Finite Group Databasebeta

← C2×#20273104 #27350447

Subgroups — C2×#20273104 #27350447

Order 13568.

1714 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×D4×D53C211yesmaximal, characteristic
2C2×#16810675C211yesmaximal, characteristic, complemented
3C2×#16810674C211yesmaximal, characteristic, complemented
4#20273104C218yesmaximal, direct product factor
5C2×#16810352C211yesmaximal, characteristic, complemented
6C2×D4⋊#88322C211yesmaximal, characteristic, complemented
7C2×#16810350C211yesmaximal, characteristic, complemented
8C2×C53×D4⋊C8C211yesmaximal, characteristic, complemented
9C22×D4×D53C411yescharacteristic
10C23×C4×D53—21no—
11C2×C4×D212C2211yescharacteristic
12C2×D4×Dic53C411yescharacteristic
13C2×#13133179C411yescharacteristic
14C2×C4×C22⋊D53—21no—
15C2×#13133177—21no—
16C4×D4×D53—22no—
17C4×D4×D53—22no—
18C4×D4×D53C414yes—
19C4×D4×D53C2214yes—
20C2×C4⋊C4×D53C411yescharacteristic
21C2×C22⋊C4×D53—21no—
22C2×C42×D53C2211yescharacteristic
23#16810675—22no—
24#16810675C2214yescomplemented
25#16810674—22no—
26#16810674C2214yescomplemented
27C2×C4⋊#88322C2211yescharacteristic, complemented
28C2×#13133146—21no—
29C2×#13133129—21no—
30C2×#13133129—21no—
31C2×C4×#88322C2211yescharacteristic, complemented
32C2×C8×Dic53—21no—
33#16810352C2214yescomplemented
34D4⋊#88322C2214yescomplemented
35#16810350C2214yescomplemented
36C2×C4×D4×C53C2211yescharacteristic
37C2×C4⋊C8×C53C2211yescharacteristic, complemented
38C2×C4×C8×C53C2211yescharacteristic, complemented
39C53×D4⋊C8C2214yescomplemented
40C24×D53—21no—
41C22×D212C2×C411yescharacteristic
42C22×C22⋊D53—21no—
43C2×D4×D53—22no—
44C2×D4×D53—24no—
45C2×D4×D53C818yescomplemented
46C2×D4×D53C2×C414yes—
47C23×Dic53—21no—
48C22×C4×D53—21no—
49C22×C4×D53—21no—
50C22×C4×D53—21no—
51C22×C4×D53—24no—
52C22×C4×D53—24no—
53C22×C4×D53—44no—
54C22×C4×D53—44no—
55C22×C4×D53C2×C411yescharacteristic
56C22×C4×D53D411yescharacteristic
57C22×C4×D53D411yescharacteristic
58C22×#88322—21no—
59C22×#88322—21no—
60C4×D212—21no—
61C4×D212—21no—
62C4×D212C2×C412yes—
63C4×D212C2312yes—
64D4×Dic53—22no—
65D4×Dic53C2×C414yes—
66#13133179—22no—
67#13133179C2×C414yes—
68C4×C22⋊D53—24no—
69C4×C22⋊D53—24no—
70#13133177—24no—
71#13133177—24no—
72C2×C4⋊Dic53C2×C411yescharacteristic
73C2×#11719919—21no—
74C4⋊C4×D53—22no—
75C4⋊C4×D53C2×C414yes—
76C2×C22⋊Dic53—21no—
77C2×#11719921—21no—
78C2×#11719921—21no—
79C22⋊C4×D53—24no—
80C22⋊C4×D53—24no—
81C2×C4×Dic53C2×C411yescharacteristic
82C2×C4×Dic53D411yescharacteristic
83C2×C4×Dic53D411yescharacteristic
84C42×D53C2×C412yes—
85C42×D53D412yes—
86C42×D53D412yes—
87C42×D53C2312yes—
88C4⋊#88322—21no—
89C4⋊#88322C2312yescomplemented
90#13133146—24no—
91#13133129—24no—
92#13133129—24no—
93C4×#88322—21no—
94C4×#88322C2312yescomplemented
95C8×Dic53—24no—
96C22×D4×C53C2×C411yescharacteristic
97C23×C4×C53—21no—
98C22×C8×C53—21no—
99C22×C8×C53—21no—
100C4×D4×C53—21no—
101C4×D4×C53—21no—
102C4×D4×C53C2×C412yes—
103C4×D4×C53C2312yes—
104C2×C4⋊C4×C53C2×C411yescharacteristic
105C2×C22⋊C4×C53—21no—
106C2×C42×C53C2311yescharacteristic
107C4⋊C8×C53—21no—
108C4⋊C8×C53C2312yescomplemented
109C4×C8×C53—21no—
110C4×C8×C53C2312yescomplemented
111C24×C53—21no—
112C23×D53—21no—
113C23×D53—28no—
114C23×D53—24no—
115C23×D53—44no—
116C23×D53—44no—
117C23×D53C22⋊C411yescharacteristic
118C2×D4×C53—21no—
119C2×D4×C53—22no—
120C2×D4×C53C2×C814yes—
121C2×D4×C53C22×C412yes—
122C22×C4×C53—21no—
123C22×C4×C53—22no—
124C22×C4×C53—22no—
125C22×C4×C53—42no—
126C22×C4×C53—42no—
127C22×C4×C53C22×C411yescharacteristic
128C22×C4×C53C2×D411yescharacteristic
129C22×C4×C53C2×D411yescharacteristic
130C2×C22⋊D53—28no—
131C2×C22⋊D53—24no—
132C22×Dic53—21no—
133C22×Dic53—24no—
134C22×Dic53—44no—
135C22×Dic53C22⋊C411yescharacteristic
136C22×Dic53C22⋊C411yescharacteristic
137C22×Dic53C22⋊C411yescharacteristic
138D4×D53—216no—
139D4×D53—28no—
140D4×D53C2×C8116yescomplemented
141C2×D212—21no—
142C2×D212—22no—
143C2×D212C2×C814yes—
144C2×D212C22×C412yes—
145C2×C4×D53—22no—
146C2×C4×D53—22no—
147C2×C4×D53—24no—
148C2×C4×D53—22no—
149C2×C4×D53—22no—
150C2×C4×D53—24no—
151C2×C4×D53—48no—
152C2×C4×D53—48no—
153C2×C4×D53—416no—
154C2×C4×D53—42no—
155C2×C4×D53—44no—
156C2×C4×D53—44no—
157C2×C4×D53—42no—
158C2×C4×D53—42no—
159C2×C4×D53C22⋊C412yes—
160C2×C4×D53C22⋊C412yes—
161C2×C4×D53C22⋊C412yes—
162C2×C4×D53C2×C814yes—
163C2×C4×D53M1614yes—
164C2×C4×D53C22×C412yes—
165C2×C4×D53C2×D412yes—
166C2×C4×D53C2×D412yes—
167C2×C8×C53—22no—
168C2×C8×C53—22no—
169C2×C8×C53—42no—
170C2×C8×C53—42no—
171C4⋊C4×C53—21no—
172C4⋊C4×C53C22×C412yes—
173C22⋊C4×C53—22no—
174C22⋊C4×C53—22no—
175C42×C53C22×C411yescharacteristic
176C42×C53C2×D411yescharacteristic
177C42×C53C2×D411yescharacteristic
178C42×C53C2411yescharacteristic
179C22⋊Dic53—24no—
180#11719921—22no—
181#11719921—22no—
182#11719921—24no—
183C4⋊Dic53—21no—
184C4⋊Dic53C22×C412yes—
185#11719919—24no—
186C4×Dic53C22⋊C412yes—
187C4×Dic53C22⋊C412yes—
188C4×Dic53C22⋊C412yes—
189C4×Dic53C22×C412yes—
190C4×Dic53C2×D412yes—
191C4×Dic53C2×D412yes—
192C2×#88322—22no—
193C2×#88322—22no—
194C2×#88322—42no—
195C2×#88322—42no—
196C23×C53—22no—
197C23×C53—24no—
198C23×C53—42no—
199C23×C53—42no—
200C23×C53C2×C22⋊C411yescharacteristic
201C22×D53—22no—
202C22×D53—22no—
203C22×D53—22no—
204C22×D53—24no—
205C22×D53—216no—
206C22×D53—48no—
207C22×D53—48no—
208C22×D53—432no—
209C22×D53—42no—
210C22×D53—42no—
211C22×D53C22⋊C814yes—
212C22×D53C2×C22⋊C412yes—
213D4×C53—22no—
214D4×C53—24no—
215D4×C53C22×C814yes—
216C2×C4×C53—22no—
217C2×C4×C53—22no—
218C2×C4×C53—42no—
219C2×C4×C53—42no—
220C2×C4×C53—42no—
221C2×C4×C53—44no—
222C2×C4×C53C2×C22⋊C411yescharacteristic
223C2×C4×C53C2×C22⋊C411yescharacteristic
224C2×C4×C53C2×C22⋊C411yescharacteristic
225C2×C4×C53C22×C812yes—
226C2×C4×C53C2×M1612yes—
227C2×C4×C53AGL(1,ℤ/8)12yes—
228C2×C4×C53AGL(1,ℤ/8)12yes—
229C2×C4×C53C23×C411yescharacteristic
230C2×C4×C53C22×D411yescharacteristic
231C2×C4×C53C22×D411yescharacteristic
232C22⋊D53—216no—
233C2×Dic53—22no—
234C2×Dic53—22no—
235C2×Dic53—22no—
236C2×Dic53—48no—
237C2×Dic53—42no—
238C2×Dic53C22⋊C814yes—
239C2×Dic53C2×C22⋊C412yes—
240C2×Dic53C2×C22⋊C412yes—
241C2×Dic53C2×C22⋊C412yes—
242D212—22no—
243D212—24no—
244D212C22×C814yes—
245C4×D53—22no—
246C4×D53—22no—
247C4×D53—24no—
248C4×D53—42no—
249C4×D53—42no—
250C4×D53—48no—
251C4×D53—48no—
252C4×D53—42no—
253C4×D53—42no—
254C4×D53—44no—
255C4×D53C22⋊C818yes—
256C4×D53C22×C814yes—
257C4×D53C2×M1614yes—
258C8×C53—42no—
259C8×C53—42no—
260#88322—42no—
261#88322—42no—
262C2×#1478—531nomaximal, Sylow
263C22×C53—24no—
264C22×C53—42no—
265C22×C53—42no—
266C22×C53—48no—
267C22×C53C2×C22⋊C812yes—
268C22×C53#41812yes—
269C22×C53#42112yes—
270C22×C53C22×C22⋊C411yescharacteristic
271C2×D53—22no—
272C2×D53—22no—
273C2×D53—28no—
274C2×D53—24no—
275C2×D53—432no—
276C2×D53—42no—
277C2×D53—42no—
278C2×D53—48no—
279C2×D53C2×C22⋊C814yes—
280C4×C53—21no—
281C4×C53—21no—
282C4×C53—42no—
283C4×C53C2×C22⋊C812yes—
284C4×C53#41812yes—
285C4×C53C23×C811yescharacteristic
286C4×C53C22×M1611yescharacteristic
287C4×C53C2×AGL(1,ℤ/8)11yescharacteristic
288C4×C53C2×AGL(1,ℤ/8)11yescharacteristic
289Dic53—22no—
290Dic53—22no—
291Dic53—42no—
292Dic53C2×C22⋊C814yes—
293C22×C4×D4—531no—
294C2×#431—531no—
295C2×#431—531no—
296#1478—538no—
297C2×D4⋊C8—531no—
298C2×D4⋊C8—531no—
299C2×D4⋊C8—531no—
300C2×D4⋊C8D5311yescharacteristic, complemented
301C2×C53—44no—
302C2×C53#147814yescomplemented
303C2×C53C22×C22⋊C811yescharacteristic
304C2×C53C2×#41811yescharacteristic
305C2×C53C2×#42111yescharacteristic
306D53—24no—
307D53—44no—
308C23×D4—531no—
309C24×C4—1061no—
310C2×C4×D4—531no—
311C2×C4×D4—534no—
312C2×C4×D4—534no—
313C2×C4×D4—531no—
314C2×C4×D4—531no—
315C2×C4×D4—1061no—
316C2×C4×D4—1062no—
317C2×C4×D4—1062no—
318C2×C4×D4—1061no—
319C2×C4×D4C2×D5311yescharacteristic
320C22×C4⋊C4—531no—
321C22×C22⋊C4—1061no—
322C22×C42—531no—
323#431—534no—
324#431—534no—
325#431—1062no—
326#431—1062no—
327C2×C4⋊C8—531no—
328C2×C4⋊C8—1061no—
329C2×C4⋊C8C2×D5311yescharacteristic, complemented
330C2×C22⋊C8—1061no—
331C2×C22⋊C8—1061no—
332C2×C4×C8—531no—
333C2×C4×C8—1061no—
334C2×C4×C8C2×D5311yescharacteristic, complemented
335D4⋊C8—534no—
336D4⋊C8—534no—
337D4⋊C8—534no—
338D4⋊C8C2×D5314yescomplemented
339C53C2×#147811yescharacteristic, complemented, Sylow
340C25—1061no—
341C22×D4—534no—
342C22×D4—538no—
343C22×D4—531no—
344C22×D4—1062no—
345C22×D4—1064no—
346C22×D4—1061no—
347C22×D4C4×D5311yescharacteristic
348C23×C4—21no—
349C23×C4—531no—
350C23×C4—531no—
351C23×C4—531no—
352C23×C4—1061no—
353C23×C4—1064no—
354C23×C4—1064no—
355C23×C4—1061no—
356C23×C4—1061no—
357C23×C4—1061no—
358C23×C4—2124no—
359C23×C4—2124no—
360C22×C8—21no—
361C22×C8—21no—
362C22×C8—1061no—
363C22×C8—1061no—
364C4×D4—21no—
365C4×D4—21no—
366C4×D4—534no—
367C4×D4—534no—
368C4×D4—532no—
369C4×D4—532no—
370C4×D4—1064no—
371C4×D4—1064no—
372C4×D4—1062no—
373C4×D4—1064no—
374C4×D4—1064no—
375C4×D4—1062no—
376C4×D4—1061no—
377C4×D4—1061no—
378C4×D4C4×D5312yes—
379C4×D4C22×D5312yes—
380C2×C4⋊C4—531no—
381C2×C4⋊C4—534no—
382C2×C4⋊C4—1062no—
383C2×C4⋊C4—1061no—
384C2×C4⋊C4C4×D5311yescharacteristic
385C2×C22⋊C4—21no—
386C2×C22⋊C4—1064no—
387C2×C22⋊C4—1064no—
388C2×C22⋊C4—1061no—
389C2×C22⋊C4—1061no—
390C2×C22⋊C4—1061no—
391C2×C42—531no—
392C2×C42—531no—
393C2×C42—531no—
394C2×C42—532no—
395C2×C42—532no—
396C2×C42—532no—
397C2×C42—532no—
398C2×C42C22×D5311yescharacteristic
399C4⋊C8—21no—
400C4⋊C8—532no—
401C4⋊C8—1064no—
402C4⋊C8—1061no—
403C4⋊C8C22×D5312yescomplemented
404C22⋊C8—1064no—
405C22⋊C8—1064no—
406C4×C8—21no—
407C4×C8—532no—
408C4×C8—1064no—
409C4×C8—1061no—
410C4×C8C22×D5312yescomplemented
411C24—21no—
412C24—531no—
413C24—1064no—
414C24—1068no—
415C24—1061no—
416C24—2124no—
417C24—2124no—
418C2×D4—21no—
419C2×D4—22no—
420C2×D4—5316no—
421C2×D4—534no—
422C2×D4—532no—
423C2×D4—10616no—
424C2×D4—1068no—
425C2×D4—1064no—
426C2×D4—1068no—
427C2×D4—1061no—
428C2×D4—1062no—
429C2×D4C8×D5314yes—
430C2×D4C2×C4×D5312yes—
431C22×C4—21no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—531no—
437C22×C4—532no—
438C22×C4—534no—
439C22×C4—534no—
440C22×C4—532no—
441C22×C4—531no—
442C22×C4—531no—
443C22×C4—532no—
444C22×C4—532no—
445C22×C4—532no—
446C22×C4—532no—
447C22×C4—1064no—
448C22×C4—1064no—
449C22×C4—1064no—
450C22×C4—1062no—
451C22×C4—1062no—
452C22×C4—1061no—
453C22×C4—1062no—
454C22×C4—1062no—
455C22×C4—2122no—
456C22×C4—2124no—
457C22×C4—21216no—
458C22×C4—2128no—
459C22×C4—2128no—
460C22×C4—2124no—
461C22×C4—2124no—
462C22×C4—2122no—
463C22×C4—2122no—
464C22×C4C2×C4×D5311yescharacteristic
465C22×C4D4×D5311yescharacteristic
466C22×C4D4×D5311yescharacteristic
467C2×C8—22no—
468C2×C8—22no—
469C2×C8—42no—
470C2×C8—42no—
471C2×C8—1062no—
472C2×C8—1062no—
473C2×C8—2122no—
474C2×C8—2122no—
475C4⋊C4—21no—
476C4⋊C4—532no—
477C4⋊C4—1064no—
478C4⋊C4—1061no—
479C4⋊C4C2×C4×D5312yes—
480C22⋊C4—22no—
481C22⋊C4—22no—
482C22⋊C4—1064no—
483C22⋊C4—1064no—
484C22⋊C4—1062no—
485C22⋊C4—1062no—
486C42—532no—
487C42—532no—
488C42—532no—
489C42—532no—
490C42—532no—
491C42—532no—
492C42C2×C4×D5311yescharacteristic
493C42D4×D5311yescharacteristic
494C42D4×D5311yescharacteristic
495C42C23×D5311yescharacteristic
496C23—22no—
497C23—24no—
498C23—42no—
499C23—42no—
500C23—532no—
501C23—534no—
502C23—10616no—
503C23—1062no—
504C23—1062no—
505C23—1062no—
506C23—1064no—
507C23—2128no—
508C23—2128no—
509C23—21232no—
510C23—2122no—
511C23—2122no—
512C23C22⋊C4×D5311yescharacteristic
513D4—24no—
514D4—22no—
515D4—534no—
516D4—10616no—
517D4—1064no—
518D4—1062no—
519D4C2×C8×D5314yes—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—42no—
523C2×C4—44no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—532no—
527C2×C4—534no—
528C2×C4—538no—
529C2×C4—534no—
530C2×C4—534no—
531C2×C4—532no—
532C2×C4—532no—
533C2×C4—1062no—
534C2×C4—1062no—
535C2×C4—1064no—
536C2×C4—1062no—
537C2×C4—1062no—
538C2×C4—1062no—
539C2×C4—2122no—
540C2×C4—2122no—
541C2×C4—2128no—
542C2×C4—2128no—
543C2×C4—2128no—
544C2×C4—2122no—
545C2×C4—2124no—
546C2×C4—2122no—
547C2×C4—2122no—
548C2×C4C22⋊C4×D5311yescharacteristic
549C2×C4C22⋊C4×D5311yescharacteristic
550C2×C4C22⋊C4×D5311yescharacteristic
551C2×C4C2×C8×D5312yes—
552C2×C4M16×D5312yes—
553C2×C4#1313324112yes—
554C2×C4#1313324412yes—
555C2×C4C22×C4×D5311yescharacteristic
556C2×C4C2×D4×D5311yescharacteristic
557C2×C4C2×D4×D5311yescharacteristic
558C8—42no—
559C8—42no—
560C8—2122no—
561C8—2122no—
562C22—24no—
563C22—42no—
564C22—42no—
565C22—48no—
566C22—534no—
567C22—1062no—
568C22—1062no—
569C22—1068no—
570C22—1064no—
571C22—21232no—
572C22—2122no—
573C22—2122no—
574C22—2128no—
575C22C22⋊C8×D5312yes—
576C22#1681062612yes—
577C22#1681063812yes—
578C22C2×C22⋊C4×D5311yescharacteristic
579C4—21no—
580C4—21no—
581C4—42no—
582C4—534no—
583C4—1062no—
584C4—1062no—
585C4—2122no—
586C4C22⋊C8×D5312yes—
587C4#1681062612yes—
588C4C22×C8×D5311yescharacteristic
589C4C2×M16×D5311yescharacteristic
590C4C2×#1313324111yescharacteristic
591C4C2×#1313324411yescharacteristic
592C2—44no—
593C2—1064no—
594C2—2124no—
595C2#2027310414yesdirect product factor
596C2C2×C22⋊C8×D5311yescharacteristic
597C2C2×#1681062611yescharacteristic
598C2C2×#1681063811yescharacteristic
subgroupG/H
centerC22×C4D4×D53
commutatorC4×C53C23×C8
FrattiniC42C23×D53
FittingC2×C53×D4⋊C8C2
socleC23×C53C2×C22⋊C4
prime radicalC22×D4×D53C4
cosocleC42×C53C24
hypercenterC2×D4⋊C8D53
hypocenterC53C2×#1478
LCSG ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22×C4 ◁ C2×C42 ◁ C2×D4⋊C8

Complements

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

NH ≅ G/Ntype
2. C2×#16810675C2semidirect
3. C2×#16810674C2semidirect
4. #20273104C2direct (N × H)
5. C2×#16810352C2semidirect
6. C2×D4⋊#88322C2semidirect
7. C2×#16810350C2semidirect
8. C2×C53×D4⋊C8C2semidirect
24. #16810675C22semidirect
26. #16810674C22semidirect
27. C2×C4⋊#88322C22semidirect
31. C2×C4×#88322C22semidirect
33. #16810352C22semidirect
34. D4⋊#88322C22semidirect
35. #16810350C22semidirect
37. C2×C4⋊C8×C53C22semidirect
38. C2×C4×C8×C53C22semidirect
39. C53×D4⋊C8C22semidirect
45. C2×D4×D53C8semidirect
89. C4⋊#88322C23semidirect
94. C4×#88322C23semidirect
108. C4⋊C8×C53C23semidirect
110. C4×C8×C53C23semidirect
140. D4×D53C2×C8semidirect
300. C2×D4⋊C8D53semidirect
302. C2×C53#1478semidirect
329. C2×C4⋊C8C2×D53semidirect
334. C2×C4×C8C2×D53semidirect
338. D4⋊C8C2×D53semidirect
339. C53C2×#1478semidirect
403. C4⋊C8C22×D53semidirect
410. C4×C8C22×D53semidirect
595. C2#20273104direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C4×D4×D532
C2×#168106752
C2×#168106742
#202731042
C2×#168103522
C2×D4⋊#883222
C2×#168103502
C2×C53×D4⋊C82
C2×#147853