Finite Group Databasebeta

← #38763030

Subgroups — #38763030

Order 25408.

330 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 107 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×#20263730C211yesmaximal, characteristic, complemented
2C22×C4×D397C211yesmaximal, characteristic, complemented
3#27163079C211yesmaximal, characteristic, complemented
4#27163074C212yesmaximal, complemented
5#27163069C212yesmaximal, complemented
6#27163066C211yesmaximal, characteristic, complemented
7#27163064C211yesmaximal, characteristic, complemented
8#27163058C212yesmaximal, complemented
9#27163056C212yesmaximal, complemented
10#27163052C211yesmaximal, characteristic, complemented
11#123×C397C211yesmaximal, characteristic, complemented
12C23×D397C2211yescharacteristic
13#20263730—24no—
14C2×Dic794C2211yescharacteristic, complemented
15C2×D1588C2212yes—
16C2×C22⋊D397C2211yescharacteristic
17C2×C22⋊D397C2212yes—
18C2×C22⋊D397C2211yescharacteristic, complemented
19C22×Dic397C2211yescharacteristic
20C22×Dic397C2211yescharacteristic, complemented
21C2×C4×D397—24no—
22C2×C4×D397C2212yes—
23C2×C4×D397C2212yes—
24C2×C4×D397C2211yescharacteristic, complemented
25#20263710C2212yescomplemented
26#20263710C2212yescomplemented
27C22⋊Dic397C2211yescharacteristic
28#20263708C2212yes—
29#20263708C2212yes—
30#20263708C2212yescomplemented
31C4×Dic397C2212yescomplemented
32C2×D4×C397C2211yescharacteristic
33C22×C4×C397C2211yescharacteristic
34C4⋊C4×C397C2212yescomplemented
35C22⋊C4×C397C2211yescharacteristic
36C22⋊C4×C397C2212yescomplemented
37C22×D397—22no—
38C22×D397—24no—
39C22×D397C2311yescharacteristic
40C22×D397C2312yes—
41Dic794—22no—
42D1588—24no—
43C22⋊D397—22no—
44C22⋊D397—24no—
45C22⋊D397—22no—
46C2×Dic397—22no—
47C2×Dic397D414yescomplemented
48C2×Dic397C2312yes—
49C2×Dic397C2311yescharacteristic
50C2×Dic397C2311yescharacteristic
51C2×Dic397C2311yescharacteristic
52C4×D397—24no—
53C4×D397—24no—
54C4×D397—22no—
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—28no—
68C2×D397—22no—
69C2×D397—22no—
70C2×D397—21no—
71C2×D397—22no—
72C2×D397#4114yescomplemented
73C4×C397—22no—
74C4×C397—21no—
75C4×C397—22no—
76Dic397—21no—
77Dic397—22no—
78Dic397C2×D412yes—
79Dic397C2×D412yes—
80C2×C397—22no—
81C2×C397—41no—
82C2×C397C22×D411yescharacteristic
83C2×C397C2×#4112yes—
84D397—24no—
85D397—41no—
86C397#52511yescharacteristic, complemented, Sylow
87#525—3971nomaximal, Sylow
88C2×#41—3971no—
89C23×C4—3971no—
90#123—3971no—
91#123D39711yescharacteristic, complemented
92C22⋊Q8—3972no—
93#121—3972no—
94#120—3971no—
95#120—3971no—
96C4×D4—3972no—
97C4×D4—3972no—
98#117—3971no—
99C24—3971no—
100#41—7944no—
101C2×Q8—3971no—
102C2×D4—3972no—
103C2×D4—3971no—
104C2×D4—3971no—
105C2×D4—3972no—
106C2×D4C2×D39711yescharacteristic
107C22×C4—7944no—
108C22×C4—3971no—
109C22×C4—3972no—
110C22×C4—3972no—
111C22×C4—3971no—
112C22×C4—3971no—
113C22×C4C2×D39711yescharacteristic
114C4⋊C4—3972no—
115C4⋊C4—3972no—
116C4⋊C4C2×D39712yescomplemented
117C22⋊C4—3971no—
118C22⋊C4—3972no—
119C22⋊C4—3972no—
120C22⋊C4—3972no—
121C22⋊C4C2×D39711yescharacteristic
122C22⋊C4C2×D39712yescomplemented
123C42—3972no—
124C23—7944no—
125C23—7942no—
126C23—3971no—
127C23—3972no—
128C23C22×D39711yescharacteristic
129C23C22×D39711yescharacteristic
130Q8—7942no—
131D4—22no—
132D4—7944no—
133D4—7944no—
134D4—7942no—
135D4—7942no—
136C2×C4—22no—
137C2×C4—7944no—
138C2×C4—7944no—
139C2×C4—7942no—
140C2×C4—7942no—
141C2×C4—3972no—
142C2×C4—3971no—
143C2×C4—3974no—
144C2×C4—3971no—
145C2×C4—3971no—
146C2×C4C22×D39712yes—
147C2×C4C22×D39711yescharacteristic
148C2×C4C22×D39712yes—
149C22—22no—
150C22—22no—
151C22—21no—
152C22—7948no—
153C22—7942no—
154C22—7942no—
155C22—7942no—
156C22—7941no—
157C22—3974no—
158C22D4×D39712yes—
159C22C23×D39711yescharacteristic
160C4—22no—
161C4—22no—
162C4—21no—
163C4—7942no—
164C4—7941no—
165C4—3972no—
166C4—3972no—
167C2—22no—
168C2—41no—
169C2—7944no—
170C2—15881no—
171C2C2×D4×D39711yescharacteristic
172C2#41×D39712yes—
subgroupG/H
center, FrattiniC22C23×D397
commutator, socle, cosocleC22×C397C24
Fitting#123×C397C2
hypercenter#123D397
hypocenterC397#525
LCSG ▷ C22×C397 ▷ C397
UCSC1 ◁ C22 ◁ #123

Complements

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

NH ≅ G/Ntype
1. C2×#20263730C2semidirect
2. C22×C4×D397C2semidirect
3. #27163079C2semidirect
4. #27163074C2semidirect
5. #27163069C2semidirect
6. #27163066C2semidirect
7. #27163064C2semidirect
8. #27163058C2semidirect
9. #27163056C2semidirect
10. #27163052C2semidirect
11. #123×C397C2semidirect
14. C2×Dic794C22semidirect
18. C2×C22⋊D397C22semidirect
20. C22×Dic397C22semidirect
24. C2×C4×D397C22semidirect
25. #20263710C22semidirect
26. #20263710C22semidirect
30. #20263708C22semidirect
31. C4×Dic397C22semidirect
34. C4⋊C4×C397C22semidirect
36. C22⋊C4×C397C22semidirect
47. C2×Dic397D4semidirect
72. C2×D397#41semidirect
86. C397#525semidirect
91. #123D397semidirect
116. C4⋊C4C2×D397semidirect
122. C22⋊C4C2×D397semidirect

Maximal subgroups

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

subgroupindex
C2×#202637302
C22×C4×D3972
#271630792
#271630742
#271630692
#271630662
#271630642
#271630582
#271630562
#271630522
#123×C3972
#525397