Finite Group Databasebeta

← C7×2−1+4⋊C3 #10621609

Subgroups — C7×2−1+4⋊C3 #10621609

Order 672.

92 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#41⋊C3C211yesmaximal, characteristic, complemented
2C2×C7×SL(2,𝔽3)C212yesmaximal, complemented
3C7×2−1+4C311yesmaximal, characteristic, complemented, Hall
4C3×C7×D4—41nomaximal
5C7×SL(2,𝔽3)C2211yescharacteristic
6C7×#41—31no—
7C7×#41—32no—
8C7×#41C2×C311yescharacteristic, complemented
9C2×C7×Q8—31no—
10C2×C7×Q8C2×C312yescomplemented
112−1+4⋊C3C711yesmaximal, characteristic, direct product factor, Hall
12C22×C3×C7—42no—
13C3×C4×C7—41no—
14C7×Q8—32no—
15C7×Q8—31no—
16C7×Q8C22×C311yescharacteristic
17C7×D4—31no—
18C7×D4—32no—
19C7×D4A411yescharacteristic
20C2×C4×C7—32no—
21C2×C4×C7—32no—
22C2×C4×C7—31no—
23#41⋊C3C2×C711yescharacteristic, complemented
24C2×SL(2,𝔽3)C2×C712yescomplemented
25C2×C3×C7—41no—
26C2×C3×C7—82no—
272−1+4C3×C711yescharacteristic, complemented, Sylow
28C22×C7—31no—
29C22×C7C2×A412yes—
30C4×C7—32no—
31C4×C7—31no—
32C4×C7C2×A411yescharacteristic
33C3×D4—41no—
34SL(2,𝔽3)C22×C711yescharacteristic
35C3×C7—41noHall
36#41—31no—
37#41—32no—
38#41C2×C3×C711yescharacteristic, complemented
39C2×Q8—31no—
40C2×Q8C2×C3×C712yescomplemented
41C2×C7—22no—
42C2×C7—61no—
43C2×C7C22×A411yescharacteristic
44C22×C3—42no—
45C3×C4—41no—
46Q8—32no—
47Q8—31no—
48Q8C22×C3×C711yescharacteristic
49D4—31no—
50D4—32no—
51D4C7×A411yescharacteristic
52C2×C4—32no—
53C2×C4—32no—
54C2×C4—31no—
55C72−1+4⋊C311yescharacteristic, direct product factor, Sylow
56C2×C3—41no—
57C2×C3—82no—
58C22—31no—
59C22C2×C7×A412yes—
60C4—31no—
61C4—32no—
62C4C2×C7×A411yescharacteristic
63C3—41noSylow
64C2—22no—
65C2—61no—
66C2C22×C7×A411yescharacteristic
subgroupG/H
center, socleC2×C7C22×A4
commutator, cosocle, hypocenterQ8C22×C3×C7
FrattiniC2C22×C7×A4
FittingC7×2−1+4C3
hypercenterC7×D4A4
LCSG ▷ Q8
UCSC1 ◁ C2×C7 ◁ C7×D4

Complements

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

NH ≅ G/Ntype
1. C7×#41⋊C3C2semidirect
2. C2×C7×SL(2,𝔽3)C2semidirect
3. C7×2−1+4C3semidirect
8. C7×#41C2×C3semidirect
10. C2×C7×Q8C2×C3semidirect
11. 2−1+4⋊C3C7direct (N × H)
23. #41⋊C3C2×C7semidirect
24. C2×SL(2,𝔽3)C2×C7semidirect
27. 2−1+4C3×C7semidirect
38. #41C2×C3×C7semidirect
40. C2×Q8C2×C3×C7semidirect
55. C72−1+4⋊C3direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#41⋊C32
C2×C7×SL(2,𝔽3)2
C7×2−1+43
C3×C7×D44
2−1+4⋊C37