Finite Group Databasebeta

← D53×#1949 #27387542

Subgroups — D53×#1949 #27387542

Order 13568.

1142 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#107×D53C211yesmaximal, characteristic
2C2×#106×D53C211yesmaximal, characteristic
3C2×C4⋊C8×D53C211yesmaximal, characteristic
4#382×D53C211yesmaximal, characteristic
5C4×C4⋊C4×D53C211yesmaximal, characteristic
6#20275486C211yesmaximal, characteristic, complemented
7#20275485C211yesmaximal, characteristic, complemented
8#20275484C211yesmaximal, characteristic, complemented
9#20275483C211yesmaximal, characteristic, complemented
10#20275482C211yesmaximal, characteristic, complemented
11#20275481C211yesmaximal, characteristic, complemented
12#20275480C211yesmaximal, characteristic, complemented
13#340×D53C211yesmaximal, characteristic
14#340×D53C211yesmaximal, characteristic
15C53×#1949C211yesmaximal, characteristic, complemented
16C22×C8×D53C2211yescharacteristic
17C22×C8×D53C2211yescharacteristic
18C2×C4⋊C4×D53C2211yescharacteristic
19C2×C4⋊C4×D53C2211yescharacteristic
20C2×C4⋊C4×D53C2211yescharacteristic
21C2×C4⋊C4×D53C2211yescharacteristic
22C2×C42×D53—21no—
23C2×C42×D53C2211yescharacteristic
24C2×#13133151C2211yescharacteristic
25C2×#13133150C2211yescharacteristic
26C2×#13133150C2211yescharacteristic
27#107×D53—24no—
28C2×#13133149C2211yescharacteristic
29C2×#13133148C2211yescharacteristic
30C2×#13133148C2211yescharacteristic
31#106×D53—24no—
32C2×C4⋊#88322C2211yescharacteristic
33C2×#13133146C2211yescharacteristic
34C2×#13133146C2211yescharacteristic
35C4⋊C8×D53C418yescomplemented
36#16810489C2211yescharacteristic
37#16810488C2211yescharacteristic
38#16810488C2211yescharacteristic
39#16810487—21no—
40#16810487—21no—
41C4⋊C4×Dic53C2211yescharacteristic
42C4⋊C4×Dic53C2211yescharacteristic
43C4×#11719919—21no—
44C4×C4⋊Dic53C2211yescharacteristic
45#16810474—21no—
46#95×D53—21no—
47#95×D53—21no—
48#95×D53—21no—
49#16810386C2211yescharacteristic
50#16810386C2211yescharacteristic
51#16810385C2211yescharacteristic
52#16810385C2211yescharacteristic
53#16810385C2211yescharacteristic
54#16810385C2211yescharacteristic
55C2×#107×C53C2211yescharacteristic
56C2×#106×C53C2211yescharacteristic
57C2×C4⋊C8×C53C2211yescharacteristic
58C53×#382C2211yescharacteristic
59C4×C4⋊C4×C53C2211yescharacteristic
60C53×#340C2211yescharacteristic
61C53×#340C2211yescharacteristic
62C22×C4×D53—21no—
63C22×C4×D53—21no—
64C22×C4×D53—21no—
65C22×C4×D53—21no—
66C22×C4×D53D411yescharacteristic
67C22×C4×D53D411yescharacteristic
68C22×C4×D53C2311yescharacteristic
69C22×#88322C2311yescharacteristic
70C22×#88322C2311yescharacteristic
71C2×C8×D53—22no—
72C2×C8×D53—22no—
73C2×C8×D53—22no—
74C2×C8×D53—22no—
75C2×C8×D53C2×C414yes—
76C2×C8×D53C2×C414yes—
77C2×C4⋊Dic53C2311yescharacteristic
78C2×C4⋊Dic53C2311yescharacteristic
79C2×C4⋊Dic53C2311yescharacteristic
80C2×C4⋊Dic53C2311yescharacteristic
81C2×#11719919—21no—
82C2×#11719919—21no—
83C2×#11719919—21no—
84C2×#11719919—21no—
85C4⋊C4×D53—22no—
86C4⋊C4×D53—22no—
87C4⋊C4×D53—22no—
88C4⋊C4×D53—22no—
89C4⋊C4×D53—22no—
90C4⋊C4×D53—22no—
91C4⋊C4×D53D414yes—
92C4⋊C4×D53Q814yes—
93C2×C4×Dic53—21no—
94C2×C4×Dic53—21no—
95C2×C4×Dic53—21no—
96C2×C4×Dic53D411yescharacteristic
97C2×C4×Dic53D411yescharacteristic
98C2×C4×Dic53C2311yescharacteristic
99C42×D53—44no—
100C42×D53C2×C414yes—
101C42×D53Q814yes—
102#13133151—22no—
103#13133150—22no—
104#13133150—22no—
105#13133149—22no—
106#13133148—22no—
107#13133148—22no—
108C4⋊#88322C2×C414yescomplemented
109#13133146C2×C414yescomplemented
110#13133146C2×C414yescomplemented
111#13133124—21no—
112#13133124—21no—
113#13133124—21no—
114#13133124—21no—
115#13133124—21no—
116#13133124—21no—
117#13133124—21no—
118#13133124—21no—
119#13133124—21no—
120C22×C8×C53C2311yescharacteristic
121C22×C8×C53C2311yescharacteristic
122C2×C4⋊C4×C53C2311yescharacteristic
123C2×C4⋊C4×C53C2311yescharacteristic
124C2×C4⋊C4×C53C2311yescharacteristic
125C2×C4⋊C4×C53C2311yescharacteristic
126C2×C42×C53—21no—
127C2×C42×C53C2311yescharacteristic
128#107×C53—22no—
129#106×C53—22no—
130C4⋊C8×C53C2×C414yescomplemented
131#95×C53—21no—
132#95×C53—21no—
133#95×C53—21no—
134C23×D53C2×D411yescharacteristic
135C22×C4×C53—21no—
136C22×C4×C53—21no—
137C22×C4×C53—21no—
138C22×C4×C53—21no—
139C22×C4×C53C2×D411yescharacteristic
140C22×C4×C53C2×D411yescharacteristic
141C22×C4×C53C2411yescharacteristic
142C22×Dic53—21no—
143C22×Dic53—21no—
144C22×Dic53—21no—
145C22×Dic53—21no—
146C22×Dic53C2×D411yescharacteristic
147C22×Dic53C2×D411yescharacteristic
148C22×Dic53C2×D411yescharacteristic
149C2×C4×D53—22no—
150C2×C4×D53—22no—
151C2×C4×D53—22no—
152C2×C4×D53—22no—
153C2×C4×D53—24no—
154C2×C4×D53—24no—
155C2×C4×D53—42no—
156C2×C4×D53—42no—
157C2×C4×D53—42no—
158C2×C4×D53—42no—
159C2×C4×D53—42no—
160C2×C4×D53—42no—
161C2×C4×D53—42no—
162C2×C4×D53—42no—
163C2×C4×D53—42no—
164C2×C4×D53—42no—
165C2×C4×D53C4⋊C414yes—
166C2×C4×D53C4⋊C414yes—
167C2×C4×D53C22×C412yes—
168C2×C4×D53C2×D412yes—
169C2×C4×D53C2×Q812yes—
170C2×C4×D53C2×Q812yes—
171C2×C4×D53#4112yes—
172C2×C4×D53#4112yes—
173C2×C8×C53—21no—
174C2×C8×C53—21no—
175C2×C8×C53—21no—
176C2×C8×C53—21no—
177C2×C8×C53C22×C412yes—
178C2×C8×C53C22×C412yes—
179C4⋊C4×C53—21no—
180C4⋊C4×C53—21no—
181C4⋊C4×C53—21no—
182C4⋊C4×C53—21no—
183C4⋊C4×C53—21no—
184C4⋊C4×C53—21no—
185C4⋊C4×C53C2×D412yes—
186C4⋊C4×C53C2×Q812yes—
187C42×C53—42no—
188C42×C53C22×C412yes—
189C42×C53C2×Q812yes—
190C4⋊Dic53—21no—
191C4⋊Dic53—21no—
192C4⋊Dic53—21no—
193C4⋊Dic53—21no—
194C4⋊Dic53—21no—
195C4⋊Dic53—21no—
196C4⋊Dic53C2×D412yes—
197C4⋊Dic53C2×Q812yes—
198#11719919—24no—
199#11719919—24no—
200#11719919—42no—
201#11719919—42no—
202C4×Dic53—42no—
203C4×Dic53—42no—
204C4×Dic53—42no—
205C4×Dic53C4⋊C414yes—
206C4×Dic53C4⋊C414yes—
207C4×Dic53C22×C412yes—
208C4×Dic53C2×Q812yes—
209C2×#88322—21no—
210C2×#88322—21no—
211C2×#88322—21no—
212C2×#88322—21no—
213C2×#88322C22×C412yes—
214C2×#88322C22×C412yes—
215C8×D53—28no—
216C8×D53—28no—
217C23×C53C22×D411yescharacteristic
218C22×D53C2×C4⋊C412yes—
219C22×D53#12112yes—
220C22×D53C22⋊Q812yes—
221C22×D53#13512yes—
222C22×D53#13512yes—
223C22×D53AGL(1,ℤ/8)12yes—
224C22×D53#13712yes—
225C2×C4×C53—21no—
226C2×C4×C53—21no—
227C2×C4×C53—21no—
228C2×C4×C53—21no—
229C2×C4×C53—22no—
230C2×C4×C53—22no—
231C2×C4×C53—41no—
232C2×C4×C53—41no—
233C2×C4×C53—41no—
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×C53C2×C4⋊C412yes—
242C2×C4×C53C2×C4⋊C412yes—
243C2×C4×C53C23×C411yescharacteristic
244C2×C4×C53C22×D411yescharacteristic
245C2×C4×C53C22×Q811yescharacteristic
246C2×C4×C53C22×Q811yescharacteristic
247C2×C4×C53C2×#4111yescharacteristic
248C2×C4×C53C2×#4111yescharacteristic
249C2×Dic53—21no—
250C2×Dic53—21no—
251C2×Dic53—21no—
252C2×Dic53—21no—
253C2×Dic53—22no—
254C2×Dic53—22no—
255C2×Dic53—41no—
256C2×Dic53—41no—
257C2×Dic53—41no—
258C2×Dic53—41no—
259C2×Dic53—41no—
260C2×Dic53—41no—
261C2×Dic53—41no—
262C2×Dic53—41no—
263C2×Dic53—41no—
264C2×Dic53—41no—
265C2×Dic53C2×C4⋊C412yes—
266C2×Dic53C2×C4⋊C412yes—
267C2×Dic53C2×C4⋊C412yes—
268C2×Dic53#12112yes—
269C2×Dic53C22⋊Q812yes—
270C4×D53—28no—
271C4×D53—28no—
272C4×D53—48no—
273C4×D53—48no—
274C4×D53—84no—
275C4×D53—84no—
276C4×D53C4×D414yes—
277C4×D53C4×Q814yes—
278C4×D53#12514yes—
279C4×D53C4⋊Q814yes—
280C8×C53—22no—
281C8×C53—22no—
282#88322—22no—
283#88322—22no—
284C2×#1949—531nomaximal, Sylow
285C22×C53C22×C4⋊C411yescharacteristic
286C22×C53C2×#12111yescharacteristic
287C22×C53C2×C22⋊Q811yescharacteristic
288C22×C53C2×#13511yescharacteristic
289C22×C53C2×#13511yescharacteristic
290C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
291C22×C53C2×#13711yescharacteristic
292C2×D53#38914yes—
293C2×D53#42714yes—
294C2×D53#42814yes—
295C2×D53#46314yes—
296C2×D53#46414yes—
297C2×D53#47814yes—
298C2×D53#47914yes—
299C4×C53—22no—
300C4×C53—22no—
301C4×C53—42no—
302C4×C53—42no—
303C4×C53—81no—
304C4×C53—81no—
305C4×C53C2×C4×D411yescharacteristic
306C4×C53C2×C4×Q811yescharacteristic
307C4×C53C2×#12511yescharacteristic
308C4×C53C2×C4⋊Q811yescharacteristic
309Dic53—22no—
310Dic53—22no—
311Dic53—42no—
312Dic53—42no—
313Dic53—81no—
314Dic53—81no—
315Dic53#38914yes—
316C22×#107—531no—
317C22×#106—531no—
318C22×C4⋊C8—531no—
319C2×#382—531no—
320C2×C4×C4⋊C4—531no—
321#1949—531no—
322#1949—531no—
323#1949—531no—
324#1949—531no—
325#1949—531no—
326#1949—531no—
327#1949—531no—
328#1949D5311yescharacteristic, direct product factor
329C2×#340—531no—
330C2×#340—531no—
331C2×C53C2×#38911yescharacteristic
332C2×C53C2×#42711yescharacteristic
333C2×C53C2×#42811yescharacteristic
334C2×C53C2×#46311yescharacteristic
335C2×C53C2×#46411yescharacteristic
336C2×C53C2×#47811yescharacteristic
337C2×C53C2×#47911yescharacteristic
338D53#194918yesdirect product factor
339C23×C8—531no—
340C23×C8—531no—
341C22×C4⋊C4—531no—
342C22×C4⋊C4—531no—
343C22×C4⋊C4—531no—
344C22×C4⋊C4—531no—
345C22×C42—531no—
346C22×C42—1061no—
347C2×#107—531no—
348C2×#107—531no—
349C2×#107—531no—
350C2×#107—1064no—
351C2×#107C2×D5311yescharacteristic
352C2×#106—531no—
353C2×#106—531no—
354C2×#106—531no—
355C2×#106—1064no—
356C2×#106C2×D5311yescharacteristic
357C2×C4⋊C8—531no—
358C2×C4⋊C8—531no—
359C2×C4⋊C8—531no—
360C2×C4⋊C8—538no—
361C2×C4⋊C8C2×D5311yescharacteristic
362#382—531no—
363#382—531no—
364#382—531no—
365#382—1061no—
366#382—1061no—
367#382C2×D5311yescharacteristic
368C4×C4⋊C4—531no—
369C4×C4⋊C4—531no—
370C4×C4⋊C4—531no—
371C4×C4⋊C4—1061no—
372C4×C4⋊C4—1061no—
373C4×C4⋊C4C2×D5311yescharacteristic
374C2×#95—1061no—
375C2×#95—1061no—
376C2×#95—1061no—
377#340—531no—
378#340—531no—
379#340—531no—
380#340—531no—
381#340—531no—
382#340—531no—
383#340C2×D5311yescharacteristic
384#340C2×D5311yescharacteristic
385C53C2×#194911yescharacteristic, complemented, Sylow
386C23×C4—531no—
387C23×C4—531no—
388C23×C4—531no—
389C23×C4—1061no—
390C23×C4—1061no—
391C23×C4—1061no—
392C23×C4—1061no—
393C22×C8—531no—
394C22×C8—531no—
395C22×C8—534no—
396C22×C8—534no—
397C22×C8—1062no—
398C22×C8—1062no—
399C22×C8—1062no—
400C22×C8—1062no—
401C22×C8C22×D5311yescharacteristic
402C22×C8C22×D5311yescharacteristic
403C2×C4⋊C4—531no—
404C2×C4⋊C4—531no—
405C2×C4⋊C4—534no—
406C2×C4⋊C4—534no—
407C2×C4⋊C4—531no—
408C2×C4⋊C4—531no—
409C2×C4⋊C4—1062no—
410C2×C4⋊C4—1062no—
411C2×C4⋊C4—1062no—
412C2×C4⋊C4—1062no—
413C2×C4⋊C4—1062no—
414C2×C4⋊C4—1062no—
415C2×C4⋊C4—1061no—
416C2×C4⋊C4—1061no—
417C2×C4⋊C4—1061no—
418C2×C4⋊C4—1061no—
419C2×C4⋊C4C22×D5311yescharacteristic
420C2×C4⋊C4C22×D5311yescharacteristic
421C2×C4⋊C4C22×D5311yescharacteristic
422C2×C4⋊C4C22×D5311yescharacteristic
423C2×C42—21no—
424C2×C42—531no—
425C2×C42—531no—
426C2×C42—531no—
427C2×C42—534no—
428C2×C42—534no—
429C2×C42—1061no—
430C2×C42—1061no—
431C2×C42—1061no—
432C2×C42—2124no—
433C2×C42C22×D5311yescharacteristic
434#107—22no—
435#107—1062no—
436#107—1062no—
437#107—1062no—
438#106—22no—
439#106—1062no—
440#106—1062no—
441#106—1062no—
442C4⋊C8—534no—
443C4⋊C8—534no—
444C4⋊C8—534no—
445C4⋊C8C4×D5314yescomplemented
446#95—21no—
447#95—21no—
448#95—21no—
449#95—1061no—
450#95—1061no—
451#95—1061no—
452#95—1061no—
453#95—1061no—
454#95—1061no—
455#95—1061no—
456#95—1061no—
457#95—1061no—
458C24—531no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—531no—
464C22×C4—532no—
465C22×C4—532no—
466C22×C4—532no—
467C22×C4—532no—
468C22×C4—532no—
469C22×C4—532no—
470C22×C4—531no—
471C22×C4—531no—
472C22×C4—534no—
473C22×C4—534no—
474C22×C4—1062no—
475C22×C4—1062no—
476C22×C4—1062no—
477C22×C4—1062no—
478C22×C4—1064no—
479C22×C4—1064no—
480C22×C4—1061no—
481C22×C4—1061no—
482C22×C4—1061no—
483C22×C4—1061no—
484C22×C4—2122no—
485C22×C4—2122no—
486C22×C4—2122no—
487C22×C4—2122no—
488C22×C4—2122no—
489C22×C4—2122no—
490C22×C4—2122no—
491C22×C4—2122no—
492C22×C4—2122no—
493C22×C4—2122no—
494C22×C4D4×D5311yescharacteristic
495C22×C4D4×D5311yescharacteristic
496C22×C4C23×D5311yescharacteristic
497C2×C8—21no—
498C2×C8—21no—
499C2×C8—21no—
500C2×C8—21no—
501C2×C8—532no—
502C2×C8—532no—
503C2×C8—1068no—
504C2×C8—1068no—
505C2×C8—1061no—
506C2×C8—1061no—
507C2×C8—1061no—
508C2×C8—1061no—
509C2×C8C2×C4×D5312yes—
510C2×C8C2×C4×D5312yes—
511C4⋊C4—21no—
512C4⋊C4—21no—
513C4⋊C4—21no—
514C4⋊C4—21no—
515C4⋊C4—21no—
516C4⋊C4—21no—
517C4⋊C4—532no—
518C4⋊C4—532no—
519C4⋊C4—1064no—
520C4⋊C4—1064no—
521C4⋊C4—1061no—
522C4⋊C4—1061no—
523C4⋊C4—1061no—
524C4⋊C4—1061no—
525C4⋊C4—1061no—
526C4⋊C4—1061no—
527C4⋊C4—2122no—
528C4⋊C4—2122no—
529C4⋊C4D4×D5312yes—
530C4⋊C4Q8×D5312yes—
531C42—42no—
532C42—534no—
533C42—534no—
534C42—532no—
535C42—532no—
536C42—2122no—
537C42—2122no—
538C42—2122no—
539C42C2×C4×D5312yes—
540C42Q8×D5312yes—
541C23—532no—
542C23—532no—
543C23—532no—
544C23—532no—
545C23—532no—
546C23—532no—
547C23—532no—
548C23C2×D4×D5311yescharacteristic
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—532no—
566C2×C4—532no—
567C2×C4—532no—
568C2×C4—534no—
569C2×C4—534no—
570C2×C4—534no—
571C2×C4—534no—
572C2×C4—532no—
573C2×C4—532no—
574C2×C4—1068no—
575C2×C4—1068no—
576C2×C4—1061no—
577C2×C4—1061no—
578C2×C4—1061no—
579C2×C4—1061no—
580C2×C4—1062no—
581C2×C4—1062no—
582C2×C4—4244no—
583C2×C4—4244no—
584C2×C4—2128no—
585C2×C4—2128no—
586C2×C4—2121no—
587C2×C4—2121no—
588C2×C4—2121no—
589C2×C4—2121no—
590C2×C4—2121no—
591C2×C4—2121no—
592C2×C4—2121no—
593C2×C4—2121no—
594C2×C4—2121no—
595C2×C4—2121no—
596C2×C4C4⋊C4×D5312yes—
597C2×C4C4⋊C4×D5312yes—
598C2×C4C22×C4×D5311yescharacteristic
599C2×C4C2×D4×D5311yescharacteristic
600C2×C4C2×Q8×D5311yescharacteristic
601C2×C4C2×Q8×D5311yescharacteristic
602C2×C4#41×D5311yescharacteristic
603C2×C4#41×D5311yescharacteristic
604C8—22no—
605C8—22no—
606C8—1062no—
607C8—1062no—
608C22—534no—
609C22—534no—
610C22—534no—
611C22—534no—
612C22—534no—
613C22—534no—
614C22—534no—
615C22C2×C4⋊C4×D5311yescharacteristic
616C22#121×D5311yescharacteristic
617C22C22⋊Q8×D5311yescharacteristic
618C22#135×D5311yescharacteristic
619C22#135×D5311yescharacteristic
620C22AGL(1,ℤ/8)×D5311yescharacteristic
621C22#137×D5311yescharacteristic
622C4—22no—
623C4—22no—
624C4—42no—
625C4—42no—
626C4—81no—
627C4—81no—
628C4—534no—
629C4—1062no—
630C4—1062no—
631C4—4241no—
632C4—4241no—
633C4—2122no—
634C4—2122no—
635C4C4×D4×D5311yescharacteristic
636C4C4×Q8×D5311yescharacteristic
637C4#125×D5311yescharacteristic
638C4C4⋊Q8×D5311yescharacteristic
639C2—538no—
640C2#389×D5311yescharacteristic
641C2#427×D5311yescharacteristic
642C2#428×D5311yescharacteristic
643C2#463×D5311yescharacteristic
644C2#464×D5311yescharacteristic
645C2#478×D5311yescharacteristic
646C2#479×D5311yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#1949C2
socleC23×C53C22×D4
prime radicalC23×D53C2×D4
cosocleC22×C4×C53C24
hypercenter#1949D53
hypocenterC53C2×#1949
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1949

Complements

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

NH ≅ G/Ntype
6. #20275486C2semidirect
7. #20275485C2semidirect
8. #20275484C2semidirect
9. #20275483C2semidirect
10. #20275482C2semidirect
11. #20275481C2semidirect
12. #20275480C2semidirect
15. C53×#1949C2semidirect
35. C4⋊C8×D53C4semidirect
108. C4⋊#88322C2×C4semidirect
109. #13133146C2×C4semidirect
110. #13133146C2×C4semidirect
130. C4⋊C8×C53C2×C4semidirect
328. #1949D53direct (N × H)
338. D53#1949direct (N × H)
385. C53C2×#1949semidirect
445. C4⋊C8C4×D53semidirect

Maximal subgroups

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

subgroupindex
C2×#107×D532
C2×#106×D532
C2×C4⋊C8×D532
#382×D532
C4×C4⋊C4×D532
#202754862
#202754852
#202754842
#202754832
#202754822
#202754812
#202754802
#340×D532
C53×#19492
C2×#194953