Finite Group Databasebeta

← C2×#20273260 #27351350

Subgroups — C2×#20273260 #27351350

Order 13568.

1302 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811168C211yesmaximal, characteristic
2C2×#16810677C211yesmaximal, characteristic, complemented
3C2×#16810490C211yesmaximal, characteristic, complemented
4#20273260C218yesmaximal, direct product factor
5C2×#16810358C211yesmaximal, characteristic, complemented
6C2×#16810357C211yesmaximal, characteristic, complemented
7C2×#16810356C211yesmaximal, characteristic, complemented
8C2×C53×#327C211yesmaximal, characteristic, complemented
9C22×#11719947—21no—
10#16811168—22no—
11#16811168—22no—
12#16811168C414yescomplemented
13#16811168C2214yes—
14C2×#13133216C411yescharacteristic, complemented
15C2×#13133213C411yescharacteristic, complemented
16C2×#13133203—21no—
17C2×#13133191C2211yescharacteristic
18C2×#13133189—21no—
19C2×D4×Dic53C411yescharacteristic, complemented
20C2×#13133179C411yescharacteristic, complemented
21C2×#13133177—21no—
22C22×C4×Dic53C2211yescharacteristic
23#16810677—22no—
24#16810677C2214yescomplemented
25C2×#13133146—21no—
26C2×C22⋊#88322C2211yescharacteristic
27C2×#13133127—21no—
28#16810490—22no—
29#16810490C2214yescomplemented
30C2×#13133124—21no—
31C2×#13133124—21no—
32C2×#13133124C2211yescharacteristic
33#16810358C2214yescomplemented
34#16810357C2214yescomplemented
35#16810356C2214yescomplemented
36C2×#121×C53C2211yescharacteristic
37C2×C22⋊C8×C53C2211yescharacteristic
38C2×#95×C53C2211yescharacteristic
39C53×#327C2214yescomplemented
40C2×#11719947—24no—
41C2×#11719947—24no—
42C2×#11719947—44no—
43C2×#11719947—44no—
44C22×Dic106—21no—
45C22×D212C2×C411yescharacteristic
46C22×C22⋊D53—21no—
47C22×C22⋊D53—21no—
48C22×C22⋊D53—21no—
49C23×Dic53—21no—
50C23×Dic53—21no—
51C23×Dic53D411yescharacteristic
52C23×Dic53D411yescharacteristic
53C22×C4×D53—21no—
54C22×#88322—21no—
55#13133216—22no—
56#13133216C2×C414yescomplemented
57#13133213—22no—
58#13133213C2×C414yescomplemented
59#13133203—24no—
60#13133203—24no—
61#13133191—21no—
62#13133191—21no—
63#13133191C2×C412yes—
64#13133191C2312yes—
65#13133189—24no—
66#13133189—24no—
67D4×Dic53—22no—
68D4×Dic53C2×C414yescomplemented
69#13133179—22no—
70#13133179C2×C414yescomplemented
71#13133177—24no—
72#13133177—24no—
73C2×C4⋊Dic53C2×C411yescharacteristic
74C2×#11719919—21no—
75C2×C22⋊Dic53—21no—
76C2×#11719921—21no—
77C2×#11719921—21no—
78C2×C4×Dic53C2×C411yescharacteristic
79C2×C4×Dic53C2×C411yescharacteristic
80C2×C4×Dic53C2×C412yes—
81C2×C4×Dic53D411yescharacteristic
82C2×C4×Dic53D411yescharacteristic
83C2×C4×Dic53D412yes—
84C2×C4×Dic53D412yes—
85C2×C4×Dic53C2312yes—
86#13133146—24no—
87C22⋊#88322—21no—
88C22⋊#88322C2312yes—
89#13133127—24no—
90#13133124—21no—
91#13133124—24no—
92#13133124—24no—
93#13133124C2312yes—
94C22×D4×C53—21no—
95C22×D4×C53C2×C411yescharacteristic
96C23×C4×C53—21no—
97C23×C4×C53C2311yescharacteristic
98C22×C8×C53—21no—
99#121×C53—21no—
100#121×C53—21no—
101#121×C53C2×C412yes—
102#121×C53C2312yes—
103C2×C4⋊C4×C53C2×C411yescharacteristic
104C2×C22⋊C4×C53—21no—
105C22⋊C8×C53—21no—
106C22⋊C8×C53C2312yes—
107#95×C53—21no—
108#95×C53C2312yes—
109C24×C53—21no—
110C24×C53C2×D411yescharacteristic
111C23×D53—21no—
112C2×D4×C53—22no—
113C2×D4×C53—22no—
114C2×D4×C53—21no—
115C2×D4×C53—42no—
116C2×D4×C53—42no—
117C2×D4×C53—42no—
118C2×D4×C53C22×C412yes—
119C22×C4×C53—21no—
120C22×C4×C53—22no—
121C22×C4×C53—42no—
122C22×C4×C53—42no—
123C22×C4×C53—42no—
124C22×C4×C53—42no—
125C22×C4×C53C22×C411yescharacteristic
126C22×C4×C53C22×C411yescharacteristic
127C22×C4×C53C2×D411yescharacteristic
128C22×C4×C53C2×D411yescharacteristic
129C22×C4×C53C2×D411yescharacteristic
130C22×C4×C53C2411yescharacteristic
131C2×C22⋊D53—22no—
132C2×C22⋊D53—22no—
133C2×C22⋊D53—24no—
134C2×C22⋊D53—24no—
135C2×C22⋊D53—44no—
136C2×C22⋊D53—44no—
137C2×C22⋊D53—42no—
138C2×C22⋊D53—42no—
139C22×Dic53—22no—
140C22×Dic53—22no—
141C22×Dic53—21no—
142C22×Dic53—24no—
143C22×Dic53—22no—
144C22×Dic53—42no—
145C22×Dic53—44no—
146C22×Dic53—42no—
147C22×Dic53—42no—
148C22×Dic53—42no—
149C22×Dic53C22⋊C411yescharacteristic
150C22×Dic53C22⋊C411yescharacteristic
151C22×Dic53C22⋊C411yescharacteristic
152C22×Dic53C22⋊C411yescharacteristic
153C22×Dic53C22⋊C412yes—
154C22×Dic53C22⋊C412yes—
155C22×Dic53C2×D412yes—
156C22×Dic53C2×D412yes—
157#11719947—48no—
158#11719947—48no—
159#11719947—416no—
160C2×D212—21no—
161C2×D212—42no—
162C2×D212C22×C412yes—
163C2×C4×D53—24no—
164C2×C4×D53—44no—
165C2×Dic106—24no—
166C2×Dic106—44no—
167C2×C8×C53—22no—
168C2×C8×C53—42no—
169C4⋊C4×C53—21no—
170C4⋊C4×C53C22×C412yes—
171C22⋊C4×C53—22no—
172C22⋊C4×C53—22no—
173C22⋊Dic53—24no—
174#11719921—22no—
175#11719921—22no—
176#11719921—24no—
177C4⋊Dic53—21no—
178C4⋊Dic53C22×C412yes—
179#11719919—24no—
180C4×Dic53C22⋊C412yes—
181C4×Dic53C22⋊C412yes—
182C4×Dic53C22⋊C412yes—
183C4×Dic53C22⋊C412yes—
184C4×Dic53C22×C412yes—
185C4×Dic53C22×C412yes—
186C4×Dic53C2×D412yes—
187C4×Dic53C2×D412yes—
188C2×#88322—22no—
189C2×#88322—42no—
190C23×C53—22no—
191C23×C53—22no—
192C23×C53—22no—
193C23×C53—42no—
194C23×C53—42no—
195C23×C53—42no—
196C23×C53C2×C22⋊C411yescharacteristic
197C23×C53C2×C22⋊C411yescharacteristic
198C23×C53AGL(1,ℤ/8)12yes—
199C23×C53AGL(1,ℤ/8)12yes—
200C23×C53C22×D411yescharacteristic
201C22×D53—22no—
202C22×D53—42no—
203C22×D53—42no—
204C22×D53—42no—
205D4×C53—42no—
206D4×C53—42no—
207D4×C53—42no—
208D4×C53—42no—
209D4×C53—44no—
210C2×C4×C53—22no—
211C2×C4×C53—22no—
212C2×C4×C53—22no—
213C2×C4×C53—41no—
214C2×C4×C53—41no—
215C2×C4×C53—41no—
216C2×C4×C53—41no—
217C2×C4×C53—42no—
218C2×C4×C53—42no—
219C2×C4×C53—42no—
220C2×C4×C53—42no—
221C2×C4×C53—84no—
222C2×C4×C53C2×C22⋊C411yescharacteristic
223C2×C4×C53C2×C22⋊C411yescharacteristic
224C2×C4×C53C23×C411yescharacteristic
225C2×C4×C53C22×D411yescharacteristic
226C22⋊D53—48no—
227C22⋊D53—48no—
228C22⋊D53—42no—
229C22⋊D53—42no—
230C22⋊D53—44no—
231C2×Dic53—22no—
232C2×Dic53—22no—
233C2×Dic53—22no—
234C2×Dic53—22no—
235C2×Dic53—24no—
236C2×Dic53—22no—
237C2×Dic53—22no—
238C2×Dic53—22no—
239C2×Dic53—42no—
240C2×Dic53—42no—
241C2×Dic53—48no—
242C2×Dic53—42no—
243C2×Dic53—42no—
244C2×Dic53—42no—
245C2×Dic53—42no—
246C2×Dic53—84no—
247C2×Dic53C2×C22⋊C412yes—
248C2×Dic53C2×C22⋊C412yes—
249C2×Dic53C2×C22⋊C412yes—
250C2×Dic53C2×C22⋊C412yes—
251D212—42no—
252D212—42no—
253C4×D53—48no—
254Dic106—48no—
255C8×C53—42no—
256#88322—42no—
257C2×#1514—531nomaximal, Sylow
258C22×C53—21no—
259C22×C53—21no—
260C22×C53—21no—
261C22×C53—21no—
262C22×C53—24no—
263C22×C53—24no—
264C22×C53—42no—
265C22×C53—42no—
266C22×C53—42no—
267C22×C53—84no—
268C22×C53#41012yes—
269C22×C53#41812yes—
270C22×C53#41812yes—
271C22×C53#42112yes—
272C22×C53C22×C22⋊C411yescharacteristic
273C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
274C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
275C2×D53—42no—
276C2×D53—42no—
277C2×D53—42no—
278C2×D53—84no—
279C4×C53—21no—
280C4×C53—21no—
281C4×C53—21no—
282C4×C53—21no—
283C4×C53—42no—
284C4×C53—82no—
285Dic53—22no—
286Dic53—22no—
287Dic53—22no—
288Dic53—22no—
289Dic53—42no—
290Dic53—82no—
291C2×#532—531no—
292C2×#432—531no—
293C2×#383—531no—
294#1514—538no—
295C2×#327—531no—
296C2×#327—531no—
297C2×#327—531no—
298C2×#327D5311yescharacteristic, complemented
299C2×C53—21no—
300C2×C53—21no—
301C2×C53—22no—
302C2×C53—82no—
303C2×C53#151414yescomplemented
304C2×C53C2×#41011yescharacteristic
305C2×C53C2×#41811yescharacteristic
306C2×C53C2×#42111yescharacteristic
307D53—82no—
308C22×#41—1061no—
309#532—534no—
310#532—534no—
311#532—1062no—
312#532—1062no—
313C2×C4⋊Q8—531no—
314C2×#127—531no—
315C2×#124—1061no—
316C2×#121—531no—
317C2×#121—1061no—
318C2×#121C2×D5311yescharacteristic
319C2×C4×D4—531no—
320C2×C4×D4—531no—
321C2×C4×D4—1061no—
322C22×C42—531no—
323#432—534no—
324#432—1062no—
325C2×C4⋊C8—1061no—
326C2×C22⋊C8—531no—
327C2×C22⋊C8C2×D5311yescharacteristic
328C2×#97—1061no—
329#383—534no—
330#383—1062no—
331C2×#95—531no—
332C2×#95—1061no—
333C2×#95—1061no—
334C2×#95C2×D5311yescharacteristic
335#327—534no—
336#327—534no—
337#327—534no—
338#327C2×D5314yescomplemented
339C53C2×#151411yescharacteristic, complemented, Sylow
340C2×#41—1064no—
341C2×#41—1064no—
342C2×#41—2124no—
343C2×#41—2124no—
344C22×Q8—1061no—
345C22×D4—21no—
346C22×D4—531no—
347C22×D4—1061no—
348C22×D4—1061no—
349C22×D4—1061no—
350C22×D4C4×D5311yescharacteristic
351C23×C4—21no—
352C23×C4—531no—
353C23×C4—531no—
354C23×C4—1061no—
355C23×C4—1061no—
356C23×C4—1061no—
357C23×C4C22×D5311yescharacteristic
358C22×C8—21no—
359C22×C8—1061no—
360C4⋊Q8—534no—
361C4⋊Q8—1062no—
362#127—534no—
363#127—1062no—
364#124—1064no—
365#124—1064no—
366#121—21no—
367#121—21no—
368#121—532no—
369#121—532no—
370#121—1064no—
371#121—1064no—
372#121—1061no—
373#121—1061no—
374#121C4×D5312yes—
375#121C22×D5312yes—
376C4×D4—534no—
377C4×D4—534no—
378C4×D4—1064no—
379C4×D4—1064no—
380C4×D4—1062no—
381C4×D4—1062no—
382C2×C4⋊C4—531no—
383C2×C4⋊C4—1061no—
384C2×C4⋊C4C4×D5311yescharacteristic
385C2×C22⋊C4—21no—
386C2×C22⋊C4—1061no—
387C2×C22⋊C4—1061no—
388C2×C22⋊C4—1061no—
389C2×C42—531no—
390C2×C42—531no—
391C2×C42—532no—
392C2×C42—531no—
393C2×C42—531no—
394C2×C42—532no—
395C2×C42—532no—
396C2×C42—532no—
397C4⋊C8—1064no—
398C22⋊C8—21no—
399C22⋊C8—532no—
400C22⋊C8—1061no—
401C22⋊C8C22×D5312yes—
402#97—1064no—
403#95—21no—
404#95—532no—
405#95—1064no—
406#95—1064no—
407#95—1061no—
408#95C22×D5312yes—
409C24—21no—
410C24—1061no—
411C24D4×D5311yescharacteristic
412#41—21216no—
413#41—2128no—
414#41—2128no—
415C2×Q8—1064no—
416C2×Q8—2124no—
417C2×D4—22no—
418C2×D4—22no—
419C2×D4—21no—
420C2×D4—42no—
421C2×D4—42no—
422C2×D4—42no—
423C2×D4—532no—
424C2×D4—1064no—
425C2×D4—1064no—
426C2×D4—1062no—
427C2×D4—1062no—
428C2×D4—1061no—
429C2×D4—2122no—
430C2×D4—2124no—
431C2×D4—2124no—
432C2×D4—2122no—
433C2×D4—2122no—
434C2×D4C2×C4×D5312yes—
435C22×C4—21no—
436C22×C4—22no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—532no—
442C22×C4—531no—
443C22×C4—531no—
444C22×C4—531no—
445C22×C4—531no—
446C22×C4—532no—
447C22×C4—532no—
448C22×C4—532no—
449C22×C4—1064no—
450C22×C4—1064no—
451C22×C4—1062no—
452C22×C4—1062no—
453C22×C4—1061no—
454C22×C4—1062no—
455C22×C4—2124no—
456C22×C4—2122no—
457C22×C4—2124no—
458C22×C4—2122no—
459C22×C4—2122no—
460C22×C4—2122no—
461C22×C4C2×C4×D5311yescharacteristic
462C22×C4C2×C4×D5311yescharacteristic
463C22×C4D4×D5311yescharacteristic
464C22×C4D4×D5311yescharacteristic
465C22×C4D4×D5311yescharacteristic
466C22×C4C23×D5311yescharacteristic
467C2×C8—22no—
468C2×C8—42no—
469C2×C8—1062no—
470C2×C8—2122no—
471C4⋊C4—21no—
472C4⋊C4—532no—
473C4⋊C4—1064no—
474C4⋊C4—1061no—
475C4⋊C4C2×C4×D5312yes—
476C22⋊C4—22no—
477C22⋊C4—22no—
478C22⋊C4—1064no—
479C22⋊C4—1064no—
480C22⋊C4—1062no—
481C22⋊C4—1062no—
482C42—532no—
483C42—532no—
484C42—532no—
485C42—532no—
486C42—532no—
487C42—532no—
488C42—532no—
489C42—532no—
490C23—22no—
491C23—22no—
492C23—22no—
493C23—42no—
494C23—42no—
495C23—42no—
496C23—1062no—
497C23—2122no—
498C23—2122no—
499C23—2122no—
500C23C22⋊C4×D5311yescharacteristic
501C23C22⋊C4×D5311yescharacteristic
502C23#1313324112yes—
503C23#1313324412yes—
504C23C2×D4×D5311yescharacteristic
505Q8—2128no—
506D4—42no—
507D4—42no—
508D4—44no—
509D4—42no—
510D4—42no—
511D4—2122no—
512D4—2122no—
513D4—2128no—
514D4—2128no—
515D4—2124no—
516D4—2122no—
517D4—2122no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—84no—
530C2×C4—532no—
531C2×C4—532no—
532C2×C4—532no—
533C2×C4—532no—
534C2×C4—1062no—
535C2×C4—1062no—
536C2×C4—1064no—
537C2×C4—1062no—
538C2×C4—1062no—
539C2×C4—1062no—
540C2×C4—1062no—
541C2×C4—1062no—
542C2×C4—4244no—
543C2×C4—2128no—
544C2×C4—2122no—
545C2×C4—2122no—
546C2×C4—2128no—
547C2×C4—2122no—
548C2×C4—2122no—
549C2×C4—2122no—
550C2×C4—2122no—
551C2×C4C22⋊C4×D5311yescharacteristic
552C2×C4C22⋊C4×D5311yescharacteristic
553C2×C4C22×C4×D5311yescharacteristic
554C2×C4C2×D4×D5311yescharacteristic
555C8—42no—
556C8—2122no—
557C22—21no—
558C22—21no—
559C22—24no—
560C22—24no—
561C22—21no—
562C22—21no—
563C22—42no—
564C22—42no—
565C22—42no—
566C22—84no—
567C22—4244no—
568C22—2122no—
569C22—2122no—
570C22—2122no—
571C22#1681059512yes—
572C22#1681062612yes—
573C22#1681062612yes—
574C22#1681063812yes—
575C22C2×C22⋊C4×D5311yescharacteristic
576C22C2×#1313324111yescharacteristic
577C22C2×#1313324411yescharacteristic
578C4—21no—
579C4—21no—
580C4—21no—
581C4—21no—
582C4—42no—
583C4—82no—
584C4—1062no—
585C4—1062no—
586C4—1062no—
587C4—1062no—
588C4—4242no—
589C4—2122no—
590C2—21no—
591C2—21no—
592C2—22no—
593C2—82no—
594C2—4242no—
595C2#2027326014yesdirect product factor
596C2C2×#1681059511yescharacteristic
597C2C2×#1681062611yescharacteristic
598C2C2×#1681063811yescharacteristic
subgroupG/H
centerC23C22⋊C4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC2×C53×#327C2
socleC23×C53C2×C22⋊C4
prime radicalC2×#16811168C2
cosocleC22×C4×C53C24
hypercenterC2×#327D53
hypocenterC53C2×#1514
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#327

Complements

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

NH ≅ G/Ntype
2. C2×#16810677C2semidirect
3. C2×#16810490C2semidirect
4. #20273260C2direct (N × H)
5. C2×#16810358C2semidirect
6. C2×#16810357C2semidirect
7. C2×#16810356C2semidirect
8. C2×C53×#327C2semidirect
12. #16811168C4semidirect
14. C2×#13133216C4semidirect
15. C2×#13133213C4semidirect
19. C2×D4×Dic53C4semidirect
20. C2×#13133179C4semidirect
24. #16810677C22semidirect
29. #16810490C22semidirect
33. #16810358C22semidirect
34. #16810357C22semidirect
35. #16810356C22semidirect
39. C53×#327C22semidirect
56. #13133216C2×C4semidirect
58. #13133213C2×C4semidirect
68. D4×Dic53C2×C4semidirect
70. #13133179C2×C4semidirect
298. C2×#327D53semidirect
303. C2×C53#1514semidirect
338. #327C2×D53semidirect
339. C53C2×#1514semidirect
595. C2#20273260direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168111682
C2×#168106772
C2×#168104902
#202732602
C2×#168103582
C2×#168103572
C2×#168103562
C2×C53×#3272
C2×#151453