Finite Group Databasebeta

← C2×#10614849 #12305857

Subgroups — C2×#10614849 #12305857

Order 1280.

3666 conjugacy classes of subgroups; only the 667 normal subgroups are shown.

normal subgroups only

This group has 3666 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×C5×#126C211yesmaximal, characteristic, complemented
2C22×#4020C211yesmaximal, characteristic, complemented
3#10614849C2116yesmaximal, direct product factor
4C2×#65280C216yesmaximal, complemented
5C2×#65249C216yesmaximal, complemented
6C2×#65226C211yesmaximal, characteristic, complemented
7C23×C22⋊D5C2214yes—
8C23×D20C2216yes—
9C2×C5×#126—28no—
10C2×C5×#126C2218yescomplemented
11C22×C5×C4⋊C4C2213yes—
12C22×C5×C22⋊C4C2213yes—
13C22×C42×C5C2211yescharacteristic
14C22×#685C22112yes—
15C2×#4020—28no—
16C2×#4020C2218yescomplemented
17C2×#3963C2216yes—
18#65280C22148yescomplemented
19#65249C22148yescomplemented
20#65226C2218yescomplemented
21C2×#2818—51nomaximal, Sylow
22C25×C5C2311yescharacteristic
23C24×D5C2314yes—
24C23×C4×C5C2313yes—
25C23×C4×C5C2313yes—
26C22×C22⋊D5—424no—
27C22×C22⋊D5C23116yes—
28C23×Dic5C2314yes—
29C22×D20—224no—
30C22×D20—424no—
31C22×D20C23124yes—
32C5×#126—232no—
33C2×C5×C4⋊C4—212no—
34C2×C5×C4⋊C4C23112yes—
35C2×C5×C22⋊C4—212no—
36C2×C5×C22⋊C4C23112yes—
37C2×C42×C5D418yescomplemented
38C2×C42×C5C2314yes—
39C2×#685—424no—
40C2×#685C23148yes—
41#4020—232no—
42#3963C23124yes—
43C22×#127—51no—
44C22×#126D511yescharacteristic, complemented
45#2818—516no—
46C2×#394—56no—
47C2×#392—56no—
48C2×#383—51no—
49C24×C5—22no—
50C24×C5—212no—
51C24×C5C2412yes—
52C24×C5C2411yescharacteristic
53C23×D5—28no—
54C23×D5—224no—
55C23×D5—224no—
56C23×D5C2418yes—
57C22×C4×C5—212no—
58C22×C4×C5—26no—
59C22×C4×C5—212no—
60C22×C4×C5C2×D4112yes—
61C22×C4×C5C2416yes—
62C22×C4×C5C2416yes—
63C2×C22⋊D5—448no—
64C2×C22⋊D5—464no—
65C22×Dic5—224no—
66C22×Dic5C2418yes—
67C2×D20—248no—
68C2×D20—448no—
69C2×D20—496no—
70C5×C4⋊C4—224no—
71C5×C22⋊C4—224no—
72C42×C5C2×D4116yescomplemented
73#685—448no—
74C23×D4—54no—
75C23×D4—56no—
76C2×#127—58no—
77C2×#127—108no—
78C2×#126—28no—
79C2×#126C2×D518yescomplemented
80C22×C4⋊C4C2×D513yes—
81C22×C22⋊C4—512no—
82C22×C22⋊C4C2×D513yes—
83C22×C42C2×D511yescharacteristic
84#394—548no—
85#392—548no—
86#383—58no—
87C2×#95—56no—
88C23×C5—22no—
89C23×C5—212no—
90C23×C5—412no—
91C23×C5—416no—
92C23×C5C22×D412yes—
93C23×C5H(5,𝔽2)112yes—
94C23×C5C2511yescharacteristic
95C22×D5—28no—
96C22×D5—224no—
97C22×D5—224no—
98C22×D5—448no—
99C22×D5—416no—
100C22×D5—448no—
101C2×C4×C5—224no—
102C2×C4×C5—212no—
103C2×C4×C5—26no—
104C2×C4×C5—212no—
105C2×C4×C5—412no—
106C2×C4×C5C22×D4112yes—
107C22⋊D5—464no—
108C2×Dic5—224no—
109C2×Dic5—416no—
110D20—496no—
111C25—54no—
112C25C22×D511yescharacteristic
113C22×D4—516no—
114C22×D4—524no—
115C22×D4—1024no—
116C22×D4—2024no—
117C22×D4—2024no—
118C23×C4—54no—
119C23×C4C22×D513yes—
120C23×C4C22×D513yes—
121#127—1032no—
122#126—232no—
123C2×C4⋊C4—212no—
124C2×C4⋊C4C22×D5112yes—
125C2×C22⋊C4—212no—
126C2×C22⋊C4—548no—
127C2×C22⋊C4—2024no—
128C2×C22⋊C4C22×D5112yes—
129C2×C42C22⋊D518yescomplemented
130C2×C42C22×D514yes—
131#95—524no—
132C22×C5—46no—
133C22×C5—48no—
134C22×C5—88no—
135C22×C5#535112yes—
136C22×C5#560116yes—
137C22×C5C23×D411yescharacteristic
138C22×C5C2×H(5,𝔽2)16yes—
139C2×D5—424no—
140C2×D5—48no—
141C2×D5—424no—
142C2×D5—832no—
143C4×C5—212no—
144C4×C5—46no—
145Dic5—48no—
146C24—212no—
147C24—22no—
148C24—58no—
149C24—1024no—
150C24—1024no—
151C24—108no—
152C24C23×D512yes—
153C24C23×D511yescharacteristic
154C2×D4—1048no—
155C2×D4—2048no—
156C2×D4—2096no—
157C2×D4—2048no—
158C2×D4—2064no—
159C22×C4—212no—
160C22×C4—212no—
161C22×C4—26no—
162C22×C4—58no—
163C22×C4—1024no—
164C22×C4C2×C22⋊D5112yes—
165C22×C4C23×D516yes—
166C22×C4C23×D516yes—
167C4⋊C4—224no—
168C22⋊C4—224no—
169C22⋊C4—2048no—
170C42C2×C22⋊D5116yescomplemented
171C2×C5—82no—
172C2×C5#281818yescomplemented
173C2×C5C2×#53513yes—
174C2×C5C2×#56014yes—
175D5—88no—
176C23—212no—
177C23—22no—
178C23—412no—
179C23—416no—
180C23—1024no—
181C23—1024no—
182C23—108no—
183C23—2048no—
184C23—2016no—
185C23—2048no—
186C23#4149112yes—
187C23C22×C22⋊D512yes—
188C23C24×D511yescharacteristic
189D4—2096no—
190D4—2064no—
191C2×C4—224no—
192C2×C4—212no—
193C2×C4—212no—
194C2×C4—26no—
195C2×C4—412no—
196C2×C4—1024no—
197C2×C4—2016no—
198C2×C4C22×C22⋊D5112yes—
199C5C2×#281811yescharacteristic, complemented, Sylow
200C22—46no—
201C22—48no—
202C22—88no—
203C22—2024no—
204C22—208no—
205C22—2024no—
206C22—4032no—
207C22#66045116yes—
208C22#66164112yes—
209C22C2×#414916yes—
210C22C23×C22⋊D511yescharacteristic
211C4—212no—
212C4—46no—
213C4—208no—
214C2—82no—
215C2—408no—
216C2#1061484918yesdirect product factor
217C2C2×#6604514yes—
218C2C2×#6616413yes—
subgroupG/H
centerC24C23×D5
commutator, cosocleC23×C5C25
FrattiniC23C24×D5
FittingC22×C5×#126C2
socleC24×C5C24
hypercenterC22×#126D5
hypocenterC5C2×#2818
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C24 ◁ C22×#126

Complements

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

NH ≅ G/Ntype
1. C22×C5×#126C2semidirect
2. C22×#4020C2semidirect
3. #10614849C2direct (N × H)
4. C2×#65280C2semidirect
5. C2×#65249C2semidirect
6. C2×#65226C2semidirect
10. C2×C5×#126C22semidirect
16. C2×#4020C22semidirect
18. #65280C22semidirect
19. #65249C22semidirect
20. #65226C22semidirect
37. C2×C42×C5D4semidirect
44. C22×#126D5semidirect
72. C42×C5C2×D4semidirect
79. C2×#126C2×D5semidirect
129. C2×C42C22⋊D5semidirect
170. C42C2×C22⋊D5semidirect
172. C2×C5#2818semidirect
199. C5C2×#2818semidirect
216. C2#10614849direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C5×#1262
C22×#40202
#106148492
C2×#652802
C2×#652492
C2×#652262
C2×#28185