Finite Group Databasebeta

← C2×#10618651 #13007571

Subgroups — C2×#10618651 #13007571

Order 1280.

7118 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C5×D4×Q8C211yesmaximal, characteristic, complemented
2C2×#66151C212yesmaximal, complemented
3C2×#66146C212yesmaximal, complemented
4#10618651C2132yesmaximal, direct product factor
5C2×#66066C213yesmaximal, complemented
6C2×#66062C213yesmaximal, complemented
7C2×#65975C216yesmaximal, complemented
8C2×#65977C216yesmaximal, complemented
9C2×Q8×Dic10C211yesmaximal, characteristic, complemented
10C2×C4×#713C211yesmaximal, characteristic, complemented
11C2×#65891C213yesmaximal, complemented
12C2×#65897C213yesmaximal, complemented
13C23×C5×Q8C2212yes—
14C22×#713C2211yescharacteristic
15C5×D4×Q8C22116yescomplemented
16C2×C5×C4⋊Q8C2213yes—
17C2×C5×C22⋊Q8C2216yes—
18C2×C4×C5×Q8C2211yescharacteristic
19C2×C4×C5×D4C2213yes—
20#66151C22132yescomplemented
21C2×#4093C2216yes—
22C2×#4092C2216yes—
23#66146C22132yescomplemented
24C2×Q8×Dic5C2216yes—
25C2×Q8×Dic5C2211yescharacteristic
26C2×Q8×Dic5C2216yes—
27C2×Q8×Dic5C2212yes—
28C2×#4084C2213yes—
29C2×#4082C2211yescharacteristic
30C2×#4076C2216yes—
31C2×C4×C22⋊D5C2216yes—
32C2×#4072C2216yes—
33C2×#4072C2216yes—
34#66066C22148yescomplemented
35#66062C22148yescomplemented
36#65977C22196yescomplemented
37#65975C22196yescomplemented
38C4×#713C22116yescomplemented
39Q8×Dic10C22116yescomplemented
40#65897C22148yescomplemented
41#65891C22148yescomplemented
42C2×#4044C22112yes—
43C2×#4043C2216yes—
44C2×#4043C2213yes—
45C2×#4039C2213yes—
46C2×#4038C2213yes—
47C2×#4038C2216yes—
48C2×#4036C2213yes—
49C2×#4034C2216yes—
50C2×#4033C2216yes—
51C2×#4031C22112yes—
52C2×#4029C2216yes—
53C2×#4025C2216yes—
54C2×C4×D20C2213yes—
55C2×C42×D5C2213yes—
56C2×#4015C2213yes—
57C2×C4×Dic10C2213yes—
58C2×#3518—51nomaximal, Sylow
59C22×C5×Q8—216no—
60C22×C5×Q8C2316yes—
61C22×C5×Q8C23116yes—
62C22×C5×Q8C2311yescharacteristic
63C22×C5×D4C2311yescharacteristic
64C23×C4×C5C2316yes—
65C22×C22⋊D5C2316yes—
66C2×#713—28no—
67C2×#713C2318yes—
68C22×D20C2313yes—
69C22×C4×D5C2313yes—
70C22×C4×D5C2313yes—
71C22×C4×D5C2313yes—
72C22×Dic10C2311yescharacteristic
73C22×Dic10C2316yes—
74C5×C4⋊Q8C23124yes—
75C5×C22⋊Q8C23148yes—
76C4×C5×Q8C2318yes—
77C4×C5×D4C23124yes—
78C2×C5×C4⋊C4C2316yes—
79C2×C5×C4⋊C4C2313yes—
80C2×C5×C4⋊C4C2313yes—
81C2×C5×C22⋊C4C2316yes—
82C2×C42×C5C2313yes—
83C2×C22⋊Dic5C2312yes—
84C2×C22⋊Dic5C2316yes—
85C2×C22⋊Dic5C2312yes—
86#4093C23148yes—
87#4092C23148yes—
88Q8×Dic5C23148yes—
89Q8×Dic5C23148yes—
90Q8×Dic5C2318yes—
91Q8×Dic5C23116yes—
92#4084C23124yes—
93#4082C2318yes—
94#4076C23148yes—
95C4×C22⋊D5C23148yes—
96C2×#685C2316yes—
97C2×#685C2316yes—
98C2×#685C23112yes—
99#4072C23148yes—
100#4072C23148yes—
101C2×C4⋊Dic5C2313yes—
102C2×C4⋊Dic5C2316yes—
103C2×C4⋊Dic5C2313yes—
104C2×C4⋊Dic5C2316yes—
105C2×C4⋊Dic5C2316yes—
106C2×#683C2316yes—
107C2×#683C2316yes—
108C2×C4×Dic5C2316yes—
109C2×C4×Dic5C2311yescharacteristic
110C2×C4×Dic5C2313yes—
111C2×C4×Dic5C2313yes—
112C2×C4×Dic5C23112yes—
113#4044C23196yes—
114#4043C23148yes—
115#4043C23124yes—
116#4039C23124yes—
117#4038C23148yes—
118#4038C23124yes—
119#4036C23124yes—
120#4034C23148yes—
121#4033C23148yes—
122#4031C23196yes—
123#4029C23148yes—
124#4025C23148yes—
125C4×D20C23124yes—
126C42×D5C23124yes—
127#4015C23124yes—
128C4×Dic10C23124yes—
129#3518—532no—
130C2×Q82—51no—
131C2×#556—53no—
132C2×#556—56no—
133C2×#555—53no—
134C2×D4×Q8D511yescharacteristic, complemented
135C2×#548—52no—
136C2×#542—56no—
137C2×#533—53no—
138C2×#529—53no—
139C2×#519—52no—
140C2×C4×#41—51no—
141C24×C5C2412yes—
142C23×D5C2413yes—
143C2×C5×Q8—264no—
144C2×C5×Q8—224no—
145C2×C5×Q8#4118yes—
146C2×C5×Q8C24124yes—
147C2×C5×Q8C2414yes—
148C2×C5×D4—24no—
149C2×C5×D4C2414yes—
150C22×C4×C5—224no—
151C22×C4×C5C2413yes—
152C22×C4×C5C2416yes—
153C22×C4×C5C24124yes—
154C22×C4×C5C2413yes—
155C22×C4×C5C2411yescharacteristic
156C2×C22⋊D5—412no—
157C2×C22⋊D5C24124yes—
158C22×Dic5C2412yes—
159C22×Dic5C2411yescharacteristic
160C22×Dic5C2411yescharacteristic
161C22×Dic5C2413yes—
162C22×Dic5C2416yes—
163#713—232no—
164C2×D20—212no—
165C2×D20C24112yes—
166C2×C4×D5—212no—
167C2×C4×D5—212no—
168C2×C4×D5—46no—
169C2×C4×D5C24112yes—
170C2×C4×D5C24112yes—
171C2×C4×D5C24112yes—
172C2×Dic10—24no—
173C2×Dic10—412no—
174C2×Dic10C2414yes—
175C2×Dic10C24124yes—
176C5×C4⋊C4C24112yes—
177C5×C4⋊C4C24112yes—
178C5×C4⋊C4C24124yes—
179C5×C22⋊C4C24124yes—
180C42×C5C24112yes—
181C22⋊Dic5C2418yes—
182C22⋊Dic5C24124yes—
183C22⋊Dic5C2418yes—
184#685C24124yes—
185#685C24124yes—
186#685C24148yes—
187C4⋊Dic5C24112yes—
188C4⋊Dic5C24124yes—
189C4⋊Dic5C24112yes—
190C4⋊Dic5C24124yes—
191C4⋊Dic5C24124yes—
192#683C24124yes—
193#683C24124yes—
194C4×Dic5C24124yes—
195C4×Dic5C2414yes—
196C4×Dic5C24112yes—
197C4×Dic5C24112yes—
198C4×Dic5C24148yes—
199C22×#41—51no—
200C23×Q8C2×D512yes—
201Q82—516no—
202#556—548no—
203#556—596no—
204#555—548no—
205D4×Q8C2×D5116yescomplemented
206#548—532no—
207#542—596no—
208#533—548no—
209C2×C4⋊Q8—56no—
210C2×C4⋊Q8—53no—
211C2×C4⋊Q8C2×D513yes—
212#529—548no—
213C2×#126—512no—
214C2×#126—56no—
215C2×#125—53no—
216C2×#124—56no—
217C2×#124—51no—
218C2×#124—512no—
219C2×#123—56no—
220C2×C22⋊Q8—56no—
221C2×C22⋊Q8—53no—
222C2×C22⋊Q8—56no—
223C2×C22⋊Q8C2×D516yes—
224C2×#121—53no—
225C2×#121—56no—
226#519—532no—
227C4×#41—516no—
228C2×C4×Q8—51no—
229C2×C4×Q8—56no—
230C2×C4×Q8—52no—
231C2×C4×Q8—56no—
232C2×C4×Q8—56no—
233C2×C4×Q8—53no—
234C2×C4×Q8C2×D511yescharacteristic
235C2×C4×D4—56no—
236C2×C4×D4—53no—
237C2×C4×D4—53no—
238C2×C4×D4C2×D513yes—
239C2×#117—56no—
240C2×#117—56no—
241C2×#117—53no—
242C2×#117—56no—
243C22×C42—53no—
244C23×C5—24no—
245C23×C5—24no—
246C23×C5—24no—
247C23×C5C2514yes—
248C23×C5C2511yescharacteristic
249C22×D5—26no—
250C22×D5—26no—
251C22×D5—26no—
252C22×D5C2516yes—
253C5×Q8—248no—
254C5×Q8C2×#41116yes—
255C5×D4—28no—
256C2×C4×C5—212no—
257C2×C4×C5—26no—
258C2×C4×C5—248no—
259C2×C4×C5C2×#41112yes—
260C2×C4×C52−1+414yes—
261C2×C4×C5C25112yes—
262C2×C4×C5C2516yes—
263C2×C4×C5C2516yes—
264C2×C4×C5C2512yes—
265C22⋊D5—424no—
266C2×Dic5—22no—
267C2×Dic5—24no—
268C2×Dic5—26no—
269C2×Dic5—212no—
270C2×Dic5—22no—
271C2×Dic5C2516yes—
272C2×Dic5C2512yes—
273C2×Dic5C2512yes—
274C2×Dic5C2514yes—
275C2×Dic5C25112yes—
276D20—224no—
277C4×D5—224no—
278C4×D5—224no—
279C4×D5—412no—
280Dic10—28no—
281Dic10—424no—
282C2×#41—58no—
283C2×#41—108no—
284C22×Q8—216no—
285C22×Q8—51no—
286C22×Q8—56no—
287C22×Q8C22×D516yes—
288C22×Q8C22×D5116yes—
289C22×Q8C22×D511yescharacteristic
290C22×D4—56no—
291C22×D4—53no—
292C22×D4C22×D511yescharacteristic
293C23×C4—53no—
294C23×C4—53no—
295C23×C4—53no—
296C23×C4C22×D516yes—
297C4⋊Q8—548no—
298C4⋊Q8—524no—
299C4⋊Q8C22×D5124yes—
300#126—596no—
301#126—548no—
302#125—524no—
303#124—548no—
304#124—58no—
305#124—596no—
306#123—548no—
307C22⋊Q8—548no—
308C22⋊Q8—524no—
309C22⋊Q8—548no—
310C22⋊Q8C22×D5148yes—
311#121—524no—
312#121—548no—
313C4×Q8—58no—
314C4×Q8—548no—
315C4×Q8—516no—
316C4×Q8—548no—
317C4×Q8—548no—
318C4×Q8—524no—
319C4×Q8C22×D518yes—
320C4×D4—548no—
321C4×D4—524no—
322C4×D4—524no—
323C4×D4C22×D5124yes—
324#117—548no—
325#117—548no—
326#117—524no—
327#117—548no—
328C2×C4⋊C4—53no—
329C2×C4⋊C4—53no—
330C2×C4⋊C4—56no—
331C2×C4⋊C4—56no—
332C2×C4⋊C4—56no—
333C2×C4⋊C4—56no—
334C2×C4⋊C4—56no—
335C2×C4⋊C4C22×D513yes—
336C2×C4⋊C4C22×D513yes—
337C2×C4⋊C4C22×D516yes—
338C2×C22⋊C4—56no—
339C2×C22⋊C4—52no—
340C2×C22⋊C4—52no—
341C2×C22⋊C4—56no—
342C2×C22⋊C4—512no—
343C2×C22⋊C4—56no—
344C2×C22⋊C4C22×D516yes—
345C2×C42—53no—
346C2×C42—512no—
347C2×C42—53no—
348C2×C42—56no—
349C2×C42—51no—
350C2×C42—524no—
351C2×C42C22×D513yes—
352C22×C5—24no—
353C22×C5—24no—
354C22×C5—24no—
355C22×C5—48no—
356C22×C5C22×#4112yes—
357C22×C5C2×2−1+412yes—
358C22×C5#58512yes—
359C22×C5C2611yescharacteristic
360C2×D5—26no—
361C2×D5—26no—
362C2×D5—26no—
363C2×D5—412no—
364C4×C5—212no—
365C4×C5—26no—
366C4×C5C22×#41112yes—
367C4×C5C2×2−1+414yes—
368Dic5—22no—
369Dic5—24no—
370Dic5—26no—
371Dic5—212no—
372Dic5—22no—
373C24—53no—
374C24C23×D512yes—
375#41—1032no—
376C2×Q8—264no—
377C2×Q8—224no—
378C2×Q8—54no—
379C2×Q8—524no—
380C2×Q8—104no—
381C2×Q8—2012no—
382C2×Q8#71118yes—
383C2×Q8C23×D5124yes—
384C2×Q8C23×D514yes—
385C2×D4—24no—
386C2×D4—524no—
387C2×D4—512no—
388C2×D4—1012no—
389C2×D4—2012no—
390C2×D4C23×D514yes—
391C22×C4—224no—
392C22×C4—53no—
393C22×C4—51no—
394C22×C4—51no—
395C22×C4—52no—
396C22×C4—56no—
397C22×C4—512no—
398C22×C4—512no—
399C22×C4—512no—
400C22×C4—1012no—
401C22×C4—1012no—
402C22×C4—206no—
403C22×C4C23×D5124yes—
404C22×C4C23×D516yes—
405C22×C4C23×D513yes—
406C22×C4C23×D513yes—
407C22×C4C23×D511yescharacteristic
408C4⋊C4—512no—
409C4⋊C4—512no—
410C4⋊C4—524no—
411C4⋊C4—524no—
412C4⋊C4—524no—
413C4⋊C4—524no—
414C4⋊C4—524no—
415C4⋊C4C23×D5112yes—
416C4⋊C4C23×D5112yes—
417C4⋊C4C23×D5124yes—
418C22⋊C4—524no—
419C22⋊C4—58no—
420C22⋊C4—58no—
421C22⋊C4—524no—
422C22⋊C4—548no—
423C22⋊C4—524no—
424C22⋊C4C23×D5124yes—
425C42—54no—
426C42—512no—
427C42—512no—
428C42—548no—
429C42—524no—
430C42C23×D5112yes—
431C2×C5—44no—
432C2×C5#351814yescomplemented
433C2×C5C23×#4111yescharacteristic
434C2×C5C22×2−1+411yescharacteristic
435C2×C5C2×#58511yescharacteristic
436D5—46no—
437C23—24no—
438C23—24no—
439C23—24no—
440C23—56no—
441C23—106no—
442C23—106no—
443C23—106no—
444C23C24×D514yes—
445C23C24×D511yescharacteristic
446Q8—248no—
447Q8—108no—
448Q8—2024no—
449Q8C2×#711116yes—
450D4—28no—
451D4—1024no—
452D4—2024no—
453C2×C4—212no—
454C2×C4—26no—
455C2×C4—248no—
456C2×C4—56no—
457C2×C4—52no—
458C2×C4—52no—
459C2×C4—54no—
460C2×C4—512no—
461C2×C4—102no—
462C2×C4—106no—
463C2×C4—104no—
464C2×C4—102no—
465C2×C4—1012no—
466C2×C4—1024no—
467C2×C4—1024no—
468C2×C4—2012no—
469C2×C4C2×#711112yes—
470C2×C4#414714yes—
471C2×C4C24×D5112yes—
472C2×C4C24×D516yes—
473C2×C4C24×D516yes—
474C2×C4C24×D512yes—
475C5C2×#351811yescharacteristic, complemented, Sylow
476C22—24no—
477C22—24no—
478C22—24no—
479C22—48no—
480C22—106no—
481C22—106no—
482C22—106no—
483C22—2012no—
484C22C22×#71112yes—
485C22C2×#414712yes—
486C22#6628712yes—
487C22C25×D511yescharacteristic
488C4—212no—
489C4—26no—
490C4—102no—
491C4—106no—
492C4—104no—
493C4—102no—
494C4—1012no—
495C4C22×#711112yes—
496C4C2×#414714yes—
497C2—44no—
498C2—206no—
499C2#1061865114yesdirect product factor
500C2C23×#71111yescharacteristic
501C2C22×#414711yescharacteristic
502C2C2×#6628711yescharacteristic
subgroupG/H
centerC23C24×D5
commutator, cosocleC22×C5C26
FrattiniC22C25×D5
FittingC2×C5×D4×Q8C2
socleC23×C5C25
hypercenterC2×D4×Q8D5
hypocenterC5C2×#3518
LCSG ▷ C22×C5 ▷ C5
UCSC1 ◁ C23 ◁ C2×D4×Q8

Complements

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

NH ≅ G/Ntype
1. C2×C5×D4×Q8C2semidirect
2. C2×#66151C2semidirect
3. C2×#66146C2semidirect
4. #10618651C2direct (N × H)
5. C2×#66066C2semidirect
6. C2×#66062C2semidirect
7. C2×#65975C2semidirect
8. C2×#65977C2semidirect
9. C2×Q8×Dic10C2semidirect
10. C2×C4×#713C2semidirect
11. C2×#65891C2semidirect
12. C2×#65897C2semidirect
15. C5×D4×Q8C22semidirect
20. #66151C22semidirect
23. #66146C22semidirect
34. #66066C22semidirect
35. #66062C22semidirect
36. #65977C22semidirect
37. #65975C22semidirect
38. C4×#713C22semidirect
39. Q8×Dic10C22semidirect
40. #65897C22semidirect
41. #65891C22semidirect
134. C2×D4×Q8D5semidirect
205. D4×Q8C2×D5semidirect
432. C2×C5#3518semidirect
475. C5C2×#3518semidirect
499. C2#10618651direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C5×D4×Q82
C2×#661512
C2×#661462
#106186512
C2×#660662
C2×#660622
C2×#659752
C2×#659772
C2×Q8×Dic102
C2×C4×#7132
C2×#658912
C2×#658972
C2×#35185