Finite Group Databasebeta

← C42×D52 #13105081

Subgroups — C42×D52 #13105081

Order 1600.

1104 conjugacy classes of subgroups; only the 372 normal subgroups are shown.

normal subgroups only

This group has 1104 conjugacy classes of subgroups — too many to draw. Only its 372 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
1C2×C4×D52C213yesmaximal
2C42×C5⋊D5C211yesmaximal, characteristic, complemented
3C42×C5×D5C212yesmaximal, complemented
4C4×#6036C213yesmaximal, complemented
5C4×D5×Dic5C216yesmaximal, complemented
6C22×D52C2211yescharacteristic
7C2×C4×C5⋊D5C2213yes—
8C2×C4×C5×D5C2216yes—
9C2×#6036C2213yes—
10C2×D5×Dic5C2216yes—
11C4×D52C4124yesdirect product factor
12C42×C52C2211yescharacteristic, complemented
13C4×#1039C2213yescomplemented
14C4×C5×Dic5C2216yescomplemented
15Dic52C2216yescomplemented
16C2×C42×D5—52nomaximal
17C22×C5⋊D5C2311yescharacteristic
18C22×C5×D5C2312yes—
19C2×D52C2×C4112yes—
20C2×C4×C52C2313yes—
21C2×#1039C2313yes—
22C4×C5⋊D5C2×C4112yescomplemented
23C2×C5×Dic5C2316yes—
24C4×C5×D5C2×C4124yescomplemented
25#6036C2×C4112yescomplemented
26D5×Dic5C2×C4124yescomplemented
27C22×C4×D5—56no—
28C2×C42×C5—52no—
29C2×C4×Dic5—56no—
30C42×D5—56no—
31C42×D5—52no—
32C42×D5—56no—
33C42×D5—102no—
34C42×D5D512yesdirect product factor
35C22×C52C2411yescharacteristic
36C2×C5⋊D5C22×C416yes—
37C2×C5×D5C22×C4112yes—
38D52C42116yesdirect product factor
39C4×C52C22×C416yescomplemented
40#1039C22×C416yescomplemented
41C5×Dic5C22×C4112yescomplemented
42C23×D5—52no—
43C22×C4×C5—56no—
44C22×Dic5—56no—
45C2×C4×D5—56no—
46C2×C4×D5—56no—
47C2×C4×D5—56no—
48C2×C4×D5—548no—
49C2×C4×D5—106no—
50C2×C4×D5C2×D516yes—
51C42×C5—22no—
52C42×C5—56no—
53C42×C5C2×D512yescomplemented
54C4×Dic5—56no—
55C4×Dic5—512no—
56C4×Dic5—106no—
57C4×Dic5C2×D516yescomplemented
58C22×C42—251noSylow
59C2×C52C23×C413yes—
60C5⋊D5C2×C4214yescomplemented
61C5×D5C2×C4218yescomplemented
62C23×C5—52no—
63C22×D5—52no—
64C22×D5—524no—
65C22×D5—102no—
66C22×D5C22×D512yes—
67C2×C4×C5—26no—
68C2×C4×C5—56no—
69C2×C4×C5—524no—
70C2×C4×C5C22×D516yes—
71C2×Dic5—56no—
72C2×Dic5—524no—
73C2×Dic5—106no—
74C2×Dic5C22×D516yes—
75C4×D5—524no—
76C4×D5—524no—
77C4×D5—524no—
78C4×D5—1024no—
79C4×D5C4×D5124yesdirect product factor
80C23×C4—253no—
81C2×C42—52no—
82C2×C42—253no—
83C2×C42—256no—
84C2×C42—251no—
85C52C22×C4211yescharacteristic, complemented, Sylow
86C22×C5—22no—
87C22×C5—512no—
88C22×C5C23×D512yes—
89C2×D5—512no—
90C2×D5—532no—
91C2×D5—1012no—
92C2×D5C2×C4×D5112yes—
93C4×C5—212no—
94C4×C5—512no—
95C4×C5C2×C4×D5112yescomplemented
96Dic5—512no—
97Dic5—1012no—
98Dic5C2×C4×D5112yescomplemented
99C24—251no—
100C22×C4—56no—
101C22×C4—253no—
102C22×C4—2524no—
103C22×C4—256no—
104C22×C4—253no—
105C42—56no—
106C42—256no—
107C42—253no—
108C42D5211yescharacteristic, direct product factor
109C2×C5—26no—
110C2×C5—58no—
111C2×C5C22×C4×D516yes—
112D5—58no—
113D5—108no—
114D5C42×D518yesdirect product factor
115C23—52no—
116C23—2512no—
117C23—251no—
118C2×C4—56no—
119C2×C4—524no—
120C2×C4—2512no—
121C2×C4—2524no—
122C2×C4—253no—
123C2×C4—2512no—
124C2×C4C2×D5213yes—
125C5—22no—
126C5C2×C42×D512yescomplemented
127C22—512no—
128C22—2516no—
129C22—256no—
130C22C22×D5211yescharacteristic
131C4—512no—
132C4—256no—
133C4C4×D5216yesdirect product factor
134C2—58no—
135C2—254no—
136C2C2×C4×D5213yes—
subgroupG/H
center, hypercenterC42D52
commutator, hypocenterC52C22×C42
FrattiniC22C22×D52
FittingC42×C52C22
socle, cosocleC22×C52C24
prime radicalC22×D52C22
LCSG ▷ C52
UCSC1 ◁ C42

Complements

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

NH ≅ G/Ntype
2. C42×C5⋊D5C2semidirect
3. C42×C5×D5C2semidirect
4. C4×#6036C2semidirect
5. C4×D5×Dic5C2semidirect
11. C4×D52C4direct (N × H)
12. C42×C52C22semidirect
13. C4×#1039C22semidirect
14. C4×C5×Dic5C22semidirect
15. Dic52C22semidirect
22. C4×C5⋊D5C2×C4semidirect
24. C4×C5×D5C2×C4semidirect
25. #6036C2×C4semidirect
26. D5×Dic5C2×C4semidirect
34. C42×D5D5direct (N × H)
38. D52C42direct (N × H)
39. C4×C52C22×C4semidirect
40. #1039C22×C4semidirect
41. C5×Dic5C22×C4semidirect
53. C42×C5C2×D5semidirect
57. C4×Dic5C2×D5semidirect
60. C5⋊D5C2×C42semidirect
61. C5×D5C2×C42semidirect
79. C4×D5C4×D5direct (N × H)
85. C52C22×C42semidirect
95. C4×C5C2×C4×D5semidirect
98. Dic5C2×C4×D5semidirect
108. C42D52direct (N × H)
114. D5C42×D5direct (N × H)
126. C5C2×C42×D5semidirect
133. C4C4×D52direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C4×D522
C42×C5⋊D52
C42×C5×D52
C4×#60362
C4×D5×Dic52
C2×C42×D55