Finite Group Databasebeta

← #72268656

Subgroups — #72268656

Order 3240.

172 conjugacy classes of subgroups; only the 30 normal subgroups are shown.

normal subgroups only

This group has 172 conjugacy classes of subgroups — too many to draw. Only its 30 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
1D5×#4176C211yesmaximal, characteristic
2#13108192C211yesmaximal, characteristic, complemented
3#13108185C211yesmaximal, characteristic, complemented
4#11761932—31nomaximal
5D3×C9⋊AGL(1,𝔽5)—31nomaximal
6C5⋊#4176C411yescharacteristic, complemented
7C5×#4176C411yescharacteristic, complemented
8D5×#731C2211yescharacteristic
9#4176⋊C4—51nomaximal, Hall
10D5×#272D311yescharacteristic
11D3×C9×D5—31no—
12C3⋊(C9⋊AGL(1,𝔽5))D311yescharacteristic, complemented
133−1+2⋊AGL(1,𝔽5)—61no—
14(C3⋊ΓL(1,𝔽16))⋊C3—31no—
15(C3⋊ΓL(1,𝔽16))⋊C3—31no—
16C3×C9⋊AGL(1,𝔽5)—31no—
17C5×#731C2×C411yescharacteristic, complemented, Hall
18D3×ΓL(1,𝔽16)—91no—
19D3×ΓL(1,𝔽16)—91no—
20C2×C9⋊AGL(1,𝔽5)—91no—
21C2×#4176—51no—
22#66331—51no—
23#66324—51no—
24C5⋊#272Dic311yescharacteristic
25C5×#272Dic311yescharacteristic
26C9×D15—31no—
27C5×D3×C9—31no—
28D5×3−1+2—21no—
29D5×H(3,𝔽3)C2×D311yescharacteristic
30C3×C9×D5C2×D311yescharacteristic
31#6199—151no—
32D3×Dic9—151no—
33C3×D3×D5—31no—
34C3×D3×D5—91no—
35C3⋊ΓL(1,𝔽16)—31no—
36C3⋊ΓL(1,𝔽16)—31no—
37C3×ΓL(1,𝔽16)—91no—
38C3×ΓL(1,𝔽16)—91no—
39C2×C9×D5—91no—
40C9⋊AGL(1,𝔽5)—31no—
41C9⋊AGL(1,𝔽5)—61no—
42C9⋊AGL(1,𝔽5)—181no—
43C2×#731—51no—
44#4176—51no—
45#4176AGL(1,𝔽5)11yescharacteristic, complemented
46C5×3−1+2—21no—
47C5×H(3,𝔽3)C2×Dic311yescharacteristic
48C3×C5×C9C4×D311yescharacteristic, complemented
49C2×ΓL(1,𝔽16)—271no—
50D3×AGL(1,𝔽5)—271no—
51D3×AGL(1,𝔽5)—271no—
52C2×#272—51no—
53C2×D3×C9—151no—
54C3⋊Dic9—51no—
553−1+2⋊C4—301no—
56(C3⋊Dic3)⋊C3—151no—
57(C3⋊Dic3)⋊C3—151no—
58C3×Dic9—151no—
59C3×D15—31no—
60C3×D15—91no—
61C3×C5×D3—31no—
62C3×C5×D3—91no—
63C32×D5—31no—
64C32×D5D3211yescharacteristic
65C2×C5×C9—91no—
66C9×D5—61no—
67C9×D5—91no—
68C9×D5—121no—
69#731C2×AGL(1,𝔽5)11yescharacteristic, complemented, Sylow
70D3×Dic3—451no—
71D3×Dic3—451no—
72C2×Dic9—451no—
73C2×C3×D5—91no—
74D3×D5—31no—
75D3×D5—181no—
76ΓL(1,𝔽16)—91no—
77ΓL(1,𝔽16)—91no—
78ΓL(1,𝔽16)—541no—
79ΓL(1,𝔽16)—541no—
80C3×AGL(1,𝔽5)—271no—
81C3×AGL(1,𝔽5)—271no—
82C2×3−1+2—101no—
83C2×H(3,𝔽3)—51no—
84C2×C3×C9—51no—
85#272—51no—
86#272ΓL(1,𝔽16)11yescharacteristic
87D3×C9—31no—
88D3×C9—151no—
89C32×C5—31no—
90C32×C5D3×Dic311yescharacteristic
91C5×C9—61no—
92C5×C9—121no—
93C2×AGL(1,𝔽5)—811noHall
94C2×C3×D3—151no—
95C2×C3×D3—451no—
96C3⋊Dic3—151no—
97C3⋊Dic3—151no—
98C3×Dic3—451no—
99C3×Dic3—451no—
100C22×C9—451no—
101Dic9—151no—
102Dic9—301no—
103Dic9—901no—
104C2×C3×C5—91no—
105D15—31no—
106D15—181no—
107C3×D5—61no—
108C3×D5—91no—
109C3×D5—361no—
110C3×D5((C3⋊D3)⋊C3)⋊C211yescharacteristic
111C5×D3—31no—
112C5×D3—181no—
1133−1+2—21no—
114H(3,𝔽3)C2×ΓL(1,𝔽16)11yescharacteristic
115C3×C9D3×AGL(1,𝔽5)11yescharacteristic, complemented
116C2×Dic3—1351no—
117C4×D3—1351no—
118C4×D3—1351no—
119C2×D5—91no—
120AGL(1,𝔽5)—271no—
121AGL(1,𝔽5)—271no—
122C2×C32—51no—
123C2×C32—151no—
124C3×D3—31no—
125C3×D3—91no—
126C3×D3—151no—
127C3×D3—451no—
128C2×C9—91no—
129C2×C9—301no—
130C2×C9—451no—
131C2×C9—601no—
132C3×C5—61no—
133C3×C5—361no—
134C3×C5((C3⋊D3)⋊C3)⋊C411yescharacteristic
135C22×C3—451no—
136C2×D3—151no—
137C2×D3—901no—
138C3×C4—1351no—
139C3×C4—1351no—
140Dic3—2701no—
141Dic3—451no—
142Dic3—451no—
143Dic3—1351no—
144C2×C5—91no—
145D5—91no—
146D5#4176⋊C211yescharacteristic
147C32—31no—
148C32D3×ΓL(1,𝔽16)11yescharacteristic
149C9—61no—
150C9—121no—
151C2×C4—4051noSylow
152C2×C3—51no—
153C2×C3—91no—
154C2×C3—151no—
155C2×C3—451no—
156C2×C3—901no—
157D3—31no—
158D3—151no—
159D3—181no—
160D3—901no—
161C5#4176⋊C411yescharacteristic, complemented, Sylow
162C22—451no—
163C4—1351no—
164C4—1351no—
165C3—31no—
166C3—181no—
167C3((C3⋊D3)⋊C3)⋊AGL(1,𝔽5)11yescharacteristic
168C2—51no—
169C2—91no—
170C2—451no—
subgroupG/H
center, hypercenterC1G
commutator, Fitting, hypocenterC5×#731C2×C4
FrattiniC32D3×ΓL(1,𝔽16)
socleC3×C5((C3⋊D3)⋊C3)⋊C4
prime radicalD5×#272D3
cosocleD5×#731C22
LCSG ▷ C5×#731
UCSC1

Complements

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

NH ≅ G/Ntype
2. #13108192C2semidirect
3. #13108185C2semidirect
6. C5⋊#4176C4semidirect
7. C5×#4176C4semidirect
12. C3⋊(C9⋊AGL(1,𝔽5))D3semidirect
17. C5×#731C2×C4semidirect
45. #4176AGL(1,𝔽5)semidirect
48. C3×C5×C9C4×D3semidirect
69. #731C2×AGL(1,𝔽5)semidirect
115. C3×C9D3×AGL(1,𝔽5)semidirect
161. C5#4176⋊C4semidirect

Maximal subgroups

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

subgroupindex
D5×#41762
#131081922
#131081852
#117619323
D3×C9⋊AGL(1,𝔽5)3
#4176⋊C45