Finite Group Databasebeta

← C4×#16810920 #27391692

Subgroups — C4×#16810920 #27391692

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
1C2×#16810920C211yesmaximal, characteristic
2C42×D212C211yesmaximal, characteristic
3#20275673C211yesmaximal, characteristic, complemented
4#20275481C211yesmaximal, characteristic, complemented
5#20275304C211yesmaximal, characteristic, complemented
6#20275290C211yesmaximal, characteristic, complemented
7#20274975C211yesmaximal, characteristic
8#20274961C211yesmaximal, characteristic
9#20274722C211yesmaximal, characteristic
10C4×#13133150C211yesmaximal, characteristic, complemented
11C4×#13133148C211yesmaximal, characteristic, complemented
12C4×C4⋊#88322C211yesmaximal, characteristic, complemented
13C4×#13133137C211yesmaximal, characteristic
14C4×#13133137C211yesmaximal, characteristic
15C4×#125×C53C211yesmaximal, characteristic, complemented
16C2×C4×D212C2211yescharacteristic
17C2×C4×D212C2211yescharacteristic
18C2×C4×D212C2211yescharacteristic
19C2×C42×D53—21no—
20#16810920C418yesdirect product factor
21C2×#13133150C2211yescharacteristic
22C2×#13133150C2211yescharacteristic
23C2×#13133148C2211yescharacteristic
24C2×#13133148C2211yescharacteristic
25C2×C4⋊#88322C2211yescharacteristic
26C2×C4⋊#88322C2211yescharacteristic
27C2×#13133137C2211yescharacteristic
28C2×#13133137C2211yescharacteristic
29C2×#13133137C2211yescharacteristic
30C2×#13133137C2211yescharacteristic
31C2×C4×#88322C2211yescharacteristic
32C2×C4×#88322C2211yescharacteristic
33C4×C4⋊Dic53C2211yescharacteristic
34C4×C4⋊Dic53C2211yescharacteristic
35C4×C4⋊Dic53C2211yescharacteristic
36C4×#11719921—21no—
37C4×#11719921—21no—
38C4×#11719921—21no—
39#16810385C2211yescharacteristic
40#16810385C2211yescharacteristic
41#16810385C2211yescharacteristic
42#16810385C2211yescharacteristic
43#16810385C2211yescharacteristic
44#16810385C2211yescharacteristic
45#16810385C2211yescharacteristic
46#16810385C2211yescharacteristic
47#16810378C2211yescharacteristic
48#16810378C2211yescharacteristic
49C2×#125×C53C2211yescharacteristic
50C53×#389C2211yescharacteristic
51C53×#389C2211yescharacteristic
52C53×#387—21no—
53C53×#387—21no—
54C53×#382C2211yescharacteristic
55C4×C4⋊C4×C53—21no—
56C4×C4⋊C4×C53—21no—
57C4×C4⋊C4×C53C2211yescharacteristic
58C4×C4⋊C4×C53C2211yescharacteristic
59C43×C53C2211yescharacteristic
60C22×D212C2311yescharacteristic
61C22×C4×D53—21no—
62C22×C4×D53—21no—
63C22×C4×D53—21no—
64C22×#88322C2311yescharacteristic
65C22×#88322C2311yescharacteristic
66C22×#88322C2311yescharacteristic
67C22×#88322C2311yescharacteristic
68C4×D212—22no—
69C4×D212—22no—
70C4×D212—22no—
71C4×D212—22no—
72C4×D212C2×C414yes—
73C4×D212Q814yes—
74C2×C4⋊Dic53C2311yescharacteristic
75C2×C4⋊Dic53C2311yescharacteristic
76C2×C4⋊Dic53C2311yescharacteristic
77C2×#11719921—21no—
78C2×#11719921—21no—
79C2×#11719921—21no—
80C2×C4×Dic53—21no—
81C2×C4×Dic53—21no—
82C2×C4×Dic53—21no—
83C42×D53—44no—
84#13133150—22no—
85#13133150C2×C414yescomplemented
86#13133148—22no—
87#13133148C2×C414yescomplemented
88C4⋊#88322—22no—
89C4⋊#88322C2×C414yescomplemented
90#13133137—22no—
91#13133137—22no—
92#13133137C2×C414yes—
93#13133137C2×C414yes—
94C4×#88322—22no—
95C4×#88322—22no—
96#13133124—21no—
97#13133124—21no—
98#13133124—21no—
99#125×C53—22no—
100#125×C53C2×C414yescomplemented
101C2×C4⋊C4×C53—21no—
102C2×C4⋊C4×C53—21no—
103C2×C4⋊C4×C53—21no—
104C2×C4⋊C4×C53—21no—
105C2×C4⋊C4×C53C2311yescharacteristic
106C2×C4⋊C4×C53C2311yescharacteristic
107C2×C4⋊C4×C53C2311yescharacteristic
108C2×C4⋊C4×C53C2311yescharacteristic
109C2×C42×C53—21no—
110C2×C42×C53—21no—
111C2×C42×C53D411yescharacteristic
112C2×C42×C53D411yescharacteristic
113C2×C42×C53D411yescharacteristic
114C2×C42×C53D411yescharacteristic
115C2×C42×C53C2311yescharacteristic
116C2×C42×C53C2311yescharacteristic
117C2×C42×C53C2311yescharacteristic
118#95×C53—21no—
119#95×C53—21no—
120#95×C53—21no—
121#95×C53—21no—
122#95×C53—21no—
123#95×C53—21no—
124C23×D53—21no—
125C22×C4×C53—21no—
126C22×C4×C53—21no—
127C22×C4×C53—21no—
128C22×C4×C53—21no—
129C22×C4×C53C2×D411yescharacteristic
130C22×C4×C53C2×D411yescharacteristic
131C22×C4×C53C2×D411yescharacteristic
132C22×C4×C53C2×D411yescharacteristic
133C22×C4×C53C2×D411yescharacteristic
134C22×C4×C53C2×D411yescharacteristic
135C22×C4×C53C2411yescharacteristic
136C22×Dic53—21no—
137C22×Dic53—21no—
138C22×Dic53—21no—
139C2×D212—21no—
140C2×D212—21no—
141C2×D212—21no—
142C2×D212C22×C412yes—
143C2×D212C2×Q812yes—
144C2×D212#4112yes—
145C2×C4×D53—24no—
146C2×C4×D53—24no—
147C2×C4×D53—24no—
148C2×C4×D53—42no—
149C2×C4×D53—42no—
150C2×C4×D53—42no—
151C2×C4×D53—42no—
152C2×C4×D53—42no—
153C2×C4×D53—42no—
154C4⋊C4×C53—24no—
155C4⋊C4×C53—24no—
156C4⋊C4×C53—21no—
157C4⋊C4×C53—21no—
158C4⋊C4×C53—21no—
159C4⋊C4×C53—21no—
160C4⋊C4×C53—21no—
161C4⋊C4×C53—21no—
162C4⋊C4×C53—42no—
163C4⋊C4×C53—42no—
164C4⋊C4×C53C22×C412yes—
165C4⋊C4×C53C22×C412yes—
166C42×C53—22no—
167C42×C53—22no—
168C42×C53—22no—
169C42×C53—21no—
170C42×C53—21no—
171C42×C53—41no—
172C42×C53—42no—
173C42×C53—42no—
174C42×C53C22×C412yes—
175C42×C53C2×Q812yes—
176C42×C53#4112yes—
177C42×C53#4112yes—
178#11719921—24no—
179#11719921—24no—
180#11719921—24no—
181C4⋊Dic53—21no—
182C4⋊Dic53—21no—
183C4⋊Dic53—21no—
184C4⋊Dic53C22×C412yes—
185C4⋊Dic53C2×Q812yes—
186C4⋊Dic53#4112yes—
187C4×Dic53—42no—
188C4×Dic53—42no—
189C4×Dic53—42no—
190C2×#88322—21no—
191C2×#88322—21no—
192C2×#88322—21no—
193C2×#88322—21no—
194C2×#88322—21no—
195C2×#88322—21no—
196C2×#88322—21no—
197C2×#88322—21no—
198C2×#88322—21no—
199C2×#88322—21no—
200C2×#88322C22×C412yes—
201C2×#88322C22×C412yes—
202C23×C53C22×D411yescharacteristic
203C22×D53—22no—
204C22×D53—22no—
205C22×D53—22no—
206C22×D53—41no—
207C22×D53—41no—
208C22×D53—41no—
209C22×D53—41no—
210C2×C4×C53—21no—
211C2×C4×C53—21no—
212C2×C4×C53—21no—
213C2×C4×C53—21no—
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—21no—
221C2×C4×C53—21no—
222C2×C4×C53—21no—
223C2×C4×C53—41no—
224C2×C4×C53—41no—
225C2×C4×C53—41no—
226C2×C4×C53—41no—
227C2×C4×C53—41no—
228C2×C4×C53—41no—
229C2×C4×C53—41no—
230C2×C4×C53—41no—
231C2×C4×C53—41no—
232C2×C4×C53—41no—
233C2×C4×C53C4×D412yes—
234C2×C4×C53C4×D412yes—
235C2×C4×C53C22⋊Q812yes—
236C2×C4×C53C22⋊Q812yes—
237C2×C4×C53#13512yes—
238C2×C4×C53#13512yes—
239C2×C4×C53AGL(1,ℤ/8)12yes—
240C2×C4×C53C23×C411yescharacteristic
241C2×C4×C53C22×Q811yescharacteristic
242C2×C4×C53C2×#4111yescharacteristic
243C2×C4×C53C2×#4111yescharacteristic
244C2×C4×C53C2×#4111yescharacteristic
245C2×C4×C53C2×#4111yescharacteristic
246C2×Dic53—22no—
247C2×Dic53—22no—
248C2×Dic53—22no—
249C2×Dic53—41no—
250C2×Dic53—41no—
251C2×Dic53—41no—
252C2×Dic53—41no—
253C2×Dic53—41no—
254C2×Dic53—41no—
255D212—22no—
256D212—22no—
257D212—22no—
258D212C4×Q814yes—
259C4×D53—48no—
260C4×D53—48no—
261C4×D53—48no—
262#88322—22no—
263#88322—22no—
264#88322—41no—
265#88322—41no—
266C4×#478—531nomaximal, Sylow
267C22×C53C2×C4×D411yescharacteristic
268C22×C53C2×C22⋊Q811yescharacteristic
269C22×C53#52511yescharacteristic
270C22×C53C2×#13511yescharacteristic
271C22×C53C2×#13511yescharacteristic
272C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
273C22×C53#57511yescharacteristic
274C2×D53—24no—
275C2×D53—42no—
276C2×D53—42no—
277C2×D53—42no—
278C2×D53—42no—
279C2×D53—42no—
280C2×D53—42no—
281C4×C53—22no—
282C4×C53—22no—
283C4×C53—22no—
284C4×C53—22no—
285C4×C53—41no—
286C4×C53—42no—
287C4×C53—42no—
288C4×C53—81no—
289C4×C53—81no—
290C4×C53#47814yescomplemented
291C4×C53C2×C4×Q811yescharacteristic
292C4×C53C4×#4111yescharacteristic
293C4×C53#52911yescharacteristic
294C4×C53#53311yescharacteristic
295Dic53—42no—
296Dic53—42no—
297Dic53—42no—
298C2×#478—531no—
299C4×#125D5311yescharacteristic, complemented
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×C53C4×C22⋊Q811yescharacteristic
314C2×C53C4×#13511yescharacteristic
315C2×C53C4×AGL(1,ℤ/8)11yescharacteristic
316C2×C53C2×#47811yescharacteristic
317C2×C53#307611yescharacteristic
318C2×C53#310211yescharacteristic
319C2×C53#310511yescharacteristic
320D53—44no—
321C2×#125C2×D5311yescharacteristic
322C2×C4×D4—531no—
323C2×C4×D4—531no—
324C2×C4×D4—531no—
325C22×C42—1061no—
326#478—538no—
327C2×#107—531no—
328C2×#107—531no—
329C2×#106—531no—
330C2×#106—531no—
331C2×C4⋊C8—531no—
332C2×C4⋊C8—531no—
333C2×#102—531no—
334C2×#102—531no—
335C2×#102—531no—
336C2×#102—531no—
337C2×C4×C8—531no—
338C2×C4×C8—531no—
339#389C2×D5311yescharacteristic
340#389C2×D5311yescharacteristic
341#387—21no—
342#387—21no—
343#382C2×D5311yescharacteristic
344C4×C4⋊C4—21no—
345C4×C4⋊C4—21no—
346C4×C4⋊C4—531no—
347C4×C4⋊C4—531no—
348C4×C4⋊C4—531no—
349C4×C4⋊C4C2×D5311yescharacteristic
350C4×C4⋊C4C2×D5311yescharacteristic
351C4×C22⋊C4—1061no—
352C4×C22⋊C4—1061no—
353C4×C22⋊C4—1061no—
354C43C2×D5311yescharacteristic
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×D4—531no—
367C23×C4—1061no—
368C23×C4—1061no—
369C23×C4—1061no—
370C22×C8—531no—
371C22×C8—531no—
372C22×C8—531no—
373C22×C8—531no—
374#125—22no—
375#125C4×D5314yescomplemented
376C4×D4—534no—
377C4×D4—534no—
378C4×D4—1062no—
379C4×D4—1062no—
380C4×D4—1062no—
381C4×D4—1062no—
382C2×C4⋊C4—21no—
383C2×C4⋊C4—21no—
384C2×C4⋊C4—21no—
385C2×C4⋊C4—21no—
386C2×C4⋊C4—531no—
387C2×C4⋊C4—531no—
388C2×C4⋊C4—531no—
389C2×C4⋊C4C22×D5311yescharacteristic
390C2×C4⋊C4C22×D5311yescharacteristic
391C2×C4⋊C4C22×D5311yescharacteristic
392C2×C4⋊C4C22×D5311yescharacteristic
393C2×C22⋊C4—1061no—
394C2×C22⋊C4—1061no—
395C2×C22⋊C4—1061no—
396C2×C42—21no—
397C2×C42—21no—
398C2×C42—1061no—
399C2×C42—1061no—
400C2×C42—1061no—
401C2×C42—2124no—
402C2×C42C22⋊D5311yescharacteristic
403C2×C42C22⋊D5311yescharacteristic
404C2×C42C22⋊D5311yescharacteristic
405C2×C42C22⋊D5311yescharacteristic
406C2×C42C22×D5311yescharacteristic
407C2×C42C22×D5311yescharacteristic
408C2×C42C22×D5311yescharacteristic
409#107—534no—
410#107—1062no—
411#106—534no—
412#106—1062no—
413C4⋊C8—534no—
414C4⋊C8—1062no—
415#102—534no—
416#102—534no—
417#102—1062no—
418#102—1062no—
419C4×C8—1062no—
420C4×C8—1062no—
421#95—21no—
422#95—21no—
423#95—21no—
424#95—21no—
425#95—21no—
426#95—21no—
427#95—1061no—
428#95—1061no—
429#95—1061no—
430C24—1061no—
431C2×D4—532no—
432C2×D4—532no—
433C2×D4—532no—
434C2×D4—1061no—
435C2×D4—1061no—
436C2×D4—1061no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—1064no—
442C22×C4—1061no—
443C22×C4—1061no—
444C22×C4—1061no—
445C22×C4—1064no—
446C22×C4—1064no—
447C22×C4—2122no—
448C22×C4—2122no—
449C22×C4—2122no—
450C22×C4—2122no—
451C22×C4—2122no—
452C22×C4—2122no—
453C22×C4C2×C22⋊D5311yescharacteristic
454C22×C4C2×C22⋊D5311yescharacteristic
455C22×C4C2×C22⋊D5311yescharacteristic
456C22×C4C2×C22⋊D5311yescharacteristic
457C22×C4C2×C22⋊D5311yescharacteristic
458C22×C4C2×C22⋊D5311yescharacteristic
459C22×C4C23×D5311yescharacteristic
460C2×C8—532no—
461C2×C8—532no—
462C2×C8—1061no—
463C2×C8—1061no—
464C2×C8—1061no—
465C2×C8—1061no—
466C2×C8—1061no—
467C2×C8—1061no—
468C2×C8—1061no—
469C2×C8—1061no—
470C2×C8—1061no—
471C2×C8—1061no—
472C4⋊C4—24no—
473C4⋊C4—24no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—42no—
481C4⋊C4—42no—
482C4⋊C4—532no—
483C4⋊C4—532no—
484C4⋊C4—532no—
485C4⋊C4—1061no—
486C4⋊C4—1061no—
487C4⋊C4—1061no—
488C4⋊C4C2×C4×D5312yes—
489C4⋊C4C2×C4×D5312yes—
490C22⋊C4—1064no—
491C22⋊C4—1064no—
492C22⋊C4—1064no—
493C42—22no—
494C42—22no—
495C42—22no—
496C42—21no—
497C42—21no—
498C42—41no—
499C42—42no—
500C42—42no—
501C42—2122no—
502C42—2122no—
503C42—2122no—
504C42C2×C4×D5312yes—
505C42#1171994512yes—
506C42Q8×D5312yes—
507C42#1171994912yes—
508C23—1062no—
509C23—1062no—
510C23—1062no—
511C23—2121no—
512C23—2121no—
513C23—2121no—
514C23—2121no—
515C23C22×C22⋊D5311yescharacteristic
516D4—534no—
517D4—1062no—
518D4—1062no—
519D4—1062no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—1062no—
544C2×C4—1062no—
545C2×C4—1062no—
546C2×C4—2128no—
547C2×C4—2128no—
548C2×C4—2121no—
549C2×C4—2121no—
550C2×C4—2121no—
551C2×C4—2121no—
552C2×C4—2121no—
553C2×C4—2121no—
554C2×C4—2128no—
555C2×C4C4×C22⋊D5312yes—
556C2×C4C4×C22⋊D5312yes—
557C2×C4#1313319312yes—
558C2×C4#1313319312yes—
559C2×C4#1313323712yes—
560C2×C4#1313323712yes—
561C2×C4#1313324212yes—
562C2×C4C22×C4×D5311yescharacteristic
563C2×C4C2×Q8×D5311yescharacteristic
564C2×C4#41×D5311yescharacteristic
565C2×C4#41×D5311yescharacteristic
566C2×C4C2×#1171994511yescharacteristic
567C2×C4C2×#1171994911yescharacteristic
568C8—1062no—
569C8—1062no—
570C8—2121no—
571C8—2121no—
572C22—1064no—
573C22—2122no—
574C22—2122no—
575C22—2122no—
576C22—2122no—
577C22—2122no—
578C22—2122no—
579C22C2×C4×C22⋊D5311yescharacteristic
580C22C2×#1313319311yescharacteristic
581C22#1681113011yescharacteristic
582C22C2×#1313323711yescharacteristic
583C22C2×#1313323711yescharacteristic
584C22C2×#1313324211yescharacteristic
585C22#1681142111yescharacteristic
586C4—22no—
587C4—22no—
588C4—22no—
589C4—22no—
590C4—41no—
591C4—42no—
592C4—42no—
593C4—81no—
594C4—81no—
595C4—2122no—
596C4—2122no—
597C4—2122no—
598C4#1681092014yesdirect product factor
599C4C4×Q8×D5311yescharacteristic
600C4C4×#1171994911yescharacteristic
601C4#1681116011yescharacteristic
602C4#1681117711yescharacteristic
603C2—2124no—
604C2C4×#1313319311yescharacteristic
605C2C4×#1313323711yescharacteristic
606C2C4×#1313324211yescharacteristic
607C2C2×#1681092011yescharacteristic
608C2#2028449711yescharacteristic
609C2#2028472711yescharacteristic
610C2#2028475811yescharacteristic
subgroupG/H
centerC22×C4C2×C22⋊D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC4×#125×C53C2
socleC23×C53C22×D4
prime radicalC22×D212C23
cosocleC22×C4×C53C24
hypercenterC4×#125D53
hypocenterC53C4×#478
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22×C4 ◁ C43 ◁ C4×#125

Complements

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

NH ≅ G/Ntype
3. #20275673C2semidirect
4. #20275481C2semidirect
5. #20275304C2semidirect
6. #20275290C2semidirect
10. C4×#13133150C2semidirect
11. C4×#13133148C2semidirect
12. C4×C4⋊#88322C2semidirect
15. C4×#125×C53C2semidirect
20. #16810920C4direct (N × H)
85. #13133150C2×C4semidirect
87. #13133148C2×C4semidirect
89. C4⋊#88322C2×C4semidirect
100. #125×C53C2×C4semidirect
290. C4×C53#478semidirect
299. C4×#125D53semidirect
365. C53C4×#478semidirect
375. #125C4×D53semidirect
598. C4#16810920direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168109202
C42×D2122
#202756732
#202754812
#202753042
#202752902
#202749752
#202749612
#202747222
C4×#131331502
C4×#131331482
C4×C4⋊#883222
C4×#131331372
C4×#125×C532
C4×#47853