Finite Group Databasebeta

← D7×#876 #13057983

Subgroups — D7×#876 #13057983

Order 1344.

492 conjugacy classes of subgroups; only the 140 normal subgroups are shown.

normal subgroups only

This group has 492 conjugacy classes of subgroups — too many to draw. Only its 140 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
1#10621359C211yesmaximal, characteristic, complemented
2C7×#876C211yesmaximal, characteristic, complemented
3C3×C42×D7C211yesmaximal, characteristic, complemented
4D7×(C2×D3)⋊C4C213yesmaximal
5D7×Dic3⋊C4C213yesmaximal, complemented
6#10620949C213yesmaximal, complemented
7#10620934C213yesmaximal, complemented
8D7×#126—31nomaximal, Hall
9C22×D3×D7C2211yescharacteristic
10C2×C3×C4×D7C2213yes—
11C2×Dic3×D7C2213yes—
12C3×C42×C7C2211yescharacteristic, complemented
13#66631C2213yes—
14#66629C2213yescomplemented
15C7×(C2×D3)⋊C4C2213yes—
16C7×Dic3⋊C4C2213yescomplemented
17C3×C4×Dic7C2213yescomplemented
18Dic3⋊Dic7C2213yescomplemented
19#66577C2213yescomplemented
20#66575C2213yes—
21(C2×D3)⋊Dic7C2213yes—
22C7×#126—31no—
23#6461—33no—
24D7×C4⋊C4—33no—
25D7×C22⋊C4—33no—
26#6442—33no—
27#6439—31no—
28C42×D7D311yescharacteristic, complemented
29C2×#876—71nomaximal, Hall
30C22×D21C2311yescharacteristic
31C22×D3×C7C2311yescharacteristic
32C22×C3×D7C2311yescharacteristic
33C2×D3×D7—26no—
34C2×C3×C4×C7C2313yes—
35C2×Dic21C2313yes—
36C2×C7×Dic3C2313yes—
37C2×C3×Dic7C2313yes—
38C3×C4×D7—26no—
39Dic3×D7—26no—
40C23×D7—31no—
41C2×C4×D7—33no—
42C2×C4×D7C2×D313yes—
43C7×C4⋊C4—33no—
44C7×C22⋊C4—33no—
45C42×C7C2×D311yescharacteristic, complemented
46C22⋊Dic7—33no—
47(C2×D7)⋊C4—33no—
48(C2×D7)⋊C4—33no—
49C4⋊Dic7—33no—
50Dic7⋊C4—33no—
51Dic7⋊C4—33no—
52C4×Dic7C2×D313yescomplemented
53C2×C3×C42—71no—
54C2×(C2×D3)⋊C4—73no—
55C2×Dic3⋊C4—73no—
56#876—71no—
57#876—73no—
58#876—73no—
59#876D711yescharacteristic, direct product factor
60C22×C3×C7C2411yescharacteristic
61C2×D21—23no—
62C2×D3×C7—23no—
63C2×C3×D7#4116yes—
64D3×D7—44no—
65C3×C4×C7—23no—
66Dic21—23no—
67C3×Dic7—23no—
68C7×Dic3—23no—
69C2×#126—211noSylow
70C23×C7—31no—
71C22×D7—31no—
72C22×D7—66no—
73C22×D7C22×D311yescharacteristic
74C2×C4×C7—33no—
75C2×C4×C7C22×D313yes—
76C2×Dic7—33no—
77C2×Dic7C22×D313yes—
78C4×D7—26no—
79C4×D7—66no—
80C23×D3—71no—
81C22×C3×C4—73no—
82C22×Dic3—73no—
83C3×C42—73no—
84C3×C42C2×D711yescharacteristic, complemented
85(C2×D3)⋊C4—73no—
86(C2×D3)⋊C4—73no—
87(C2×D3)⋊C4—73no—
88(C2×D3)⋊C4C2×D713yes—
89Dic3⋊C4—73no—
90Dic3⋊C4—73no—
91Dic3⋊C4—73no—
92Dic3⋊C4C2×D713yescomplemented
93C2×C3×C7C2×#4113yes—
94D21—41no—
95C3×D7#12614yescomplemented
96D3×C7—41no—
97#126—31no—
98#126—213no—
99#126—213no—
100#126—211no—
101C2×C4⋊C4—213no—
102C2×C22⋊C4—213no—
103C2×C42—71no—
104C22×C7—63no—
105C22×C7C23×D311yescharacteristic
106C2×D7—63no—
107C2×D7—124no—
108C2×D7#23516yes—
109C4×C7—23no—
110C4×C7—63no—
111Dic7—23no—
112Dic7—63no—
113C23×C3—71no—
114C22×D3—71no—
115C22×D3—146no—
116C22×D3C22×D711yescharacteristic
117C2×C3×C4—73no—
118C2×C3×C4—146no—
119C2×C3×C4C22×D713yes—
120C2×Dic3—73no—
121C2×Dic3—146no—
122C2×Dic3C22×D713yes—
123C3×C7C2×#12611yescharacteristic, complemented, Hall
124C24—211no—
125C22×C4—73no—
126C22×C4—213no—
127C4⋊C4—33no—
128C4⋊C4—213no—
129C4⋊C4—213no—
130C4⋊C4—213no—
131C22⋊C4—33no—
132C22⋊C4—213no—
133C22⋊C4—213no—
134C22⋊C4—213no—
135C42—73no—
136C42D3×D711yescharacteristic, complemented
137C2×C7—121no—
138C2×C7C2×#23513yes—
139D7—121no—
140D7#87614yesdirect product factor
141C22×C3—76no—
142C22×C3C23×D711yescharacteristic
143C2×D3—23no—
144C2×D3—143no—
145C2×D3—284no—
146C3×C4—23no—
147C3×C4—143no—
148Dic3—23no—
149Dic3—143no—
150C23—31no—
151C23—71no—
152C23—211no—
153C23—426no—
154C2×C4—33no—
155C2×C4—73no—
156C2×C4—146no—
157C2×C4—213no—
158C2×C4—426no—
159C2×C4C2×D3×D713yes—
160C7C2×#87611yescharacteristic, complemented, Sylow
161C2×C3—74no—
162C2×C3D7×#4113yes—
163D3—41no—
164D3—281no—
165C22—63no—
166C22—76no—
167C22—423no—
168C22—844no—
169C22C22×D3×D711yescharacteristic
170C4—23no—
171C4—63no—
172C4—143no—
173C4—423no—
174C3D7×#12611yescharacteristic, complemented, Sylow
175C2—74no—
176C2—121no—
177C2—841no—
178C2D7×#23513yes—
subgroupG/H
center, FrattiniC22C22×D3×D7
commutator, socle, cosocleC22×C3×C7C24
FittingC3×C42×C7C22
prime radicalC22×D3×D7C22
hypercenterC42D3×D7
hypocenterC3×C7C2×#126
LCSG ▷ C22×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
1. #10621359C2semidirect
2. C7×#876C2semidirect
3. C3×C42×D7C2semidirect
5. D7×Dic3⋊C4C2semidirect
6. #10620949C2semidirect
7. #10620934C2semidirect
12. C3×C42×C7C22semidirect
14. #66629C22semidirect
16. C7×Dic3⋊C4C22semidirect
17. C3×C4×Dic7C22semidirect
18. Dic3⋊Dic7C22semidirect
19. #66577C22semidirect
28. C42×D7D3semidirect
45. C42×C7C2×D3semidirect
52. C4×Dic7C2×D3semidirect
59. #876D7direct (N × H)
84. C3×C42C2×D7semidirect
92. Dic3⋊C4C2×D7semidirect
95. C3×D7#126semidirect
123. C3×C7C2×#126semidirect
136. C42D3×D7semidirect
140. D7#876direct (N × H)
160. C7C2×#876semidirect
174. C3D7×#126semidirect

Maximal subgroups

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

subgroupindex
#106213592
C7×#8762
C3×C42×D72
D7×(C2×D3)⋊C42
D7×Dic3⋊C42
#106209492
#106209342
D7×#1263
C2×#8767