Finite Group Databasebeta

← #123×D397 #38763022

Subgroups — #123×D397 #38763022

Order 25408.

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

normal subgroups only

This group has 342 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
1C2×D4×D397C211yesmaximal, characteristic
2C22×C4×D397C211yesmaximal, characteristic, complemented
3#27163081C212yesmaximal, complemented
4#27163080C211yesmaximal, characteristic, complemented
5#27163079C211yesmaximal, characteristic, complemented
6#27163078C212yesmaximal, complemented
7#27163077C211yesmaximal, characteristic, complemented
8C4⋊C4×D397C212yesmaximal, complemented
9C22⋊C4×D397C212yesmaximal, complemented
10C22⋊C4×D397C211yesmaximal, characteristic, complemented
11#123×C397C211yesmaximal, characteristic, complemented
12C23×D397C2211yescharacteristic
13C23×D397C2211yescharacteristic
14C2×D1588C2211yescharacteristic
15C2×C22⋊D397C2211yescharacteristic
16C2×C22⋊D397C2211yescharacteristic
17D4×D397—24no—
18C22×Dic397C2211yescharacteristic, complemented
19C2×C4×D397—24no—
20C2×C4×D397C2212yes—
21C2×C4×D397C2212yes—
22C2×C4×D397C2211yescharacteristic
23C2×C4×D397C2212yescomplemented
24C4⋊Dic397C2212yescomplemented
25#20263710C2212yescomplemented
26#20263710C2212yescomplemented
27C22⋊Dic397C2212yescomplemented
28C22⋊Dic397C2211yescharacteristic, complemented
29#20263708C2212yescomplemented
30#20263708C2212yescomplemented
31#20263708C2212yescomplemented
32C2×D4×C397C2211yescharacteristic
33C22×C4×C397C2211yescharacteristic, complemented
34C4⋊C4×C397C2212yescomplemented
35C22⋊C4×C397C2212yescomplemented
36C22⋊C4×C397C2211yescharacteristic, complemented
37C22×D397—24no—
38C22×D397—22no—
39C22×D397—24no—
40C22×D397D414yes—
41C22×D397C2311yescharacteristic
42C22×D397C2311yescharacteristic
43C22×D397C2311yescharacteristic
44D1588—22no—
45C22⋊D397—22no—
46C22⋊D397—22no—
47C2×Dic397—22no—
48C2×Dic397C2312yes—
49C2×Dic397C2312yes—
50C2×Dic397C2311yescharacteristic
51C4×D397—24no—
52C4×D397—24no—
53C4×D397—22no—
54C4×D397—24no—
55C23×C397C2311yescharacteristic
56C23×C397C2311yescharacteristic
57D4×C397—22no—
58C2×C4×C397—22no—
59C2×C4×C397C2312yes—
60C2×C4×C397C2312yes—
61C2×C4×C397C2311yescharacteristic
62C22×C397—22no—
63C22×C397—21no—
64C22×C397—22no—
65C22×C397C2×D412yes—
66C22×C397C2411yescharacteristic
67C2×D397—22no—
68C2×D397—28no—
69C2×D397—21no—
70C2×D397—22no—
71C2×D397—44no—
72C2×D397C2×D412yes—
73C2×D397C2×D412yes—
74C2×D397#4114yes—
75C4×C397—22no—
76C4×C397—21no—
77C4×C397—22no—
78Dic397—22no—
79Dic397—21no—
80Dic397—22no—
81C2×C397—22no—
82C2×C397—41no—
83C2×C397C22×D411yescharacteristic
84C2×C397C2×#4112yes—
85D397—22no—
86D397—41no—
87D397#12314yesdirect product factor
88C397C2×#12311yescharacteristic, complemented, Sylow
89C2×#123—3971nomaximal, Sylow
90C22×D4—3971no—
91C23×C4—3971no—
92#123—3971no—
93#123—3971no—
94#123—3972no—
95#123—3972no—
96#123—3971no—
97#123D39711yescharacteristic, direct product factor
98C2×C4⋊C4—3972no—
99C2×C22⋊C4—3971no—
100C2×C22⋊C4—3972no—
101C24—3971no—
102C24—3971no—
103C2×D4—7944no—
104C2×D4—3971no—
105C2×D4—3971no—
106C2×D4—3971no—
107C2×D4C2×D39711yescharacteristic
108C22×C4—7944no—
109C22×C4—3971no—
110C22×C4—3972no—
111C22×C4—3972no—
112C22×C4—3971no—
113C22×C4—3972no—
114C22×C4C2×D39711yescharacteristic, complemented
115C4⋊C4—3972no—
116C4⋊C4—3972no—
117C4⋊C4—3972no—
118C4⋊C4C2×D39712yescomplemented
119C22⋊C4—3971no—
120C22⋊C4—3972no—
121C22⋊C4—3972no—
122C22⋊C4—3972no—
123C22⋊C4—3972no—
124C22⋊C4C2×D39711yescharacteristic, complemented
125C22⋊C4C2×D39712yescomplemented
126C23—7944no—
127C23—7942no—
128C23—7944no—
129C23—3971no—
130C23—3974no—
131C23—3971no—
132C23—3971no—
133C23C22×D39711yescharacteristic
134C23C22×D39711yescharacteristic
135D4—22no—
136D4—7942no—
137D4—7942no—
138D4—7942no—
139C2×C4—22no—
140C2×C4—7944no—
141C2×C4—7944no—
142C2×C4—7944no—
143C2×C4—7942no—
144C2×C4—7942no—
145C2×C4—3972no—
146C2×C4—3972no—
147C2×C4—3971no—
148C2×C4C22×D39712yes—
149C2×C4C22×D39712yes—
150C2×C4C22×D39711yescharacteristic
151C22—22no—
152C22—22no—
153C22—21no—
154C22—7948no—
155C22—7942no—
156C22—7942no—
157C22—7941no—
158C22—15884no—
159C22—3972no—
160C22—3974no—
161C22—3972no—
162C22D4×D39712yes—
163C22C23×D39711yescharacteristic
164C4—22no—
165C4—22no—
166C4—21no—
167C4—7942no—
168C4—7942no—
169C4—7941no—
170C2—22no—
171C2—41no—
172C2—7942no—
173C2—15881no—
174C2—3974no—
175C2C2×D4×D39711yescharacteristic
176C2#41×D39712yes—
subgroupG/H
center, FrattiniC22C23×D397
commutator, socle, cosocleC22×C397C24
Fitting#123×C397C2
prime radicalC2×D4×D397C2
hypercenter#123D397
hypocenterC397C2×#123
LCSG ▷ C22×C397 ▷ C397
UCSC1 ◁ C22 ◁ #123

Complements

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

NH ≅ G/Ntype
2. C22×C4×D397C2semidirect
3. #27163081C2semidirect
4. #27163080C2semidirect
5. #27163079C2semidirect
6. #27163078C2semidirect
7. #27163077C2semidirect
8. C4⋊C4×D397C2semidirect
9. C22⋊C4×D397C2semidirect
10. C22⋊C4×D397C2semidirect
11. #123×C397C2semidirect
18. C22×Dic397C22semidirect
23. C2×C4×D397C22semidirect
24. C4⋊Dic397C22semidirect
25. #20263710C22semidirect
26. #20263710C22semidirect
27. C22⋊Dic397C22semidirect
28. C22⋊Dic397C22semidirect
29. #20263708C22semidirect
30. #20263708C22semidirect
31. #20263708C22semidirect
33. C22×C4×C397C22semidirect
34. C4⋊C4×C397C22semidirect
35. C22⋊C4×C397C22semidirect
36. C22⋊C4×C397C22semidirect
87. D397#123direct (N × H)
88. C397C2×#123semidirect
97. #123D397direct (N × H)
114. C22×C4C2×D397semidirect
118. C4⋊C4C2×D397semidirect
124. C22⋊C4C2×D397semidirect
125. C22⋊C4C2×D397semidirect

Maximal subgroups

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

subgroupindex
C2×D4×D3972
C22×C4×D3972
#271630812
#271630802
#271630792
#271630782
#271630772
C4⋊C4×D3972
C22⋊C4×D3972
#123×C3972
C2×#123397