Finite Group Databasebeta

← #90280

Subgroups — #90280

Order 448.

346 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 346 conjugacy classes of subgroups — too many to draw. Only its 103 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⋊D7C211yesmaximal, characteristic, complemented
2C2×D4×D7C211yesmaximal, characteristic, complemented
3C7×#120C211yesmaximal, characteristic, complemented
4#6516C211yesmaximal, characteristic, complemented
5#6501C212yesmaximal, complemented
6#6500C212yesmaximal, complemented
7#6498C211yesmaximal, characteristic, complemented
8#6449C211yesmaximal, characteristic, complemented
9#6447C212yesmaximal, complemented
10#6446C212yesmaximal, complemented
11D7×C22⋊C4C211yesmaximal, characteristic, complemented
12C24×C7C2211yescharacteristic, complemented
13C23×D7C2211yescharacteristic
14C23×D7C2211yescharacteristic, complemented
15C2×C7×D4C2211yescharacteristic
16C2×C7×D4C2212yescomplemented
17C2×C22⋊D7—24no—
18C2×C22⋊D7C2212yes—
19C2×C22⋊D7C2211yescharacteristic
20C2×C22⋊D7C2212yescomplemented
21C2×C22⋊D7C2211yescharacteristic, complemented
22C2×C22⋊D7C2212yescomplemented
23C22×Dic7C2211yescharacteristic, complemented
24D4×D7—24no—
25C2×D28C2211yescharacteristic, complemented
26C2×C4×D7C2211yescharacteristic, complemented
27C2×C4×D7C2212yescomplemented
28C7×C22⋊C4C2211yescharacteristic, complemented
29C7×C22⋊C4C2212yescomplemented
30C22⋊Dic7C2211yescharacteristic, complemented
31C22⋊Dic7C2212yescomplemented
32C22⋊Dic7C2212yescomplemented
33(C2×D7)⋊C4C2212yescomplemented
34(C2×D7)⋊C4C2212yescomplemented
35C4⋊Dic7C2212yescomplemented
36Dic7⋊C4C2212yescomplemented
37#534—71nomaximal, Sylow
38C23×C7—22no—
39C23×C7—42no—
40C23×C7C2311yescharacteristic
41C23×C7C2312yes—
42C23×C7C2311yescharacteristic
43C22×D7—22no—
44C22×D7—42no—
45C22×D7—42no—
46C22×D7D414yescomplemented
47C22×D7C2311yescharacteristic
48C22×D7C2311yescharacteristic
49C22×D7C2311yescharacteristic
50C7×D4—22no—
51C7×D4—42no—
52C2×C4×C7C2311yescharacteristic
53C2×C4×C7C2312yes—
54C22⋊D7—22no—
55C22⋊D7—22no—
56C22⋊D7—24no—
57C22⋊D7—24no—
58C22⋊D7—42no—
59C2×Dic7—22no—
60C2×Dic7C2312yes—
61C2×Dic7C2312yes—
62C2×Dic7C2311yescharacteristic
63D28—22no—
64C4×D7—22no—
65C4×D7—42no—
66C22×D4—71no—
67C22×D4—71no—
68#123—71no—
69#123—72no—
70#123—71no—
71#121—72no—
72#121—72no—
73#120—71no—
74#120—72no—
75#120D711yescharacteristic, complemented
76C2×C22⋊C4—71no—
77C22×C7—22no—
78C22×C7—22no—
79C22×C7—21no—
80C22×C7—22no—
81C22×C7—22no—
82C22×C7—22no—
83C22×C7—44no—
84C22×C7C2×D412yes—
85C22×C7C2411yescharacteristic
86C2×D7—22no—
87C2×D7—22no—
88C2×D7—21no—
89C2×D7—22no—
90C2×D7—44no—
91C2×D7—44no—
92C2×D7C2×D412yes—
93C2×D7C2×D412yes—
94C4×C7—21no—
95C4×C7—22no—
96Dic7—21no—
97Dic7—22no—
98Dic7—22no—
99C24—71no—
100C24—71no—
101C24C2×D711yescharacteristic, complemented
102C2×D4—72no—
103C2×D4—71no—
104C2×D4—71no—
105C2×D4—72no—
106C2×D4—72no—
107C2×D4—71no—
108C2×D4—144no—
109C2×D4—144no—
110C2×D4C2×D711yescharacteristic
111C2×D4C2×D712yescomplemented
112C22×C4—71no—
113C22×C4—71no—
114C22×C4—72no—
115C4⋊C4—72no—
116C4⋊C4—72no—
117C22⋊C4—71no—
118C22⋊C4—72no—
119C22⋊C4—72no—
120C22⋊C4—72no—
121C22⋊C4—72no—
122C22⋊C4C2×D711yescharacteristic, complemented
123C22⋊C4C2×D712yescomplemented
124C2×C7—22no—
125C2×C7—41no—
126C2×C7—42no—
127C2×C7C22×D411yescharacteristic
128C2×C7H(5,𝔽2)12yes—
129D7—22no—
130D7—22no—
131D7—41no—
132C23—22no—
133C23—42no—
134C23—71no—
135C23—71no—
136C23—74no—
137C23—71no—
138C23—142no—
139C23—282no—
140C23—282no—
141C23C22×D711yescharacteristic
142C23C22×D712yes—
143C23C22×D711yescharacteristic
144D4—22no—
145D4—42no—
146D4—142no—
147D4—142no—
148D4—144no—
149D4—144no—
150D4—142no—
151D4—282no—
152C2×C4—72no—
153C2×C4—71no—
154C2×C4—72no—
155C2×C4—142no—
156C2×C4—142no—
157C2×C4—282no—
158C2×C4C22×D712yes—
159C2×C4C22×D711yescharacteristic
160C7#53411yescharacteristic, complemented, Sylow
161C22—22no—
162C22—21no—
163C22—22no—
164C22—22no—
165C22—22no—
166C22—22no—
167C22—44no—
168C22—72no—
169C22—72no—
170C22—142no—
171C22—142no—
172C22—141no—
173C22—142no—
174C22—284no—
175C22—284no—
176C22D4×D712yes—
177C22C23×D711yescharacteristic
178C4—21no—
179C4—22no—
180C4—141no—
181C4—142no—
182C4—142no—
183C2—22no—
184C2—42no—
185C2—41no—
186C2—142no—
187C2—142no—
188C2—281no—
189C2C2×D4×D711yescharacteristic
190C2#654812yes—
subgroupG/H
center, FrattiniC22C23×D7
commutator, socle, cosocleC22×C7C24
FittingC7×#120C2
hypercenter#120D7
hypocenterC7#534
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. #6516C2semidirect
5. #6501C2semidirect
6. #6500C2semidirect
7. #6498C2semidirect
8. #6449C2semidirect
9. #6447C2semidirect
10. #6446C2semidirect
11. D7×C22⋊C4C2semidirect
12. C24×C7C22semidirect
14. C23×D7C22semidirect
16. C2×C7×D4C22semidirect
20. C2×C22⋊D7C22semidirect
21. C2×C22⋊D7C22semidirect
22. C2×C22⋊D7C22semidirect
23. C22×Dic7C22semidirect
25. C2×D28C22semidirect
26. C2×C4×D7C22semidirect
27. C2×C4×D7C22semidirect
28. C7×C22⋊C4C22semidirect
29. C7×C22⋊C4C22semidirect
30. C22⋊Dic7C22semidirect
31. C22⋊Dic7C22semidirect
32. C22⋊Dic7C22semidirect
33. (C2×D7)⋊C4C22semidirect
34. (C2×D7)⋊C4C22semidirect
35. C4⋊Dic7C22semidirect
36. Dic7⋊C4C22semidirect
46. C22×D7D4semidirect
75. #120D7semidirect
101. C24C2×D7semidirect
111. C2×D4C2×D7semidirect
122. C22⋊C4C2×D7semidirect
123. C22⋊C4C2×D7semidirect
160. C7#534semidirect

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
#65162
#65012
#65002
#64982
#64492
#64472
#64462
D7×C22⋊C42
#5347