Finite Group Databasebeta

← #37559608

Subgroups — #37559608

Order 23872.

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#26045837C211yesmaximal, characteristic, complemented
2#26045761C212yesmaximal, complemented
3#26045707C211yesmaximal, characteristic, complemented
4#26045706C212yesmaximal, complemented
5#120×C373C211yesmaximal, characteristic, complemented
6#20238610—21no—
7#20238609—22no—
8C2×C22⋊D373—22no—
9C2×C22⋊D373—21no—
10C2×C22⋊D373C2212yes—
11D4×D373—22no—
12C22⋊Dic373—22no—
13C22⋊Dic373C2212yescomplemented
14C24×C373C2211yescharacteristic, complemented
15C2×D4×C373—21no—
16C2×D4×C373C2211yescharacteristic
17C22⋊C4×C373—21no—
18C22⋊C4×C373C2211yescharacteristic, complemented
19C22×D373—21no—
20C22×D373D412yescomplemented
21Dic746—21no—
22D1492—21no—
23C22⋊D373—22no—
24C22⋊D373—22no—
25C22⋊D373—22no—
26C22⋊D373—44no—
27C2×Dic373—21no—
28C2×Dic373—22no—
29C2×Dic373D412yescomplemented
30C4×D373—22no—
31C23×C373—22no—
32C23×C373—21no—
33C23×C373—42no—
34C23×C373D411yescharacteristic, complemented
35C23×C373C2311yescharacteristic
36D4×C373—22no—
37D4×C373—42no—
38C2×C4×C373—21no—
39C2×C4×C373D411yescharacteristic, complemented
40C22×C373—22no—
41C22×C373—24no—
42C22×C373—21no—
43C22×C373—41no—
44C22×C373—41no—
45C22×C373—42no—
46C22×C373—41no—
47C22×C373—41no—
48C22×C373C2×D412yes—
49C22×C373C2×D411yescharacteristic
50C2×D373—22no—
51C2×D373—42no—
52C2×D373—81no—
53C4×C373—21no—
54C4×C373—41no—
55Dic373—22no—
56Dic373—42no—
57C2×C373—22no—
58C2×C373—21no—
59C2×C373—41no—
60C2×C373—42no—
61C2×C373#12011yescharacteristic
62D373—42no—
63C373B(4,𝔽2)11yescharacteristic, complemented, Sylow
64B(4,𝔽2)—3731nomaximal, Sylow
65H(5,𝔽2)—3731no—
66#120—3732no—
67#120D37311yescharacteristic, complemented
68#99—3731no—
69#99—3732no—
70C24C2×D37311yescharacteristic, complemented
71#41—7462no—
72#41—7461no—
73C2×D4—21no—
74C2×D4—3732no—
75C2×D4—7462no—
76C2×D4—7462no—
77C2×D4—7461no—
78C2×D4C2×D37311yescharacteristic
79C22⋊C4—21no—
80C22⋊C4—3732no—
81C22⋊C4—7462no—
82C22⋊C4C2×D37311yescharacteristic, complemented
83C23—22no—
84C23—21no—
85C23—42no—
86C23—3732no—
87C23—7461no—
88C23C22⋊D37311yescharacteristic, complemented
89C23C22×D37311yescharacteristic
90Q8—7461no—
91D4—22no—
92D4—42no—
93D4—14924no—
94D4—7461no—
95D4—7462no—
96D4—7462no—
97D4—7462no—
98C2×C4—21no—
99C2×C4—3732no—
100C2×C4—7462no—
101C2×C4—7462no—
102C2×C4—7461no—
103C2×C4C22⋊D37311yescharacteristic, complemented
104C22—22no—
105C22—24no—
106C22—21no—
107C22—41no—
108C22—41no—
109C22—41no—
110C22—42no—
111C22—41no—
112C22—29841no—
113C22—14922no—
114C22—7462no—
115C22D4×D37312yes—
116C22C2×C22⋊D37311yescharacteristic
117C4—21no—
118C4—41no—
119C4—14922no—
120C4—7462no—
121C2—21no—
122C2—22no—
123C2—42no—
124C2—41no—
125C2—14922no—
126C2#2604576011yescharacteristic
subgroupG/H
centerC2#26045760
commutator, cosocleC23×C373C23
FrattiniC23C22×D373
Fitting#120×C373C2
socleC2×C373#120
hypercenter#120D373
hypocenterC373B(4,𝔽2)
LCSG ▷ C23×C373 ▷ C2×C373 ▷ C373
UCSC1 ◁ C2 ◁ C23 ◁ #120

Complements

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

NH ≅ G/Ntype
1. #26045837C2semidirect
2. #26045761C2semidirect
3. #26045707C2semidirect
4. #26045706C2semidirect
5. #120×C373C2semidirect
13. C22⋊Dic373C22semidirect
14. C24×C373C22semidirect
18. C22⋊C4×C373C22semidirect
20. C22×D373D4semidirect
29. C2×Dic373D4semidirect
34. C23×C373D4semidirect
39. C2×C4×C373D4semidirect
63. C373B(4,𝔽2)semidirect
67. #120D373semidirect
70. C24C2×D373semidirect
82. C22⋊C4C2×D373semidirect
88. C23C22⋊D373semidirect
103. C2×C4C22⋊D373semidirect

Maximal subgroups

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

subgroupindex
#260458372
#260457612
#260457072
#260457062
#120×C3732
B(4,𝔽2)373