Finite Group Databasebeta

← C2×C7×2−1+4⋊C3 #13062307

Subgroups — C2×C7×2−1+4⋊C3 #13062307

Order 1344.

408 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 408 conjugacy classes of subgroups — too many to draw. Only its 102 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
1C7×2−1+4⋊C3C214yesmaximal, direct product factor
2C2×C7×#41⋊C3C211yesmaximal, characteristic, complemented
3C22×C7×SL(2,𝔽3)C212yesmaximal, complemented
4C2×C7×2−1+4C311yesmaximal, characteristic, complemented, Hall
5C2×C3×C7×D4—41nomaximal
6C7×#41⋊C3C2212yescomplemented
7C2×C7×SL(2,𝔽3)C2211yescharacteristic
8C2×C7×SL(2,𝔽3)C2214yescomplemented
9C7×2−1+4—34no—
10C7×2−1+4C2×C314yescomplemented
11C2×C7×#41—31no—
12C2×C7×#41—32no—
13C2×C7×#41C2×C311yescharacteristic, complemented
14C22×C7×Q8—31no—
15C22×C7×Q8C2×C312yescomplemented
16C2×2−1+4⋊C3C711yesmaximal, characteristic, direct product factor, Hall
17C23×C3×C7—42no—
18C3×C7×D4—44no—
19C2×C3×C4×C7—41no—
20C7×SL(2,𝔽3)C2311yescharacteristic
21C7×#41—38no—
22C7×#41—34no—
23C7×#41—34no—
24C7×#41—38no—
25C7×#41—32no—
26C7×#41C22×C312yescomplemented
27C2×C7×Q8—34no—
28C2×C7×Q8—34no—
29C2×C7×Q8—31no—
30C2×C7×Q8—32no—
31C2×C7×Q8—34no—
32C2×C7×Q8C22×C311yescharacteristic
33C2×C7×Q8C22×C314yescomplemented
34C2×C7×D4—32no—
35C2×C7×D4—31no—
36C2×C7×D4A411yescharacteristic
37C22×C4×C7—32no—
38C22×C4×C7—32no—
39C22×C4×C7—31no—
402−1+4⋊C3C2×C714yesdirect product factor
41C2×#41⋊C3C2×C711yescharacteristic, complemented
42C22×SL(2,𝔽3)C2×C712yescomplemented
43C22×C3×C7—44no—
44C22×C3×C7—41no—
45C22×C3×C7—84no—
46C3×C4×C7—42no—
47C2×2−1+4C3×C711yescharacteristic, complemented, Sylow
48C23×C7—31no—
49C23×C7C2×A412yes—
50C7×Q8—31no—
51C7×Q8—34no—
52C7×Q8—34no—
53C7×Q8—34no—
54C7×Q8C23×C311yescharacteristic
55C7×D4—32no—
56C7×D4—32no—
57C7×D4—38no—
58C7×D4C2×A414yes—
59C2×C4×C7—38no—
60C2×C4×C7—34no—
61C2×C4×C7—34no—
62C2×C4×C7—32no—
63C2×C4×C7—31no—
64C2×C4×C7—32no—
65C2×C4×C7—32no—
66C2×C4×C7C2×A411yescharacteristic
67C2×C3×D4—41no—
68#41⋊C3C22×C712yescomplemented
69C2×SL(2,𝔽3)C22×C711yescharacteristic
70C2×SL(2,𝔽3)C22×C714yescomplemented
71C2×C3×C7—42no—
72C2×C3×C7—41no—
73C2×C3×C7—84no—
742−1+4—34no—
752−1+4C2×C3×C714yescomplemented
76C2×#41—31no—
77C2×#41—32no—
78C2×#41C2×C3×C711yescharacteristic, complemented
79C22×Q8—31no—
80C22×Q8C2×C3×C712yescomplemented
81C22×C7—24no—
82C22×C7—32no—
83C22×C7—62no—
84C22×C7C22×A414yes—
85C22×C7C22×A411yescharacteristic
86C4×C7—34no—
87C4×C7—31no—
88C4×C7—31no—
89C4×C7C22×A412yes—
90C23×C3—42no—
91C3×D4—44no—
92C2×C3×C4—41no—
93SL(2,𝔽3)C23×C711yescharacteristic
94C3×C7—41noHall
95#41—38no—
96#41—34no—
97#41—34no—
98#41—38no—
99#41—32no—
100#41C22×C3×C712yescomplemented
101C2×Q8—34no—
102C2×Q8—34no—
103C2×Q8—31no—
104C2×Q8—32no—
105C2×Q8—34no—
106C2×Q8C22×C3×C711yescharacteristic
107C2×Q8C22×C3×C714yescomplemented
108C2×D4—31no—
109C2×D4—32no—
110C2×D4C7×A411yescharacteristic
111C22×C4—32no—
112C22×C4—32no—
113C22×C4—31no—
114C2×C7—24no—
115C2×C7—62no—
116C2×C72−1+4⋊C312yesdirect product factor
117C2×C7C23×A411yescharacteristic
118C22×C3—41no—
119C22×C3—44no—
120C22×C3—84no—
121C3×C4—42no—
122C23—31no—
123C23C2×C7×A412yes—
124Q8—31no—
125Q8—34no—
126Q8—34no—
127Q8—34no—
128Q8C23×C3×C711yescharacteristic
129D4—32no—
130D4—32no—
131D4—38no—
132D4C2×C7×A414yes—
133C2×C4—38no—
134C2×C4—32no—
135C2×C4—31no—
136C2×C4—34no—
137C2×C4—34no—
138C2×C4—32no—
139C2×C4—32no—
140C2×C4C2×C7×A411yescharacteristic
141C7C2×2−1+4⋊C311yescharacteristic, direct product factor, Sylow
142C2×C3—42no—
143C2×C3—41no—
144C2×C3—84no—
145C22—24no—
146C22—32no—
147C22—62no—
148C22C22×C7×A414yes—
149C22C22×C7×A411yescharacteristic
150C4—31no—
151C4—34no—
152C4—31no—
153C4C22×C7×A412yes—
154C3—41noSylow
155C2—24no—
156C2—62no—
157C2C7×2−1+4⋊C312yesdirect product factor
158C2C23×C7×A411yescharacteristic
subgroupG/H
center, socleC22×C7C22×A4
commutator, cosocle, hypocenterQ8C23×C3×C7
FrattiniC2C23×C7×A4
FittingC2×C7×2−1+4C3
hypercenterC2×C7×D4A4
LCSG ▷ Q8
UCSC1 ◁ C22×C7 ◁ C2×C7×D4

Complements

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

NH ≅ G/Ntype
1. C7×2−1+4⋊C3C2direct (N × H)
2. C2×C7×#41⋊C3C2semidirect
3. C22×C7×SL(2,𝔽3)C2semidirect
4. C2×C7×2−1+4C3semidirect
6. C7×#41⋊C3C22semidirect
8. C2×C7×SL(2,𝔽3)C22semidirect
10. C7×2−1+4C2×C3semidirect
13. C2×C7×#41C2×C3semidirect
15. C22×C7×Q8C2×C3semidirect
16. C2×2−1+4⋊C3C7direct (N × H)
26. C7×#41C22×C3semidirect
33. C2×C7×Q8C22×C3semidirect
40. 2−1+4⋊C3C2×C7direct (N × H)
41. C2×#41⋊C3C2×C7semidirect
42. C22×SL(2,𝔽3)C2×C7semidirect
47. C2×2−1+4C3×C7semidirect
68. #41⋊C3C22×C7semidirect
70. C2×SL(2,𝔽3)C22×C7semidirect
75. 2−1+4C2×C3×C7semidirect
78. C2×#41C2×C3×C7semidirect
80. C22×Q8C2×C3×C7semidirect
100. #41C22×C3×C7semidirect
107. C2×Q8C22×C3×C7semidirect
116. C2×C72−1+4⋊C3direct (N × H)
141. C7C2×2−1+4⋊C3direct (N × H)
157. C2C7×2−1+4⋊C3direct (N × H)

Maximal subgroups

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

subgroupindex
C7×2−1+4⋊C32
C2×C7×#41⋊C32
C22×C7×SL(2,𝔽3)2
C2×C7×2−1+43
C2×C3×C7×D44
C2×2−1+4⋊C37