Finite Group Databasebeta

← C2×#20273138 #27350484

Subgroups — C2×#20273138 #27350484

Order 13568.

1714 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×D4×D53C211yesmaximal, characteristic
2C2×#16810674C211yesmaximal, characteristic, complemented
3C2×#16810645C211yesmaximal, characteristic, complemented
4#20273138C218yesmaximal, 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—
11C22×#11719912—21no—
12C2×C4×D212C2211yescharacteristic
13C2×D4×Dic53C411yescharacteristic
14C2×#13133179C411yescharacteristic
15C2×C4×C22⋊D53—21no—
16C2×#13133177—21no—
17C4×D4×D53—22no—
18C4×D4×D53—22no—
19C4×D4×D53C414yes—
20C4×D4×D53C2214yes—
21C2×C4⋊C4×D53C411yescharacteristic
22C2×C22⋊C4×D53—21no—
23C2×C42×D53C2211yescharacteristic
24#16810674—22no—
25#16810674C2214yescomplemented
26#16810645—22no—
27#16810645C2214yescomplemented
28C2×C4⋊#88322C2211yescharacteristic, complemented
29C2×#13133146—21no—
30C2×#13133146—21no—
31C2×#13133129—21no—
32C2×C4×#88322C2211yescharacteristic, complemented
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—24no—
44C2×D4×D53—24no—
45C2×D4×D53—22no—
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
58C2×#11719912—24no—
59C2×#11719912—44no—
60C22×#88322—21no—
61C22×#88322—21no—
62C4×D212—21no—
63C4×D212—21no—
64C4×D212C2×C412yes—
65C4×D212C2312yes—
66D4×Dic53—22no—
67D4×Dic53C2×C414yes—
68#13133179—22no—
69#13133179C2×C414yes—
70C4×C22⋊D53—24no—
71C4×C22⋊D53—24no—
72#13133177—24no—
73#13133177—24no—
74C2×C4⋊Dic53C2×C411yescharacteristic
75C2×#11719919—21no—
76C4⋊C4×D53—22no—
77C4⋊C4×D53C2×C414yes—
78C2×C22⋊Dic53—21no—
79C2×#11719921—21no—
80C2×#11719921—21no—
81C22⋊C4×D53—24no—
82C22⋊C4×D53—24no—
83C2×C4×Dic53C2×C411yescharacteristic
84C2×C4×Dic53D411yescharacteristic
85C2×C4×Dic53D411yescharacteristic
86C42×D53C2×C412yes—
87C42×D53D412yes—
88C42×D53D412yes—
89C42×D53C2312yes—
90C4⋊#88322—21no—
91C4⋊#88322C2312yescomplemented
92#13133146—24no—
93#13133146—24no—
94#13133129—24no—
95C4×#88322—21no—
96C4×#88322C2312yescomplemented
97C22×D4×C53C2×C411yescharacteristic
98C23×C4×C53—21no—
99C22×C8×C53—21no—
100C22×C8×C53—21no—
101C4×D4×C53—21no—
102C4×D4×C53—21no—
103C4×D4×C53C2×C412yes—
104C4×D4×C53C2312yes—
105C2×C4⋊C4×C53C2×C411yescharacteristic
106C2×C22⋊C4×C53—21no—
107C2×C42×C53C2311yescharacteristic
108C4⋊C8×C53—21no—
109C4⋊C8×C53C2312yescomplemented
110C4×C8×C53—21no—
111C4×C8×C53C2312yescomplemented
112C24×C53—21no—
113C23×D53—28no—
114C23×D53—21no—
115C23×D53—24no—
116C23×D53—44no—
117C23×D53—44no—
118C23×D53C22⋊C411yescharacteristic
119C2×D4×C53—21no—
120C2×D4×C53—22no—
121C2×D4×C53M1614yescomplemented
122C2×D4×C53C22×C412yes—
123C22×C4×C53—21no—
124C22×C4×C53—22no—
125C22×C4×C53—22no—
126C22×C4×C53—42no—
127C22×C4×C53—42no—
128C22×C4×C53C22×C411yescharacteristic
129C22×C4×C53C2×D411yescharacteristic
130C22×C4×C53C2×D411yescharacteristic
131C2×C22⋊D53—28no—
132C2×C22⋊D53—24no—
133C22×Dic53—21no—
134C22×Dic53—24no—
135C22×Dic53—44no—
136C22×Dic53C22⋊C411yescharacteristic
137C22×Dic53C22⋊C411yescharacteristic
138C22×Dic53C22⋊C411yescharacteristic
139D4×D53—216no—
140D4×D53—28no—
141D4×D53—28no—
142C2×D212—21no—
143C2×D212—22no—
144C2×D212M1614yescomplemented
145C2×D212C22×C412yes—
146C2×C4×D53—22no—
147C2×C4×D53—22no—
148C2×C4×D53—24no—
149C2×C4×D53—22no—
150C2×C4×D53—22no—
151C2×C4×D53—24no—
152C2×C4×D53—42no—
153C2×C4×D53—44no—
154C2×C4×D53—42no—
155C2×C4×D53—42no—
156C2×C4×D53—44no—
157C2×C4×D53—416no—
158C2×C4×D53—48no—
159C2×C4×D53—48no—
160C2×C4×D53C22⋊C412yes—
161C2×C4×D53C22⋊C412yes—
162C2×C4×D53C22⋊C412yes—
163C2×C4×D53D814yes—
164C2×C4×D53SD1614yes—
165C2×C4×D53C22×C412yes—
166C2×C4×D53C2×D412yes—
167C2×C4×D53C2×D412yes—
168C2×C8×C53—22no—
169C2×C8×C53—22no—
170C2×C8×C53—42no—
171C2×C8×C53—42no—
172C4⋊C4×C53—21no—
173C4⋊C4×C53C22×C412yes—
174C22⋊C4×C53—22no—
175C22⋊C4×C53—22no—
176C42×C53C22×C411yescharacteristic
177C42×C53C2×D411yescharacteristic
178C42×C53C2×D411yescharacteristic
179C42×C53C2411yescharacteristic
180C22⋊Dic53—24no—
181#11719921—22no—
182#11719921—22no—
183#11719921—24no—
184C4⋊Dic53—21no—
185C4⋊Dic53C22×C412yes—
186#11719919—24no—
187C4×Dic53C22⋊C412yes—
188C4×Dic53C22⋊C412yes—
189C4×Dic53C22⋊C412yes—
190C4×Dic53C22×C412yes—
191C4×Dic53C2×D412yes—
192C4×Dic53C2×D412yes—
193C2×#88322—22no—
194C2×#88322—22no—
195C2×#88322—42no—
196C2×#88322—42no—
197#11719912—48no—
198C23×C53—22no—
199C23×C53—24no—
200C23×C53—42no—
201C23×C53—42no—
202C23×C53C2×C22⋊C411yescharacteristic
203C22×D53—22no—
204C22×D53—22no—
205C22×D53—216no—
206C22×D53—22no—
207C22×D53—24no—
208C22×D53—48no—
209C22×D53—432no—
210C22×D53—42no—
211C22×D53—42no—
212C22×D53—48no—
213C22×D53#10214yes—
214C22×D53C2×C22⋊C412yes—
215D4×C53—22no—
216D4×C53—24no—
217D4×C53C2×M1614yescomplemented
218C2×C4×C53—22no—
219C2×C4×C53—22no—
220C2×C4×C53—42no—
221C2×C4×C53—42no—
222C2×C4×C53—42no—
223C2×C4×C53—44no—
224C2×C4×C53C2×C22⋊C411yescharacteristic
225C2×C4×C53C2×C22⋊C411yescharacteristic
226C2×C4×C53C2×C22⋊C411yescharacteristic
227C2×C4×C53C2×M1612yes—
228C2×C4×C53C2×M1612yes—
229C2×C4×C53C2×D812yes—
230C2×C4×C53C2×SD1612yes—
231C2×C4×C53C23×C411yescharacteristic
232C2×C4×C53C22×D411yescharacteristic
233C2×C4×C53C22×D411yescharacteristic
234C22⋊D53—216no—
235C2×Dic53—22no—
236C2×Dic53—22no—
237C2×Dic53—22no—
238C2×Dic53—42no—
239C2×Dic53—48no—
240C2×Dic53#10214yes—
241C2×Dic53C2×C22⋊C412yes—
242C2×Dic53C2×C22⋊C412yes—
243C2×Dic53C2×C22⋊C412yes—
244D212—22no—
245D212—24no—
246D212C2×M1614yescomplemented
247C4×D53—22no—
248C4×D53—22no—
249C4×D53—24no—
250C4×D53—42no—
251C4×D53—42no—
252C4×D53—48no—
253C4×D53—42no—
254C4×D53—42no—
255C4×D53—44no—
256C4×D53—48no—
257C4×D53#10218yes—
258C4×D53C2×D814yes—
259C4×D53C2×SD1614yes—
260C8×C53—42no—
261C8×C53—42no—
262#88322—42no—
263#88322—42no—
264C2×#1486—531nomaximal, Sylow
265C22×C53—24no—
266C22×C53—42no—
267C22×C53—42no—
268C22×C53—48no—
269C22×C53#40712yes—
270C22×C53C2×#10212yes—
271C22×C53#42112yes—
272C22×C53C22×C22⋊C411yescharacteristic
273C2×D53—22no—
274C2×D53—22no—
275C2×D53—28no—
276C2×D53—24no—
277C2×D53—432no—
278C2×D53—42no—
279C2×D53—42no—
280C2×D53—48no—
281C2×D53C2×#10214yes—
282C4×C53—21no—
283C4×C53—21no—
284C4×C53—42no—
285C4×C53#40712yes—
286C4×C53C2×#10212yes—
287C4×C53C22×M1611yescharacteristic
288C4×C53C22×M1611yescharacteristic
289C4×C53C22×D811yescharacteristic
290C4×C53C22×SD1611yescharacteristic
291Dic53—22no—
292Dic53—22no—
293Dic53—42no—
294Dic53C2×#10214yes—
295C22×C4×D4—531no—
296C2×#431—531no—
297C2×#423—531no—
298#1486—538no—
299C2×D4⋊C8—531no—
300C2×D4⋊C8—531no—
301C2×D4⋊C8—531no—
302C2×D4⋊C8D5311yescharacteristic, complemented
303C2×C53—44no—
304C2×C53#148614yescomplemented
305C2×C53C2×#40711yescharacteristic
306C2×C53C22×#10211yescharacteristic
307C2×C53C2×#42111yescharacteristic
308D53—24no—
309D53—44no—
310C23×D4—531no—
311C24×C4—1061no—
312C22×M16—1061no—
313C2×C4×D4—531no—
314C2×C4×D4—534no—
315C2×C4×D4—534no—
316C2×C4×D4—531no—
317C2×C4×D4—531no—
318C2×C4×D4—1061no—
319C2×C4×D4—1062no—
320C2×C4×D4—1062no—
321C2×C4×D4—1061no—
322C2×C4×D4C2×D5311yescharacteristic
323C22×C4⋊C4—531no—
324C22×C22⋊C4—1061no—
325C22×C42—531no—
326#431—534no—
327#431—1062no—
328#423—534no—
329#423—1062no—
330C2×C4⋊C8—531no—
331C2×C4⋊C8—1061no—
332C2×C4⋊C8—1061no—
333C2×C4⋊C8C2×D5311yescharacteristic, complemented
334C2×C22⋊C8—1061no—
335C2×C4×C8—531no—
336C2×C4×C8C2×D5311yescharacteristic, complemented
337D4⋊C8—534no—
338D4⋊C8—534no—
339D4⋊C8—534no—
340D4⋊C8C2×D5314yescomplemented
341C53C2×#148611yescharacteristic, complemented, Sylow
342C25—1061no—
343C22×D4—534no—
344C22×D4—531no—
345C22×D4—1062no—
346C22×D4—1061no—
347C22×D4—1064no—
348C22×D4—1064no—
349C22×D4C4×D5311yescharacteristic
350C23×C4—21no—
351C23×C4—531no—
352C23×C4—531no—
353C23×C4—531no—
354C23×C4—1061no—
355C23×C4—1064no—
356C23×C4—1064no—
357C23×C4—1061no—
358C23×C4—1061no—
359C23×C4—1061no—
360C23×C4—2124no—
361C23×C4—2124no—
362C2×M16—1064no—
363C2×M16—2124no—
364C22×C8—21no—
365C22×C8—21no—
366C22×C8—1061no—
367C22×C8—1061no—
368C4×D4—21no—
369C4×D4—21no—
370C4×D4—534no—
371C4×D4—534no—
372C4×D4—532no—
373C4×D4—532no—
374C4×D4—1064no—
375C4×D4—1064no—
376C4×D4—1062no—
377C4×D4—1064no—
378C4×D4—1064no—
379C4×D4—1062no—
380C4×D4—1061no—
381C4×D4—1061no—
382C4×D4C4×D5312yes—
383C4×D4C22×D5312yes—
384C2×C4⋊C4—531no—
385C2×C4⋊C4—534no—
386C2×C4⋊C4—1062no—
387C2×C4⋊C4—1061no—
388C2×C4⋊C4C4×D5311yescharacteristic
389C2×C22⋊C4—21no—
390C2×C22⋊C4—1064no—
391C2×C22⋊C4—1064no—
392C2×C22⋊C4—1061no—
393C2×C22⋊C4—1061no—
394C2×C22⋊C4—1061no—
395C2×C42—531no—
396C2×C42—531no—
397C2×C42—531no—
398C2×C42—532no—
399C2×C42—532no—
400C2×C42—532no—
401C2×C42—532no—
402C2×C42C22×D5311yescharacteristic
403C4⋊C8—21no—
404C4⋊C8—532no—
405C4⋊C8—1064no—
406C4⋊C8—1064no—
407C4⋊C8—1061no—
408C4⋊C8C22×D5312yescomplemented
409C22⋊C8—1064no—
410C4×C8—21no—
411C4×C8—532no—
412C4×C8—1061no—
413C4×C8C22×D5312yescomplemented
414C24—21no—
415C24—531no—
416C24—1064no—
417C24—1068no—
418C24—1061no—
419C24—2124no—
420C24—2124no—
421C2×D4—21no—
422C2×D4—22no—
423C2×D4—534no—
424C2×D4—532no—
425C2×D4—1064no—
426C2×D4—1068no—
427C2×D4—10616no—
428C2×D4—1068no—
429C2×D4—1068no—
430C2×D4—1061no—
431C2×D4—1062no—
432C2×D4#1171991214yescomplemented
433C2×D4C2×C4×D5312yes—
434C22×C4—21no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—531no—
440C22×C4—532no—
441C22×C4—534no—
442C22×C4—534no—
443C22×C4—532no—
444C22×C4—531no—
445C22×C4—531no—
446C22×C4—532no—
447C22×C4—532no—
448C22×C4—532no—
449C22×C4—532no—
450C22×C4—1064no—
451C22×C4—1064no—
452C22×C4—1064no—
453C22×C4—1062no—
454C22×C4—1062no—
455C22×C4—1061no—
456C22×C4—1062no—
457C22×C4—1062no—
458C22×C4—2122no—
459C22×C4—2124no—
460C22×C4—21216no—
461C22×C4—2128no—
462C22×C4—2128no—
463C22×C4—2124no—
464C22×C4—2124no—
465C22×C4—2122no—
466C22×C4—2122no—
467C22×C4C2×C4×D5311yescharacteristic
468C22×C4D4×D5311yescharacteristic
469C22×C4D4×D5311yescharacteristic
470M16—2128no—
471C2×C8—22no—
472C2×C8—22no—
473C2×C8—42no—
474C2×C8—42no—
475C2×C8—1062no—
476C2×C8—1062no—
477C2×C8—2122no—
478C2×C8—2122no—
479C4⋊C4—21no—
480C4⋊C4—532no—
481C4⋊C4—1064no—
482C4⋊C4—1061no—
483C4⋊C4C2×C4×D5312yes—
484C22⋊C4—22no—
485C22⋊C4—22no—
486C22⋊C4—1064no—
487C22⋊C4—1064no—
488C22⋊C4—1062no—
489C22⋊C4—1062no—
490C42—532no—
491C42—532no—
492C42—532no—
493C42—532no—
494C42—532no—
495C42—532no—
496C42C2×C4×D5311yescharacteristic
497C42D4×D5311yescharacteristic
498C42D4×D5311yescharacteristic
499C42C23×D5311yescharacteristic
500C23—22no—
501C23—24no—
502C23—42no—
503C23—42no—
504C23—532no—
505C23—534no—
506C23—10616no—
507C23—1062no—
508C23—1062no—
509C23—1062no—
510C23—1064no—
511C23—2128no—
512C23—2128no—
513C23—21232no—
514C23—2122no—
515C23—2122no—
516C23C22⋊C4×D5311yescharacteristic
517D4—24no—
518D4—22no—
519D4—534no—
520D4—10616no—
521D4—1064no—
522D4—1062no—
523D4C2×#1171991214yescomplemented
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—42no—
527C2×C4—44no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—532no—
531C2×C4—534no—
532C2×C4—538no—
533C2×C4—534no—
534C2×C4—534no—
535C2×C4—532no—
536C2×C4—532no—
537C2×C4—1062no—
538C2×C4—1062no—
539C2×C4—1064no—
540C2×C4—1062no—
541C2×C4—1062no—
542C2×C4—1062no—
543C2×C4—2122no—
544C2×C4—2122no—
545C2×C4—2128no—
546C2×C4—2128no—
547C2×C4—2128no—
548C2×C4—2122no—
549C2×C4—2124no—
550C2×C4—2122no—
551C2×C4—2122no—
552C2×C4C22⋊C4×D5311yescharacteristic
553C2×C4C22⋊C4×D5311yescharacteristic
554C2×C4C22⋊C4×D5311yescharacteristic
555C2×C4M16×D5312yes—
556C2×C4C2×#1171991212yes—
557C2×C4D8×D5312yes—
558C2×C4SD16×D5312yes—
559C2×C4C22×C4×D5311yescharacteristic
560C2×C4C2×D4×D5311yescharacteristic
561C2×C4C2×D4×D5311yescharacteristic
562C8—42no—
563C8—42no—
564C8—2122no—
565C8—2122no—
566C22—24no—
567C22—42no—
568C22—42no—
569C22—48no—
570C22—534no—
571C22—1062no—
572C22—1062no—
573C22—1068no—
574C22—1064no—
575C22—21232no—
576C22—2122no—
577C22—2122no—
578C22—2128no—
579C22#1681058512yes—
580C22#102×D5312yes—
581C22#1681063812yes—
582C22C2×C22⋊C4×D5311yescharacteristic
583C4—21no—
584C4—21no—
585C4—42no—
586C4—534no—
587C4—1062no—
588C4—1062no—
589C4—2122no—
590C4#1681058512yes—
591C4#102×D5312yes—
592C4C2×M16×D5311yescharacteristic
593C4C22×#1171991211yescharacteristic
594C4C2×D8×D5311yescharacteristic
595C4C2×SD16×D5311yescharacteristic
596C2—44no—
597C2—1064no—
598C2—2124no—
599C2#2027313814yesdirect product factor
600C2C2×#1681058511yescharacteristic
601C2C2×#102×D5311yescharacteristic
602C2C2×#1681063811yescharacteristic
subgroupG/H
centerC22×C4D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC42C23×D53
FittingC2×C53×D4⋊C8C2
socleC23×C53C2×C22⋊C4
prime radicalC22×D4×D53C4
cosocleC42×C53C24
hypercenterC2×D4⋊C8D53
hypocenterC53C2×#1486
LCSG ▷ C2×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×#16810674C2semidirect
3. C2×#16810645C2semidirect
4. #20273138C2direct (N × H)
5. C2×#16810352C2semidirect
6. C2×D4⋊#88322C2semidirect
7. C2×#16810350C2semidirect
8. C2×C53×D4⋊C8C2semidirect
25. #16810674C22semidirect
27. #16810645C22semidirect
28. C2×C4⋊#88322C22semidirect
32. C2×C4×#88322C22semidirect
33. #16810352C22semidirect
34. D4⋊#88322C22semidirect
35. #16810350C22semidirect
37. C2×C4⋊C8×C53C22semidirect
38. C2×C4×C8×C53C22semidirect
39. C53×D4⋊C8C22semidirect
91. C4⋊#88322C23semidirect
96. C4×#88322C23semidirect
109. C4⋊C8×C53C23semidirect
111. C4×C8×C53C23semidirect
121. C2×D4×C53M16semidirect
144. C2×D212M16semidirect
217. D4×C53C2×M16semidirect
246. D212C2×M16semidirect
302. C2×D4⋊C8D53semidirect
304. C2×C53#1486semidirect
333. C2×C4⋊C8C2×D53semidirect
336. C2×C4×C8C2×D53semidirect
340. D4⋊C8C2×D53semidirect
341. C53C2×#1486semidirect
408. C4⋊C8C22×D53semidirect
413. C4×C8C22×D53semidirect
432. C2×D4#11719912semidirect
523. D4C2×#11719912semidirect
599. C2#20273138direct (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×#168106742
C2×#168106452
#202731382
C2×#168103522
C2×D4⋊#883222
C2×#168103502
C2×C53×D4⋊C82
C2×#148653