Finite Group Databasebeta

← #11752138

Subgroups — #11752138

Order 960.

160 conjugacy classes of subgroups; only the 31 normal subgroups are shown.

normal subgroups only

This group has 160 conjugacy classes of subgroups — too many to draw. Only its 31 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
1C22×C5×SL(2,𝔽3)C211yesmaximal, characteristic
2C2×C5⋊#226C211yesmaximal, characteristic, complemented
3#91184C211yesmaximal, characteristic, complemented
4#65517—31nomaximal, Hall
5C2×C5×SL(2,𝔽3)—21no—
6C2×C5×SL(2,𝔽3)C2211yescharacteristic
7C5⋊#226—21no—
8#6719—41nomaximal
9#5419—51nomaximal, Hall
10C22×C5×Q8D311yescharacteristic
11C2×#689—31no—
12C2×#689—31no—
13#4070—31no—
14#3967—31no—
15#3967—31no—
16C22⋊#168—31no—
17C23×C3×C5—41no—
18C2×Dic15—41no—
19C2×Dic15—41no—
20C5×SL(2,𝔽3)D411yescharacteristic
21C22×SL(2,𝔽3)D511yescharacteristic
22C2×#226—51no—
23#859—51no—
24C2×C5×Q8—21no—
25C2×C5×Q8—31no—
26C2×C5×Q8—61no—
27C2×C5×Q8C2×D311yescharacteristic
28C22×C4×C5—31no—
29C2×Dic10—31no—
30C22⋊Dic5—61no—
31#689—61no—
32#689—62no—
33#689—61no—
34C4⋊Dic5—31no—
35#683—61no—
36C2×#168—31no—
37C2×#168—31no—
38#451—151noSylow
39C22×C3×C5—42no—
40C22×C3×C5—41no—
41C22×C3×C5—81no—
42C22×C3×C5—81no—
43Dic15—81no—
44Dic15—81no—
45C2×SL(2,𝔽3)—21no—
46C2×SL(2,𝔽3)C2×D511yescharacteristic
47#226—101no—
48#217—201no—
49C23×C5S411yescharacteristic
50C5×Q8—31no—
51C5×Q8—32no—
52C5×Q8—61no—
53C5×Q8#6711yescharacteristic
54C2×C4×C5—31no—
55C2×C4×C5—31no—
56C2×C4×C5—61no—
57C2×C4×C5—61no—
58C2×Dic5—61no—
59C2×Dic5—61no—
60Dic10—61no—
61Dic10—61no—
62#168—61no—
63#168—61no—
64C22×Q8D1511yescharacteristic
65C2×Q16—151no—
66C2×Q16—151no—
67C22⋊Q8—151no—
68#103—151no—
69#103—151no—
70C22⋊C8—151no—
71C2×C3×C5—41no—
72C2×C3×C5—41no—
73C2×C3×C5—41no—
74C2×C3×C5—82no—
75C23×C3—41no—
76C2×Dic3—201no—
77C2×Dic3—201no—
78SL(2,𝔽3)C22⋊D511yescharacteristic
79C22×C5—21no—
80C22×C5—21no—
81C22×C5#22612yescomplemented
82C22×C5C2×S411yescharacteristic
83C4×C5—31no—
84C4×C5—31no—
85C4×C5—61no—
86Dic5—121no—
87Dic5—121no—
88C2×Q8—21no—
89C2×Q8—31no—
90C2×Q8—61no—
91C2×Q8—151no—
92C2×Q8C2×D1511yescharacteristic
93C22×C4—31no—
94Q16—301no—
95Q16—302no—
96Q16—301no—
97C2×C8—151no—
98C2×C8—151no—
99C4⋊C4—151no—
100C4⋊C4—301no—
101C22⋊C4—301no—
102C3×C5—41noHall
103C22×C3—42no—
104C22×C3—41no—
105C22×C3—81no—
106C22×C3—81no—
107Dic3—401no—
108Dic3—401no—
109C2×C5—22no—
110C2×C5C2×#22611yescharacteristic
111C2×C5#98311yescharacteristic
112C2×C5GL(2,ℤ/4)11yescharacteristic
113C23C5⋊S411yescharacteristic
114Q8—31no—
115Q8—32no—
116Q8—61no—
117Q8—301no—
118Q8—301no—
119Q8#122011yescharacteristic
120C2×C4—31no—
121C2×C4—31no—
122C2×C4—61no—
123C2×C4—61no—
124C2×C4—301no—
125C2×C4—301no—
126C8—301no—
127C8—301no—
128C2×C3—41no—
129C2×C3—41no—
130C2×C3—41no—
131C2×C3—82no—
132C5#541911yescharacteristic, complemented, Sylow
133C22—21no—
134C22—21no—
135C22C5⋊#22612yescomplemented
136C22C2×C5⋊S411yescharacteristic
137C4—31no—
138C4—31no—
139C4—61no—
140C4—601no—
141C4—601no—
142C3—41noSylow
143C2—22no—
144C2C2×C5⋊#22611yescharacteristic
145C2#9195311yescharacteristic
146C2#9195811yescharacteristic
subgroupG/H
center, FrattiniC22C2×C5⋊S4
commutator, cosocleC2×C5×SL(2,𝔽3)C22
FittingC22×C5×Q8D3
socleC22×C5C2×S4
prime radicalC22×C5×SL(2,𝔽3)C2
hypercenterC23C5⋊S4
hypocenterC5×SL(2,𝔽3)D4
LCSG ▷ C2×C5×SL(2,𝔽3) ▷ C5×SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C23

Complements

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

NH ≅ G/Ntype
2. C2×C5⋊#226C2semidirect
3. #91184C2semidirect
81. C22×C5#226semidirect
132. C5#5419semidirect
135. C22C5⋊#226semidirect

Maximal subgroups

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

subgroupindex
C22×C5×SL(2,𝔽3)2
C2×C5⋊#2262
#911842
#655173
#67194
#54195