Finite Group Databasebeta

← #90495

Subgroups — #90495

Order 448.

346 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 346 conjugacy classes of subgroups — too many to draw. Only its 111 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×C7×D4C211yesmaximal, characteristic, complemented
2C22×C22⋊D7C211yesmaximal, characteristic, complemented
3#6516C212yesmaximal, complemented
4C2×C22⋊Dic7C211yesmaximal, characteristic, complemented
5#6502C214yesmaximal, complemented
6#6500C212yesmaximal, complemented
7#6498C212yesmaximal, complemented
8#6492C212yesmaximal, complemented
9C24×C7C2211yescharacteristic
10C24×C7C2211yescharacteristic, complemented
11C23×D7C2211yescharacteristic, complemented
12C2×C7×D4—24no—
13C2×C7×D4C2212yes—
14C2×C7×D4C2212yescomplemented
15C22×C4×C7C2211yescharacteristic, complemented
16C2×C22⋊D7—24no—
17C2×C22⋊D7C2212yes—
18C2×C22⋊D7C2212yescomplemented
19C2×C22⋊D7C2214yescomplemented
20C22×Dic7C2211yescharacteristic, complemented
21C22×Dic7C2212yescomplemented
22C22⋊Dic7C2212yescomplemented
23C22⋊Dic7C2212yescomplemented
24C22⋊Dic7C2214yescomplemented
25(C2×D7)⋊C4C2214yescomplemented
26Dic7⋊C4C2214yescomplemented
27#534—71nomaximal, Sylow
28C23×C7—22no—
29C23×C7—42no—
30C23×C7—42no—
31C23×C7D414yescomplemented
32C23×C7C2312yes—
33C23×C7C2311yescharacteristic
34C23×C7C2311yescharacteristic
35C23×C7C2311yescharacteristic
36C22×D7—22no—
37C22×D7—42no—
38C22×D7C2312yes—
39C7×D4—24no—
40C7×D4—24no—
41C2×C4×C7—22no—
42C2×C4×C7C2312yes—
43C22⋊D7—24no—
44C22⋊D7—24no—
45C22⋊D7—44no—
46C2×Dic7—22no—
47C2×Dic7—42no—
48C2×Dic7C2312yes—
49C2×Dic7C2314yes—
50C22×D4—71no—
51C22×D4D711yescharacteristic, complemented
52#123—72no—
53#123—72no—
54#121—74no—
55#120—72no—
56#120—72no—
57C2×C22⋊C4—71no—
58C22×C7—22no—
59C22×C7—22no—
60C22×C7—22no—
61C22×C7—22no—
62C22×C7—24no—
63C22×C7—44no—
64C22×C7—44no—
65C22×C7C2×D412yes—
66C22×C7C2×D412yes—
67C22×C7C2×D412yes—
68C22×C7C2411yescharacteristic
69C2×D7—22no—
70C2×D7—24no—
71C2×D7—44no—
72C4×C7—22no—
73Dic7—22no—
74Dic7—24no—
75C24—71no—
76C24C2×D711yescharacteristic
77C24C2×D711yescharacteristic, complemented
78C2×D4—24no—
79C2×D4—72no—
80C2×D4—72no—
81C2×D4—74no—
82C2×D4—144no—
83C2×D4C2×D712yes—
84C2×D4C2×D712yescomplemented
85C22×C4—71no—
86C22×C4—72no—
87C22×C4C2×D711yescharacteristic, complemented
88C4⋊C4—74no—
89C22⋊C4—72no—
90C22⋊C4—72no—
91C22⋊C4—74no—
92C22⋊C4—74no—
93C2×C7—22no—
94C2×C7—22no—
95C2×C7—22no—
96C2×C7—42no—
97C2×C7C22×D411yescharacteristic
98C2×C7H(5,𝔽2)12yes—
99D7—42no—
100C23—22no—
101C23—42no—
102C23—42no—
103C23—72no—
104C23—142no—
105C23—282no—
106C23C22⋊D714yescomplemented
107C23C22×D712yes—
108C23C22×D711yescharacteristic
109C23C22×D711yescharacteristic
110C23C22×D711yescharacteristic
111D4—24no—
112D4—24no—
113D4—144no—
114D4—144no—
115D4—284no—
116C2×C4—22no—
117C2×C4—74no—
118C2×C4—72no—
119C2×C4—142no—
120C2×C4—282no—
121C2×C4C22×D712yes—
122C7#53411yescharacteristic, complemented, Sylow
123C22—22no—
124C22—22no—
125C22—22no—
126C22—24no—
127C22—22no—
128C22—44no—
129C22—44no—
130C22—144no—
131C22—142no—
132C22—284no—
133C22C2×C22⋊D712yes—
134C22C2×C22⋊D712yes—
135C22C2×C22⋊D712yes—
136C22C23×D711yescharacteristic
137C4—22no—
138C4—144no—
139C4—142no—
140C2—22no—
141C2—22no—
142C2—22no—
143C2—42no—
144C2—282no—
145C2#654812yes—
146C2C22×C22⋊D711yescharacteristic
subgroupG/H
center, FrattiniC22C23×D7
commutator, socle, cosocleC22×C7C24
FittingC22×C7×D4C2
hypercenterC22×D4D7
hypocenterC7#534
LCSG ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×D4

Complements

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

NH ≅ G/Ntype
1. C22×C7×D4C2semidirect
2. C22×C22⋊D7C2semidirect
3. #6516C2semidirect
4. C2×C22⋊Dic7C2semidirect
5. #6502C2semidirect
6. #6500C2semidirect
7. #6498C2semidirect
8. #6492C2semidirect
10. C24×C7C22semidirect
11. C23×D7C22semidirect
14. C2×C7×D4C22semidirect
15. C22×C4×C7C22semidirect
18. C2×C22⋊D7C22semidirect
19. C2×C22⋊D7C22semidirect
20. C22×Dic7C22semidirect
21. C22×Dic7C22semidirect
22. C22⋊Dic7C22semidirect
23. C22⋊Dic7C22semidirect
24. C22⋊Dic7C22semidirect
25. (C2×D7)⋊C4C22semidirect
26. Dic7⋊C4C22semidirect
31. C23×C7D4semidirect
51. C22×D4D7semidirect
77. C24C2×D7semidirect
84. C2×D4C2×D7semidirect
87. C22×C4C2×D7semidirect
106. C23C22⋊D7semidirect
122. C7#534semidirect

Maximal subgroups

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

subgroupindex
C22×C7×D42
C22×C22⋊D72
#65162
C2×C22⋊Dic72
#65022
#65002
#64982
#64922
#5347