Finite Group Databasebeta

← C2×#20273252 #27351333

Subgroups — C2×#20273252 #27351333

Order 13568.

1098 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811159C211yesmaximal, characteristic
2C2×#16810677C211yesmaximal, characteristic, complemented
3C2×#16810487C211yesmaximal, characteristic, complemented
4#20273252C218yesmaximal, direct product factor
5C2×#16810360C211yesmaximal, characteristic, complemented
6C2×#16810358C211yesmaximal, characteristic
7C2×#16810356C211yesmaximal, characteristic
8C2×C53×#328C211yesmaximal, characteristic, complemented
9#16811159—22no—
10#16811159—22no—
11#16811159C414yescomplemented
12#16811159C2214yes—
13C2×#13133212—21no—
14C2×#13133208C411yescharacteristic, complemented
15C2×#13133206C411yescharacteristic, complemented
16C2×#13133201—21no—
17C2×#13133191C2211yescharacteristic
18C2×Q8×Dic53C411yescharacteristic, complemented
19C2×#13133179C411yescharacteristic, complemented
20C2×#13133177—21no—
21C2×#13133175—21no—
22C22×#11719919—21no—
23C22×C4×Dic53C2211yescharacteristic
24#16810677—22no—
25#16810677C2214yescomplemented
26C2×#13133146—21no—
27C2×C22⋊#88322C2211yescharacteristic
28C2×#13133127—21no—
29#16810487—22no—
30#16810487C2214yescomplemented
31C2×#13133124—21no—
32C2×#13133124C2211yescharacteristic
33#16810360C2214yescomplemented
34#16810358C2214yes—
35#16810356C2214yes—
36C2×C22⋊Q8×C53C2211yescharacteristic
37C2×C22⋊C8×C53C2211yescharacteristic
38C2×#95×C53C2211yescharacteristic
39C53×#328C2214yescomplemented
40C22×D212C2×C411yescharacteristic
41C22×C22⋊D53—21no—
42C23×Dic53—21no—
43C23×Dic53D411yescharacteristic
44C23×Dic53D411yescharacteristic
45C22×C4×D53—21no—
46C22×#88322—21no—
47#13133212—24no—
48#13133212—24no—
49#13133208—22no—
50#13133208C2×C414yescomplemented
51#13133206—22no—
52#13133206C2×C414yescomplemented
53#13133201—24no—
54#13133201—24no—
55#13133191—21no—
56#13133191—21no—
57#13133191C2×C412yes—
58#13133191C2312yes—
59Q8×Dic53—22no—
60Q8×Dic53C2×C414yescomplemented
61#13133179—22no—
62#13133179C2×C414yescomplemented
63#13133177—24no—
64#13133177—24no—
65#13133175—24no—
66#13133175—24no—
67C2×C4⋊Dic53—21no—
68C2×C4⋊Dic53—21no—
69C2×C4⋊Dic53C2×C411yescharacteristic
70C2×#11719919—21no—
71C2×#11719919—24no—
72C2×#11719919—44no—
73C2×#11719921—21no—
74C2×#11719921—21no—
75C2×#11719921—21no—
76C2×#11719921—21no—
77C2×C4×Dic53—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#13133124C2312yes—
93C22×Q8×C53C2×C411yescharacteristic
94C23×C4×C53—21no—
95C23×C4×C53C2311yescharacteristic
96C22×C8×C53—21no—
97C22⋊Q8×C53—21no—
98C22⋊Q8×C53—21no—
99C22⋊Q8×C53C2×C412yes—
100C22⋊Q8×C53C2312yes—
101C2×C4⋊C4×C53—21no—
102C2×C4⋊C4×C53C2×C411yescharacteristic
103C2×C22⋊C4×C53—21no—
104C22⋊C8×C53—21no—
105C22⋊C8×C53C2312yes—
106#95×C53—21no—
107#95×C53C2312yes—
108C24×C53C2×D411yescharacteristic
109C23×D53—21no—
110C2×Q8×C53—21no—
111C2×Q8×C53—42no—
112C2×Q8×C53C22×C412yes—
113C22×C4×C53—21no—
114C22×C4×C53—21no—
115C22×C4×C53—22no—
116C22×C4×C53—42no—
117C22×C4×C53—42no—
118C22×C4×C53—42no—
119C22×C4×C53—42no—
120C22×C4×C53C22×C411yescharacteristic
121C22×C4×C53C22×C411yescharacteristic
122C22×C4×C53C2×D411yescharacteristic
123C22×C4×C53C2×D411yescharacteristic
124C22×C4×C53C2×D411yescharacteristic
125C22×C4×C53C2411yescharacteristic
126C2×C22⋊D53—22no—
127C2×C22⋊D53—22no—
128C2×C22⋊D53—42no—
129C2×C22⋊D53—42no—
130C22×Dic53—22no—
131C22×Dic53—22no—
132C22×Dic53—21no—
133C22×Dic53—22no—
134C22×Dic53—42no—
135C22×Dic53—42no—
136C22×Dic53—42no—
137C22×Dic53—42no—
138C22×Dic53C22⋊C411yescharacteristic
139C22×Dic53C22⋊C411yescharacteristic
140C22×Dic53C22⋊C411yescharacteristic
141C22×Dic53C22⋊C411yescharacteristic
142C22×Dic53C22⋊C412yes—
143C22×Dic53C22⋊C412yes—
144C22×Dic53C2×D412yes—
145C22×Dic53C2×D412yes—
146C2×D212—21no—
147C2×D212—42no—
148C2×D212C22×C412yes—
149C2×C4×D53—24no—
150C2×C4×D53—44no—
151C2×C8×C53—22no—
152C2×C8×C53—42no—
153C4⋊C4×C53—22no—
154C4⋊C4×C53—21no—
155C4⋊C4×C53—22no—
156C4⋊C4×C53C22×C412yes—
157C22⋊C4×C53—22no—
158C22⋊C4×C53—22no—
159#11719921—22no—
160#11719921—22no—
161#11719921—24no—
162#11719921—24no—
163#11719921—24no—
164C4⋊Dic53—21no—
165C4⋊Dic53—24no—
166C4⋊Dic53—24no—
167C4⋊Dic53C22×C412yes—
168#11719919—24no—
169#11719919—48no—
170C4×Dic53—24no—
171C4×Dic53C22⋊C412yes—
172C4×Dic53C22⋊C412yes—
173C4×Dic53C22⋊C412yes—
174C4×Dic53C22⋊C412yes—
175C4×Dic53C22×C412yes—
176C4×Dic53C22×C412yes—
177C4×Dic53C2×D412yes—
178C4×Dic53C2×D412yes—
179C2×#88322—22no—
180C2×#88322—42no—
181C23×C53—22no—
182C23×C53—22no—
183C23×C53C2×C22⋊C411yescharacteristic
184C23×C53C2×C22⋊C411yescharacteristic
185C23×C53AGL(1,ℤ/8)12yes—
186C23×C53#13712yes—
187C23×C53C22×D411yescharacteristic
188C22×D53—22no—
189C22×D53—42no—
190C22×D53—42no—
191C22×D53—42no—
192Q8×C53—42no—
193Q8×C53—42no—
194C2×C4×C53—22no—
195C2×C4×C53—22no—
196C2×C4×C53—22no—
197C2×C4×C53—22no—
198C2×C4×C53—41no—
199C2×C4×C53—41no—
200C2×C4×C53—41no—
201C2×C4×C53—41no—
202C2×C4×C53—42no—
203C2×C4×C53—42no—
204C2×C4×C53—42no—
205C2×C4×C53—42no—
206C2×C4×C53—42no—
207C2×C4×C53—84no—
208C2×C4×C53C2×C22⋊C411yescharacteristic
209C2×C4×C53C2×C22⋊C411yescharacteristic
210C2×C4×C53C23×C411yescharacteristic
211C2×C4×C53C22×D411yescharacteristic
212C22⋊D53—42no—
213C22⋊D53—42no—
214C22⋊D53—44no—
215C2×Dic53—22no—
216C2×Dic53—22no—
217C2×Dic53—24no—
218C2×Dic53—22no—
219C2×Dic53—22no—
220C2×Dic53—22no—
221C2×Dic53—22no—
222C2×Dic53—22no—
223C2×Dic53—42no—
224C2×Dic53—42no—
225C2×Dic53—42no—
226C2×Dic53—42no—
227C2×Dic53—42no—
228C2×Dic53—42no—
229C2×Dic53—84no—
230C2×Dic53C2×C22⋊C412yes—
231C2×Dic53C2×C22⋊C412yes—
232C2×Dic53C2×C22⋊C412yes—
233C2×Dic53C2×C22⋊C412yes—
234D212—42no—
235D212—42no—
236C4×D53—48no—
237C8×C53—42no—
238#88322—42no—
239C2×#1512—531nomaximal, Sylow
240C22×C53—21no—
241C22×C53—21no—
242C22×C53—21no—
243C22×C53—21no—
244C22×C53—24no—
245C22×C53—24no—
246C22×C53#41012yes—
247C22×C53#41712yes—
248C22×C53#41712yes—
249C22×C53#42112yes—
250C22×C53C22×C22⋊C411yescharacteristic
251C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
252C22×C53C2×#13711yescharacteristic
253C2×D53—42no—
254C2×D53—42no—
255C2×D53—42no—
256C2×D53—84no—
257C4×C53—21no—
258C4×C53—21no—
259C4×C53—21no—
260C4×C53—21no—
261C4×C53—42no—
262C4×C53—42no—
263C4×C53—82no—
264Dic53—22no—
265Dic53—22no—
266Dic53—22no—
267Dic53—22no—
268Dic53—42no—
269Dic53—82no—
270C2×#529—531no—
271C2×#432—531no—
272C2×#382—531no—
273#1512—538no—
274C2×#328—531no—
275C2×#328D5311yescharacteristic, complemented
276C2×#327—531no—
277C2×#327—531no—
278C2×C53—21no—
279C2×C53—21no—
280C2×C53—22no—
281C2×C53#151214yescomplemented
282C2×C53C2×#41011yescharacteristic
283C2×C53C2×#41711yescharacteristic
284C2×C53C2×#42111yescharacteristic
285D53—82no—
286#529—534no—
287#529—534no—
288#529—1062no—
289#529—1062no—
290C2×#126—1061no—
291C2×#125—531no—
292C2×#124—531no—
293C2×#123—1061no—
294C2×C22⋊Q8C2×D5311yescharacteristic
295C2×#121—531no—
296C2×C4×Q8—531no—
297C2×C4×D4—531no—
298C2×C4×D4—1061no—
299C2×#117—1061no—
300C22×C4⋊C4—1061no—
301C22×C42—531no—
302#432—534no—
303#432—1062no—
304C2×C4⋊C8—1061no—
305C2×C22⋊C8—531no—
306C2×C22⋊C8C2×D5311yescharacteristic
307C2×#97—1061no—
308#382—534no—
309#382—1062no—
310C2×#95—531no—
311C2×#95—1061no—
312C2×#95C2×D5311yescharacteristic
313#328—534no—
314#328C2×D5314yescomplemented
315#327—534no—
316#327—534no—
317C53C2×#151211yescharacteristic, complemented, Sylow
318C22×Q8C4×D5311yescharacteristic
319C22×D4—531no—
320C22×D4—1061no—
321C23×C4—21no—
322C23×C4—531no—
323C23×C4—531no—
324C23×C4—1061no—
325C23×C4—1061no—
326C23×C4C22×D5311yescharacteristic
327C22×C8—21no—
328C22×C8—1061no—
329#126—1064no—
330#126—1064no—
331#125—534no—
332#125—1062no—
333#124—534no—
334#124—1062no—
335#123—1064no—
336#123—1064no—
337C22⋊Q8—21no—
338C22⋊Q8—21no—
339C22⋊Q8C4×D5312yes—
340C22⋊Q8C22×D5312yes—
341#121—532no—
342#121—532no—
343#121—1061no—
344#121—1061no—
345C4×Q8—534no—
346C4×Q8—1062no—
347C4×D4—534no—
348C4×D4—1062no—
349C4×D4—1064no—
350C4×D4—1064no—
351#117—1064no—
352#117—1064no—
353C2×C4⋊C4—21no—
354C2×C4⋊C4—531no—
355C2×C4⋊C4—1061no—
356C2×C4⋊C4—1061no—
357C2×C4⋊C4—1061no—
358C2×C4⋊C4—1064no—
359C2×C4⋊C4—2124no—
360C2×C4⋊C4C4×D5311yescharacteristic
361C2×C22⋊C4—21no—
362C2×C22⋊C4—1061no—
363C2×C22⋊C4—1061no—
364C2×C22⋊C4—1061no—
365C2×C22⋊C4—1061no—
366C2×C42—531no—
367C2×C42—531no—
368C2×C42—532no—
369C2×C42—531no—
370C2×C42—531no—
371C2×C42—532no—
372C2×C42—532no—
373C2×C42—532no—
374C2×C42—1061no—
375C4⋊C8—1064no—
376C22⋊C8—21no—
377C22⋊C8—532no—
378C22⋊C8—1061no—
379C22⋊C8C22×D5312yes—
380#97—1064no—
381#95—21no—
382#95—532no—
383#95—1064no—
384#95—1061no—
385#95C22×D5312yes—
386C24—1061no—
387C24D4×D5311yescharacteristic
388C2×Q8—21no—
389C2×Q8—42no—
390C2×Q8C2×C4×D5312yes—
391C2×D4—532no—
392C2×D4—1062no—
393C2×D4—1062no—
394C2×D4—1061no—
395C2×D4—2122no—
396C2×D4—2122no—
397C2×D4—2122no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—22no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—532no—
406C22×C4—531no—
407C22×C4—531no—
408C22×C4—531no—
409C22×C4—531no—
410C22×C4—532no—
411C22×C4—532no—
412C22×C4—532no—
413C22×C4—1064no—
414C22×C4—1062no—
415C22×C4—1062no—
416C22×C4—1061no—
417C22×C4—1062no—
418C22×C4—2124no—
419C22×C4—2122no—
420C22×C4—2122no—
421C22×C4—2122no—
422C22×C4—2122no—
423C22×C4C2×C4×D5311yescharacteristic
424C22×C4C2×C4×D5311yescharacteristic
425C22×C4D4×D5311yescharacteristic
426C22×C4D4×D5311yescharacteristic
427C22×C4D4×D5311yescharacteristic
428C22×C4C23×D5311yescharacteristic
429C2×C8—22no—
430C2×C8—42no—
431C2×C8—1062no—
432C2×C8—2122no—
433C4⋊C4—22no—
434C4⋊C4—21no—
435C4⋊C4—22no—
436C4⋊C4—532no—
437C4⋊C4—1064no—
438C4⋊C4—1064no—
439C4⋊C4—1064no—
440C4⋊C4—1061no—
441C4⋊C4—2128no—
442C4⋊C4C2×C4×D5312yes—
443C22⋊C4—22no—
444C22⋊C4—22no—
445C22⋊C4—1064no—
446C22⋊C4—1064no—
447C22⋊C4—1064no—
448C22⋊C4—1062no—
449C22⋊C4—1062no—
450C42—532no—
451C42—532no—
452C42—532no—
453C42—532no—
454C42—532no—
455C42—532no—
456C42—532no—
457C42—532no—
458C42—1064no—
459C23—22no—
460C23—22no—
461C23—1062no—
462C23—2122no—
463C23—2122no—
464C23—2122no—
465C23C22⋊C4×D5311yescharacteristic
466C23C22⋊C4×D5311yescharacteristic
467C23#1313324412yes—
468C23#1313324912yes—
469C23C2×D4×D5311yescharacteristic
470Q8—42no—
471Q8—42no—
472D4—2122no—
473D4—2122no—
474D4—2124no—
475D4—2122no—
476D4—2122no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—84no—
491C2×C4—532no—
492C2×C4—532no—
493C2×C4—532no—
494C2×C4—532no—
495C2×C4—1064no—
496C2×C4—1062no—
497C2×C4—1062no—
498C2×C4—1062no—
499C2×C4—1062no—
500C2×C4—1062no—
501C2×C4—1062no—
502C2×C4—1062no—
503C2×C4—4244no—
504C2×C4—2128no—
505C2×C4—2122no—
506C2×C4—2122no—
507C2×C4—2122no—
508C2×C4—2122no—
509C2×C4—2122no—
510C2×C4—2122no—
511C2×C4C22⋊C4×D5311yescharacteristic
512C2×C4C22⋊C4×D5311yescharacteristic
513C2×C4C22×C4×D5311yescharacteristic
514C2×C4C2×D4×D5311yescharacteristic
515C8—42no—
516C8—2122no—
517C22—21no—
518C22—21no—
519C22—24no—
520C22—24no—
521C22—21no—
522C22—21no—
523C22—4244no—
524C22—2122no—
525C22—2122no—
526C22—2122no—
527C22#1681059512yes—
528C22#1681062312yes—
529C22#1681062312yes—
530C22#1681063812yes—
531C22C2×C22⋊C4×D5311yescharacteristic
532C22C2×#1313324411yescharacteristic
533C22C2×#1313324911yescharacteristic
534C4—21no—
535C4—21no—
536C4—21no—
537C4—21no—
538C4—42no—
539C4—42no—
540C4—82no—
541C4—1062no—
542C4—1062no—
543C4—1062no—
544C4—1062no—
545C4—4242no—
546C4—2122no—
547C2—21no—
548C2—22no—
549C2—21no—
550C2—4242no—
551C2#2027325214yesdirect product factor
552C2C2×#1681059511yescharacteristic
553C2C2×#1681062311yescharacteristic
554C2C2×#1681063811yescharacteristic
subgroupG/H
centerC23C22⋊C4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC2×C53×#328C2
socleC23×C53C2×C22⋊C4
prime radicalC2×#13133191C22
cosocleC22×C4×C53C24
hypercenterC2×#328D53
hypocenterC53C2×#1512
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#328

Complements

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

NH ≅ G/Ntype
2. C2×#16810677C2semidirect
3. C2×#16810487C2semidirect
4. #20273252C2direct (N × H)
5. C2×#16810360C2semidirect
8. C2×C53×#328C2semidirect
11. #16811159C4semidirect
14. C2×#13133208C4semidirect
15. C2×#13133206C4semidirect
18. C2×Q8×Dic53C4semidirect
19. C2×#13133179C4semidirect
25. #16810677C22semidirect
30. #16810487C22semidirect
33. #16810360C22semidirect
39. C53×#328C22semidirect
50. #13133208C2×C4semidirect
52. #13133206C2×C4semidirect
60. Q8×Dic53C2×C4semidirect
62. #13133179C2×C4semidirect
275. C2×#328D53semidirect
281. C2×C53#1512semidirect
314. #328C2×D53semidirect
317. C53C2×#1512semidirect
551. C2#20273252direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168111592
C2×#168106772
C2×#168104872
#202732522
C2×#168103602
C2×#168103582
C2×#168103562
C2×C53×#3282
C2×#151253