Finite Group Databasebeta

← C4×#65928 #12262005

Subgroups — C4×#65928 #12262005

Order 1280.

2664 conjugacy classes of subgroups.

The subgroup lattice has 460 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#65928C211yesmaximal, characteristic
2C4×C5×#120C211yesmaximal, characteristic, complemented
3C4×#4099C212yesmaximal, complemented
4#10614311C211yesmaximal, characteristic, complemented
5C4×#4083C213yesmaximal, complemented
6C4×#4082C213yesmaximal, complemented
7#10606613C213yesmaximal, complemented
8#10606415C213yesmaximal, complemented
9#10606133C216yesmaximal, complemented
10#10605582C212yesmaximal, complemented
11C4×#4031C216yesmaximal, complemented
12C24×C4×C5C2211yescharacteristic, complemented
13C23×C4×D5C2211yescharacteristic, complemented
14C2×C5×#120C2211yescharacteristic
15C2×C4×C5×D4C2213yes—
16C2×#4099C2212yes—
17C2×#4083C2213yes—
18C2×#4082C2213yes—
19C2×C4×C22⋊D5C2216yes—
20#65928C4116yesdirect product factor
21C2×#4031C2216yes—
22C2×C4×Dic10C2213yescomplemented
23C5×#386C2213yes—
24C5×#385C2213yescomplemented
25C5×#379C2211yescharacteristic, complemented
26C4×C5×C22⋊C4C2213yescomplemented
27#65536C2212yescomplemented
28#65510C2213yes—
29#65509C2213yes—
30#65502C2216yes—
31#65499C2216yescomplemented
32C4×C22⋊Dic5C2213yescomplemented
33C4×C22⋊Dic5C2216yescomplemented
34#65278C2216yescomplemented
35#65266C2213yescomplemented
36#65263C2213yescomplemented
37#65246C2216yes—
38#65245C2216yes—
39#65244C2213yescomplemented
40#65240C2216yescomplemented
41#65235C2213yescomplemented
42#65234C2216yescomplemented
43#65234C2216yescomplemented
44#65233C2216yescomplemented
45C4×#685C2216yescomplemented
46C42×Dic5C2213yescomplemented
47#64959C2216yescomplemented
48#64958C2216yescomplemented
49#64939C2216yescomplemented
50#64935C2216yescomplemented
51C4×H(3,𝔽4)—51nomaximal, Sylow
52C25×C5C2311yescharacteristic
53C24×D5C2311yescharacteristic
54C22×C5×D4C2313yes—
55C23×C4×C5—46no—
56C23×C4×C5C2313yes—
57C23×C4×C5C2311yescharacteristic
58C23×C4×C5C2313yes—
59C22×C22⋊D5C2316yes—
60C23×Dic5C2311yescharacteristic
61C22×C4×D5—46no—
62C22×C4×D5C2312yes—
63C22×C4×D5C2312yes—
64C22×Dic10C2313yes—
65C5×#120C2×C418yescomplemented
66C4×C5×D4—46no—
67C2×C5×C4⋊C4C2313yes—
68C2×C5×C22⋊C4C2313yes—
69C2×C5×C22⋊C4C2313yes—
70C2×C5×C22⋊C4C2313yes—
71C2×C5×C22⋊C4C2313yes—
72C2×C42×C5C2313yes—
73#4099C2×C4116yescomplemented
74C2×C22⋊Dic5C2316yes—
75C2×C22⋊Dic5C2316yes—
76C2×C22⋊Dic5C2313yes—
77C2×C22⋊Dic5C2316yes—
78C2×C22⋊Dic5C2313yes—
79#4083C2×C4124yescomplemented
80#4082C2×C4124yescomplemented
81C4×C22⋊D5—412no—
82C2×#685C2316yes—
83C2×#685C2316yes—
84C2×#685C2316yes—
85C2×C4⋊Dic5C2313yes—
86C2×#683C2316yes—
87C2×#683C2316yes—
88C2×C4×Dic5C2316yes—
89C2×C4×Dic5C2316yes—
90C2×C4×Dic5C2313yes—
91C2×C4×Dic5C2313yes—
92#4031C2×C4148yescomplemented
93C4×Dic10—46no—
94C5×#95C2316yes—
95#3963C2316yes—
96#3963C2316yes—
97#3963C2316yes—
98#3963C2316yes—
99#3963C2316yes—
100C2×H(3,𝔽4)—51no—
101#2379—53no—
102#2379—56no—
103#2375—53no—
104#2375—52no—
105#2375—51no—
106C4×#124—56no—
107C4×#124—53no—
108C4×#120—52no—
109C4×#120—53no—
110C4×#120D511yescharacteristic, complemented
111C24×C5—43no—
112C24×C5—43no—
113C24×C5C22×C414yescomplemented
114C24×C5C2411yescharacteristic
115C24×C5C2413yes—
116C23×D5—43no—
117C23×D5—43no—
118C23×D5C22×C414yescomplemented
119C23×D5C2412yes—
120C2×C5×D4—43no—
121C2×C5×D4—43no—
122C2×C5×D4C22×C4112yes—
123C22×C4×C5—26no—
124C22×C4×C5—26no—
125C22×C4×C5—212no—
126C22×C4×C5—43no—
127C22×C4×C5—46no—
128C22×C4×C5—416no—
129C22×C4×C5C2413yes—
130C22×C4×C5C2411yescharacteristic
131C22×C4×C5C2413yes—
132C22×C4×C5C2413yes—
133C22×C4×C5C2411yescharacteristic
134C2×C22⋊D5—46no—
135C2×C22⋊D5—46no—
136C2×C22⋊D5C22×C4124yes—
137C22×Dic5—43no—
138C22×Dic5C2416yes—
139C22×Dic5C2412yes—
140C22×Dic5C2416yes—
141C2×C4×D5—212no—
142C2×C4×D5—46no—
143C2×C4×D5—416no—
144C2×Dic10—43no—
145C2×Dic10—43no—
146C2×Dic10C22×C4112yescomplemented
147C5×C4⋊C4—43no—
148C5×C22⋊C4—43no—
149C5×C22⋊C4—43no—
150C5×C22⋊C4—43no—
151C5×C22⋊C4C22×C4112yescomplemented
152C42×C5—26no—
153C22⋊Dic5—46no—
154C22⋊Dic5—43no—
155C22⋊Dic5—46no—
156C22⋊Dic5C22×C4124yescomplemented
157C22⋊Dic5C22×C4112yescomplemented
158#685—46no—
159#685—46no—
160#685C22×C4124yescomplemented
161C4⋊Dic5—43no—
162#683—46no—
163#683—46no—
164C4×Dic5—212no—
165C4×Dic5—46no—
166C4×Dic5—43no—
167C4×Dic5C22×C4112yescomplemented
168C24×C4—51no—
169C24×C4C2×D511yescharacteristic, complemented
170H(3,𝔽4)—516no—
171C2×#124—53no—
172C2×#124—56no—
173C2×#120—52no—
174C2×#120—53no—
175C2×#120C2×D511yescharacteristic
176C2×C4×Q8—53no—
177C2×C4×D4—56no—
178C2×C4×D4C2×D513yes—
179#391—53no—
180#391—56no—
181#390—53no—
182#390—56no—
183#388—56no—
184#388—56no—
185#388—56no—
186#388—56no—
187#388—56no—
188#388—56no—
189#386—53no—
190#386—56no—
191#386—56no—
192#386C2×D513yes—
193#385—56no—
194#385—56no—
195#385—53no—
196#385C2×D513yescomplemented
197#382—53no—
198#382—56no—
199#379—52no—
200#379—53no—
201#379C2×D511yescharacteristic, complemented
202C4×C22⋊C4—56no—
203C4×C22⋊C4—53no—
204C4×C22⋊C4—56no—
205C4×C22⋊C4C2×D513yescomplemented
206C43—53no—
207C23×C5—26no—
208C23×C5—23no—
209C23×C5—23no—
210C23×C5—26no—
211C23×C5—23no—
212C23×C5—23no—
213C23×C5—412no—
214C23×C5—44no—
215C23×C5—412no—
216C23×C5C23×C412yes—
217C23×C5C23×C416yes—
218C23×C5C2511yescharacteristic
219C22×D5—26no—
220C22×D5—26no—
221C22×D5—412no—
222C22×D5—44no—
223C22×D5—412no—
224C22×D5C23×C414yes—
225C5×D4—412no—
226C2×C4×C5—23no—
227C2×C4×C5—23no—
228C2×C4×C5—23no—
229C2×C4×C5—26no—
230C2×C4×C5—23no—
231C2×C4×C5—23no—
232C2×C4×C5—26no—
233C2×C4×C5—412no—
234C2×C4×C5—44no—
235C2×C4×C5—412no—
236C2×C4×C5C23×C416yes—
237C2×C4×C5H(5,𝔽2)16yes—
238C22⋊D5—424no—
239C2×Dic5—26no—
240C2×Dic5—26no—
241C2×Dic5—26no—
242C2×Dic5—26no—
243C2×Dic5—26no—
244C2×Dic5—26no—
245C2×Dic5—44no—
246C2×Dic5C23×C4112yes—
247C4×D5—48no—
248C4×D5—48no—
249Dic10—412no—
250C25—51no—
251C25C22×D511yescharacteristic
252C22×Q8—53no—
253C22×D4—56no—
254C22×D4C22×D513yes—
255C23×C4—46no—
256C23×C4—51no—
257C23×C4—52no—
258C23×C4—52no—
259C23×C4—206no—
260C23×C4C22×D513yes—
261C23×C4C22×D511yescharacteristic
262C23×C4C22×D513yes—
263#124—524no—
264#124—548no—
265#120—516no—
266#120—524no—
267#120C4×D518yescomplemented
268C4×Q8—206no—
269C4×D4—46no—
270C4×D4—2012no—
271C2×C4⋊C4—56no—
272C2×C4⋊C4—56no—
273C2×C4⋊C4—53no—
274C2×C4⋊C4C22×D513yes—
275C2×C22⋊C4—53no—
276C2×C22⋊C4—56no—
277C2×C22⋊C4—56no—
278C2×C22⋊C4—56no—
279C2×C22⋊C4—53no—
280C2×C22⋊C4—56no—
281C2×C22⋊C4—56no—
282C2×C22⋊C4—56no—
283C2×C22⋊C4C22×D513yes—
284C2×C22⋊C4C22×D513yes—
285C2×C22⋊C4C22×D513yes—
286C2×C22⋊C4C22×D513yes—
287C2×C42—53no—
288C2×C42—56no—
289C2×C42—56no—
290C2×C42—53no—
291C2×C42C22×D513yes—
292#95—56no—
293#95—56no—
294#95—56no—
295#95—56no—
296#95—56no—
297#95C22×D516yes—
298C22×C5—212no—
299C22×C5—26no—
300C22×C5—26no—
301C22×C5—416no—
302C22×C5—41no—
303C22×C5—43no—
304C22×C5—43no—
305C22×C5—46no—
306C22×C5—43no—
307C22×C5C24×C411yescharacteristic
308C22×C5C2×H(5,𝔽2)13yes—
309C22×C5#58513yes—
310C2×D5—212no—
311C2×D5—416no—
312C2×D5—42no—
313C2×D5—46no—
314C4×C5—26no—
315C4×C5—41no—
316C4×C5—43no—
317C4×C5—43no—
318C4×C5H(3,𝔽4)14yescomplemented
319Dic5—212no—
320Dic5—42no—
321Dic5—46no—
322C24—43no—
323C24—43no—
324C24—52no—
325C24—54no—
326C24—203no—
327C24—203no—
328C24C2×C4×D514yescomplemented
329C24C23×D511yescharacteristic
330C24C23×D513yes—
331C2×Q8—512no—
332C2×Q8—203no—
333C2×Q8—203no—
334C2×D4—43no—
335C2×D4—43no—
336C2×D4—524no—
337C2×D4—206no—
338C2×D4—206no—
339C2×D4C2×C4×D5112yes—
340C22×C4—26no—
341C22×C4—26no—
342C22×C4—212no—
343C22×C4—46no—
344C22×C4—43no—
345C22×C4—416no—
346C22×C4—56no—
347C22×C4—52no—
348C22×C4—56no—
349C22×C4—1012no—
350C22×C4—206no—
351C22×C4—2016no—
352C22×C4—203no—
353C22×C4C23×D511yescharacteristic
354C22×C4C23×D513yes—
355C22×C4C23×D513yes—
356C22×C4C23×D513yes—
357C22×C4C23×D511yescharacteristic
358C4⋊C4—43no—
359C4⋊C4—206no—
360C4⋊C4—206no—
361C4⋊C4—203no—
362C22⋊C4—43no—
363C22⋊C4—43no—
364C22⋊C4—43no—
365C22⋊C4—512no—
366C22⋊C4—524no—
367C22⋊C4—524no—
368C22⋊C4—206no—
369C22⋊C4—206no—
370C22⋊C4—206no—
371C22⋊C4—203no—
372C22⋊C4—206no—
373C22⋊C4C2×C4×D5112yescomplemented
374C42—26no—
375C42—512no—
376C42—1012no—
377C42—203no—
378C42—206no—
379C2×C5—42no—
380C2×C5—46no—
381C2×C5C4×H(5,𝔽2)13yes—
382C2×C5C2×H(3,𝔽4)11yescharacteristic
383C2×C5#352913yes—
384D5—44no—
385C23—23no—
386C23—23no—
387C23—26no—
388C23—23no—
389C23—23no—
390C23—26no—
391C23—412no—
392C23—44no—
393C23—412no—
394C23—54no—
395C23—106no—
396C23—106no—
397C23—2012no—
398C23—204no—
399C23—2012no—
400C23C22×C4×D512yes—
401C23C22×C4×D516yes—
402C23C24×D511yescharacteristic
403Q8—2012no—
404D4—412no—
405D4—2024no—
406C2×C4—23no—
407C2×C4—23no—
408C2×C4—23no—
409C2×C4—23no—
410C2×C4—23no—
411C2×C4—26no—
412C2×C4—26no—
413C2×C4—412no—
414C2×C4—412no—
415C2×C4—44no—
416C2×C4—512no—
417C2×C4—106no—
418C2×C4—106no—
419C2×C4—106no—
420C2×C4—106no—
421C2×C4—106no—
422C2×C4—106no—
423C2×C4—208no—
424C2×C4—204no—
425C2×C4—208no—
426C2×C4C22×C4×D516yes—
427C2×C4#414416yes—
428C5C4×H(3,𝔽4)11yescharacteristic, complemented, Sylow
429C22—26no—
430C22—26no—
431C22—212no—
432C22—416no—
433C22—43no—
434C22—46no—
435C22—43no—
436C22—43no—
437C22—41no—
438C22—1012no—
439C22—2016no—
440C22—206no—
441C22—202no—
442C22C23×C4×D511yescharacteristic
443C22C2×#414413yes—
444C22#6628313yes—
445C4—26no—
446C4—43no—
447C4—43no—
448C4—41no—
449C4—1012no—
450C4—206no—
451C4—202no—
452C4#6592814yesdirect product factor
453C2—46no—
454C2—42no—
455C2—204no—
456C2C4×#414413yes—
457C2C2×#6592811yescharacteristic
458C2#1061794913yes—
subgroupG/H
centerC22×C4C23×D5
commutatorC22×C5C24×C4
FrattiniC23C24×D5
FittingC4×C5×#120C2
socle, cosocleC23×C5C25
prime radicalC2×#65928C2
hypercenterC4×#120D5
hypocenterC5C4×H(3,𝔽4)
LCSG ▷ C22×C5 ▷ C5
UCSC1 ◁ C22×C4 ◁ C4×#120

Complements

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

NH ≅ G/Ntype
2. C4×C5×#120C2semidirect
3. C4×#4099C2semidirect
4. #10614311C2semidirect
5. C4×#4083C2semidirect
6. C4×#4082C2semidirect
7. #10606613C2semidirect
8. #10606415C2semidirect
9. #10606133C2semidirect
10. #10605582C2semidirect
11. C4×#4031C2semidirect
12. C24×C4×C5C22semidirect
13. C23×C4×D5C22semidirect
20. #65928C4direct (N × H)
22. C2×C4×Dic10C22semidirect
24. C5×#385C22semidirect
25. C5×#379C22semidirect
26. C4×C5×C22⋊C4C22semidirect
27. #65536C22semidirect
31. #65499C22semidirect
32. C4×C22⋊Dic5C22semidirect
33. C4×C22⋊Dic5C22semidirect
34. #65278C22semidirect
35. #65266C22semidirect
36. #65263C22semidirect
39. #65244C22semidirect
40. #65240C22semidirect
41. #65235C22semidirect
42. #65234C22semidirect
43. #65234C22semidirect
44. #65233C22semidirect
45. C4×#685C22semidirect
46. C42×Dic5C22semidirect
47. #64959C22semidirect
48. #64958C22semidirect
49. #64939C22semidirect
50. #64935C22semidirect
65. C5×#120C2×C4semidirect
73. #4099C2×C4semidirect
79. #4083C2×C4semidirect
80. #4082C2×C4semidirect
92. #4031C2×C4semidirect
110. C4×#120D5semidirect
113. C24×C5C22×C4semidirect
118. C23×D5C22×C4semidirect
146. C2×Dic10C22×C4semidirect
151. C5×C22⋊C4C22×C4semidirect
156. C22⋊Dic5C22×C4semidirect
157. C22⋊Dic5C22×C4semidirect
160. #685C22×C4semidirect
167. C4×Dic5C22×C4semidirect
169. C24×C4C2×D5semidirect
196. #385C2×D5semidirect
201. #379C2×D5semidirect
205. C4×C22⋊C4C2×D5semidirect
267. #120C4×D5semidirect
318. C4×C5H(3,𝔽4)semidirect
328. C24C2×C4×D5semidirect
373. C22⋊C4C2×C4×D5semidirect
428. C5C4×H(3,𝔽4)semidirect
452. C4#65928direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#659282
C4×C5×#1202
C4×#40992
#106143112
C4×#40832
C4×#40822
#106066132
#106064152
#106061332
#106055822
C4×#40312
C4×H(3,𝔽4)5