Finite Group Databasebeta

← C2×#20273250 #27351331

Subgroups — C2×#20273250 #27351331

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×#16811154C211yesmaximal, characteristic
2C2×#16810677C211yesmaximal, characteristic, complemented
3C2×#16810487C211yesmaximal, characteristic, complemented
4#20273250C218yesmaximal, direct product factor
5C2×#16810361C211yesmaximal, characteristic, complemented
6C2×#16810359C211yesmaximal, characteristic, complemented
7C2×#16810357C211yesmaximal, characteristic
8C2×C53×#327C211yesmaximal, characteristic
9#16811154—22no—
10#16811154—22no—
11#16811154C414yescomplemented
12#16811154C2214yes—
13C2×#13133210—21no—
14C2×#13133208C411yescharacteristic, complemented
15C2×#13133204C411yescharacteristic, complemented
16C2×#13133198—21no—
17C2×#13133196C2211yescharacteristic
18C2×#13133182C411yescharacteristic, complemented
19C2×D4×Dic53—21no—
20C2×D4×Dic53C411yescharacteristic, complemented
21C2×#13133173—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#16810361C2214yescomplemented
34#16810359C2214yescomplemented
35#16810357C2214yes—
36C2×#121×C53C2211yescharacteristic
37C2×C22⋊C8×C53C2211yescharacteristic
38C2×#95×C53C2211yescharacteristic
39C53×#327C2214yes—
40C22×Dic106C2×C411yescharacteristic
41C23×Dic53—21no—
42C23×Dic53—21no—
43C23×Dic53D411yescharacteristic
44C23×Dic53D411yescharacteristic
45C22×#88322—21no—
46#13133210—24no—
47#13133210—24no—
48#13133208—22no—
49#13133208C2×C414yescomplemented
50#13133204—22no—
51#13133204C2×C414yescomplemented
52#13133198—24no—
53#13133198—24no—
54#13133196—21no—
55#13133196—21no—
56#13133196C2×C412yes—
57#13133196C2312yes—
58#13133182—22no—
59#13133182C2×C414yescomplemented
60D4×Dic53—22no—
61D4×Dic53—24no—
62D4×Dic53—24no—
63D4×Dic53C2×C414yescomplemented
64#13133173—24no—
65#13133173—24no—
66C2×C4⋊Dic53—21no—
67C2×C4⋊Dic53C2×C411yescharacteristic
68C2×#11719919—21no—
69C2×#11719919—21no—
70C2×#11719919—21no—
71C2×#11719919—24no—
72C2×#11719919—44no—
73C2×C22⋊Dic53—21no—
74C2×C22⋊Dic53—21no—
75C2×C22⋊Dic53—21no—
76C2×C22⋊Dic53—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×D4×C53—21no—
94C22×D4×C53C2×C411yescharacteristic
95C23×C4×C53—21no—
96C23×C4×C53C2311yescharacteristic
97C22×C8×C53—21no—
98#121×C53—21no—
99#121×C53—21no—
100#121×C53C2×C412yes—
101#121×C53C2312yes—
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×C53—21no—
109C24×C53C2×D411yescharacteristic
110C2×D4×C53—22no—
111C2×D4×C53—22no—
112C2×D4×C53—21no—
113C2×D4×C53—42no—
114C2×D4×C53—42no—
115C2×D4×C53—42no—
116C2×D4×C53C22×C412yes—
117C22×C4×C53—21no—
118C22×C4×C53—22no—
119C22×C4×C53—42no—
120C22×C4×C53—42no—
121C22×C4×C53—42no—
122C22×C4×C53—42no—
123C22×C4×C53C22×C411yescharacteristic
124C22×C4×C53C22×C411yescharacteristic
125C22×C4×C53C2×D411yescharacteristic
126C22×C4×C53C2×D411yescharacteristic
127C22×C4×C53C2×D411yescharacteristic
128C22×C4×C53C2411yescharacteristic
129C22×Dic53—22no—
130C22×Dic53—22no—
131C22×Dic53—21no—
132C22×Dic53—21no—
133C22×Dic53—24no—
134C22×Dic53—22no—
135C22×Dic53—42no—
136C22×Dic53—44no—
137C22×Dic53—42no—
138C22×Dic53—42no—
139C22×Dic53—42no—
140C22×Dic53C22⋊C411yescharacteristic
141C22×Dic53C22⋊C411yescharacteristic
142C22×Dic53C22⋊C411yescharacteristic
143C22×Dic53C22⋊C411yescharacteristic
144C22×Dic53C22⋊C412yes—
145C22×Dic53C22⋊C412yes—
146C22×Dic53C2×D412yes—
147C22×Dic53C2×D412yes—
148C2×Dic106—21no—
149C2×Dic106—42no—
150C2×Dic106C22×C412yes—
151C2×C8×C53—22no—
152C2×C8×C53—42no—
153C4⋊C4×C53—21no—
154C4⋊C4×C53C22×C412yes—
155C22⋊C4×C53—22no—
156C22⋊C4×C53—22no—
157C22⋊Dic53—22no—
158C22⋊Dic53—22no—
159C22⋊Dic53—24no—
160C22⋊Dic53—24no—
161C22⋊Dic53—24no—
162C4⋊Dic53—21no—
163C4⋊Dic53—24no—
164C4⋊Dic53C22×C412yes—
165#11719919—22no—
166#11719919—22no—
167#11719919—24no—
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×C53—22no—
184C23×C53—42no—
185C23×C53—42no—
186C23×C53—42no—
187C23×C53C2×C22⋊C411yescharacteristic
188C23×C53C2×C22⋊C411yescharacteristic
189C23×C53AGL(1,ℤ/8)12yes—
190C23×C53#13712yes—
191C23×C53C22×D411yescharacteristic
192D4×C53—42no—
193D4×C53—42no—
194D4×C53—42no—
195D4×C53—42no—
196D4×C53—44no—
197C2×C4×C53—22no—
198C2×C4×C53—22no—
199C2×C4×C53—22no—
200C2×C4×C53—41no—
201C2×C4×C53—41no—
202C2×C4×C53—41no—
203C2×C4×C53—41no—
204C2×C4×C53—42no—
205C2×C4×C53—42no—
206C2×C4×C53—42no—
207C2×C4×C53—42no—
208C2×C4×C53—84no—
209C2×C4×C53C2×C22⋊C411yescharacteristic
210C2×C4×C53C2×C22⋊C411yescharacteristic
211C2×C4×C53C23×C411yescharacteristic
212C2×C4×C53C22×D411yescharacteristic
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—22no—
222C2×Dic53—42no—
223C2×Dic53—42no—
224C2×Dic53—48no—
225C2×Dic53—42no—
226C2×Dic53—42no—
227C2×Dic53—42no—
228C2×Dic53—42no—
229C2×Dic53—42no—
230C2×Dic53—84no—
231C2×Dic53C2×C22⋊C412yes—
232C2×Dic53C2×C22⋊C412yes—
233C2×Dic53C2×C22⋊C412yes—
234C2×Dic53C2×C22⋊C412yes—
235Dic106—42no—
236Dic106—42no—
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—42no—
247C22×C53—42no—
248C22×C53—42no—
249C22×C53—84no—
250C22×C53#41012yes—
251C22×C53#41712yes—
252C22×C53#41712yes—
253C22×C53#42112yes—
254C22×C53C22×C22⋊C411yescharacteristic
255C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
256C22×C53C2×#13711yescharacteristic
257C4×C53—21no—
258C4×C53—21no—
259C4×C53—21no—
260C4×C53—21no—
261C4×C53—42no—
262C4×C53—82no—
263Dic53—22no—
264Dic53—22no—
265Dic53—22no—
266Dic53—22no—
267Dic53—42no—
268Dic53—42no—
269Dic53—82no—
270C2×#529—531no—
271C2×#432—531no—
272C2×#382—531no—
273#1512—538no—
274C2×#328—531no—
275C2×#328—531no—
276C2×#327—531no—
277C2×#327D5311yescharacteristic
278C2×C53—21no—
279C2×C53—21no—
280C2×C53—22no—
281C2×C53—82no—
282C2×C53#151214yescomplemented
283C2×C53C2×#41011yescharacteristic
284C2×C53C2×#41711yescharacteristic
285C2×C53C2×#42111yescharacteristic
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⋊Q8—531no—
295C2×#121C2×D5311yescharacteristic
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#328—534no—
315#327—534no—
316#327C2×D5314yes—
317C53C2×#151211yescharacteristic, complemented, Sylow
318C22×Q8—531no—
319C22×D4—21no—
320C22×D4C4×D5311yescharacteristic
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—532no—
338C22⋊Q8—532no—
339C22⋊Q8—1061no—
340C22⋊Q8—1061no—
341#121—21no—
342#121—21no—
343#121C4×D5312yes—
344#121C22×D5312yes—
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—531no—
354C2×C4⋊C4—1061no—
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—21no—
387C24D4×D5311yescharacteristic
388C2×Q8—532no—
389C2×Q8—1061no—
390C2×Q8—2122no—
391C2×D4—22no—
392C2×D4—22no—
393C2×D4—21no—
394C2×D4—42no—
395C2×D4—42no—
396C2×D4—42no—
397C2×D4C2×C4×D5312yes—
398C22×C4—21no—
399C22×C4—22no—
400C22×C4—42no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—532no—
405C22×C4—531no—
406C22×C4—531no—
407C22×C4—531no—
408C22×C4—531no—
409C22×C4—532no—
410C22×C4—532no—
411C22×C4—532no—
412C22×C4—1064no—
413C22×C4—1062no—
414C22×C4—1062no—
415C22×C4—1061no—
416C22×C4—1061no—
417C22×C4—1062no—
418C22×C4—2122no—
419C22×C4—2124no—
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—21no—
434C4⋊C4—532no—
435C4⋊C4—1064no—
436C4⋊C4—1064no—
437C4⋊C4—1064no—
438C4⋊C4—1061no—
439C4⋊C4—1062no—
440C4⋊C4—1062no—
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—22no—
462C23—42no—
463C23—42no—
464C23—42no—
465C23C22⋊C4×D5311yescharacteristic
466C23C22⋊C4×D5311yescharacteristic
467C23#1313324112yes—
468C23#1313324612yes—
469C23C2×D4×D5311yescharacteristic
470Q8—2122no—
471Q8—2122no—
472D4—42no—
473D4—42no—
474D4—44no—
475D4—42no—
476D4—42no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—84no—
489C2×C4—532no—
490C2×C4—532no—
491C2×C4—532no—
492C2×C4—532no—
493C2×C4—1064no—
494C2×C4—1062no—
495C2×C4—1062no—
496C2×C4—1062no—
497C2×C4—1062no—
498C2×C4—1062no—
499C2×C4—1062no—
500C2×C4—1062no—
501C2×C4—1062no—
502C2×C4—4244no—
503C2×C4—2122no—
504C2×C4—2122no—
505C2×C4—2128no—
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—42no—
524C22—42no—
525C22—42no—
526C22—84no—
527C22#1681059512yes—
528C22#1681062112yes—
529C22#1681062112yes—
530C22#1681063812yes—
531C22C2×C22⋊C4×D5311yescharacteristic
532C22C2×#1313324111yescharacteristic
533C22C2×#1313324611yescharacteristic
534C4—21no—
535C4—21no—
536C4—21no—
537C4—21no—
538C4—42no—
539C4—82no—
540C4—1062no—
541C4—1062no—
542C4—1062no—
543C4—1062no—
544C4—4242no—
545C4—2122no—
546C4—2122no—
547C2—21no—
548C2—22no—
549C2—21no—
550C2—82no—
551C2#2027325014yesdirect product factor
552C2C2×#1681059511yescharacteristic
553C2C2×#1681062111yescharacteristic
554C2C2×#1681063811yescharacteristic
subgroupG/H
centerC23C22⋊C4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC2×C53×#327C2
socleC23×C53C2×C22⋊C4
prime radicalC2×#121×C53C22
cosocleC22×C4×C53C24
hypercenterC2×#327D53
hypocenterC53C2×#1512
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×#16810487C2semidirect
4. #20273250C2direct (N × H)
5. C2×#16810361C2semidirect
6. C2×#16810359C2semidirect
11. #16811154C4semidirect
14. C2×#13133208C4semidirect
15. C2×#13133204C4semidirect
18. C2×#13133182C4semidirect
20. C2×D4×Dic53C4semidirect
25. #16810677C22semidirect
30. #16810487C22semidirect
33. #16810361C22semidirect
34. #16810359C22semidirect
49. #13133208C2×C4semidirect
51. #13133204C2×C4semidirect
59. #13133182C2×C4semidirect
63. D4×Dic53C2×C4semidirect
282. C2×C53#1512semidirect
317. C53C2×#1512semidirect
551. C2#20273250direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168111542
C2×#168106772
C2×#168104872
#202732502
C2×#168103612
C2×#168103592
C2×#168103572
C2×C53×#3272
C2×#151253