Finite Group Databasebeta

← C2×C42×#1044 #72242733

Subgroups — C2×C42×#1044 #72242733

Order 3200.

3042 conjugacy classes of subgroups; only the 798 normal subgroups are shown.

normal subgroups only

This group has 3042 conjugacy classes of subgroups — too many to draw. Only its 798 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×C4×#1044C216yesmaximal
2C2×C42×C5⋊D5C211yesmaximal, characteristic
3C42×#1044C218yesmaximal, direct product factor
4C23×#1044C2214yes—
5C22×C4×C5⋊D5C2213yes—
6C2×C4×#1044C4148yesdirect product factor
7C2×C4×#1044C22124yes—
8C2×C42×C52C411yescharacteristic, complemented
9C2×C4×#1039C413yescomplemented
10C42×C5⋊D5C414yescomplemented
11C42×C5⋊D5C2214yes—
12C2×C42×AGL(1,𝔽5)—52nomaximal
13C23×C5⋊D5C2311yescharacteristic
14C22×#1044C2×C4148yes—
15C22×#1044C2318yes—
16C22×C4×C52C2×C413yes—
17C22×#1039C2×C413yes—
18C2×C4×C5⋊D5C2×C416yes—
19C2×C4×C5⋊D5C2×C4124yes—
20C2×C4×C5⋊D5C2316yes—
21C4×#1044C2×C4196yesdirect product factor
22C42×C52C2×C414yescomplemented
23C4×#1039C2×C4112yescomplemented
24C22×C4×AGL(1,𝔽5)—512no—
25C2×C42×D5—52no—
26C2×C42×D5—102no—
27C42×AGL(1,𝔽5)—516no—
28C23×C52C22×C411yescharacteristic
29C22×C5⋊D5C22×C411yescharacteristic
30C22×C5⋊D5C22×C4112yes—
31C22×C5⋊D5C2411yescharacteristic
32C2×#1044C42164yesdirect product factor
33C2×#1044C22×C4148yes—
34C2×C4×C52C42112yescomplemented
35C2×C4×C52C22×C416yes—
36C2×#1039C42112yescomplemented
37C2×#1039C22×C416yes—
38C4×C5⋊D5C42124yescomplemented
39C4×C5⋊D5C22×C4124yes—
40C23×AGL(1,𝔽5)—58no—
41C22×C4×D5—56no—
42C22×C4×D5—106no—
43C2×C4×AGL(1,𝔽5)—596no—
44C2×C4×AGL(1,𝔽5)—548no—
45C2×C42×C5—22no—
46C2×C42×C5AGL(1,𝔽5)12yescomplemented
47C2×C4×Dic5—56no—
48C2×C4×Dic5—106no—
49C42×D5—58no—
50C42×D5—58no—
51C42×D5—108no—
52C42×D5—108no—
53C2×C43—251noSylow
54C22×C52C2×C4216yes—
55C22×C52C23×C411yescharacteristic
56C2×C5⋊D5C2×C4216yes—
57C2×C5⋊D5C2×C42116yes—
58C2×C5⋊D5C23×C416yes—
59#1044C2×C42132yesdirect product factor
60C4×C52C2×C42112yescomplemented
61#1039C2×C42112yescomplemented
62C23×D5—52no—
63C23×D5—102no—
64C22×AGL(1,𝔽5)—516no—
65C22×AGL(1,𝔽5)—596no—
66C22×C4×C5—26no—
67C22×C4×C5C2×AGL(1,𝔽5)16yes—
68C22×Dic5—56no—
69C22×Dic5—106no—
70C2×C4×D5—548no—
71C2×C4×D5—512no—
72C2×C4×D5—512no—
73C2×C4×D5—1048no—
74C2×C4×D5—1012no—
75C2×C4×D5—1012no—
76C4×AGL(1,𝔽5)—5192no—
77C42×C5—28no—
78C42×C5C2×AGL(1,𝔽5)18yescomplemented
79C4×Dic5—524no—
80C4×Dic5—1024no—
81C22×C42—256no—
82C22×C42—251no—
83C43—258no—
84C2×C52C4314yescomplemented
85C2×C52C22×C4213yes—
86C5⋊D5C4314yescomplemented
87C5⋊D5C22×C4214yes—
88C23×C5—22no—
89C23×C5C22×AGL(1,𝔽5)12yes—
90C22×D5—524no—
91C22×D5—52no—
92C22×D5—52no—
93C22×D5—102no—
94C22×D5—1024no—
95C22×D5—102no—
96C2×AGL(1,𝔽5)—5128no—
97C2×AGL(1,𝔽5)—596no—
98C2×C4×C5—224no—
99C2×C4×C5—212no—
100C2×C4×C5C4×AGL(1,𝔽5)124yescomplemented
101C2×C4×C5C22×AGL(1,𝔽5)112yes—
102C2×Dic5—524no—
103C2×Dic5—512no—
104C2×Dic5—1024no—
105C2×Dic5—1012no—
106C4×D5—548no—
107C4×D5—548no—
108C4×D5—1048no—
109C4×D5—1048no—
110C23×C4—254no—
111C23×C4—253no—
112C2×C42—2524no—
113C2×C42—2548no—
114C2×C42—253no—
115C2×C42—254no—
116C2×C42—254no—
117C2×C42#104411yescharacteristic, direct product factor
118C52C2×C4311yescharacteristic, complemented, Sylow
119C22×C5—212no—
120C22×C5—22no—
121C22×C5C2×C4×AGL(1,𝔽5)112yes—
122C22×C5C23×AGL(1,𝔽5)12yes—
123C2×D5—532no—
124C2×D5—512no—
125C2×D5—512no—
126C2×D5—1012no—
127C2×D5—1032no—
128C2×D5—1012no—
129AGL(1,𝔽5)—564no—
130C4×C5—224no—
131C4×C5C2×C4×AGL(1,𝔽5)124yescomplemented
132Dic5—524no—
133Dic5—1024no—
134C24—251no—
135C22×C4—258no—
136C22×C4—2548no—
137C22×C4—253no—
138C22×C4—2524no—
139C22×C4—256no—
140C22×C4—256no—
141C22×C4C2×#104413yes—
142C42—2596no—
143C42—2512no—
144C42C2×#104414yesdirect product factor
145C2×C5—26no—
146C2×C5—28no—
147C2×C5C42×AGL(1,𝔽5)18yescomplemented
148C2×C5C22×C4×AGL(1,𝔽5)16yes—
149D5—58no—
150D5—58no—
151D5—108no—
152D5—108no—
153C23—2512no—
154C23—251no—
155C23—251no—
156C23C22×#104411yescharacteristic
157C2×C4—2548no—
158C2×C4—2564no—
159C2×C4—2512no—
160C2×C4—256no—
161C2×C4—2524no—
162C2×C4—2524no—
163C2×C4C4×#1044112yesdirect product factor
164C2×C4C22×#104416yes—
165C5—22no—
166C5C2×C42×AGL(1,𝔽5)12yescomplemented
167C22—256no—
168C22—256no—
169C22—2516no—
170C22C2×C4×#104416yes—
171C22C23×#104411yescharacteristic
172C4—2532no—
173C4—2512no—
174C4C2×C4×#1044112yesdirect product factor
175C2—254no—
176C2—254no—
177C2C42×#104414yesdirect product factor
178C2C22×C4×#104413yes—
subgroupG/H
center, hypercenterC2×C42#1044
commutator, hypocenterC52C2×C43
FrattiniC22C23×#1044
FittingC2×C42×C52C4
socleC23×C52C22×C4
prime radicalC23×C5⋊D5C23
cosocleC22×C5⋊D5C24
LCSG ▷ C52
UCSC1 ◁ C2×C42

Complements

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

NH ≅ G/Ntype
3. C42×#1044C2direct (N × H)
6. C2×C4×#1044C4direct (N × H)
8. C2×C42×C52C4semidirect
9. C2×C4×#1039C4semidirect
10. C42×C5⋊D5C4semidirect
21. C4×#1044C2×C4direct (N × H)
22. C42×C52C2×C4semidirect
23. C4×#1039C2×C4semidirect
32. C2×#1044C42direct (N × H)
34. C2×C4×C52C42semidirect
36. C2×#1039C42semidirect
38. C4×C5⋊D5C42semidirect
46. C2×C42×C5AGL(1,𝔽5)semidirect
59. #1044C2×C42direct (N × H)
60. C4×C52C2×C42semidirect
61. #1039C2×C42semidirect
78. C42×C5C2×AGL(1,𝔽5)semidirect
84. C2×C52C43semidirect
86. C5⋊D5C43semidirect
100. C2×C4×C5C4×AGL(1,𝔽5)semidirect
117. C2×C42#1044direct (N × H)
118. C52C2×C43semidirect
131. C4×C5C2×C4×AGL(1,𝔽5)semidirect
144. C42C2×#1044direct (N × H)
147. C2×C5C42×AGL(1,𝔽5)semidirect
163. C2×C4C4×#1044direct (N × H)
166. C5C2×C42×AGL(1,𝔽5)semidirect
174. C4C2×C4×#1044direct (N × H)
177. C2C42×#1044direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C4×#10442
C2×C42×C5⋊D52
C42×#10442
C2×C42×AGL(1,𝔽5)5