Finite Group Databasebeta

← C2×#20276167 #27413012

Subgroups — C2×#20276167 #27413012

Order 13568.

906 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810650C211yesmaximal, characteristic, complemented
2C2×#16810641C211yesmaximal, characteristic
3C2×#16810617C211yesmaximal, characteristic
4C2×#16810605C211yesmaximal, characteristic
5#20276167C218yesmaximal, direct product factor
6C2×#16810393C211yesmaximal, characteristic, complemented
7C2×#16810387C211yesmaximal, characteristic, complemented
8C2×C53×H(3,ℤ/4)C211yesmaximal, characteristic, complemented
9C22×#11719945C2211yescharacteristic
10C22×#11719917C2211yescharacteristic
11C22×#11719917C2211yescharacteristic
12C23×#88322C2211yescharacteristic
13#16810650—22no—
14#16810650C2214yescomplemented
15C2×C4⋊#88322—22no—
16#16810641—22no—
17#16810641C2214yes—
18C2×#13133145—22no—
19#16810617—22no—
20#16810617C2214yes—
21C2×#13133140—21no—
22C2×#13133137—21no—
23#16810605—22no—
24#16810605C2214yes—
25C2×#13133135—21no—
26C2×#13133133—21no—
27#16810393C2214yescomplemented
28#16810387C2214yescomplemented
29C22×M16×C53C2211yescharacteristic
30C2×#117×C53C2211yescharacteristic
31C2×#117×C53C2211yescharacteristic
32C53×H(3,ℤ/4)C2214yescomplemented
33C2×#11719945—21no—
34C2×#11719945—21no—
35C2×#11719945—21no—
36C2×#11719945—21no—
37C2×#11719945—21no—
38C2×#11719945—21no—
39C2×#11719945—21no—
40C2×#11719945—21no—
41C2×#11719945—21no—
42C2×#11719945—21no—
43C2×#11719945D412yes—
44C2×#11719945C2312yes—
45C22×Dic106D411yescharacteristic
46C22×D212D411yescharacteristic
47C22×C22⋊D53—21no—
48C22×C4×D53—21no—
49C2×#11719917—21no—
50C2×#11719917—21no—
51C2×#11719917—21no—
52C2×#11719917—21no—
53C2×#11719917—21no—
54C2×#11719917—21no—
55C2×#11719917—21no—
56C2×#11719917—21no—
57C2×#11719917—21no—
58C2×#11719917—21no—
59C2×#11719917C2312yes—
60C2×#11719917C2312yes—
61C22×#88322—21no—
62C22×#88322—21no—
63C22×#88322—21no—
64C22×#88322—21no—
65C22×#88322—21no—
66C22×#88322—21no—
67C22×#88322—21no—
68C22×#88322—21no—
69C22×#88322C2312yes—
70C4⋊#88322—28no—
71#13133145—28no—
72#13133140—24no—
73#13133137—24no—
74#13133135—24no—
75#13133133—24no—
76C23×C4×C53C2311yescharacteristic
77C2×M16×C53—21no—
78C2×M16×C53—21no—
79C2×M16×C53—21no—
80C2×M16×C53—21no—
81C2×M16×C53—21no—
82C2×M16×C53C2312yes—
83C22×C8×C53—21no—
84#117×C53—21no—
85#117×C53—21no—
86#117×C53—21no—
87#117×C53—21no—
88#117×C53—21no—
89#117×C53D412yes—
90#117×C53C2312yes—
91#117×C53C2312yes—
92C2×C4⋊C4×C53—21no—
93C2×C4⋊C4×C53—21no—
94C2×C22⋊C4×C53—21no—
95C2×C22⋊C4×C53—21no—
96C2×C42×C53—21no—
97C2×C42×C53D412yes—
98C24×C53#4111yescharacteristic
99C23×D53—21no—
100C22×C4×C53—22no—
101C22×C4×C53—41no—
102C22×C4×C53C2×D411yescharacteristic
103C22×C4×C53C2×D411yescharacteristic
104C22×C4×C53C2×D411yescharacteristic
105C22×C4×C53C2×D411yescharacteristic
106C22×C4×C53#4111yescharacteristic
107C22×C4×C53#4111yescharacteristic
108C22×C4×C53#4111yescharacteristic
109C22×C4×C53#4111yescharacteristic
110C22×C4×C53#4111yescharacteristic
111C22×C4×C53#4111yescharacteristic
112C22×C4×C53#4111yescharacteristic
113C22×C4×C53#4111yescharacteristic
114C22×C4×C53#4111yescharacteristic
115C22×C4×C53C2411yescharacteristic
116C2×C22⋊D53—22no—
117C2×C22⋊D53—22no—
118C2×C22⋊D53—41no—
119C2×C22⋊D53—41no—
120C2×C22⋊D53—41no—
121C2×C22⋊D53—41no—
122C22×Dic53—21no—
123#11719945—22no—
124#11719945—22no—
125#11719945—22no—
126#11719945—22no—
127#11719945—22no—
128#11719945—22no—
129#11719945—22no—
130#11719945—22no—
131#11719945—41no—
132#11719945—41no—
133#11719945—41no—
134#11719945—41no—
135#11719945—41no—
136#11719945—41no—
137#11719945—41no—
138#11719945—41no—
139C2×D212—21no—
140C2×D212—21no—
141C2×D212—21no—
142C2×D212—21no—
143C2×D212—21no—
144C2×D212C2×D412yes—
145C2×C4×D53—22no—
146C2×C4×D53—22no—
147C2×C4×D53—41no—
148C2×C4×D53—41no—
149C2×C4×D53—41no—
150C2×C4×D53—41no—
151C2×Dic106—21no—
152C2×Dic106—21no—
153C2×Dic106—21no—
154C2×Dic106—21no—
155C2×Dic106—21no—
156C2×Dic106C2×D412yes—
157M16×C53—22no—
158M16×C53—22no—
159M16×C53—41no—
160M16×C53—41no—
161C2×C8×C53—22no—
162C2×C8×C53—41no—
163C2×C8×C53—41no—
164C4⋊C4×C53—22no—
165C4⋊C4×C53—22no—
166C4⋊C4×C53—22no—
167C4⋊C4×C53—41no—
168C22⋊C4×C53—22no—
169C22⋊C4×C53—22no—
170C22⋊C4×C53—22no—
171C22⋊C4×C53—41no—
172C42×C53—22no—
173C42×C53—22no—
174C42×C53—41no—
175C42×C53C2×D414yes—
176#11719917—22no—
177#11719917—22no—
178#11719917—22no—
179#11719917—22no—
180#11719917—41no—
181#11719917—41no—
182#11719917—41no—
183#11719917—41no—
184C2×#88322—22no—
185C2×#88322—22no—
186C2×#88322—22no—
187C2×#88322—22no—
188C2×#88322—22no—
189C2×#88322—41no—
190C2×#88322—41no—
191C2×#88322—41no—
192C2×#88322—41no—
193C2×#88322—41no—
194C2×#88322—41no—
195C2×#88322—41no—
196C2×#88322—41no—
197C23×C53—42no—
198C23×C53#12311yescharacteristic
199C23×C53#12311yescharacteristic
200C23×C53#12311yescharacteristic
201C23×C53#12311yescharacteristic
202C23×C53#12611yescharacteristic
203C23×C53#12611yescharacteristic
204C23×C53C2×#4111yescharacteristic
205C22×D53—22no—
206C22×D53—41no—
207C22×D53—41no—
208C22×D53—82no—
209C2×C4×C53—24no—
210C2×C4×C53—44no—
211C2×C4×C53—42no—
212C2×C4×C53—82no—
213C2×C4×C53#12111yescharacteristic
214C2×C4×C53#12111yescharacteristic
215C2×C4×C53#12111yescharacteristic
216C2×C4×C53#12111yescharacteristic
217C2×C4×C53#12311yescharacteristic
218C2×C4×C53#12311yescharacteristic
219C2×C4×C53#12311yescharacteristic
220C2×C4×C53#12311yescharacteristic
221C2×C4×C53#12311yescharacteristic
222C2×C4×C53#12311yescharacteristic
223C2×C4×C53#12311yescharacteristic
224C2×C4×C53#12311yescharacteristic
225C2×C4×C53#12411yescharacteristic
226C2×C4×C53#12411yescharacteristic
227C2×C4×C53#12411yescharacteristic
228C2×C4×C53#12411yescharacteristic
229C2×C4×C53#12611yescharacteristic
230C2×C4×C53#12611yescharacteristic
231C2×C4×C53#12611yescharacteristic
232C2×C4×C53#12611yescharacteristic
233C2×C4×C53#12611yescharacteristic
234C2×C4×C53#12611yescharacteristic
235C2×C4×C53C22×D411yescharacteristic
236C2×C4×C53C22×D411yescharacteristic
237C2×C4×C53C2×#4111yescharacteristic
238C2×C4×C53C2×#4111yescharacteristic
239C2×C4×C53C2×#4111yescharacteristic
240C2×C4×C53C2×#4111yescharacteristic
241C22⋊D53—42no—
242C22⋊D53—42no—
243C22⋊D53—42no—
244C22⋊D53—42no—
245C2×Dic53—22no—
246C2×Dic53—41no—
247C2×Dic53—41no—
248D212—22no—
249D212—22no—
250D212—41no—
251D212—41no—
252C4×D53—42no—
253C4×D53—42no—
254C4×D53—42no—
255C4×D53—42no—
256Dic106—22no—
257Dic106—22no—
258Dic106—41no—
259Dic106—41no—
260C8×C53—42no—
261#88322—42no—
262#88322—42no—
263#88322—42no—
264C2×#2080—531nomaximal, Sylow
265C22×C53—22no—
266C22×C53—22no—
267C22×C53—22no—
268C22×C53—41no—
269C22×C53—41no—
270C22×C53—41no—
271C22×C53—41no—
272C22×C53#39711yescharacteristic
273C22×C53#39711yescharacteristic
274C22×C53#39711yescharacteristic
275C22×C53#39711yescharacteristic
276C22×C53C2×#12311yescharacteristic
277C22×C53C2×#12311yescharacteristic
278C22×C53C2×#12611yescharacteristic
279C2×D53—42no—
280C2×D53—82no—
281C2×D53—82no—
282C2×D53—82no—
283C4×C53—44no—
284C4×C53—82no—
285C4×C53#39711yescharacteristic
286C4×C53#39711yescharacteristic
287C4×C53#39711yescharacteristic
288C4×C53#39711yescharacteristic
289C4×C53C2×#12111yescharacteristic
290C4×C53C2×#12311yescharacteristic
291C4×C53C2×#12411yescharacteristic
292C4×C53C2×#12611yescharacteristic
293Dic53—42no—
294C2×#424—531no—
295C2×#421—531no—
296C2×#416—531no—
297C2×#413—531no—
298#2080—538no—
299C2×#344—531no—
300C2×#341—531no—
301C2×H(3,ℤ/4)D5311yescharacteristic, complemented
302C2×C53—21no—
303C2×C53—21no—
304C2×C53—21no—
305C2×C53—21no—
306C2×C53—21no—
307C2×C53—21no—
308C2×C53#208012yescomplemented
309C2×C53C2×#39711yescharacteristic
310D53—82no—
311C22×#41—531no—
312C22×M16—531no—
313C22×M16—531no—
314C22×M16C2×D5311yescharacteristic
315C23×C8—531no—
316C2×#117C2×D5311yescharacteristic
317C2×#117C2×D5311yescharacteristic
318#424—534no—
319#424—1062no—
320C2×C4⋊C8—1062no—
321#421—534no—
322#421—1062no—
323C2×#104—1062no—
324#416—534no—
325#416—1062no—
326C2×#103—1061no—
327C2×#102—1061no—
328#413—534no—
329#413—1062no—
330C2×#101—1061no—
331C2×#100—1061no—
332#344—534no—
333#341—534no—
334H(3,ℤ/4)C2×D5314yescomplemented
335C53C2×#208011yescharacteristic, complemented, Sylow
336C2×#41—532no—
337C2×#41—532no—
338C2×#41—1061no—
339C2×#41—1061no—
340C2×#41—1061no—
341C2×#41—1061no—
342C2×#41—1061no—
343C2×#41—1061no—
344C2×#41—1061no—
345C2×#41—1061no—
346C2×#41—1061no—
347C2×#41—1061no—
348C22×Q8—531no—
349C22×D4—531no—
350C22×D4—1061no—
351C23×C4—1061no—
352C23×C4C22×D5311yescharacteristic
353C2×M16—21no—
354C2×M16—21no—
355C2×M16—21no—
356C2×M16—21no—
357C2×M16—21no—
358C2×M16—532no—
359C2×M16—532no—
360C2×M16—1061no—
361C2×M16—1061no—
362C2×M16—1061no—
363C2×M16—1061no—
364C2×M16—1061no—
365C2×M16—1061no—
366C2×M16—1061no—
367C2×M16—1061no—
368C2×M16—1061no—
369C2×M16—1061no—
370C2×M16C22×D5312yes—
371C22×C8—21no—
372C22×C8—532no—
373C22×C8—1061no—
374C22×C8—1061no—
375C22×C8—1061no—
376C22×C8—1061no—
377C22×C8—1061no—
378C22×C8—1061no—
379C22×C8—1061no—
380C22×C8—1061no—
381#117—21no—
382#117—21no—
383#117—21no—
384#117—21no—
385#117—21no—
386#117D21212yes—
387#117C22×D5312yes—
388#117C22×D5312yes—
389C2×C4⋊C4—21no—
390C2×C4⋊C4—21no—
391C2×C22⋊C4—21no—
392C2×C22⋊C4—21no—
393C2×C42—21no—
394C2×C42D21212yes—
395C4⋊C8—1068no—
396#104—1068no—
397#103—1064no—
398#102—1064no—
399#101—1064no—
400#100—1064no—
401C24—1061no—
402C24#1171994711yescharacteristic
403#41—1062no—
404#41—1062no—
405#41—1062no—
406#41—1062no—
407#41—1062no—
408#41—1062no—
409#41—1062no—
410#41—1062no—
411#41—2121no—
412#41—2121no—
413#41—2121no—
414#41—2121no—
415#41—2121no—
416#41—2121no—
417#41—2121no—
418#41—2121no—
419C2×Q8—532no—
420C2×Q8—1061no—
421C2×Q8—1061no—
422C2×Q8—1061no—
423C2×Q8—1061no—
424C2×Q8—1061no—
425C2×D4—532no—
426C2×D4—1062no—
427C2×D4—1062no—
428C2×D4—1061no—
429C2×D4—1061no—
430C2×D4—1061no—
431C2×D4—1061no—
432C2×D4—1061no—
433C2×D4—2121no—
434C2×D4—2121no—
435C2×D4—2121no—
436C2×D4—2121no—
437C22×C4—22no—
438C22×C4—41no—
439C22×C4—1061no—
440C22×C4—1062no—
441C22×C4—1062no—
442C22×C4—2121no—
443C22×C4—2121no—
444C22×C4—2121no—
445C22×C4—2121no—
446C22×C4C2×D21211yescharacteristic
447C22×C4C2×D21211yescharacteristic
448C22×C4#1171994511yescharacteristic
449C22×C4#1171994511yescharacteristic
450C22×C4#1171994511yescharacteristic
451C22×C4#1171994511yescharacteristic
452C22×C4D4×D5311yescharacteristic
453C22×C4D4×D5311yescharacteristic
454C22×C4#1171994711yescharacteristic
455C22×C4#1171994711yescharacteristic
456C22×C4#1171994711yescharacteristic
457C22×C4#1171994911yescharacteristic
458C22×C4#1171994911yescharacteristic
459C22×C4C23×D5311yescharacteristic
460M16—22no—
461M16—22no—
462M16—41no—
463M16—41no—
464M16—1062no—
465M16—1062no—
466M16—1062no—
467M16—1062no—
468M16—2121no—
469M16—2121no—
470M16—2121no—
471M16—2121no—
472C2×C8—22no—
473C2×C8—41no—
474C2×C8—41no—
475C2×C8—1062no—
476C2×C8—1062no—
477C2×C8—1062no—
478C2×C8—1062no—
479C2×C8—1062no—
480C2×C8—2121no—
481C2×C8—2121no—
482C2×C8—2121no—
483C2×C8—2121no—
484C2×C8—2121no—
485C2×C8—2121no—
486C2×C8—2121no—
487C2×C8—2121no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—41no—
492C22⋊C4—22no—
493C22⋊C4—22no—
494C22⋊C4—22no—
495C22⋊C4—41no—
496C42—22no—
497C42—22no—
498C42—41no—
499C42C2×D21214yes—
500C23—42no—
501C23—1062no—
502C23—4242no—
503C23—2121no—
504C23—2121no—
505C23#1313319911yescharacteristic
506C23#1313319911yescharacteristic
507C23#1313320111yescharacteristic
508C23#1313320111yescharacteristic
509C23#1313321211yescharacteristic
510C23#1313321211yescharacteristic
511C23C2×#1171994711yescharacteristic
512Q8—1062no—
513Q8—1062no—
514Q8—2121no—
515Q8—2121no—
516D4—1062no—
517D4—1062no—
518D4—2122no—
519D4—2122no—
520D4—2122no—
521D4—2122no—
522D4—2121no—
523D4—2121no—
524C2×C4—24no—
525C2×C4—44no—
526C2×C4—42no—
527C2×C4—82no—
528C2×C4—1062no—
529C2×C4—2121no—
530C2×C4—2121no—
531C2×C4—2122no—
532C2×C4—2122no—
533C2×C4—2122no—
534C2×C4—2122no—
535C2×C4#1313319211yescharacteristic
536C2×C4#1313319211yescharacteristic
537C2×C4#1313319211yescharacteristic
538C2×C4#1313319211yescharacteristic
539C2×C4#1313319911yescharacteristic
540C2×C4#1313319911yescharacteristic
541C2×C4#1313319911yescharacteristic
542C2×C4#1313319911yescharacteristic
543C2×C4#1313319911yescharacteristic
544C2×C4#1313319911yescharacteristic
545C2×C4#1313320111yescharacteristic
546C2×C4#1313320111yescharacteristic
547C2×C4#1313320511yescharacteristic
548C2×C4#1313320511yescharacteristic
549C2×C4#1313320511yescharacteristic
550C2×C4#1313320511yescharacteristic
551C2×C4#1313321211yescharacteristic
552C2×C4#1313321211yescharacteristic
553C2×C4#1313321211yescharacteristic
554C2×C4#1313321211yescharacteristic
555C2×C4#1313321211yescharacteristic
556C2×C4#1313321211yescharacteristic
557C2×C4C2×D4×D5311yescharacteristic
558C2×C4C22×D21211yescharacteristic
559C2×C4C2×#1171994711yescharacteristic
560C2×C4C2×#1171994511yescharacteristic
561C2×C4C2×#1171994511yescharacteristic
562C2×C4C2×#1171994911yescharacteristic
563C8—42no—
564C8—2122no—
565C8—2122no—
566C8—2122no—
567C22—22no—
568C22—22no—
569C22—22no—
570C22—41no—
571C22—41no—
572C22—41no—
573C22—41no—
574C22—4242no—
575C22—4242no—
576C22—4242no—
577C22—2122no—
578C22#1681055011yescharacteristic
579C22#1681055011yescharacteristic
580C22#1681055011yescharacteristic
581C22#1681055011yescharacteristic
582C22C2×#1313319911yescharacteristic
583C22C2×#1313320111yescharacteristic
584C22C2×#1313321211yescharacteristic
585C4—44no—
586C4—82no—
587C4—2122no—
588C4#1681055011yescharacteristic
589C4#1681055011yescharacteristic
590C4#1681055011yescharacteristic
591C4#1681055011yescharacteristic
592C4C2×#1313319211yescharacteristic
593C4C2×#1313319911yescharacteristic
594C4C2×#1313320511yescharacteristic
595C4C2×#1313321211yescharacteristic
596C2—21no—
597C2—21no—
598C2—21no—
599C2—21no—
600C2—21no—
601C2—21no—
602C2—4242no—
603C2#2027616712yesdirect product factor
604C2C2×#1681055011yescharacteristic
subgroupG/H
centerC2×C4#13133205
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC2×C53×H(3,ℤ/4)C2
socleC22×C53#397
prime radicalC22×#11719945C22
hypercenterC2×H(3,ℤ/4)D53
hypocenterC53C2×#2080
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C2×H(3,ℤ/4)

Complements

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

NH ≅ G/Ntype
1. C2×#16810650C2semidirect
5. #20276167C2direct (N × H)
6. C2×#16810393C2semidirect
7. C2×#16810387C2semidirect
8. C2×C53×H(3,ℤ/4)C2semidirect
14. #16810650C22semidirect
27. #16810393C22semidirect
28. #16810387C22semidirect
32. C53×H(3,ℤ/4)C22semidirect
301. C2×H(3,ℤ/4)D53semidirect
308. C2×C53#2080semidirect
334. H(3,ℤ/4)C2×D53semidirect
335. C53C2×#2080semidirect
603. C2#20276167direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168106502
C2×#168106412
C2×#168106172
C2×#168106052
#202761672
C2×#168103932
C2×#168103872
C2×C53×H(3,ℤ/4)2
C2×#208053