Finite Group Databasebeta

← D53×#1803 #27383547

Subgroups — D53×#1803 #27383547

Order 13568.

1598 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×Q8×D53C211yesmaximal, characteristic
2C2×C4⋊C8×D53C211yesmaximal, characteristic
3C2×#103×D53C212yesmaximal
4#384×D53C211yesmaximal, characteristic
5#20274721C211yesmaximal, characteristic, complemented
6#20274720C211yesmaximal, characteristic, complemented
7#20274719C212yesmaximal, complemented
8#20274718C212yesmaximal, complemented
9#20274717C211yesmaximal, characteristic, complemented
10#340×D53C212yesmaximal
11C53×#1803C211yesmaximal, characteristic, complemented
12C22×Q8×D53C2211yescharacteristic
13C22×C8×D53C2212yes—
14C2×Q8×Dic53C2211yescharacteristic
15C2×C4×Dic106—21no—
16C2×C4×Dic106C2211yescharacteristic
17C2×#13133182—21no—
18C2×#13133182C2211yescharacteristic
19C4×Q8×D53—24no—
20C4×Q8×D53C418yescomplemented
21C2×C4⋊C4×D53—21no—
22C2×C4⋊C4×D53—21no—
23C2×C4⋊C4×D53—21no—
24C2×C4⋊C4×D53C2212yes—
25C2×C4⋊C4×D53C2211yescharacteristic
26C2×C42×D53—21no—
27C2×C42×D53C2211yescharacteristic
28C2×C4⋊#88322C2211yescharacteristic
29C2×#13133146C2212yes—
30C4⋊C8×D53—24no—
31C2×#13133142C2212yes—
32C2×#13133141C2212yes—
33C2×#13133140C2212yes—
34#103×D53—28no—
35#16810493C2211yescharacteristic
36#16810492—21no—
37#16810492—21no—
38#16810492C2212yes—
39#16810386C2212yes—
40#16810385C2212yes—
41#16810385C2212yes—
42C2×C4×Q8×C53C2211yescharacteristic
43C2×C4⋊C8×C53C2211yescharacteristic
44C2×#103×C53C2212yes—
45C53×#384C2211yescharacteristic
46C53×#340C2212yes—
47C22×Dic106—21no—
48C22×Dic106C2311yescharacteristic
49C2×Q8×D53—22no—
50C2×Q8×D53—22no—
51C2×Q8×D53—22no—
52C2×Q8×D53C2×C414yes—
53C2×Q8×D53D414yes—
54C2×Q8×D53Q814yes—
55C22×C4×D53—21no—
56C22×C4×D53—21no—
57C22×C4×D53—22no—
58C22×C4×D53D411yescharacteristic
59C22×C4×D53D411yescharacteristic
60C22×C4×D53C2311yescharacteristic
61C22×#88322C2312yes—
62C2×C8×D53—24no—
63C2×C8×D53—24no—
64C2×C8×D53—24no—
65Q8×Dic53—22no—
66Q8×Dic53C2×C414yescomplemented
67C4×Dic106—22no—
68C4×Dic106—28no—
69C4×Dic106C2×C414yescomplemented
70#13133182—28no—
71#13133182—22no—
72#13133182C2×C414yescomplemented
73C2×C4⋊Dic53—21no—
74C2×C4⋊Dic53—21no—
75C2×C4⋊Dic53—21no—
76C2×C4⋊Dic53C2312yes—
77C2×C4⋊Dic53C2311yescharacteristic
78C2×#11719919—21no—
79C2×#11719919—21no—
80C2×#11719919—21no—
81C2×#11719919—22no—
82C2×#11719919—22no—
83C2×#11719919—22no—
84C4⋊C4×D53—28no—
85C4⋊C4×D53—22no—
86C4⋊C4×D53—24no—
87C4⋊C4×D53—24no—
88C4⋊C4×D53—44no—
89C4⋊C4×D53—44no—
90C4⋊C4×D53C2×C414yes—
91C2×C4×Dic53—21no—
92C2×C4×Dic53—21no—
93C2×C4×Dic53—21no—
94C2×C4×Dic53D411yescharacteristic
95C2×C4×Dic53D411yescharacteristic
96C2×C4×Dic53C2311yescharacteristic
97C42×D53—28no—
98C42×D53C2×C414yes—
99C42×D53D414yes—
100C4⋊#88322—22no—
101#13133146—24no—
102#13133142—24no—
103#13133141—24no—
104#13133140—24no—
105C22×Q8×C53C2311yescharacteristic
106C22×C8×C53C2312yes—
107C4×Q8×C53—22no—
108C4×Q8×C53C2×C414yescomplemented
109C2×C4⋊C4×C53—21no—
110C2×C4⋊C4×C53—21no—
111C2×C4⋊C4×C53—21no—
112C2×C4⋊C4×C53C2312yes—
113C2×C4⋊C4×C53C2311yescharacteristic
114C2×C42×C53—21no—
115C2×C42×C53C2311yescharacteristic
116C4⋊C8×C53—22no—
117#103×C53—24no—
118C23×D53C2×D411yescharacteristic
119C2×Q8×C53—21no—
120C2×Q8×C53—21no—
121C2×Q8×C53—21no—
122C2×Q8×C53C22×C412yes—
123C2×Q8×C53C2×D412yes—
124C2×Q8×C53C2×Q812yes—
125C22×C4×C53—21no—
126C22×C4×C53—21no—
127C22×C4×C53—22no—
128C22×C4×C53C2×D411yescharacteristic
129C22×C4×C53C2×D411yescharacteristic
130C22×C4×C53C2411yescharacteristic
131C22×Dic53—21no—
132C22×Dic53—21no—
133C22×Dic53—22no—
134C22×Dic53C2×D411yescharacteristic
135C22×Dic53C2×D411yescharacteristic
136C22×Dic53C2×D411yescharacteristic
137Q8×D53—28no—
138Q8×D53—28no—
139Q8×D53—28no—
140Q8×D53C4⋊C4116yescomplemented
141C2×C4×D53—24no—
142C2×C4×D53—24no—
143C2×C4×D53—24no—
144C2×C4×D53—22no—
145C2×C4×D53—22no—
146C2×C4×D53—24no—
147C2×C4×D53—42no—
148C2×C4×D53—42no—
149C2×C4×D53—44no—
150C2×C4×D53—44no—
151C2×C4×D53—44no—
152C2×C4×D53C22⋊C414yes—
153C2×C4×D53C22⋊C414yes—
154C2×C4×D53SD1614yes—
155C2×C4×D53Q1614yes—
156C2×C4×D53C22×C412yes—
157C2×C4×D53C2×D412yes—
158C2×C4×D53C2×D412yes—
159C2×C4×D53C2×Q812yes—
160C2×C4×D53#4112yes—
161C2×C4×D53#4112yes—
162C2×Dic106—21no—
163C2×Dic106—21no—
164C2×Dic106—21no—
165C2×Dic106—24no—
166C2×Dic106—24no—
167C2×Dic106—24no—
168C2×Dic106C22×C412yes—
169C2×Dic106C2×D412yes—
170C2×Dic106C2×Q812yes—
171C2×C8×C53—22no—
172C2×C8×C53—22no—
173C2×C8×C53—22no—
174C4⋊C4×C53—24no—
175C4⋊C4×C53—21no—
176C4⋊C4×C53—22no—
177C4⋊C4×C53—22no—
178C4⋊C4×C53—42no—
179C4⋊C4×C53—42no—
180C4⋊C4×C53C22×C412yes—
181C42×C53—24no—
182C42×C53C22×C412yes—
183C42×C53C2×D412yes—
184C4⋊Dic53—21no—
185C4⋊Dic53—24no—
186C4⋊Dic53—22no—
187C4⋊Dic53—22no—
188C4⋊Dic53—42no—
189C4⋊Dic53—42no—
190C4⋊Dic53C22×C412yes—
191#11719919—24no—
192#11719919—24no—
193#11719919—24no—
194#11719919—44no—
195#11719919—44no—
196#11719919—44no—
197C4×Dic53—24no—
198C4×Dic53—24no—
199C4×Dic53—24no—
200C4×Dic53C22⋊C414yes—
201C4×Dic53C22⋊C414yes—
202C4×Dic53C22×C412yes—
203C4×Dic53C2×D412yes—
204C2×#88322—22no—
205C2×#88322—22no—
206C2×#88322—22no—
207C8×D53—48no—
208C23×C53C22×D411yescharacteristic
209C22×D53C2×C22⋊C412yes—
210C22×D53#12112yes—
211C22×D53C22⋊Q812yes—
212C22×D53C2×SD1612yes—
213C22×D53C2×Q1612yes—
214C22×D53AGL(1,ℤ/8)12yes—
215C22×D53#13712yes—
216Q8×C53—22no—
217Q8×C53—22no—
218Q8×C53—22no—
219Q8×C53C2×C4⋊C414yescomplemented
220C2×C4×C53—22no—
221C2×C4×C53—22no—
222C2×C4×C53—22no—
223C2×C4×C53—21no—
224C2×C4×C53—21no—
225C2×C4×C53—22no—
226C2×C4×C53—41no—
227C2×C4×C53—41no—
228C2×C4×C53—42no—
229C2×C4×C53—42no—
230C2×C4×C53—42no—
231C2×C4×C53C2×C22⋊C412yes—
232C2×C4×C53C2×C22⋊C412yes—
233C2×C4×C53C2×SD1612yes—
234C2×C4×C53C2×Q1612yes—
235C2×C4×C53C23×C411yescharacteristic
236C2×C4×C53C22×D411yescharacteristic
237C2×C4×C53C22×D411yescharacteristic
238C2×C4×C53C22×Q811yescharacteristic
239C2×C4×C53C2×#4111yescharacteristic
240C2×C4×C53C2×#4111yescharacteristic
241C2×Dic53—22no—
242C2×Dic53—22no—
243C2×Dic53—22no—
244C2×Dic53—21no—
245C2×Dic53—21no—
246C2×Dic53—22no—
247C2×Dic53—41no—
248C2×Dic53—41no—
249C2×Dic53—42no—
250C2×Dic53—42no—
251C2×Dic53—42no—
252C2×Dic53C2×C22⋊C412yes—
253C2×Dic53C2×C22⋊C412yes—
254C2×Dic53C2×C22⋊C412yes—
255C2×Dic53#12112yes—
256C2×Dic53C22⋊Q812yes—
257C2×Dic53C2×SD1612yes—
258C2×Dic53C2×Q1612yes—
259C4×D53—216no—
260C4×D53—28no—
261C4×D53—48no—
262C4×D53—88no—
263C4×D53#103116yescomplemented
264C4×D53C2×C4⋊C414yes—
265C4×D53#11714yes—
266C4×D53#12114yes—
267C4×D53C22⋊Q814yes—
268Dic106—22no—
269Dic106—22no—
270Dic106—22no—
271Dic106—216no—
272Dic106C2×C4⋊C414yescomplemented
273C8×C53—42no—
274#88322—42no—
275C2×#1803—531nomaximal, Sylow
276C22×C53C22×C22⋊C411yescharacteristic
277C22×C53C2×#12111yescharacteristic
278C22×C53C2×C22⋊Q811yescharacteristic
279C22×C53C22×SD1611yescharacteristic
280C22×C53C22×Q1611yescharacteristic
281C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
282C22×C53C2×#13711yescharacteristic
283C2×D53#38014yes—
284C2×D53C2×#10314yes—
285C2×D53#41714yes—
286C2×D53#46014yes—
287C2×D53#46214yes—
288C2×D53#47514yes—
289C2×D53#47714yes—
290C4×C53—22no—
291C4×C53—24no—
292C4×C53—42no—
293C4×C53—82no—
294C4×C53C2×#10314yescomplemented
295C4×C53C22×C4⋊C411yescharacteristic
296C4×C53C2×#11711yescharacteristic
297C4×C53C2×#12111yescharacteristic
298C4×C53C2×C22⋊Q811yescharacteristic
299Dic53—22no—
300Dic53—24no—
301Dic53—42no—
302Dic53—82no—
303Dic53#38014yes—
304Dic53C2×#10314yescomplemented
305C22×C4×Q8—531no—
306C22×C4⋊C8—531no—
307C22×#103—532no—
308C2×#384—531no—
309#1803—531no—
310#1803—532no—
311#1803—531no—
312#1803—532no—
313#1803—531no—
314#1803D5311yescharacteristic, direct product factor
315C2×#340—532no—
316C2×C53C2×#38011yescharacteristic
317C2×C53C22×#10311yescharacteristic
318C2×C53C2×#41711yescharacteristic
319C2×C53C2×#46011yescharacteristic
320C2×C53C2×#46211yescharacteristic
321C2×C53C2×#47511yescharacteristic
322C2×C53C2×#47711yescharacteristic
323D53#180318yesdirect product factor
324C23×Q8—531no—
325C23×C8—532no—
326C2×C4×Q8—531no—
327C2×C4×Q8—538no—
328C2×C4×Q8—531no—
329C2×C4×Q8—531no—
330C2×C4×Q8—1061no—
331C2×C4×Q8—1061no—
332C2×C4×Q8—1064no—
333C2×C4×Q8C2×D5311yescharacteristic
334C22×C4⋊C4—531no—
335C22×C4⋊C4—532no—
336C22×C4⋊C4—1061no—
337C22×C4⋊C4—1061no—
338C22×C4⋊C4—1061no—
339C22×C42—531no—
340C22×C42—1061no—
341C2×C4⋊C8—532no—
342C2×C4⋊C8—531no—
343C2×C4⋊C8—1064no—
344C2×C4⋊C8C2×D5311yescharacteristic
345C2×#103—532no—
346C2×#103—532no—
347C2×#103—532no—
348C2×#103—1068no—
349C2×#103C2×D5312yes—
350#384—532no—
351#384—531no—
352#384—1061no—
353#384—1061no—
354#384C2×D5311yescharacteristic
355#340—532no—
356#340—532no—
357#340—532no—
358#340C2×D5312yes—
359C53C2×#180311yescharacteristic, complemented, Sylow
360C22×Q8—534no—
361C22×Q8—534no—
362C22×Q8—534no—
363C22×Q8—531no—
364C22×Q8—1061no—
365C22×Q8—1062no—
366C22×Q8—1062no—
367C22×Q8—1062no—
368C22×Q8C22×D5311yescharacteristic
369C23×C4—531no—
370C23×C4—531no—
371C23×C4—531no—
372C23×C4—1061no—
373C23×C4—1061no—
374C23×C4—1062no—
375C22×C8—532no—
376C22×C8—1064no—
377C22×C8—1064no—
378C22×C8—1064no—
379C22×C8C22×D5312yes—
380C4×Q8—22no—
381C4×Q8—534no—
382C4×Q8—534no—
383C4×Q8—534no—
384C4×Q8—1068no—
385C4×Q8—1068no—
386C4×Q8—1062no—
387C4×Q8—1062no—
388C4×Q8—1062no—
389C4×Q8C4×D5314yescomplemented
390C2×C4⋊C4—21no—
391C2×C4⋊C4—21no—
392C2×C4⋊C4—21no—
393C2×C4⋊C4—531no—
394C2×C4⋊C4—534no—
395C2×C4⋊C4—532no—
396C2×C4⋊C4—1062no—
397C2×C4⋊C4—1068no—
398C2×C4⋊C4—1064no—
399C2×C4⋊C4—1064no—
400C2×C4⋊C4—1061no—
401C2×C4⋊C4—1061no—
402C2×C4⋊C4—1061no—
403C2×C4⋊C4—1061no—
404C2×C4⋊C4—1062no—
405C2×C4⋊C4—1062no—
406C2×C4⋊C4—1062no—
407C2×C4⋊C4—1061no—
408C2×C4⋊C4—1061no—
409C2×C4⋊C4—2124no—
410C2×C4⋊C4—2124no—
411C2×C4⋊C4C22×D5312yes—
412C2×C4⋊C4C22×D5311yescharacteristic
413C2×C42—21no—
414C2×C42—531no—
415C2×C42—531no—
416C2×C42—531no—
417C2×C42—534no—
418C2×C42—534no—
419C2×C42—1068no—
420C2×C42—1061no—
421C2×C42—1061no—
422C2×C42—1061no—
423C2×C42C22×D5311yescharacteristic
424C4⋊C8—22no—
425C4⋊C8—1064no—
426C4⋊C8—1062no—
427#103—24no—
428#103—1064no—
429#103—1064no—
430#103—1064no—
431C24—531no—
432C2×Q8—21no—
433C2×Q8—21no—
434C2×Q8—21no—
435C2×Q8—5316no—
436C2×Q8—532no—
437C2×Q8—532no—
438C2×Q8—532no—
439C2×Q8—1064no—
440C2×Q8—1064no—
441C2×Q8—1064no—
442C2×Q8—1068no—
443C2×Q8—1068no—
444C2×Q8—1068no—
445C2×Q8—1061no—
446C2×Q8—1061no—
447C2×Q8—1061no—
448C2×Q8C2×C4×D5312yes—
449C2×Q8D4×D5312yes—
450C2×Q8Q8×D5312yes—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—22no—
454C22×C4—531no—
455C22×C4—531no—
456C22×C4—534no—
457C22×C4—534no—
458C22×C4—534no—
459C22×C4—532no—
460C22×C4—532no—
461C22×C4—532no—
462C22×C4—532no—
463C22×C4—532no—
464C22×C4—532no—
465C22×C4—531no—
466C22×C4—534no—
467C22×C4—1064no—
468C22×C4—1062no—
469C22×C4—1062no—
470C22×C4—1064no—
471C22×C4—1064no—
472C22×C4—1064no—
473C22×C4—1061no—
474C22×C4—1061no—
475C22×C4—1062no—
476C22×C4—2124no—
477C22×C4—2124no—
478C22×C4—2124no—
479C22×C4—2122no—
480C22×C4—2122no—
481C22×C4D4×D5311yescharacteristic
482C22×C4D4×D5311yescharacteristic
483C22×C4C23×D5311yescharacteristic
484C2×C8—22no—
485C2×C8—22no—
486C2×C8—22no—
487C2×C8—1062no—
488C2×C8—1062no—
489C2×C8—1062no—
490C2×C8—2128no—
491C4⋊C4—24no—
492C4⋊C4—21no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—42no—
496C4⋊C4—42no—
497C4⋊C4—532no—
498C4⋊C4—1064no—
499C4⋊C4—1064no—
500C4⋊C4—1064no—
501C4⋊C4—1064no—
502C4⋊C4—1061no—
503C4⋊C4—1062no—
504C4⋊C4—1062no—
505C4⋊C4—2122no—
506C4⋊C4—2124no—
507C4⋊C4—2124no—
508C4⋊C4—2124no—
509C4⋊C4—2122no—
510C4⋊C4C2×C4×D5312yes—
511C42—24no—
512C42—534no—
513C42—534no—
514C42—532no—
515C42—532no—
516C42—1064no—
517C42—1064no—
518C42—1064no—
519C42C2×C4×D5312yes—
520C42D4×D5312yes—
521C23—532no—
522C23—532no—
523C23—532no—
524C23—532no—
525C23—532no—
526C23—532no—
527C23—532no—
528C23C2×D4×D5311yescharacteristic
529Q8—22no—
530Q8—22no—
531Q8—22no—
532Q8—534no—
533Q8—10616no—
534Q8—1062no—
535Q8—1062no—
536Q8—1062no—
537Q8C4⋊C4×D5314yescomplemented
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—22no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—532no—
550C2×C4—532no—
551C2×C4—532no—
552C2×C4—532no—
553C2×C4—532no—
554C2×C4—532no—
555C2×C4—534no—
556C2×C4—534no—
557C2×C4—534no—
558C2×C4—534no—
559C2×C4—5316no—
560C2×C4—532no—
561C2×C4—1068no—
562C2×C4—10616no—
563C2×C4—1062no—
564C2×C4—1062no—
565C2×C4—1062no—
566C2×C4—1061no—
567C2×C4—1061no—
568C2×C4—1062no—
569C2×C4—4248no—
570C2×C4—2128no—
571C2×C4—2122no—
572C2×C4—2122no—
573C2×C4—2122no—
574C2×C4—2121no—
575C2×C4—2121no—
576C2×C4C22⋊C4×D5312yes—
577C2×C4C22⋊C4×D5312yes—
578C2×C4SD16×D5312yes—
579C2×C4Q16×D5312yes—
580C2×C4C22×C4×D5311yescharacteristic
581C2×C4C2×D4×D5311yescharacteristic
582C2×C4C2×D4×D5311yescharacteristic
583C2×C4C2×Q8×D5311yescharacteristic
584C2×C4#41×D5311yescharacteristic
585C2×C4#41×D5311yescharacteristic
586C8—42no—
587C8—2122no—
588C22—534no—
589C22—534no—
590C22—534no—
591C22—534no—
592C22—534no—
593C22—534no—
594C22—534no—
595C22C2×C22⋊C4×D5311yescharacteristic
596C22#121×D5311yescharacteristic
597C22C22⋊Q8×D5311yescharacteristic
598C22C2×SD16×D5311yescharacteristic
599C22C2×Q16×D5311yescharacteristic
600C22AGL(1,ℤ/8)×D5311yescharacteristic
601C22#137×D5311yescharacteristic
602C4—22no—
603C4—24no—
604C4—42no—
605C4—82no—
606C4—534no—
607C4—534no—
608C4—1062no—
609C4—1064no—
610C4—4242no—
611C4—2122no—
612C4#103×D5314yescomplemented
613C4C2×C4⋊C4×D5311yescharacteristic
614C4#117×D5311yescharacteristic
615C4#121×D5311yescharacteristic
616C4C22⋊Q8×D5311yescharacteristic
617C2—538no—
618C2#380×D5311yescharacteristic
619C2C2×#103×D5311yescharacteristic
620C2#417×D5311yescharacteristic
621C2#460×D5311yescharacteristic
622C2#462×D5311yescharacteristic
623C2#475×D5311yescharacteristic
624C2#477×D5311yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#1803C2
socleC23×C53C22×D4
prime radicalC23×D53C2×D4
cosocleC22×C4×C53C24
hypercenter#1803D53
hypocenterC53C2×#1803
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1803

Complements

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

NH ≅ G/Ntype
5. #20274721C2semidirect
6. #20274720C2semidirect
7. #20274719C2semidirect
8. #20274718C2semidirect
9. #20274717C2semidirect
11. C53×#1803C2semidirect
20. C4×Q8×D53C4semidirect
66. Q8×Dic53C2×C4semidirect
69. C4×Dic106C2×C4semidirect
72. #13133182C2×C4semidirect
108. C4×Q8×C53C2×C4semidirect
140. Q8×D53C4⋊C4semidirect
219. Q8×C53C2×C4⋊C4semidirect
263. C4×D53#103semidirect
272. Dic106C2×C4⋊C4semidirect
294. C4×C53C2×#103semidirect
304. Dic53C2×#103semidirect
314. #1803D53direct (N × H)
323. D53#1803direct (N × H)
359. C53C2×#1803semidirect
389. C4×Q8C4×D53semidirect
537. Q8C4⋊C4×D53semidirect
612. C4#103×D53semidirect

Maximal subgroups

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

subgroupindex
C2×C4×Q8×D532
C2×C4⋊C8×D532
C2×#103×D532
#384×D532
#202747212
#202747202
#202747192
#202747182
#202747172
#340×D532
C53×#18032
C2×#180353