Finite Group Databasebeta

← C23×C4⋊#1043 #72221000

Subgroups — C23×C4⋊#1043 #72221000

Order 3200.

9952 conjugacy classes of subgroups; only the 1814 normal subgroups are shown.

normal subgroups only

This group has 9952 conjugacy classes of subgroups — too many to draw. Only its 1814 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
1C24×#1043C212yesmaximal
2C23×C4×C5⋊D5C211yesmaximal, characteristic
3C22×C4⋊#1043C2128yesmaximal, direct product factor
4C24×C5⋊D5C2211yescharacteristic
5C23×#1043—216no—
6C23×#1043C22128yes—
7C23×C4×C52C411yescharacteristic, complemented
8C23×#1039C411yescharacteristic, complemented
9C22×C4×C5⋊D5C4114yescomplemented
10C22×C4×C5⋊D5C22114yes—
11C2×C4⋊#1043C221112yesdirect product factor
12C23×C4⋊AGL(1,𝔽5)—56nomaximal
13C24×C52C2×C411yescharacteristic
14C23×C5⋊D5C2×C417yes—
15C23×C5⋊D5D418yes—
16C23×C5⋊D5Q818yes—
17C23×C5⋊D5C2317yes—
18C22×#1043—2112no—
19C22×#1043C23156yes—
20C22×C4×C52C2×C4114yescomplemented
21C22×#1039C2×C4114yescomplemented
22C2×C4×C5⋊D5C2×C4184yescomplemented
23C2×C4×C5⋊D5C23128yes—
24C4⋊#1043C23164yesdirect product factor
25C24×AGL(1,𝔽5)—512no—
26C23×C4×D5—56no—
27C22×C4⋊AGL(1,𝔽5)—5168no—
28C23×C52C4⋊C418yescomplemented
29C23×C52C22×C417yes—
30C22×C5⋊D5C4⋊C4156yescomplemented
31C22×C5⋊D5C22×C4121yes—
32C22×C5⋊D5C2×D4128yes—
33C22×C5⋊D5C2×Q8128yes—
34C22×C5⋊D5C2417yes—
35C2×#1043—2112no—
36C2×#1043C24116yes—
37C2×C4×C52C22×C4128yescomplemented
38C2×#1039C22×C4128yescomplemented
39C4×C5⋊D5C22×C4156yescomplemented
40C4×C5⋊D5C2418yes—
41C24×D5—56no—
42C23×AGL(1,𝔽5)—5168no—
43C23×AGL(1,𝔽5)—1096no—
44C23×C4×C5AGL(1,𝔽5)16yescomplemented
45C23×Dic5—56no—
46C22×C4×D5—584no—
47C22×C4×D5—584no—
48C2×C4⋊AGL(1,𝔽5)—5672no—
49C23×C4⋊C4—251noSylow
50C22×C52C2×C4⋊C4128yescomplemented
51C22×C52C23×C417yes—
52C2×C5⋊D5C2×C4⋊C4184yescomplemented
53C2×C5⋊D5C23×C417yes—
54C2×C5⋊D5C22×D4114yes—
55C2×C5⋊D5C22×Q8114yes—
56C2×C5⋊D5C2511yescharacteristic
57#1043—216no—
58C4×C52C23×C418yescomplemented
59#1039C23×C418yescomplemented
60C24×C5C2×AGL(1,𝔽5)16yes—
61C23×D5—548no—
62C23×D5—548no—
63C23×D5—542no—
64C23×D5—542no—
65C22×AGL(1,𝔽5)—5336no—
66C22×AGL(1,𝔽5)—10672no—
67C22×C4×C5C2×AGL(1,𝔽5)184yescomplemented
68C22×Dic5—584no—
69C2×C4×D5—5168no—
70C2×C4×D5—5504no—
71C4⋊AGL(1,𝔽5)—5384no—
72C24×C4—251no—
73C24×C4—252no—
74C22×C4⋊C4—2528no—
75C2×C52C22×C4⋊C4114yescomplemented
76C2×C52C24×C411yescharacteristic
77C5⋊D5C22×C4⋊C4114yescomplemented
78C5⋊D5C23×D411yescharacteristic
79C5⋊D5C23×Q811yescharacteristic
80C23×C5C4⋊AGL(1,𝔽5)148yescomplemented
81C23×C5C22×AGL(1,𝔽5)142yes—
82C22×D5—5336no—
83C22×D5—5168no—
84C22×D5—5168no—
85C22×D5—5126no—
86C22×D5—542no—
87C2×AGL(1,𝔽5)—596no—
88C2×AGL(1,𝔽5)—10672no—
89C2×C4×C5C22×AGL(1,𝔽5)1168yescomplemented
90C2×Dic5—5168no—
91C4×D5—5336no—
92C4×D5—548no—
93C25—251no—
94C23×C4—251no—
95C23×C4—2514no—
96C23×C4—2514no—
97C23×C4—2528no—
98C23×C4—5016no—
99C23×C4#104311yescharacteristic, complemented
100C2×C4⋊C4—25112no—
101C52C23×C4⋊C411yescharacteristic, complemented, Sylow
102C22×C5C2×C4⋊AGL(1,𝔽5)1168yescomplemented
103C22×C5C23×AGL(1,𝔽5)142yes—
104C2×D5—5504no—
105C2×D5—584no—
106C2×D5—584no—
107C2×D5—542no—
108C2×D5—56no—
109AGL(1,𝔽5)—1096no—
110C4×C5C23×AGL(1,𝔽5)148yescomplemented
111Dic5—548no—
112C24—258no—
113C24—258no—
114C24—257no—
115C24—257no—
116C24C2×#104311yescharacteristic
117C22×C4—2514no—
118C22×C4—2528no—
119C22×C4—2584no—
120C22×C4—2556no—
121C22×C4—50112no—
122C22×C4C2×#1043114yescomplemented
123C4⋊C4—2564no—
124C2×C5C22×C4⋊AGL(1,𝔽5)184yescomplemented
125C2×C5C24×AGL(1,𝔽5)16yes—
126D5—584no—
127D5—56no—
128D5—56no—
129C23—2528no—
130C23—2556no—
131C23—2528no—
132C23—257no—
133C23—2521no—
134C23C4⋊#104318yesdirect product factor
135C23C22×#104317yes—
136C2×C4—2528no—
137C2×C4—258no—
138C2×C4—2556no—
139C2×C4—2516no—
140C2×C4—50112no—
141C2×C4C22×#1043128yescomplemented
142C5C23×C4⋊AGL(1,𝔽5)16yescomplemented
143C22—2514no—
144C22—2584no—
145C22—2514no—
146C22—251no—
147C22—257no—
148C22C2×C4⋊#1043128yesdirect product factor
149C22C23×#104317yes—
150C4—258no—
151C4—5016no—
152C4C23×#104318yescomplemented
153C2—251no—
154C2—2514no—
155C2—251no—
156C2C22×C4⋊#1043114yesdirect product factor
157C2C24×#104311yescharacteristic
subgroupG/H
centerC24C2×#1043
commutatorC2×C52C24×C4
FrattiniC2C24×#1043
FittingC23×C4×C52C4
socleC24×C52C2×C4
prime radicalC24×C5⋊D5C22
cosocleC2×C5⋊D5C25
hypercenterC23×C4#1043
hypocenterC52C23×C4⋊C4
LCSG ▷ C2×C52 ▷ C52
UCSC1 ◁ C24 ◁ C23×C4

Complements

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

NH ≅ G/Ntype
3. C22×C4⋊#1043C2direct (N × H)
7. C23×C4×C52C4semidirect
8. C23×#1039C4semidirect
9. C22×C4×C5⋊D5C4semidirect
11. C2×C4⋊#1043C22direct (N × H)
20. C22×C4×C52C2×C4semidirect
21. C22×#1039C2×C4semidirect
22. C2×C4×C5⋊D5C2×C4semidirect
24. C4⋊#1043C23direct (N × H)
28. C23×C52C4⋊C4semidirect
30. C22×C5⋊D5C4⋊C4semidirect
37. C2×C4×C52C22×C4semidirect
38. C2×#1039C22×C4semidirect
39. C4×C5⋊D5C22×C4semidirect
44. C23×C4×C5AGL(1,𝔽5)semidirect
50. C22×C52C2×C4⋊C4semidirect
52. C2×C5⋊D5C2×C4⋊C4semidirect
58. C4×C52C23×C4semidirect
59. #1039C23×C4semidirect
67. C22×C4×C5C2×AGL(1,𝔽5)semidirect
75. C2×C52C22×C4⋊C4semidirect
77. C5⋊D5C22×C4⋊C4semidirect
80. C23×C5C4⋊AGL(1,𝔽5)semidirect
89. C2×C4×C5C22×AGL(1,𝔽5)semidirect
99. C23×C4#1043semidirect
101. C52C23×C4⋊C4semidirect
102. C22×C5C2×C4⋊AGL(1,𝔽5)semidirect
110. C4×C5C23×AGL(1,𝔽5)semidirect
122. C22×C4C2×#1043semidirect
124. C2×C5C22×C4⋊AGL(1,𝔽5)semidirect
134. C23C4⋊#1043direct (N × H)
141. C2×C4C22×#1043semidirect
142. C5C23×C4⋊AGL(1,𝔽5)semidirect
148. C22C2×C4⋊#1043direct (N × H)
152. C4C23×#1043semidirect
156. C2C22×C4⋊#1043direct (N × H)

Maximal subgroups

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

subgroupindex
C24×#10432
C23×C4×C5⋊D52
C22×C4⋊#10432
C23×C4⋊AGL(1,𝔽5)5