Finite Group Databasebeta

← C2×#10618924 #13007559

Subgroups — C2×#10618924 #13007559

Order 1280.

7854 conjugacy classes of subgroups.

The subgroup lattice has 516 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C5×#557C211yesmaximal, characteristic, complemented
2#10618924C2132yesmaximal, direct product factor
3C2×#66060C213yesmaximal, complemented
4C2×#66059C213yesmaximal, complemented
5C2×#66029C216yesmaximal, complemented
6C2×#66032C216yesmaximal, complemented
7C2×#65914C213yesmaximal, complemented
8C2×#65911C213yesmaximal, complemented
9C2×#65911C211yesmaximal, characteristic, complemented
10C2×#65911C212yesmaximal, complemented
11C2×#65908C212yesmaximal, complemented
12C2×C4×#713C211yesmaximal, characteristic, complemented
13C22×#713C2211yescharacteristic
14C22×#713C2212yes—
15C22×#713C2216yes—
16C5×#557C22116yescomplemented
17C2×C5×C4⋊Q8C2213yes—
18C2×C5×#125C2216yes—
19C2×C4×C5×Q8C2211yescharacteristic
20C2×C4×C5×Q8C2212yes—
21C2×C4×C5×Q8C2213yes—
22C2×#4093C2213yes—
23C2×#4092C2213yes—
24C2×Q8×Dic5C2211yescharacteristic
25#66060C22148yescomplemented
26#66059C22148yescomplemented
27#66032C22196yescomplemented
28#66029C22196yescomplemented
29#65914C22148yescomplemented
30#65911C22116yescomplemented
31#65911C22132yescomplemented
32#65911C22148yescomplemented
33#65908C22132yescomplemented
34C4×#713C22116yescomplemented
35C2×#4044C2216yes—
36C2×#4041C2216yes—
37C2×#4041C2213yes—
38C2×#4041C2216yes—
39C2×#4041C22112yes—
40C2×#4041C2216yes—
41C2×#4041C2216yes—
42C2×#4040C22112yes—
43C2×#4040C2216yes—
44C2×#4039C2213yes—
45C2×#4039C2216yes—
46C2×#4039C2216yes—
47C2×#4038C2213yes—
48C2×D5×C4⋊C4C2216yes—
49C2×#4036C2211yescharacteristic
50C2×#4020C2216yes—
51C2×#4020C2213yes—
52C2×C4×D20C2213yes—
53C2×C4×D20C2213yes—
54C2×C4×D20C2216yes—
55C2×C4×D20C2216yes—
56C2×#4018C2216yes—
57C2×C42×D5C2213yes—
58C2×#3517—51nomaximal, Sylow
59C22×C5×Q8C2313yes—
60C22×C5×Q8C2311yescharacteristic
61C2×#713—28no—
62C2×#713—216no—
63C2×#713—424no—
64C2×#713C2318yes—
65C2×#713C23116yes—
66C2×#713C23148yes—
67C22×D20C2313yes—
68C22×D20C2316yes—
69C22×D20C2313yes—
70C22×D20C2316yes—
71C22×D20C2316yes—
72C22×D20C23112yes—
73C22×C4×D5C2313yes—
74C22×C4×D5C2313yes—
75C22×C4×D5C2316yes—
76C22×C4×D5C2313yes—
77C22×C4×D5C23112yes—
78C5×C4⋊Q8C23124yes—
79C5×#125C23148yes—
80C4×C5×Q8C2318yes—
81C4×C5×Q8C23116yes—
82C4×C5×Q8C23124yes—
83C2×C5×C4⋊C4C2311yescharacteristic
84C2×C5×C4⋊C4C2316yes—
85C2×C5×C4⋊C4C2313yes—
86C2×C5×C4⋊C4C2316yes—
87C2×C5×C4⋊C4C2316yes—
88C2×C42×C5C2313yes—
89C2×C42×C5C2316yes—
90#4093C23124yes—
91#4092C23124yes—
92Q8×Dic5C2318yes—
93C2×#685C2316yes—
94C2×#685C2316yes—
95C2×#685C2316yes—
96C2×#685C2316yes—
97C2×#685C2316yes—
98C2×#685C2316yes—
99C2×C4⋊Dic5C2312yes—
100C2×C4⋊Dic5C2311yescharacteristic
101C2×C4⋊Dic5C2313yes—
102C2×#683C2312yes—
103C2×#683C2316yes—
104C2×C4×Dic5C2311yescharacteristic
105C2×C4×Dic5C2313yes—
106C2×C4×Dic5C2313yes—
107#4044C23148yes—
108#4041C23148yes—
109#4041C23148yes—
110#4041C23148yes—
111#4041C23124yes—
112#4041C23196yes—
113#4041C23148yes—
114#4040C23196yes—
115#4040C23148yes—
116#4039C23148yes—
117#4039C23148yes—
118#4039C23124yes—
119#4038C23124yes—
120D5×C4⋊C4C23148yes—
121#4036C2318yes—
122#4020C23124yes—
123#4020C23148yes—
124C4×D20C23124yes—
125C4×D20C23124yes—
126C4×D20C23148yes—
127C4×D20C23148yes—
128#4018C23148yes—
129C42×D5C23124yes—
130#3517—532no—
131C2×#557D511yescharacteristic, complemented
132C2×#553—56no—
133C2×#550—53no—
134C2×#550—51no—
135C2×#550—52no—
136C2×#547—53no—
137C2×#540—56no—
138C2×#532—53no—
139C2×#529—53no—
140C2×#520—52no—
141C2×C4×#41—51no—
142C23×D5C2413yes—
143C23×D5C2416yes—
144C2×C5×Q8—212no—
145C2×C5×Q8#4118yescomplemented
146C2×C5×Q8C24112yes—
147C2×C5×Q8C2414yes—
148C22×C4×C5C2412yes—
149C22×C4×C5C2413yes—
150C22×C4×C5C2416yes—
151C22×C4×C5C2413yes—
152C22×C4×C5C2411yescharacteristic
153C22×Dic5C2413yes—
154C22×Dic5C2411yescharacteristic
155C22×Dic5C2411yescharacteristic
156C22×Dic5C2412yes—
157#713—232no—
158#713—264no—
159#713—496no—
160C2×D20—212no—
161C2×D20—224no—
162C2×D20—212no—
163C2×D20—412no—
164C2×D20—412no—
165C2×D20—424no—
166C2×D20C24112yes—
167C2×D20C24112yes—
168C2×D20C24124yes—
169C2×D20C24148yes—
170C2×D20C24124yes—
171C2×D20C24124yes—
172C2×C4×D5—212no—
173C2×C4×D5—224no—
174C2×C4×D5—212no—
175C2×C4×D5—46no—
176C2×C4×D5—424no—
177C2×C4×D5C24112yes—
178C2×C4×D5C24112yes—
179C2×C4×D5C24124yes—
180C2×C4×D5C24148yes—
181C2×C4×D5C24112yes—
182C5×C4⋊C4C2414yes—
183C5×C4⋊C4C24112yes—
184C5×C4⋊C4C24124yes—
185C5×C4⋊C4C24124yes—
186C5×C4⋊C4C24124yes—
187C42×C5C24124yes—
188C42×C5C24112yes—
189#685C24124yes—
190#685C24124yes—
191#685C24124yes—
192#685C24124yes—
193#685C24124yes—
194#685C24124yes—
195C4⋊Dic5C2418yes—
196C4⋊Dic5C2414yes—
197C4⋊Dic5C24112yes—
198#683C2418yes—
199#683C24124yes—
200C4×Dic5C2414yes—
201C4×Dic5C24112yes—
202C4×Dic5C24112yes—
203C22×#41—51no—
204C22×#41—52no—
205C22×#41—56no—
206#557C2×D5116yescomplemented
207#553—596no—
208#550—548no—
209#550—516no—
210#550—532no—
211#547—548no—
212#540—596no—
213#532—548no—
214C2×C4⋊Q8C2×D513yes—
215C2×#127—53no—
216C2×#127—56no—
217#529—548no—
218C2×#126—56no—
219C2×#125—51no—
220C2×#125C2×D516yes—
221C2×#124—53no—
222C2×#123—56no—
223C2×#123—512no—
224C2×C22⋊Q8—53no—
225C2×#121—53no—
226C2×#121—56no—
227C2×#121—56no—
228C2×#121—56no—
229C2×#121—512no—
230C2×#121—56no—
231#520—532no—
232C4×#41—516no—
233C2×C4×Q8—51no—
234C2×C4×Q8C2×D511yescharacteristic
235C2×C4×Q8C2×D512yes—
236C2×C4×Q8C2×D513yes—
237C2×C4×D4—53no—
238C2×C4×D4—56no—
239C2×C4×D4—56no—
240C2×C4×D4—53no—
241C2×C4×D4—56no—
242C2×C4×D4—53no—
243C2×C4×D4—56no—
244C2×#117—53no—
245C2×#117—56no—
246C22×C4⋊C4—56no—
247C22×C42—53no—
248C23×C5C2511yescharacteristic
249C22×D5—26no—
250C22×D5—212no—
251C22×D5—26no—
252C22×D5—212no—
253C22×D5—26no—
254C22×D5—212no—
255C22×D5C2516yes—
256C22×D5C25112yes—
257C5×Q8—224no—
258C5×Q8C2×#41116yescomplemented
259C2×C4×C5—212no—
260C2×C4×C5—26no—
261C2×C4×C5—24no—
262C2×C4×C5C2×#41112yes—
263C2×C4×C5H(5,𝔽2)14yes—
264C2×C4×C5C2516yes—
265C2×C4×C5C2514yes—
266C2×C4×C5C25112yes—
267C2×C4×C5C2516yes—
268C2×C4×C5C2512yes—
269C2×Dic5—22no—
270C2×Dic5—24no—
271C2×Dic5—26no—
272C2×Dic5—22no—
273C2×Dic5C2512yes—
274C2×Dic5C2512yes—
275C2×Dic5C2516yes—
276C2×Dic5C2514yes—
277D20—224no—
278D20—248no—
279D20—224no—
280D20—424no—
281D20—424no—
282D20—448no—
283C4×D5—224no—
284C4×D5—248no—
285C4×D5—224no—
286C4×D5—412no—
287C4×D5—448no—
288C2×#41—58no—
289C2×#41—516no—
290C2×#41—548no—
291C2×#41—108no—
292C2×#41—1016no—
293C2×#41—2024no—
294C22×Q8C22×D513yes—
295C22×Q8C22×D511yescharacteristic
296C22×D4—53no—
297C22×D4—56no—
298C22×D4—53no—
299C22×D4—56no—
300C22×D4—56no—
301C22×D4—512no—
302C23×C4—56no—
303C23×C4—53no—
304C23×C4—53no—
305C23×C4—512no—
306C23×C4—53no—
307C4⋊Q8C22×D5124yes—
308#127—524no—
309#127—548no—
310#126—548no—
311#125—58no—
312#125C22×D5148yes—
313#124—524no—
314#123—548no—
315#123—596no—
316C22⋊Q8—524no—
317#121—524no—
318#121—548no—
319#121—548no—
320#121—548no—
321#121—596no—
322#121—548no—
323C4×Q8—58no—
324C4×Q8C22×D518yes—
325C4×Q8C22×D5116yes—
326C4×Q8C22×D5124yes—
327C4×D4—524no—
328C4×D4—548no—
329C4×D4—548no—
330C4×D4—524no—
331C4×D4—548no—
332C4×D4—524no—
333C4×D4—548no—
334#117—524no—
335#117—548no—
336C2×C4⋊C4—53no—
337C2×C4⋊C4—56no—
338C2×C4⋊C4—52no—
339C2×C4⋊C4—51no—
340C2×C4⋊C4—52no—
341C2×C4⋊C4—548no—
342C2×C4⋊C4C22×D516yes—
343C2×C4⋊C4C22×D513yes—
344C2×C4⋊C4C22×D516yes—
345C2×C4⋊C4C22×D511yescharacteristic
346C2×C4⋊C4C22×D516yes—
347C2×C22⋊C4—56no—
348C2×C22⋊C4—56no—
349C2×C22⋊C4—56no—
350C2×C22⋊C4—56no—
351C2×C22⋊C4—56no—
352C2×C22⋊C4—56no—
353C2×C42—53no—
354C2×C42—53no—
355C2×C42—51no—
356C2×C42—524no—
357C2×C42C22×D516yes—
358C2×C42C22×D513yes—
359C22×C5C22×#4112yes—
360C22×C5C2×H(5,𝔽2)12yes—
361C22×C5#58512yes—
362C22×C5C2611yescharacteristic
363C2×D5—26no—
364C2×D5—212no—
365C2×D5—26no—
366C2×D5—212no—
367C2×D5—26no—
368C2×D5—212no—
369C2×D5—412no—
370C2×D5—424no—
371C4×C5—212no—
372C4×C5—26no—
373C4×C5—24no—
374C4×C5C22×#41112yes—
375C4×C5C2×H(5,𝔽2)14yes—
376Dic5—22no—
377Dic5—24no—
378Dic5—26no—
379Dic5—22no—
380C24—53no—
381C24—56no—
382#41—1032no—
383#41—1064no—
384#41—2096no—
385C2×Q8—212no—
386C2×Q8#71318yescomplemented
387C2×Q8C23×D5112yes—
388C2×Q8C23×D514yes—
389C2×D4—512no—
390C2×D4—524no—
391C2×D4—512no—
392C2×D4—524no—
393C2×D4—524no—
394C2×D4—548no—
395C2×D4—1012no—
396C2×D4—1024no—
397C2×D4—1012no—
398C2×D4—2024no—
399C2×D4—2012no—
400C2×D4—2012no—
401C22×C4—51no—
402C22×C4—51no—
403C22×C4—53no—
404C22×C4—52no—
405C22×C4—512no—
406C22×C4—512no—
407C22×C4—524no—
408C22×C4—512no—
409C22×C4—548no—
410C22×C4—1024no—
411C22×C4—1012no—
412C22×C4—1012no—
413C22×C4—2024no—
414C22×C4—206no—
415C22×C4C23×D513yes—
416C22×C4C23×D512yes—
417C22×C4C23×D516yes—
418C22×C4C23×D513yes—
419C22×C4C23×D511yescharacteristic
420C4⋊C4—512no—
421C4⋊C4—524no—
422C4⋊C4—58no—
423C4⋊C4—54no—
424C4⋊C4—58no—
425C4⋊C4C23×D5124yes—
426C4⋊C4C23×D5112yes—
427C4⋊C4C23×D5124yes—
428C4⋊C4C23×D514yes—
429C4⋊C4C23×D5124yes—
430C22⋊C4—524no—
431C22⋊C4—524no—
432C22⋊C4—524no—
433C22⋊C4—524no—
434C22⋊C4—524no—
435C22⋊C4—524no—
436C42—54no—
437C42—512no—
438C42—512no—
439C42C23×D5112yes—
440C42C23×D5124yes—
441C2×C5#351714yescomplemented
442C2×C5C23×#4111yescharacteristic
443C2×C5C22×H(5,𝔽2)11yescharacteristic
444C2×C5C2×#58511yescharacteristic
445D5—46no—
446D5—412no—
447C23—56no—
448C23—512no—
449C23—106no—
450C23—1012no—
451C23—106no—
452C23—1012no—
453C23—106no—
454C23—1012no—
455C23C24×D511yescharacteristic
456Q8—224no—
457Q8C2×#713116yescomplemented
458D4—1024no—
459D4—1048no—
460D4—1024no—
461D4—2048no—
462D4—2024no—
463D4—2024no—
464C2×C4—212no—
465C2×C4—26no—
466C2×C4—24no—
467C2×C4—52no—
468C2×C4—52no—
469C2×C4—56no—
470C2×C4—54no—
471C2×C4—102no—
472C2×C4—104no—
473C2×C4—106no—
474C2×C4—102no—
475C2×C4—1024no—
476C2×C4—1048no—
477C2×C4—1024no—
478C2×C4—2048no—
479C2×C4—2012no—
480C2×C4C2×#713112yes—
481C2×C4#414914yes—
482C2×C4C24×D516yes—
483C2×C4C24×D514yes—
484C2×C4C24×D5112yes—
485C2×C4C24×D516yes—
486C2×C4C24×D512yes—
487C5C2×#351711yescharacteristic, complemented, Sylow
488C22—106no—
489C22—1012no—
490C22—106no—
491C22—1012no—
492C22—106no—
493C22—1012no—
494C22—2012no—
495C22—2024no—
496C22C22×#71312yes—
497C22C2×#414912yes—
498C22#6628712yes—
499C22C25×D511yescharacteristic
500C4—212no—
501C4—26no—
502C4—24no—
503C4—102no—
504C4—104no—
505C4—106no—
506C4—102no—
507C4C22×#713112yes—
508C4C2×#414914yes—
509C2—206no—
510C2—2012no—
511C2#1061892414yesdirect product factor
512C2C23×#71311yescharacteristic
513C2C22×#414911yescharacteristic
514C2C2×#6628711yescharacteristic
subgroupG/H
centerC23C24×D5
commutator, cosocleC22×C5C26
FrattiniC22C25×D5
FittingC2×C5×#557C2
socleC23×C5C25
hypercenterC2×#557D5
hypocenterC5C2×#3517
LCSG ▷ C22×C5 ▷ C5
UCSC1 ◁ C23 ◁ C2×#557

Complements

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

NH ≅ G/Ntype
1. C2×C5×#557C2semidirect
2. #10618924C2direct (N × H)
3. C2×#66060C2semidirect
4. C2×#66059C2semidirect
5. C2×#66029C2semidirect
6. C2×#66032C2semidirect
7. C2×#65914C2semidirect
8. C2×#65911C2semidirect
9. C2×#65911C2semidirect
10. C2×#65911C2semidirect
11. C2×#65908C2semidirect
12. C2×C4×#713C2semidirect
16. C5×#557C22semidirect
25. #66060C22semidirect
26. #66059C22semidirect
27. #66032C22semidirect
28. #66029C22semidirect
29. #65914C22semidirect
30. #65911C22semidirect
31. #65911C22semidirect
32. #65911C22semidirect
33. #65908C22semidirect
34. C4×#713C22semidirect
131. C2×#557D5semidirect
145. C2×C5×Q8#41semidirect
206. #557C2×D5semidirect
258. C5×Q8C2×#41semidirect
386. C2×Q8#713semidirect
441. C2×C5#3517semidirect
457. Q8C2×#713semidirect
487. C5C2×#3517semidirect
511. C2#10618924direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C5×#5572
#106189242
C2×#660602
C2×#660592
C2×#660292
C2×#660322
C2×#659142
C2×#659112
C2×#659082
C2×C4×#7132
C2×#35175