Finite Group Databasebeta

← C7×#519⋊C3 #13060746

Subgroups — C7×#519⋊C3 #13060746

Order 1344.

172 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 172 conjugacy classes of subgroups — too many to draw. Only its 54 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×C7×SL(2,𝔽3)C211yesmaximal, characteristic
2C4×C7×SL(2,𝔽3)C212yesmaximal, complemented
3C7×#519C311yesmaximal, characteristic, complemented, Hall
4C2×C7×SL(2,𝔽3)C412yescomplemented
5C2×C7×SL(2,𝔽3)C2211yescharacteristic
6C3×C7×C22⋊C4—41nomaximal
7C22×C7×Q8C2×C311yescharacteristic
8C4×C7×Q8—32no—
9C4×C7×Q8C2×C312yescomplemented
10C7×#117—31no—
11C7×#117—31no—
12#519⋊C3C711yesmaximal, characteristic, direct product factor, Hall
13C23×C3×C7—41no—
14C2×C3×C4×C7—42no—
15C7×SL(2,𝔽3)C2×C411yescharacteristic
16C2×C7×Q8—31no—
17C2×C7×Q8—32no—
18C2×C7×Q8C3×C412yescomplemented
19C2×C7×Q8C22×C311yescharacteristic
20C22×C4×C7—31no—
21C7×C4⋊C4—32no—
22C7×C4⋊C4—32no—
23C7×C22⋊C4—31no—
24C7×C22⋊C4A411yescharacteristic
25C42×C7—32no—
26C42×C7—32no—
27C22×SL(2,𝔽3)C2×C711yescharacteristic
28C4×SL(2,𝔽3)C2×C712yescomplemented
29C22×C3×C7—42no—
30C22×C3×C7—41no—
31C22×C3×C7—82no—
32C3×C4×C7—82no—
33#519C3×C711yescharacteristic, complemented, Sylow
34C23×C7C2×A411yescharacteristic
35C7×Q8—32no—
36C7×Q8—31no—
37C7×Q8—32no—
38C7×Q8C2×C3×C411yescharacteristic
39C2×C4×C7—32no—
40C2×C4×C7—32no—
41C2×C4×C7—31no—
42C2×C4×C7—31no—
43C2×C4×C7—32no—
44C2×C4×C7C2×A412yes—
45C2×SL(2,𝔽3)C4×C712yescomplemented
46C2×SL(2,𝔽3)C22×C711yescharacteristic
47C3×C22⋊C4—41no—
48C2×C3×C7—41no—
49C2×C3×C7—42no—
50C2×C3×C7—82no—
51C22×Q8C2×C3×C711yescharacteristic
52C4×Q8—32no—
53C4×Q8C2×C3×C712yescomplemented
54#117—31no—
55#117—31no—
56C22×C7—22no—
57C22×C7C4×A412yes—
58C22×C7C22×A411yescharacteristic
59C4×C7—22no—
60C4×C7—31no—
61C4×C7—32no—
62C4×C7—31no—
63C4×C7—62no—
64C23×C3—41no—
65C2×C3×C4—42no—
66SL(2,𝔽3)C2×C4×C711yescharacteristic
67C3×C7—41noHall
68C2×Q8—31no—
69C2×Q8—32no—
70C2×Q8C3×C4×C712yescomplemented
71C2×Q8C22×C3×C711yescharacteristic
72C22×C4—31no—
73C4⋊C4—32no—
74C4⋊C4—32no—
75C22⋊C4—31no—
76C22⋊C4C7×A411yescharacteristic
77C42—32no—
78C42—32no—
79C2×C7—22no—
80C2×C7C2×C4×A411yescharacteristic
81C2×C72−1+4⋊C312yes—
82C22×C3—42no—
83C22×C3—41no—
84C22×C3—82no—
85C3×C4—82no—
86C23C2×C7×A411yescharacteristic
87Q8—32no—
88Q8—31no—
89Q8—32no—
90Q8C2×C3×C4×C711yescharacteristic
91C2×C4—32no—
92C2×C4—32no—
93C2×C4—31no—
94C2×C4—31no—
95C2×C4—32no—
96C2×C4C2×C7×A412yes—
97C7#519⋊C311yescharacteristic, direct product factor, Sylow
98C2×C3—41no—
99C2×C3—42no—
100C2×C3—82no—
101C22—22no—
102C22C4×C7×A412yes—
103C22C22×C7×A411yescharacteristic
104C4—22no—
105C4—31no—
106C4—31no—
107C4—32no—
108C4—62no—
109C3—41noSylow
110C2—22no—
111C2C2×C4×C7×A411yescharacteristic
112C2C7×2−1+4⋊C312yes—
subgroupG/H
center, socleC22×C7C22×A4
commutator, hypocenterQ8C2×C3×C4×C7
FrattiniC22C22×C7×A4
FittingC7×#519C3
prime radicalC22×C7×SL(2,𝔽3)C2
cosocleC2×Q8C22×C3×C7
hypercenterC7×C22⋊C4A4
LCSG ▷ Q8
UCSC1 ◁ C22×C7 ◁ C7×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C4×C7×SL(2,𝔽3)C2semidirect
3. C7×#519C3semidirect
4. C2×C7×SL(2,𝔽3)C4semidirect
9. C4×C7×Q8C2×C3semidirect
12. #519⋊C3C7direct (N × H)
18. C2×C7×Q8C3×C4semidirect
28. C4×SL(2,𝔽3)C2×C7semidirect
33. #519C3×C7semidirect
45. C2×SL(2,𝔽3)C4×C7semidirect
53. C4×Q8C2×C3×C7semidirect
70. C2×Q8C3×C4×C7semidirect
97. C7#519⋊C3direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C7×SL(2,𝔽3)2
C4×C7×SL(2,𝔽3)2
C7×#5193
C3×C7×C22⋊C44
#519⋊C37