Finite Group Databasebeta

← #13107788

Subgroups — #13107788

Order 1600.

703 conjugacy classes of subgroups; only the 225 normal subgroups are shown.

normal subgroups only

This group has 703 conjugacy classes of subgroups — too many to draw. Only its 225 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×C5×D4×D5C211yesmaximal, characteristic
2C2×C22⋊#1042C214yesmaximal, complemented
3D4×#1042C218yesmaximal, complemented
4#11717425C212yesmaximal, complemented
5C23×C5×D5C2212yes—
6C22×#1042C2218yescomplemented
7C2×C52×D4C411yescharacteristic, complemented
8C2×C5×C22⋊D5C412yescomplemented
9C5×D4×D5C414yescomplemented
10C5×D4×D5C2214yes—
11C2×C5×D20C411yescharacteristic, complemented
12C2×C4×C5×D5C2211yescharacteristic
13C22⋊#1042C2214yes—
14C22⋊#1042C2218yescomplemented
15C4⋊#1042C2214yescomplemented
16C4×#1042C2214yescomplemented
17#66258—51nomaximal
18#66132—51nomaximal
19C23×C52C2×C412yescomplemented
20C22×C5×D5—44no—
21C22×C5×D5C2×C412yescomplemented
22C22×C5×D5C2×C414yescomplemented
23C22×C5×D5C2311yescharacteristic
24C22×C5×D5C2314yes—
25C2×#1042—48no—
26C2×#1042C2318yes—
27C52×D4C2×C414yescomplemented
28C2×C4×C52C2×C411yescharacteristic, complemented
29C5×C22⋊D5C2×C418yescomplemented
30C2×C5×Dic5C2×C411yescharacteristic, complemented
31C5×D20C2×C414yescomplemented
32C4×C5×D5C2×C412yescomplemented
33C4×C5×D5C2312yes—
34C22×C5×D4—51no—
35C2×D4×D5D511yescharacteristic
36C2×C22⋊AGL(1,𝔽5)—54no—
37D4×AGL(1,𝔽5)—58no—
38#4130—52no—
39C2×C22⋊Dic5—54no—
40D4×Dic5—58no—
41#4072—52no—
42C22×C52—42no—
43C22×C52C22×C411yescharacteristic
44C22×C52C22×C414yes—
45C2×C5×D5—28no—
46C2×C5×D5—22no—
47C2×C5×D5—42no—
48C2×C5×D5—44no—
49C2×C5×D5C22×C411yescharacteristic
50C2×C5×D5C22×C414yes—
51C2×C5×D5C2411yescharacteristic
52#1042—28no—
53C4×C52C22×C412yes—
54C5×Dic5C22×C412yes—
55C24×C5—52no—
56C23×D5C2×D512yes—
57C22×AGL(1,𝔽5)—58no—
58C2×C5×D4—41no—
59C2×C5×D4—52no—
60C2×C5×D4—54no—
61C2×C5×D4—54no—
62C2×C5×D4—51no—
63C2×C5×D4Dic511yescharacteristic, complemented
64C2×C5×D4AGL(1,𝔽5)11yescharacteristic, complemented
65C22×C4×C5—51no—
66C2×C22⋊D5Dic512yescomplemented
67C22×Dic5—58no—
68D4×D5Dic514yescomplemented
69D4×D5C2×D514yes—
70C2×D20Dic511yescharacteristic, complemented
71C2×C4×D5C2×D511yescharacteristic
72C22⋊AGL(1,𝔽5)—54no—
73C22⋊AGL(1,𝔽5)—58no—
74C4⋊AGL(1,𝔽5)—54no—
75C4×AGL(1,𝔽5)—54no—
76C22⋊Dic5—54no—
77C22⋊Dic5—58no—
78C4⋊Dic5—54no—
79C4×Dic5—54no—
80#518—251noSylow
81C2×C52—21no—
82C2×C52—24no—
83C2×C52C23×C411yescharacteristic
84C5×D5—21no—
85C5×D5—24no—
86C5×D5H(5,𝔽2)12yes—
87C23×C5—42no—
88C23×C5—51no—
89C23×C5—54no—
90C23×C5—54no—
91C23×C5—52no—
92C23×C5—204no—
93C23×C5C2×Dic512yescomplemented
94C23×C5C2×AGL(1,𝔽5)12yescomplemented
95C22×D5—44no—
96C22×D5C2×Dic512yescomplemented
97C22×D5C2×Dic514yescomplemented
98C22×D5C22×D511yescharacteristic
99C22×D5C22×D514yes—
100C2×AGL(1,𝔽5)—58no—
101C2×AGL(1,𝔽5)—208no—
102C5×D4—44no—
103C5×D4—58no—
104C5×D4—54no—
105C5×D4C2×Dic514yescomplemented
106C5×D4C2×AGL(1,𝔽5)14yescomplemented
107C2×C4×C5—41no—
108C2×C4×C5—51no—
109C2×C4×C5—52no—
110C2×C4×C5—52no—
111C2×C4×C5C2×Dic511yescharacteristic, complemented
112C2×C4×C5C2×AGL(1,𝔽5)11yescharacteristic, complemented
113C22⋊D5C2×Dic518yescomplemented
114C2×Dic5—58no—
115C2×Dic5—208no—
116C2×Dic5C2×Dic511yescharacteristic, complemented
117D20C2×Dic514yescomplemented
118C4×D5C2×Dic512yescomplemented
119C4×D5C22×D512yes—
120C22×D4—51no—
121C4×D4—258no—
122#117—252no—
123C2×C22⋊C4—254no—
124C52#51811yescharacteristic, complemented, Sylow
125C22×C5—42no—
126C22×C5—42no—
127C22×C5—41no—
128C22×C5—44no—
129C22×C5—51no—
130C22×C5—51no—
131C22×C5—54no—
132C22×C5—84no—
133C22×C5—108no—
134C22×C5—102no—
135C22×C5—202no—
136C22×C5—204no—
137C22×C5C22×Dic511yescharacteristic
138C22×C5C22×Dic514yes—
139C22×C5C22×AGL(1,𝔽5)11yescharacteristic
140C22×C5C22×AGL(1,𝔽5)14yes—
141C2×D5—28no—
142C2×D5—22no—
143C2×D5—44no—
144C2×D5—42no—
145C2×D5C22×Dic511yescharacteristic
146C2×D5C22×Dic514yes—
147C2×D5C23×D511yescharacteristic
148AGL(1,𝔽5)—108no—
149C4×C5—42no—
150C4×C5—52no—
151C4×C5C22×Dic512yes—
152C4×C5C22×AGL(1,𝔽5)12yes—
153Dic5—108no—
154Dic5C22×Dic512yes—
155C24—52no—
156C2×D4—52no—
157C2×D4—54no—
158C2×D4—54no—
159C2×D4—51no—
160C2×D4#104211yescharacteristic, complemented
161C22×C4—51no—
162C22×C4—258no—
163C4⋊C4—254no—
164C22⋊C4—254no—
165C22⋊C4—258no—
166C42—254no—
167C2×C5—21no—
168C2×C5—24no—
169C2×C5—21no—
170C2×C5—24no—
171C2×C5—42no—
172C2×C5—41no—
173C2×C5—52no—
174C2×C5—84no—
175C2×C5—101no—
176C2×C5—104no—
177C2×C5C23×Dic511yescharacteristic
178C2×C5C23×AGL(1,𝔽5)11yescharacteristic
179D5—21no—
180D5—24no—
181D5#414412yes—
182C23—51no—
183C23—54no—
184C23—54no—
185C23—52no—
186C23—204no—
187C23C2×#104212yescomplemented
188D4—58no—
189D4—54no—
190D4C2×#104214yescomplemented
191C2×C4—51no—
192C2×C4—52no—
193C2×C4—52no—
194C2×C4—258no—
195C2×C4—1008no—
196C2×C4C2×#104211yescharacteristic, complemented
197C5—41no—
198C5#6613211yescharacteristic, complemented
199C5#6625811yescharacteristic, complemented
200C22—42no—
201C22—51no—
202C22—51no—
203C22—54no—
204C22—108no—
205C22—102no—
206C22—202no—
207C22—204no—
208C22C22×#104211yescharacteristic
209C22C22×#104214yes—
210C4—52no—
211C4—508no—
212C4C22×#104212yes—
213C2—21no—
214C2—24no—
215C2—52no—
216C2—101no—
217C2—104no—
218C2C23×#104211yescharacteristic
subgroupG/H
center, FrattiniC2C23×#1042
commutator, socleC2×C52C23×C4
FittingC2×C52×D4C4
prime radicalC2×C5×D4×D5C2
cosocleC2×C5×D5C24
hypercenterC2×D4#1042
hypocenterC52#518
LCSG ▷ C2×C52 ▷ C52
UCSC1 ◁ C2 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
2. C2×C22⋊#1042C2semidirect
3. D4×#1042C2semidirect
4. #11717425C2semidirect
6. C22×#1042C22semidirect
7. C2×C52×D4C4semidirect
8. C2×C5×C22⋊D5C4semidirect
9. C5×D4×D5C4semidirect
11. C2×C5×D20C4semidirect
14. C22⋊#1042C22semidirect
15. C4⋊#1042C22semidirect
16. C4×#1042C22semidirect
19. C23×C52C2×C4semidirect
21. C22×C5×D5C2×C4semidirect
22. C22×C5×D5C2×C4semidirect
27. C52×D4C2×C4semidirect
28. C2×C4×C52C2×C4semidirect
29. C5×C22⋊D5C2×C4semidirect
30. C2×C5×Dic5C2×C4semidirect
31. C5×D20C2×C4semidirect
32. C4×C5×D5C2×C4semidirect
63. C2×C5×D4Dic5semidirect
64. C2×C5×D4AGL(1,𝔽5)semidirect
66. C2×C22⋊D5Dic5semidirect
68. D4×D5Dic5semidirect
70. C2×D20Dic5semidirect
93. C23×C5C2×Dic5semidirect
94. C23×C5C2×AGL(1,𝔽5)semidirect
96. C22×D5C2×Dic5semidirect
97. C22×D5C2×Dic5semidirect
105. C5×D4C2×Dic5semidirect
106. C5×D4C2×AGL(1,𝔽5)semidirect
111. C2×C4×C5C2×Dic5semidirect
112. C2×C4×C5C2×AGL(1,𝔽5)semidirect
113. C22⋊D5C2×Dic5semidirect
116. C2×Dic5C2×Dic5semidirect
117. D20C2×Dic5semidirect
118. C4×D5C2×Dic5semidirect
124. C52#518semidirect
160. C2×D4#1042semidirect
187. C23C2×#1042semidirect
190. D4C2×#1042semidirect
196. C2×C4C2×#1042semidirect
198. C5#66132semidirect
199. C5#66258semidirect

Maximal subgroups

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

subgroupindex
C2×C5×D4×D52
C2×C22⋊#10422
D4×#10422
#117174252
#662585
#661325