Finite Group Databasebeta

← #37559980

Subgroups — #37559980

Order 23872.

346 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 346 conjugacy classes of subgroups — too many to draw. Only its 103 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×D4×D373C212yesmaximal, complemented
2#26045777C212yesmaximal, complemented
3#26045774C212yesmaximal, complemented
4#26045766C211yesmaximal, characteristic, complemented
5#26045764C212yesmaximal, complemented
6#26045762C212yesmaximal, complemented
7#26045760C212yesmaximal, complemented
8C2×#20238587C211yesmaximal, characteristic, complemented
9#121×C373C211yesmaximal, characteristic, complemented
10C23×D373C2211yescharacteristic
11C23×D373C2212yescomplemented
12C2×D1492C2212yes—
13C2×D1492C2211yescharacteristic, complemented
14C2×C22⋊D373C2212yes—
15C2×C22⋊D373C2212yescomplemented
16C2×C22⋊D373C2212yescomplemented
17D4×D373—28no—
18C22×Dic373C2211yescharacteristic, complemented
19C2×C4×D373C2212yescomplemented
20C4⋊Dic373C2211yescharacteristic, complemented
21#20238589C2212yescomplemented
22C22⋊Dic373C2212yescomplemented
23#20238587C2212yescomplemented
24#20238587C2212yescomplemented
25#20238587C2212yescomplemented
26#20238587C2212yescomplemented
27C2×D4×C373C2212yescomplemented
28C2×D4×C373C2211yescharacteristic, complemented
29C22×C4×C373C2211yescharacteristic, complemented
30C4⋊C4×C373C2211yescharacteristic, complemented
31C22⋊C4×C373C2212yescomplemented
32C22×D373—24no—
33C22×D373—41no—
34C22×D373—41no—
35C22×D373—42no—
36C22×D373—42no—
37C22×D373D414yescomplemented
38C22×D373C2312yes—
39C22×D373C2312yes—
40D1492—24no—
41D1492—41no—
42C22⋊D373—24no—
43C22⋊D373—24no—
44C22⋊D373—42no—
45C2×Dic373—41no—
46C2×Dic373C2312yes—
47C2×Dic373C2312yes—
48C4×D373—24no—
49C23×C373C2312yes—
50C23×C373C2311yescharacteristic
51D4×C373—24no—
52D4×C373—41no—
53C2×C4×C373—41no—
54C2×C4×C373C2312yes—
55C2×C4×C373C2311yescharacteristic
56C2×C4×C373C2311yescharacteristic
57C22×C373—21no—
58C22×C373—21no—
59C22×C373—22no—
60C22×C373—22no—
61C22×C373—22no—
62C22×C373C2×D412yes—
63C22×C373C2411yescharacteristic
64C2×D373—22no—
65C2×D373—22no—
66C2×D373—22no—
67C2×D373—22no—
68C2×D373—22no—
69C2×D373—44no—
70C2×D373—48no—
71C2×D373C2×D414yes—
72C4×C373—22no—
73C4×C373—21no—
74C4×C373—21no—
75Dic373—22no—
76Dic373—22no—
77C2×C373—22no—
78C2×C373—42no—
79C2×C373C22×D411yescharacteristic
80C2×C373H(5,𝔽2)11yescharacteristic
81C2×C373H(5,𝔽2)11yescharacteristic
82D373—24no—
83D373—42no—
84C373#53411yescharacteristic, complemented, Sylow
85#534—3731nomaximal, Sylow
86C22×D4—3732no—
87#123—3732no—
88#123—3732no—
89#121—3732no—
90#121—3731no—
91#121D37311yescharacteristic, complemented
92#120—3732no—
93#120—3732no—
94C2×C22⋊C4—3731no—
95C24—3731no—
96C24—3732no—
97C2×D4—3732no—
98C2×D4—3732no—
99C2×D4—3732no—
100C2×D4—3731no—
101C2×D4—3732no—
102C2×D4—7468no—
103C2×D4C2×D37312yescomplemented
104C2×D4C2×D37311yescharacteristic, complemented
105C22×C4—3731no—
106C22×C4—3732no—
107C22×C4C2×D37311yescharacteristic, complemented
108C4⋊C4—3732no—
109C4⋊C4—3731no—
110C4⋊C4C2×D37311yescharacteristic, complemented
111C22⋊C4—3732no—
112C22⋊C4—3732no—
113C22⋊C4—3732no—
114C22⋊C4—3732no—
115C22⋊C4—3732no—
116C22⋊C4C2×D37312yescomplemented
117C23—3734no—
118C23—3732no—
119C23—3732no—
120C23—14921no—
121C23—14921no—
122C23—14922no—
123C23—14922no—
124C23—7464no—
125C23C22×D37312yes—
126C23C22×D37311yescharacteristic
127D4—24no—
128D4—41no—
129D4—14921no—
130D4—14922no—
131D4—7464no—
132D4—7464no—
133D4—7464no—
134C2×C4—41no—
135C2×C4—3732no—
136C2×C4—3732no—
137C2×C4—14921no—
138C2×C4—7464no—
139C2×C4C22×D37311yescharacteristic
140C2×C4C22×D37311yescharacteristic
141C2×C4C22×D37312yes—
142C22—21no—
143C22—21no—
144C22—22no—
145C22—22no—
146C22—22no—
147C22—3734no—
148C22—14924no—
149C22—14928no—
150C22—7462no—
151C22—7462no—
152C22—7462no—
153C22—7462no—
154C22—7462no—
155C22D4×D37312yes—
156C22C23×D37311yescharacteristic
157C4—22no—
158C4—21no—
159C4—21no—
160C4—7462no—
161C4—7462no—
162C2—22no—
163C2—42no—
164C2—14922no—
165C2—7464no—
166C2C2×D4×D37311yescharacteristic
167C2#2604583711yescharacteristic
168C2#2604583811yescharacteristic
subgroupG/H
center, FrattiniC22C23×D373
commutator, socle, cosocleC22×C373C24
Fitting#121×C373C2
hypercenter#121D373
hypocenterC373#534
LCSG ▷ C22×C373 ▷ C373
UCSC1 ◁ C22 ◁ #121

Complements

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

NH ≅ G/Ntype
1. C2×D4×D373C2semidirect
2. #26045777C2semidirect
3. #26045774C2semidirect
4. #26045766C2semidirect
5. #26045764C2semidirect
6. #26045762C2semidirect
7. #26045760C2semidirect
8. C2×#20238587C2semidirect
9. #121×C373C2semidirect
11. C23×D373C22semidirect
13. C2×D1492C22semidirect
15. C2×C22⋊D373C22semidirect
16. C2×C22⋊D373C22semidirect
18. C22×Dic373C22semidirect
19. C2×C4×D373C22semidirect
20. C4⋊Dic373C22semidirect
21. #20238589C22semidirect
22. C22⋊Dic373C22semidirect
23. #20238587C22semidirect
24. #20238587C22semidirect
25. #20238587C22semidirect
26. #20238587C22semidirect
27. C2×D4×C373C22semidirect
28. C2×D4×C373C22semidirect
29. C22×C4×C373C22semidirect
30. C4⋊C4×C373C22semidirect
31. C22⋊C4×C373C22semidirect
37. C22×D373D4semidirect
84. C373#534semidirect
91. #121D373semidirect
103. C2×D4C2×D373semidirect
104. C2×D4C2×D373semidirect
107. C22×C4C2×D373semidirect
110. C4⋊C4C2×D373semidirect
116. C22⋊C4C2×D373semidirect

Maximal subgroups

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

subgroupindex
C2×D4×D3732
#260457772
#260457742
#260457662
#260457642
#260457622
#260457602
C2×#202385872
#121×C3732
#534373