Finite Group Databasebeta

← #90281

Subgroups — #90281

Order 448.

428 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 428 conjugacy classes of subgroups — too many to draw. Only its 115 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
1C22×C22⋊D7C212yesmaximal, complemented
2C2×D4×D7C212yesmaximal, complemented
3C7×#120C211yesmaximal, characteristic, complemented
4#6503C211yesmaximal, characteristic, complemented
5#6502C212yesmaximal, complemented
6#6500C211yesmaximal, characteristic, complemented
7#6447C212yesmaximal, complemented
8#6445C212yesmaximal, complemented
9(C22⋊D7)⋊C4C212yesmaximal, complemented
10C24×C7C2211yescharacteristic, complemented
11C23×D7C2212yescomplemented
12C23×D7C2211yescharacteristic, complemented
13C2×C7×D4C2212yescomplemented
14C2×C7×D4C2211yescharacteristic, complemented
15C2×C22⋊D7—28no—
16C2×C22⋊D7C2212yes—
17C2×C22⋊D7C2212yescomplemented
18C2×C22⋊D7C2212yescomplemented
19C2×C22⋊D7C2212yescomplemented
20C2×C22⋊D7C2212yescomplemented
21C22×Dic7C2212yescomplemented
22D4×D7—28no—
23C2×D28C2212yescomplemented
24C2×D28C2211yescharacteristic, complemented
25C2×C4×D7C2212yescomplemented
26C7×C22⋊C4C2212yescomplemented
27C7×C22⋊C4C2211yescharacteristic, complemented
28C22⋊Dic7C2211yescharacteristic, complemented
29(C2×D7)⋊C4C2212yescomplemented
30(C2×D7)⋊C4C2212yescomplemented
31Dic7⋊C4C2212yescomplemented
32C4×Dic7C2211yescharacteristic, complemented
33D42—71nomaximal, Sylow
34C23×C7—24no—
35C23×C7—41no—
36C23×C7C2311yescharacteristic
37C23×C7C2311yescharacteristic
38C23×C7C2312yes—
39C22×D7—24no—
40C22×D7—24no—
41C22×D7—24no—
42C22×D7—41no—
43C22×D7—42no—
44C22×D7C2312yes—
45C22×D7C2312yes—
46C7×D4—24no—
47C7×D4—41no—
48C2×C4×C7C2311yescharacteristic
49C2×C4×C7C2312yes—
50C22⋊D7—24no—
51C22⋊D7—24no—
52C22⋊D7—24no—
53C22⋊D7—24no—
54C22⋊D7D418yescomplemented
55C2×Dic7—24no—
56C2×Dic7C2312yes—
57C2×Dic7C2312yes—
58D28—24no—
59D28—41no—
60C4×D7—24no—
61C22×D4—72no—
62C22×D4—72no—
63#127—71no—
64#121—72no—
65#121—72no—
66#120—71no—
67#120—72no—
68#120D711yescharacteristic, complemented
69C4×D4—72no—
70C22×C7—22no—
71C22×C7—22no—
72C22×C7—22no—
73C22×C7—22no—
74C22×C7—21no—
75C22×C7—21no—
76C22×C7—44no—
77C22×C7C2×D414yescomplemented
78C22×C7C2411yescharacteristic
79C2×D7—22no—
80C2×D7—22no—
81C2×D7—22no—
82C2×D7—22no—
83C2×D7—22no—
84C2×D7—44no—
85C2×D7—48no—
86C2×D7C2×D414yescomplemented
87C4×C7—22no—
88C4×C7—21no—
89Dic7—22no—
90Dic7C2×D414yescomplemented
91C24—71no—
92C24—72no—
93C24C2×D711yescharacteristic, complemented
94C2×D4—72no—
95C2×D4—72no—
96C2×D4—72no—
97C2×D4—72no—
98C2×D4—72no—
99C2×D4—72no—
100C2×D4—71no—
101C2×D4—148no—
102C2×D4—148no—
103C2×D4C2×D712yescomplemented
104C2×D4C2×D711yescharacteristic, complemented
105C22×C4—72no—
106C22×C4—72no—
107C4⋊C4—72no—
108C22⋊C4—71no—
109C22⋊C4—72no—
110C22⋊C4—72no—
111C22⋊C4C2×D712yescomplemented
112C22⋊C4C2×D711yescharacteristic, complemented
113C42—71no—
114C2×C7—24no—
115C2×C7—41no—
116C2×C7—41no—
117C2×C7C22×D412yes—
118C2×C7H(5,𝔽2)11yescharacteristic
119D7—24no—
120D7—42no—
121C23—24no—
122C23—41no—
123C23—72no—
124C23—72no—
125C23—144no—
126C23—144no—
127C23—144no—
128C23—282no—
129C23—281no—
130C23C22×D711yescharacteristic
131C23C22×D711yescharacteristic
132C23C22×D712yes—
133D4—24no—
134D4—41no—
135D4—78no—
136D4—144no—
137D4—144no—
138D4—144no—
139D4—144no—
140D4—144no—
141D4—281no—
142C2×C4—72no—
143C2×C4—72no—
144C2×C4—144no—
145C2×C4—144no—
146C2×C4C22×D711yescharacteristic
147C2×C4C22×D712yes—
148C7D4211yescharacteristic, complemented, Sylow
149C22—22no—
150C22—22no—
151C22—22no—
152C22—22no—
153C22—21no—
154C22—21no—
155C22—44no—
156C22—74no—
157C22—142no—
158C22—142no—
159C22—142no—
160C22—142no—
161C22—142no—
162C22—288no—
163C22—284no—
164C22D4×D714yescomplemented
165C22C23×D711yescharacteristic
166C4—22no—
167C4—21no—
168C4—74no—
169C4—142no—
170C2—24no—
171C2—41no—
172C2—41no—
173C2—144no—
174C2—282no—
175C2C2×D4×D712yes—
176C2#654811yescharacteristic
subgroupG/H
center, FrattiniC22C23×D7
commutator, socle, cosocleC22×C7C24
FittingC7×#120C2
hypercenter#120D7
hypocenterC7D42
LCSG ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ #120

Complements

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

NH ≅ G/Ntype
1. C22×C22⋊D7C2semidirect
2. C2×D4×D7C2semidirect
3. C7×#120C2semidirect
4. #6503C2semidirect
5. #6502C2semidirect
6. #6500C2semidirect
7. #6447C2semidirect
8. #6445C2semidirect
9. (C22⋊D7)⋊C4C2semidirect
10. C24×C7C22semidirect
11. C23×D7C22semidirect
12. C23×D7C22semidirect
13. C2×C7×D4C22semidirect
14. C2×C7×D4C22semidirect
17. C2×C22⋊D7C22semidirect
18. C2×C22⋊D7C22semidirect
19. C2×C22⋊D7C22semidirect
20. C2×C22⋊D7C22semidirect
21. C22×Dic7C22semidirect
23. C2×D28C22semidirect
24. C2×D28C22semidirect
25. C2×C4×D7C22semidirect
26. C7×C22⋊C4C22semidirect
27. C7×C22⋊C4C22semidirect
28. C22⋊Dic7C22semidirect
29. (C2×D7)⋊C4C22semidirect
30. (C2×D7)⋊C4C22semidirect
31. Dic7⋊C4C22semidirect
32. C4×Dic7C22semidirect
54. C22⋊D7D4semidirect
68. #120D7semidirect
77. C22×C7C2×D4semidirect
86. C2×D7C2×D4semidirect
90. Dic7C2×D4semidirect
93. C24C2×D7semidirect
103. C2×D4C2×D7semidirect
104. C2×D4C2×D7semidirect
111. C22⋊C4C2×D7semidirect
112. C22⋊C4C2×D7semidirect
148. C7D42semidirect
164. C22D4×D7semidirect

Maximal subgroups

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

subgroupindex
C22×C22⋊D72
C2×D4×D72
C7×#1202
#65032
#65022
#65002
#64472
#64452
(C22⋊D7)⋊C42
D427