Finite Group Databasebeta

← #11752113

Subgroups — #11752113

Order 960.

154 conjugacy classes of subgroups; only the 29 normal subgroups are shown.

normal subgroups only

This group has 154 conjugacy classes of subgroups — too many to draw. Only its 29 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×C5⋊GL(2,𝔽3)C211yesmaximal, characteristic
2C4×C5×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
3#91184C211yesmaximal, characteristic, complemented
4#65315—31nomaximal, Hall
5C2×C5×SL(2,𝔽3)C2211yescharacteristic
6C5⋊GL(2,𝔽3)C412yescomplemented
7#6714—41nomaximal
8#5394—51nomaximal, Hall
9C4×C5×Q8D311yescharacteristic, complemented
10C2×#688—31no—
11C4×D20—31no—
12#3967—31no—
13#3941—31no—
14#3939—31no—
15#3935—31no—
16C22×D15—41no—
17C2×C3×C4×C5—41no—
18C2×Dic15—41no—
19C5×SL(2,𝔽3)C2×C411yescharacteristic, complemented
20C2×GL(2,𝔽3)—51no—
21C4×SL(2,𝔽3)D511yescharacteristic, complemented
22#859—51no—
23C2×C5×Q8C2×D311yescharacteristic
24C2×D20—31no—
25C2×C4×D5—61no—
26C5×C4⋊C4—31no—
27C42×C5—31no—
28#688—32no—
29#688—32no—
30#685—61no—
31C4⋊Dic5—31no—
32C2×#168—31no—
33C2×#168—31no—
34#440—151noSylow
35C22×C3×C5—41no—
36C2×D15—42no—
37C2×D15—81no—
38C2×D15—81no—
39C3×C4×C5—81no—
40Dic15—81no—
41C2×SL(2,𝔽3)C2×D511yescharacteristic
42GL(2,𝔽3)—52no—
43(C2×D3)⋊C4—201no—
44C22×D5—61no—
45C5×Q8—31no—
46C5×Q8C4×D311yescharacteristic
47C2×C4×C5—31no—
48C2×C4×C5—31no—
49C2×C4×C5S411yescharacteristic
50C2×Dic5—61no—
51D20—32no—
52D20—61no—
53C4×D5—122no—
54#168—32no—
55#168—61no—
56C2×SD16—151no—
57C4×Q8D1511yescharacteristic, complemented
58C4×D4—151no—
59#107—151no—
60#103—151no—
61#102—151no—
62#97—151no—
63C2×C3×C5—41no—
64C2×C3×C5—41no—
65C2×C3×C5—41no—
66D15—82no—
67C22×D3—201no—
68C2×C3×C4—41no—
69C2×Dic3—201no—
70SL(2,𝔽3)C4×D511yescharacteristic, complemented
71C22×C5C2×S411yescharacteristic
72C2×D5—62no—
73C2×D5—121no—
74C2×D5—121no—
75C4×C5—21no—
76C4×C5—31no—
77C4×C5—31no—
78C4×C5—61no—
79Dic5—121no—
80C2×Q8C2×D1511yescharacteristic
81C2×D4—151no—
82C22×C4—301no—
83SD16—152no—
84SD16—152no—
85C2×C8—151no—
86C2×C8—151no—
87C4⋊C4—31no—
88C4⋊C4—151no—
89C22⋊C4—301no—
90C42—31no—
91C3×C5—41noHall
92C22×C3—41no—
93C2×D3—202no—
94C2×D3—401no—
95C2×D3—401no—
96C3×C4—81no—
97Dic3—401no—
98C2×C5C4×S411yescharacteristic
99C2×C5#98311yescharacteristic
100C2×C5#98611yescharacteristic
101D5—122no—
102C23—301no—
103Q8—31no—
104Q8C4×D1511yescharacteristic
105D4—152no—
106D4—301no—
107C2×C4—31no—
108C2×C4—31no—
109C2×C4—301no—
110C2×C4—602no—
111C2×C4C5⋊S411yescharacteristic
112C8—152no—
113C8—301no—
114C2×C3—41no—
115C2×C3—41no—
116C2×C3—41no—
117D3—402no—
118C5#539411yescharacteristic, complemented, Sylow
119C22—302no—
120C22—601no—
121C22—601no—
122C22C2×C5⋊S411yescharacteristic
123C4—21no—
124C4—31no—
125C4—31no—
126C4—61no—
127C4—601no—
128C3—41noSylow
129C2—602no—
130C2C4×C5⋊S411yescharacteristic
131C2#9195311yescharacteristic
132C2#9195611yescharacteristic
subgroupG/H
center, FrattiniC22C2×C5⋊S4
commutator, hypocenterC5×SL(2,𝔽3)C2×C4
FittingC4×C5×Q8D3
socleC22×C5C2×S4
prime radicalC2×C5⋊GL(2,𝔽3)C2
cosocleC2×C5×SL(2,𝔽3)C22
hypercenterC2×C4C5⋊S4
LCSG ▷ C5×SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. C4×C5×SL(2,𝔽3)C2semidirect
3. #91184C2semidirect
6. C5⋊GL(2,𝔽3)C4semidirect
9. C4×C5×Q8D3semidirect
19. C5×SL(2,𝔽3)C2×C4semidirect
21. C4×SL(2,𝔽3)D5semidirect
57. C4×Q8D15semidirect
70. SL(2,𝔽3)C4×D5semidirect
118. C5#5394semidirect

Maximal subgroups

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

subgroupindex
C2×C5⋊GL(2,𝔽3)2
C4×C5×SL(2,𝔽3)2
#911842
#653153
#67144
#53945