Finite Group Databasebeta

← #38762710

Subgroups — #38762710

Order 25408.

198 conjugacy classes of subgroups; only the 39 normal subgroups are shown.

normal subgroups only

This group has 198 conjugacy classes of subgroups — too many to draw. Only its 39 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
1#27163141C211yesmaximal, characteristic, complemented
2#27163066C212yesmaximal, complemented
3#27163064C211yesmaximal, characteristic, complemented
4#27163010C212yesmaximal, complemented
5#99×C397C211yesmaximal, characteristic, complemented
6C23×D397C2211yescharacteristic, complemented
7#20263731—21no—
8#20263730—22no—
9C2×D1588—22no—
10C2×C22⋊D397—21no—
11C2×C22⋊D397—21no—
12C2×C22⋊D397C2212yes—
13D4×D397—22no—
14C22⋊Dic397C2211yescharacteristic, complemented
15#20263708—22no—
16#20263708—21no—
17C2×D4×C397C2211yescharacteristic
18C22⋊C4×C397C2212yescomplemented
19C22×D397—21no—
20C22×D397—22no—
21C22×D397—21no—
22C22×D397—42no—
23C22×D397D412yescomplemented
24Dic794—21no—
25D1588—21no—
26D1588—44no—
27C22⋊D397—22no—
28C22⋊D397—22no—
29C22⋊D397—22no—
30C22⋊D397—42no—
31C2×Dic397—21no—
32C2×Dic397—21no—
33C2×Dic397D412yescomplemented
34C4×D397—22no—
35C23×C397D411yescharacteristic, complemented
36C23×C397C2311yescharacteristic
37D4×C397—22no—
38C2×C4×C397—22no—
39C2×C4×C397D411yescharacteristic, complemented
40C22×C397—21no—
41C22×C397—41no—
42C22×C397—41no—
43C22×C397C2×D412yes—
44C22×C397C2×D411yescharacteristic
45C2×D397—22no—
46C2×D397—24no—
47C2×D397—22no—
48C2×D397—41no—
49C2×D397—42no—
50C2×D397—41no—
51C2×D397—42no—
52C2×D397—81no—
53C4×C397—21no—
54C4×C397—42no—
55Dic397—22no—
56Dic397—41no—
57C2×C397—21no—
58C2×C397—22no—
59C2×C397—41no—
60C2×C397#12011yescharacteristic
61D397—42no—
62D397—42no—
63C397B(4,𝔽2)11yescharacteristic, complemented, Sylow
64B(4,𝔽2)—3971nomaximal, Sylow
65H(5,𝔽2)—3971no—
66#120—3971no—
67#120—3972no—
68#99—3972no—
69#99D39711yescharacteristic, complemented
70C24—3971no—
71#41—7942no—
72#41—7941no—
73C2×D4—7942no—
74C2×D4—7942no—
75C2×D4—7941no—
76C2×D4—7941no—
77C2×D4—3972no—
78C2×D4C2×D39711yescharacteristic
79C22⋊C4—7942no—
80C22⋊C4—7941no—
81C22⋊C4—3971no—
82C22⋊C4C2×D39712yescomplemented
83C23—7941no—
84C23—7942no—
85C23—7941no—
86C23—15882no—
87C23—3972no—
88C23D158811yescharacteristic, complemented
89C23C22×D39711yescharacteristic
90Q8—7941no—
91D4—22no—
92D4—7941no—
93D4—7942no—
94D4—7942no—
95D4—7942no—
96D4—15884no—
97D4—15882no—
98C2×C4—22no—
99C2×C4—7942no—
100C2×C4—7941no—
101C2×C4—7941no—
102C2×C4—3972no—
103C2×C4D158811yescharacteristic, complemented
104C22—21no—
105C22—41no—
106C22—41no—
107C22—7944no—
108C22—7942no—
109C22—7942no—
110C22—15882no—
111C22—15882no—
112C22—15881no—
113C22—15881no—
114C22—31761no—
115C22D4×D39712yes—
116C22C2×D158811yescharacteristic
117C4—21no—
118C4—42no—
119C4—7942no—
120C4—15881no—
121C2—22no—
122C2—21no—
123C2—41no—
124C2—15882no—
125C2—15882no—
126C2#2716306611yescharacteristic
subgroupG/H
centerC2#27163066
commutator, cosocleC23×C397C23
FrattiniC23C22×D397
Fitting#99×C397C2
socleC2×C397#120
hypercenter#99D397
hypocenterC397B(4,𝔽2)
LCSG ▷ C23×C397 ▷ C2×C397 ▷ C397
UCSC1 ◁ C2 ◁ C23 ◁ #99

Complements

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

NH ≅ G/Ntype
1. #27163141C2semidirect
2. #27163066C2semidirect
3. #27163064C2semidirect
4. #27163010C2semidirect
5. #99×C397C2semidirect
6. C23×D397C22semidirect
14. C22⋊Dic397C22semidirect
18. C22⋊C4×C397C22semidirect
23. C22×D397D4semidirect
33. C2×Dic397D4semidirect
35. C23×C397D4semidirect
39. C2×C4×C397D4semidirect
63. C397B(4,𝔽2)semidirect
69. #99D397semidirect
82. C22⋊C4C2×D397semidirect
88. C23D1588semidirect
103. C2×C4D1588semidirect

Maximal subgroups

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

subgroupindex
#271631412
#271630662
#271630642
#271630102
#99×C3972
B(4,𝔽2)397