Finite Group Databasebeta

← #11756722

Subgroups — #11756722

Order 960.

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

normal subgroups only

This group has 162 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
1C2×D5×SL(2,𝔽3)C211yesmaximal, characteristic
2#91886C212yesmaximal, complemented
3#65782—31nomaximal, Hall
4C2×C5×SL(2,𝔽3)C411yescharacteristic, complemented
5#6763—41nomaximal
6D5×SL(2,𝔽3)C411yescharacteristic, complemented
7D5×SL(2,𝔽3)C2211yescharacteristic
8#5420—51nomaximal, Hall
9C2×Q8×D5D311yescharacteristic
10#4130—31no—
11D5⋊M16—31no—
12#4009—32no—
13#4009—32no—
14C22×C3×D5—41no—
15C2×ΓL(1,𝔽16)—42no—
16C5×SL(2,𝔽3)C2×C411yescharacteristic
17C22×SL(2,𝔽3)—51no—
18#859—52no—
19C2×C5×Q8Dic311yescharacteristic, complemented
20Q8×D5—31no—
21Q8×D5—31no—
22Q8×D5Dic311yescharacteristic, complemented
23Q8×D5C2×D311yescharacteristic
24C2×C4×D5—31no—
25C2×Dic10—31no—
26C22⋊AGL(1,𝔽5)—61no—
27#704—61no—
28C4⋊AGL(1,𝔽5)—32no—
29C4×AGL(1,𝔽5)—61no—
30#700—61no—
31D5⋊C8—32no—
32#419—151noSylow
33C22×C3×C5—41no—
34C2×C3×D5—41no—
35C2×C3×D5—41no—
36C2×C3×D5—82no—
37ΓL(1,𝔽16)—82no—
38C2×SL(2,𝔽3)—51no—
39C2×SL(2,𝔽3)—51no—
40C2×SL(2,𝔽3)AGL(1,𝔽5)11yescharacteristic, complemented
41#217—201no—
42C22×D5S411yescharacteristic
43C2×AGL(1,𝔽5)—62no—
44C5×Q8—31no—
45C5×Q8C2×Dic311yescharacteristic
46C2×C4×C5—31no—
47C2×Dic5—31no—
48C4×D5—31no—
49C4×D5—31no—
50C4×D5—31no—
51C4×D5—31no—
52Dic10—31no—
53Dic10—31no—
54Dic10—61no—
55#170—62no—
56C22×Q8—51no—
57C2×M16—151no—
58#117—151no—
59#103—152no—
60#103—152no—
61C2×C3×C5—41no—
62C2×C3×C5—81no—
63C3×D5—42no—
64C3×D5—81no—
65C23×C3—201no—
66C2×Dic3—202no—
67SL(2,𝔽3)C2×AGL(1,𝔽5)11yescharacteristic
68C22×C5SL(2,ℤ/4)11yescharacteristic
69C2×D5—22no—
70C2×D5SL(2,ℤ/4)11yescharacteristic
71C2×D5C2×S411yescharacteristic
72AGL(1,𝔽5)—122no—
73C4×C5—31no—
74C4×C5—31no—
75Dic5—31no—
76Dic5—31no—
77C2×Q8—51no—
78C2×Q8—51no—
79C2×Q8—151no—
80C2×Q8—151no—
81C2×Q8—151no—
82C2×Q8ΓL(1,𝔽16)11yescharacteristic, complemented
83C22×C4—151no—
84M16—301no—
85M16—301no—
86C2×C8—152no—
87C4⋊C4—152no—
88C22⋊C4—301no—
89C42—301no—
90C3×C5—41noHall
91C22×C3—41no—
92C22×C3—201no—
93C22×C3—201no—
94C22×C3—402no—
95Dic3—402no—
96C2×C5—21no—
97C2×C5C2×SL(2,ℤ/4)11yescharacteristic
98D5—21no—
99D5#98312yes—
100C23—51no—
101Q8—31no—
102Q8—151no—
103Q8—151no—
104Q8—301no—
105Q8C2×ΓL(1,𝔽16)11yescharacteristic
106C2×C4—31no—
107C2×C4—151no—
108C2×C4—151no—
109C2×C4—151no—
110C2×C4—151no—
111C2×C4—151no—
112C2×C4—302no—
113C8—302no—
114C2×C3—41no—
115C2×C3—81no—
116C2×C3—202no—
117C2×C3—401no—
118C5#542011yescharacteristic, complemented, Sylow
119C22—51no—
120C22—51no—
121C22—102no—
122C22A4⋊AGL(1,𝔽5)11yescharacteristic
123C4—31no—
124C4—31no—
125C4—151no—
126C4—151no—
127C4—602no—
128C3—41noSylow
129C2—21no—
130C2—52no—
131C2—101no—
132C2C2×A4⋊AGL(1,𝔽5)11yescharacteristic
subgroupG/H
center, FrattiniC2C2×A4⋊AGL(1,𝔽5)
commutator, hypocenterC5×SL(2,𝔽3)C2×C4
FittingC2×C5×Q8Dic3
socleC2×C5C2×SL(2,ℤ/4)
prime radicalC2×D5×SL(2,𝔽3)C2
cosocleD5×SL(2,𝔽3)C22
hypercenterC22A4⋊AGL(1,𝔽5)
LCSG ▷ C5×SL(2,𝔽3)
UCSC1 ◁ C2 ◁ C22

Complements

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

NH ≅ G/Ntype
2. #91886C2semidirect
4. C2×C5×SL(2,𝔽3)C4semidirect
6. D5×SL(2,𝔽3)C4semidirect
19. C2×C5×Q8Dic3semidirect
22. Q8×D5Dic3semidirect
40. C2×SL(2,𝔽3)AGL(1,𝔽5)semidirect
82. C2×Q8ΓL(1,𝔽16)semidirect
118. C5#5420semidirect

Maximal subgroups

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

subgroupindex
C2×D5×SL(2,𝔽3)2
#918862
#657823
#67634
#54205