Finite Group Databasebeta

← #38762994

Subgroups — #38762994

Order 25408.

338 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 338 conjugacy classes of subgroups — too many to draw. Only its 151 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
1C2×D4×D397C211yesmaximal, characteristic
2C4×D1588C211yesmaximal, characteristic, complemented
3D4×Dic397C211yesmaximal, characteristic, complemented
4#27163058C211yesmaximal, characteristic, complemented
5C4×C22⋊D397C212yesmaximal, complemented
6#27163056C212yesmaximal, complemented
7#27163054C211yesmaximal, characteristic, complemented
8#27163053C211yesmaximal, characteristic, complemented
9C2×#20263708C212yesmaximal, complemented
10C22⋊C4×D397C212yesmaximal, complemented
11C4×D4×C397C211yesmaximal, characteristic, complemented
12C23×D397C2212yes—
13C2×D1588C2211yescharacteristic
14C2×C22⋊D397C2212yes—
15D4×D397C418yescomplemented
16C22×Dic397C2212yescomplemented
17C2×C4×D397C2211yescharacteristic
18C2×C4×D397C2212yescomplemented
19C2×C4×D397C2212yescomplemented
20C4⋊Dic397C2211yescharacteristic, complemented
21#20263710C2211yescharacteristic, complemented
22#20263710C2211yescharacteristic, complemented
23C22⋊Dic397C2212yescomplemented
24#20263708C2211yescharacteristic
25#20263708C2212yes—
26#20263708C2211yescharacteristic
27#20263708C2212yescomplemented
28#20263708C2212yescomplemented
29C4×Dic397C2211yescharacteristic, complemented
30C4×Dic397C2212yescomplemented
31C2×D4×C397C2211yescharacteristic
32C22×C4×C397C2212yescomplemented
33C4⋊C4×C397C2211yescharacteristic, complemented
34C22⋊C4×C397C2212yescomplemented
35C42×C397C2211yescharacteristic, complemented
36C22×D397—42no—
37C22×D397—42no—
38C22×D397C2×C418yescomplemented
39C22×D397C2311yescharacteristic
40C22×D397C2312yes—
41D1588C2×C414yescomplemented
42C22⋊D397C2×C418yescomplemented
43C2×Dic397—42no—
44C2×Dic397C2311yescharacteristic
45C2×Dic397C2312yes—
46C2×Dic397C2311yescharacteristic
47C2×Dic397C2311yescharacteristic
48C4×D397—42no—
49C4×D397—42no—
50C4×D397C2×C414yescomplemented
51C23×C397C2312yes—
52D4×C397C2×C414yescomplemented
53C2×C4×C397—42no—
54C2×C4×C397C2311yescharacteristic
55C2×C4×C397C2311yescharacteristic
56C2×C4×C397C2312yes—
57C2×C4×C397C2311yescharacteristic
58C22×C397—22no—
59C22×C397—22no—
60C22×C397C22×C414yes—
61C22×C397C2411yescharacteristic
62C2×D397—22no—
63C2×D397—22no—
64C2×D397—21no—
65C2×D397—21no—
66C2×D397—48no—
67C2×D397C22×C412yes—
68C2×D397C22×C414yes—
69C4×C397—21no—
70C4×C397—22no—
71C4×C397—21no—
72C4×C397C22×C412yes—
73Dic397—21no—
74Dic397—22no—
75Dic397—21no—
76Dic397C22×C412yes—
77C2×C397—24no—
78C2×C397C23×C411yescharacteristic
79C2×C397H(5,𝔽2)11yescharacteristic
80C2×C397H(5,𝔽2)11yescharacteristic
81D397—22no—
82D397—24no—
83C397#51811yescharacteristic, complemented, Sylow
84#518—3971nomaximal, Sylow
85C22×D4—3971no—
86C4×D4—3971no—
87C4×D4—3972no—
88C4×D4—3971no—
89C4×D4—3972no—
90C4×D4—3971no—
91C4×D4D39711yescharacteristic, complemented
92#117—3971no—
93#117—3971no—
94C2×C22⋊C4—3972no—
95C2×C22⋊C4—3972no—
96C24—3972no—
97C2×D4—3972no—
98C2×D4—3978no—
99C2×D4—3971no—
100C2×D4C2×D39711yescharacteristic
101C22×C4—3972no—
102C22×C4—3972no—
103C22×C4—3972no—
104C22×C4—3971no—
105C22×C4C2×D39712yescomplemented
106C4⋊C4—3971no—
107C4⋊C4—3971no—
108C4⋊C4—3971no—
109C4⋊C4C2×D39711yescharacteristic, complemented
110C22⋊C4—3972no—
111C22⋊C4—3972no—
112C22⋊C4—3971no—
113C22⋊C4—3971no—
114C22⋊C4—3972no—
115C22⋊C4—3972no—
116C22⋊C4C2×D39712yescomplemented
117C42—3971no—
118C42—3972no—
119C42C2×D39711yescharacteristic, complemented
120C23—15882no—
121C23—15882no—
122C23—3971no—
123C23—3972no—
124C23—3978no—
125C23C22×D39712yes—
126D4—3978no—
127D4—3974no—
128D4C4×D39714yescomplemented
129C2×C4—42no—
130C2×C4—15882no—
131C2×C4—15882no—
132C2×C4—15882no—
133C2×C4—3971no—
134C2×C4—3971no—
135C2×C4—3972no—
136C2×C4—3971no—
137C2×C4—3974no—
138C2×C4C22×D39712yes—
139C2×C4C22×D39711yescharacteristic
140C2×C4C22×D39711yescharacteristic
141C2×C4C22×D39711yescharacteristic
142C22—22no—
143C22—22no—
144C22—7942no—
145C22—7941no—
146C22—7942no—
147C22—7941no—
148C22—15888no—
149C22—3972no—
150C22—3974no—
151C22C2×C4×D39714yes—
152C22C23×D39711yescharacteristic
153C4—21no—
154C4—22no—
155C4—21no—
156C4—7941no—
157C4—7942no—
158C4—7941no—
159C4—3972no—
160C4C2×C4×D39712yes—
161C2—24no—
162C2—7942no—
163C2—7944no—
164C2C22×C4×D39711yescharacteristic
165C2#2716314111yescharacteristic
166C2#2716314211yescharacteristic
subgroupG/H
center, FrattiniC22C23×D397
commutatorC2×C397C23×C4
FittingC4×D4×C397C2
socle, cosocleC22×C397C24
prime radicalC2×D4×D397C2
hypercenterC4×D4D397
hypocenterC397#518
LCSG ▷ C2×C397 ▷ C397
UCSC1 ◁ C22 ◁ C4×D4

Complements

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

NH ≅ G/Ntype
2. C4×D1588C2semidirect
3. D4×Dic397C2semidirect
4. #27163058C2semidirect
5. C4×C22⋊D397C2semidirect
6. #27163056C2semidirect
7. #27163054C2semidirect
8. #27163053C2semidirect
9. C2×#20263708C2semidirect
10. C22⋊C4×D397C2semidirect
11. C4×D4×C397C2semidirect
15. D4×D397C4semidirect
16. C22×Dic397C22semidirect
18. C2×C4×D397C22semidirect
19. C2×C4×D397C22semidirect
20. C4⋊Dic397C22semidirect
21. #20263710C22semidirect
22. #20263710C22semidirect
23. C22⋊Dic397C22semidirect
27. #20263708C22semidirect
28. #20263708C22semidirect
29. C4×Dic397C22semidirect
30. C4×Dic397C22semidirect
32. C22×C4×C397C22semidirect
33. C4⋊C4×C397C22semidirect
34. C22⋊C4×C397C22semidirect
35. C42×C397C22semidirect
38. C22×D397C2×C4semidirect
41. D1588C2×C4semidirect
42. C22⋊D397C2×C4semidirect
50. C4×D397C2×C4semidirect
52. D4×C397C2×C4semidirect
83. C397#518semidirect
91. C4×D4D397semidirect
105. C22×C4C2×D397semidirect
109. C4⋊C4C2×D397semidirect
116. C22⋊C4C2×D397semidirect
119. C42C2×D397semidirect
128. D4C4×D397semidirect

Maximal subgroups

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

subgroupindex
C2×D4×D3972
C4×D15882
D4×Dic3972
#271630582
C4×C22⋊D3972
#271630562
#271630542
#271630532
C2×#202637082
C22⋊C4×D3972
C4×D4×C3972
#518397