Finite Group Databasebeta

← #72231092

Subgroups — #72231092

Order 2400.

596 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 596 conjugacy classes of subgroups — too many to draw. Only its 103 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×C3×D52C211yesmaximal, characteristic
2C2×D5⋊ΓL(1,𝔽16)C212yesmaximal, complemented
3#11925540C211yesmaximal, characteristic, complemented
4#11925533C211yesmaximal, characteristic, complemented
5#11925350C212yesmaximal, complemented
6D5⋊(C22⋊AGL(1,𝔽5))—31nomaximal, Hall
7C22×C3×C5⋊D5C2211yescharacteristic
8C22×C3×C5×D5C412yescomplemented
9C2×C3×D52—24no—
10C2×C3×D52C412yescomplemented
11C2×C3×D52C2211yescharacteristic
12C2×C3×D52C2211yescharacteristic
13C2×C3⋊#1044C2212yescomplemented
14C2×C3⋊#1043C2212yescomplemented
15D5⋊ΓL(1,𝔽16)—24no—
16C2×#6763—52nomaximal
17C22×D52D311yescharacteristic
18C2×D5⋊AGL(1,𝔽5)—32no—
19C22⋊#1044—31no—
20C22⋊#1043—31no—
21#87831—32no—
22C22×C3×C52C2×C411yescharacteristic, complemented
23C2×C3×C5⋊D5—22no—
24C2×C3×C5⋊D5C2×C411yescharacteristic, complemented
25C2×C3×C5⋊D5C2311yescharacteristic
26C2×C3×C5×D5—24no—
27C2×C3×C5×D5C2×C412yes—
28C2×C3×C5×D5C2×C412yescomplemented
29C3×D52—22no—
30C3×D52—24no—
31C3×D52D414yes—
32C3⋊#1044—22no—
33C3⋊#1043—22no—
34C23×C3×D5—52no—
35C22×ΓL(1,𝔽16)—54no—
36#6763—52no—
37#6763—52no—
38#6763—54no—
39#6763—102no—
40C22×C5⋊D5C2×D311yescharacteristic
41C22×C5×D5Dic312yescomplemented
42C2×D52—24no—
43C2×D52Dic312yescomplemented
44C2×D52C2×D311yescharacteristic
45C2×D52C2×D311yescharacteristic
46C2×#1044—32no—
47C2×#1043—32no—
48D5⋊AGL(1,𝔽5)—64no—
49C2×C22⋊AGL(1,𝔽5)—152no—
50C2×C3×C52—21no—
51C2×C3×C52C22×C411yescharacteristic
52C3×C5⋊D5—21no—
53C3×C5⋊D5C2×D412yes—
54C3×C5×D5—22no—
55C3×C5×D5C22⋊C414yescomplemented
56C23×C3×C5—52no—
57C22×C3×D5—52no—
58C22×C3×D5—52no—
59C22×C3×D5—52no—
60C22×C3×D5—52no—
61C22×C3×D5—52no—
62C22×C3×D5—108no—
63C22×C3×D5—102no—
64C22×C3×D5AGL(1,𝔽5)12yescomplemented
65C2×ΓL(1,𝔽16)—54no—
66C2×ΓL(1,𝔽16)—54no—
67C2×ΓL(1,𝔽16)—108no—
68C2×ΓL(1,𝔽16)—104no—
69C22×C52C2×Dic311yescharacteristic, complemented
70C2×C5⋊D5—22no—
71C2×C5⋊D5C2×Dic311yescharacteristic, complemented
72C2×C5⋊D5C22×D311yescharacteristic
73C2×C5×D5—24no—
74C2×C5×D5C2×Dic312yes—
75C2×C5×D5C2×Dic312yescomplemented
76D52—24no—
77D52—22no—
78D52#6714yes—
79#1044—62no—
80#1043—62no—
81C2×#217—251noHall
82C23×D5—52no—
83C22×AGL(1,𝔽5)—154no—
84C22⋊AGL(1,𝔽5)—154no—
85C22⋊AGL(1,𝔽5)—152no—
86C22⋊AGL(1,𝔽5)—152no—
87C22⋊AGL(1,𝔽5)—302no—
88C3×C52C2×C22⋊C411yescharacteristic, complemented, Hall
89C22×C3×C5—22no—
90C22×C3×C5—52no—
91C22×C3×C5—52no—
92C22×C3×C5—104no—
93C22×C3×C5C2×AGL(1,𝔽5)12yescomplemented
94C2×C3×D5—24no—
95C2×C3×D5—52no—
96C2×C3×D5—58no—
97C2×C3×D5—52no—
98C2×C3×D5—102no—
99C2×C3×D5—102no—
100C2×C3×D5—104no—
101C2×C3×D5—104no—
102C2×C3×D5—104no—
103C2×C3×D5—104no—
104C2×C3×D5—204no—
105C2×C3×D5C2×AGL(1,𝔽5)12yes—
106C2×C3×D5C2×AGL(1,𝔽5)12yescomplemented
107ΓL(1,𝔽16)—104no—
108ΓL(1,𝔽16)—104no—
109ΓL(1,𝔽16)—204no—
110C2×C52—21no—
111C2×C52C22×Dic311yescharacteristic
112C5⋊D5—21no—
113C5⋊D5C2×#6712yes—
114C5×D5—22no—
115C5×D5#21714yescomplemented
116C24×C3—251no—
117C22×Dic3—252no—
118#217—251no—
119#217—252no—
120#217—251no—
121C23×C5—52no—
122C22×D5—52no—
123C22×D5—52no—
124C22×D5—52no—
125C22×D5—52no—
126C22×D5—52no—
127C22×D5—102no—
128C22×D5—108no—
129C22×D5ΓL(1,𝔽16)12yescomplemented
130C2×AGL(1,𝔽5)—154no—
131C2×AGL(1,𝔽5)—154no—
132C2×AGL(1,𝔽5)—304no—
133C2×AGL(1,𝔽5)—308no—
134C2×C22⋊C4—751noSylow
135C2×C3×C5—22no—
136C2×C3×C5—22no—
137C2×C3×C5—42no—
138C2×C3×C5—54no—
139C2×C3×C5—102no—
140C2×C3×C5C22×AGL(1,𝔽5)12yes—
141C3×D5—22no—
142C3×D5—54no—
143C3×D5—104no—
144C3×D5—102no—
145C3×D5—202no—
146C3×D5C22⋊AGL(1,𝔽5)14yescomplemented
147C52C2×#21711yescharacteristic, complemented, Sylow
148C23×C3—52no—
149C23×C3—251no—
150C23×C3—251no—
151C23×C3—252no—
152C23×C3—251no—
153C23×C3—504no—
154C2×Dic3—252no—
155C2×Dic3—252no—
156C2×Dic3—504no—
157C22×C5—22no—
158C22×C5—52no—
159C22×C5—52no—
160C22×C5—104no—
161C22×C5C2×ΓL(1,𝔽16)12yescomplemented
162C2×D5—24no—
163C2×D5—52no—
164C2×D5—58no—
165C2×D5—52no—
166C2×D5—102no—
167C2×D5—102no—
168C2×D5—104no—
169C2×D5—104no—
170C2×D5—104no—
171C2×D5—104no—
172C2×D5—204no—
173C2×D5C2×ΓL(1,𝔽16)12yes—
174C2×D5C2×ΓL(1,𝔽16)12yescomplemented
175AGL(1,𝔽5)—304no—
176AGL(1,𝔽5)—304no—
177AGL(1,𝔽5)—604no—
178C24—251no—
179C22×C4—752no—
180C22⋊C4—751no—
181C22⋊C4—751no—
182C22⋊C4—752no—
183C3×C5—22no—
184C3×C5C2×C22⋊AGL(1,𝔽5)12yescomplemented
185C22×C3—52no—
186C22×C3—52no—
187C22×C3—104no—
188C22×C3—254no—
189C22×C3—251no—
190C22×C3—251no—
191C22×C3—502no—
192C22×C3—504no—
193C22×C3—502no—
194C22×C3D5⋊AGL(1,𝔽5)11yescharacteristic, complemented
195Dic3—502no—
196Dic3—502no—
197C2×C5—22no—
198C2×C5—22no—
199C2×C5—42no—
200C2×C5—54no—
201C2×C5—102no—
202C2×C5C22×ΓL(1,𝔽16)12yes—
203D5—22no—
204D5—54no—
205D5—104no—
206D5—102no—
207D5—202no—
208D5#676314yescomplemented
209C23—52no—
210C23—251no—
211C23—251no—
212C23—252no—
213C23—251no—
214C23—504no—
215C2×C4—752no—
216C2×C4—752no—
217C2×C4—1504no—
218C2×C3—21no—
219C2×C3—54no—
220C2×C3—102no—
221C2×C3—252no—
222C2×C3—501no—
223C2×C3C2×D5⋊AGL(1,𝔽5)11yescharacteristic
224C5—22no—
225C5C2×#676312yescomplemented
226C22—52no—
227C22—52no—
228C22—104no—
229C22—254no—
230C22—251no—
231C22—251no—
232C22—502no—
233C22—504no—
234C22—502no—
235C22D5⋊ΓL(1,𝔽16)11yescharacteristic, complemented
236C4—1502no—
237C4—1502no—
238C3D5⋊(C22⋊AGL(1,𝔽5))11yescharacteristic, complemented, Sylow
239C2—21no—
240C2—54no—
241C2—102no—
242C2—252no—
243C2—501no—
244C2C2×D5⋊ΓL(1,𝔽16)11yescharacteristic
subgroupG/H
center, FrattiniC2C2×D5⋊ΓL(1,𝔽16)
commutator, socleC2×C3×C52C22×C4
FittingC22×C3×C52C2×C4
prime radicalC22×C3×D52C2
cosocleC2×C3×C5⋊D5C23
hypercenterC22D5⋊ΓL(1,𝔽16)
hypocenterC3×C52C2×C22⋊C4
LCSG ▷ C2×C3×C52 ▷ C3×C52
UCSC1 ◁ C2 ◁ C22

Complements

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

NH ≅ G/Ntype
2. C2×D5⋊ΓL(1,𝔽16)C2semidirect
3. #11925540C2semidirect
4. #11925533C2semidirect
5. #11925350C2semidirect
8. C22×C3×C5×D5C4semidirect
10. C2×C3×D52C4semidirect
13. C2×C3⋊#1044C22semidirect
14. C2×C3⋊#1043C22semidirect
22. C22×C3×C52C2×C4semidirect
24. C2×C3×C5⋊D5C2×C4semidirect
28. C2×C3×C5×D5C2×C4semidirect
41. C22×C5×D5Dic3semidirect
43. C2×D52Dic3semidirect
55. C3×C5×D5C22⋊C4semidirect
64. C22×C3×D5AGL(1,𝔽5)semidirect
69. C22×C52C2×Dic3semidirect
71. C2×C5⋊D5C2×Dic3semidirect
75. C2×C5×D5C2×Dic3semidirect
88. C3×C52C2×C22⋊C4semidirect
93. C22×C3×C5C2×AGL(1,𝔽5)semidirect
106. C2×C3×D5C2×AGL(1,𝔽5)semidirect
115. C5×D5#217semidirect
129. C22×D5ΓL(1,𝔽16)semidirect
146. C3×D5C22⋊AGL(1,𝔽5)semidirect
147. C52C2×#217semidirect
161. C22×C5C2×ΓL(1,𝔽16)semidirect
174. C2×D5C2×ΓL(1,𝔽16)semidirect
184. C3×C5C2×C22⋊AGL(1,𝔽5)semidirect
194. C22×C3D5⋊AGL(1,𝔽5)semidirect
208. D5#6763semidirect
225. C5C2×#6763semidirect
235. C22D5⋊ΓL(1,𝔽16)semidirect
238. C3D5⋊(C22⋊AGL(1,𝔽5))semidirect

Maximal subgroups

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

subgroupindex
C22×C3×D522
C2×D5⋊ΓL(1,𝔽16)2
#119255402
#119255332
#119253502
D5⋊(C22⋊AGL(1,𝔽5))3
C2×#67635