Finite Group Databasebeta

← C2×C3×#6729 #13088308

Subgroups — C2×C3×#6729 #13088308

Order 1440.

174 conjugacy classes of subgroups; only the 20 normal subgroups are shown.

normal subgroups only

This group has 174 conjugacy classes of subgroups — too many to draw. Only its 20 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×SL(2,𝔽5)C211yesmaximal, characteristic, complemented
2C3×#6729C212yesmaximal, direct product factor
3C2×#6729C311yesmaximal, characteristic, direct product factor
4C3×SL(2,𝔽5)C2211yescharacteristic, complemented
5C2×C3×GL(2,𝔽3)—51nomaximal, Hall
6C2×C3×#170—61nomaximal
7C2×SL(2,𝔽5)C2×C311yescharacteristic, complemented
8#6729C2×C312yesdirect product factor
9C2×C3×#67—101nomaximal
10C2×C3×SL(2,𝔽3)—51no—
11C3×GL(2,𝔽3)—52no—
12C2×C3×Dic5—61no—
13C3×#170—62no—
14SL(2,𝔽5)C22×C311yescharacteristic, complemented
15C2×GL(2,𝔽3)—51no—
16C2×C3×SD16—151no—
17C2×#170—61no—
18C23×C32—101no—
19C22×C3×D3—101no—
20C3×#67—102no—
21C3×#67—102no—
22C2×C3×Dic3—101no—
23C3×SL(2,𝔽3)—51no—
24C22×C3×C5—61no—
25C3×Dic5—61no—
26C3×Dic5—61no—
27C2×C3×Q8—51no—
28C2×C3×D4—151no—
29C2×#67—101no—
30C2×SL(2,𝔽3)—51no—
31C2×SL(2,𝔽3)—101no—
32GL(2,𝔽3)—52no—
33C3×SD16—152no—
34C3×SD16—152no—
35C2×C3×C8—151no—
36C2×Dic5—61no—
37#170—62no—
38C22×C32—102no—
39C22×C32—101no—
40C22×C32—202no—
41C2×C3×D3—102no—
42C2×C3×D3—202no—
43C3×Dic3—101no—
44C3×Dic3—101no—
45C2×SD16—151noSylow
46C2×C3×C5—62no—
47C2×C3×C5—61no—
48C23×C3—101no—
49C23×C3—101no—
50C23×C3—201no—
51C22×D3—101no—
52C3×Q8—51no—
53C3×Q8—151no—
54C3×D4—152no—
55C3×D4—301no—
56C2×C3×C4—151no—
57#67—102no—
58#67—102no—
59C2×Dic3—101no—
60SL(2,𝔽3)—51no—
61SL(2,𝔽3)—101no—
62C3×C8—152no—
63C22×C5—61no—
64Dic5—61no—
65Dic5—61no—
66C2×C32—102no—
67C2×C32—101no—
68C2×C32—202no—
69C3×D3—202no—
70C2×Q8—51no—
71C2×D4—151no—
72SD16—152no—
73SD16—152no—
74C2×C8—151no—
75C3×C5—61no—
76C22×C3—102no—
77C22×C3—102no—
78C22×C3—101no—
79C22×C3—202no—
80C22×C3—202no—
81C22×C3—204no—
82C22×C3—202no—
83C22×C3—201no—
84C22×C3S511yescharacteristic
85C2×D3—102no—
86C2×D3—202no—
87C3×C4—151no—
88C3×C4—151no—
89Dic3—101no—
90Dic3—101no—
91C2×C5—61no—
92C2×C5—62no—
93C32—101noSylow
94C23—101no—
95Q8—51no—
96Q8—151no—
97D4—152no—
98D4—301no—
99C2×C4—151no—
100C8—152no—
101C2×C3—101no—
102C2×C3—102no—
103C2×C3—202no—
104C2×C3—204no—
105C2×C3—202no—
106C2×C3—201no—
107C2×C3—202no—
108C2×C3#672912yesdirect product factor
109C2×C3C2×S511yescharacteristic
110D3—202no—
111C5—61noSylow
112C22—102no—
113C22—202no—
114C22C3×S511yescharacteristic
115C4—151no—
116C4—151no—
117C3—101no—
118C3—201no—
119C3C2×#672911yescharacteristic, direct product factor
120C2—202no—
121C2C3×#672912yesdirect product factor
122C2C2×C3×S511yescharacteristic
subgroupG/H
center, Fitting, socle, hypercenterC22×C3S5
commutator, cosocle, hypocenterSL(2,𝔽5)C22×C3
FrattiniC2C2×C3×S5
LCSG ▷ SL(2,𝔽5)
UCSC1 ◁ C22×C3

Complements

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

NH ≅ G/Ntype
1. C2×C3×SL(2,𝔽5)C2semidirect
2. C3×#6729C2direct (N × H)
3. C2×#6729C3direct (N × H)
4. C3×SL(2,𝔽5)C22semidirect
7. C2×SL(2,𝔽5)C2×C3semidirect
8. #6729C2×C3direct (N × H)
14. SL(2,𝔽5)C22×C3semidirect
108. C2×C3#6729direct (N × H)
119. C3C2×#6729direct (N × H)
121. C2C3×#6729direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×SL(2,𝔽5)2
C3×#67292
C2×#67293
C2×C3×GL(2,𝔽3)5
C2×C3×#1706
C2×C3×#6710