Finite Group Databasebeta

← C2×#20274943 #27384948

Subgroups — C2×#20274943 #27384948

Order 13568.

1566 conjugacy classes of subgroups.

The subgroup lattice has 598 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811106C211yesmaximal, characteristic
2C2×#16810665C211yesmaximal, characteristic, complemented
3C2×#16810624C211yesmaximal, characteristic
4C2×#16810493C211yesmaximal, characteristic, complemented
5#20274943C218yesmaximal, direct product factor
6C2×#16810391C211yesmaximal, characteristic, complemented
7C2×#16810383C211yesmaximal, characteristic, complemented
8C22×#104×C53C211yesmaximal, characteristic
9C22×#11719917C2211yescharacteristic
10#16811106—22no—
11#16811106—22no—
12#16811106—22no—
13#16811106C2214yes—
14C2×Q8×Dic53—21no—
15C2×D4×Dic53—22no—
16C2×D4×Dic53—21no—
17C2×#13133176—21no—
18C2×#13133176C2211yescharacteristic
19C22×C4⋊Dic53—21no—
20C22×C4⋊Dic53—22no—
21C22×C4⋊Dic53C2211yescharacteristic
22C22×C4⋊Dic53C2211yescharacteristic
23#16810665—22no—
24#16810665C2214yescomplemented
25C2×#13133151—21no—
26C2×#13133149—21no—
27#16810624—22no—
28#16810624C2214yes—
29C2×#13133141—21no—
30C2×#13133138—21no—
31#16810493—22no—
32#16810493C2214yescomplemented
33#16810391C2214yescomplemented
34#16810383C2214yescomplemented
35C22×#41×C53C2211yescharacteristic
36C22×M16×C53C2211yescharacteristic
37C22×C42×C53C2211yescharacteristic
38C2×#104×C53C418yescomplemented
39C2×#104×C53C2214yes—
40C23×Dic53—22no—
41C23×Dic53—21no—
42C23×Dic53D412yes—
43C2×#11719917—21no—
44C2×#11719917—22no—
45C2×#11719917—22no—
46C2×#11719917C2312yes—
47C22×#88322—21no—
48Q8×Dic53—24no—
49Q8×Dic53—24no—
50D4×Dic53—24no—
51D4×Dic53—28no—
52D4×Dic53—24no—
53D4×Dic53—28no—
54#13133176—24no—
55#13133176—24no—
56#13133176—21no—
57#13133176—21no—
58#13133176D412yes—
59#13133176C2312yes—
60C2×C4⋊Dic53—22no—
61C2×C4⋊Dic53—24no—
62C2×C4⋊Dic53—24no—
63C2×C4⋊Dic53—21no—
64C2×C4⋊Dic53—21no—
65C2×C4⋊Dic53—21no—
66C2×C4⋊Dic53—21no—
67C2×C4⋊Dic53—21no—
68C2×C4⋊Dic53—28no—
69C2×C4⋊Dic53—21no—
70C2×C4⋊Dic53—21no—
71C2×C4⋊Dic53—21no—
72C2×C4⋊Dic53—21no—
73C2×C4⋊Dic53—48no—
74C2×C4⋊Dic53D411yescharacteristic
75C2×C4⋊Dic53D411yescharacteristic
76C2×C4⋊Dic53D412yes—
77C2×C4⋊Dic53C2312yes—
78C2×C4⋊Dic53C2312yes—
79C2×C22⋊Dic53—22no—
80C2×C22⋊Dic53—21no—
81C2×C4×Dic53—22no—
82C2×C4×Dic53—21no—
83#13133151—24no—
84#13133149—24no—
85#13133141—24no—
86#13133138—24no—
87C2×#41×C53—21no—
88C2×#41×C53—21no—
89C2×#41×C53—22no—
90C2×#41×C53—22no—
91C2×#41×C53C2×C414yes—
92C2×#41×C53D412yes—
93C2×#41×C53Q814yes—
94C2×#41×C53C2312yes—
95C22×Q8×C53D411yescharacteristic
96C22×D4×C53—21no—
97C22×D4×C53D411yescharacteristic
98C23×C4×C53—21no—
99C23×C4×C53—21no—
100C23×C4×C53C2311yescharacteristic
101C2×M16×C53—21no—
102C2×M16×C53—22no—
103C2×M16×C53C2×C414yescomplemented
104C2×M16×C53C2312yes—
105C22×C8×C53—21no—
106C2×C42×C53—21no—
107C2×C42×C53—22no—
108C2×C42×C53C2×C414yescomplemented
109C2×C42×C53C2312yes—
110#104×C53C2×C4116yescomplemented
111C24×C53—21no—
112C24×C53C2×D411yescharacteristic
113#41×C53—24no—
114#41×C53—24no—
115#41×C53—24no—
116#41×C53—24no—
117#41×C53—24no—
118#41×C53—22no—
119#41×C53—22no—
120#41×C53C4⋊C418yescomplemented
121#41×C53C22×C414yes—
122#41×C53C2×Q814yes—
123C2×Q8×C53—21no—
124C2×Q8×C53—22no—
125C2×Q8×C53C4⋊C414yescomplemented
126C2×Q8×C53C2×D412yes—
127C2×D4×C53—24no—
128C2×D4×C53—22no—
129C2×D4×C53—24no—
130C2×D4×C53—22no—
131C2×D4×C53—21no—
132C2×D4×C53—22no—
133C2×D4×C53C4⋊C414yescomplemented
134C2×D4×C53C2×D412yes—
135C22×C4×C53—24no—
136C22×C4×C53—24no—
137C22×C4×C53—24no—
138C22×C4×C53—21no—
139C22×C4×C53—22no—
140C22×C4×C53—24no—
141C22×C4×C53—22no—
142C22×C4×C53—24no—
143C22×C4×C53—22no—
144C22×C4×C53C22×C412yes—
145C22×C4×C53C2×D411yescharacteristic
146C22×C4×C53C2×D411yescharacteristic
147C22×C4×C53C2×D411yescharacteristic
148C22×C4×C53C2×D411yescharacteristic
149C22×C4×C53C2×D411yescharacteristic
150C22×C4×C53C2×Q812yes—
151C22×C4×C53#4112yes—
152C22×C4×C53#4112yes—
153C22×C4×C53C2411yescharacteristic
154C22×Dic53—21no—
155C22×Dic53—22no—
156C22×Dic53—21no—
157C22×Dic53—28no—
158C22×Dic53—22no—
159C22×Dic53—22no—
160C22×Dic53—44no—
161C22×Dic53—44no—
162C22×Dic53—41no—
163C22×Dic53—41no—
164C22×Dic53—84no—
165C22×Dic53C2×D414yes—
166M16×C53—24no—
167M16×C53—22no—
168M16×C53C22×C414yescomplemented
169C2×C8×C53—24no—
170C2×C8×C53—22no—
171C42×C53—24no—
172C42×C53—22no—
173C42×C53C22×C414yescomplemented
174C22⋊Dic53—28no—
175C22⋊Dic53—22no—
176C22⋊Dic53—22no—
177C4⋊Dic53—28no—
178C4⋊Dic53—21no—
179C4⋊Dic53—22no—
180C4⋊Dic53—22no—
181C4⋊Dic53—21no—
182C4⋊Dic53—22no—
183C4⋊Dic53—22no—
184C4⋊Dic53—22no—
185C4⋊Dic53—22no—
186C4⋊Dic53—416no—
187C4⋊Dic53—41no—
188C4⋊Dic53—41no—
189C4⋊Dic53C2×D412yes—
190C4⋊Dic53C2×D412yes—
191C4×Dic53—28no—
192C4×Dic53—22no—
193C4×Dic53—22no—
194#11719917—22no—
195#11719917—22no—
196#11719917—42no—
197C2×#88322—22no—
198C2×#88322—42no—
199C23×C53—22no—
200C23×C53—22no—
201C23×C53—22no—
202C23×C53—24no—
203C23×C53—42no—
204C23×C53—42no—
205C23×C53C4×D412yes—
206C23×C53#12011yescharacteristic
207C23×C53#12011yescharacteristic
208C23×C53C22⋊Q812yes—
209C23×C53C22×D411yescharacteristic
210Q8×C53—22no—
211Q8×C53—24no—
212Q8×C53C2×C4⋊C414yescomplemented
213D4×C53—24no—
214D4×C53—24no—
215D4×C53—28no—
216D4×C53—22no—
217D4×C53—24no—
218D4×C53C2×C4⋊C414yescomplemented
219C2×C4×C53—216no—
220C2×C4×C53—24no—
221C2×C4×C53—24no—
222C2×C4×C53—24no—
223C2×C4×C53—22no—
224C2×C4×C53—24no—
225C2×C4×C53—24no—
226C2×C4×C53—24no—
227C2×C4×C53—28no—
228C2×C4×C53C2×C4⋊C412yes—
229C2×C4×C53C2×C4⋊C412yes—
230C2×C4×C53C4×D412yes—
231C2×C4×C53C4×D412yes—
232C2×C4×C53C4×D412yes—
233C2×C4×C53#12011yescharacteristic
234C2×C4×C53#12011yescharacteristic
235C2×C4×C53C22⋊Q812yes—
236C2×C4×C53C22⋊Q812yes—
237C2×C4×C53C22⋊Q812yes—
238C2×C4×C53#12312yes—
239C2×C4×C53#12312yes—
240C2×C4×C53C23×C411yescharacteristic
241C2×C4×C53C22×D411yescharacteristic
242C2×C4×C53C22×D411yescharacteristic
243C2×C4×C53C22×Q811yescharacteristic
244C2×C4×C53C2×#4111yescharacteristic
245C2×C4×C53C2×#4111yescharacteristic
246C2×Dic53—22no—
247C2×Dic53—24no—
248C2×Dic53—22no—
249C2×Dic53—44no—
250C2×Dic53—42no—
251C2×Dic53—42no—
252C2×Dic53—44no—
253C2×Dic53—44no—
254C2×Dic53—42no—
255C2×Dic53—82no—
256C2×Dic53—88no—
257C2×Dic53—82no—
258C2×Dic53—82no—
259C8×C53—24no—
260#88322—42no—
261C2×#1859—531nomaximal, Sylow
262C22×C53—24no—
263C22×C53—24no—
264C22×C53—21no—
265C22×C53—21no—
266C22×C53—21no—
267C22×C53—21no—
268C22×C53—24no—
269C22×C53—48no—
270C22×C53—42no—
271C22×C53—42no—
272C22×C53#38512yes—
273C22×C53#38512yes—
274C22×C53#45312yes—
275C22×C53#45612yes—
276C22×C53C2×C4×D411yescharacteristic
277C22×C53C2×#12011yescharacteristic
278C22×C53C2×C22⋊Q811yescharacteristic
279C4×C53—28no—
280C4×C53—24no—
281C4×C53#38512yes—
282C4×C53#38512yes—
283C4×C53C22×C4⋊C411yescharacteristic
284C4×C53C2×C4×D411yescharacteristic
285C4×C53C2×C22⋊Q811yescharacteristic
286C4×C53C2×#12311yescharacteristic
287Dic53—44no—
288Dic53—42no—
289Dic53—42no—
290Dic53—82no—
291C2×#520—531no—
292C2×#428—531no—
293C22×#104D5311yescharacteristic
294C2×#417—531no—
295C2×#384—531no—
296#1859—538no—
297C2×#343—531no—
298C2×#339—531no—
299C2×C53—21no—
300C2×C53—21no—
301C2×C53—22no—
302C2×C53—44no—
303C2×C53#185914yescomplemented
304C2×C53C2×#38511yescharacteristic
305C2×C53C2×#45311yescharacteristic
306C2×C53C2×#45611yescharacteristic
307C22×#41C2×D5311yescharacteristic
308C22×M16—531no—
309C22×M16C2×D5311yescharacteristic
310#520—534no—
311#520—1062no—
312#520—1062no—
313#520—1062no—
314C2×C4×Q8—1061no—
315C2×C4×D4—1062no—
316C2×C4×D4—1061no—
317C2×#117—531no—
318C2×#117—1061no—
319C22×C4⋊C4—531no—
320C22×C4⋊C4—531no—
321C22×C4⋊C4—1061no—
322C22×C4⋊C4—1062no—
323C22×C42C2×D5311yescharacteristic
324#428—534no—
325#428—1062no—
326C2×#107—1061no—
327C2×#106—1061no—
328C2×#104Dic5318yescomplemented
329C2×#104C2×D5314yes—
330#417—534no—
331#417—1062no—
332C2×#103—1061no—
333C2×#102—1061no—
334#384—534no—
335#384—1062no—
336#343—534no—
337#339—534no—
338C53C2×#185911yescharacteristic, complemented, Sylow
339C2×#41—21no—
340C2×#41—21no—
341C2×#41—22no—
342C2×#41—22no—
343C2×#41Dic10614yes—
344C2×#41D21212yes—
345C2×#41C2×Dic5314yes—
346C2×#41C22×D5312yes—
347C22×Q8D21211yescharacteristic
348C22×D4—21no—
349C22×D4D21211yescharacteristic
350C23×C4—21no—
351C23×C4—21no—
352C23×C4—532no—
353C23×C4—1062no—
354C23×C4—1061no—
355C23×C4C22×D5311yescharacteristic
356C2×M16—21no—
357C2×M16—22no—
358C2×M16—532no—
359C2×M16—1061no—
360C2×M16—1062no—
361C2×M16—1062no—
362C2×M16C2×Dic5314yescomplemented
363C2×M16C22×D5312yes—
364C22×C8—21no—
365C22×C8—1061no—
366C4×Q8—1064no—
367C4×Q8—1064no—
368C4×D4—1068no—
369C4×D4—1068no—
370C4×D4—1064no—
371C4×D4—1064no—
372#117—532no—
373#117—532no—
374#117—1064no—
375#117—1064no—
376#117—1061no—
377#117—1061no—
378C2×C4⋊C4—531no—
379C2×C4⋊C4—531no—
380C2×C4⋊C4—532no—
381C2×C4⋊C4—532no—
382C2×C4⋊C4—532no—
383C2×C4⋊C4—1064no—
384C2×C4⋊C4—1062no—
385C2×C4⋊C4—1064no—
386C2×C4⋊C4—1061no—
387C2×C4⋊C4—1061no—
388C2×C4⋊C4—1061no—
389C2×C4⋊C4—1061no—
390C2×C4⋊C4—1061no—
391C2×C4⋊C4—1061no—
392C2×C4⋊C4—1061no—
393C2×C4⋊C4—1061no—
394C2×C4⋊C4—1068no—
395C2×C4⋊C4—1061no—
396C2×C4⋊C4—2128no—
397C2×C22⋊C4—1062no—
398C2×C22⋊C4—1061no—
399C2×C42—21no—
400C2×C42—22no—
401C2×C42—1062no—
402C2×C42—1061no—
403C2×C42C2×Dic5314yescomplemented
404C2×C42C22×D5312yes—
405#107—1064no—
406#106—1064no—
407#104C2×Dic53116yescomplemented
408#103—1064no—
409#102—1064no—
410C24—21no—
411C24D4×D5311yescharacteristic
412#41—24no—
413#41—24no—
414#41—24no—
415#41—24no—
416#41—24no—
417#41—22no—
418#41—22no—
419#41C4⋊Dic5318yescomplemented
420#41C2×Dic10614yes—
421#41C22×Dic5314yes—
422C2×Q8—21no—
423C2×Q8—22no—
424C2×Q8C4⋊Dic5314yescomplemented
425C2×Q8C2×D21212yes—
426C2×D4—24no—
427C2×D4—22no—
428C2×D4—24no—
429C2×D4—22no—
430C2×D4—21no—
431C2×D4—22no—
432C2×D4C4⋊Dic5314yescomplemented
433C2×D4C2×D21212yes—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—21no—
438C22×C4—22no—
439C22×C4—24no—
440C22×C4—22no—
441C22×C4—24no—
442C22×C4—22no—
443C22×C4—534no—
444C22×C4—1068no—
445C22×C4—1062no—
446C22×C4—1061no—
447C22×C4—1061no—
448C22×C4—1062no—
449C22×C4—1062no—
450C22×C4—4244no—
451C22×C4—2124no—
452C22×C4—2121no—
453C22×C4—2121no—
454C22×C4—2124no—
455C22×C4C2×Dic10612yes—
456C22×C4C2×D21211yescharacteristic
457C22×C4C2×D21211yescharacteristic
458C22×C4D4×D5311yescharacteristic
459C22×C4D4×D5311yescharacteristic
460C22×C4D4×D5311yescharacteristic
461C22×C4#1171994712yes—
462C22×C4#1171994712yes—
463C22×C4C22×Dic5312yes—
464C22×C4C23×D5311yescharacteristic
465M16—24no—
466M16—22no—
467M16—1062no—
468M16—1062no—
469M16—2122no—
470M16C22×Dic5314yescomplemented
471C2×C8—24no—
472C2×C8—22no—
473C2×C8—1062no—
474C2×C8—2122no—
475C4⋊C4—532no—
476C4⋊C4—532no—
477C4⋊C4—1068no—
478C4⋊C4—1061no—
479C4⋊C4—1062no—
480C4⋊C4—1062no—
481C4⋊C4—1061no—
482C4⋊C4—1062no—
483C4⋊C4—1062no—
484C4⋊C4—1062no—
485C4⋊C4—1062no—
486C4⋊C4—21216no—
487C4⋊C4—2121no—
488C4⋊C4—2121no—
489C22⋊C4—1068no—
490C22⋊C4—1062no—
491C22⋊C4—1062no—
492C42—24no—
493C42—22no—
494C42—1068no—
495C42—1062no—
496C42—1062no—
497C42C22×Dic5314yescomplemented
498C23—22no—
499C23—22no—
500C23—24no—
501C23—22no—
502C23—42no—
503C23—42no—
504C23D4×Dic5312yes—
505C23#1313318711yescharacteristic
506C23#1313318711yescharacteristic
507C23#1313319412yes—
508C23C2×D4×D5311yescharacteristic
509Q8—24no—
510Q8—22no—
511Q8C2×C4⋊Dic5314yescomplemented
512D4—28no—
513D4—24no—
514D4—24no—
515D4—24no—
516D4—22no—
517D4C2×C4⋊Dic5314yescomplemented
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—216no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—22no—
524C2×C4—28no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—1062no—
528C2×C4—1062no—
529C2×C4—1064no—
530C2×C4—4242no—
531C2×C4—4242no—
532C2×C4—4242no—
533C2×C4—4248no—
534C2×C4—2124no—
535C2×C4—2122no—
536C2×C4—2122no—
537C2×C4—2124no—
538C2×C4—2122no—
539C2×C4—2124no—
540C2×C4C2×C4⋊Dic5312yes—
541C2×C4C2×C4⋊Dic5312yes—
542C2×C4D4×Dic5312yes—
543C2×C4D4×Dic5312yes—
544C2×C4D4×Dic5312yes—
545C2×C4#1313318711yescharacteristic
546C2×C4#1313318711yescharacteristic
547C2×C4#1313319412yes—
548C2×C4#1313319412yes—
549C2×C4#1313319412yes—
550C2×C4#1313320112yes—
551C2×C4#1313320112yes—
552C2×C4C23×Dic5311yescharacteristic
553C2×C4C2×D4×D5311yescharacteristic
554C2×C4C22×D21211yescharacteristic
555C2×C4C22×Dic10611yescharacteristic
556C2×C4C2×#1171994711yescharacteristic
557C2×C4C2×#1171994711yescharacteristic
558C8—24no—
559C8—2122no—
560C22—21no—
561C22—21no—
562C22—21no—
563C22—21no—
564C22—24no—
565C22—24no—
566C22—24no—
567C22—42no—
568C22—42no—
569C22—48no—
570C22#1681049812yes—
571C22#1681049812yes—
572C22#1681079212yes—
573C22#1681080712yes—
574C22C2×D4×Dic5311yescharacteristic
575C22C2×#1313318711yescharacteristic
576C22C2×#1313319411yescharacteristic
577C4—28no—
578C4—24no—
579C4—4242no—
580C4—2122no—
581C4—2124no—
582C4—2122no—
583C4#1681049812yes—
584C4#1681049812yes—
585C4C22×C4⋊Dic5311yescharacteristic
586C4C2×D4×Dic5311yescharacteristic
587C4C2×#1313319411yescharacteristic
588C4C2×#1313320111yescharacteristic
589C2—21no—
590C2—22no—
591C2—21no—
592C2—44no—
593C2#2027494314yesdirect product factor
594C2C2×#1681049811yescharacteristic
595C2C2×#1681079211yescharacteristic
596C2C2×#1681080711yescharacteristic
subgroupG/H
centerC23#13133187
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC22×#104×C53C2
socleC23×C53#120
prime radicalC22×#41×C53C22
cosocleC22×C4×C53C24
hypercenterC22×#104D53
hypocenterC53C2×#1859
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×#104

Complements

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

NH ≅ G/Ntype
2. C2×#16810665C2semidirect
4. C2×#16810493C2semidirect
5. #20274943C2direct (N × H)
6. C2×#16810391C2semidirect
7. C2×#16810383C2semidirect
24. #16810665C22semidirect
32. #16810493C22semidirect
33. #16810391C22semidirect
34. #16810383C22semidirect
38. C2×#104×C53C4semidirect
103. C2×M16×C53C2×C4semidirect
108. C2×C42×C53C2×C4semidirect
110. #104×C53C2×C4semidirect
120. #41×C53C4⋊C4semidirect
125. C2×Q8×C53C4⋊C4semidirect
133. C2×D4×C53C4⋊C4semidirect
168. M16×C53C22×C4semidirect
173. C42×C53C22×C4semidirect
212. Q8×C53C2×C4⋊C4semidirect
218. D4×C53C2×C4⋊C4semidirect
303. C2×C53#1859semidirect
328. C2×#104Dic53semidirect
338. C53C2×#1859semidirect
362. C2×M16C2×Dic53semidirect
403. C2×C42C2×Dic53semidirect
407. #104C2×Dic53semidirect
419. #41C4⋊Dic53semidirect
424. C2×Q8C4⋊Dic53semidirect
432. C2×D4C4⋊Dic53semidirect
470. M16C22×Dic53semidirect
497. C42C22×Dic53semidirect
511. Q8C2×C4⋊Dic53semidirect
517. D4C2×C4⋊Dic53semidirect
593. C2#20274943direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168111062
C2×#168106652
C2×#168106242
C2×#168104932
#202749432
C2×#168103912
C2×#168103832
C22×#104×C532
C2×#185953