Finite Group Databasebeta

← C2×C3×C5⋊GL(2,𝔽3) #13089060

Subgroups — C2×C3×C5⋊GL(2,𝔽3) #13089060

Order 1440.

218 conjugacy classes of subgroups; only the 42 normal subgroups are shown.

normal subgroups only

This group has 218 conjugacy classes of subgroups — too many to draw. Only its 42 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×C3×C5×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
2C3×C5⋊GL(2,𝔽3)C212yesmaximal, direct product factor
3C2×C5⋊GL(2,𝔽3)C311yesmaximal, characteristic, direct product factor
4C2×C3×#688—31nomaximal
5C22×C3×D15—41nomaximal
6C3×C5×SL(2,𝔽3)C2211yescharacteristic, complemented
7C2×C3×GL(2,𝔽3)—51nomaximal, Hall
8C2×C3×C5×Q8D311yescharacteristic, complemented
9C2×C3×D20—31no—
10C2×C5×SL(2,𝔽3)—21no—
11C2×C5×SL(2,𝔽3)C2×C311yescharacteristic, complemented
12C5⋊GL(2,𝔽3)C2×C312yesdirect product factor
13C3×#688—32no—
14C3×#688—32no—
15C2×C3×#168—31no—
16C22×C32×C5—41no—
17C2×C3×D15—42no—
18C2×C3×D15—44no—
19C2×#688—31noHall
20C2×C3×SL(2,𝔽3)D511yescharacteristic, complemented
21C3×GL(2,𝔽3)—52no—
22C22×D15—41no—
23C22×C3×D5—61no—
24C3×C5×Q8—31no—
25C3×C5×Q8C2×D311yescharacteristic, complemented
26C2×C3×C4×C5—31no—
27C3×D20—32no—
28C3×D20—61no—
29C5×SL(2,𝔽3)—21no—
30C5×SL(2,𝔽3)C22×C311yescharacteristic, complemented
31C3×#168—32no—
32C2×GL(2,𝔽3)—51no—
33C2×C3×SD16—151no—
34C2×C32×C5—42no—
35C2×C32×C5—41no—
36C3×D15—44no—
37C2×C5×Q8C3×D311yescharacteristic, complemented
38C2×D20—31no—
39#688—32no—
40#688—32no—
41C2×#168—31no—
42C22×C3×D3—201no—
43C3×SL(2,𝔽3)C2×D511yescharacteristic, complemented
44C22×C3×C5—41no—
45C22×C3×C5—81no—
46C22×C3×C5S411yescharacteristic
47C2×D15—42no—
48C2×D15—44no—
49C2×C3×D5—62no—
50C2×C3×D5—122no—
51C3×C4×C5—31no—
52C3×C4×C5—31no—
53C2×C3×Q8D1511yescharacteristic, complemented
54C2×C3×D4—151no—
55C2×SL(2,𝔽3)—21no—
56C2×SL(2,𝔽3)C3×D511yescharacteristic, complemented
57GL(2,𝔽3)—52no—
58C3×SD16—152no—
59C3×SD16—152no—
60C2×C3×C8—151no—
61C32×C5—41noHall
62C22×D5—61no—
63C5×Q8—31no—
64C5×Q8C2×C3×D311yescharacteristic, complemented
65C2×C4×C5—31no—
66D20—32no—
67D20—61no—
68#168—32no—
69C22×C32—41no—
70C2×C3×D3—202no—
71C2×C3×D3—204no—
72C2×SD16—151noSylow
73C2×C3×C5—41no—
74C2×C3×C5—42no—
75C2×C3×C5—81no—
76C2×C3×C5—82no—
77C2×C3×C5GL(2,𝔽3)12yescomplemented
78C2×C3×C5C2×S411yescharacteristic
79D15—44no—
80C3×D5—122no—
81C23×C3—301no—
82C22×D3—201no—
83C3×Q8—31no—
84C3×Q8C2×D1511yescharacteristic, complemented
85C3×D4—152no—
86C3×D4—301no—
87C2×C3×C4—31no—
88SL(2,𝔽3)—21no—
89SL(2,𝔽3)C2×C3×D511yescharacteristic, complemented
90C3×C8—152no—
91C22×C5C3×S411yescharacteristic
92C2×D5—62no—
93C2×D5—122no—
94C4×C5—31no—
95C4×C5—31no—
96C2×C32—42no—
97C2×C32—41no—
98C3×D3—204no—
99C2×Q8C3×D1511yescharacteristic, complemented
100C2×D4—151no—
101SD16—152no—
102SD16—152no—
103C2×C8—151no—
104C3×C5—41no—
105C3×C5—81no—
106C3×C5C2×GL(2,𝔽3)11yescharacteristic, complemented
107C22×C3—41no—
108C22×C3—81no—
109C22×C3—302no—
110C22×C3—602no—
111C22×C3C5⋊S411yescharacteristic
112C2×D3—202no—
113C2×D3—204no—
114C3×C4—31no—
115C3×C4—31no—
116C2×C5C3×GL(2,𝔽3)12yescomplemented
117C2×C5C2×C3×S411yescharacteristic
118D5—122no—
119C32—41noSylow
120C23—301no—
121Q8—31no—
122Q8C2×C3×D1511yescharacteristic, complemented
123D4—152no—
124D4—301no—
125C2×C4—31no—
126C8—152no—
127C2×C3—42no—
128C2×C3—41no—
129C2×C3—82no—
130C2×C3—81no—
131C2×C3—602no—
132C2×C3C5⋊GL(2,𝔽3)12yesdirect product factor
133C2×C3C2×C5⋊S411yescharacteristic
134D3—204no—
135C5C2×C3×GL(2,𝔽3)11yescharacteristic, complemented, Sylow
136C22—302no—
137C22—602no—
138C22C3×C5⋊S411yescharacteristic
139C4—31no—
140C4—31no—
141C3—41no—
142C3—81no—
143C3C2×C5⋊GL(2,𝔽3)11yescharacteristic, direct product factor
144C2—602no—
145C2C3×C5⋊GL(2,𝔽3)12yesdirect product factor
146C2C2×C3×C5⋊S411yescharacteristic
subgroupG/H
center, hypercenterC22×C3C5⋊S4
commutator, cosocle, hypocenterC5×SL(2,𝔽3)C22×C3
FrattiniC2C2×C3×C5⋊S4
FittingC2×C3×C5×Q8D3
socleC22×C3×C5S4
LCSG ▷ C5×SL(2,𝔽3)
UCSC1 ◁ C22×C3

Complements

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

NH ≅ G/Ntype
1. C2×C3×C5×SL(2,𝔽3)C2semidirect
2. C3×C5⋊GL(2,𝔽3)C2direct (N × H)
3. C2×C5⋊GL(2,𝔽3)C3direct (N × H)
6. C3×C5×SL(2,𝔽3)C22semidirect
8. C2×C3×C5×Q8D3semidirect
11. C2×C5×SL(2,𝔽3)C2×C3semidirect
12. C5⋊GL(2,𝔽3)C2×C3direct (N × H)
20. C2×C3×SL(2,𝔽3)D5semidirect
25. C3×C5×Q8C2×D3semidirect
30. C5×SL(2,𝔽3)C22×C3semidirect
37. C2×C5×Q8C3×D3semidirect
43. C3×SL(2,𝔽3)C2×D5semidirect
53. C2×C3×Q8D15semidirect
56. C2×SL(2,𝔽3)C3×D5semidirect
64. C5×Q8C2×C3×D3semidirect
77. C2×C3×C5GL(2,𝔽3)semidirect
84. C3×Q8C2×D15semidirect
89. SL(2,𝔽3)C2×C3×D5semidirect
99. C2×Q8C3×D15semidirect
106. C3×C5C2×GL(2,𝔽3)semidirect
116. C2×C5C3×GL(2,𝔽3)semidirect
122. Q8C2×C3×D15semidirect
132. C2×C3C5⋊GL(2,𝔽3)direct (N × H)
135. C5C2×C3×GL(2,𝔽3)semidirect
143. C3C2×C5⋊GL(2,𝔽3)direct (N × H)
145. C2C3×C5⋊GL(2,𝔽3)direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×C5×SL(2,𝔽3)2
C3×C5⋊GL(2,𝔽3)2
C2×C5⋊GL(2,𝔽3)3
C2×C3×#6883
C22×C3×D154
C2×C3×GL(2,𝔽3)5