Finite Group Databasebeta

← #89974

Subgroups — #89974

Order 448.

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

normal subgroups only

This group has 152 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#6553C211yesmaximal, characteristic
2C7×#137C211yesmaximal, characteristic, complemented
3#6510C211yesmaximal, characteristic, complemented
4#6505C211yesmaximal, characteristic, complemented
5#6411C211yesmaximal, characteristic, complemented
6#6399C211yesmaximal, characteristic, complemented
7#6397C211yesmaximal, characteristic, complemented
8C7×#41C2211yescharacteristic
9C2×C7×Q8C2211yescharacteristic, complemented
10#1161—21no—
11D4×D7—22no—
12D4×D7—21no—
13#1157—21no—
14#1157C2211yescharacteristic
15C2×D28—21no—
16C2×D28C2211yescharacteristic
17C7×Q16—21no—
18C7×SD16—21no—
19C7×M16C2211yescharacteristic, complemented
20#1144—21no—
21#1143—21no—
22(C2×D7)⋊C4—22no—
23C4×Dic7C2211yescharacteristic, complemented
24#1136C2211yescharacteristic, complemented
25#455—71nomaximal, Sylow
26C22×D7—22no—
27C22×D7D412yes—
28C7×Q8—21no—
29C7×Q8—21no—
30C7×Q8D411yescharacteristic
31C7×D4—21no—
32C7×D4D411yescharacteristic
33C2×C4×C7—21no—
34C2×C4×C7—21no—
35C2×C4×C7C2311yescharacteristic
36C22⋊D7—22no—
37C22⋊D7—21no—
38C2×Dic7—21no—
39D28—22no—
40D28—21no—
41D28—21no—
42D28D411yescharacteristic
43C4×D7—22no—
44C4×D7—21no—
45Dic14D411yescharacteristic
46C7×C8—21no—
47C7⋊C8—21no—
48H(5,𝔽2)—71no—
49#137—71no—
50#137D711yescharacteristic, complemented
51#124—71no—
52#104—71no—
53#104—71no—
54#100—71no—
55C22×C7—21no—
56C22×C7C2×D411yescharacteristic
57C2×D7—22no—
58C2×D7—21no—
59C2×D7—42no—
60C2×D7—82no—
61C4×C7—21no—
62C4×C7—41no—
63C4×C7C2×D411yescharacteristic
64C4×C7C2×D411yescharacteristic
65Dic7—21no—
66Dic7—41no—
67#41—71no—
68#41—141no—
69#41—141no—
70#41C2×D711yescharacteristic
71C2×Q8C2×D711yescharacteristic, complemented
72C2×D4—71no—
73C2×D4—141no—
74C2×D4—142no—
75C2×D4—141no—
76Q16—21no—
77Q16—141no—
78SD16—21no—
79SD16—141no—
80M16—71no—
81M16C2×D711yescharacteristic, complemented
82C22⋊C4—142no—
83C42—71no—
84C2×C7—21no—
85C2×C7—41no—
86C2×C7#12011yescharacteristic
87D7—42no—
88D7—41no—
89C23—72no—
90C23—142no—
91Q8—21no—
92Q8—21no—
93Q8—71no—
94Q8C22⋊D711yescharacteristic
95D4—21no—
96D4—71no—
97D4—142no—
98D4—142no—
99D4—141no—
100D4—141no—
101D4—141no—
102D4C22⋊D711yescharacteristic
103C2×C4—21no—
104C2×C4—21no—
105C2×C4—142no—
106C2×C4—141no—
107C2×C4—141no—
108C2×C4C22×D711yescharacteristic
109C8—21no—
110C8—141no—
111C7#45511yescharacteristic, complemented, Sylow
112C22—21no—
113C22—141no—
114C22—142no—
115C22—282no—
116C22—562no—
117C22D4×D711yescharacteristic
118C4—21no—
119C4—41no—
120C4—141no—
121C4—281no—
122C4D4×D711yescharacteristic
123C4C2×C22⋊D711yescharacteristic
124C2—21no—
125C2—41no—
126C2—282no—
127C2—281no—
128C2#650011yescharacteristic
subgroupG/H
centerC2#6500
commutator, cosocleC2×C4×C7C23
FrattiniC2×C4C22×D7
FittingC7×#137C2
socleC2×C7#120
prime radical#6553C2
hypercenter#137D7
hypocenterC7#455
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C2 ◁ C2×C4 ◁ #137

Complements

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

NH ≅ G/Ntype
2. C7×#137C2semidirect
3. #6510C2semidirect
4. #6505C2semidirect
5. #6411C2semidirect
6. #6399C2semidirect
7. #6397C2semidirect
9. C2×C7×Q8C22semidirect
19. C7×M16C22semidirect
23. C4×Dic7C22semidirect
24. #1136C22semidirect
50. #137D7semidirect
71. C2×Q8C2×D7semidirect
81. M16C2×D7semidirect
111. C7#455semidirect

Maximal subgroups

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

subgroupindex
#65532
C7×#1372
#65102
#65052
#64112
#63992
#63972
#4557