Finite Group Databasebeta

← #90283

Subgroups — #90283

Order 448.

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

normal subgroups only

This group has 334 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×#1159C211yesmaximal, characteristic, complemented
3C7×#120C211yesmaximal, characteristic, complemented
4#6516C211yesmaximal, characteristic, complemented
5#6503C212yesmaximal, complemented
6#6502C212yesmaximal, complemented
7#6500C211yesmaximal, characteristic, complemented
8#6448C212yesmaximal, complemented
9#6447C212yesmaximal, complemented
10#6445C211yesmaximal, characteristic, complemented
11#6440C211yesmaximal, characteristic, complemented
12C24×C7C2211yescharacteristic
13C23×D7C2211yescharacteristic
14C2×C7×D4C2211yescharacteristic
15C2×C7×D4C2212yescomplemented
16C2×C22⋊D7—24no—
17C2×C22⋊D7C2211yescharacteristic
18C2×C22⋊D7C2212yes—
19C2×C22⋊D7C2212yes—
20C2×C22⋊D7C2211yescharacteristic, complemented
21C2×C22⋊D7C2212yescomplemented
22C2×C22⋊D7C2212yescomplemented
23C22×Dic7C2211yescharacteristic
24C22×Dic7C2211yescharacteristic, complemented
25#1159—24no—
26C2×D28C2212yescomplemented
27C2×C4×D7C2211yescharacteristic, complemented
28C2×Dic14C2211yescharacteristic, complemented
29C7×C22⋊C4C2211yescharacteristic
30C7×C22⋊C4C2212yescomplemented
31C22⋊Dic7C2211yescharacteristic
32C22⋊Dic7C2212yescomplemented
33(C2×D7)⋊C4C2212yescomplemented
34(C2×D7)⋊C4C2212yescomplemented
35Dic7⋊C4C2212yescomplemented
36C4×Dic7C2212yescomplemented
37#535—71nomaximal, Sylow
38C23×C7—22no—
39C23×C7—42no—
40C23×C7C2312yes—
41C23×C7C2311yescharacteristic
42C23×C7C2311yescharacteristic
43C22×D7—22no—
44C22×D7—42no—
45C22×D7C2312yes—
46C22×D7C2311yescharacteristic
47C7×D4—22no—
48C7×D4—42no—
49C2×C4×C7C2312yes—
50C2×C4×C7C2311yescharacteristic
51C22⋊D7—22no—
52C22⋊D7—22no—
53C22⋊D7—24no—
54C22⋊D7—24no—
55C22⋊D7—42no—
56C22⋊D7—42no—
57C2×Dic7—22no—
58C2×Dic7D414yescomplemented
59C2×Dic7C2312yes—
60C2×Dic7C2311yescharacteristic
61C2×Dic7C2311yescharacteristic
62C2×Dic7C2311yescharacteristic
63D28—42no—
64C4×D7—22no—
65Dic14—22no—
66C2×#41—71no—
67C22×D4—71no—
68#127—72no—
69#124—72no—
70#121—72no—
71#121—72no—
72#120—71no—
73#120—71no—
74#120—71no—
75#120D711yescharacteristic, complemented
76#117—71no—
77C22×C7—22no—
78C22×C7—21no—
79C22×C7—22no—
80C22×C7—22no—
81C22×C7—22no—
82C22×C7—22no—
83C22×C7—44no—
84C22×C7C2×D412yes—
85C22×C7C2411yescharacteristic
86C2×D7—21no—
87C2×D7—22no—
88C2×D7—22no—
89C2×D7—22no—
90C2×D7—22no—
91C2×D7—44no—
92C4×C7—21no—
93C4×C7—22no—
94Dic7—21no—
95Dic7—22no—
96Dic7C2×D412yes—
97Dic7C2×D412yes—
98C24—71no—
99C24C2×D711yescharacteristic
100#41—144no—
101C2×Q8—71no—
102C2×D4—71no—
103C2×D4—72no—
104C2×D4—72no—
105C2×D4—71no—
106C2×D4—72no—
107C2×D4—72no—
108C2×D4—72no—
109C2×D4—144no—
110C2×D4C2×D711yescharacteristic
111C2×D4C2×D712yescomplemented
112C22×C4—71no—
113C22×C4—71no—
114C22×C4—71no—
115C4⋊C4—72no—
116C22⋊C4—71no—
117C22⋊C4—72no—
118C22⋊C4—72no—
119C22⋊C4—72no—
120C22⋊C4C2×D711yescharacteristic
121C22⋊C4C2×D712yescomplemented
122C42—72no—
123C2×C7—22no—
124C2×C7—41no—
125C2×C7—42no—
126C2×C7C22×D411yescharacteristic
127C2×C7H(5,𝔽2)12yes—
128D7—41no—
129D7—42no—
130C23—22no—
131C23—42no—
132C23—72no—
133C23—71no—
134C23—142no—
135C23—282no—
136C23C22×D712yes—
137C23C22×D711yescharacteristic
138C23C22×D711yescharacteristic
139Q8—142no—
140D4—22no—
141D4—42no—
142D4—144no—
143D4—144no—
144D4—142no—
145D4—142no—
146D4—282no—
147D4—282no—
148D4—282no—
149C2×C4—74no—
150C2×C4—71no—
151C2×C4—71no—
152C2×C4—72no—
153C2×C4—71no—
154C2×C4—142no—
155C2×C4—142no—
156C2×C4C22×D712yes—
157C2×C4C22×D711yescharacteristic
158C7#53511yescharacteristic, complemented, Sylow
159C22—22no—
160C22—21no—
161C22—22no—
162C22—22no—
163C22—22no—
164C22—22no—
165C22—44no—
166C22—141no—
167C22—142no—
168C22—142no—
169C22—142no—
170C22—142no—
171C22—284no—
172C22D4×D712yes—
173C22C23×D711yescharacteristic
174C4—21no—
175C4—22no—
176C4—72no—
177C4—72no—
178C4—141no—
179C4—142no—
180C2—22no—
181C2—42no—
182C2—41no—
183C2—282no—
184C2—281no—
185C2C2×D4×D711yescharacteristic
186C2#654812yes—
subgroupG/H
center, FrattiniC22C23×D7
commutator, socle, cosocleC22×C7C24
FittingC7×#120C2
hypercenter#120D7
hypocenterC7#535
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×#1159C2semidirect
3. C7×#120C2semidirect
4. #6516C2semidirect
5. #6503C2semidirect
6. #6502C2semidirect
7. #6500C2semidirect
8. #6448C2semidirect
9. #6447C2semidirect
10. #6445C2semidirect
11. #6440C2semidirect
15. C2×C7×D4C22semidirect
20. C2×C22⋊D7C22semidirect
21. C2×C22⋊D7C22semidirect
22. C2×C22⋊D7C22semidirect
24. C22×Dic7C22semidirect
26. C2×D28C22semidirect
27. C2×C4×D7C22semidirect
28. C2×Dic14C22semidirect
30. C7×C22⋊C4C22semidirect
32. C22⋊Dic7C22semidirect
33. (C2×D7)⋊C4C22semidirect
34. (C2×D7)⋊C4C22semidirect
35. Dic7⋊C4C22semidirect
36. C4×Dic7C22semidirect
58. C2×Dic7D4semidirect
75. #120D7semidirect
111. C2×D4C2×D7semidirect
121. C22⋊C4C2×D7semidirect
158. C7#535semidirect

Maximal subgroups

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

subgroupindex
C22×C22⋊D72
C2×#11592
C7×#1202
#65162
#65032
#65022
#65002
#64482
#64472
#64452
#64402
#5357