Finite Group Databasebeta

← #37559983

Subgroups — #37559983

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
1C22×D1492C211yesmaximal, characteristic, complemented
2C2×D4×D373C211yesmaximal, characteristic, complemented
3#26045777C212yesmaximal, complemented
4#26045775C211yesmaximal, characteristic, complemented
5#26045767C212yesmaximal, complemented
6#26045765C212yesmaximal, complemented
7#26045762C211yesmaximal, characteristic, complemented
8#26045762C212yesmaximal, complemented
9#26045760C211yesmaximal, characteristic, complemented
10C22⋊C4×D373C211yesmaximal, characteristic, complemented
11#123×C373C211yesmaximal, characteristic, complemented
12C23×D373C2211yescharacteristic
13C23×D373C2211yescharacteristic, complemented
14C23×D373C2211yescharacteristic, complemented
15C2×D1492—24no—
16C2×D1492C2212yes—
17C2×D1492C2212yescomplemented
18C2×D1492C2211yescharacteristic, complemented
19C2×D1492C2212yescomplemented
20C2×C22⋊D373C2211yescharacteristic
21C2×C22⋊D373C2211yescharacteristic, complemented
22C2×C22⋊D373C2212yescomplemented
23D4×D373—24no—
24C2×C4×D373C2211yescharacteristic, complemented
25C2×C4×D373C2212yescomplemented
26#20238589C2212yescomplemented
27C22⋊Dic373C2211yescharacteristic, complemented
28#20238587C2212yescomplemented
29#20238587C2212yescomplemented
30#20238587C2212yescomplemented
31#20238587C2212yescomplemented
32C2×D4×C373C2211yescharacteristic
33C22×C4×C373C2211yescharacteristic, complemented
34C4⋊C4×C373C2212yescomplemented
35C22⋊C4×C373C2211yescharacteristic, complemented
36C22⋊C4×C373C2212yescomplemented
37C22×D373—22no—
38C22×D373—22no—
39C22×D373—42no—
40C22×D373—42no—
41C22×D373—42no—
42C22×D373D414yescomplemented
43C22×D373C2311yescharacteristic
44C22×D373C2312yes—
45C22×D373C2311yescharacteristic
46C22×D373C2311yescharacteristic
47D1492—22no—
48D1492—24no—
49D1492—24no—
50D1492—42no—
51C22⋊D373—22no—
52C22⋊D373—22no—
53C22⋊D373—42no—
54C2×Dic373C2311yescharacteristic
55C2×Dic373C2312yes—
56C4×D373—22no—
57C4×D373—42no—
58C23×C373C2311yescharacteristic
59C23×C373C2311yescharacteristic
60D4×C373—22no—
61C2×C4×C373—22no—
62C2×C4×C373C2312yes—
63C2×C4×C373C2311yescharacteristic
64C2×C4×C373C2312yes—
65C22×C373—22no—
66C22×C373—21no—
67C22×C373—22no—
68C22×C373C2×D412yes—
69C22×C373C2411yescharacteristic
70C2×D373—22no—
71C2×D373—22no—
72C2×D373—21no—
73C2×D373—22no—
74C2×D373—22no—
75C2×D373—22no—
76C2×D373—22no—
77C2×D373—44no—
78C2×D373—44no—
79C2×D373—44no—
80C2×D373C2×D412yes—
81C2×D373C2×D412yes—
82C4×C373—22no—
83C4×C373—21no—
84C4×C373—22no—
85Dic373—21no—
86Dic373—22no—
87C2×C373—22no—
88C2×C373—41no—
89C2×C373C22×D411yescharacteristic
90C2×C373H(5,𝔽2)12yes—
91D373—22no—
92D373—22no—
93D373—41no—
94D373—42no—
95C373#53411yescharacteristic, complemented, Sylow
96#534—3731nomaximal, Sylow
97C22×D4—3731no—
98C22×D4—3731no—
99#123—3731no—
100#123—3732no—
101#123D37311yescharacteristic, complemented
102#121—3732no—
103#121—3732no—
104#120—3731no—
105#120—3732no—
106#120—3731no—
107C2×C22⋊C4—3731no—
108C24—3731no—
109C24—3731no—
110C24—3731no—
111C2×D4—3732no—
112C2×D4—3731no—
113C2×D4—3731no—
114C2×D4—3732no—
115C2×D4—3732no—
116C2×D4—3731no—
117C2×D4—3732no—
118C2×D4—7464no—
119C2×D4—7464no—
120C2×D4C2×D37311yescharacteristic
121C22×C4—3732no—
122C22×C4—3731no—
123C22×C4C2×D37311yescharacteristic, complemented
124C4⋊C4—3732no—
125C4⋊C4C2×D37312yescomplemented
126C22⋊C4—3731no—
127C22⋊C4—3732no—
128C22⋊C4—3732no—
129C22⋊C4—3732no—
130C22⋊C4—3732no—
131C22⋊C4C2×D37311yescharacteristic, complemented
132C22⋊C4C2×D37312yescomplemented
133C23—3734no—
134C23—3731no—
135C23—3731no—
136C23—3732no—
137C23—3731no—
138C23—14922no—
139C23—14922no—
140C23—14922no—
141C23—7462no—
142C23—7462no—
143C23C22×D37311yescharacteristic
144C23C22×D37311yescharacteristic
145D4—22no—
146D4—14922no—
147D4—14922no—
148D4—7464no—
149D4—7464no—
150D4—7462no—
151D4—7462no—
152D4—7462no—
153C2×C4—22no—
154C2×C4—3732no—
155C2×C4—3731no—
156C2×C4—14922no—
157C2×C4—7462no—
158C2×C4C22×D37312yes—
159C2×C4C22×D37311yescharacteristic
160C2×C4C22×D37312yes—
161C22—22no—
162C22—21no—
163C22—22no—
164C22—3732no—
165C22—3732no—
166C22—14924no—
167C22—14924no—
168C22—14924no—
169C22—7461no—
170C22—7462no—
171C22—7462no—
172C22—7462no—
173C22—7462no—
174C22—7462no—
175C22—7462no—
176C22D4×D37312yes—
177C22C23×D37311yescharacteristic
178C4—22no—
179C4—21no—
180C4—22no—
181C4—7461no—
182C4—7462no—
183C2—22no—
184C2—41no—
185C2—14922no—
186C2—14921no—
187C2—7462no—
188C2—7462no—
189C2C2×D4×D37311yescharacteristic
190C2#2604583812yes—
subgroupG/H
center, FrattiniC22C23×D373
commutator, socle, cosocleC22×C373C24
Fitting#123×C373C2
hypercenter#123D373
hypocenterC373#534
LCSG ▷ C22×C373 ▷ C373
UCSC1 ◁ C22 ◁ #123

Complements

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

NH ≅ G/Ntype
1. C22×D1492C2semidirect
2. C2×D4×D373C2semidirect
3. #26045777C2semidirect
4. #26045775C2semidirect
5. #26045767C2semidirect
6. #26045765C2semidirect
7. #26045762C2semidirect
8. #26045762C2semidirect
9. #26045760C2semidirect
10. C22⋊C4×D373C2semidirect
11. #123×C373C2semidirect
13. C23×D373C22semidirect
14. C23×D373C22semidirect
17. C2×D1492C22semidirect
18. C2×D1492C22semidirect
19. C2×D1492C22semidirect
21. C2×C22⋊D373C22semidirect
22. C2×C22⋊D373C22semidirect
24. C2×C4×D373C22semidirect
25. C2×C4×D373C22semidirect
26. #20238589C22semidirect
27. C22⋊Dic373C22semidirect
28. #20238587C22semidirect
29. #20238587C22semidirect
30. #20238587C22semidirect
31. #20238587C22semidirect
33. C22×C4×C373C22semidirect
34. C4⋊C4×C373C22semidirect
35. C22⋊C4×C373C22semidirect
36. C22⋊C4×C373C22semidirect
42. C22×D373D4semidirect
95. C373#534semidirect
101. #123D373semidirect
123. C22×C4C2×D373semidirect
125. C4⋊C4C2×D373semidirect
131. C22⋊C4C2×D373semidirect
132. C22⋊C4C2×D373semidirect

Maximal subgroups

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

subgroupindex
C22×D14922
C2×D4×D3732
#260457772
#260457752
#260457672
#260457652
#260457622
#260457602
C22⋊C4×D3732
#123×C3732
#534373