Finite Group Databasebeta

← C2×#20273239 #27351218

Subgroups — C2×#20273239 #27351218

Order 13568.

1206 conjugacy classes of subgroups.

The subgroup lattice has 560 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811159C211yesmaximal, characteristic
2C2×#16810677C211yesmaximal, characteristic, complemented
3C2×#16810474C211yesmaximal, characteristic, complemented
4#20273239C218yesmaximal, 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×Dic53—21no—
24C22×C4×Dic53C2211yescharacteristic
25#16810677—22no—
26#16810677C2214yescomplemented
27C2×#13133146—21no—
28C2×C22⋊#88322C2211yescharacteristic
29C2×#13133127—21no—
30#16810474—22no—
31#16810474C2214yescomplemented
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—28no—
71C2×#11719919—21no—
72C2×#11719919—24no—
73C2×#11719921—21no—
74C2×#11719921—21no—
75C2×#11719921—21no—
76C2×#11719921—21no—
77C2×C4×Dic53—21no—
78C2×C4×Dic53—24no—
79C2×C4×Dic53—44no—
80C2×C4×Dic53C2×C411yescharacteristic
81C2×C4×Dic53C2×C411yescharacteristic
82C2×C4×Dic53C2×C412yes—
83C2×C4×Dic53D411yescharacteristic
84C2×C4×Dic53D411yescharacteristic
85C2×C4×Dic53D412yes—
86C2×C4×Dic53D412yes—
87C2×C4×Dic53C2312yes—
88#13133146—24no—
89C22⋊#88322—21no—
90C22⋊#88322C2312yes—
91#13133127—24no—
92#13133124—21no—
93#13133124C2312yes—
94C22×Q8×C53C2×C411yescharacteristic
95C23×C4×C53—21no—
96C23×C4×C53C2311yescharacteristic
97C22×C8×C53—21no—
98C22⋊Q8×C53—21no—
99C22⋊Q8×C53—21no—
100C22⋊Q8×C53C2×C412yes—
101C22⋊Q8×C53C2312yes—
102C2×C4⋊C4×C53—21no—
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×C53C2×D411yescharacteristic
110C23×D53—21no—
111C2×Q8×C53—21no—
112C2×Q8×C53—42no—
113C2×Q8×C53C22×C412yes—
114C22×C4×C53—24no—
115C22×C4×C53—21no—
116C22×C4×C53—21no—
117C22×C4×C53—22no—
118C22×C4×C53—42no—
119C22×C4×C53—42no—
120C22×C4×C53—42no—
121C22×C4×C53C22×C411yescharacteristic
122C22×C4×C53C22×C411yescharacteristic
123C22×C4×C53C2×D411yescharacteristic
124C22×C4×C53C2×D411yescharacteristic
125C22×C4×C53C2×D411yescharacteristic
126C22×C4×C53C2411yescharacteristic
127C2×C22⋊D53—22no—
128C2×C22⋊D53—22no—
129C2×C22⋊D53—42no—
130C2×C22⋊D53—42no—
131C22×Dic53—22no—
132C22×Dic53—22no—
133C22×Dic53—24no—
134C22×Dic53—24no—
135C22×Dic53—21no—
136C22×Dic53—22no—
137C22×Dic53—42no—
138C22×Dic53—42no—
139C22×Dic53C22⋊C411yescharacteristic
140C22×Dic53C22⋊C411yescharacteristic
141C22×Dic53C22⋊C411yescharacteristic
142C22×Dic53C22⋊C412yes—
143C22×Dic53C22⋊C412yes—
144C22×Dic53C22⋊C411yescharacteristic
145C22×Dic53C2×D412yes—
146C22×Dic53C2×D412yes—
147C2×D212—21no—
148C2×D212—42no—
149C2×D212C22×C412yes—
150C2×C4×D53—24no—
151C2×C4×D53—44no—
152C2×C8×C53—22no—
153C2×C8×C53—42no—
154C4⋊C4×C53—22no—
155C4⋊C4×C53—21no—
156C4⋊C4×C53—22no—
157C4⋊C4×C53C22×C412yes—
158C22⋊C4×C53—22no—
159C22⋊C4×C53—22no—
160#11719921—22no—
161#11719921—22no—
162#11719921—24no—
163#11719921—24no—
164#11719921—24no—
165C4⋊Dic53—21no—
166C4⋊Dic53—24no—
167C4⋊Dic53—24no—
168C4⋊Dic53C22×C412yes—
169#11719919—216no—
170#11719919—24no—
171C4×Dic53—24no—
172C4×Dic53—48no—
173C4×Dic53C22⋊C412yes—
174C4×Dic53C22⋊C412yes—
175C4×Dic53C22⋊C412yes—
176C4×Dic53C22⋊C412yes—
177C4×Dic53C22×C412yes—
178C4×Dic53C22×C412yes—
179C4×Dic53C2×D412yes—
180C4×Dic53C2×D412yes—
181C2×#88322—22no—
182C2×#88322—42no—
183C23×C53—22no—
184C23×C53—22no—
185C23×C53C2×C22⋊C411yescharacteristic
186C23×C53C2×C22⋊C411yescharacteristic
187C23×C53AGL(1,ℤ/8)12yes—
188C23×C53#13712yes—
189C23×C53C22×D411yescharacteristic
190C22×D53—22no—
191C22×D53—42no—
192C22×D53—42no—
193C22×D53—42no—
194Q8×C53—42no—
195Q8×C53—42no—
196C2×C4×C53—22no—
197C2×C4×C53—22no—
198C2×C4×C53—24no—
199C2×C4×C53—22no—
200C2×C4×C53—22no—
201C2×C4×C53—41no—
202C2×C4×C53—41no—
203C2×C4×C53—41no—
204C2×C4×C53—41no—
205C2×C4×C53—42no—
206C2×C4×C53—48no—
207C2×C4×C53—42no—
208C2×C4×C53—42no—
209C2×C4×C53—42no—
210C2×C4×C53C2×C22⋊C411yescharacteristic
211C2×C4×C53C2×C22⋊C411yescharacteristic
212C2×C4×C53C23×C411yescharacteristic
213C2×C4×C53C22×D411yescharacteristic
214C22⋊D53—42no—
215C22⋊D53—42no—
216C22⋊D53—44no—
217C2×Dic53—22no—
218C2×Dic53—22no—
219C2×Dic53—24no—
220C2×Dic53—22no—
221C2×Dic53—22no—
222C2×Dic53—22no—
223C2×Dic53—24no—
224C2×Dic53—24no—
225C2×Dic53—24no—
226C2×Dic53—22no—
227C2×Dic53—42no—
228C2×Dic53—48no—
229C2×Dic53—42no—
230C2×Dic53—42no—
231C2×Dic53#10414yes—
232C2×Dic53C2×C22⋊C412yes—
233C2×Dic53C2×C22⋊C412yes—
234C2×Dic53C2×C22⋊C412yes—
235C2×Dic53C2×C22⋊C412yes—
236D212—42no—
237D212—42no—
238C4×D53—48no—
239C8×C53—42no—
240#88322—42no—
241C2×#1509—531nomaximal, Sylow
242C22×C53—21no—
243C22×C53—21no—
244C22×C53—21no—
245C22×C53—21no—
246C22×C53—24no—
247C22×C53—24no—
248C22×C53#41012yes—
249C22×C53#41712yes—
250C22×C53#41712yes—
251C22×C53C2×#10412yes—
252C22×C53C22×C22⋊C411yescharacteristic
253C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
254C22×C53C2×#13711yescharacteristic
255C2×D53—42no—
256C2×D53—42no—
257C2×D53—42no—
258C2×D53—84no—
259C4×C53—21no—
260C4×C53—21no—
261C4×C53—21no—
262C4×C53—21no—
263C4×C53—44no—
264C4×C53—42no—
265C4×C53—42no—
266Dic53—22no—
267Dic53—22no—
268Dic53—22no—
269Dic53—44no—
270Dic53—42no—
271Dic53C2×#10414yes—
272C2×#529—531no—
273C2×#432—531no—
274C2×C4×C4⋊C4—531no—
275#1509—538no—
276C2×#328—531no—
277C2×#328D5311yescharacteristic, complemented
278C2×#327—531no—
279C2×#327—531no—
280C2×C53—21no—
281C2×C53—21no—
282C2×C53—22no—
283C2×C53#150914yescomplemented
284C2×C53C2×#41011yescharacteristic
285C2×C53C2×#41711yescharacteristic
286C2×C53C22×#10411yescharacteristic
287D53—82no—
288#529—534no—
289#529—534no—
290#529—1062no—
291#529—1062no—
292C2×#126—1061no—
293C2×#125—531no—
294C2×#124—531no—
295C2×#123—1061no—
296C2×C22⋊Q8C2×D5311yescharacteristic
297C2×#121—531no—
298C2×C4×Q8—531no—
299C2×C4×D4—531no—
300C2×C4×D4—1061no—
301C2×#117—1061no—
302C22×C4⋊C4—1061no—
303C22×C42—531no—
304C22×C42—1061no—
305#432—534no—
306#432—1062no—
307C2×C4⋊C8—1061no—
308C2×C22⋊C8—531no—
309C2×C22⋊C8C2×D5311yescharacteristic
310C2×#97—1061no—
311C4×C4⋊C4—534no—
312C4×C4⋊C4—1062no—
313C2×#95—531no—
314C2×#95C2×D5311yescharacteristic
315#328—534no—
316#328C2×D5314yescomplemented
317#327—534no—
318#327—534no—
319C53C2×#150911yescharacteristic, complemented, Sylow
320C22×Q8C4×D5311yescharacteristic
321C22×D4—531no—
322C22×D4—1061no—
323C23×C4—21no—
324C23×C4—531no—
325C23×C4—531no—
326C23×C4—1061no—
327C23×C4—1061no—
328C23×C4C22×D5311yescharacteristic
329C22×C8—21no—
330C22×C8—1061no—
331#126—1064no—
332#126—1064no—
333#125—534no—
334#125—1062no—
335#124—534no—
336#124—1062no—
337#123—1064no—
338#123—1064no—
339C22⋊Q8—21no—
340C22⋊Q8—21no—
341C22⋊Q8C4×D5312yes—
342C22⋊Q8C22×D5312yes—
343#121—532no—
344#121—532no—
345#121—1061no—
346#121—1061no—
347C4×Q8—534no—
348C4×Q8—1062no—
349C4×D4—534no—
350C4×D4—1062no—
351C4×D4—1064no—
352C4×D4—1064no—
353#117—1064no—
354#117—1064no—
355C2×C4⋊C4—21no—
356C2×C4⋊C4—531no—
357C2×C4⋊C4—1064no—
358C2×C4⋊C4—1061no—
359C2×C4⋊C4—1061no—
360C2×C4⋊C4—1061no—
361C2×C4⋊C4—1068no—
362C2×C4⋊C4C4×D5311yescharacteristic
363C2×C22⋊C4—21no—
364C2×C22⋊C4—1061no—
365C2×C22⋊C4—1061no—
366C2×C22⋊C4—1061no—
367C2×C22⋊C4—1061no—
368C2×C42—531no—
369C2×C42—531no—
370C2×C42—532no—
371C2×C42—531no—
372C2×C42—531no—
373C2×C42—532no—
374C2×C42—532no—
375C2×C42—532no—
376C2×C42—1061no—
377C2×C42—1064no—
378C2×C42—2124no—
379C4⋊C8—1064no—
380C22⋊C8—21no—
381C22⋊C8—532no—
382C22⋊C8—1061no—
383C22⋊C8C22×D5312yes—
384#97—1064no—
385#95—21no—
386#95—532no—
387#95—1061no—
388#95C22×D5312yes—
389C24—1061no—
390C24D4×D5311yescharacteristic
391C2×Q8—21no—
392C2×Q8—42no—
393C2×Q8C2×C4×D5312yes—
394C2×D4—532no—
395C2×D4—1062no—
396C2×D4—1062no—
397C2×D4—1061no—
398C2×D4—2122no—
399C2×D4—2122no—
400C2×D4—2122no—
401C22×C4—24no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—22no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4—42no—
408C22×C4—531no—
409C22×C4—532no—
410C22×C4—532no—
411C22×C4—531no—
412C22×C4—531no—
413C22×C4—531no—
414C22×C4—532no—
415C22×C4—532no—
416C22×C4—1064no—
417C22×C4—1064no—
418C22×C4—1062no—
419C22×C4—1062no—
420C22×C4—1064no—
421C22×C4—1061no—
422C22×C4—1062no—
423C22×C4—2124no—
424C22×C4—2122no—
425C22×C4—2122no—
426C22×C4C2×C4×D5311yescharacteristic
427C22×C4C2×C4×D5311yescharacteristic
428C22×C4D4×D5311yescharacteristic
429C22×C4D4×D5311yescharacteristic
430C22×C4D4×D5311yescharacteristic
431C22×C4C23×D5311yescharacteristic
432C2×C8—22no—
433C2×C8—42no—
434C2×C8—1062no—
435C2×C8—2122no—
436C4⋊C4—22no—
437C4⋊C4—21no—
438C4⋊C4—22no—
439C4⋊C4—532no—
440C4⋊C4—1064no—
441C4⋊C4—1064no—
442C4⋊C4—1064no—
443C4⋊C4—10616no—
444C4⋊C4—1061no—
445C4⋊C4C2×C4×D5312yes—
446C22⋊C4—22no—
447C22⋊C4—22no—
448C22⋊C4—1064no—
449C22⋊C4—1064no—
450C22⋊C4—1064no—
451C22⋊C4—1062no—
452C22⋊C4—1062no—
453C42—532no—
454C42—532no—
455C42—532no—
456C42—532no—
457C42—532no—
458C42—532no—
459C42—532no—
460C42—532no—
461C42—1064no—
462C42—2128no—
463C23—22no—
464C23—22no—
465C23—1062no—
466C23—2122no—
467C23—2122no—
468C23—2122no—
469C23C22⋊C4×D5311yescharacteristic
470C23C22⋊C4×D5311yescharacteristic
471C23#1313324412yes—
472C23#1313324912yes—
473C23C2×D4×D5311yescharacteristic
474Q8—42no—
475Q8—42no—
476D4—2122no—
477D4—2122no—
478D4—2124no—
479D4—2122no—
480D4—2122no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—24no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—42no—
491C2×C4—48no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—532no—
496C2×C4—534no—
497C2×C4—532no—
498C2×C4—532no—
499C2×C4—532no—
500C2×C4—1064no—
501C2×C4—1064no—
502C2×C4—1064no—
503C2×C4—1062no—
504C2×C4—1062no—
505C2×C4—1062no—
506C2×C4—1062no—
507C2×C4—1062no—
508C2×C4—1064no—
509C2×C4—1062no—
510C2×C4—2128no—
511C2×C4—2122no—
512C2×C4—2128no—
513C2×C4—2122no—
514C2×C4—2122no—
515C2×C4C22⋊C4×D5311yescharacteristic
516C2×C4C22⋊C4×D5311yescharacteristic
517C2×C4C22×C4×D5311yescharacteristic
518C2×C4C2×D4×D5311yescharacteristic
519C8—42no—
520C8—2122no—
521C22—21no—
522C22—21no—
523C22—24no—
524C22—24no—
525C22—21no—
526C22—21no—
527C22—4244no—
528C22—2122no—
529C22—2122no—
530C22—2122no—
531C22#1681059512yes—
532C22#1681062312yes—
533C22#1681062312yes—
534C22#104×D5312yes—
535C22C2×C22⋊C4×D5311yescharacteristic
536C22C2×#1313324411yescharacteristic
537C22C2×#1313324911yescharacteristic
538C4—21no—
539C4—21no—
540C4—21no—
541C4—21no—
542C4—44no—
543C4—42no—
544C4—42no—
545C4—534no—
546C4—1062no—
547C4—1062no—
548C4—1062no—
549C4—2124no—
550C4—2122no—
551C2—21no—
552C2—22no—
553C2—21no—
554C2—4242no—
555C2#2027323914yesdirect product factor
556C2C2×#1681059511yescharacteristic
557C2C2×#1681062311yescharacteristic
558C2C2×#104×D5311yescharacteristic
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×#1509
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×#16810474C2semidirect
4. #20273239C2direct (N × H)
5. C2×#16810360C2semidirect
8. C2×C53×#328C2semidirect
11. #16811159C4semidirect
14. C2×#13133208C4semidirect
15. C2×#13133206C4semidirect
18. C2×Q8×Dic53C4semidirect
19. C2×#13133179C4semidirect
26. #16810677C22semidirect
31. #16810474C22semidirect
33. #16810360C22semidirect
39. C53×#328C22semidirect
50. #13133208C2×C4semidirect
52. #13133206C2×C4semidirect
60. Q8×Dic53C2×C4semidirect
62. #13133179C2×C4semidirect
277. C2×#328D53semidirect
283. C2×C53#1509semidirect
316. #328C2×D53semidirect
319. C53C2×#1509semidirect
555. C2#20273239direct (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×#168104742
#202732392
C2×#168103602
C2×#168103582
C2×#168103562
C2×C53×#3282
C2×#150953