Finite Group Databasebeta

← #38763081

Subgroups — #38763081

Order 25408.

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×C22⋊D397C211yesmaximal, characteristic, complemented
2#27163079C212yesmaximal, complemented
3#27163077C212yesmaximal, complemented
4#27163068C214yesmaximal, complemented
5#27163065C212yesmaximal, complemented
6#27163064C212yesmaximal, complemented
7C2×C22⋊Dic397C211yesmaximal, characteristic, complemented
8C22×D4×C397C211yesmaximal, characteristic, complemented
9C23×D397C2211yescharacteristic, complemented
10C2×C22⋊D397—24no—
11C2×C22⋊D397C2212yes—
12C2×C22⋊D397C2212yescomplemented
13C2×C22⋊D397C2214yescomplemented
14C22×Dic397C2211yescharacteristic, complemented
15C22×Dic397C2212yescomplemented
16#20263710C2214yescomplemented
17C22⋊Dic397C2212yescomplemented
18C22⋊Dic397C2212yescomplemented
19C22⋊Dic397C2214yescomplemented
20#20263708C2214yescomplemented
21C24×C397C2211yescharacteristic
22C24×C397C2211yescharacteristic, complemented
23C2×D4×C397—24no—
24C2×D4×C397C2212yes—
25C2×D4×C397C2212yescomplemented
26C22×C4×C397C2211yescharacteristic, complemented
27C22×D397—22no—
28C22×D397—42no—
29C22×D397C2312yes—
30C22⋊D397—24no—
31C22⋊D397—24no—
32C22⋊D397—44no—
33C2×Dic397—22no—
34C2×Dic397—42no—
35C2×Dic397C2314yes—
36C2×Dic397C2312yes—
37C23×C397—22no—
38C23×C397—42no—
39C23×C397—42no—
40C23×C397D414yescomplemented
41C23×C397C2312yes—
42C23×C397C2311yescharacteristic
43C23×C397C2311yescharacteristic
44C23×C397C2311yescharacteristic
45D4×C397—24no—
46D4×C397—24no—
47C2×C4×C397—22no—
48C2×C4×C397C2312yes—
49C22×C397—22no—
50C22×C397—22no—
51C22×C397—22no—
52C22×C397—22no—
53C22×C397—24no—
54C22×C397—44no—
55C22×C397—44no—
56C22×C397C2×D412yes—
57C22×C397C2×D412yes—
58C22×C397C2×D412yes—
59C22×C397C2411yescharacteristic
60C2×D397—22no—
61C2×D397—24no—
62C2×D397—44no—
63C4×C397—22no—
64Dic397—22no—
65Dic397—24no—
66C2×C397—22no—
67C2×C397—22no—
68C2×C397—22no—
69C2×C397—42no—
70C2×C397C22×D411yescharacteristic
71C2×C397H(5,𝔽2)12yes—
72D397—42no—
73C397#53411yescharacteristic, complemented, Sylow
74#534—3971nomaximal, Sylow
75C22×D4—3971no—
76C22×D4D39711yescharacteristic, complemented
77#123—3972no—
78#123—3972no—
79#121—3974no—
80#120—3972no—
81#120—3972no—
82C2×C22⋊C4—3971no—
83C24—3971no—
84C24C2×D39711yescharacteristic
85C24C2×D39711yescharacteristic, complemented
86C2×D4—24no—
87C2×D4—7944no—
88C2×D4—3972no—
89C2×D4—3972no—
90C2×D4—3974no—
91C2×D4C2×D39712yes—
92C2×D4C2×D39712yescomplemented
93C22×C4—3971no—
94C22×C4—3972no—
95C22×C4C2×D39711yescharacteristic, complemented
96C4⋊C4—3974no—
97C22⋊C4—3972no—
98C22⋊C4—3972no—
99C22⋊C4—3974no—
100C22⋊C4—3974no—
101C23—22no—
102C23—42no—
103C23—42no—
104C23—7942no—
105C23—15882no—
106C23—3972no—
107C23C22⋊D39714yescomplemented
108C23C22×D39712yes—
109C23C22×D39711yescharacteristic
110C23C22×D39711yescharacteristic
111C23C22×D39711yescharacteristic
112D4—24no—
113D4—24no—
114D4—7944no—
115D4—7944no—
116D4—15884no—
117C2×C4—22no—
118C2×C4—7942no—
119C2×C4—15882no—
120C2×C4—3974no—
121C2×C4—3972no—
122C2×C4C22×D39712yes—
123C22—22no—
124C22—22no—
125C22—22no—
126C22—24no—
127C22—22no—
128C22—44no—
129C22—44no—
130C22—7944no—
131C22—7942no—
132C22—15884no—
133C22C2×C22⋊D39712yes—
134C22C2×C22⋊D39712yes—
135C22C2×C22⋊D39712yes—
136C22C23×D39711yescharacteristic
137C4—22no—
138C4—7944no—
139C4—7942no—
140C2—22no—
141C2—22no—
142C2—22no—
143C2—42no—
144C2—15882no—
145C2C22×C22⋊D39711yescharacteristic
146C2#2716314112yes—
subgroupG/H
center, FrattiniC22C23×D397
commutator, socle, cosocleC22×C397C24
FittingC22×D4×C397C2
hypercenterC22×D4D397
hypocenterC397#534
LCSG ▷ C22×C397 ▷ C397
UCSC1 ◁ C22 ◁ C22×D4

Complements

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

NH ≅ G/Ntype
1. C22×C22⋊D397C2semidirect
2. #27163079C2semidirect
3. #27163077C2semidirect
4. #27163068C2semidirect
5. #27163065C2semidirect
6. #27163064C2semidirect
7. C2×C22⋊Dic397C2semidirect
8. C22×D4×C397C2semidirect
9. C23×D397C22semidirect
12. C2×C22⋊D397C22semidirect
13. C2×C22⋊D397C22semidirect
14. C22×Dic397C22semidirect
15. C22×Dic397C22semidirect
16. #20263710C22semidirect
17. C22⋊Dic397C22semidirect
18. C22⋊Dic397C22semidirect
19. C22⋊Dic397C22semidirect
20. #20263708C22semidirect
22. C24×C397C22semidirect
25. C2×D4×C397C22semidirect
26. C22×C4×C397C22semidirect
40. C23×C397D4semidirect
73. C397#534semidirect
76. C22×D4D397semidirect
85. C24C2×D397semidirect
92. C2×D4C2×D397semidirect
95. C22×C4C2×D397semidirect
107. C23C22⋊D397semidirect

Maximal subgroups

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

subgroupindex
C22×C22⋊D3972
#271630792
#271630772
#271630682
#271630652
#271630642
C2×C22⋊Dic3972
C22×D4×C3972
#534397