Finite Group Databasebeta

← D7×#121 #90295

Subgroups — D7×#121 #90295

Order 448.

426 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 426 conjugacy classes of subgroups — too many to draw. Only its 121 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×D7C212yesmaximal, complemented
2C2×D4×D7C211yesmaximal, characteristic, complemented
3C22×C4×D7C211yesmaximal, characteristic, complemented
4C7×#121C211yesmaximal, characteristic, complemented
5#6502C212yesmaximal, complemented
6#6501C211yesmaximal, characteristic, complemented
7#6493C211yesmaximal, characteristic, complemented
8#6458C211yesmaximal, characteristic, complemented
9D7×C4⋊C4C211yesmaximal, characteristic, complemented
10#6447C212yesmaximal, complemented
11D7×C22⋊C4C212yesmaximal, complemented
12C23×D7C2212yes—
13C23×D7C2211yescharacteristic
14C2×C7×D4C2211yescharacteristic, complemented
15C2×C7×D4C2212yescomplemented
16C22×C4×C7C2211yescharacteristic, complemented
17C2×C22⋊D7C2212yescomplemented
18C2×C22⋊D7C2212yescomplemented
19C2×C22⋊D7C2212yescomplemented
20C22×Dic7C2211yescharacteristic, complemented
21D4×D7—28no—
22D4×D7—24no—
23C2×D28C2212yescomplemented
24C2×D28C2211yescharacteristic, complemented
25C2×C4×D7—24no—
26C2×C4×D7C2211yescharacteristic
27C2×C4×D7C2212yes—
28C2×C4×D7C2211yescharacteristic
29C2×C4×D7C2211yescharacteristic, complemented
30C2×C4×D7C2211yescharacteristic, complemented
31C7×C4⋊C4C2211yescharacteristic, complemented
32C7×C22⋊C4C2212yescomplemented
33C22⋊Dic7C2212yescomplemented
34(C2×D7)⋊C4C2212yescomplemented
35(C2×D7)⋊C4C2212yescomplemented
36C4⋊Dic7C2211yescharacteristic, complemented
37Dic7⋊C4C2212yescomplemented
38C2×#121—71nomaximal, Sylow
39C23×C7C2312yes—
40C23×C7C2311yescharacteristic
41C22×D7—22no—
42C22×D7—22no—
43C22×D7—24no—
44C22×D7—24no—
45C22×D7—24no—
46C22×D7D414yescomplemented
47C22×D7C2312yes—
48C22×D7C2311yescharacteristic
49C22×D7C2311yescharacteristic
50C7×D4—24no—
51C7×D4—22no—
52C2×C4×C7—22no—
53C2×C4×C7C2312yes—
54C2×C4×C7C2311yescharacteristic
55C2×C4×C7C2311yescharacteristic
56C22⋊D7—24no—
57C22⋊D7—24no—
58C22⋊D7—24no—
59C2×Dic7—22no—
60C2×Dic7C2312yes—
61C2×Dic7C2311yescharacteristic
62C2×Dic7C2311yescharacteristic
63D28—24no—
64D28—22no—
65C4×D7—22no—
66C4×D7—24no—
67C4×D7—22no—
68C4×D7—22no—
69C4×D7D414yescomplemented
70C22×D4—71no—
71C22×D4—72no—
72C23×C4—71no—
73#121—71no—
74#121—72no—
75#121—71no—
76#121—71no—
77#121—72no—
78#121D711yescharacteristic, direct product factor
79C2×C4⋊C4—71no—
80C2×C22⋊C4—72no—
81C22×C7—21no—
82C22×C7—21no—
83C22×C7—22no—
84C22×C7—22no—
85C22×C7—22no—
86C22×C7C2×D412yescomplemented
87C22×C7C2411yescharacteristic
88C2×D7—21no—
89C2×D7—21no—
90C2×D7—28no—
91C2×D7—22no—
92C2×D7—22no—
93C2×D7—22no—
94C2×D7—48no—
95C2×D7C2×D412yes—
96C2×D7C2×D412yes—
97C2×D7C2×D412yescomplemented
98C2×D7#4112yes—
99C4×C7—21no—
100C4×C7—22no—
101C4×C7C2×D412yescomplemented
102Dic7—21no—
103Dic7—22no—
104Dic7C2×D412yescomplemented
105C24—71no—
106C24—72no—
107C2×D4—72no—
108C2×D4—72no—
109C2×D4—72no—
110C2×D4—72no—
111C2×D4—71no—
112C2×D4—144no—
113C2×D4—148no—
114C2×D4C2×D712yescomplemented
115C2×D4C2×D711yescharacteristic, complemented
116C22×C4—71no—
117C22×C4—71no—
118C22×C4—71no—
119C22×C4—72no—
120C22×C4—71no—
121C22×C4—71no—
122C22×C4—144no—
123C22×C4C2×D711yescharacteristic, complemented
124C4⋊C4—72no—
125C4⋊C4—71no—
126C4⋊C4C2×D711yescharacteristic, complemented
127C22⋊C4—72no—
128C22⋊C4—72no—
129C22⋊C4—72no—
130C22⋊C4C2×D712yescomplemented
131C2×C7—22no—
132C2×C7—42no—
133C2×C7C22×D411yescharacteristic
134C2×C7C22×D411yescharacteristic
135C2×C7C2×#4111yescharacteristic
136D7—22no—
137D7—42no—
138D7#12114yesdirect product factor
139C23—71no—
140C23—74no—
141C23—71no—
142C23—72no—
143C23—142no—
144C23—142no—
145C23—144no—
146C23—144no—
147C23—144no—
148C23C22×D712yes—
149C23C22×D711yescharacteristic
150D4—24no—
151D4—22no—
152D4—144no—
153D4—144no—
154D4—144no—
155D4—144no—
156D4—142no—
157C2×C4—22no—
158C2×C4—71no—
159C2×C4—74no—
160C2×C4—71no—
161C2×C4—72no—
162C2×C4—142no—
163C2×C4—142no—
164C2×C4—144no—
165C2×C4—142no—
166C2×C4—142no—
167C2×C4C22×D711yescharacteristic
168C2×C4C22×D712yes—
169C2×C4C22×D711yescharacteristic
170C7C2×#12111yescharacteristic, complemented, Sylow
171C22—21no—
172C22—21no—
173C22—22no—
174C22—22no—
175C22—22no—
176C22—72no—
177C22—72no—
178C22—72no—
179C22—72no—
180C22—148no—
181C22—141no—
182C22—141no—
183C22—142no—
184C22—142no—
185C22—142no—
186C22—288no—
187C22D4×D712yescomplemented
188C22C23×D711yescharacteristic
189C4—21no—
190C4—22no—
191C4—72no—
192C4—141no—
193C4—142no—
194C4D4×D712yescomplemented
195C2—22no—
196C2—42no—
197C2—74no—
198C2—142no—
199C2—282no—
200C2C2×D4×D711yescharacteristic
201C2C2×D4×D711yescharacteristic
202C2D7×#4111yescharacteristic
subgroupG/H
center, FrattiniC22C23×D7
commutator, socle, cosocleC22×C7C24
FittingC7×#121C2
hypercenter#121D7
hypocenterC7C2×#121
LCSG ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ #121

Complements

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

NH ≅ G/Ntype
1. C2×D4×D7C2semidirect
2. C2×D4×D7C2semidirect
3. C22×C4×D7C2semidirect
4. C7×#121C2semidirect
5. #6502C2semidirect
6. #6501C2semidirect
7. #6493C2semidirect
8. #6458C2semidirect
9. D7×C4⋊C4C2semidirect
10. #6447C2semidirect
11. D7×C22⋊C4C2semidirect
14. C2×C7×D4C22semidirect
15. C2×C7×D4C22semidirect
16. C22×C4×C7C22semidirect
17. C2×C22⋊D7C22semidirect
18. C2×C22⋊D7C22semidirect
19. C2×C22⋊D7C22semidirect
20. C22×Dic7C22semidirect
23. C2×D28C22semidirect
24. C2×D28C22semidirect
29. C2×C4×D7C22semidirect
30. C2×C4×D7C22semidirect
31. C7×C4⋊C4C22semidirect
32. C7×C22⋊C4C22semidirect
33. C22⋊Dic7C22semidirect
34. (C2×D7)⋊C4C22semidirect
35. (C2×D7)⋊C4C22semidirect
36. C4⋊Dic7C22semidirect
37. Dic7⋊C4C22semidirect
46. C22×D7D4semidirect
69. C4×D7D4semidirect
78. #121D7direct (N × H)
86. C22×C7C2×D4semidirect
97. C2×D7C2×D4semidirect
101. C4×C7C2×D4semidirect
104. Dic7C2×D4semidirect
114. C2×D4C2×D7semidirect
115. C2×D4C2×D7semidirect
123. C22×C4C2×D7semidirect
126. C4⋊C4C2×D7semidirect
130. C22⋊C4C2×D7semidirect
138. D7#121direct (N × H)
170. C7C2×#121semidirect
187. C22D4×D7semidirect
194. C4D4×D7semidirect

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×#1212
#65022
#65012
#64932
#64582
D7×C4⋊C42
#64472
D7×C22⋊C42
C2×#1217