Finite Group Databasebeta

← #89804

Subgroups — #89804

Order 448.

198 conjugacy classes of subgroups; only the 39 normal subgroups are shown.

normal subgroups only

This group has 198 conjugacy classes of subgroups — too many to draw. Only its 39 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
1#6548C211yesmaximal, characteristic, complemented
2C7×#120C211yesmaximal, characteristic, complemented
3#6516C212yesmaximal, complemented
4#6408C211yesmaximal, characteristic, complemented
5#6380C212yesmaximal, complemented
6C24×C7C2211yescharacteristic, complemented
7C2×C7×D4—21no—
8C2×C7×D4C2211yescharacteristic
9C2×C22⋊D7—21no—
10C2×C22⋊D7—22no—
11C2×C22⋊D7C2212yes—
12#1159—22no—
13D4×D7—22no—
14#1157—21no—
15C7×C22⋊C4—21no—
16C7×C22⋊C4C2211yescharacteristic, complemented
17C22⋊Dic7—22no—
18C22⋊Dic7C2212yescomplemented
19B(4,𝔽2)—71nomaximal, Sylow
20C23×C7—22no—
21C23×C7—21no—
22C23×C7—42no—
23C23×C7D411yescharacteristic, complemented
24C23×C7C2311yescharacteristic
25C22×D7—21no—
26C22×D7D412yescomplemented
27C7×D4—22no—
28C7×D4—42no—
29C2×C4×C7—21no—
30C2×C4×C7D411yescharacteristic, complemented
31C22⋊D7—22no—
32C22⋊D7—22no—
33C22⋊D7—22no—
34C22⋊D7—44no—
35C2×Dic7—21no—
36C2×Dic7—22no—
37C2×Dic7D412yescomplemented
38D28—21no—
39C4×D7—22no—
40Dic14—21no—
41H(5,𝔽2)—71no—
42#120—72no—
43#120D711yescharacteristic, complemented
44#99—71no—
45#99—72no—
46C22×C7—24no—
47C22×C7—22no—
48C22×C7—21no—
49C22×C7—41no—
50C22×C7—41no—
51C22×C7—42no—
52C22×C7—41no—
53C22×C7—41no—
54C22×C7C2×D412yes—
55C22×C7C2×D411yescharacteristic
56C2×D7—22no—
57C2×D7—42no—
58C2×D7—81no—
59C4×C7—21no—
60C4×C7—41no—
61Dic7—22no—
62Dic7—42no—
63C24C2×D711yescharacteristic, complemented
64#41—142no—
65#41—141no—
66C2×D4—21no—
67C2×D4—72no—
68C2×D4—142no—
69C2×D4—141no—
70C2×D4—142no—
71C2×D4C2×D711yescharacteristic
72C22⋊C4—21no—
73C22⋊C4—72no—
74C22⋊C4—142no—
75C22⋊C4C2×D711yescharacteristic, complemented
76C2×C7—22no—
77C2×C7—21no—
78C2×C7—41no—
79C2×C7—42no—
80C2×C7#12011yescharacteristic
81D7—42no—
82C23—22no—
83C23—21no—
84C23—42no—
85C23—72no—
86C23—141no—
87C23C22⋊D711yescharacteristic, complemented
88C23C22×D711yescharacteristic
89Q8—141no—
90D4—22no—
91D4—42no—
92D4—141no—
93D4—142no—
94D4—142no—
95D4—142no—
96D4—284no—
97C2×C4—21no—
98C2×C4—72no—
99C2×C4—142no—
100C2×C4—141no—
101C2×C4—142no—
102C2×C4C22⋊D711yescharacteristic, complemented
103C7B(4,𝔽2)11yescharacteristic, complemented, Sylow
104C22—22no—
105C22—24no—
106C22—21no—
107C22—41no—
108C22—41no—
109C22—41no—
110C22—42no—
111C22—41no—
112C22—142no—
113C22—282no—
114C22—561no—
115C22D4×D712yes—
116C22C2×C22⋊D711yescharacteristic
117C4—21no—
118C4—41no—
119C4—142no—
120C4—282no—
121C2—21no—
122C2—22no—
123C2—42no—
124C2—41no—
125C2—282no—
126C2#650011yescharacteristic
subgroupG/H
centerC2#6500
commutator, cosocleC23×C7C23
FrattiniC23C22×D7
FittingC7×#120C2
socleC2×C7#120
hypercenter#120D7
hypocenterC7B(4,𝔽2)
LCSG ▷ C23×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C2 ◁ C23 ◁ #120

Complements

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

NH ≅ G/Ntype
1. #6548C2semidirect
2. C7×#120C2semidirect
3. #6516C2semidirect
4. #6408C2semidirect
5. #6380C2semidirect
6. C24×C7C22semidirect
16. C7×C22⋊C4C22semidirect
18. C22⋊Dic7C22semidirect
23. C23×C7D4semidirect
26. C22×D7D4semidirect
30. C2×C4×C7D4semidirect
37. C2×Dic7D4semidirect
43. #120D7semidirect
63. C24C2×D7semidirect
75. C22⋊C4C2×D7semidirect
87. C23C22⋊D7semidirect
102. C2×C4C22⋊D7semidirect
103. C7B(4,𝔽2)semidirect

Maximal subgroups

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

subgroupindex
#65482
C7×#1202
#65162
#64082
#63802
B(4,𝔽2)7