Finite Group Databasebeta

← #13104111

Subgroups — #13104111

Order 1600.

648 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 648 conjugacy classes of subgroups — too many to draw. Only its 107 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×#6036C211yesmaximal, characteristic
2C2×C22⋊#1044C211yesmaximal, characteristic
3C2×C22⋊#1043C211yesmaximal, characteristic
4C2×#87830C212yesmaximal, complemented
5#11716776C212yesmaximal, complemented
6C23×C5⋊D5C2211yescharacteristic
7C22×#1044C2212yes—
8C22×#1043C2212yes—
9C22×C5×Dic5C412yescomplemented
10C2×#6036—24no—
11C2×#6036C412yescomplemented
12C2×#6036C2211yescharacteristic
13C2×#6036C2211yescharacteristic
14C22⋊#1044C414yescomplemented
15C22⋊#1043C414yescomplemented
16#87830—24no—
17C4×C22⋊AGL(1,𝔽5)—52nomaximal
18C23×C52C2×C411yescharacteristic
19C22×C5⋊D5—22no—
20C22×C5⋊D5—22no—
21C22×C5⋊D5C2×C411yescharacteristic
22C22×C5⋊D5C2×C412yes—
23C22×C5⋊D5C2×C412yes—
24C22×C5⋊D5C2311yescharacteristic
25C2×#1044—22no—
26C2×#1044—22no—
27C2×#1044C2×C414yescomplemented
28C2×#1043—22no—
29C2×#1043—22no—
30C2×#1043C2×C414yescomplemented
31C2×C5×Dic5—24no—
32C2×C5×Dic5C2×C412yes—
33C2×C5×Dic5C2×C412yes—
34#6036—22no—
35#6036—24no—
36#6036D414yes—
37C22×C4×D5—52no—
38C2×C22⋊AGL(1,𝔽5)—52no—
39C2×C22⋊AGL(1,𝔽5)—52no—
40C2×C22⋊AGL(1,𝔽5)—102no—
41C2×C4×AGL(1,𝔽5)—54no—
42(C2×AGL(1,𝔽5))⋊C4—54no—
43C22×C52—21no—
44C22×C52—21no—
45C22×C52C4212yescomplemented
46C22×C52C22×C411yescharacteristic
47C2×C5⋊D5—21no—
48C2×C5⋊D5—21no—
49C2×C5⋊D5—24no—
50C2×C5⋊D5—24no—
51C2×C5⋊D5C4212yescomplemented
52C2×C5⋊D5C22×C411yescharacteristic
53C2×C5⋊D5C22×C411yescharacteristic
54C2×C5⋊D5C2×D412yes—
55C2×C5⋊D5#4112yes—
56#1044—24no—
57#1043—24no—
58C5×Dic5—22no—
59C5×Dic5C22⋊C414yescomplemented
60C23×D5—52no—
61C23×D5—102no—
62C22×AGL(1,𝔽5)—54no—
63C22×AGL(1,𝔽5)—54no—
64C22×AGL(1,𝔽5)—104no—
65C22×C4×C5—52no—
66C22×Dic5AGL(1,𝔽5)12yescomplemented
67C2×C4×D5—52no—
68C2×C4×D5—52no—
69C2×C4×D5—52no—
70C2×C4×D5—52no—
71C2×C4×D5—108no—
72C22⋊AGL(1,𝔽5)—58no—
73C22⋊AGL(1,𝔽5)—58no—
74C22⋊AGL(1,𝔽5)—108no—
75C4×AGL(1,𝔽5)—108no—
76C4×C22⋊C4—251noSylow
77C2×C52—22no—
78C2×C52C2×C4211yescharacteristic
79C2×C52C2×C22⋊C411yescharacteristic
80C2×C52#11711yescharacteristic
81C5⋊D5—22no—
82C5⋊D5C4×D412yes—
83C5⋊D5C4×D412yes—
84C23×C5—22no—
85C23×C5C2×AGL(1,𝔽5)12yes—
86C22×D5—52no—
87C22×D5—54no—
88C22×D5—54no—
89C22×D5—52no—
90C22×D5—102no—
91C22×D5—104no—
92C22×D5—104no—
93C22×D5—102no—
94C22×D5—104no—
95C22×D5—104no—
96C22×D5—204no—
97C22×D5—204no—
98C2×AGL(1,𝔽5)—58no—
99C2×AGL(1,𝔽5)—58no—
100C2×AGL(1,𝔽5)—108no—
101C2×AGL(1,𝔽5)—104no—
102C2×AGL(1,𝔽5)—104no—
103C2×AGL(1,𝔽5)—104no—
104C2×AGL(1,𝔽5)—104no—
105C2×AGL(1,𝔽5)—204no—
106C2×AGL(1,𝔽5)—204no—
107C2×C4×C5—52no—
108C2×C4×C5—52no—
109C2×C4×C5—104no—
110C2×Dic5—24no—
111C2×Dic5C2×AGL(1,𝔽5)12yes—
112C2×Dic5C2×AGL(1,𝔽5)12yes—
113C4×D5—58no—
114C4×D5—104no—
115C4×D5—104no—
116C4×D5—104no—
117C23×C4—251no—
118C2×C22⋊C4—251no—
119C2×C22⋊C4—251no—
120C2×C42—252no—
121#95—252no—
122C52C4×C22⋊C411yescharacteristic, complemented, Sylow
123C22×C5—22no—
124C22×C5—22no—
125C22×C5—24no—
126C22×C5—22no—
127C22×C5—42no—
128C22×C5—42no—
129C22×C5C4×AGL(1,𝔽5)14yescomplemented
130C22×C5C22×AGL(1,𝔽5)12yes—
131C2×D5—52no—
132C2×D5—52no—
133C2×D5—54no—
134C2×D5—54no—
135C2×D5—54no—
136C2×D5—102no—
137C2×D5—102no—
138C2×D5—104no—
139C2×D5—104no—
140C2×D5—104no—
141C2×D5—108no—
142C2×D5—108no—
143C2×D5—102no—
144C2×D5—102no—
145C2×D5—208no—
146C2×D5—208no—
147C2×D5—202no—
148C2×D5—202no—
149AGL(1,𝔽5)—108no—
150AGL(1,𝔽5)—108no—
151AGL(1,𝔽5)—208no—
152C4×C5—54no—
153C4×C5—102no—
154Dic5—22no—
155Dic5C22⋊AGL(1,𝔽5)14yescomplemented
156C24—251no—
157C22×C4—52no—
158C22×C4—251no—
159C22×C4—251no—
160C22×C4—252no—
161C22×C4—252no—
162C22×C4—252no—
163C22×C4—504no—
164C22⋊C4—254no—
165C22⋊C4—254no—
166C42—504no—
167C2×C5—24no—
168C2×C5—22no—
169C2×C5—22no—
170C2×C5—22no—
171C2×C5—44no—
172C2×C5C2×C4×AGL(1,𝔽5)12yes—
173C2×C5#413012yes—
174C2×C5C2×C22⋊AGL(1,𝔽5)12yes—
175D5—54no—
176D5—54no—
177D5—104no—
178D5—104no—
179D5—104no—
180D5—204no—
181C23—251no—
182C23—252no—
183C23—252no—
184C23—251no—
185C23—502no—
186C23—502no—
187C23D5⋊AGL(1,𝔽5)11yescharacteristic
188C2×C4—52no—
189C2×C4—52no—
190C2×C4—104no—
191C2×C4—254no—
192C2×C4—254no—
193C2×C4—254no—
194C2×C4—504no—
195C2×C4—502no—
196C2×C4—502no—
197C2×C4—502no—
198C2×C4—502no—
199C2×C4—502no—
200C5—22no—
201C5C4×C22⋊AGL(1,𝔽5)12yescomplemented
202C22—21no—
203C22—21no—
204C22—252no—
205C22—251no—
206C22—251no—
207C22—252no—
208C22—252no—
209C22—504no—
210C22—504no—
211C22—501no—
212C22—501no—
213C22#8783012yescomplemented
214C22C2×D5⋊AGL(1,𝔽5)11yescharacteristic
215C4—54no—
216C4—102no—
217C4—504no—
218C4—504no—
219C2—22no—
220C2—252no—
221C2—252no—
222C2—502no—
223C2C2×#8783011yescharacteristic
224C2#1171736711yescharacteristic
225C2D5⋊(C22⋊AGL(1,𝔽5))11yescharacteristic
subgroupG/H
center, FrattiniC22C2×D5⋊AGL(1,𝔽5)
commutatorC2×C52C2×C42
FittingC23×C52C2×C4
socleC22×C52C22×C4
prime radicalC23×C5⋊D5C22
cosocleC22×C5⋊D5C23
hypercenterC23D5⋊AGL(1,𝔽5)
hypocenterC52C4×C22⋊C4
LCSG ▷ C2×C52 ▷ C52
UCSC1 ◁ C22 ◁ C23

Complements

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

NH ≅ G/Ntype
4. C2×#87830C2semidirect
5. #11716776C2semidirect
9. C22×C5×Dic5C4semidirect
11. C2×#6036C4semidirect
14. C22⋊#1044C4semidirect
15. C22⋊#1043C4semidirect
27. C2×#1044C2×C4semidirect
30. C2×#1043C2×C4semidirect
45. C22×C52C42semidirect
51. C2×C5⋊D5C42semidirect
59. C5×Dic5C22⋊C4semidirect
66. C22×Dic5AGL(1,𝔽5)semidirect
122. C52C4×C22⋊C4semidirect
129. C22×C5C4×AGL(1,𝔽5)semidirect
155. Dic5C22⋊AGL(1,𝔽5)semidirect
201. C5C4×C22⋊AGL(1,𝔽5)semidirect
213. C22#87830semidirect

Maximal subgroups

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

subgroupindex
C22×#60362
C2×C22⋊#10442
C2×C22⋊#10432
C2×#878302
#117167762
C4×C22⋊AGL(1,𝔽5)5