Finite Group Databasebeta

← #37560162

Subgroups — #37560162

Order 23872.

252 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 91 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#26045788C211yesmaximal, characteristic, complemented
2#26045787C212yesmaximal, complemented
3#26045774C212yesmaximal, complemented
4#26045767C212yesmaximal, complemented
5#26045766C212yesmaximal, complemented
6#26045765C212yesmaximal, complemented
7#26045762C212yesmaximal, complemented
8#26045760C211yesmaximal, characteristic, complemented
9#123×C373C211yesmaximal, characteristic, complemented
10C23×D373C2211yescharacteristic
11C2×D1492C2212yes—
12C2×D1492C2212yes—
13C2×D1492C2211yescharacteristic, complemented
14C2×C22⋊D373C2212yes—
15C2×C22⋊D373C2212yes—
16C2×C22⋊D373C2212yescomplemented
17C2×C4×D373C2212yescomplemented
18C4⋊Dic373C2212yescomplemented
19#20238589C2212yescomplemented
20C22⋊Dic373C2211yescharacteristic
21#20238587C2212yes—
22#20238587C2212yes—
23#20238587C2212yes—
24#20238587C2212yescomplemented
25C4×Dic373C2211yescharacteristic, complemented
26C2×D4×C373C2211yescharacteristic
27C22×C4×C373C2211yescharacteristic, complemented
28C4⋊C4×C373C2212yescomplemented
29C22⋊C4×C373C2212yes—
30C22⋊C4×C373C2211yescharacteristic, complemented
31C22×D373—42no—
32C22×D373—41no—
33C22×D373C2312yes—
34C22×D373C2312yes—
35D1492—42no—
36D1492—42no—
37D1492—41no—
38C22⋊D373—42no—
39C22⋊D373—42no—
40C22⋊D373—42no—
41C2×Dic373C2312yes—
42C2×Dic373C2312yes—
43C4×D373—42no—
44C23×C373C2311yescharacteristic
45C23×C373C2311yescharacteristic
46D4×C373—41no—
47C2×C4×C373—41no—
48C2×C4×C373C2312yes—
49C2×C4×C373C2312yes—
50C2×C4×C373C2311yescharacteristic
51C22×C373—21no—
52C22×C373—21no—
53C22×C373—22no—
54C22×C373—22no—
55C22×C373C2411yescharacteristic
56C2×D373—22no—
57C2×D373—22no—
58C2×D373—22no—
59C2×D373—22no—
60C2×D373—22no—
61C2×D373—22no—
62C2×D373—44no—
63C4×C373—22no—
64C4×C373—21no—
65C4×C373—22no—
66Dic373—22no—
67Dic373—22no—
68C2×C373—41no—
69C2×C373—41no—
70C2×C373H(5,𝔽2)11yescharacteristic
71C2×C373H(5,𝔽2)12yes—
72D373—42no—
73D373—42no—
74C373#56011yescharacteristic, complemented, Sylow
75#560—3731nomaximal, Sylow
76#127—3731no—
77#126—3732no—
78#123—3732no—
79#123D37311yescharacteristic, complemented
80#121—3732no—
81#121—3732no—
82#121—3732no—
83#120—3731no—
84#120—3732no—
85C24—3731no—
86C2×D4—3732no—
87C2×D4—3732no—
88C2×D4—3732no—
89C2×D4—3732no—
90C2×D4—3732no—
91C2×D4—3731no—
92C2×D4C2×D37311yescharacteristic
93C22×C4—3732no—
94C22×C4C2×D37311yescharacteristic, complemented
95C4⋊C4—3732no—
96C4⋊C4—3732no—
97C4⋊C4C2×D37312yescomplemented
98C22⋊C4—3731no—
99C22⋊C4—3732no—
100C22⋊C4—3732no—
101C22⋊C4—3732no—
102C22⋊C4—3732no—
103C22⋊C4C2×D37312yes—
104C22⋊C4C2×D37311yescharacteristic, complemented
105C42—3731no—
106C23—3732no—
107C23—3732no—
108C23—14921no—
109C23—14922no—
110C23C22×D37311yescharacteristic
111C23C22×D37311yescharacteristic
112D4—41no—
113D4—14922no—
114D4—14921no—
115D4—14922no—
116D4—14922no—
117D4—14922no—
118D4—14922no—
119C2×C4—41no—
120C2×C4—3732no—
121C2×C4—3732no—
122C2×C4—14922no—
123C2×C4C22×D37311yescharacteristic
124C2×C4C22×D37312yes—
125C2×C4C22×D37312yes—
126C22—21no—
127C22—21no—
128C22—22no—
129C22—22no—
130C22—14924no—
131C22—7462no—
132C22—7462no—
133C22—7462no—
134C22—7462no—
135C22—7462no—
136C22—7462no—
137C22C23×D37311yescharacteristic
138C4—22no—
139C4—21no—
140C4—22no—
141C4—7462no—
142C4—7462no—
143C2—41no—
144C2—41no—
145C2—14922no—
146C2—14922no—
147C2#2604583711yescharacteristic
148C2#2604583812yes—
subgroupG/H
center, FrattiniC22C23×D373
commutator, socle, cosocleC22×C373C24
Fitting#123×C373C2
hypercenter#123D373
hypocenterC373#560
LCSG ▷ C22×C373 ▷ C373
UCSC1 ◁ C22 ◁ #123

Complements

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

NH ≅ G/Ntype
1. #26045788C2semidirect
2. #26045787C2semidirect
3. #26045774C2semidirect
4. #26045767C2semidirect
5. #26045766C2semidirect
6. #26045765C2semidirect
7. #26045762C2semidirect
8. #26045760C2semidirect
9. #123×C373C2semidirect
13. C2×D1492C22semidirect
16. C2×C22⋊D373C22semidirect
17. C2×C4×D373C22semidirect
18. C4⋊Dic373C22semidirect
19. #20238589C22semidirect
24. #20238587C22semidirect
25. C4×Dic373C22semidirect
27. C22×C4×C373C22semidirect
28. C4⋊C4×C373C22semidirect
30. C22⋊C4×C373C22semidirect
74. C373#560semidirect
79. #123D373semidirect
94. C22×C4C2×D373semidirect
97. C4⋊C4C2×D373semidirect
104. C22⋊C4C2×D373semidirect

Maximal subgroups

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

subgroupindex
#260457882
#260457872
#260457742
#260457672
#260457662
#260457652
#260457622
#260457602
#123×C3732
#560373