Finite Group Databasebeta

← C2×#20273291 #27351412

Subgroups — C2×#20273291 #27351412

Order 13568.

1074 conjugacy classes of subgroups.

The subgroup lattice has 552 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811159C211yesmaximal, characteristic
2C2×#16810673C211yesmaximal, characteristic, complemented
3C2×#16810490C211yesmaximal, characteristic, complemented
4#20273291C218yesmaximal, 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×C4×Dic53C2211yescharacteristic
23#16810673—22no—
24#16810673C2214yescomplemented
25C2×#13133146—21no—
26C2×C22⋊#88322C2211yescharacteristic
27C2×C8×Dic53—21no—
28#16810490—22no—
29#16810490C2214yescomplemented
30C2×#13133124—21no—
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×#11719921—21no—
72C2×#11719921—21no—
73C2×#11719921—21no—
74C2×#11719921—21no—
75C2×C4×Dic53—21no—
76C2×C4×Dic53C2×C411yescharacteristic
77C2×C4×Dic53C2×C411yescharacteristic
78C2×C4×Dic53C2×C412yes—
79C2×C4×Dic53D411yescharacteristic
80C2×C4×Dic53D411yescharacteristic
81C2×C4×Dic53D412yes—
82C2×C4×Dic53D412yes—
83C2×C4×Dic53C2312yes—
84#13133146—24no—
85C22⋊#88322—21no—
86C22⋊#88322C2312yes—
87C8×Dic53—24no—
88#13133124—21no—
89#13133124—24no—
90#13133124—24no—
91#13133124C2312yes—
92C22×Q8×C53C2×C411yescharacteristic
93C23×C4×C53—21no—
94C23×C4×C53C2311yescharacteristic
95C22×C8×C53—21no—
96C22⋊Q8×C53—21no—
97C22⋊Q8×C53—21no—
98C22⋊Q8×C53C2×C412yes—
99C22⋊Q8×C53C2312yes—
100C2×C4⋊C4×C53—21no—
101C2×C4⋊C4×C53C2×C411yescharacteristic
102C2×C22⋊C4×C53—21no—
103C22⋊C8×C53—21no—
104C22⋊C8×C53C2312yes—
105#95×C53—21no—
106#95×C53C2312yes—
107C24×C53C2×D411yescharacteristic
108C23×D53—21no—
109C2×Q8×C53—21no—
110C2×Q8×C53—42no—
111C2×Q8×C53C22×C412yes—
112C22×C4×C53—21no—
113C22×C4×C53—21no—
114C22×C4×C53—22no—
115C22×C4×C53—42no—
116C22×C4×C53—42no—
117C22×C4×C53—42no—
118C22×C4×C53—42no—
119C22×C4×C53C22×C411yescharacteristic
120C22×C4×C53C22×C411yescharacteristic
121C22×C4×C53C2×D411yescharacteristic
122C22×C4×C53C2×D411yescharacteristic
123C22×C4×C53C2×D411yescharacteristic
124C22×C4×C53C2411yescharacteristic
125C2×C22⋊D53—22no—
126C2×C22⋊D53—22no—
127C2×C22⋊D53—42no—
128C2×C22⋊D53—42no—
129C22×Dic53—22no—
130C22×Dic53—22no—
131C22×Dic53—21no—
132C22×Dic53—22no—
133C22×Dic53—42no—
134C22×Dic53—42no—
135C22×Dic53—42no—
136C22×Dic53—42no—
137C22×Dic53C22⋊C411yescharacteristic
138C22×Dic53C22⋊C411yescharacteristic
139C22×Dic53C22⋊C411yescharacteristic
140C22×Dic53C22⋊C411yescharacteristic
141C22×Dic53C22⋊C412yes—
142C22×Dic53C22⋊C412yes—
143C22×Dic53C2×D412yes—
144C22×Dic53C2×D412yes—
145C2×D212—21no—
146C2×D212—42no—
147C2×D212C22×C412yes—
148C2×C4×D53—24no—
149C2×C4×D53—44no—
150C2×C8×C53—22no—
151C2×C8×C53—42no—
152C4⋊C4×C53—22no—
153C4⋊C4×C53—21no—
154C4⋊C4×C53—22no—
155C4⋊C4×C53C22×C412yes—
156C22⋊C4×C53—22no—
157C22⋊C4×C53—22no—
158#11719921—22no—
159#11719921—22no—
160#11719921—24no—
161#11719921—24no—
162#11719921—24no—
163C4⋊Dic53—21no—
164C4⋊Dic53—24no—
165C4⋊Dic53—24no—
166C4⋊Dic53C22×C412yes—
167#11719919—24no—
168C4×Dic53—24no—
169C4×Dic53C22⋊C412yes—
170C4×Dic53C22⋊C412yes—
171C4×Dic53C22⋊C412yes—
172C4×Dic53C22⋊C412yes—
173C4×Dic53C22×C412yes—
174C4×Dic53C22×C412yes—
175C4×Dic53C2×D412yes—
176C4×Dic53C2×D412yes—
177C2×#88322—22no—
178C2×#88322—42no—
179C23×C53—22no—
180C23×C53—22no—
181C23×C53C2×C22⋊C411yescharacteristic
182C23×C53C2×C22⋊C411yescharacteristic
183C23×C53#13512yes—
184C23×C53#13512yes—
185C23×C53C22×D411yescharacteristic
186C22×D53—22no—
187C22×D53—42no—
188C22×D53—42no—
189C22×D53—42no—
190Q8×C53—42no—
191Q8×C53—42no—
192C2×C4×C53—22no—
193C2×C4×C53—22no—
194C2×C4×C53—22no—
195C2×C4×C53—22no—
196C2×C4×C53—41no—
197C2×C4×C53—41no—
198C2×C4×C53—41no—
199C2×C4×C53—41no—
200C2×C4×C53—42no—
201C2×C4×C53—42no—
202C2×C4×C53—42no—
203C2×C4×C53—42no—
204C2×C4×C53—42no—
205C2×C4×C53—84no—
206C2×C4×C53C2×C22⋊C411yescharacteristic
207C2×C4×C53C2×C22⋊C411yescharacteristic
208C2×C4×C53C23×C411yescharacteristic
209C2×C4×C53C22×D411yescharacteristic
210C22⋊D53—42no—
211C22⋊D53—42no—
212C22⋊D53—44no—
213C2×Dic53—22no—
214C2×Dic53—22no—
215C2×Dic53—24no—
216C2×Dic53—22no—
217C2×Dic53—22no—
218C2×Dic53—22no—
219C2×Dic53—22no—
220C2×Dic53—22no—
221C2×Dic53—42no—
222C2×Dic53—42no—
223C2×Dic53—42no—
224C2×Dic53—42no—
225C2×Dic53—42no—
226C2×Dic53—42no—
227C2×Dic53—84no—
228C2×Dic53C2×C22⋊C412yes—
229C2×Dic53C2×C22⋊C412yes—
230C2×Dic53C2×C22⋊C412yes—
231C2×Dic53C2×C22⋊C412yes—
232D212—42no—
233D212—42no—
234C4×D53—48no—
235C8×C53—42no—
236#88322—42no—
237C2×#1521—531nomaximal, Sylow
238C22×C53—21no—
239C22×C53—21no—
240C22×C53—21no—
241C22×C53—21no—
242C22×C53—24no—
243C22×C53—24no—
244C22×C53#41012yes—
245C22×C53#41612yes—
246C22×C53#41612yes—
247C22×C53#42112yes—
248C22×C53C22×C22⋊C411yescharacteristic
249C22×C53C2×#13511yescharacteristic
250C22×C53C2×#13511yescharacteristic
251C2×D53—42no—
252C2×D53—42no—
253C2×D53—42no—
254C2×D53—84no—
255C4×C53—21no—
256C4×C53—21no—
257C4×C53—21no—
258C4×C53—21no—
259C4×C53—42no—
260C4×C53—42no—
261C4×C53—82no—
262Dic53—22no—
263Dic53—22no—
264Dic53—22no—
265Dic53—22no—
266Dic53—42no—
267Dic53—82no—
268C2×#529—531no—
269C2×#431—531no—
270C2×#383—531no—
271#1521—538no—
272C2×#328—531no—
273C2×#328D5311yescharacteristic, complemented
274C2×#327—531no—
275C2×#327—531no—
276C2×C53—21no—
277C2×C53—21no—
278C2×C53—22no—
279C2×C53#152114yescomplemented
280C2×C53C2×#41011yescharacteristic
281C2×C53C2×#41611yescharacteristic
282C2×C53C2×#42111yescharacteristic
283D53—82no—
284#529—534no—
285#529—534no—
286#529—1062no—
287#529—1062no—
288C2×#126—1061no—
289C2×#125—531no—
290C2×#124—531no—
291C2×#123—1061no—
292C2×C22⋊Q8C2×D5311yescharacteristic
293C2×#121—531no—
294C2×C4×Q8—531no—
295C2×C4×D4—531no—
296C2×C4×D4—1061no—
297C2×#117—1061no—
298C22×C42—531no—
299#431—534no—
300#431—1062no—
301C2×C4⋊C8—1061no—
302C2×C22⋊C8—531no—
303C2×C22⋊C8C2×D5311yescharacteristic
304C2×C4×C8—1061no—
305#383—534no—
306#383—1062no—
307C2×#95—531no—
308C2×#95—1061no—
309C2×#95—1061no—
310C2×#95C2×D5311yescharacteristic
311#328—534no—
312#328C2×D5314yescomplemented
313#327—534no—
314#327—534no—
315C53C2×#152111yescharacteristic, complemented, Sylow
316C22×Q8C4×D5311yescharacteristic
317C22×D4—531no—
318C22×D4—1061no—
319C23×C4—21no—
320C23×C4—531no—
321C23×C4—531no—
322C23×C4—1061no—
323C23×C4—1061no—
324C23×C4C22×D5311yescharacteristic
325C22×C8—21no—
326C22×C8—1061no—
327#126—1064no—
328#126—1064no—
329#125—534no—
330#125—1062no—
331#124—534no—
332#124—1062no—
333#123—1064no—
334#123—1064no—
335C22⋊Q8—21no—
336C22⋊Q8—21no—
337C22⋊Q8C4×D5312yes—
338C22⋊Q8C22×D5312yes—
339#121—532no—
340#121—532no—
341#121—1061no—
342#121—1061no—
343C4×Q8—534no—
344C4×Q8—1062no—
345C4×D4—534no—
346C4×D4—1062no—
347C4×D4—1064no—
348C4×D4—1064no—
349#117—1064no—
350#117—1064no—
351C2×C4⋊C4—21no—
352C2×C4⋊C4—531no—
353C2×C4⋊C4—1061no—
354C2×C4⋊C4—1061no—
355C2×C4⋊C4—1061no—
356C2×C4⋊C4C4×D5311yescharacteristic
357C2×C22⋊C4—21no—
358C2×C22⋊C4—1061no—
359C2×C22⋊C4—1061no—
360C2×C22⋊C4—1061no—
361C2×C22⋊C4—1061no—
362C2×C42—531no—
363C2×C42—531no—
364C2×C42—532no—
365C2×C42—531no—
366C2×C42—531no—
367C2×C42—532no—
368C2×C42—532no—
369C2×C42—532no—
370C2×C42—1061no—
371C4⋊C8—1064no—
372C22⋊C8—21no—
373C22⋊C8—532no—
374C22⋊C8—1061no—
375C22⋊C8C22×D5312yes—
376C4×C8—1064no—
377#95—21no—
378#95—532no—
379#95—1064no—
380#95—1064no—
381#95—1061no—
382#95C22×D5312yes—
383C24—1061no—
384C24D4×D5311yescharacteristic
385C2×Q8—21no—
386C2×Q8—42no—
387C2×Q8C2×C4×D5312yes—
388C2×D4—532no—
389C2×D4—1062no—
390C2×D4—1062no—
391C2×D4—1061no—
392C2×D4—2122no—
393C2×D4—2122no—
394C2×D4—2122no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—22no—
398C22×C4—42no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4—42no—
402C22×C4—532no—
403C22×C4—531no—
404C22×C4—531no—
405C22×C4—531no—
406C22×C4—531no—
407C22×C4—532no—
408C22×C4—532no—
409C22×C4—532no—
410C22×C4—1064no—
411C22×C4—1062no—
412C22×C4—1062no—
413C22×C4—1061no—
414C22×C4—1062no—
415C22×C4—2124no—
416C22×C4—2122no—
417C22×C4—2122no—
418C22×C4—2122no—
419C22×C4—2122no—
420C22×C4C2×C4×D5311yescharacteristic
421C22×C4C2×C4×D5311yescharacteristic
422C22×C4D4×D5311yescharacteristic
423C22×C4D4×D5311yescharacteristic
424C22×C4D4×D5311yescharacteristic
425C22×C4C23×D5311yescharacteristic
426C2×C8—22no—
427C2×C8—42no—
428C2×C8—1062no—
429C2×C8—2122no—
430C4⋊C4—22no—
431C4⋊C4—21no—
432C4⋊C4—22no—
433C4⋊C4—532no—
434C4⋊C4—1064no—
435C4⋊C4—1064no—
436C4⋊C4—1064no—
437C4⋊C4—1061no—
438C4⋊C4C2×C4×D5312yes—
439C22⋊C4—22no—
440C22⋊C4—22no—
441C22⋊C4—1064no—
442C22⋊C4—1064no—
443C22⋊C4—1064no—
444C22⋊C4—1062no—
445C22⋊C4—1062no—
446C42—532no—
447C42—532no—
448C42—532no—
449C42—532no—
450C42—532no—
451C42—532no—
452C42—532no—
453C42—532no—
454C42—1064no—
455C23—22no—
456C23—22no—
457C23—1062no—
458C23—2122no—
459C23—2122no—
460C23—2122no—
461C23C22⋊C4×D5311yescharacteristic
462C23C22⋊C4×D5311yescharacteristic
463C23#1313323812yes—
464C23#1313324012yes—
465C23C2×D4×D5311yescharacteristic
466Q8—42no—
467Q8—42no—
468D4—2122no—
469D4—2122no—
470D4—2124no—
471D4—2122no—
472D4—2122no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—84no—
487C2×C4—532no—
488C2×C4—532no—
489C2×C4—532no—
490C2×C4—532no—
491C2×C4—1064no—
492C2×C4—1062no—
493C2×C4—1062no—
494C2×C4—1062no—
495C2×C4—1062no—
496C2×C4—1062no—
497C2×C4—1062no—
498C2×C4—1062no—
499C2×C4—4244no—
500C2×C4—2128no—
501C2×C4—2122no—
502C2×C4—2122no—
503C2×C4—2122no—
504C2×C4—2122no—
505C2×C4—2122no—
506C2×C4—2122no—
507C2×C4C22⋊C4×D5311yescharacteristic
508C2×C4C22⋊C4×D5311yescharacteristic
509C2×C4C22×C4×D5311yescharacteristic
510C2×C4C2×D4×D5311yescharacteristic
511C8—42no—
512C8—2122no—
513C22—21no—
514C22—21no—
515C22—21no—
516C22—21no—
517C22—24no—
518C22—24no—
519C22—4244no—
520C22—2122no—
521C22—2122no—
522C22—2122no—
523C22#1681059512yes—
524C22#1681061812yes—
525C22#1681061812yes—
526C22#1681063812yes—
527C22C2×C22⋊C4×D5311yescharacteristic
528C22C2×#1313323811yescharacteristic
529C22C2×#1313324011yescharacteristic
530C4—21no—
531C4—21no—
532C4—21no—
533C4—21no—
534C4—42no—
535C4—42no—
536C4—82no—
537C4—1062no—
538C4—1062no—
539C4—1062no—
540C4—1062no—
541C4—4242no—
542C4—2122no—
543C2—21no—
544C2—21no—
545C2—22no—
546C2—4242no—
547C2#2027329114yesdirect product factor
548C2C2×#1681059511yescharacteristic
549C2C2×#1681061811yescharacteristic
550C2C2×#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×#1521
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×#16810673C2semidirect
3. C2×#16810490C2semidirect
4. #20273291C2direct (N × H)
5. C2×#16810360C2semidirect
8. C2×C53×#328C2semidirect
11. #16811159C4semidirect
14. C2×#13133208C4semidirect
15. C2×#13133206C4semidirect
18. C2×Q8×Dic53C4semidirect
19. C2×#13133179C4semidirect
24. #16810673C22semidirect
29. #16810490C22semidirect
33. #16810360C22semidirect
39. C53×#328C22semidirect
50. #13133208C2×C4semidirect
52. #13133206C2×C4semidirect
60. Q8×Dic53C2×C4semidirect
62. #13133179C2×C4semidirect
273. C2×#328D53semidirect
279. C2×C53#1521semidirect
312. #328C2×D53semidirect
315. C53C2×#1521semidirect
547. C2#20273291direct (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×#168106732
C2×#168104902
#202732912
C2×#168103602
C2×#168103582
C2×#168103562
C2×C53×#3282
C2×#152153