Finite Group Databasebeta

← C22×C5×#686 #13106952

Subgroups — C22×C5×#686 #13106952

Order 1600.

912 conjugacy classes of subgroups; only the 254 normal subgroups are shown.

normal subgroups only

This group has 912 conjugacy classes of subgroups — too many to draw. Only its 254 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×C52×D4C211yesmaximal, characteristic, complemented
2C22×C5×D20C211yesmaximal, characteristic, complemented
3C2×C5×#686C2112yesmaximal, direct product factor
4C22×C5×#168C211yesmaximal, characteristic, complemented
5C24×C52—21no—
6C23×C5×D5—21no—
7C2×C52×D4—23no—
8C2×C52×D4C2216yescomplemented
9C22×C4×C52C2211yescharacteristic
10C2×C5×D20—23no—
11C2×C5×D20C2216yescomplemented
12C5×#686C22116yesdirect product factor
13C2×C5×#168C2216yescomplemented
14C22×C5×D8—51nomaximal
15C22×#686C511yesmaximal, characteristic, direct product factor
16C23×C52—26no—
17C23×C52—44no—
18C23×C52D411yescharacteristic
19C22×C5×D5—26no—
20C22×C5×D5—44no—
21C52×D4—26no—
22C52×D4C2314yescomplemented
23C2×C4×C52D413yes—
24C2×C4×C52C2313yes—
25C5×D20—26no—
26C5×D20C2314yescomplemented
27C5×#168C2314yescomplemented
28C22×C5×D4—22no—
29C22×C5×D4—51no—
30C22×C5×D4D511yescharacteristic, complemented
31C22×C5×D4C2×C511yescharacteristic, complemented
32C22×D20C2×C511yescharacteristic, complemented
33C2×C5×D8—512no—
34C22×C5×C8—51no—
35C2×#686C2×C5112yesdirect product factor
36C22×#168C2×C511yescharacteristic, complemented
37C22×C52—24no—
38C22×C52—412no—
39C22×C52D814yescomplemented
40C22×C52C2×D413yes—
41C2×C5×D5—24no—
42C2×C5×D5—412no—
43C4×C52C2×D413yes—
44C4×C52C2411yescharacteristic
45C24×C5—24no—
46C24×C5—21no—
47C24×C5—21no—
48C24×C5—101no—
49C23×D5—21no—
50C2×C5×D4—212no—
51C2×C5×D4—23no—
52C2×C5×D4—23no—
53C2×C5×D4—212no—
54C2×C5×D4—56no—
55C2×C5×D4—103no—
56C2×C5×D4C2×D516yescomplemented
57C2×C5×D4C22×C516yescomplemented
58C22×C4×C5—22no—
59C22×C4×C5C2×D511yescharacteristic
60C22×C4×C5C22×C511yescharacteristic
61C2×D20—23no—
62C2×D20C22×C516yescomplemented
63C5×D8—516no—
64C2×C5×C8—56no—
65#686C22×C5116yesdirect product factor
66C2×#168C22×C516yescomplemented
67C22×D8—51noSylow
68C2×C52—44no—
69C2×C52C2×D816yescomplemented
70C2×C52C22×D411yescharacteristic
71C5×D5—44no—
72C23×C5—224no—
73C23×C5—26no—
74C23×C5—26no—
75C23×C5—22no—
76C23×C5—416no—
77C23×C5—44no—
78C23×C5—44no—
79C23×C5—106no—
80C23×C5—204no—
81C23×C5C22⋊D511yescharacteristic
82C23×C5C5×D411yescharacteristic
83C22×D5—26no—
84C22×D5—44no—
85C5×D4—224no—
86C5×D4—26no—
87C5×D4—26no—
88C5×D4—28no—
89C5×D4—54no—
90C5×D4—106no—
91C5×D4C22×D514yescomplemented
92C5×D4C23×C514yescomplemented
93C2×C4×C5—26no—
94C2×C4×C5—26no—
95C2×C4×C5C22⋊D513yes—
96C2×C4×C5C5×D413yes—
97C2×C4×C5C22×D513yes—
98C2×C4×C5C23×C513yes—
99D20—26no—
100D20C23×C514yescomplemented
101C5×C8—54no—
102#168C23×C514yescomplemented
103C22×D4—51no—
104C22×D4C5×D511yescharacteristic, complemented
105C2×D8—512no—
106C22×C8—51no—
107C52C22×D811yescharacteristic, complemented, Sylow
108C22×C5—216no—
109C22×C5—24no—
110C22×C5—24no—
111C22×C5—26no—
112C22×C5—28no—
113C22×C5—448no—
114C22×C5—412no—
115C22×C5—412no—
116C22×C5—104no—
117C22×C5—2012no—
118C22×C5#68614yesdirect product factor
119C22×C5C5×D814yescomplemented
120C22×C5C2×C22⋊D513yes—
121C22×C5C2×C5×D413yes—
122C2×D5—24no—
123C2×D5—412no—
124C4×C5—26no—
125C4×C5—22no—
126C4×C5C2×C22⋊D513yes—
127C4×C5C2×C5×D413yes—
128C4×C5C23×D511yescharacteristic
129C4×C5C24×C511yescharacteristic
130C24—21no—
131C24—101no—
132C2×D4—23no—
133C2×D4—56no—
134C2×D4—103no—
135C2×D4C2×C5×D516yescomplemented
136C22×C4C2×C5×D511yescharacteristic
137D8—516no—
138C2×C8—56no—
139C2×C5—212no—
140C2×C5—22no—
141C2×C5—416no—
142C2×C5—44no—
143C2×C5—44no—
144C2×C5—204no—
145C2×C5C2×#68616yesdirect product factor
146C2×C5C2×C5×D816yescomplemented
147C2×C5C22×C22⋊D511yescharacteristic
148C2×C5C22×C5×D411yescharacteristic
149D5—44no—
150C23—26no—
151C23—44no—
152C23—106no—
153C23—204no—
154C23C5×C22⋊D511yescharacteristic
155D4—26no—
156D4—54no—
157D4—106no—
158D4C22×C5×D514yescomplemented
159C2×C4C5×C22⋊D513yes—
160C2×C4C22×C5×D513yes—
161C8—54no—
162C5—22no—
163C5C22×#68611yescharacteristic, direct product factor
164C5C22×C5×D811yescharacteristic, complemented
165C22—24no—
166C22—412no—
167C22—104no—
168C22—2012no—
169C22C5×#68614yesdirect product factor
170C22C2×C5×C22⋊D513yes—
171C4C2×C5×C22⋊D513yes—
172C4C23×C5×D511yescharacteristic
173C2—44no—
174C2—204no—
175C2C2×C5×#68616yesdirect product factor
176C2C22×C5×C22⋊D511yescharacteristic
subgroupG/H
centerC23×C5C22⋊D5
commutator, cosocleC4×C5C24×C5
FrattiniC4C23×C5×D5
FittingC22×C52×D4C2
socleC23×C52D4
hypercenterC22×C5×D4D5
hypocenterC5C22×C5×D8
LCSG ▷ C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23×C5 ◁ C22×C4×C5 ◁ C22×C5×D4

Complements

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

NH ≅ G/Ntype
1. C22×C52×D4C2semidirect
2. C22×C5×D20C2semidirect
3. C2×C5×#686C2direct (N × H)
4. C22×C5×#168C2semidirect
8. C2×C52×D4C22semidirect
11. C2×C5×D20C22semidirect
12. C5×#686C22direct (N × H)
13. C2×C5×#168C22semidirect
15. C22×#686C5direct (N × H)
22. C52×D4C23semidirect
26. C5×D20C23semidirect
27. C5×#168C23semidirect
30. C22×C5×D4D5semidirect
31. C22×C5×D4C2×C5semidirect
32. C22×D20C2×C5semidirect
35. C2×#686C2×C5direct (N × H)
36. C22×#168C2×C5semidirect
39. C22×C52D8semidirect
56. C2×C5×D4C2×D5semidirect
57. C2×C5×D4C22×C5semidirect
62. C2×D20C22×C5semidirect
65. #686C22×C5direct (N × H)
66. C2×#168C22×C5semidirect
69. C2×C52C2×D8semidirect
91. C5×D4C22×D5semidirect
92. C5×D4C23×C5semidirect
100. D20C23×C5semidirect
102. #168C23×C5semidirect
104. C22×D4C5×D5semidirect
107. C52C22×D8semidirect
118. C22×C5#686direct (N × H)
119. C22×C5C5×D8semidirect
135. C2×D4C2×C5×D5semidirect
145. C2×C5C2×#686direct (N × H)
146. C2×C5C2×C5×D8semidirect
158. D4C22×C5×D5semidirect
163. C5C22×#686direct (N × H)
164. C5C22×C5×D8semidirect
169. C22C5×#686direct (N × H)
175. C2C2×C5×#686direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C52×D42
C22×C5×D202
C2×C5×#6862
C22×C5×#1682
C22×C5×D85
C22×#6865