Finite Group Databasebeta

← C2×#10605591 #12305418

Subgroups — C2×#10605591 #12305418

Order 1280.

4438 conjugacy classes of subgroups; only the 667 normal subgroups are shown.

normal subgroups only

This group has 4438 conjugacy classes of subgroups — too many to draw. Only its 667 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#4099C212yesmaximal, complemented
2C2×C5×#379C211yesmaximal, characteristic, complemented
3C2×#65249C213yesmaximal, complemented
4C2×#65242C213yesmaximal, complemented
5C2×#65250C216yesmaximal, complemented
6#10605591C2116yesmaximal, direct product factor
7C26×C5C2211yescharacteristic
8C23×C22⋊D5C2216yes—
9C23×D20C2211yescharacteristic, complemented
10C22×C5×C22⋊C4C2213yes—
11C22×C5×C22⋊C4C2213yes—
12C2×#4099—216no—
13C2×#4099C22116yescomplemented
14C22×C22⋊Dic5C2216yes—
15C22×#685C2216yescomplemented
16C22×C4⋊Dic5C2213yescomplemented
17C5×#379C2218yescomplemented
18C2×#3963C2213yescomplemented
19C2×#3963C2213yescomplemented
20#65250C22148yescomplemented
21#65249C22124yescomplemented
22#65242C22124yescomplemented
23C2×#2812—51nomaximal, Sylow
24C25×C5—46no—
25C25×C5—46no—
26C25×C5D418yescomplemented
27C25×C5C2313yes—
28C25×C5C2314yes—
29C24×D5C2312yes—
30C23×C4×C5C2313yes—
31C23×C4×C5C2311yescharacteristic
32C22×C22⋊D5—224no—
33C22×C22⋊D5—412no—
34C22×C22⋊D5—412no—
35C22×C22⋊D5C23124yes—
36C23×Dic5C2316yes—
37C22×D20—46no—
38C22×D20C2314yescomplemented
39C2×C5×C22⋊C4—46no—
40C2×C5×C22⋊C4—46no—
41C2×C5×C22⋊C4C23112yes—
42C2×C5×C22⋊C4C23112yes—
43#4099—264no—
44C2×C22⋊Dic5—224no—
45C2×C22⋊Dic5C23124yes—
46C2×#685—412no—
47C2×#685C23124yescomplemented
48C2×C4⋊Dic5—46no—
49C2×C4⋊Dic5C23112yescomplemented
50#3963C23112yescomplemented
51#3963C23112yescomplemented
52C22×#120—52no—
53#2812—516no—
54C2×#397—53no—
55C2×#394—56no—
56C2×#392—53no—
57C2×#379D511yescharacteristic, complemented
58C24×C5—26no—
59C24×C5—26no—
60C24×C5—212no—
61C24×C5—212no—
62C24×C5—448no—
63C24×C5—412no—
64C24×C5—412no—
65C24×C5—448no—
66C24×C5—416no—
67C24×C5C2×D4112yes—
68C24×C5C2×D4116yescomplemented
69C24×C5C2416yes—
70C24×C5C2411yescharacteristic
71C23×D5—24no—
72C23×D5—212no—
73C23×D5—212no—
74C23×D5C2414yes—
75C22×C4×C5—26no—
76C22×C4×C5—26no—
77C22×C4×C5—212no—
78C22×C4×C5C2412yes—
79C22×C4×C5C2416yes—
80C2×C22⋊D5—248no—
81C2×C22⋊D5—424no—
82C2×C22⋊D5—424no—
83C2×C22⋊D5—496no—
84C22×Dic5—212no—
85C22×Dic5—212no—
86C22×Dic5—212no—
87C22×Dic5C24112yes—
88C2×D20—412no—
89C2×D20—416no—
90C5×C22⋊C4—412no—
91C5×C22⋊C4—412no—
92C22⋊Dic5—248no—
93#685—424no—
94C4⋊Dic5—412no—
95C26C2×D511yescharacteristic
96C23×D4—56no—
97C23×D4—51no—
98C2×#120—516no—
99C2×#120—1016no—
100C22×C4⋊C4—53no—
101C22×C22⋊C4—56no—
102C22×C22⋊C4—56no—
103C22×C22⋊C4C2×D513yes—
104C22×C22⋊C4C2×D513yes—
105#397—524no—
106#394—548no—
107#392—524no—
108#379C2×D518yescomplemented
109C2×#95—53no—
110C2×#95—53no—
111C23×C5—248no—
112C23×C5—224no—
113C23×C5—26no—
114C23×C5—26no—
115C23×C5—212no—
116C23×C5—212no—
117C23×C5—448no—
118C23×C5—448no—
119C23×C5—416no—
120C23×C5—4128no—
121C23×C5—424no—
122C23×C5—412no—
123C23×C5—412no—
124C23×C5C22×D4112yes—
125C23×C5C22×D412yes—
126C23×C5H(5,𝔽2)16yes—
127C23×C5H(5,𝔽2)16yes—
128C23×C5C2511yescharacteristic
129C22×D5—24no—
130C22×D5—212no—
131C22×D5—212no—
132C22×D5—424no—
133C22×D5—424no—
134C22×D5—48no—
135C2×C4×C5—26no—
136C2×C4×C5—212no—
137C2×C4×C5—26no—
138C2×C4×C5—412no—
139C2×C4×C5—44no—
140C22⋊D5—496no—
141C2×Dic5—212no—
142C2×Dic5—212no—
143C2×Dic5—212no—
144C2×Dic5—424no—
145D20—416no—
146C25—46no—
147C25—46no—
148C25—52no—
149C25D2018yescomplemented
150C25C22×D513yes—
151C25C22×D514yes—
152C22×D4—524no—
153C22×D4—54no—
154C22×D4—1024no—
155C22×D4—206no—
156C22×D4—2012no—
157C22×D4—2012no—
158C23×C4—56no—
159C23×C4C22×D513yes—
160C23×C4C22×D511yescharacteristic
161#120—1064no—
162C2×C4⋊C4—512no—
163C2×C4⋊C4—206no—
164C2×C22⋊C4—46no—
165C2×C22⋊C4—46no—
166C2×C22⋊C4—524no—
167C2×C22⋊C4—524no—
168C2×C22⋊C4—1024no—
169C2×C22⋊C4—2012no—
170C2×C22⋊C4C22×D5112yes—
171C2×C22⋊C4C22×D5112yes—
172#95—512no—
173#95—512no—
174C22×C5—224no—
175C22×C5—212no—
176C22×C5—464no—
177C22×C5—412no—
178C22×C5—46no—
179C22×C5—46no—
180C22×C5—448no—
181C22×C5#534112yes—
182C22×C5#560112yes—
183C22×C5H(3,𝔽4)14yes—
184C22×C5C23×D411yescharacteristic
185C22×C5C2×H(5,𝔽2)13yes—
186C22×C5C2×H(5,𝔽2)13yes—
187C2×D5—412no—
188C2×D5—412no—
189C2×D5—44no—
190C2×D5—816no—
191C4×C5—46no—
192C4×C5—42no—
193Dic5—412no—
194C24—26no—
195C24—26no—
196C24—212no—
197C24—212no—
198C24—448no—
199C24—412no—
200C24—412no—
201C24—416no—
202C24—448no—
203C24—54no—
204C24—104no—
205C24—1012no—
206C24—1012no—
207C24C2×D20112yes—
208C24C2×D20116yescomplemented
209C24C23×D516yes—
210C24C23×D511yescharacteristic
211C2×D4—1048no—
212C2×D4—2012no—
213C2×D4—2016no—
214C2×D4—2024no—
215C2×D4—2024no—
216C2×D4—2096no—
217C22×C4—212no—
218C22×C4—26no—
219C22×C4—26no—
220C22×C4—512no—
221C22×C4—1012no—
222C22×C4—1012no—
223C22×C4—1012no—
224C22×C4C23×D512yes—
225C22×C4C23×D516yes—
226C4⋊C4—2012no—
227C22⋊C4—412no—
228C22⋊C4—412no—
229C22⋊C4—1048no—
230C22⋊C4—2024no—
231C2×C5—412no—
232C2×C5#281218yescomplemented
233C2×C5C2×#53413yes—
234C2×C5C2×#56013yes—
235C2×C5C2×H(3,𝔽4)11yescharacteristic
236D5—84no—
237C23—248no—
238C23—224no—
239C23—26no—
240C23—26no—
241C23—212no—
242C23—212no—
243C23—448no—
244C23—4128no—
245C23—416no—
246C23—448no—
247C23—412no—
248C23—412no—
249C23—424no—
250C23—104no—
251C23—1012no—
252C23—1012no—
253C23—2024no—
254C23—208no—
255C23—2024no—
256C23C22×D20112yes—
257C23C22×D2012yes—
258C23#414416yes—
259C23#414416yes—
260C23C24×D511yescharacteristic
261D4—2016no—
262D4—2096no—
263C2×C4—212no—
264C2×C4—26no—
265C2×C4—26no—
266C2×C4—44no—
267C2×C4—412no—
268C2×C4—1012no—
269C2×C4—1012no—
270C2×C4—1012no—
271C2×C4—2024no—
272C5C2×#281211yescharacteristic, complemented, Sylow
273C22—224no—
274C22—212no—
275C22—464no—
276C22—448no—
277C22—46no—
278C22—46no—
279C22—412no—
280C22—204no—
281C22—2012no—
282C22—2012no—
283C22—4016no—
284C22#65827112yes—
285C22#6592814yes—
286C22#65929112yes—
287C22C23×D2011yescharacteristic
288C22C2×#414413yes—
289C22C2×#414413yes—
290C4—42no—
291C4—46no—
292C4—2012no—
293C2—412no—
294C2—404no—
295C2#1060559118yesdirect product factor
296C2C2×#6582713yes—
297C2C2×#6592811yescharacteristic
298C2C2×#6592913yes—
subgroupG/H
centerC24C23×D5
commutator, cosocleC23×C5C25
FrattiniC23C24×D5
FittingC2×C5×#379C2
socleC24×C5C24
hypercenterC2×#379D5
hypocenterC5C2×#2812
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C24 ◁ C2×#379

Complements

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

NH ≅ G/Ntype
1. C22×#4099C2semidirect
2. C2×C5×#379C2semidirect
3. C2×#65249C2semidirect
4. C2×#65242C2semidirect
5. C2×#65250C2semidirect
6. #10605591C2direct (N × H)
9. C23×D20C22semidirect
13. C2×#4099C22semidirect
15. C22×#685C22semidirect
16. C22×C4⋊Dic5C22semidirect
17. C5×#379C22semidirect
18. C2×#3963C22semidirect
19. C2×#3963C22semidirect
20. #65250C22semidirect
21. #65249C22semidirect
22. #65242C22semidirect
26. C25×C5D4semidirect
38. C22×D20C23semidirect
47. C2×#685C23semidirect
49. C2×C4⋊Dic5C23semidirect
50. #3963C23semidirect
51. #3963C23semidirect
57. C2×#379D5semidirect
68. C24×C5C2×D4semidirect
108. #379C2×D5semidirect
149. C25D20semidirect
208. C24C2×D20semidirect
232. C2×C5#2812semidirect
272. C5C2×#2812semidirect
295. C2#10605591direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#40992
C2×C5×#3792
C2×#652492
C2×#652422
C2×#652502
#106055912
C2×#28125