Finite Group Databasebeta

← #38763082

Subgroups — #38763082

Order 25408.

346 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 346 conjugacy classes of subgroups — too many to draw. Only its 103 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
2C2×D4×D397C211yesmaximal, characteristic, complemented
3#27163080C211yesmaximal, characteristic, complemented
4#27163078C212yesmaximal, complemented
5#27163077C211yesmaximal, characteristic, complemented
6#27163069C212yesmaximal, complemented
7#27163067C212yesmaximal, complemented
8#27163065C211yesmaximal, characteristic, complemented
9#27163064C212yesmaximal, complemented
10C22⋊C4×D397C211yesmaximal, characteristic, complemented
11#120×C397C211yesmaximal, characteristic, complemented
12C23×D397C2211yescharacteristic
13C23×D397C2211yescharacteristic, complemented
14C2×D1588C2211yescharacteristic, complemented
15C2×C22⋊D397—24no—
16C2×C22⋊D397C2212yes—
17C2×C22⋊D397C2211yescharacteristic
18C2×C22⋊D397C2211yescharacteristic, complemented
19C2×C22⋊D397C2212yescomplemented
20C2×C22⋊D397C2212yescomplemented
21D4×D397—24no—
22C22×Dic397C2211yescharacteristic, complemented
23C2×C4×D397C2211yescharacteristic, complemented
24C2×C4×D397C2212yescomplemented
25C4⋊Dic397C2212yescomplemented
26#20263710C2212yescomplemented
27C22⋊Dic397C2211yescharacteristic, complemented
28C22⋊Dic397C2212yescomplemented
29C22⋊Dic397C2212yescomplemented
30#20263708C2212yescomplemented
31#20263708C2212yescomplemented
32C24×C397C2211yescharacteristic, complemented
33C2×D4×C397C2211yescharacteristic
34C2×D4×C397C2212yescomplemented
35C22⋊C4×C397C2211yescharacteristic, complemented
36C22⋊C4×C397C2212yescomplemented
37C22×D397—22no—
38C22×D397—42no—
39C22×D397—42no—
40C22×D397D414yescomplemented
41C22×D397C2311yescharacteristic
42C22×D397C2311yescharacteristic
43C22×D397C2311yescharacteristic
44D1588—22no—
45C22⋊D397—22no—
46C22⋊D397—22no—
47C22⋊D397—24no—
48C22⋊D397—24no—
49C22⋊D397—42no—
50C2×Dic397—22no—
51C2×Dic397C2312yes—
52C2×Dic397C2312yes—
53C2×Dic397C2311yescharacteristic
54C4×D397—22no—
55C4×D397—42no—
56C23×C397—22no—
57C23×C397—42no—
58C23×C397C2312yes—
59C23×C397C2311yescharacteristic
60C23×C397C2311yescharacteristic
61D4×C397—22no—
62D4×C397—42no—
63C2×C4×C397C2311yescharacteristic
64C2×C4×C397C2312yes—
65C22×C397—22no—
66C22×C397—21no—
67C22×C397—22no—
68C22×C397—22no—
69C22×C397—22no—
70C22×C397—22no—
71C22×C397—44no—
72C22×C397C2×D412yes—
73C22×C397C2411yescharacteristic
74C2×D397—22no—
75C2×D397—22no—
76C2×D397—21no—
77C2×D397—22no—
78C2×D397—44no—
79C2×D397—44no—
80C2×D397C2×D412yes—
81C2×D397C2×D412yes—
82C4×C397—21no—
83C4×C397—22no—
84Dic397—21no—
85Dic397—22no—
86Dic397—22no—
87C2×C397—22no—
88C2×C397—41no—
89C2×C397—42no—
90C2×C397C22×D411yescharacteristic
91C2×C397H(5,𝔽2)12yes—
92D397—22no—
93D397—22no—
94D397—41no—
95C397#53411yescharacteristic, complemented, Sylow
96#534—3971nomaximal, Sylow
97C22×D4—3971no—
98C22×D4—3971no—
99#123—3971no—
100#123—3972no—
101#123—3971no—
102#121—3972no—
103#121—3972no—
104#120—3971no—
105#120—3972no—
106#120D39711yescharacteristic, complemented
107C2×C22⋊C4—3971no—
108C24—3971no—
109C24—3971no—
110C24C2×D39711yescharacteristic, complemented
111C2×D4—7944no—
112C2×D4—7944no—
113C2×D4—3972no—
114C2×D4—3971no—
115C2×D4—3971no—
116C2×D4—3972no—
117C2×D4—3972no—
118C2×D4—3971no—
119C2×D4C2×D39711yescharacteristic
120C2×D4C2×D39712yescomplemented
121C22×C4—3971no—
122C22×C4—3971no—
123C22×C4—3972no—
124C4⋊C4—3972no—
125C4⋊C4—3972no—
126C22⋊C4—3971no—
127C22⋊C4—3972no—
128C22⋊C4—3972no—
129C22⋊C4—3972no—
130C22⋊C4—3972no—
131C22⋊C4C2×D39711yescharacteristic, complemented
132C22⋊C4C2×D39712yescomplemented
133C23—22no—
134C23—42no—
135C23—7942no—
136C23—15882no—
137C23—15882no—
138C23—3971no—
139C23—3971no—
140C23—3971no—
141C23—3974no—
142C23C22×D39711yescharacteristic
143C23C22×D39712yes—
144C23C22×D39711yescharacteristic
145D4—22no—
146D4—42no—
147D4—7942no—
148D4—7942no—
149D4—7944no—
150D4—7944no—
151D4—7942no—
152D4—15882no—
153C2×C4—7942no—
154C2×C4—7942no—
155C2×C4—15882no—
156C2×C4—3972no—
157C2×C4—3972no—
158C2×C4—3971no—
159C2×C4C22×D39712yes—
160C2×C4C22×D39711yescharacteristic
161C22—22no—
162C22—22no—
163C22—22no—
164C22—22no—
165C22—21no—
166C22—22no—
167C22—44no—
168C22—7942no—
169C22—7942no—
170C22—7942no—
171C22—7941no—
172C22—15884no—
173C22—15884no—
174C22—3972no—
175C22—3972no—
176C22D4×D39712yes—
177C22C23×D39711yescharacteristic
178C4—22no—
179C4—21no—
180C4—7942no—
181C4—7941no—
182C4—7942no—
183C2—22no—
184C2—42no—
185C2—41no—
186C2—7942no—
187C2—7942no—
188C2—15881no—
189C2C2×D4×D39711yescharacteristic
190C2#2716314112yes—
subgroupG/H
center, FrattiniC22C23×D397
commutator, socle, cosocleC22×C397C24
Fitting#120×C397C2
hypercenter#120D397
hypocenterC397#534
LCSG ▷ C22×C397 ▷ C397
UCSC1 ◁ C22 ◁ #120

Complements

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

NH ≅ G/Ntype
1. C22×C22⋊D397C2semidirect
2. C2×D4×D397C2semidirect
3. #27163080C2semidirect
4. #27163078C2semidirect
5. #27163077C2semidirect
6. #27163069C2semidirect
7. #27163067C2semidirect
8. #27163065C2semidirect
9. #27163064C2semidirect
10. C22⋊C4×D397C2semidirect
11. #120×C397C2semidirect
13. C23×D397C22semidirect
14. C2×D1588C22semidirect
18. C2×C22⋊D397C22semidirect
19. C2×C22⋊D397C22semidirect
20. C2×C22⋊D397C22semidirect
22. C22×Dic397C22semidirect
23. C2×C4×D397C22semidirect
24. C2×C4×D397C22semidirect
25. C4⋊Dic397C22semidirect
26. #20263710C22semidirect
27. C22⋊Dic397C22semidirect
28. C22⋊Dic397C22semidirect
29. C22⋊Dic397C22semidirect
30. #20263708C22semidirect
31. #20263708C22semidirect
32. C24×C397C22semidirect
34. C2×D4×C397C22semidirect
35. C22⋊C4×C397C22semidirect
36. C22⋊C4×C397C22semidirect
40. C22×D397D4semidirect
95. C397#534semidirect
106. #120D397semidirect
110. C24C2×D397semidirect
120. C2×D4C2×D397semidirect
131. C22⋊C4C2×D397semidirect
132. C22⋊C4C2×D397semidirect

Maximal subgroups

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

subgroupindex
C22×C22⋊D3972
C2×D4×D3972
#271630802
#271630782
#271630772
#271630692
#271630672
#271630652
#271630642
C22⋊C4×D3972
#120×C3972
#534397