Finite Group Databasebeta

← #120×D397 #38763006

Subgroups — #120×D397 #38763006

Order 25408.

662 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 662 conjugacy classes of subgroups — too many to draw. Only its 131 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
1C24×D397C211yesmaximal, characteristic, complemented
2C2×D4×D397C213yesmaximal, complemented
3#27163066C213yesmaximal, complemented
4#27163065C211yesmaximal, characteristic, complemented
5#27163064C213yesmaximal, complemented
6C22⋊C4×D397C213yesmaximal, complemented
7#120×C397C211yesmaximal, characteristic, complemented
8C23×D397—212no—
9C23×D397C2213yes—
10C23×D397C2213yescomplemented
11C23×D397C2211yescharacteristic, complemented
12C2×D1588C2213yescomplemented
13C2×C22⋊D397C2213yescomplemented
14C2×C22⋊D397C2213yescomplemented
15D4×D397—212no—
16C2×C4×D397C2213yescomplemented
17C22⋊Dic397C2213yescomplemented
18#20263708C2216yescomplemented
19C24×C397C2211yescharacteristic, complemented
20C2×D4×C397C2213yescomplemented
21C22⋊C4×C397C2213yescomplemented
22C22×D397—232no—
23C22×D397—26no—
24C22×D397—212no—
25C22×D397—212no—
26C22×D397—26no—
27C22×D397D4112yescomplemented
28C22×D397C2313yes—
29C22×D397C2311yescharacteristic
30C22×D397C2311yescharacteristic, complemented
31D1588—26no—
32C22⋊D397—26no—
33C22⋊D397—26no—
34C2×Dic397C2313yescomplemented
35C4×D397—26no—
36C23×C397—26no—
37C23×C397C2313yes—
38C23×C397C2311yescharacteristic, complemented
39D4×C397—26no—
40C2×C4×C397C2313yescomplemented
41C22×C397—28no—
42C22×C397—26no—
43C22×C397—23no—
44C22×C397C2×D416yescomplemented
45C22×C397C2411yescharacteristic
46C2×D397—224no—
47C2×D397—26no—
48C2×D397—224no—
49C2×D397—23no—
50C2×D397—44no—
51C2×D397C2×D416yes—
52C2×D397C2×D416yescomplemented
53C4×C397—23no—
54Dic397—23no—
55C2×C397—26no—
56C2×C397—41no—
57C2×C397C22×D413yes—
58D397—26no—
59D397—41no—
60D397#12014yesdirect product factor
61C397C2×#12011yescharacteristic, complemented, Sylow
62C2×#120—3971nomaximal, Sylow
63C25—3971no—
64C22×D4—3973no—
65#120—3971no—
66#120—3973no—
67#120—3973no—
68#120D39711yescharacteristic, direct product factor
69C2×C22⋊C4—3973no—
70C24—79412no—
71C24—3973no—
72C24—3971no—
73C24—3973no—
74C24C2×D39711yescharacteristic, complemented
75C2×D4—79412no—
76C2×D4—3973no—
77C2×D4—3973no—
78C2×D4—3973no—
79C2×D4C2×D39713yescomplemented
80C22×C4—3973no—
81C22⋊C4—3973no—
82C22⋊C4—3976no—
83C22⋊C4C2×D39713yescomplemented
84C23—26no—
85C23—79432no—
86C23—79412no—
87C23—79412no—
88C23—7946no—
89C23—7946no—
90C23—3971no—
91C23—39712no—
92C23—3973no—
93C23—3971no—
94C23C22×D39713yes—
95C23C22×D39711yescharacteristic, complemented
96D4—26no—
97D4—7946no—
98D4—7946no—
99D4—7946no—
100C2×C4—7946no—
101C2×C4—3973no—
102C2×C4C22×D39713yescomplemented
103C22—28no—
104C22—26no—
105C22—23no—
106C22—79424no—
107C22—79424no—
108C22—7946no—
109C22—7943no—
110C22—15884no—
111C22—3976no—
112C22—3976no—
113C22D4×D39716yescomplemented
114C22C23×D39711yescharacteristic
115C4—23no—
116C4—7943no—
117C2—26no—
118C2—41no—
119C2—7946no—
120C2—15881no—
121C2—3974no—
122C2C2×D4×D39713yes—
subgroupG/H
center, FrattiniC22C23×D397
commutator, socle, cosocleC22×C397C24
Fitting#120×C397C2
hypercenter#120D397
hypocenterC397C2×#120
LCSG ▷ C22×C397 ▷ C397
UCSC1 ◁ C22 ◁ #120

Complements

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

NH ≅ G/Ntype
1. C24×D397C2semidirect
2. C2×D4×D397C2semidirect
3. #27163066C2semidirect
4. #27163065C2semidirect
5. #27163064C2semidirect
6. C22⋊C4×D397C2semidirect
7. #120×C397C2semidirect
10. C23×D397C22semidirect
11. C23×D397C22semidirect
12. C2×D1588C22semidirect
13. C2×C22⋊D397C22semidirect
14. C2×C22⋊D397C22semidirect
16. C2×C4×D397C22semidirect
17. C22⋊Dic397C22semidirect
18. #20263708C22semidirect
19. C24×C397C22semidirect
20. C2×D4×C397C22semidirect
21. C22⋊C4×C397C22semidirect
27. C22×D397D4semidirect
30. C22×D397C23semidirect
34. C2×Dic397C23semidirect
38. C23×C397C23semidirect
40. C2×C4×C397C23semidirect
44. C22×C397C2×D4semidirect
52. C2×D397C2×D4semidirect
60. D397#120direct (N × H)
61. C397C2×#120semidirect
68. #120D397direct (N × H)
74. C24C2×D397semidirect
79. C2×D4C2×D397semidirect
83. C22⋊C4C2×D397semidirect
95. C23C22×D397semidirect
102. C2×C4C22×D397semidirect
113. C22D4×D397semidirect

Maximal subgroups

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

subgroupindex
C24×D3972
C2×D4×D3972
#271630662
#271630652
#271630642
C22⋊C4×D3972
#120×C3972
C2×#120397