Finite Group Databasebeta

← C2×#10617754 #13007558

Subgroups — C2×#10617754 #13007558

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×C4×C5×#41C211yesmaximal, characteristic, complemented
2C2×#66168C212yesmaximal, complemented
3#10617754C2132yesmaximal, direct product factor
4C2×#65904C211yesmaximal, characteristic, complemented
5C2×#65911C211yesmaximal, characteristic, complemented
6C2×#65908C212yesmaximal, complemented
7C2×#65885C213yesmaximal, complemented
8C2×#65896C216yesmaximal, complemented
9C2×#65887C213yesmaximal, complemented
10C2×#65864C216yesmaximal, complemented
11C2×#65811C213yesmaximal, complemented
12C2×#65813C213yesmaximal, complemented
13C22×C5×#41C2211yescharacteristic
14C22×#713C2212yes—
15C22×#709C2216yes—
16C4×C5×#41C22116yescomplemented
17C2×C4×C5×Q8C2211yescharacteristic
18C2×C4×C5×Q8C2211yescharacteristic
19C2×C4×C5×D4C2213yes—
20C2×C4×C5×D4C2213yes—
21C2×C5×#117C2213yes—
22C22×C42×C5C2213yes—
23#66168C22132yescomplemented
24C2×Q8×Dic5C2212yes—
25C2×#4086C2216yes—
26C2×#4084C2216yes—
27C2×#4076C2216yes—
28C2×#4076C2213yes—
29C2×#4076C2216yes—
30C2×#4075C2216yes—
31C2×C4×C22⋊D5C2216yes—
32C2×#4070C2213yes—
33#65911C22116yescomplemented
34#65908C22132yescomplemented
35#65904C22116yescomplemented
36#65896C22196yescomplemented
37#65887C22148yescomplemented
38#65885C22148yescomplemented
39#65864C22196yescomplemented
40C2×#4041C2216yes—
41C2×#4039C2216yes—
42C2×D5×C4⋊C4C2216yes—
43C2×#4036C2216yes—
44C2×#4030C22112yes—
45C2×#4029C22112yes—
46#65813C22148yescomplemented
47C2×#4022C2216yes—
48#65811C22148yescomplemented
49C2×#4021C2213yes—
50C2×#4020C2213yes—
51C2×C4×D20C2216yes—
52C2×C4×D20C2216yes—
53C2×C4×D20C2213yes—
54C2×#4018C2216yes—
55C2×#4016C2211yescharacteristic
56C2×#4015C2213yes—
57C2×C4×Dic10C2213yes—
58C2×#3517—51nomaximal, Sylow
59C2×C5×#41—28no—
60C2×C5×#41C2318yes—
61C22×C5×Q8C2311yescharacteristic
62C22×C5×D4C2313yes—
63C23×C4×C5C2313yes—
64C23×C4×C5C2313yes—
65C23×C4×C5C2313yes—
66C22×C22⋊D5C23112yes—
67C22×C22⋊D5C2316yes—
68C2×#713—216no—
69C2×#713C23116yes—
70C2×#709—424no—
71C2×#709C23148yes—
72C22×D20C2313yes—
73C22×D20C2316yes—
74C22×D20C2316yes—
75C22×C4×D5C2316yes—
76C22×C4×D5C23112yes—
77C22×Dic10C2313yes—
78C4×C5×Q8C2318yes—
79C4×C5×Q8C2318yes—
80C4×C5×D4C23124yes—
81C4×C5×D4C23124yes—
82C5×#117C23124yes—
83C2×C5×C4⋊C4C2313yes—
84C2×C5×C4⋊C4C2313yes—
85C2×C5×C22⋊C4C2316yes—
86C2×C42×C5C2313yes—
87C2×C42×C5C2311yescharacteristic
88C2×C42×C5C2313yes—
89C2×C42×C5C2313yes—
90C2×C42×C5C23124yes—
91C2×C22⋊Dic5C2316yes—
92Q8×Dic5C23116yes—
93#4086C23148yes—
94#4084C23148yes—
95#4076C23148yes—
96#4076C23148yes—
97#4076C23124yes—
98#4075C23148yes—
99C4×C22⋊D5C23148yes—
100C2×#685C2316yes—
101C2×#685C2316yes—
102C2×#685C2316yes—
103C2×#685C2316yes—
104C2×C4⋊Dic5C2311yescharacteristic
105C2×C4⋊Dic5C2316yes—
106C2×C4⋊Dic5C2316yes—
107C2×C4⋊Dic5C2311yescharacteristic
108#4070C23124yes—
109C2×#683C2316yes—
110C2×#683C2312yes—
111C2×#683C2312yes—
112C2×#683C2316yes—
113C2×C4×Dic5C2316yes—
114#4041C23148yes—
115#4039C23148yes—
116D5×C4⋊C4C23148yes—
117#4036C23148yes—
118#4030C23196yes—
119#4029C23196yes—
120#4022C23148yes—
121#4021C23124yes—
122#4020C23124yes—
123C4×D20C23148yes—
124C4×D20C23148yes—
125C4×D20C23124yes—
126#4018C23148yes—
127#4016C2318yes—
128#4015C23124yes—
129C4×Dic10C23124yes—
130#3517—532no—
131C2×#557—51no—
132C2×#553—56no—
133C2×#550—52no—
134C2×#550—51no—
135C2×#550—53no—
136C2×#547—53no—
137C2×#540—56no—
138C2×#532—53no—
139C2×#529—53no—
140C2×#520—52no—
141C2×C4×#41D511yescharacteristic, complemented
142C24×C5C2413yes—
143C23×D5C2416yes—
144C5×#41—232no—
145C2×C5×Q8#4118yescomplemented
146C2×C5×Q8C2414yes—
147C2×C5×D4—212no—
148C2×C5×D4C24112yes—
149C22×C4×C5—212no—
150C22×C4×C5—212no—
151C22×C4×C5—46no—
152C22×C4×C5C2411yescharacteristic
153C22×C4×C5C2411yescharacteristic
154C22×C4×C5C2413yes—
155C22×C4×C5C2413yes—
156C22×C4×C5C24112yes—
157C22×C4×C5C24112yes—
158C22×C4×C5C24112yes—
159C22×C4×C5C2413yes—
160C22×C4×C5C2411yescharacteristic
161C2×C22⋊D5—424no—
162C2×C22⋊D5—412no—
163C2×C22⋊D5C24148yes—
164C2×C22⋊D5C24124yes—
165C22×Dic5C2412yes—
166C22×Dic5C2416yes—
167C22×Dic5C2412yes—
168#713—264no—
169#709—496no—
170C2×D20—224no—
171C2×D20—212no—
172C2×D20—412no—
173C2×D20C24112yes—
174C2×D20C24124yes—
175C2×D20C24124yes—
176C2×C4×D5—224no—
177C2×C4×D5—424no—
178C2×C4×D5C24124yes—
179C2×C4×D5C24148yes—
180C2×Dic10—212no—
181C2×Dic10C24112yes—
182C5×C4⋊C4C24112yes—
183C5×C4⋊C4C24112yes—
184C5×C22⋊C4C24124yes—
185C42×C5C2414yes—
186C42×C5C24112yes—
187C42×C5C24112yes—
188C42×C5C24112yes—
189C22⋊Dic5C24124yes—
190#685C24124yes—
191#685C24124yes—
192#685C24124yes—
193#685C24124yes—
194C4⋊Dic5C2414yes—
195C4⋊Dic5C24124yes—
196C4⋊Dic5C24124yes—
197C4⋊Dic5C2414yes—
198#683C24124yes—
199#683C2418yes—
200#683C2418yes—
201#683C24124yes—
202C4×Dic5C24124yes—
203C22×#41—52no—
204C22×#41—56no—
205C22×#41C2×D511yescharacteristic
206#557—516no—
207#553—596no—
208#550—532no—
209#550—516no—
210#550—548no—
211#547—548no—
212#540—596no—
213#532—548no—
214C2×C4⋊Q8—53no—
215C2×#127—56no—
216C2×#127—53no—
217#529—548no—
218C2×#126—56no—
219C2×#125—56no—
220C2×#125—51no—
221C2×#124—53no—
222C2×#123—56no—
223C2×#123—512no—
224C2×C22⋊Q8—53no—
225C2×#121—56no—
226C2×#121—53no—
227C2×#121—56no—
228C2×#121—56no—
229C2×#121—56no—
230C2×#121—512no—
231#520—532no—
232C4×#41C2×D5116yescomplemented
233C2×C4×Q8—52no—
234C2×C4×Q8—53no—
235C2×C4×Q8C2×D511yescharacteristic
236C2×C4×Q8C2×D511yescharacteristic
237C2×C4×D4—56no—
238C2×C4×D4—56no—
239C2×C4×D4—56no—
240C2×C4×D4—53no—
241C2×C4×D4—56no—
242C2×C4×D4C2×D513yes—
243C2×C4×D4C2×D513yes—
244C2×#117—56no—
245C2×#117C2×D513yes—
246C22×C4⋊C4—56no—
247C22×C42C2×D513yes—
248C23×C5—26no—
249C23×C5—26no—
250C23×C5—26no—
251C23×C5C2516yes—
252C23×C5C2511yescharacteristic
253C22×D5—212no—
254C22×D5—212no—
255C22×D5—212no—
256C22×D5C25112yes—
257C5×Q8C2×#41116yescomplemented
258C5×D4—224no—
259C2×C4×C5—22no—
260C2×C4×C5—26no—
261C2×C4×C5—26no—
262C2×C4×C5—22no—
263C2×C4×C5—224no—
264C2×C4×C5—224no—
265C2×C4×C5—412no—
266C2×C4×C5C2×#41112yes—
267C2×C4×C5H(5,𝔽2)14yes—
268C2×C4×C5C2516yes—
269C2×C4×C5C2512yes—
270C2×C4×C5C2512yes—
271C2×C4×C5C2516yes—
272C2×C4×C5C2516yes—
273C2×C4×C5C2512yes—
274C22⋊D5—448no—
275C22⋊D5—424no—
276C2×Dic5—24no—
277C2×Dic5—212no—
278C2×Dic5—24no—
279C2×Dic5C2514yes—
280C2×Dic5C25112yes—
281C2×Dic5C2514yes—
282D20—248no—
283D20—224no—
284D20—424no—
285C4×D5—248no—
286C4×D5—448no—
287Dic10—224no—
288C2×#41—28no—
289C2×#41—516no—
290C2×#41—548no—
291C2×#41—1016no—
292C2×#41—2024no—
293C2×#41C22×D518yes—
294C22×Q8—53no—
295C22×Q8C22×D511yescharacteristic
296C22×D4—56no—
297C22×D4—512no—
298C22×D4—56no—
299C22×D4—53no—
300C22×D4—56no—
301C22×D4C22×D513yes—
302C23×C4—56no—
303C23×C4—512no—
304C23×C4C22×D513yes—
305C23×C4C22×D513yes—
306C23×C4C22×D513yes—
307C4⋊Q8—524no—
308#127—548no—
309#127—524no—
310#126—548no—
311#125—548no—
312#125—58no—
313#124—524no—
314#123—548no—
315#123—596no—
316C22⋊Q8—524no—
317#121—548no—
318#121—524no—
319#121—548no—
320#121—548no—
321#121—548no—
322#121—596no—
323C4×Q8—516no—
324C4×Q8—524no—
325C4×Q8C22×D518yes—
326C4×Q8C22×D518yes—
327C4×D4—548no—
328C4×D4—548no—
329C4×D4—548no—
330C4×D4—524no—
331C4×D4—548no—
332C4×D4C22×D5124yes—
333C4×D4C22×D5124yes—
334#117—548no—
335#117C22×D5124yes—
336C2×C4⋊C4—56no—
337C2×C4⋊C4—56no—
338C2×C4⋊C4—52no—
339C2×C4⋊C4—51no—
340C2×C4⋊C4—52no—
341C2×C4⋊C4—56no—
342C2×C4⋊C4—51no—
343C2×C4⋊C4—56no—
344C2×C4⋊C4—548no—
345C2×C4⋊C4C22×D513yes—
346C2×C4⋊C4C22×D513yes—
347C2×C22⋊C4—56no—
348C2×C22⋊C4—56no—
349C2×C22⋊C4—56no—
350C2×C22⋊C4—56no—
351C2×C22⋊C4—56no—
352C2×C22⋊C4C22×D516yes—
353C2×C42—56no—
354C2×C42C22×D513yes—
355C2×C42C22×D513yes—
356C2×C42C22×D513yes—
357C2×C42C22×D5124yes—
358C2×C42C22×D511yescharacteristic
359C22×C5—26no—
360C22×C5—26no—
361C22×C5—26no—
362C22×C5—412no—
363C22×C5C22×#4112yes—
364C22×C5C2×H(5,𝔽2)12yes—
365C22×C5#58512yes—
366C22×C5C2611yescharacteristic
367C2×D5—212no—
368C2×D5—212no—
369C2×D5—212no—
370C2×D5—424no—
371C4×C5—22no—
372C4×C5—26no—
373C4×C5—26no—
374C4×C5—22no—
375C4×C5C22×#41112yes—
376C4×C5C2×H(5,𝔽2)14yes—
377Dic5—24no—
378Dic5—212no—
379Dic5—24no—
380C24—56no—
381C24C23×D513yes—
382#41—232no—
383#41—1064no—
384#41—2096no—
385C2×Q8—512no—
386C2×Q8—1012no—
387C2×Q8#70918yescomplemented
388C2×Q8C23×D514yes—
389C2×D4—212no—
390C2×D4—524no—
391C2×D4—548no—
392C2×D4—524no—
393C2×D4—512no—
394C2×D4—524no—
395C2×D4—1024no—
396C2×D4—1012no—
397C2×D4—2012no—
398C2×D4—2024no—
399C2×D4—2012no—
400C2×D4C23×D5112yes—
401C22×C4—212no—
402C22×C4—212no—
403C22×C4—46no—
404C22×C4—52no—
405C22×C4—56no—
406C22×C4—52no—
407C22×C4—524no—
408C22×C4—548no—
409C22×C4—1024no—
410C22×C4—2024no—
411C22×C4C23×D5112yes—
412C22×C4C23×D5112yes—
413C22×C4C23×D511yescharacteristic
414C22×C4C23×D511yescharacteristic
415C22×C4C23×D513yes—
416C22×C4C23×D513yes—
417C22×C4C23×D5112yes—
418C22×C4C23×D513yes—
419C22×C4C23×D511yescharacteristic
420C4⋊C4—524no—
421C4⋊C4—524no—
422C4⋊C4—58no—
423C4⋊C4—54no—
424C4⋊C4—58no—
425C4⋊C4—524no—
426C4⋊C4—54no—
427C4⋊C4—524no—
428C4⋊C4C23×D5112yes—
429C4⋊C4C23×D5112yes—
430C22⋊C4—524no—
431C22⋊C4—524no—
432C22⋊C4—524no—
433C22⋊C4—524no—
434C22⋊C4—524no—
435C22⋊C4C23×D5124yes—
436C42—524no—
437C42C23×D514yes—
438C42C23×D5112yes—
439C42C23×D5112yes—
440C42C23×D5112yes—
441C2×C5—46no—
442C2×C5#351714yescomplemented
443C2×C5C23×#4111yescharacteristic
444C2×C5C22×H(5,𝔽2)11yescharacteristic
445C2×C5C2×#58511yescharacteristic
446D5—412no—
447C23—26no—
448C23—26no—
449C23—26no—
450C23—512no—
451C23—1012no—
452C23—1012no—
453C23—1012no—
454C23C24×D516yes—
455C23C24×D511yescharacteristic
456Q8—1024no—
457Q8C2×#709116yescomplemented
458D4—224no—
459D4—1048no—
460D4—1024no—
461D4—2024no—
462D4—2048no—
463D4—2024no—
464C2×C4—22no—
465C2×C4—26no—
466C2×C4—22no—
467C2×C4—26no—
468C2×C4—224no—
469C2×C4—224no—
470C2×C4—412no—
471C2×C4—54no—
472C2×C4—512no—
473C2×C4—54no—
474C2×C4—104no—
475C2×C4—1012no—
476C2×C4—104no—
477C2×C4—1048no—
478C2×C4—2048no—
479C2×C4C2×#709112yes—
480C2×C4#414914yes—
481C2×C4C24×D512yes—
482C2×C4C24×D512yes—
483C2×C4C24×D516yes—
484C2×C4C24×D516yes—
485C2×C4C24×D516yes—
486C2×C4C24×D512yes—
487C5C2×#351711yescharacteristic, complemented, Sylow
488C22—26no—
489C22—26no—
490C22—26no—
491C22—412no—
492C22—1012no—
493C22—1012no—
494C22—1012no—
495C22—2024no—
496C22C22×#70912yes—
497C22C2×#414912yes—
498C22#6628312yes—
499C22C25×D511yescharacteristic
500C4—22no—
501C4—26no—
502C4—22no—
503C4—26no—
504C4—104no—
505C4—1012no—
506C4—104no—
507C4C22×#709112yes—
508C4C2×#414914yes—
509C2—46no—
510C2—2012no—
511C2#1061775414yesdirect product factor
512C2C23×#70911yescharacteristic
513C2C22×#414911yescharacteristic
514C2C2×#6628311yescharacteristic
subgroupG/H
centerC23C24×D5
commutator, cosocleC22×C5C26
FrattiniC22C25×D5
FittingC2×C4×C5×#41C2
socleC23×C5C25
hypercenterC2×C4×#41D5
hypocenterC5C2×#3517
LCSG ▷ C22×C5 ▷ C5
UCSC1 ◁ C23 ◁ C2×C4×#41

Complements

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

NH ≅ G/Ntype
1. C2×C4×C5×#41C2semidirect
2. C2×#66168C2semidirect
3. #10617754C2direct (N × H)
4. C2×#65904C2semidirect
5. C2×#65911C2semidirect
6. C2×#65908C2semidirect
7. C2×#65885C2semidirect
8. C2×#65896C2semidirect
9. C2×#65887C2semidirect
10. C2×#65864C2semidirect
11. C2×#65811C2semidirect
12. C2×#65813C2semidirect
16. C4×C5×#41C22semidirect
23. #66168C22semidirect
33. #65911C22semidirect
34. #65908C22semidirect
35. #65904C22semidirect
36. #65896C22semidirect
37. #65887C22semidirect
38. #65885C22semidirect
39. #65864C22semidirect
46. #65813C22semidirect
48. #65811C22semidirect
141. C2×C4×#41D5semidirect
145. C2×C5×Q8#41semidirect
232. C4×#41C2×D5semidirect
257. C5×Q8C2×#41semidirect
387. C2×Q8#709semidirect
442. C2×C5#3517semidirect
457. Q8C2×#709semidirect
487. C5C2×#3517semidirect
511. C2#10617754direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C4×C5×#412
C2×#661682
#106177542
C2×#659042
C2×#659112
C2×#659082
C2×#658852
C2×#658962
C2×#658872
C2×#658642
C2×#658112
C2×#658132
C2×#35175