Finite Group Databasebeta

← #72231372

Subgroups — #72231372

Order 3600.

346 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 346 conjugacy classes of subgroups — too many to draw. Only its 62 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×C3×C3⋊#1044C211yesmaximal, characteristic
2C2×C3×C3⋊#1043C211yesmaximal, characteristic
3#14268268C211yesmaximal, characteristic
4#11925359—31nomaximal
5#11925354—31nomaximal
6C2×C32×C5⋊D5C2211yescharacteristic
7C3×C3⋊#1044C412yescomplemented
8C3×C3⋊#1043C412yescomplemented
9C5×C3⋊Dic15C412yescomplemented
10ΓL(1,𝔽16)⋊Dic3—52nomaximal
11C2×C3⋊#1044D311yescharacteristic
12C2×C3⋊#1043D311yescharacteristic
13C2×C3×#1044—31no—
14C2×C3×#1043—31no—
15#10597126—31no—
16#10597126—31no—
17#10597126—61no—
18C2×C32×C52C2×C411yescharacteristic
19C32×C5⋊D5C2×C411yescharacteristic
20C32×C5⋊D5C2×C411yescharacteristic
21#87830—91noHall
22C2×C3×ΓL(1,𝔽16)—52no—
23C2×C3×ΓL(1,𝔽16)—52no—
24C2×C3×ΓL(1,𝔽16)—102no—
25D5×C3⋊Dic3—52no—
26C2×C3×C5⋊D5—21no—
27C2×C3×C5⋊D5C2×D311yescharacteristic
28C2×C3×C5⋊D5C2×D311yescharacteristic
29C3⋊#1044Dic312yescomplemented
30C3⋊#1043Dic312yescomplemented
31C3×#1044—32no—
32C3×#1043—32no—
33C5×Dic15—32no—
34C5×Dic15—32no—
35C5×Dic15—62no—
36Dic3×AGL(1,𝔽5)—152no—
37Dic3×AGL(1,𝔽5)—152no—
38C32×C52C4211yescharacteristic, complemented, Hall
39C2×#1044—31no—
40C2×#1043—31no—
41#6036—91no—
42C2×C32×D5—52no—
43C2×C32×D5—102no—
44C3×ΓL(1,𝔽16)—54no—
45C3×ΓL(1,𝔽16)—54no—
46C3×ΓL(1,𝔽16)—104no—
47C3⋊Dic15AGL(1,𝔽5)12yescomplemented
48C5×C3⋊Dic3—52no—
49C2×C3×C52—21no—
50C2×C3×C52C4×D311yescharacteristic
51C2×C3×C52C4×D311yescharacteristic
52C3×C5⋊D5—21no—
53C3×C5⋊D5—21no—
54C3×C5⋊D5C4×D311yescharacteristic
55C3×C5⋊D5C4×D311yescharacteristic
56C3×C5⋊D5C2×Dic311yescharacteristic
57C3×C5⋊D5C2×Dic311yescharacteristic
58Dic32—251noHall
59C2×ΓL(1,𝔽16)—52no—
60C2×ΓL(1,𝔽16)—52no—
61C2×ΓL(1,𝔽16)—102no—
62C2×C3×AGL(1,𝔽5)—152no—
63C2×C3×AGL(1,𝔽5)—152no—
64C2×C3×AGL(1,𝔽5)—302no—
65D5×Dic3—152no—
66D5×Dic3—152no—
67D5×Dic3—302no—
68C2×C5⋊D5D3211yescharacteristic
69#1044—32no—
70#1043—32no—
71C5×Dic5—92no—
72C2×C32×C5—22no—
73C2×C32×C5C2×AGL(1,𝔽5)12yes—
74C32×D5—52no—
75C32×D5—52no—
76C32×D5—102no—
77C32×D5—102no—
78C4×AGL(1,𝔽5)—452no—
79C3×C52—21no—
80C3×C52C4×Dic311yescharacteristic, complemented
81C3×C52C4×Dic311yescharacteristic, complemented
82C2×C3⋊Dic3—251no—
83C2×C3×Dic3—251no—
84C2×C3×Dic3—251no—
85C2×C3×D5—52no—
86C2×C3×D5—52no—
87C2×C3×D5—102no—
88C2×C3×D5—102no—
89C2×C3×D5—102no—
90C2×C3×D5—202no—
91ΓL(1,𝔽16)—54no—
92ΓL(1,𝔽16)—54no—
93ΓL(1,𝔽16)—104no—
94C3×AGL(1,𝔽5)—154no—
95C3×AGL(1,𝔽5)—154no—
96C3×AGL(1,𝔽5)—304no—
97Dic15—32no—
98Dic15—32no—
99Dic15—62no—
100C5×Dic3—152no—
101C5×Dic3—152no—
102C5×Dic3—302no—
103C2×C52#62411yescharacteristic
104C5⋊D5D3×Dic311yescharacteristic
105C5⋊D5D3×Dic311yescharacteristic
106C4×Dic3—751no—
107C4×Dic3—751no—
108C32×C5—22no—
109C32×C5C4×AGL(1,𝔽5)12yescomplemented
110C2×AGL(1,𝔽5)—152no—
111C2×AGL(1,𝔽5)—152no—
112C2×AGL(1,𝔽5)—302no—
113C4×D5—452no—
114C22×C32—251no—
115C3⋊Dic3—52no—
116C3×Dic3—252no—
117C3×Dic3—252no—
118C2×C3×C5—22no—
119C2×C3×C5—22no—
120C2×C3×C5—22no—
121C2×C3×C5—42no—
122C2×C3×C5D3×AGL(1,𝔽5)12yes—
123C2×C3×C5D3×AGL(1,𝔽5)12yes—
124C3×D5—52no—
125C3×D5—52no—
126C3×D5—52no—
127C3×D5—52no—
128C3×D5—102no—
129C3×D5—102no—
130C3×D5—102no—
131C3×D5—102no—
132C3×D5—102no—
133C3×D5—102no—
134C3×D5—202no—
135C3×D5—202no—
136C52Dic3211yescharacteristic, complemented, Sylow
137C2×C3×C4—751no—
138C2×C3×C4—751no—
139C2×Dic3—251no—
140C2×Dic3—251no—
141C2×Dic3—751no—
142C2×Dic3—751no—
143C2×Dic3—1501no—
144C2×D5—52no—
145C2×D5—102no—
146AGL(1,𝔽5)—154no—
147AGL(1,𝔽5)—154no—
148AGL(1,𝔽5)—304no—
149C4×C5—452no—
150Dic5—92no—
151C2×C32—251no—
152C2×C32—251no—
153C2×C32D5⋊AGL(1,𝔽5)11yescharacteristic
154C42—2251noSylow
155C3×C5—22no—
156C3×C5—22no—
157C3×C5—22no—
158C3×C5—42no—
159C3×C5Dic3×AGL(1,𝔽5)12yescomplemented
160C3×C5Dic3×AGL(1,𝔽5)12yescomplemented
161C22×C3—251no—
162C22×C3—251no—
163C22×C3—501no—
164C3×C4—752no—
165C3×C4—752no—
166Dic3—152no—
167Dic3—152no—
168Dic3—252no—
169Dic3—252no—
170Dic3—302no—
171C2×C5—22no—
172C2×C5ΓL(1,𝔽16)⋊D312yes—
173D5—52no—
174D5—52no—
175D5—102no—
176D5—102no—
177C32#8783011yescharacteristic, complemented, Sylow
178C2×C4—751no—
179C2×C4—751no—
180C2×C4—2251no—
181C2×C3—21no—
182C2×C3—251no—
183C2×C3—251no—
184C2×C3—251no—
185C2×C3—251no—
186C2×C3—501no—
187C2×C3—501no—
188C2×C3#1059718211yescharacteristic
189C2×C3#1059718311yescharacteristic
190C5—22no—
191C5ΓL(1,𝔽16)⋊Dic312yescomplemented
192C22—251no—
193C4—452no—
194C4—752no—
195C4—752no—
196C3—21no—
197C3#1192535411yescharacteristic, complemented
198C3#1192535911yescharacteristic, complemented
199C2—251no—
200C2—251no—
201C2#1426843811yescharacteristic
subgroupG/H
center, Frattini, hypercenterC2#14268438
commutator, hypocenterC32×C52C42
Fitting, socleC2×C32×C52C2×C4
prime radical, cosocleC2×C32×C5⋊D5C22
LCSG ▷ C32×C52
UCSC1 ◁ C2

Complements

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

NH ≅ G/Ntype
7. C3×C3⋊#1044C4semidirect
8. C3×C3⋊#1043C4semidirect
9. C5×C3⋊Dic15C4semidirect
29. C3⋊#1044Dic3semidirect
30. C3⋊#1043Dic3semidirect
38. C32×C52C42semidirect
47. C3⋊Dic15AGL(1,𝔽5)semidirect
80. C3×C52C4×Dic3semidirect
81. C3×C52C4×Dic3semidirect
109. C32×C5C4×AGL(1,𝔽5)semidirect
136. C52Dic32semidirect
159. C3×C5Dic3×AGL(1,𝔽5)semidirect
160. C3×C5Dic3×AGL(1,𝔽5)semidirect
177. C32#87830semidirect
191. C5ΓL(1,𝔽16)⋊Dic3semidirect
197. C3#11925354semidirect
198. C3#11925359semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C3⋊#10442
C2×C3×C3⋊#10432
#142682682
#119253593
#119253543
ΓL(1,𝔽16)⋊Dic35