42no—
212Q8×C53—41no—
213Q8×C53—41no—
214Q8×C53—82no—
215Q8×C53#10312yescomplemented
216Q8×C53#10412yescomplemented
217D4×C53—22no—
218D4×C53—22no—
219D4×C53—21no—
220D4×C53#10312yescomplemented
221C2×C4×C53—21no—
222C2×C4×C53—21no—
223C2×C4×C53—21no—
224C2×C4×C53—21no—
225C2×C4×C53—22no—
226C2×C4×C53—22no—
227C2×C4×C53—21no—
228C2×C4×C53—21no—
229C2×C4×C53—21no—
230C2×C4×C53—22no—
231C2×C4×C53—21no—
232C2×C4×C53—22no—
233C2×C4×C53—21no—
234C2×C4×C53—41no—
235C2×C4×C53—42no—
236C2×C4×C53—44no—
237C2×C4×C53—41no—
238C2×C4×C53—41no—
239C2×C4×C53—42no—
240C2×C4×C53—81no—
241C2×C4×C53—81no—
242C2×C4×C53—82no—
243C2×C4×C53—81no—
244C2×C4×C53—81no—
245C2×C4×C53C2×C22⋊C411yescharacteristic
246C2×C4×C53C2×C22⋊C411yescharacteristic
247C2×C4×C53C2×C22⋊C411yescharacteristic
248C2×C4×C53#12011yescharacteristic
249C2×C4×C53#12011yescharacteristic
250C2×C4×C53#12011yescharacteristic
251C2×C4×C53C2×SD1611yescharacteristic
252C2×C4×C53C2×Q1611yescharacteristic
253C2×C4×C53#13711yescharacteristic
254C2×C4×C53#13711yescharacteristic
255C8×C53—42no—
256C8×C53—81no—
257C8×C53—81no—
258#8439C5311yesmaximal, characteristic, direct product factor, Sylow
259C22×C53—21no—
260C22×C53—21no—
261C22×C53—22no—
262C22×C53—41no—
263C22×C53—41no—
264C22×C53#37911yescharacteristic
265C22×C53#44811yescharacteristic
266C22×C53#45111yescharacteristic
267C4×C53—21no—
268C4×C53—21no—
269C4×C53—22no—
270C4×C53—21no—
271C4×C53—22no—
272C4×C53—41no—
273C4×C53—42no—
274C4×C53—41no—
275C4×C53—41no—
276C4×C53—41no—
277C4×C53—42no—
278C4×C53—41no—
279C4×C53—82no—
280C4×C53C2×#10311yescharacteristic
281C4×C53#41911yescharacteristic
282C4×C53C2×#10411yescharacteristic
283C4×C53#42111yescharacteristic
284C4×C53#44811yescharacteristic
285C4×C53#45111yescharacteristic
286Q8×#41C2×C5311yescharacteristic
287#1806C2×C5311yescharacteristic
288C4×#103C2×C5311yescharacteristic, complemented
289#1503C2×C5311yescharacteristic, complemented
290#1493C2×C5311yescharacteristic, complemented
291#1487C2×C5311yescharacteristic, complemented
292#1476C2×C5311yescharacteristic
293C2×C53—21no—
294C2×C53—21no—
295C2×C53—42no—
296C2×C53#178711yescharacteristic
297C2×C53#178911yescharacteristic
298C2×C53#179411yescharacteristic
299Q82C4×C5311yescharacteristic, complemented
300#557—21no—
301#557C4×C5311yescharacteristic, complemented
302#554—22no—
303#554—21no—
304#554—21no—
305#554C4×C5311yescharacteristic, complemented
306D4×Q8—21no—
307D4×Q8C4×C5311yescharacteristic, complemented
308#533—22no—
309#533—21no—
310#533—21no—
311#533C22×C5311yescharacteristic
312C4×#41—21no—
313C4×#41C22×C5311yescharacteristic
314C2×C4×Q8—21no—
315C2×C4×Q8C22×C5311yescharacteristic
316#431C22×C5311yescharacteristic
317#423C22×C5311yescharacteristic
318C2×#104—21no—
319C2×#103—21no—
320C2×C4×C8C22×C5311yescharacteristic
321C4×C4⋊C4C22×C5311yescharacteristic
322#340—21no—
323#339—21no—
324#328—22no—
325Q8⋊C8—21no—
326Q8⋊C8—21no—
327Q8⋊C8—21no—
328Q8⋊C8—22no—
329D4⋊C8—21no—
330C53#843911yescharacteristic, direct product factor, Sylow
331C2×#41D4×C5311yescharacteristic
332C22×Q8—21no—
333C22×Q8D4×C5311yescharacteristic
334C2×M16—21no—
335C22×C8—21no—
336C4⋊Q8—22no—
337C4⋊Q8—21no—
338C4⋊Q8—22no—
339C4⋊Q8—21no—
340C4⋊Q8—21no—
341C4⋊Q8—21no—
342C4⋊Q8C2×C4×C5311yescharacteristic
343C4⋊Q8C2×C4×C5311yescharacteristic
344#125—22no—
345#125—22no—
346#125—22no—
347#125—21no—
348#125—21no—
349#125—21no—
350C22⋊Q8—22no—
351C22⋊Q8—22no—
352C22⋊Q8—22no—
353C22⋊Q8—22no—
354C22⋊Q8—22no—
355C22⋊Q8—21no—
356C22⋊Q8—22no—
357C22⋊Q8—21no—
358C22⋊Q8—21no—
359C22⋊Q8—22no—
360C22⋊Q8D4×C5312yes—
361C4×Q8—22no—
362C4×Q8—22no—
363C4×Q8—22no—
364C4×Q8—21no—
365C4×Q8—21no—
366C4×Q8—21no—
367C4×Q8—21no—
368C4×Q8—21no—
369C4×Q8—21no—
370C4×Q8—21no—
371C4×Q8C2×C4×C5311yescharacteristic
372C4×Q8C2×C4×C5311yescharacteristic
373C4×Q8C2×C4×C5311yescharacteristic
374C4×Q8D4×C5311yescharacteristic
375C4×Q8D4×C5311yescharacteristic
376C4×Q8D4×C5311yescharacteristic
377C4×Q8D4×C5312yes—
378C4×D4—22no—
379C4×D4—22no—
380C4×D4—21no—
381C4×D4—21no—
382C4×D4—21no—
383C4×D4—21no—
384C4×D4C2×C4×C5311yescharacteristic
385C4×D4D4×C5311yescharacteristic
386#117—22no—
387#117—21no—
388#117—21no—
389#117D4×C5311yescharacteristic
390C2×C4⋊C4—22no—
391C2×C4⋊C4—21no—
392C2×C4⋊C4—21no—
393C2×C4⋊C4—21no—
394C2×C4⋊C4D4×C5311yescharacteristic
395C2×C42—22no—
396C2×C42—21no—
397C2×C42—21no—
398C2×C42—21no—
399C2×C42C23×C5311yescharacteristic
400C4⋊C8—21no—
401C4⋊C8—21no—
402C4⋊C8—21no—
403#104—42no—
404#103—24no—
405C22⋊C8—21no—
406C4×C8—21no—
407C4×C8—21no—
408C4×C8—21no—
409#95—21no—
410#41—21no—
411#41—21no—
412#41SD16×C5312yes—
413#41Q16×C5312yes—
414C2×Q8—22no—
415C2×Q8—22no—
416C2×Q8—21no—
417C2×Q8—21no—
418C2×Q8—22no—
419C2×Q8—41no—
420C2×Q8—42no—
421C2×Q8—42no—
422C2×Q8C22⋊C4×C5311yescharacteristic
423C2×Q8C22⋊C4×C5311yescharacteristic
424C2×Q8C22⋊C4×C5311yescharacteristic
425C2×Q8C22⋊C4×C5312yes—
426C2×D4—21no—
427C2×D4C22⋊C4×C5311yescharacteristic
428C22×C4—21no—
429C22×C4—22no—
430C22×C4—21no—
431C22×C4—22no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—41no—
436C22×C4C2×D4×C5311yescharacteristic
437C22×C4C2×D4×C5311yescharacteristic
438C22×C4C2×D4×C5311yescharacteristic
439M16—41no—
440M16—41no—
441C2×C8—22no—
442C2×C8—41no—
443C2×C8—41no—
444C2×C8—41no—
445C2×C8—41no—
446C4⋊C4—22no—
447C4⋊C4—22no—
448C4⋊C4—22no—
449C4⋊C4—22no—
450C4⋊C4—22no—
451C4⋊C4—22no—
452C4⋊C4—22no—
453C4⋊C4—22no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—22no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—22no—
466C4⋊C4—22no—
467C4⋊C4—41no—
468C4⋊C4C22⋊C4×C5311yescharacteristic
469C4⋊C4C22⋊C4×C5311yescharacteristic
470C4⋊C4C22⋊C4×C5311yescharacteristic
471C4⋊C4C22⋊C4×C5311yescharacteristic
472C4⋊C4C22⋊C4×C5312yes—
473C22⋊C4—22no—
474C22⋊C4—22no—
475C22⋊C4—22no—
476C22⋊C4—21no—
477C22⋊C4—21no—
478C22⋊C4—21no—
479C42—22no—
480C42—22no—
481C42—21no—
482C42—21no—
483C42—21no—
484C42—21no—
485C42—22no—
486C42—21no—
487C42—21no—
488C42—21no—
489C42—41no—
490C42—41no—
491C42C22×C4×C5311yescharacteristic
492C42C2×D4×C5311yescharacteristic
493C42C2×D4×C5311yescharacteristic
494C42C2×D4×C5311yescharacteristic
495C23—21no—
496C23#120×C5311yescharacteristic
497Q8—21no—
498Q8—21no—
499Q8—21no—
500Q8—21no—
501Q8—42no—
502Q8—42no—
503Q8—44no—
504Q8—41no—
505Q8—41no—
506Q8—82no—
507Q8#103×C5312yescomplemented
508Q8#104×C5312yescomplemented
509D4—22no—
510D4—22no—
511D4—21no—
512D4#103×C5312yescomplemented
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—41no—
527C2×C4—42no—
528C2×C4—44no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—42no—
532C2×C4—81no—
533C2×C4—81no—
534C2×C4—82no—
535C2×C4—81no—
536C2×C4—81no—
537C2×C4C2×C22⋊C4×C5311yescharacteristic
538C2×C4C2×C22⋊C4×C5311yescharacteristic
539C2×C4C2×C22⋊C4×C5311yescharacteristic
540C2×C4#120×C5311yescharacteristic
541C2×C4#120×C5311yescharacteristic
542C2×C4#120×C5311yescharacteristic
543C2×C4C2×SD16×C5311yescharacteristic
544C2×C4C2×Q16×C5311yescharacteristic
545C2×C4#137×C5311yescharacteristic
546C2×C4#137×C5311yescharacteristic
547C8—42no—
548C8—81no—
549C8—81no—
550C22—21no—
551C22—21no—
552C22—22no—
553C22—41no—
554C22—41no—
555C22C53×#37911yescharacteristic
556C22C53×#44811yescharacteristic
557C22C53×#45111yescharacteristic
558C4—22no—
559C4—21no—
560C4—22no—
561C4—21no—
562C4—21no—
563C4—41no—
564C4—42no—
565C4—41no—
566C4—41no—
567C4—42no—
568C4—41no—
569C4—41no—
570C4—82no—
571C4C2×#103×C5311yescharacteristic
572C4C53×#41911yescharacteristic
573C4C2×#104×C5311yescharacteristic
574C4C53×#42111yescharacteristic
575C4C53×#44811yescharacteristic
576C4C53×#45111yescharacteristic
577C2—21no—
578C2—21no—
579C2—42no—
580C2C53×#178711yescharacteristic
581C2C53×#178911yescharacteristic
582C2C53×#179411yescharacteristic
subgroupG/H
centerC2×C4×C53#120
commutatorC42C22×C4×C53
Frattini, cosocleC2×C42C23×C53
socleC22×C53#379
prime radicalC2×#41×C53D4
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4×C53 ◁ C2×C42×C53 ◁ G

Complements

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

NH ≅ G/Ntype
3. C4×#103×C53C2semidirect
4. C53×#1503C2semidirect
5. C53×#1493C2semidirect
6. C53×#1487C2semidirect
8. Q82×C53C4semidirect
10. C53×#557C4semidirect
14. C53×#554C4semidirect
16. D4×Q8×C53C4semidirect
215. Q8×C53#103semidirect
216. Q8×C53#104semidirect
220. D4×C53#103semidirect
258. #8439C53direct (N × H)
288. C4×#103C2×C53semidirect
289. #1503C2×C53semidirect
290. #1493C2×C53semidirect
291. #1487C2×C53semidirect
299. Q82C4×C53semidirect
301. #557C4×C53semidirect
305. #554C4×C53semidirect
307. D4×Q8C4×C53semidirect
330. C53#8439direct (N × H)
507. Q8#103×C53semidirect
508. Q8#104×C53semidirect
512. D4#103×C53semidirect

Maximal subgroups

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

subgroupindex
Q8×#41×C532
C53×#18062
C4×#103×C532
C53×#15032
C53×#14932
C53×#14872
C53×#14762
#843953