Finite Group Databasebeta

← #72230973

Subgroups — #72230973

Order 2400.

420 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 77 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×#10597182C212yesmaximal
2C4×D15⋊D5C211yesmaximal, characteristic
3C4⋊(C3⋊#1044)C211yesmaximal, characteristic, complemented
4C3×C4⋊#1043C211yesmaximal, characteristic, complemented
5#11925356C212yesmaximal, complemented
6D5⋊(C4⋊AGL(1,𝔽5))—31nomaximal, Hall
7C2×D15⋊D5C2211yescharacteristic
8C2×C3⋊#1044C2212yes—
9C2×C3×#1043C2212yes—
10#10597182—24no—
11C4×C5×D15C412yescomplemented
12C3×C4×C5⋊D5C2211yescharacteristic
13#10597126C2211yescharacteristic
14#10597125C412yescomplemented
15D3×C4⋊AGL(1,𝔽5)—52nomaximal
16C2×D5⋊AGL(1,𝔽5)—32no—
17C4×D52—31no—
18C4⋊#1044—31no—
19C4⋊#1043D311yescharacteristic, complemented
20#87832—32no—
21C2×C5×D15C2×C412yes—
22C2×C3×C5⋊D5C2311yescharacteristic
23D15⋊D5D412yes—
24D15⋊D5Q812yes—
25C3⋊#1044—22no—
26C3×#1043—22no—
27C3×C4×C52C2×C411yescharacteristic, complemented
28C5×Dic15C2×C412yescomplemented
29C3×#1039C2×C411yescharacteristic, complemented
30C2×D3×AGL(1,𝔽5)—54no—
31C4×D3×D5—52no—
32C4⋊ΓL(1,𝔽16)—52no—
33C3×C4⋊AGL(1,𝔽5)—52no—
34C3×C4⋊AGL(1,𝔽5)—102no—
35Dic3⋊AGL(1,𝔽5)—54no—
36C2×D52—31no—
37C2×#1044—32no—
38C2×#1043C2×D312yes—
39D5⋊AGL(1,𝔽5)—64no—
40C4×C5⋊D5C2×D311yescharacteristic
41C4×C5×D5—32no—
42#6036—31no—
43D5×Dic5—32no—
44C2×C4⋊AGL(1,𝔽5)—152no—
45C2×C3×C52C22×C411yescharacteristic
46C5×D15C4⋊C414yescomplemented
47C3×C5⋊D5C2×D411yescharacteristic
48C3×C5⋊D5C2×Q811yescharacteristic
49C2×D3×D5—52no—
50C2×ΓL(1,𝔽16)—54no—
51C2×C3×AGL(1,𝔽5)—54no—
52C2×C3×AGL(1,𝔽5)—104no—
53D3×AGL(1,𝔽5)—108no—
54C4×D15AGL(1,𝔽5)12yescomplemented
55C4×C5×D3—52no—
56C3×C4×D5—52no—
57C3×C4×D5—102no—
58#1200—52no—
59D3×Dic5—52no—
60D5×Dic3—52no—
61C2×C5⋊D5C22×D311yescharacteristic
62C2×C5×D5—32no—
63D52—32no—
64D52—32no—
65#1044—62no—
66#1043—22no—
67C4×C52C4×D311yescharacteristic, complemented
68#1039C4×D311yescharacteristic, complemented
69C5×Dic5—32no—
70D3×C4⋊C4—251noHall
71C22×AGL(1,𝔽5)—154no—
72C2×C4×D5—152no—
73C4⋊AGL(1,𝔽5)—52no—
74C4⋊AGL(1,𝔽5)—102no—
75C4⋊AGL(1,𝔽5)—152no—
76C4⋊AGL(1,𝔽5)—154no—
77C3×C52C2×C4⋊C411yescharacteristic, complemented, Hall
78C2×D15C2×AGL(1,𝔽5)12yes—
79C2×C5×D3—52no—
80C2×C3×D5—52no—
81C2×C3×D5—102no—
82D3×D5—54no—
83D3×D5—54no—
84ΓL(1,𝔽16)—104no—
85C3×AGL(1,𝔽5)—104no—
86C3×AGL(1,𝔽5)—204no—
87C3×C4×C5—22no—
88C3×C4×C5C2×AGL(1,𝔽5)12yescomplemented
89Dic15C2×AGL(1,𝔽5)12yescomplemented
90C3×Dic5—52no—
91C3×Dic5—102no—
92C5×Dic3—52no—
93C2×C52C2×C4×D311yescharacteristic
94C5⋊D5D3×D411yescharacteristic
95C5⋊D5D3×Q811yescharacteristic
96C5×D5—34no—
97C2×C4×D3—251no—
98C2×C4×D3—252no—
99C3×C4⋊C4—251no—
100C4⋊Dic3—251no—
101Dic3⋊C4—252no—
102C22×D5—152no—
103C2×AGL(1,𝔽5)—54no—
104C2×AGL(1,𝔽5)—104no—
105C2×AGL(1,𝔽5)—154no—
106C2×AGL(1,𝔽5)—308no—
107C2×C4×C5—152no—
108C2×Dic5—152no—
109C4×D5—32no—
110C4×D5—52no—
111C4×D5—102no—
112C4×D5—152no—
113C4×D5—152no—
114C2×C4⋊C4—751noSylow
115C2×C3×C5—22no—
116C2×C3×C5C22×AGL(1,𝔽5)12yes—
117D15C4⋊AGL(1,𝔽5)14yescomplemented
118C3×D5—52no—
119C3×D5—52no—
120C3×D5—102no—
121C3×D5—102no—
122C5×D3—54no—
123C52D3×C4⋊C411yescharacteristic, complemented, Sylow
124C22×D3—251no—
125C2×C3×C4—251no—
126C2×C3×C4—252no—
127C2×Dic3—251no—
128C2×Dic3—252no—
129C4×D3—52no—
130C4×D3—252no—
131C4×D3—504no—
132C22×C5—152no—
133C2×D5—32no—
134C2×D5—52no—
135C2×D5—102no—
136C2×D5—154no—
137C2×D5—154no—
138AGL(1,𝔽5)—104no—
139AGL(1,𝔽5)—204no—
140AGL(1,𝔽5)—304no—
141C4×C5—22no—
142C4×C5—152no—
143C4×C5D3×AGL(1,𝔽5)12yescomplemented
144Dic5—32no—
145Dic5—52no—
146Dic5—102no—
147C22×C4—751no—
148C22×C4—752no—
149C4⋊C4—251no—
150C4⋊C4—752no—
151C4⋊C4—751no—
152C3×C5—22no—
153C3×C5C2×C4⋊AGL(1,𝔽5)12yescomplemented
154C22×C3—251no—
155C2×D3—52no—
156C2×D3—252no—
157C2×D3—252no—
158C3×C4—251no—
159C3×C4—502no—
160C3×C4D5⋊AGL(1,𝔽5)11yescharacteristic, complemented
161Dic3—52no—
162Dic3—502no—
163C2×C5—22no—
164C2×C5—154no—
165C2×C5C2×D3×AGL(1,𝔽5)12yes—
166D5—34no—
167D5—52no—
168D5—52no—
169D5—102no—
170D5—102no—
171C23—751no—
172C2×C4—152no—
173C2×C4—251no—
174C2×C4—252no—
175C2×C4—751no—
176C2×C4—752no—
177C2×C4—752no—
178C2×C4—1504no—
179C2×C3—251no—
180C2×C3—251no—
181C2×C3C2×D5⋊AGL(1,𝔽5)11yescharacteristic
182D3—54no—
183C5—22no—
184C5D3×C4⋊AGL(1,𝔽5)12yescomplemented
185C22—152no—
186C22—251no—
187C22—752no—
188C22—752no—
189C4—152no—
190C4—251no—
191C4—502no—
192C4—1502no—
193C4#1059718211yescharacteristic, complemented
194C3D5⋊(C4⋊AGL(1,𝔽5))11yescharacteristic, complemented, Sylow
195C2—154no—
196C2—251no—
197C2—251no—
198C2C2×#1059718211yescharacteristic
subgroupG/H
center, FrattiniC2C2×#10597182
commutator, socleC2×C3×C52C22×C4
FittingC3×C4×C52C2×C4
prime radicalC2×D15⋊D5C22
cosocleC2×C3×C5⋊D5C23
hypercenterC4#10597182
hypocenterC3×C52C2×C4⋊C4
LCSG ▷ C2×C3×C52 ▷ C3×C52
UCSC1 ◁ C2 ◁ C4

Complements

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

NH ≅ G/Ntype
3. C4⋊(C3⋊#1044)C2semidirect
4. C3×C4⋊#1043C2semidirect
5. #11925356C2semidirect
11. C4×C5×D15C4semidirect
14. #10597125C4semidirect
19. C4⋊#1043D3semidirect
27. C3×C4×C52C2×C4semidirect
28. C5×Dic15C2×C4semidirect
29. C3×#1039C2×C4semidirect
46. C5×D15C4⋊C4semidirect
54. C4×D15AGL(1,𝔽5)semidirect
67. C4×C52C4×D3semidirect
68. #1039C4×D3semidirect
77. C3×C52C2×C4⋊C4semidirect
88. C3×C4×C5C2×AGL(1,𝔽5)semidirect
89. Dic15C2×AGL(1,𝔽5)semidirect
117. D15C4⋊AGL(1,𝔽5)semidirect
123. C52D3×C4⋊C4semidirect
143. C4×C5D3×AGL(1,𝔽5)semidirect
153. C3×C5C2×C4⋊AGL(1,𝔽5)semidirect
160. C3×C4D5⋊AGL(1,𝔽5)semidirect
184. C5D3×C4⋊AGL(1,𝔽5)semidirect
193. C4#10597182semidirect
194. C3D5⋊(C4⋊AGL(1,𝔽5))semidirect

Maximal subgroups

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

subgroupindex
C2×#105971822
C4×D15⋊D52
C4⋊(C3⋊#1044)2
C3×C4⋊#10432
#119253562
D5⋊(C4⋊AGL(1,𝔽5))3
D3×C4⋊AGL(1,𝔽5)5