Finite Group Databasebeta

← C22×C4×#1042 #13107782

Subgroups — C22×C4×#1042 #13107782

Order 1600.

1023 conjugacy classes of subgroups; only the 421 normal subgroups are shown.

normal subgroups only

This group has 1023 conjugacy classes of subgroups — too many to draw. Only its 421 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
1C23×#1042C212yesmaximal
2C22×C4×C5×D5C211yesmaximal, characteristic
3C2×C4×#1042C2112yesmaximal, direct product factor
4C23×C5×D5C2211yescharacteristic
5C22×#1042C4116yesdirect product factor
6C22×#1042C22112yes—
7C22×C4×C52C411yescharacteristic, complemented
8C22×C5×Dic5C411yescharacteristic, complemented
9C2×C4×C5×D5C416yescomplemented
10C2×C4×C5×D5C2216yes—
11C4×#1042C22116yesdirect product factor
12C22×C4×AGL(1,𝔽5)—51nomaximal
13C22×C4×Dic5—51nomaximal
14C23×C52C2×C411yescharacteristic
15C22×C5×D5C2×C413yes—
16C22×C5×D5C2×C418yes—
17C22×C5×D5C2313yes—
18C2×#1042C2×C4148yesdirect product factor
19C2×#1042C2318yes—
20C2×C4×C52C2×C416yescomplemented
21C2×C5×Dic5C2×C416yescomplemented
22C4×C5×D5C2×C4112yescomplemented
23C4×C5×D5C2314yes—
24C23×AGL(1,𝔽5)—52no—
25C23×C4×C5—51no—
26C23×Dic5—52no—
27C22×C4×D5D511yescharacteristic
28C2×C4×AGL(1,𝔽5)—512no—
29C2×C4×Dic5—512no—
30C22×C52C4214yescomplemented
31C22×C52C22×C413yes—
32C2×C5×D5C42112yescomplemented
33C2×C5×D5C22×C4112yes—
34C2×C5×D5C22×C413yes—
35C2×C5×D5C2411yescharacteristic
36#1042C22×C4116yesdirect product factor
37C4×C52C22×C414yescomplemented
38C5×Dic5C22×C414yescomplemented
39C24×C5—51no—
40C23×D5C2×D511yescharacteristic
41C22×AGL(1,𝔽5)—516no—
42C22×AGL(1,𝔽5)—512no—
43C22×C4×C5—41no—
44C22×C4×C5—51no—
45C22×C4×C5—56no—
46C22×C4×C5—56no—
47C22×C4×C5Dic511yescharacteristic, complemented
48C22×C4×C5AGL(1,𝔽5)11yescharacteristic, complemented
49C22×Dic5—516no—
50C22×Dic5—512no—
51C22×Dic5Dic511yescharacteristic, complemented
52C2×C4×D5Dic516yescomplemented
53C2×C4×D5C2×D516yes—
54C4×AGL(1,𝔽5)—516no—
55C4×Dic5—516no—
56C22×C42—251noSylow
57C2×C52C2×C4216yescomplemented
58C2×C52C23×C411yescharacteristic
59C5×D5C2×C4216yescomplemented
60C5×D5C23×C412yes—
61C23×C5—41no—
62C23×C5—58no—
63C23×C5—53no—
64C23×C5—53no—
65C23×C5C2×Dic511yescharacteristic
66C23×C5C2×AGL(1,𝔽5)11yescharacteristic
67C22×D5C4×D518yes—
68C22×D5C2×Dic513yes—
69C22×D5C22×D513yes—
70C2×AGL(1,𝔽5)—548no—
71C2×AGL(1,𝔽5)—58no—
72C2×C4×C5—46no—
73C2×C4×C5—56no—
74C2×C4×C5—512no—
75C2×C4×C5—54no—
76C2×C4×C5C2×Dic516yescomplemented
77C2×C4×C5C2×AGL(1,𝔽5)16yescomplemented
78C2×Dic5—58no—
79C2×Dic5—548no—
80C2×Dic5C2×Dic516yescomplemented
81C4×D5C2×Dic5112yescomplemented
82C4×D5C22×D514yes—
83C23×C4—51no—
84C23×C4—252no—
85C2×C42—2512no—
86C52C22×C4211yescharacteristic, complemented, Sylow
87C22×C5—44no—
88C22×C5—43no—
89C22×C5—512no—
90C22×C5—512no—
91C22×C5—53no—
92C22×C5—51no—
93C22×C5C4×Dic514yescomplemented
94C22×C5C4×AGL(1,𝔽5)14yescomplemented
95C22×C5C22×Dic513yes—
96C22×C5C22×AGL(1,𝔽5)13yes—
97C2×D5C4×Dic5112yescomplemented
98C2×D5C2×C4×D5112yes—
99C2×D5C22×Dic513yes—
100C2×D5C23×D511yescharacteristic
101AGL(1,𝔽5)—516no—
102C4×C5—44no—
103C4×C5—54no—
104C4×C5C22×Dic514yescomplemented
105C4×C5C22×AGL(1,𝔽5)14yescomplemented
106Dic5—516no—
107Dic5C22×Dic514yescomplemented
108C24—51no—
109C22×C4—51no—
110C22×C4—56no—
111C22×C4—56no—
112C22×C4—2512no—
113C22×C4—2516no—
114C22×C4#104211yescharacteristic, direct product factor
115C42—2516no—
116C2×C5—46no—
117C2×C5—41no—
118C2×C5—52no—
119C2×C5—56no—
120C2×C5C2×C4×Dic516yescomplemented
121C2×C5C2×C4×AGL(1,𝔽5)16yescomplemented
122C2×C5C23×Dic511yescharacteristic
123C2×C5C23×AGL(1,𝔽5)11yescharacteristic
124D5C2×C4×Dic516yescomplemented
125D5C22×C4×D512yes—
126C23—58no—
127C23—53no—
128C23—53no—
129C23C2×#104211yescharacteristic
130C2×C4—56no—
131C2×C4—54no—
132C2×C4—512no—
133C2×C4—258no—
134C2×C4—2548no—
135C2×C4C2×#104216yesdirect product factor
136C5—41no—
137C5C22×C4×Dic511yescharacteristic, complemented
138C5C22×C4×AGL(1,𝔽5)11yescharacteristic, complemented
139C22—512no—
140C22—512no—
141C22—51no—
142C22—53no—
143C22C4×#104214yesdirect product factor
144C22C22×#104213yes—
145C4—54no—
146C4—2516no—
147C4C22×#104214yesdirect product factor
148C2—52no—
149C2—56no—
150C2C2×C4×#104216yesdirect product factor
151C2C23×#104211yescharacteristic
subgroupG/H
center, hypercenterC22×C4#1042
commutator, hypocenterC52C22×C42
FrattiniC2C23×#1042
FittingC22×C4×C52C4
socleC23×C52C2×C4
prime radicalC23×C5×D5C22
cosocleC2×C5×D5C24
LCSG ▷ C52
UCSC1 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
3. C2×C4×#1042C2direct (N × H)
5. C22×#1042C4direct (N × H)
7. C22×C4×C52C4semidirect
8. C22×C5×Dic5C4semidirect
9. C2×C4×C5×D5C4semidirect
11. C4×#1042C22direct (N × H)
18. C2×#1042C2×C4direct (N × H)
20. C2×C4×C52C2×C4semidirect
21. C2×C5×Dic5C2×C4semidirect
22. C4×C5×D5C2×C4semidirect
30. C22×C52C42semidirect
32. C2×C5×D5C42semidirect
36. #1042C22×C4direct (N × H)
37. C4×C52C22×C4semidirect
38. C5×Dic5C22×C4semidirect
47. C22×C4×C5Dic5semidirect
48. C22×C4×C5AGL(1,𝔽5)semidirect
51. C22×Dic5Dic5semidirect
52. C2×C4×D5Dic5semidirect
57. C2×C52C2×C42semidirect
59. C5×D5C2×C42semidirect
76. C2×C4×C5C2×Dic5semidirect
77. C2×C4×C5C2×AGL(1,𝔽5)semidirect
80. C2×Dic5C2×Dic5semidirect
81. C4×D5C2×Dic5semidirect
86. C52C22×C42semidirect
93. C22×C5C4×Dic5semidirect
94. C22×C5C4×AGL(1,𝔽5)semidirect
97. C2×D5C4×Dic5semidirect
104. C4×C5C22×Dic5semidirect
105. C4×C5C22×AGL(1,𝔽5)semidirect
107. Dic5C22×Dic5semidirect
114. C22×C4#1042direct (N × H)
120. C2×C5C2×C4×Dic5semidirect
121. C2×C5C2×C4×AGL(1,𝔽5)semidirect
124. D5C2×C4×Dic5semidirect
135. C2×C4C2×#1042direct (N × H)
137. C5C22×C4×Dic5semidirect
138. C5C22×C4×AGL(1,𝔽5)semidirect
143. C22C4×#1042direct (N × H)
147. C4C22×#1042direct (N × H)
150. C2C2×C4×#1042direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#10422
C22×C4×C5×D52
C2×C4×#10422
C22×C4×AGL(1,𝔽5)5
C22×C4×Dic55