Finite Group Databasebeta

← C23×C22⋊#1043 #72221146

Subgroups — C23×C22⋊#1043 #72221146

Order 3200.

18528 conjugacy classes of subgroups; only the 1814 normal subgroups are shown.

normal subgroups only

This group has 18528 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
1C25×C5⋊D5C211yesmaximal, characteristic
2C24×#1043C212yesmaximal, complemented
3C22×C22⋊#1043C2128yesmaximal, direct product factor
4C25×C52C411yescharacteristic, complemented
5C24×C5⋊D5—216no—
6C24×C5⋊D5C4114yescomplemented
7C24×C5⋊D5C411yescharacteristic, complemented
8C24×C5⋊D5C22114yes—
9C24×C5⋊D5C2211yescharacteristic
10C23×#1043—216no—
11C23×#1043C22128yescomplemented
12C2×C22⋊#1043C221112yesdirect product factor
13C23×C22⋊AGL(1,𝔽5)—56nomaximal
14C24×C52—28no—
15C24×C52C2×C411yescharacteristic
16C24×C52C2×C4114yescomplemented
17C23×C5⋊D5—28no—
18C23×C5⋊D5—2112no—
19C23×C5⋊D5—2112no—
20C23×C5⋊D5C2×C417yes—
21C23×C5⋊D5C2×C4184yescomplemented
22C23×C5⋊D5C2×C4114yescomplemented
23C23×C5⋊D5D4116yes—
24C23×C5⋊D5C23128yes—
25C23×C5⋊D5C2317yes—
26C22×#1043—2112no—
27C22×#1043C23156yescomplemented
28C22⋊#1043C23164yesdirect product factor
29C25×D5—56no—
30C24×AGL(1,𝔽5)—512no—
31C22×C22⋊AGL(1,𝔽5)—5168no—
32C23×C52—256no—
33C23×C52C22⋊C418yescomplemented
34C23×C52C22×C417yes—
35C23×C52C22×C4128yescomplemented
36C22×C5⋊D5—256no—
37C22×C5⋊D5—2336no—
38C22×C5⋊D5—2112no—
39C22×C5⋊D5C22⋊C4156yescomplemented
40C22×C5⋊D5C22×C4121yes—
41C22×C5⋊D5C22×C4156yescomplemented
42C22×C5⋊D5C22×C4128yescomplemented
43C22×C5⋊D5C2×D4156yes—
44C22×C5⋊D5C2418yes—
45C22×C5⋊D5C2417yes—
46C2×#1043—2112no—
47C2×#1043C24116yescomplemented
48C25×C5AGL(1,𝔽5)16yescomplemented
49C24×D5—56no—
50C24×D5—56no—
51C24×D5—584no—
52C24×D5—584no—
53C24×D5—1096no—
54C23×AGL(1,𝔽5)—5168no—
55C23×AGL(1,𝔽5)—1096no—
56C2×C22⋊AGL(1,𝔽5)—5672no—
57C23×C22⋊C4—251noSylow
58C22×C52—256no—
59C22×C52C2×C22⋊C4128yescomplemented
60C22×C52C23×C417yes—
61C22×C52C23×C418yescomplemented
62C2×C5⋊D5—256no—
63C2×C5⋊D5—2112no—
64C2×C5⋊D5—216no—
65C2×C5⋊D5C2×C22⋊C4184yescomplemented
66C2×C5⋊D5C23×C417yes—
67C2×C5⋊D5C23×C418yescomplemented
68C2×C5⋊D5C22×D4128yes—
69C2×C5⋊D5C2511yescharacteristic
70#1043—216no—
71C24×C5—248no—
72C24×C5C2×AGL(1,𝔽5)16yes—
73C24×C5C2×AGL(1,𝔽5)184yescomplemented
74C23×D5—596no—
75C23×D5—542no—
76C23×D5—542no—
77C23×D5—584no—
78C23×D5—5168no—
79C23×D5—5504no—
80C23×D5—10672no—
81C23×D5—10672no—
82C23×D5—1048no—
83C22×AGL(1,𝔽5)—5336no—
84C22×AGL(1,𝔽5)—10672no—
85C22⋊AGL(1,𝔽5)—5384no—
86C26—251no—
87C24×C4—252no—
88C22×C22⋊C4—2528no—
89C2×C52—28no—
90C2×C52C22×C22⋊C4114yescomplemented
91C2×C52C24×C411yescharacteristic
92C5⋊D5—28no—
93C5⋊D5C22×C22⋊C4114yescomplemented
94C5⋊D5C23×D412yes—
95C23×C5—2336no—
96C23×C5C22⋊AGL(1,𝔽5)148yescomplemented
97C23×C5C22×AGL(1,𝔽5)142yes—
98C23×C5C22×AGL(1,𝔽5)1168yescomplemented
99C22×D5—5336no—
100C22×D5—5336no—
101C22×D5—5126no—
102C22×D5—542no—
103C22×D5—5168no—
104C22×D5—5336no—
105C22×D5—548no—
106C22×D5—102016no—
107C22×D5—10672no—
108C22×D5—10336no—
109C2×AGL(1,𝔽5)—596no—
110C2×AGL(1,𝔽5)—10672no—
111C25—251no—
112C25—251no—
113C25—2514no—
114C25—2514no—
115C25—5016no—
116C25#104311yescharacteristic, complemented
117C23×C4—2528no—
118C23×C4—5016no—
119C2×C22⋊C4—25112no—
120C52C23×C22⋊C411yescharacteristic, complemented, Sylow
121C22×C5—2336no—
122C22×C5C2×C22⋊AGL(1,𝔽5)1168yescomplemented
123C22×C5C23×AGL(1,𝔽5)142yes—
124C22×C5C23×AGL(1,𝔽5)148yescomplemented
125C2×D5—5504no—
126C2×D5—5168no—
127C2×D5—542no—
128C2×D5—56no—
129C2×D5—548no—
130C2×D5—10672no—
131C2×D5—1096no—
132C2×D5—10336no—
133AGL(1,𝔽5)—1096no—
134C24—28no—
135C24—2516no—
136C24—257no—
137C24—257no—
138C24—2514no—
139C24—2528no—
140C24—2584no—
141C24—50112no—
142C24—50112no—
143C24—508no—
144C24C2×#104311yescharacteristic
145C24C2×#1043114yescomplemented
146C22×C4—2556no—
147C22×C4—50112no—
148C22⋊C4—2564no—
149C2×C5—248no—
150C2×C5C22×C22⋊AGL(1,𝔽5)184yescomplemented
151C2×C5C24×AGL(1,𝔽5)16yes—
152D5—584no—
153D5—512no—
154D5—1048no—
155C23—256no—
156C23—2556no—
157C23—2556no—
158C23—257no—
159C23—2521no—
160C23—2528no—
161C23—258no—
162C23—2556no—
163C23—50112no—
164C23—50336no—
165C23—5056no—
166C23C22⋊#104318yesdirect product factor
167C23C22×#104317yes—
168C23C22×#1043128yescomplemented
169C2×C4—2516no—
170C2×C4—50112no—
171C5C23×C22⋊AGL(1,𝔽5)16yescomplemented
172C22—256no—
173C22—2528no—
174C22—2584no—
175C22—251no—
176C22—257no—
177C22—258no—
178C22—5016no—
179C22—50112no—
180C22—5056no—
181C22C2×C22⋊#1043128yesdirect product factor
182C22C23×#104317yes—
183C22C23×#104318yescomplemented
184C4—5016no—
185C2—28no—
186C2—252no—
187C2—2514no—
188C2—508no—
189C2C22×C22⋊#1043114yesdirect product factor
190C2C24×#104311yescharacteristic
subgroupG/H
centerC24C2×#1043
commutatorC2×C52C24×C4
FrattiniC2C24×#1043
FittingC25×C52C4
socleC24×C52C2×C4
prime radicalC25×C5⋊D5C2
cosocleC2×C5⋊D5C25
hypercenterC25#1043
hypocenterC52C23×C22⋊C4
LCSG ▷ C2×C52 ▷ C52
UCSC1 ◁ C24 ◁ C25

Complements

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

NH ≅ G/Ntype
2. C24×#1043C2semidirect
3. C22×C22⋊#1043C2direct (N × H)
4. C25×C52C4semidirect
6. C24×C5⋊D5C4semidirect
7. C24×C5⋊D5C4semidirect
11. C23×#1043C22semidirect
12. C2×C22⋊#1043C22direct (N × H)
16. C24×C52C2×C4semidirect
21. C23×C5⋊D5C2×C4semidirect
22. C23×C5⋊D5C2×C4semidirect
27. C22×#1043C23semidirect
28. C22⋊#1043C23direct (N × H)
33. C23×C52C22⋊C4semidirect
35. C23×C52C22×C4semidirect
39. C22×C5⋊D5C22⋊C4semidirect
41. C22×C5⋊D5C22×C4semidirect
42. C22×C5⋊D5C22×C4semidirect
47. C2×#1043C24semidirect
48. C25×C5AGL(1,𝔽5)semidirect
59. C22×C52C2×C22⋊C4semidirect
61. C22×C52C23×C4semidirect
65. C2×C5⋊D5C2×C22⋊C4semidirect
67. C2×C5⋊D5C23×C4semidirect
73. C24×C5C2×AGL(1,𝔽5)semidirect
90. C2×C52C22×C22⋊C4semidirect
93. C5⋊D5C22×C22⋊C4semidirect
96. C23×C5C22⋊AGL(1,𝔽5)semidirect
98. C23×C5C22×AGL(1,𝔽5)semidirect
116. C25#1043semidirect
120. C52C23×C22⋊C4semidirect
122. C22×C5C2×C22⋊AGL(1,𝔽5)semidirect
124. C22×C5C23×AGL(1,𝔽5)semidirect
145. C24C2×#1043semidirect
150. C2×C5C22×C22⋊AGL(1,𝔽5)semidirect
166. C23C22⋊#1043direct (N × H)
168. C23C22×#1043semidirect
171. C5C23×C22⋊AGL(1,𝔽5)semidirect
181. C22C2×C22⋊#1043direct (N × H)
183. C22C23×#1043semidirect
189. C2C22×C22⋊#1043direct (N × H)

Maximal subgroups

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

subgroupindex
C25×C5⋊D52
C24×#10432
C22×C22⋊#10432
C23×C22⋊AGL(1,𝔽5)5