Finite Group Databasebeta

← C2×#20274942 #27384947

Subgroups — C2×#20274942 #27384947

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×#16811103C211yesmaximal, characteristic
2C2×#16810666C211yesmaximal, characteristic, complemented
3C22×#13133145C211yesmaximal, characteristic
4C2×#16810625C211yesmaximal, characteristic
5C2×#16810493C211yesmaximal, characteristic, complemented
6#20274942C218yesmaximal, direct product factor
7C2×#16810391C211yesmaximal, characteristic, complemented
8C2×C53×#339C211yesmaximal, characteristic, complemented
9C22×#11719945C2211yescharacteristic
10C22×#11719917C2211yescharacteristic
11#16811103—22no—
12#16811103—22no—
13#16811103—22no—
14#16811103C2214yes—
15C2×C4×Dic106—21no—
16C2×C4×D212—21no—
17C2×#13133177—22no—
18C2×#13133175—21no—
19C22×C4⋊Dic53—22no—
20C22×C4⋊Dic53C2211yescharacteristic
21C22×C4⋊Dic53C2211yescharacteristic
22C2×C4⋊C4×D53—21no—
23#16810666—22no—
24#16810666C2214yescomplemented
25C2×#13133150—21no—
26C2×#13133148—21no—
27C2×#13133145C418yescomplemented
28C2×#13133145C2214yes—
29#16810625—22no—
30#16810625C2214yes—
31C2×#13133142—21no—
32C2×#13133139—21no—
33#16810493—22no—
34#16810493C2214yescomplemented
35#16810391C2214yescomplemented
36C22×M16×C53C2211yescharacteristic
37C2×#117×C53C2211yescharacteristic
38C22×C42×C53C2211yescharacteristic
39C53×#339C2214yescomplemented
40C2×#11719945—21no—
41C2×#11719945—21no—
42C2×#11719945—22no—
43C2×#11719945—22no—
44C2×#11719945C2×C414yes—
45C2×#11719945D412yes—
46C2×#11719945Q814yes—
47C2×#11719945C2312yes—
48C22×Dic106D411yescharacteristic
49C22×D212D411yescharacteristic
50C22×C22⋊D53—21no—
51C23×Dic53—21no—
52C23×Dic53D412yes—
53C22×C4×D53—21no—
54C22×C4×D53—22no—
55C2×#11719917—21no—
56C2×#11719917—22no—
57C2×#11719917C2×C414yescomplemented
58C2×#11719917C2312yes—
59C22×#88322—21no—
60C4×Dic106—24no—
61C4×Dic106—24no—
62C4×D212—24no—
63C4×D212—24no—
64#13133177—28no—
65#13133177—28no—
66#13133175—24no—
67#13133175—24no—
68C2×C4⋊Dic53—21no—
69C2×C4⋊Dic53—21no—
70C2×C4⋊Dic53—21no—
71C2×C4⋊Dic53—21no—
72C2×C4⋊Dic53—21no—
73C2×C4⋊Dic53—28no—
74C2×C4⋊Dic53—21no—
75C2×C4⋊Dic53—21no—
76C2×C4⋊Dic53—21no—
77C2×C4⋊Dic53—21no—
78C2×C4⋊Dic53—48no—
79C2×C4⋊Dic53D412yes—
80C2×C4⋊Dic53C2312yes—
81C2×C4⋊Dic53C2312yes—
82C2×#11719919—22no—
83C4⋊C4×D53—24no—
84C4⋊C4×D53—24no—
85C2×#11719921—22no—
86C2×C4×Dic53—22no—
87#13133150—24no—
88#13133148—24no—
89#13133145C2×C4116yescomplemented
90#13133142—24no—
91#13133139—24no—
92C23×C4×C53—21no—
93C23×C4×C53C2311yescharacteristic
94C2×M16×C53—21no—
95C2×M16×C53—22no—
96C2×M16×C53—22no—
97C2×M16×C53C2312yes—
98C22×C8×C53—21no—
99#117×C53—21no—
100#117×C53—21no—
101#117×C53D412yes—
102#117×C53C2312yes—
103C2×C4⋊C4×C53D411yescharacteristic
104C2×C4⋊C4×C53D411yescharacteristic
105C2×C22⋊C4×C53—21no—
106C2×C42×C53—21no—
107C2×C42×C53—21no—
108C2×C42×C53—22no—
109C2×C42×C53C2×C414yescomplemented
110C2×C42×C53C2312yes—
111C24×C53C2×D411yescharacteristic
112C23×D53—21no—
113C22×C4×C53—24no—
114C22×C4×C53—24no—
115C22×C4×C53—24no—
116C22×C4×C53—22no—
117C22×C4×C53—21no—
118C22×C4×C53—21no—
119C22×C4×C53C22×C412yes—
120C22×C4×C53C2×D411yescharacteristic
121C22×C4×C53C2×D411yescharacteristic
122C22×C4×C53C2×D411yescharacteristic
123C22×C4×C53C2×D411yescharacteristic
124C22×C4×C53C2×D411yescharacteristic
125C22×C4×C53C2×Q812yes—
126C22×C4×C53#4112yes—
127C22×C4×C53#4112yes—
128C22×C4×C53C2411yescharacteristic
129C2×C22⋊D53—24no—
130C2×C22⋊D53—22no—
131C2×C22⋊D53—24no—
132C2×C22⋊D53—22no—
133C22×Dic53—21no—
134C22×Dic53—22no—
135C22×Dic53—22no—
136C22×Dic53—22no—
137C22×Dic53—44no—
138C22×Dic53—44no—
139C22×Dic53—41no—
140C22×Dic53—41no—
141C22×Dic53C2×D414yes—
142#11719945—24no—
143#11719945—24no—
144#11719945—24no—
145#11719945—24no—
146#11719945—24no—
147#11719945—22no—
148#11719945—22no—
149#11719945C4⋊C418yescomplemented
150#11719945C22×C414yes—
151#11719945C2×Q814yes—
152C2×D212—21no—
153C2×D212—22no—
154C2×D212C4⋊C414yescomplemented
155C2×D212C2×D412yes—
156C2×C4×D53—24no—
157C2×C4×D53—22no—
158C2×C4×D53—24no—
159C2×C4×D53—22no—
160C2×C4×D53—28no—
161C2×C4×D53—84no—
162C2×Dic106—21no—
163C2×Dic106—22no—
164C2×Dic106C4⋊C414yescomplemented
165C2×Dic106C2×D412yes—
166M16×C53—22no—
167M16×C53—22no—
168M16×C53—42no—
169C2×C8×C53—22no—
170C2×C8×C53—42no—
171C4⋊C4×C53—21no—
172C4⋊C4×C53—21no—
173C4⋊C4×C53C2×D412yes—
174C4⋊C4×C53C2×D412yes—
175C22⋊C4×C53—22no—
176C22⋊C4×C53—22no—
177C42×C53—24no—
178C42×C53—22no—
179C42×C53—22no—
180C42×C53—22no—
181C42×C53C22×C414yescomplemented
182#11719921—28no—
183C4⋊Dic53—22no—
184C4⋊Dic53—22no—
185C4⋊Dic53—22no—
186C4⋊Dic53—22no—
187C4⋊Dic53—22no—
188C4⋊Dic53—22no—
189C4⋊Dic53—416no—
190C4⋊Dic53—41no—
191C4⋊Dic53—41no—
192#11719919—28no—
193C4×Dic53—28no—
194#11719917—24no—
195#11719917—22no—
196#11719917C22×C414yescomplemented
197C2×#88322—24no—
198C2×#88322—22no—
199C23×C53—22no—
200C23×C53—22no—
201C23×C53C4×D412yes—
202C23×C53#12011yescharacteristic
203C23×C53#12011yescharacteristic
204C23×C53C22⋊Q812yes—
205C23×C53C22×D411yescharacteristic
206C22×D53—22no—
207C22×D53—24no—
208C22×D53—42no—
209C22×D53—42no—
210C2×C4×C53—216no—
211C2×C4×C53—24no—
212C2×C4×C53—24no—
213C2×C4×C53—24no—
214C2×C4×C53—22no—
215C2×C4×C53—22no—
216C2×C4×C53—42no—
217C2×C4×C53—42no—
218C2×C4×C53C2×C4⋊C412yes—
219C2×C4×C53C2×C4⋊C412yes—
220C2×C4×C53C4×D412yes—
221C2×C4×C53C4×D412yes—
222C2×C4×C53C4×D412yes—
223C2×C4×C53#12011yescharacteristic
224C2×C4×C53#12011yescharacteristic
225C2×C4×C53C22⋊Q812yes—
226C2×C4×C53C22⋊Q812yes—
227C2×C4×C53C22⋊Q812yes—
228C2×C4×C53#12312yes—
229C2×C4×C53#12312yes—
230C2×C4×C53C23×C411yescharacteristic
231C2×C4×C53C22×D411yescharacteristic
232C2×C4×C53C22×D411yescharacteristic
233C2×C4×C53C22×Q811yescharacteristic
234C2×C4×C53C2×#4111yescharacteristic
235C2×C4×C53C2×#4111yescharacteristic
236C22⋊D53—24no—
237C22⋊D53—24no—
238C22⋊D53—28no—
239C2×Dic53—22no—
240C2×Dic53—24no—
241C2×Dic53—24no—
242C2×Dic53—44no—
243C2×Dic53—44no—
244C2×Dic53—44no—
245C2×Dic53—42no—
246C2×Dic53—82no—
247C2×Dic53—82no—
248C2×Dic53—82no—
249D212—22no—
250D212—24no—
251D212C2×C4⋊C414yescomplemented
252C4×D53—24no—
253C4×D53—24no—
254C4×D53—28no—
255C4×D53—88no—
256Dic106—22no—
257Dic106—24no—
258Dic106C2×C4⋊C414yescomplemented
259C8×C53—42no—
260#88322—24no—
261C2×#1859—531nomaximal, Sylow
262C22×C53—24no—
263C22×C53—24no—
264C22×C53—21no—
265C22×C53—21no—
266C22×C53—21no—
267C22×C53—21no—
268C22×C53#38512yes—
269C22×C53#38512yes—
270C22×C53#45312yes—
271C22×C53#45612yes—
272C22×C53C2×C4×D411yescharacteristic
273C22×C53C2×#12011yescharacteristic
274C22×C53C2×C22⋊Q811yescharacteristic
275C2×D53—24no—
276C2×D53—48no—
277C2×D53—42no—
278C2×D53—42no—
279C4×C53—28no—
280C4×C53—42no—
281C4×C53—42no—
282C4×C53#38512yes—
283C4×C53#38512yes—
284C4×C53C22×C4⋊C411yescharacteristic
285C4×C53C2×C4×D411yescharacteristic
286C4×C53C2×C22⋊Q811yescharacteristic
287C4×C53C2×#12311yescharacteristic
288Dic53—24no—
289Dic53—44no—
290Dic53—82no—
291C2×#520—531no—
292C2×#428—531no—
293C22×#104—531no—
294C2×#417—531no—
295C2×#384—531no—
296#1859—538no—
297C2×#343—531no—
298C2×#339D5311yescharacteristic, complemented
299C2×C53—21no—
300C2×C53—21no—
301C2×C53—22no—
302C2×C53#185914yescomplemented
303C2×C53C2×#38511yescharacteristic
304C2×C53C2×#45311yescharacteristic
305C2×C53C2×#45611yescharacteristic
306D53—44no—
307C22×#41—531no—
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—1061no—
316C2×C4×D4—1062no—
317C2×#117—1061no—
318C2×#117C2×D5311yescharacteristic
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×#104—538no—
329C2×#104—534no—
330#417—534no—
331#417—1062no—
332C2×#103—1061no—
333C2×#102—1061no—
334#384—534no—
335#384—1062no—
336#343—534no—
337#339C2×D5314yescomplemented
338C53C2×#185911yescharacteristic, complemented, Sylow
339C2×#41—534no—
340C2×#41—532no—
341C2×#41—534no—
342C2×#41—532no—
343C2×#41—1061no—
344C2×#41—1062no—
345C2×#41—1061no—
346C2×#41—1062no—
347C22×Q8—531no—
348C22×D4—531no—
349C22×D4—1061no—
350C23×C4—21no—
351C23×C4—532no—
352C23×C4—1062no—
353C23×C4—1061no—
354C23×C4—1061no—
355C23×C4C22×D5311yescharacteristic
356C2×M16—21no—
357C2×M16—22no—
358C2×M16—22no—
359C2×M16—534no—
360C2×M16—532no—
361C2×M16—1061no—
362C2×M16—1062no—
363C2×M16C22×D5312yes—
364C22×C8—21no—
365C22×C8—1061no—
366C4×Q8—1064no—
367C4×Q8—1064no—
368C4×D4—1064no—
369C4×D4—1064no—
370C4×D4—1068no—
371C4×D4—1068no—
372#117—21no—
373#117—21no—
374#117—1064no—
375#117—1064no—
376#117D21212yes—
377#117C22×D5312yes—
378C2×C4⋊C4—532no—
379C2×C4⋊C4—532no—
380C2×C4⋊C4—532no—
381C2×C4⋊C4—1064no—
382C2×C4⋊C4—1062no—
383C2×C4⋊C4—1061no—
384C2×C4⋊C4—1061no—
385C2×C4⋊C4—1061no—
386C2×C4⋊C4—1061no—
387C2×C4⋊C4—1064no—
388C2×C4⋊C4—1061no—
389C2×C4⋊C4—1061no—
390C2×C4⋊C4—1061no—
391C2×C4⋊C4—1061no—
392C2×C4⋊C4—1068no—
393C2×C4⋊C4—1061no—
394C2×C4⋊C4—2128no—
395C2×C4⋊C4D21211yescharacteristic
396C2×C4⋊C4D21211yescharacteristic
397C2×C22⋊C4—21no—
398C2×C22⋊C4—1062no—
399C2×C42—21no—
400C2×C42—21no—
401C2×C42—22no—
402C2×C42—1062no—
403C2×C42C4×D5314yescomplemented
404C2×C42C22×D5312yes—
405#107—1064no—
406#106—1064no—
407#104—5316no—
408#103—1064no—
409#102—1064no—
410C24—1061no—
411C24D4×D5311yescharacteristic
412#41—538no—
413#41—534no—
414#41—534no—
415#41—1064no—
416#41—1064no—
417#41—1064no—
418#41—1064no—
419#41—1064no—
420#41—1062no—
421#41—1062no—
422C2×Q8—534no—
423C2×Q8—532no—
424C2×Q8—1061no—
425C2×Q8—1062no—
426C2×D4—534no—
427C2×D4—532no—
428C2×D4—1064no—
429C2×D4—1062no—
430C2×D4—1064no—
431C2×D4—1062no—
432C2×D4—1061no—
433C2×D4—1062no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—22no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—534no—
441C22×C4—1068no—
442C22×C4—1061no—
443C22×C4—1064no—
444C22×C4—1062no—
445C22×C4—1064no—
446C22×C4—1062no—
447C22×C4—1062no—
448C22×C4—1062no—
449C22×C4—1062no—
450C22×C4—4244no—
451C22×C4—2124no—
452C22×C4—2121no—
453C22×C4—2121no—
454C22×C4—2124no—
455C22×C4C2×C4×D5312yes—
456C22×C4C2×D21211yescharacteristic
457C22×C4C2×D21211yescharacteristic
458C22×C4#1171994512yes—
459C22×C4D4×D5311yescharacteristic
460C22×C4D4×D5311yescharacteristic
461C22×C4D4×D5311yescharacteristic
462C22×C4#1171994712yes—
463C22×C4Q8×D5312yes—
464C22×C4C23×D5311yescharacteristic
465M16—22no—
466M16—22no—
467M16—42no—
468M16—534no—
469M16—1064no—
470M16—1062no—
471C2×C8—22no—
472C2×C8—42no—
473C2×C8—1064no—
474C2×C8—1062no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—1068no—
478C4⋊C4—1062no—
479C4⋊C4—1062no—
480C4⋊C4—1062no—
481C4⋊C4—1062no—
482C4⋊C4—1062no—
483C4⋊C4—1062no—
484C4⋊C4—21216no—
485C4⋊C4—2121no—
486C4⋊C4—2121no—
487C4⋊C4C2×D21212yes—
488C4⋊C4C2×D21212yes—
489C22⋊C4—22no—
490C22⋊C4—22no—
491C22⋊C4—1068no—
492C42—24no—
493C42—22no—
494C42—22no—
495C42—22no—
496C42—1068no—
497C42C2×C4×D5314yescomplemented
498C23—22no—
499C23—22no—
500C23—1064no—
501C23—1062no—
502C23—2122no—
503C23—2122no—
504C23#1313317712yes—
505C23#1313318711yescharacteristic
506C23#1313318711yescharacteristic
507C23#1313319512yes—
508C23C2×D4×D5311yescharacteristic
509Q8—534no—
510Q8—1064no—
511Q8—1062no—
512D4—534no—
513D4—1068no—
514D4—1064no—
515D4—1064no—
516D4—1064no—
517D4—1062no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—216no—
521C2×C4—24no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—1064no—
527C2×C4—1062no—
528C2×C4—1068no—
529C2×C4—1064no—
530C2×C4—1064no—
531C2×C4—1064no—
532C2×C4—4248no—
533C2×C4—4242no—
534C2×C4—4242no—
535C2×C4—4242no—
536C2×C4—2124no—
537C2×C4—2122no—
538C2×C4—2124no—
539C2×C4—2124no—
540C2×C4C4⋊C4×D5312yes—
541C2×C4C4⋊C4×D5312yes—
542C2×C4#1313317712yes—
543C2×C4C4×D21212yes—
544C2×C4C4×D21212yes—
545C2×C4#1313318711yescharacteristic
546C2×C4#1313318711yescharacteristic
547C2×C4#1313319512yes—
548C2×C4#1313319712yes—
549C2×C4#1313319712yes—
550C2×C4#1313319912yes—
551C2×C4#1313319912yes—
552C2×C4C22×C4×D5311yescharacteristic
553C2×C4C2×D4×D5311yescharacteristic
554C2×C4C22×D21211yescharacteristic
555C2×C4C2×Q8×D5311yescharacteristic
556C2×C4C2×#1171994711yescharacteristic
557C2×C4C2×#1171994511yescharacteristic
558C8—42no—
559C8—1064no—
560C22—21no—
561C22—21no—
562C22—21no—
563C22—21no—
564C22—24no—
565C22—24no—
566C22—1064no—
567C22—2122no—
568C22—2122no—
569C22—2128no—
570C22#1681049712yes—
571C22#1681049712yes—
572C22#1681079212yes—
573C22#1681080712yes—
574C22C2×#1313317711yescharacteristic
575C22C2×#1313318711yescharacteristic
576C22C2×#1313319511yescharacteristic
577C4—28no—
578C4—42no—
579C4—42no—
580C4—1064no—
581C4—4242no—
582C4—2124no—
583C4#1681049712yes—
584C4#1681049712yes—
585C4C2×C4⋊C4×D5311yescharacteristic
586C4C2×C4×D21211yescharacteristic
587C4C2×#1313319711yescharacteristic
588C4C2×#1313319911yescharacteristic
589C2—21no—
590C2—22no—
591C2—21no—
592C2—2124no—
593C2#2027494214yesdirect product factor
594C2C2×#1681049711yescharacteristic
595C2C2×#1681079211yescharacteristic
596C2C2×#1681080711yescharacteristic
subgroupG/H
centerC23#13133187
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC2×C53×#339C2
socleC23×C53#120
prime radicalC22×#11719945C22
cosocleC22×C4×C53C24
hypercenterC2×#339D53
hypocenterC53C2×#1859
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#339

Complements

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

NH ≅ G/Ntype
2. C2×#16810666C2semidirect
5. C2×#16810493C2semidirect
6. #20274942C2direct (N × H)
7. C2×#16810391C2semidirect
8. C2×C53×#339C2semidirect
24. #16810666C22semidirect
27. C2×#13133145C4semidirect
34. #16810493C22semidirect
35. #16810391C22semidirect
39. C53×#339C22semidirect
57. C2×#11719917C2×C4semidirect
89. #13133145C2×C4semidirect
109. C2×C42×C53C2×C4semidirect
149. #11719945C4⋊C4semidirect
154. C2×D212C4⋊C4semidirect
164. C2×Dic106C4⋊C4semidirect
181. C42×C53C22×C4semidirect
196. #11719917C22×C4semidirect
251. D212C2×C4⋊C4semidirect
258. Dic106C2×C4⋊C4semidirect
298. C2×#339D53semidirect
302. C2×C53#1859semidirect
337. #339C2×D53semidirect
338. C53C2×#1859semidirect
403. C2×C42C4×D53semidirect
497. C42C2×C4×D53semidirect
593. C2#20274942direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168111032
C2×#168106662
C22×#131331452
C2×#168106252
C2×#168104932
#202749422
C2×#168103912
C2×C53×#3392
C2×#185953