Finite Group Databasebeta

← C2×#20274940 #27384945

Subgroups — C2×#20274940 #27384945

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×#16811104C211yesmaximal, characteristic
2C2×#16810663C211yesmaximal, characteristic, complemented
3C22×#13133145C211yesmaximal, characteristic
4C2×#16810622C211yesmaximal, characteristic
5#20274940C218yesmaximal, direct product factor
6C2×#16810392C211yesmaximal, characteristic, complemented
7C2×#16810383C211yesmaximal, characteristic, complemented
8C2×C53×#384C211yesmaximal, characteristic, complemented
9C22×#11719945C2211yescharacteristic
10C22×#11719917C2211yescharacteristic
11C22×#11719917C2211yescharacteristic
12#16811104—22no—
13#16811104—22no—
14#16811104—22no—
15#16811104C2214yes—
16C2×#13133182—21no—
17C2×#13133179—21no—
18C2×C4×C22⋊D53—22no—
19C2×#13133176C2211yescharacteristic
20C2×#13133175—21no—
21C2×C4⋊C4×D53—21no—
22#16810663—22no—
23#16810663C2214yescomplemented
24C2×#13133150—21no—
25C2×#13133148—21no—
26C2×#13133145C418yescomplemented
27C2×#13133145C2214yes—
28#16810622—22no—
29#16810622C2214yes—
30C2×#13133140—21no—
31C2×#13133137—21no—
32#16810392C2214yescomplemented
33#16810383C2214yescomplemented
34C22×C4⋊C4×C53—22no—
35C22×C4⋊C4×C53C2211yescharacteristic
36C22×C4⋊C4×C53C2211yescharacteristic
37C22×C42×C53C2211yescharacteristic
38C53×#384—22no—
39C53×#384C2214yescomplemented
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—
51C22×C4×D53—21no—
52C22×C4×D53—22no—
53C2×#11719917—21no—
54C2×#11719917—21no—
55C2×#11719917—22no—
56C2×#11719917—22no—
57C2×#11719917—22no—
58C2×#11719917C2×C414yescomplemented
59C2×#11719917C2312yes—
60C2×#11719917C2312yes—
61C22×#88322—21no—
62C22×#88322—21no—
63#13133182—24no—
64#13133182—24no—
65#13133179—24no—
66#13133179—24no—
67C4×C22⋊D53—28no—
68C4×C22⋊D53—28no—
69#13133176—21no—
70#13133176—21no—
71#13133176D412yes—
72#13133176C2312yes—
73#13133175—24no—
74#13133175—24no—
75C2×C4⋊Dic53D411yescharacteristic
76C2×C4⋊Dic53D411yescharacteristic
77C2×#11719919—22no—
78C4⋊C4×D53—24no—
79C4⋊C4×D53—24no—
80C2×C22⋊Dic53—21no—
81C2×#11719921—22no—
82C2×C4×Dic53—22no—
83C2×C4×Dic53—21no—
84#13133150—24no—
85#13133148—24no—
86#13133145C2×C4116yescomplemented
87#13133140—24no—
88#13133137—24no—
89C23×C4×C53—21no—
90C23×C4×C53—21no—
91C23×C4×C53D412yes—
92C23×C4×C53C2311yescharacteristic
93C2×C4⋊C4×C53—21no—
94C2×C4⋊C4×C53—21no—
95C2×C4⋊C4×C53—21no—
96C2×C4⋊C4×C53—21no—
97C2×C4⋊C4×C53—28no—
98C2×C4⋊C4×C53—21no—
99C2×C4⋊C4×C53—21no—
100C2×C4⋊C4×C53—21no—
101C2×C4⋊C4×C53—21no—
102C2×C4⋊C4×C53—21no—
103C2×C4⋊C4×C53—48no—
104C2×C4⋊C4×C53D412yes—
105C2×C4⋊C4×C53C2312yes—
106C2×C4⋊C4×C53C2312yes—
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—22no—
118C22×C4×C53—22no—
119C22×C4×C53—22no—
120C22×C4×C53—44no—
121C22×C4×C53—44no—
122C22×C4×C53—41no—
123C22×C4×C53—41no—
124C22×C4×C53C22×C412yes—
125C22×C4×C53C2×D414yes—
126C22×C4×C53C2×D411yescharacteristic
127C22×C4×C53C2×D411yescharacteristic
128C22×C4×C53C2×D411yescharacteristic
129C22×C4×C53C2×D411yescharacteristic
130C22×C4×C53C2×D411yescharacteristic
131C22×C4×C53C2×Q812yes—
132C22×C4×C53#4112yes—
133C22×C4×C53#4112yes—
134C22×C4×C53C2411yescharacteristic
135C2×C22⋊D53—24no—
136C2×C22⋊D53—22no—
137C2×C22⋊D53—24no—
138C2×C22⋊D53—22no—
139C22×Dic53—21no—
140C22×Dic53—21no—
141C22×Dic53—21no—
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—
166C4⋊C4×C53—22no—
167C4⋊C4×C53—22no—
168C4⋊C4×C53—22no—
169C4⋊C4×C53—22no—
170C4⋊C4×C53—22no—
171C4⋊C4×C53—22no—
172C4⋊C4×C53—416no—
173C4⋊C4×C53—41no—
174C4⋊C4×C53—41no—
175C42×C53—24no—
176C42×C53—22no—
177C42×C53C22×C414yescomplemented
178C22⋊Dic53—22no—
179C22⋊Dic53—22no—
180#11719921—28no—
181C4⋊Dic53—21no—
182C4⋊Dic53—21no—
183C4⋊Dic53C2×D412yes—
184C4⋊Dic53C2×D412yes—
185#11719919—28no—
186C4×Dic53—28no—
187C4×Dic53—22no—
188C4×Dic53—22no—
189#11719917—24no—
190#11719917—22no—
191#11719917—22no—
192#11719917—22no—
193#11719917—42no—
194#11719917C22×C414yescomplemented
195C2×#88322—24no—
196C2×#88322—22no—
197C2×#88322—22no—
198C2×#88322—42no—
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—24no—
215C2×C4×C53—44no—
216C2×C4×C53—44no—
217C2×C4×C53—44no—
218C2×C4×C53—42no—
219C2×C4×C53—82no—
220C2×C4×C53—82no—
221C2×C4×C53—82no—
222C2×C4×C53C2×C4⋊C412yes—
223C2×C4×C53C2×C4⋊C412yes—
224C2×C4×C53C4×D412yes—
225C2×C4×C53C4×D412yes—
226C2×C4×C53C4×D412yes—
227C2×C4×C53#12011yescharacteristic
228C2×C4×C53#12011yescharacteristic
229C2×C4×C53C22⋊Q812yes—
230C2×C4×C53C22⋊Q812yes—
231C2×C4×C53C22⋊Q812yes—
232C2×C4×C53#12312yes—
233C2×C4×C53#12312yes—
234C2×C4×C53C23×C411yescharacteristic
235C2×C4×C53C22×D411yescharacteristic
236C2×C4×C53C22×D411yescharacteristic
237C2×C4×C53C22×Q811yescharacteristic
238C2×C4×C53C2×#4111yescharacteristic
239C2×C4×C53C2×#4111yescharacteristic
240C22⋊D53—24no—
241C22⋊D53—24no—
242C22⋊D53—28no—
243C2×Dic53—22no—
244C2×Dic53—24no—
245C2×Dic53—22no—
246C2×Dic53—22no—
247C2×Dic53—42no—
248C2×Dic53—42no—
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
259#88322—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#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—44no—
281C4×C53—82no—
282C4×C53#38512yes—
283C4×C53#38512yes—
284C4×C53C22×C4⋊C411yescharacteristic
285C4×C53C2×C4×D411yescharacteristic
286C4×C53C2×C22⋊Q811yescharacteristic
287C4×C53C2×#12311yescharacteristic
288Dic53—24no—
289Dic53—42no—
290Dic53—42no—
291C2×#520—531no—
292C2×#428—531no—
293C22×#104—531no—
294C2×#417—531no—
295C2×#384D5311yescharacteristic, complemented
296#1859—538no—
297C2×#343—531no—
298C2×#339—531no—
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×M16—531no—
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—531no—
318C2×#117—1061no—
319C22×C4⋊C4—22no—
320C22×C4⋊C4—1061no—
321C22×C4⋊C4C2×D5311yescharacteristic
322C22×C4⋊C4C2×D5311yescharacteristic
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—22no—
335#384C2×D5314yescomplemented
336#343—534no—
337#339—534no—
338C53C2×#185911yescharacteristic, complemented, Sylow
339C2×#41—534no—
340C2×#41—532no—
341C2×#41—534no—
342C2×#41—532no—
343C2×#41—1062no—
344C2×#41—1061no—
345C2×#41—1062no—
346C2×#41—1061no—
347C22×Q8—531no—
348C22×D4—531no—
349C22×D4—1061no—
350C23×C4—21no—
351C23×C4—21no—
352C23×C4—1062no—
353C23×C4—1061no—
354C23×C4C22⋊D5312yes—
355C23×C4C22×D5311yescharacteristic
356C2×M16—534no—
357C2×M16—532no—
358C2×M16—532no—
359C2×M16—1061no—
360C2×M16—1062no—
361C2×M16—1061no—
362C2×M16—1062no—
363C2×M16—1062no—
364C22×C8—1061no—
365C22×C8—1061no—
366C4×Q8—1064no—
367C4×Q8—1064no—
368C4×D4—1064no—
369C4×D4—1064no—
370C4×D4—1068no—
371C4×D4—1068no—
372#117—532no—
373#117—532no—
374#117—1064no—
375#117—1064no—
376#117—1061no—
377#117—1061no—
378C2×C4⋊C4—21no—
379C2×C4⋊C4—21no—
380C2×C4⋊C4—21no—
381C2×C4⋊C4—21no—
382C2×C4⋊C4—21no—
383C2×C4⋊C4—21no—
384C2×C4⋊C4—21no—
385C2×C4⋊C4—21no—
386C2×C4⋊C4—28no—
387C2×C4⋊C4—21no—
388C2×C4⋊C4—48no—
389C2×C4⋊C4—531no—
390C2×C4⋊C4—531no—
391C2×C4⋊C4—1064no—
392C2×C4⋊C4—1064no—
393C2×C4⋊C4—1062no—
394C2×C4⋊C4C22⋊D5312yes—
395C2×C4⋊C4C22×D5312yes—
396C2×C4⋊C4C22×D5312yes—
397C2×C22⋊C4—1062no—
398C2×C22⋊C4—1061no—
399C2×C42—22no—
400C2×C42—21no—
401C2×C42—1062no—
402C2×C42—1061no—
403C2×C42C4×D5314yescomplemented
404C2×C42C22×D5312yes—
405#107—1064no—
406#106—1064no—
407#104—5316no—
408#103—1064no—
409#102—1064no—
410C24—1061no—
411C24C2×C22⋊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—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—44no—
442C22×C4—44no—
443C22×C4—41no—
444C22×C4—41no—
445C22×C4—1068no—
446C22×C4—1061no—
447C22×C4—1064no—
448C22×C4—1062no—
449C22×C4—1064no—
450C22×C4—1062no—
451C22×C4—1061no—
452C22×C4—1061no—
453C22×C4—4244no—
454C22×C4C2×C4×D5312yes—
455C22×C4#1171994512yes—
456C22×C4D4×D5311yescharacteristic
457C22×C4D4×D5311yescharacteristic
458C22×C4D4×D5311yescharacteristic
459C22×C4D4×D5311yescharacteristic
460C22×C4Q8×D5312yes—
461C22×C4#1171994912yes—
462C22×C4C2×C22⋊D5314yes—
463C22×C4C2×C22⋊D5311yescharacteristic
464C22×C4C23×D5311yescharacteristic
465M16—534no—
466M16—1064no—
467M16—1062no—
468M16—1062no—
469M16—1062no—
470M16—2122no—
471C2×C8—1064no—
472C2×C8—1062no—
473C2×C8—1062no—
474C2×C8—2122no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—416no—
482C4⋊C4—41no—
483C4⋊C4—41no—
484C4⋊C4—532no—
485C4⋊C4—532no—
486C4⋊C4—1068no—
487C4⋊C4—1061no—
488C4⋊C4—1061no—
489C22⋊C4—1068no—
490C22⋊C4—1062no—
491C22⋊C4—1062no—
492C42—24no—
493C42—22no—
494C42—1068no—
495C42—1062no—
496C42—1062no—
497C42C2×C4×D5314yescomplemented
498C23—22no—
499C23—22no—
500C23—1062no—
501C23—1064no—
502C23—2122no—
503C23—2122no—
504C23C4×C22⋊D5312yes—
505C23#1313318511yescharacteristic
506C23#1313318511yescharacteristic
507C23#1313319312yes—
508C23C22×C22⋊D5311yescharacteristic
509Q8—534no—
510Q8—1062no—
511Q8—1064no—
512D4—534no—
513D4—1064no—
514D4—1064no—
515D4—1068no—
516D4—1062no—
517D4—1064no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—216no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—44no—
524C2×C4—44no—
525C2×C4—44no—
526C2×C4—42no—
527C2×C4—82no—
528C2×C4—82no—
529C2×C4—82no—
530C2×C4—1062no—
531C2×C4—1064no—
532C2×C4—1064no—
533C2×C4—1064no—
534C2×C4—1068no—
535C2×C4—1062no—
536C2×C4—1062no—
537C2×C4—4248no—
538C2×C4—2122no—
539C2×C4—2122no—
540C2×C4C4⋊C4×D5312yes—
541C2×C4C4⋊C4×D5312yes—
542C2×C4C4×C22⋊D5312yes—
543C2×C4#1313317912yes—
544C2×C4#1313317912yes—
545C2×C4#1313318511yescharacteristic
546C2×C4#1313318511yescharacteristic
547C2×C4#1313319312yes—
548C2×C4#1313319512yes—
549C2×C4#1313319512yes—
550C2×C4#1313320212yes—
551C2×C4#1313320212yes—
552C2×C4C22×C4×D5311yescharacteristic
553C2×C4C2×D4×D5311yescharacteristic
554C2×C4C2×D4×D5311yescharacteristic
555C2×C4C2×Q8×D5311yescharacteristic
556C2×C4C2×#1171994511yescharacteristic
557C2×C4C2×#1171994911yescharacteristic
558C8—1064no—
559C8—2122no—
560C22—21no—
561C22—21no—
562C22—21no—
563C22—21no—
564C22—24no—
565C22—24no—
566C22—1064no—
567C22—2122no—
568C22—2122no—
569C22—2128no—
570C22#1681049412yes—
571C22#1681049412yes—
572C22#1681079012yes—
573C22#1681080512yes—
574C22C2×C4×C22⋊D5311yescharacteristic
575C22C2×#1313318511yescharacteristic
576C22C2×#1313319311yescharacteristic
577C4—28no—
578C4—44no—
579C4—82no—
580C4—1064no—
581C4—2122no—
582C4—2122no—
583C4#1681049412yes—
584C4#1681049412yes—
585C4C2×C4⋊C4×D5311yescharacteristic
586C4C2×#1313317911yescharacteristic
587C4C2×#1313319511yescharacteristic
588C4C2×#1313320211yescharacteristic
589C2—21no—
590C2—21no—
591C2—22no—
592C2—2124no—
593C2#2027494014yesdirect product factor
594C2C2×#1681049411yescharacteristic
595C2C2×#1681079011yescharacteristic
596C2C2×#1681080511yescharacteristic
subgroupG/H
centerC23#13133185
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC2×C53×#384C2
socleC23×C53#120
prime radicalC22×#11719945C22
cosocleC22×C4×C53C24
hypercenterC2×#384D53
hypocenterC53C2×#1859
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#384

Complements

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

NH ≅ G/Ntype
2. C2×#16810663C2semidirect
5. #20274940C2direct (N × H)
6. C2×#16810392C2semidirect
7. C2×#16810383C2semidirect
8. C2×C53×#384C2semidirect
23. #16810663C22semidirect
26. C2×#13133145C4semidirect
32. #16810392C22semidirect
33. #16810383C22semidirect
39. C53×#384C22semidirect
58. C2×#11719917C2×C4semidirect
86. #13133145C2×C4semidirect
109. C2×C42×C53C2×C4semidirect
149. #11719945C4⋊C4semidirect
154. C2×D212C4⋊C4semidirect
164. C2×Dic106C4⋊C4semidirect
177. C42×C53C22×C4semidirect
194. #11719917C22×C4semidirect
251. D212C2×C4⋊C4semidirect
258. Dic106C2×C4⋊C4semidirect
295. C2×#384D53semidirect
302. C2×C53#1859semidirect
335. #384C2×D53semidirect
338. C53C2×#1859semidirect
403. C2×C42C4×D53semidirect
497. C42C2×C4×D53semidirect
593. C2#20274940direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168111042
C2×#168106632
C22×#131331452
C2×#168106222
#202749402
C2×#168103922
C2×#168103832
C2×C53×#3842
C2×#185953