Finite Group Databasebeta

← D53×#1805 #27383553

Subgroups — D53×#1805 #27383553

Order 13568.

1510 conjugacy classes of subgroups.

The subgroup lattice has 576 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#382×D53C211yesmaximal, characteristic
5#20274731C211yesmaximal, characteristic, complemented
6#20274730C211yesmaximal, characteristic, complemented
7#20274729C212yesmaximal, complemented
8#20274728C212yesmaximal, complemented
9#20274727C211yesmaximal, characteristic, complemented
10#340×D53C212yesmaximal
11C53×#1805C211yesmaximal, 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×D53C2212yes—
23C2×C4⋊C4×D53C2211yescharacteristic
24C2×C42×D53—21no—
25C2×C42×D53C2211yescharacteristic
26C2×C4⋊#88322C2211yescharacteristic
27C2×#13133146C2212yes—
28C4⋊C8×D53—24no—
29C2×#13133142C2212yes—
30C2×#13133141C2212yes—
31C2×#13133140C2212yes—
32#103×D53—28no—
33#16810489C2211yescharacteristic
34#16810488C2212yes—
35#16810487—21no—
36#16810487—21no—
37#95×D53—21no—
38#95×D53—21no—
39#16810386C2212yes—
40#16810385C2212yes—
41#16810385C2212yes—
42C2×C4×Q8×C53C2211yescharacteristic
43C2×C4⋊C8×C53C2211yescharacteristic
44C2×#103×C53C2212yes—
45C53×#382C2211yescharacteristic
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⋊Dic53C2312yes—
75C2×C4⋊Dic53C2311yescharacteristic
76C2×#11719919—21no—
77C2×#11719919—21no—
78C2×#11719919—21no—
79C2×#11719919—22no—
80C4⋊C4×D53—28no—
81C4⋊C4×D53—22no—
82C4⋊C4×D53—24no—
83C4⋊C4×D53—24no—
84C4⋊C4×D53C2×C414yes—
85C2×C4×Dic53—21no—
86C2×C4×Dic53—21no—
87C2×C4×Dic53—21no—
88C2×C4×Dic53D411yescharacteristic
89C2×C4×Dic53D411yescharacteristic
90C2×C4×Dic53C2311yescharacteristic
91C42×D53—28no—
92C42×D53C2×C414yes—
93C42×D53D414yes—
94C4⋊#88322—22no—
95#13133146—24no—
96#13133142—24no—
97#13133141—24no—
98#13133140—24no—
99#13133124—21no—
100#13133124—21no—
101#13133124—22no—
102#13133124—22no—
103C22×Q8×C53C2311yescharacteristic
104C22×C8×C53C2312yes—
105C4×Q8×C53—22no—
106C4×Q8×C53C2×C414yescomplemented
107C2×C4⋊C4×C53—21no—
108C2×C4⋊C4×C53C2312yes—
109C2×C4⋊C4×C53C2311yescharacteristic
110C2×C42×C53—21no—
111C2×C42×C53C2311yescharacteristic
112C4⋊C8×C53—22no—
113#103×C53—24no—
114#95×C53—21no—
115#95×C53—21no—
116C23×D53C2×D411yescharacteristic
117C2×Q8×C53—21no—
118C2×Q8×C53—21no—
119C2×Q8×C53—21no—
120C2×Q8×C53C22×C412yes—
121C2×Q8×C53C2×D412yes—
122C2×Q8×C53C2×Q812yes—
123C22×C4×C53—21no—
124C22×C4×C53—21no—
125C22×C4×C53—22no—
126C22×C4×C53C2×D411yescharacteristic
127C22×C4×C53C2×D411yescharacteristic
128C22×C4×C53C2411yescharacteristic
129C22×Dic53—21no—
130C22×Dic53—21no—
131C22×Dic53—22no—
132C22×Dic53C2×D411yescharacteristic
133C22×Dic53C2×D411yescharacteristic
134C22×Dic53C2×D411yescharacteristic
135Q8×D53—28no—
136Q8×D53—28no—
137Q8×D53—28no—
138Q8×D53C4⋊C4116yes—
139C2×C4×D53—24no—
140C2×C4×D53—24no—
141C2×C4×D53—24no—
142C2×C4×D53—24no—
143C2×C4×D53—24no—
144C2×C4×D53—24no—
145C2×C4×D53—44no—
146C2×C4×D53—44no—
147C2×C4×D53—44no—
148C2×C4×D53—44no—
149C2×C4×D53C22⋊C414yes—
150C2×C4×D53C22⋊C414yes—
151C2×C4×D53C22×C412yes—
152C2×C4×D53C2×D412yes—
153C2×C4×D53C2×D412yes—
154C2×C4×D53C2×Q812yes—
155C2×C4×D53#4112yes—
156C2×C4×D53#4112yes—
157C2×Dic106—21no—
158C2×Dic106—21no—
159C2×Dic106—21no—
160C2×Dic106—24no—
161C2×Dic106—24no—
162C2×Dic106—24no—
163C2×Dic106C22×C412yes—
164C2×Dic106C2×D412yes—
165C2×Dic106C2×Q812yes—
166C2×C8×C53—22no—
167C2×C8×C53—22no—
168C2×C8×C53—22no—
169C4⋊C4×C53—24no—
170C4⋊C4×C53—21no—
171C4⋊C4×C53—22no—
172C4⋊C4×C53—22no—
173C4⋊C4×C53C22×C412yes—
174C42×C53—24no—
175C42×C53C22×C412yes—
176C42×C53C2×D412yes—
177C4⋊Dic53—21no—
178C4⋊Dic53—24no—
179C4⋊Dic53—22no—
180C4⋊Dic53—22no—
181C4⋊Dic53C22×C412yes—
182#11719919—24no—
183#11719919—24no—
184#11719919—24no—
185#11719919—44no—
186C4×Dic53—24no—
187C4×Dic53—24no—
188C4×Dic53—24no—
189C4×Dic53C22⋊C414yes—
190C4×Dic53C22⋊C414yes—
191C4×Dic53C22×C412yes—
192C4×Dic53C2×D412yes—
193C2×#88322—22no—
194C2×#88322—22no—
195C2×#88322—22no—
196C8×D53—48no—
197C23×C53C22×D411yescharacteristic
198C22×D53C2×C22⋊C412yes—
199C22×D53#12112yes—
200C22×D53C22⋊Q812yes—
201C22×D53#13514yes—
202C22×D53#13714yes—
203Q8×C53—22no—
204Q8×C53—22no—
205Q8×C53—22no—
206Q8×C53C2×C4⋊C414yes—
207C2×C4×C53—22no—
208C2×C4×C53—22no—
209C2×C4×C53—22no—
210C2×C4×C53—22no—
211C2×C4×C53—22no—
212C2×C4×C53—22no—
213C2×C4×C53—42no—
214C2×C4×C53—42no—
215C2×C4×C53—42no—
216C2×C4×C53—42no—
217C2×C4×C53C2×C22⋊C412yes—
218C2×C4×C53C2×C22⋊C412yes—
219C2×C4×C53C23×C411yescharacteristic
220C2×C4×C53C22×D411yescharacteristic
221C2×C4×C53C22×D411yescharacteristic
222C2×C4×C53C22×Q811yescharacteristic
223C2×C4×C53C2×#4111yescharacteristic
224C2×C4×C53C2×#4111yescharacteristic
225C2×Dic53—22no—
226C2×Dic53—22no—
227C2×Dic53—22no—
228C2×Dic53—22no—
229C2×Dic53—22no—
230C2×Dic53—22no—
231C2×Dic53—42no—
232C2×Dic53—42no—
233C2×Dic53—42no—
234C2×Dic53—42no—
235C2×Dic53C2×C22⋊C412yes—
236C2×Dic53C2×C22⋊C412yes—
237C2×Dic53C2×C22⋊C412yes—
238C2×Dic53#12112yes—
239C2×Dic53C22⋊Q812yes—
240C4×D53—216no—
241C4×D53—28no—
242C4×D53—28no—
243C4×D53—48no—
244C4×D53—88no—
245C4×D53C2×C4⋊C414yes—
246C4×D53#11714yes—
247C4×D53#12114yes—
248C4×D53C22⋊Q814yes—
249Dic106—22no—
250Dic106—22no—
251Dic106—22no—
252Dic106—216no—
253Dic106C2×C4⋊C414yes—
254C8×C53—42no—
255#88322—42no—
256C2×#1805—531nomaximal, Sylow
257C22×C53C22×C22⋊C411yescharacteristic
258C22×C53C2×#12111yescharacteristic
259C22×C53C2×C22⋊Q811yescharacteristic
260C22×C53C2×#13512yes—
261C22×C53C2×#13712yes—
262C2×D53#38014yes—
263C2×D53#41614yes—
264C2×D53#41914yes—
265C2×D53#46418yes—
266C2×D53#47918yes—
267C4×C53—22no—
268C4×C53—22no—
269C4×C53—24no—
270C4×C53—42no—
271C4×C53—82no—
272C4×C53C22×C4⋊C411yescharacteristic
273C4×C53C2×#11711yescharacteristic
274C4×C53C2×#12111yescharacteristic
275C4×C53C2×C22⋊Q811yescharacteristic
276Dic53—22no—
277Dic53—22no—
278Dic53—24no—
279Dic53—42no—
280Dic53—82no—
281Dic53#38014yes—
282C22×C4×Q8—531no—
283C22×C4⋊C8—531no—
284C22×#103—532no—
285C2×#382—531no—
286#1805—531no—
287#1805—532no—
288#1805—531no—
289#1805—532no—
290#1805—531no—
291#1805D5311yescharacteristic, direct product factor
292C2×#340—532no—
293C2×C53C2×#38011yescharacteristic
294C2×C53C2×#41611yescharacteristic
295C2×C53C2×#41911yescharacteristic
296C2×C53C2×#46412yes—
297C2×C53C2×#47912yes—
298D53#180518yesdirect product factor
299C23×Q8—531no—
300C23×C8—532no—
301C2×C4×Q8—531no—
302C2×C4×Q8—538no—
303C2×C4×Q8—531no—
304C2×C4×Q8—531no—
305C2×C4×Q8—1061no—
306C2×C4×Q8—1061no—
307C2×C4×Q8—1064no—
308C2×C4×Q8C2×D5311yescharacteristic
309C22×C4⋊C4—531no—
310C22×C4⋊C4—532no—
311C22×C4⋊C4—1061no—
312C22×C42—531no—
313C22×C42—1061no—
314C2×C4⋊C8—532no—
315C2×C4⋊C8—531no—
316C2×C4⋊C8—1064no—
317C2×C4⋊C8C2×D5311yescharacteristic
318C2×#103—532no—
319C2×#103—532no—
320C2×#103—532no—
321C2×#103—1068no—
322C2×#103C2×D5312yes—
323#382—532no—
324#382—531no—
325#382—1061no—
326#382—1061no—
327#382C2×D5311yescharacteristic
328C2×#95—1061no—
329C2×#95—1061no—
330#340—532no—
331#340—532no—
332#340—532no—
333#340C2×D5312yes—
334C53C2×#180511yescharacteristic, complemented, Sylow
335C22×Q8—534no—
336C22×Q8—534no—
337C22×Q8—534no—
338C22×Q8—531no—
339C22×Q8—1061no—
340C22×Q8—1062no—
341C22×Q8—1062no—
342C22×Q8—1062no—
343C22×Q8C22×D5311yescharacteristic
344C23×C4—531no—
345C23×C4—531no—
346C23×C4—531no—
347C23×C4—1061no—
348C23×C4—1061no—
349C23×C4—1062no—
350C22×C8—532no—
351C22×C8—1064no—
352C22×C8—1064no—
353C22×C8—1064no—
354C22×C8C22×D5312yes—
355C4×Q8—22no—
356C4×Q8—534no—
357C4×Q8—534no—
358C4×Q8—534no—
359C4×Q8—1068no—
360C4×Q8—1068no—
361C4×Q8—1062no—
362C4×Q8—1062no—
363C4×Q8—1062no—
364C4×Q8C4×D5314yescomplemented
365C2×C4⋊C4—21no—
366C2×C4⋊C4—531no—
367C2×C4⋊C4—534no—
368C2×C4⋊C4—532no—
369C2×C4⋊C4—1062no—
370C2×C4⋊C4—1068no—
371C2×C4⋊C4—1064no—
372C2×C4⋊C4—1064no—
373C2×C4⋊C4—1061no—
374C2×C4⋊C4—1061no—
375C2×C4⋊C4—1061no—
376C2×C4⋊C4—1061no—
377C2×C4⋊C4—1062no—
378C2×C4⋊C4C22×D5312yes—
379C2×C4⋊C4C22×D5311yescharacteristic
380C2×C42—21no—
381C2×C42—531no—
382C2×C42—531no—
383C2×C42—531no—
384C2×C42—534no—
385C2×C42—534no—
386C2×C42—1068no—
387C2×C42—1061no—
388C2×C42—1061no—
389C2×C42—1061no—
390C2×C42C22×D5311yescharacteristic
391C4⋊C8—22no—
392C4⋊C8—1064no—
393C4⋊C8—1062no—
394#103—24no—
395#103—1064no—
396#103—1064no—
397#103—1064no—
398#95—21no—
399#95—21no—
400#95—1062no—
401#95—1062no—
402#95—1061no—
403#95—1061no—
404C24—531no—
405C2×Q8—21no—
406C2×Q8—21no—
407C2×Q8—21no—
408C2×Q8—5316no—
409C2×Q8—532no—
410C2×Q8—532no—
411C2×Q8—532no—
412C2×Q8—1064no—
413C2×Q8—1064no—
414C2×Q8—1064no—
415C2×Q8—1068no—
416C2×Q8—1068no—
417C2×Q8—1068no—
418C2×Q8—1061no—
419C2×Q8—1061no—
420C2×Q8—1061no—
421C2×Q8C2×C4×D5312yes—
422C2×Q8D4×D5312yes—
423C2×Q8Q8×D5312yes—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—22no—
427C22×C4—531no—
428C22×C4—532no—
429C22×C4—532no—
430C22×C4—532no—
431C22×C4—532no—
432C22×C4—532no—
433C22×C4—532no—
434C22×C4—531no—
435C22×C4—531no—
436C22×C4—534no—
437C22×C4—534no—
438C22×C4—1064no—
439C22×C4—1064no—
440C22×C4—1064no—
441C22×C4—1064no—
442C22×C4—1064no—
443C22×C4—1064no—
444C22×C4—1061no—
445C22×C4—1061no—
446C22×C4—1062no—
447C22×C4—2124no—
448C22×C4—2124no—
449C22×C4—2124no—
450C22×C4—2124no—
451C22×C4D4×D5311yescharacteristic
452C22×C4D4×D5311yescharacteristic
453C22×C4C23×D5311yescharacteristic
454C2×C8—22no—
455C2×C8—22no—
456C2×C8—22no—
457C2×C8—1062no—
458C2×C8—1062no—
459C2×C8—1062no—
460C2×C8—2128no—
461C4⋊C4—24no—
462C4⋊C4—21no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—532no—
466C4⋊C4—1064no—
467C4⋊C4—1064no—
468C4⋊C4—1064no—
469C4⋊C4—1064no—
470C4⋊C4—1061no—
471C4⋊C4—1062no—
472C4⋊C4—1062no—
473C4⋊C4—2124no—
474C4⋊C4C2×C4×D5312yes—
475C42—24no—
476C42—534no—
477C42—534no—
478C42—532no—
479C42—532no—
480C42—1064no—
481C42—1064no—
482C42—1064no—
483C42C2×C4×D5312yes—
484C42D4×D5312yes—
485C23—532no—
486C23—532no—
487C23—532no—
488C23—534no—
489C23—534no—
490C23C2×D4×D5311yescharacteristic
491Q8—22no—
492Q8—22no—
493Q8—22no—
494Q8—534no—
495Q8—10616no—
496Q8—1062no—
497Q8—1062no—
498Q8—1062no—
499Q8C4⋊C4×D5314yes—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—532no—
511C2×C4—532no—
512C2×C4—532no—
513C2×C4—534no—
514C2×C4—534no—
515C2×C4—534no—
516C2×C4—534no—
517C2×C4—532no—
518C2×C4—532no—
519C2×C4—1068no—
520C2×C4—1068no—
521C2×C4—10616no—
522C2×C4—1062no—
523C2×C4—1062no—
524C2×C4—1062no—
525C2×C4—1062no—
526C2×C4—1062no—
527C2×C4—1062no—
528C2×C4—4248no—
529C2×C4—2128no—
530C2×C4—2122no—
531C2×C4—2122no—
532C2×C4—2122no—
533C2×C4—2122no—
534C2×C4C22⋊C4×D5312yes—
535C2×C4C22⋊C4×D5312yes—
536C2×C4C22×C4×D5311yescharacteristic
537C2×C4C2×D4×D5311yescharacteristic
538C2×C4C2×D4×D5311yescharacteristic
539C2×C4C2×Q8×D5311yescharacteristic
540C2×C4#41×D5311yescharacteristic
541C2×C4#41×D5311yescharacteristic
542C8—42no—
543C8—2122no—
544C22—534no—
545C22—538no—
546C22—534no—
547C22—534no—
548C22—538no—
549C22C2×C22⋊C4×D5311yescharacteristic
550C22#121×D5311yescharacteristic
551C22C22⋊Q8×D5311yescharacteristic
552C22#135×D5312yes—
553C22#137×D5312yes—
554C4—22no—
555C4—22no—
556C4—24no—
557C4—42no—
558C4—82no—
559C4—534no—
560C4—1062no—
561C4—1062no—
562C4—1064no—
563C4—4242no—
564C4—2122no—
565C4C2×C4⋊C4×D5311yescharacteristic
566C4#117×D5311yescharacteristic
567C4#121×D5311yescharacteristic
568C4C22⋊Q8×D5311yescharacteristic
569C2—538no—
570C2#380×D5311yescharacteristic
571C2#416×D5311yescharacteristic
572C2#419×D5311yescharacteristic
573C2#464×D5312yes—
574C2#479×D5312yes—
subgroupG/H
centerC23C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#1805C2
socleC23×C53C22×D4
prime radicalC23×D53C2×D4
cosocleC22×C4×C53C24
hypercenter#1805D53
hypocenterC53C2×#1805
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1805

Complements

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

NH ≅ G/Ntype
5. #20274731C2semidirect
6. #20274730C2semidirect
7. #20274729C2semidirect
8. #20274728C2semidirect
9. #20274727C2semidirect
11. C53×#1805C2semidirect
20. C4×Q8×D53C4semidirect
66. Q8×Dic53C2×C4semidirect
69. C4×Dic106C2×C4semidirect
72. #13133182C2×C4semidirect
106. C4×Q8×C53C2×C4semidirect
291. #1805D53direct (N × H)
298. D53#1805direct (N × H)
334. C53C2×#1805semidirect
364. C4×Q8C4×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
#382×D532
#202747312
#202747302
#202747292
#202747282
#202747272
#340×D532
C53×#18052
C2×#180553