Finite Group Databasebeta

← D7×#123 #90343

Subgroups — D7×#123 #90343

Order 448.

342 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 111 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×D4×D7C211yesmaximal, characteristic
2C22×C4×D7C211yesmaximal, characteristic, complemented
3C7×#123C211yesmaximal, characteristic, complemented
4#6498C211yesmaximal, characteristic, complemented
5#6492C211yesmaximal, characteristic, complemented
6#6457C212yesmaximal, complemented
7D7×C4⋊C4C212yesmaximal, complemented
8#6449C211yesmaximal, characteristic, complemented
9#6446C212yesmaximal, complemented
10D7×C22⋊C4C211yesmaximal, characteristic, complemented
11D7×C22⋊C4C212yesmaximal, complemented
12C23×D7C2211yescharacteristic
13C23×D7C2211yescharacteristic
14C2×C7×D4C2211yescharacteristic
15C22×C4×C7C2211yescharacteristic, complemented
16C2×C22⋊D7C2211yescharacteristic
17C2×C22⋊D7C2211yescharacteristic
18C22×Dic7C2211yescharacteristic, complemented
19D4×D7—24no—
20C2×D28C2211yescharacteristic
21C2×C4×D7—24no—
22C2×C4×D7C2212yes—
23C2×C4×D7C2212yes—
24C2×C4×D7C2211yescharacteristic
25C2×C4×D7C2212yescomplemented
26C7×C4⋊C4C2212yescomplemented
27C7×C22⋊C4C2212yescomplemented
28C7×C22⋊C4C2211yescharacteristic, complemented
29C22⋊Dic7C2212yescomplemented
30C22⋊Dic7C2211yescharacteristic, complemented
31(C2×D7)⋊C4C2212yescomplemented
32(C2×D7)⋊C4C2212yescomplemented
33(C2×D7)⋊C4C2212yescomplemented
34C4⋊Dic7C2212yescomplemented
35Dic7⋊C4C2212yescomplemented
36Dic7⋊C4C2212yescomplemented
37C2×#123—71nomaximal, Sylow
38C23×C7C2311yescharacteristic
39C23×C7C2311yescharacteristic
40C22×D7—24no—
41C22×D7—22no—
42C22×D7—24no—
43C22×D7D414yes—
44C22×D7C2311yescharacteristic
45C22×D7C2311yescharacteristic
46C22×D7C2311yescharacteristic
47C7×D4—22no—
48C2×C4×C7—22no—
49C2×C4×C7C2311yescharacteristic
50C2×C4×C7C2312yes—
51C2×C4×C7C2312yes—
52C22⋊D7—22no—
53C22⋊D7—22no—
54C2×Dic7—22no—
55C2×Dic7C2311yescharacteristic
56C2×Dic7C2312yes—
57C2×Dic7C2312yes—
58D28—22no—
59C4×D7—24no—
60C4×D7—24no—
61C4×D7—22no—
62C4×D7—24no—
63C22×D4—71no—
64C23×C4—71no—
65#123—71no—
66#123—71no—
67#123—72no—
68#123—72no—
69#123—71no—
70#123D711yescharacteristic, direct product factor
71C2×C4⋊C4—72no—
72C2×C22⋊C4—71no—
73C2×C22⋊C4—72no—
74C22×C7—22no—
75C22×C7—21no—
76C22×C7—22no—
77C22×C7C2×D412yes—
78C22×C7C2411yescharacteristic
79C2×D7—22no—
80C2×D7—28no—
81C2×D7—21no—
82C2×D7—22no—
83C2×D7—44no—
84C2×D7C2×D412yes—
85C2×D7C2×D412yes—
86C2×D7#4114yes—
87C4×C7—22no—
88C4×C7—21no—
89C4×C7—22no—
90Dic7—22no—
91Dic7—21no—
92Dic7—22no—
93C24—71no—
94C24—71no—
95C2×D4—71no—
96C2×D4—71no—
97C2×D4—71no—
98C2×D4—144no—
99C2×D4C2×D711yescharacteristic
100C22×C4—71no—
101C22×C4—72no—
102C22×C4—72no—
103C22×C4—71no—
104C22×C4—72no—
105C22×C4—144no—
106C22×C4C2×D711yescharacteristic, complemented
107C4⋊C4—72no—
108C4⋊C4—72no—
109C4⋊C4—72no—
110C4⋊C4C2×D712yescomplemented
111C22⋊C4—71no—
112C22⋊C4—72no—
113C22⋊C4—72no—
114C22⋊C4—72no—
115C22⋊C4—72no—
116C22⋊C4C2×D711yescharacteristic, complemented
117C22⋊C4C2×D712yescomplemented
118C2×C7—22no—
119C2×C7—41no—
120C2×C7C22×D411yescharacteristic
121C2×C7C2×#4112yes—
122D7—22no—
123D7—41no—
124D7#12314yesdirect product factor
125C23—71no—
126C23—74no—
127C23—71no—
128C23—71no—
129C23—144no—
130C23—142no—
131C23—144no—
132C23C22×D711yescharacteristic
133C23C22×D711yescharacteristic
134D4—22no—
135D4—142no—
136D4—142no—
137D4—142no—
138C2×C4—22no—
139C2×C4—72no—
140C2×C4—72no—
141C2×C4—71no—
142C2×C4—144no—
143C2×C4—144no—
144C2×C4—144no—
145C2×C4—142no—
146C2×C4—142no—
147C2×C4C22×D712yes—
148C2×C4C22×D712yes—
149C2×C4C22×D711yescharacteristic
150C7C2×#12311yescharacteristic, complemented, Sylow
151C22—22no—
152C22—21no—
153C22—22no—
154C22—72no—
155C22—74no—
156C22—72no—
157C22—148no—
158C22—142no—
159C22—141no—
160C22—142no—
161C22—284no—
162C22D4×D712yes—
163C22C23×D711yescharacteristic
164C4—22no—
165C4—21no—
166C4—22no—
167C4—142no—
168C4—141no—
169C4—142no—
170C2—22no—
171C2—41no—
172C2—74no—
173C2—142no—
174C2—281no—
175C2C2×D4×D711yescharacteristic
176C2D7×#4112yes—
subgroupG/H
center, FrattiniC22C23×D7
commutator, socle, cosocleC22×C7C24
FittingC7×#123C2
prime radicalC2×D4×D7C2
hypercenter#123D7
hypocenterC7C2×#123
LCSG ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ #123

Complements

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

NH ≅ G/Ntype
2. C22×C4×D7C2semidirect
3. C7×#123C2semidirect
4. #6498C2semidirect
5. #6492C2semidirect
6. #6457C2semidirect
7. D7×C4⋊C4C2semidirect
8. #6449C2semidirect
9. #6446C2semidirect
10. D7×C22⋊C4C2semidirect
11. D7×C22⋊C4C2semidirect
15. C22×C4×C7C22semidirect
18. C22×Dic7C22semidirect
25. C2×C4×D7C22semidirect
26. C7×C4⋊C4C22semidirect
27. C7×C22⋊C4C22semidirect
28. C7×C22⋊C4C22semidirect
29. C22⋊Dic7C22semidirect
30. C22⋊Dic7C22semidirect
31. (C2×D7)⋊C4C22semidirect
32. (C2×D7)⋊C4C22semidirect
33. (C2×D7)⋊C4C22semidirect
34. C4⋊Dic7C22semidirect
35. Dic7⋊C4C22semidirect
36. Dic7⋊C4C22semidirect
70. #123D7direct (N × H)
106. C22×C4C2×D7semidirect
110. C4⋊C4C2×D7semidirect
116. C22⋊C4C2×D7semidirect
117. C22⋊C4C2×D7semidirect
124. D7#123direct (N × H)
150. C7C2×#123semidirect

Maximal subgroups

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

subgroupindex
C2×D4×D72
C22×C4×D72
C7×#1232
#64982
#64922
#64572
D7×C4⋊C42
#64492
#64462
D7×C22⋊C42
C2×#1237