Finite Group Databasebeta

← #11756725

Subgroups — #11756725

Order 960.

158 conjugacy classes of subgroups; only the 28 normal subgroups are shown.

normal subgroups only

This group has 158 conjugacy classes of subgroups — too many to draw. Only its 28 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×#41⋊C3C211yesmaximal, characteristic
2#91886C212yesmaximal, complemented
3#65794—31nomaximal, Hall
4C5×#41⋊C3C411yescharacteristic, complemented
5C4×ΓL(1,𝔽16)—41nomaximal
6D5×SL(2,𝔽3)C2211yescharacteristic
7C5⋊(#41⋊C3)C411yescharacteristic, complemented
8#5424—51nomaximal, Hall
9D5×#41D311yescharacteristic
10#4130—31no—
11C2×D5⋊C8—31no—
12#4009—32no—
13#4007—32no—
14C2×ΓL(1,𝔽16)—42no—
15C3×C4×D5—41no—
16C5×SL(2,𝔽3)C2×C411yescharacteristic
17C2×#41⋊C3—51no—
18#859—52no—
19C5×#41Dic311yescharacteristic, complemented
20#713Dic311yescharacteristic, complemented
21Q8×D5C2×D311yescharacteristic
22#711—31no—
23D4×D5—31no—
24#709—31no—
25C2×C4×D5—31no—
26C22⋊AGL(1,𝔽5)—61no—
27C2×#170—61no—
28C4⋊AGL(1,𝔽5)—32no—
29C4×AGL(1,𝔽5)—61no—
30D5⋊C8—32no—
31D5⋊C8—61no—
32#416—151noSylow
33C2×C3×D5—41no—
34ΓL(1,𝔽16)—44no—
35C3×C4×C5—41no—
36C3×Dic5—41no—
37#41⋊C3—51no—
38#41⋊C3AGL(1,𝔽5)11yescharacteristic, complemented
39C2×SL(2,𝔽3)—51no—
40C4×Dic3—201no—
41C22×D5—31no—
42C2×AGL(1,𝔽5)—62no—
43C5×Q8C2×Dic311yescharacteristic
44C5×D4—31no—
45C2×C4×C5—31no—
46C22⋊D5—61no—
47C2×Dic5—31no—
48D20—31no—
49C4×D5—31no—
50C4×D5—31no—
51C4×D5—31no—
52C4×D5S411yescharacteristic
53Dic10—31no—
54#170—62no—
55C2×#41—51no—
56C22×C8—151no—
57#117—151no—
58#103—152no—
59#102—152no—
60C2×C3×C5—41no—
61C3×D5—42no—
62C2×C3×C4—201no—
63C2×Dic3—202no—
64SL(2,𝔽3)C2×AGL(1,𝔽5)11yescharacteristic
65C22×C5—31no—
66C2×D5—31no—
67C2×D5—62no—
68C2×D5C2×S411yescharacteristic
69AGL(1,𝔽5)—122no—
70C4×C5—31no—
71C4×C5SL(2,ℤ/4)11yescharacteristic
72Dic5—31no—
73Dic5SL(2,ℤ/4)11yescharacteristic
74#41—51no—
75#41—151no—
76#41—151no—
77#41ΓL(1,𝔽16)11yescharacteristic, complemented
78C2×Q8—51no—
79C2×D4—151no—
80C22×C4—151no—
81C2×C8—152no—
82C2×C8—301no—
83C2×C8—301no—
84C4⋊C4—152no—
85C22⋊C4—301no—
86C42—301no—
87C3×C5—41noHall
88C22×C3—201no—
89C3×C4—41no—
90C3×C4—201no—
91Dic3—204no—
92C2×C5—61no—
93C2×C5C2×SL(2,ℤ/4)11yescharacteristic
94D5—61no—
95D5#98512yes—
96C23—151no—
97Q8—151no—
98Q8C2×ΓL(1,𝔽16)11yescharacteristic
99D4—31no—
100D4—151no—
101D4—301no—
102C2×C4—31no—
103C2×C4—51no—
104C2×C4—151no—
105C2×C4—151no—
106C2×C4—151no—
107C2×C4—151no—
108C2×C4—302no—
109C8—302no—
110C2×C3—41no—
111C2×C3—202no—
112C5#542411yescharacteristic, complemented, Sylow
113C22—31no—
114C22—51no—
115C22—151no—
116C22—302no—
117C4—31no—
118C4—51no—
119C4—151no—
120C4—602no—
121C4A4⋊AGL(1,𝔽5)11yescharacteristic
122C3—41noSylow
123C2—52no—
124C2—61no—
125C2—301no—
126C2C2×A4⋊AGL(1,𝔽5)11yescharacteristic
subgroupG/H
center, hypercenterC4A4⋊AGL(1,𝔽5)
commutator, hypocenterC5×SL(2,𝔽3)C2×C4
FrattiniC2C2×A4⋊AGL(1,𝔽5)
FittingC5×#41Dic3
socleC2×C5C2×SL(2,ℤ/4)
prime radicalD5×#41⋊C3C2
cosocleD5×SL(2,𝔽3)C22
LCSG ▷ C5×SL(2,𝔽3)
UCSC1 ◁ C4

Complements

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

NH ≅ G/Ntype
2. #91886C2semidirect
4. C5×#41⋊C3C4semidirect
7. C5⋊(#41⋊C3)C4semidirect
19. C5×#41Dic3semidirect
20. #713Dic3semidirect
38. #41⋊C3AGL(1,𝔽5)semidirect
77. #41ΓL(1,𝔽16)semidirect
112. C5#5424semidirect

Maximal subgroups

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

subgroupindex
D5×#41⋊C32
#918862
#657943
C4×ΓL(1,𝔽16)4
#54245