Finite Group Databasebeta

← #38763088

Subgroups — #38763088

Order 25408.

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

normal subgroups only

This group has 334 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
1C2×#20263731C211yesmaximal, characteristic, complemented
2C2×D4×D397C211yesmaximal, characteristic, complemented
3#27163092C212yesmaximal, complemented
4#27163082C212yesmaximal, complemented
5#27163071C211yesmaximal, characteristic, complemented
6#27163070C211yesmaximal, characteristic, complemented
7#27163067C211yesmaximal, characteristic, complemented
8#27163066C212yesmaximal, complemented
9#27163064C212yesmaximal, complemented
10#27163054C211yesmaximal, characteristic, complemented
11#121×C397C211yesmaximal, characteristic, complemented
12C23×D397C2212yes—
13#20263731—24no—
14C2×Dic794C2211yescharacteristic, complemented
15C2×D1588C2211yescharacteristic
16C2×D1588C2211yescharacteristic
17C2×D1588C2212yescomplemented
18C2×C22⋊D397C2211yescharacteristic
19C2×C22⋊D397C2212yes—
20C2×C22⋊D397C2211yescharacteristic, complemented
21C2×C22⋊D397C2212yescomplemented
22C2×C22⋊D397C2212yescomplemented
23D4×D397—24no—
24C2×C4×D397C2211yescharacteristic
25C2×C4×D397C2211yescharacteristic, complemented
26C4⋊Dic397C2211yescharacteristic, complemented
27C22⋊Dic397C2212yescomplemented
28#20263708C2212yes—
29#20263708C2212yescomplemented
30#20263708C2212yescomplemented
31C4×Dic397C2212yescomplemented
32C2×D4×C397C2211yescharacteristic
33C2×D4×C397C2212yescomplemented
34C22×C4×C397C2211yescharacteristic, complemented
35C4⋊C4×C397C2211yescharacteristic, complemented
36C22⋊C4×C397C2212yescomplemented
37C22×D397—24no—
38C22×D397—42no—
39C22×D397—42no—
40C22×D397C2312yes—
41C22×D397C2311yescharacteristic
42C22×D397C2311yescharacteristic
43Dic794—22no—
44D1588—22no—
45D1588—22no—
46D1588—42no—
47C22⋊D397—22no—
48C22⋊D397—24no—
49C22⋊D397—22no—
50C22⋊D397—42no—
51C22⋊D397—42no—
52C2×Dic397C2312yes—
53C2×Dic397C2311yescharacteristic
54C2×Dic397C2311yescharacteristic
55C4×D397—22no—
56C4×D397D414yescomplemented
57C23×C397C2312yes—
58C23×C397C2311yescharacteristic
59D4×C397—22no—
60D4×C397—42no—
61C2×C4×C397—22no—
62C2×C4×C397C2311yescharacteristic
63C2×C4×C397C2312yes—
64C2×C4×C397C2311yescharacteristic
65C22×C397—21no—
66C22×C397—22no—
67C22×C397—21no—
68C22×C397—22no—
69C22×C397—21no—
70C22×C397—22no—
71C22×C397C2411yescharacteristic
72C2×D397—21no—
73C2×D397—21no—
74C2×D397—21no—
75C2×D397—22no—
76C2×D397—21no—
77C2×D397—22no—
78C2×D397—21no—
79C2×D397—22no—
80C2×D397—48no—
81C2×D397C2×D412yes—
82C4×C397—21no—
83C4×C397—22no—
84C4×C397C2×D412yes—
85Dic397—21no—
86Dic397—22no—
87Dic397C2×D412yes—
88C2×C397—42no—
89C2×C397—41no—
90C2×C397C22×D411yescharacteristic
91C2×C397H(5,𝔽2)11yescharacteristic
92C2×C397H(5,𝔽2)11yescharacteristic
93D397—22no—
94D397—41no—
95D397—42no—
96C397#53511yescharacteristic, complemented, Sylow
97#535—3971nomaximal, Sylow
98C2×#41—3971no—
99C22×D4—3971no—
100#127—3972no—
101#124—3972no—
102#121—3971no—
103#121—3971no—
104#121—3971no—
105#121D39711yescharacteristic, complemented
106#120—3972no—
107#120—3972no—
108#117—3971no—
109C24—3972no—
110#41—7944no—
111C2×Q8—3971no—
112C2×D4—7944no—
113C2×D4—3971no—
114C2×D4—3972no—
115C2×D4—3971no—
116C2×D4—3972no—
117C2×D4—3972no—
118C2×D4—3971no—
119C2×D4—3971no—
120C2×D4—3972no—
121C2×D4C2×D39711yescharacteristic
122C2×D4C2×D39712yescomplemented
123C22×C4—3971no—
124C22×C4—3971no—
125C22×C4C2×D39711yescharacteristic, complemented
126C4⋊C4—3971no—
127C4⋊C4C2×D39711yescharacteristic, complemented
128C22⋊C4—3972no—
129C22⋊C4—3972no—
130C22⋊C4—3972no—
131C22⋊C4—3972no—
132C22⋊C4C2×D39712yescomplemented
133C42—3972no—
134C23—7944no—
135C23—15882no—
136C23—15882no—
137C23—3972no—
138C23—3971no—
139C23—3971no—
140C23C22×D39712yes—
141C23C22×D39711yescharacteristic
142Q8—7942no—
143D4—22no—
144D4—42no—
145D4—7942no—
146D4—7942no—
147D4—7944no—
148D4—7942no—
149D4—7942no—
150D4—15882no—
151D4—15882no—
152D4—15882no—
153C2×C4—22no—
154C2×C4—7942no—
155C2×C4—3972no—
156C2×C4—3971no—
157C2×C4—3971no—
158C2×C4—3974no—
159C2×C4C22×D39712yes—
160C2×C4C22×D39711yescharacteristic
161C2×C4C22×D39711yescharacteristic
162C22—21no—
163C22—22no—
164C22—21no—
165C22—22no—
166C22—21no—
167C22—22no—
168C22—7941no—
169C22—7941no—
170C22—7941no—
171C22—7942no—
172C22—7941no—
173C22—7942no—
174C22—7941no—
175C22—7942no—
176C22—15888no—
177C22—3972no—
178C22C23×D39711yescharacteristic
179C4—22no—
180C4—21no—
181C4—7942no—
182C4—7941no—
183C4—3972no—
184C4D4×D39712yes—
185C2—42no—
186C2—41no—
187C2—7942no—
188C2—15882no—
189C2—15881no—
190C2C2×D4×D39711yescharacteristic
191C2#2716314111yescharacteristic
192C2#2716314211yescharacteristic
subgroupG/H
center, FrattiniC22C23×D397
commutator, socle, cosocleC22×C397C24
Fitting#121×C397C2
hypercenter#121D397
hypocenterC397#535
LCSG ▷ C22×C397 ▷ C397
UCSC1 ◁ C22 ◁ #121

Complements

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

NH ≅ G/Ntype
1. C2×#20263731C2semidirect
2. C2×D4×D397C2semidirect
3. #27163092C2semidirect
4. #27163082C2semidirect
5. #27163071C2semidirect
6. #27163070C2semidirect
7. #27163067C2semidirect
8. #27163066C2semidirect
9. #27163064C2semidirect
10. #27163054C2semidirect
11. #121×C397C2semidirect
14. C2×Dic794C22semidirect
17. C2×D1588C22semidirect
20. C2×C22⋊D397C22semidirect
21. C2×C22⋊D397C22semidirect
22. C2×C22⋊D397C22semidirect
25. C2×C4×D397C22semidirect
26. C4⋊Dic397C22semidirect
27. C22⋊Dic397C22semidirect
29. #20263708C22semidirect
30. #20263708C22semidirect
31. C4×Dic397C22semidirect
33. C2×D4×C397C22semidirect
34. C22×C4×C397C22semidirect
35. C4⋊C4×C397C22semidirect
36. C22⋊C4×C397C22semidirect
56. C4×D397D4semidirect
96. C397#535semidirect
105. #121D397semidirect
122. C2×D4C2×D397semidirect
125. C22×C4C2×D397semidirect
127. C4⋊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
C2×#202637312
C2×D4×D3972
#271630922
#271630822
#271630712
#271630702
#271630672
#271630662
#271630642
#271630542
#121×C3972
#535397