Finite Group Databasebeta

← #478×Dic53 #27391699

Subgroups — #478×Dic53 #27391699

Order 13568.

948 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#125×Dic53C211yesmaximal, characteristic, complemented
2C4×D4×Dic53C211yesmaximal, characteristic
3#20275673C211yesmaximal, characteristic, complemented
4#20275485C211yesmaximal, characteristic, complemented
5#20275309C211yesmaximal, characteristic, complemented
6#20275295C211yesmaximal, characteristic, complemented
7#20274978C211yesmaximal, characteristic
8#20274964C211yesmaximal, characteristic
9#20274725C211yesmaximal, characteristic
10#107×Dic53C211yesmaximal, characteristic, complemented
11#106×Dic53C211yesmaximal, characteristic, complemented
12C4⋊C8×Dic53C211yesmaximal, characteristic, complemented
13#102×Dic53C211yesmaximal, characteristic
14#102×Dic53C211yesmaximal, characteristic
15C2×C53×#478C211yesmaximal, characteristic
16C2×D4×Dic53C2211yescharacteristic
17C2×D4×Dic53C2211yescharacteristic
18C22×C4×Dic53—21no—
19C2×#13133151C2211yescharacteristic
20C2×#13133149C2211yescharacteristic
21C2×C4⋊#88322C2211yescharacteristic
22C2×#13133138C2211yescharacteristic
23C2×#13133138C2211yescharacteristic
24C2×C8×Dic53C2211yescharacteristic
25C2×C8×Dic53C2211yescharacteristic
26#16810522C2211yescharacteristic
27#16810522C2211yescharacteristic
28#16810509—21no—
29#16810509—21no—
30#16810489C2211yescharacteristic
31C4⋊C4×Dic53—21no—
32C4⋊C4×Dic53—21no—
33C4⋊C4×Dic53C2211yescharacteristic
34C4⋊C4×Dic53C2211yescharacteristic
35C4⋊C4×Dic53C2211yescharacteristic
36C4⋊C4×Dic53C2211yescharacteristic
37C4×C4⋊Dic53C2211yescharacteristic
38C22⋊C4×Dic53—21no—
39C22⋊C4×Dic53—21no—
40C4×C22⋊Dic53—21no—
41C42×Dic53C2211yescharacteristic
42#16810386C2211yescharacteristic
43#16810386C2211yescharacteristic
44#16810385C2211yescharacteristic
45#16810385C2211yescharacteristic
46#16810385C2211yescharacteristic
47#16810385C2211yescharacteristic
48#16810385C2211yescharacteristic
49#16810385C2211yescharacteristic
50#16810377C2211yescharacteristic
51#16810377C2211yescharacteristic
52C2×#125×C53C2211yescharacteristic
53C2×C4×D4×C53C2211yescharacteristic
54C53×#478C418yescomplemented
55C2×#107×C53C2211yescharacteristic
56C2×#106×C53C2211yescharacteristic
57C2×C4⋊C8×C53C2211yescharacteristic
58C2×#102×C53C2211yescharacteristic
59C2×#102×C53C2211yescharacteristic
60C23×Dic53—21no—
61C23×Dic53—21no—
62C22×#88322C2311yescharacteristic
63C22×#88322C2311yescharacteristic
64D4×Dic53—22no—
65D4×Dic53—22no—
66D4×Dic53—22no—
67D4×Dic53Q814yes—
68C2×C4⋊Dic53—21no—
69C2×C4⋊Dic53—21no—
70C2×C4⋊Dic53C2311yescharacteristic
71C2×C4⋊Dic53C2311yescharacteristic
72C2×C4⋊Dic53C2311yescharacteristic
73C2×C4⋊Dic53C2311yescharacteristic
74C2×C22⋊Dic53—21no—
75C2×C22⋊Dic53—21no—
76C2×C4×Dic53—21no—
77C2×C4×Dic53—21no—
78C2×C4×Dic53—21no—
79C2×C4×Dic53—21no—
80C2×C4×Dic53—21no—
81C2×C4×Dic53—44no—
82C2×C4×Dic53D411yescharacteristic
83C2×C4×Dic53D411yescharacteristic
84C2×C4×Dic53D411yescharacteristic
85C2×C4×Dic53D411yescharacteristic
86C2×C4×Dic53C2311yescharacteristic
87C2×C4×Dic53C2311yescharacteristic
88#13133151—22no—
89#13133149—22no—
90C4⋊#88322—22no—
91#13133138—22no—
92#13133138—22no—
93C8×Dic53—22no—
94C8×Dic53—22no—
95#13133124—21no—
96#13133124—21no—
97#13133124—21no—
98#13133124—21no—
99#13133124—21no—
100#13133124—21no—
101#13133124—21no—
102#13133124—21no—
103#13133124—21no—
104C22×D4×C53C2311yescharacteristic
105C23×C4×C53—21no—
106C22×C8×C53C2311yescharacteristic
107C22×C8×C53C2311yescharacteristic
108#125×C53—22no—
109#125×C53C2×C414yescomplemented
110C4×D4×C53—22no—
111C4×D4×C53C2×C414yes—
112C2×C4⋊C4×C53—21no—
113C2×C4⋊C4×C53—21no—
114C2×C4⋊C4×C53C2311yescharacteristic
115C2×C4⋊C4×C53C2311yescharacteristic
116C2×C4⋊C4×C53C2311yescharacteristic
117C2×C22⋊C4×C53—21no—
118C2×C42×C53C2311yescharacteristic
119#107×C53C2×C414yescomplemented
120#106×C53C2×C414yescomplemented
121C4⋊C8×C53C2×C414yescomplemented
122#102×C53C2×C414yes—
123#102×C53C2×C414yes—
124C24×C53—21no—
125C2×D4×C53—21no—
126C2×D4×C53—21no—
127C2×D4×C53—21no—
128C2×D4×C53C22×C412yes—
129C2×D4×C53C2×Q812yes—
130C2×D4×C53#4112yes—
131C22×C4×C53—21no—
132C22×C4×C53—21no—
133C22×C4×C53—21no—
134C22×C4×C53—24no—
135C22×C4×C53—42no—
136C22×C4×C53—42no—
137C22×C4×C53C2×D411yescharacteristic
138C22×C4×C53C2×D411yescharacteristic
139C22×C4×C53C2411yescharacteristic
140C22×Dic53—24no—
141C22×Dic53—21no—
142C22×Dic53—21no—
143C22×Dic53—24no—
144C22×Dic53—21no—
145C22×Dic53—21no—
146C22×Dic53—42no—
147C22×Dic53—42no—
148C22×Dic53—42no—
149C22×Dic53—42no—
150C22×Dic53C2×D411yescharacteristic
151C22×Dic53C2×D411yescharacteristic
152C22×Dic53C2×D411yescharacteristic
153C22×Dic53C2×D411yescharacteristic
154C2×C8×C53—21no—
155C2×C8×C53—21no—
156C2×C8×C53—21no—
157C2×C8×C53—21no—
158C2×C8×C53C22×C412yes—
159C2×C8×C53C22×C412yes—
160C4⋊C4×C53—24no—
161C4⋊C4×C53—24no—
162C4⋊C4×C53—21no—
163C4⋊C4×C53—21no—
164C4⋊C4×C53—21no—
165C4⋊C4×C53C22×C412yes—
166C4⋊C4×C53C22×C412yes—
167C4⋊C4×C53C22×C412yes—
168C22⋊C4×C53—24no—
169C42×C53C22×C412yes—
170C42×C53#4112yes—
171C22⋊Dic53—24no—
172C22⋊Dic53—24no—
173C4⋊Dic53—21no—
174C4⋊Dic53—21no—
175C4⋊Dic53—21no—
176C4⋊Dic53—21no—
177C4⋊Dic53—21no—
178C4⋊Dic53—21no—
179C4⋊Dic53—42no—
180C4⋊Dic53—42no—
181C4⋊Dic53C2×Q812yes—
182C4⋊Dic53#4112yes—
183C4×Dic53—21no—
184C4×Dic53—21no—
185C4×Dic53—22no—
186C4×Dic53—22no—
187C4×Dic53—22no—
188C4×Dic53—41no—
189C4×Dic53—42no—
190C4×Dic53—42no—
191C4×Dic53—42no—
192C4×Dic53—42no—
193C4×Dic53—42no—
194C4×Dic53C2×Q812yes—
195C4×Dic53#4112yes—
196C2×#88322—21no—
197C2×#88322—21no—
198C2×#88322—21no—
199C2×#88322—21no—
200C2×#88322—21no—
201C2×#88322—21no—
202C23×C53—22no—
203C23×C53—22no—
204C23×C53—22no—
205C23×C53—41no—
206C23×C53—41no—
207C23×C53—41no—
208C23×C53—41no—
209C23×C53C22×D411yescharacteristic
210D4×C53—22no—
211D4×C53—22no—
212D4×C53—22no—
213D4×C53C4×Q814yes—
214C2×C4×C53—21no—
215C2×C4×C53—21no—
216C2×C4×C53—21no—
217C2×C4×C53—21no—
218C2×C4×C53—22no—
219C2×C4×C53—22no—
220C2×C4×C53—22no—
221C2×C4×C53—48no—
222C2×C4×C53—41no—
223C2×C4×C53—41no—
224C2×C4×C53—41no—
225C2×C4×C53—41no—
226C2×C4×C53—41no—
227C2×C4×C53—41no—
228C2×C4×C53C4×D412yes—
229C2×C4×C53C4×D412yes—
230C2×C4×C53C23×C411yescharacteristic
231C2×C4×C53C22×Q811yescharacteristic
232C2×C4×C53C2×#4111yescharacteristic
233C2×C4×C53C2×#4111yescharacteristic
234C2×C4×C53C2×#4111yescharacteristic
235C2×C4×C53C2×#4111yescharacteristic
236C2×Dic53—21no—
237C2×Dic53—21no—
238C2×Dic53—21no—
239C2×Dic53—21no—
240C2×Dic53—22no—
241C2×Dic53—22no—
242C2×Dic53—21no—
243C2×Dic53—21no—
244C2×Dic53—21no—
245C2×Dic53—48no—
246C2×Dic53—41no—
247C2×Dic53—41no—
248C2×Dic53—41no—
249C2×Dic53—41no—
250C2×Dic53—48no—
251C2×Dic53—41no—
252C2×Dic53—41no—
253C2×Dic53—41no—
254C2×Dic53—41no—
255C2×Dic53—41no—
256C2×Dic53—41no—
257C2×Dic53C22⋊Q812yes—
258C2×Dic53C22⋊Q812yes—
259C2×Dic53#13512yes—
260C2×Dic53#13512yes—
261C2×Dic53AGL(1,ℤ/8)12yes—
262C8×C53—22no—
263C8×C53—22no—
264#88322—41no—
265#88322—41no—
266C4×#478—531nomaximal, Sylow
267C22×C53—24no—
268C22×C53—42no—
269C22×C53—42no—
270C22×C53—42no—
271C22×C53—42no—
272C22×C53—42no—
273C22×C53—42no—
274C22×C53C2×C4×D411yescharacteristic
275C22×C53C2×C22⋊Q811yescharacteristic
276C22×C53#52511yescharacteristic
277C22×C53C2×#13511yescharacteristic
278C22×C53C2×#13511yescharacteristic
279C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
280C22×C53#57511yescharacteristic
281C4×C53—22no—
282C4×C53—22no—
283C4×C53—42no—
284C4×C53—42no—
285C4×C53—42no—
286C4×C53C2×C4×Q811yescharacteristic
287C4×C53C4×#4111yescharacteristic
288C4×C53#52911yescharacteristic
289C4×C53#53311yescharacteristic
290Dic53—22no—
291Dic53—22no—
292Dic53—42no—
293Dic53—41no—
294Dic53—42no—
295Dic53—81no—
296Dic53—81no—
297Dic53#47814yesdirect product factor
298C2×#478D5311yescharacteristic
299C4×#125—531no—
300C42×D4—531no—
301#1989—531no—
302#1949—531no—
303#1921—531no—
304#1919—531no—
305#1864—531no—
306#1862—531no—
307#1804—531no—
308C4×#107—531no—
309C4×#106—531no—
310C4×C4⋊C8—531no—
311C4×#102—531no—
312C4×#102—531no—
313C2×C53—44no—
314C2×C53C4×C22⋊Q811yescharacteristic
315C2×C53C4×#13511yescharacteristic
316C2×C53C4×AGL(1,ℤ/8)11yescharacteristic
317C2×C53C2×#47811yescharacteristic
318C2×C53#307611yescharacteristic
319C2×C53#310211yescharacteristic
320C2×C53#310511yescharacteristic
321C2×#125C2×D5311yescharacteristic
322C2×C4×D4—531no—
323C2×C4×D4—531no—
324C2×C4×D4C2×D5311yescharacteristic
325C22×C42—1061no—
326#478Dic5318yesdirect product factor
327C2×#107—531no—
328C2×#107C2×D5311yescharacteristic
329C2×#106—531no—
330C2×#106C2×D5311yescharacteristic
331C2×C4⋊C8—531no—
332C2×C4⋊C8C2×D5311yescharacteristic
333C2×#102—531no—
334C2×#102—531no—
335C2×#102C2×D5311yescharacteristic
336C2×#102C2×D5311yescharacteristic
337C2×C4×C8—531no—
338C2×C4×C8—531no—
339#389—531no—
340#389—531no—
341#387—1061no—
342#387—1061no—
343#382—531no—
344C4×C4⋊C4—531no—
345C4×C4⋊C4—531no—
346C4×C4⋊C4—531no—
347C4×C4⋊C4—531no—
348C4×C4⋊C4—531no—
349C4×C4⋊C4—1061no—
350C4×C4⋊C4—1061no—
351C4×C22⋊C4—1061no—
352C4×C22⋊C4—1061no—
353C4×C22⋊C4—1061no—
354C43—531no—
355#340—531no—
356#340—531no—
357#340—531no—
358#340—531no—
359#340—531no—
360#340—531no—
361#340—531no—
362#340—531no—
363#336—531no—
364#336—531no—
365C53C4×#47811yescharacteristic, complemented, Sylow
366C22×D4C22×D5311yescharacteristic
367C23×C4—21no—
368C23×C4—1061no—
369C23×C4—1061no—
370C22×C8—531no—
371C22×C8—531no—
372C22×C8C22×D5311yescharacteristic
373C22×C8C22×D5311yescharacteristic
374#125—22no—
375#125C2×Dic5314yescomplemented
376C4×D4—22no—
377C4×D4—534no—
378C4×D4—1062no—
379C4×D4—1062no—
380C4×D4—1062no—
381C4×D4C2×Dic5314yes—
382C2×C4⋊C4—21no—
383C2×C4⋊C4—21no—
384C2×C4⋊C4—531no—
385C2×C4⋊C4—531no—
386C2×C4⋊C4—531no—
387C2×C4⋊C4—531no—
388C2×C4⋊C4—1061no—
389C2×C4⋊C4—1061no—
390C2×C4⋊C4C22×D5311yescharacteristic
391C2×C4⋊C4C22×D5311yescharacteristic
392C2×C4⋊C4C22×D5311yescharacteristic
393C2×C22⋊C4—21no—
394C2×C22⋊C4—1061no—
395C2×C22⋊C4—1061no—
396C2×C42—531no—
397C2×C42—531no—
398C2×C42—531no—
399C2×C42—531no—
400C2×C42—531no—
401C2×C42—531no—
402C2×C42—1061no—
403C2×C42—1061no—
404C2×C42—1061no—
405C2×C42—1061no—
406C2×C42—1061no—
407C2×C42—2124no—
408C2×C42C22×D5311yescharacteristic
409#107—1062no—
410#107C2×Dic5314yescomplemented
411#106—1062no—
412#106C2×Dic5314yescomplemented
413C4⋊C8—1062no—
414C4⋊C8C2×Dic5314yescomplemented
415#102—1062no—
416#102—1062no—
417#102C2×Dic5314yes—
418#102C2×Dic5314yes—
419C4×C8—1062no—
420C4×C8—1062no—
421#95—1061no—
422#95—1061no—
423#95—1061no—
424#95—1061no—
425#95—1061no—
426#95—1061no—
427#95—1061no—
428#95—1061no—
429#95—1061no—
430C24—21no—
431C2×D4—21no—
432C2×D4—21no—
433C2×D4—21no—
434C2×D4Q8×D5312yes—
435C2×D4#1171994912yes—
436C2×D4C22×Dic5312yes—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—24no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—531no—
444C22×C4—531no—
445C22×C4—531no—
446C22×C4—531no—
447C22×C4—1064no—
448C22×C4—1064no—
449C22×C4—1061no—
450C22×C4—1061no—
451C22×C4—1061no—
452C22×C4—1061no—
453C22×C4—2122no—
454C22×C4—2122no—
455C22×C4—2122no—
456C22×C4—2122no—
457C22×C4D4×D5311yescharacteristic
458C22×C4D4×D5311yescharacteristic
459C22×C4C23×D5311yescharacteristic
460C2×C8—21no—
461C2×C8—21no—
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—1061no—
465C2×C8—1061no—
466C2×C8—1061no—
467C2×C8—1061no—
468C2×C8—1061no—
469C2×C8—1061no—
470C2×C8C22×Dic5312yes—
471C2×C8C22×Dic5312yes—
472C4⋊C4—24no—
473C4⋊C4—24no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—532no—
478C4⋊C4—532no—
479C4⋊C4—1061no—
480C4⋊C4—1061no—
481C4⋊C4—1061no—
482C4⋊C4—1061no—
483C4⋊C4—1061no—
484C4⋊C4—1061no—
485C4⋊C4—2122no—
486C4⋊C4—2122no—
487C4⋊C4C22×Dic5312yes—
488C4⋊C4C22×Dic5312yes—
489C4⋊C4C22×Dic5312yes—
490C22⋊C4—24no—
491C22⋊C4—1064no—
492C22⋊C4—1064no—
493C42—532no—
494C42—532no—
495C42—1062no—
496C42—1062no—
497C42—1062no—
498C42—1061no—
499C42—1061no—
500C42—2121no—
501C42—2122no—
502C42—2122no—
503C42—2122no—
504C42—2122no—
505C42—2122no—
506C42#1171994712yes—
507C42C22×Dic5312yes—
508C23—22no—
509C23—22no—
510C23—22no—
511C23—41no—
512C23—41no—
513C23—41no—
514C23—41no—
515C23C2×D4×D5311yescharacteristic
516D4—22no—
517D4—22no—
518D4—22no—
519D4Q8×Dic5314yes—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—48no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—532no—
535C2×C4—532no—
536C2×C4—532no—
537C2×C4—532no—
538C2×C4—532no—
539C2×C4—1061no—
540C2×C4—1061no—
541C2×C4—1061no—
542C2×C4—1061no—
543C2×C4—1062no—
544C2×C4—1061no—
545C2×C4—1062no—
546C2×C4—1061no—
547C2×C4—1061no—
548C2×C4—2128no—
549C2×C4—2128no—
550C2×C4—2121no—
551C2×C4—2121no—
552C2×C4—2121no—
553C2×C4—2121no—
554C2×C4—2121no—
555C2×C4—2121no—
556C2×C4—2121no—
557C2×C4—2121no—
558C2×C4—2121no—
559C2×C4—2121no—
560C2×C4D4×Dic5312yes—
561C2×C4D4×Dic5312yes—
562C2×C4C23×Dic5311yescharacteristic
563C2×C4C2×Q8×D5311yescharacteristic
564C2×C4#41×D5311yescharacteristic
565C2×C4#41×D5311yescharacteristic
566C2×C4C2×#1171994711yescharacteristic
567C2×C4C2×#1171994911yescharacteristic
568C8—22no—
569C8—22no—
570C8—2121no—
571C8—2121no—
572C22—24no—
573C22—42no—
574C22—42no—
575C22—42no—
576C22—42no—
577C22—42no—
578C22—42no—
579C22C2×D4×Dic5311yescharacteristic
580C22C22⋊Q8×D5311yescharacteristic
581C22#1681113611yescharacteristic
582C22#135×D5311yescharacteristic
583C22#135×D5311yescharacteristic
584C22AGL(1,ℤ/8)×D5311yescharacteristic
585C22#1681141911yescharacteristic
586C4—22no—
587C4—22no—
588C4—42no—
589C4—42no—
590C4—42no—
591C4—534no—
592C4—1062no—
593C4—1062no—
594C4—4241no—
595C4—4241no—
596C4—2122no—
597C4—2121no—
598C4—2122no—
599C4C2×Q8×Dic5311yescharacteristic
600C4#41×Dic5311yescharacteristic
601C4#1681115911yescharacteristic
602C4#1681117511yescharacteristic
603C2—44no—
604C2C22⋊Q8×Dic5311yescharacteristic
605C2#135×Dic5311yescharacteristic
606C2AGL(1,ℤ/8)×Dic5311yescharacteristic
607C2#478×D5311yescharacteristic
608C2#2028450011yescharacteristic
609C2#2028473811yescharacteristic
610C2#2028476711yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC2×C53×#478C2
socleC23×C53C22×D4
prime radicalC22×D4×C53C23
cosocleC22×C4×C53C24
hypercenterC2×#478D53
hypocenterC53C4×#478
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#478

Complements

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

NH ≅ G/Ntype
1. #125×Dic53C2semidirect
3. #20275673C2semidirect
4. #20275485C2semidirect
5. #20275309C2semidirect
6. #20275295C2semidirect
10. #107×Dic53C2semidirect
11. #106×Dic53C2semidirect
12. C4⋊C8×Dic53C2semidirect
54. C53×#478C4semidirect
109. #125×C53C2×C4semidirect
119. #107×C53C2×C4semidirect
120. #106×C53C2×C4semidirect
121. C4⋊C8×C53C2×C4semidirect
297. Dic53#478direct (N × H)
326. #478Dic53direct (N × H)
365. C53C4×#478semidirect
375. #125C2×Dic53semidirect
410. #107C2×Dic53semidirect
412. #106C2×Dic53semidirect
414. C4⋊C8C2×Dic53semidirect

Maximal subgroups

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

subgroupindex
#125×Dic532
C4×D4×Dic532
#202756732
#202754852
#202753092
#202752952
#202749782
#202749642
#202747252
#107×Dic532
#106×Dic532
C4⋊C8×Dic532
#102×Dic532
C2×C53×#4782
C4×#47853