Finite Group Databasebeta

← #37559988

Subgroups — #37559988

Order 23872.

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

normal subgroups only

This group has 334 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×#20238610C211yesmaximal, characteristic, complemented
2C2×D4×D373C211yesmaximal, characteristic, complemented
3#26045788C212yesmaximal, complemented
4#26045778C212yesmaximal, complemented
5#26045767C211yesmaximal, characteristic, complemented
6#26045766C211yesmaximal, characteristic, complemented
7#26045763C211yesmaximal, characteristic, complemented
8#26045762C212yesmaximal, complemented
9#26045760C212yesmaximal, complemented
10#26045750C211yesmaximal, characteristic, complemented
11#121×C373C211yesmaximal, characteristic, complemented
12C23×D373C2212yes—
13#20238610—24no—
14C2×Dic746C2211yescharacteristic, complemented
15C2×D1492C2211yescharacteristic
16C2×D1492C2211yescharacteristic
17C2×D1492C2212yescomplemented
18C2×C22⋊D373C2211yescharacteristic
19C2×C22⋊D373C2212yes—
20C2×C22⋊D373C2211yescharacteristic, complemented
21C2×C22⋊D373C2212yescomplemented
22C2×C22⋊D373C2212yescomplemented
23D4×D373—24no—
24C2×C4×D373C2211yescharacteristic
25C2×C4×D373C2211yescharacteristic, complemented
26C4⋊Dic373C2211yescharacteristic, complemented
27C22⋊Dic373C2212yescomplemented
28#20238587C2212yes—
29#20238587C2212yescomplemented
30#20238587C2212yescomplemented
31C4×Dic373C2212yescomplemented
32C2×D4×C373C2211yescharacteristic
33C2×D4×C373C2212yescomplemented
34C22×C4×C373C2211yescharacteristic, complemented
35C4⋊C4×C373C2211yescharacteristic, complemented
36C22⋊C4×C373C2212yescomplemented
37C22×D373—24no—
38C22×D373—42no—
39C22×D373—42no—
40C22×D373C2311yescharacteristic
41C22×D373C2312yes—
42C22×D373C2311yescharacteristic
43Dic746—22no—
44D1492—22no—
45D1492—22no—
46D1492—42no—
47C22⋊D373—22no—
48C22⋊D373—24no—
49C22⋊D373—22no—
50C22⋊D373—42no—
51C22⋊D373—42no—
52C2×Dic373C2311yescharacteristic
53C2×Dic373C2312yes—
54C2×Dic373C2311yescharacteristic
55C4×D373—22no—
56C4×D373D414yescomplemented
57C23×C373C2311yescharacteristic
58C23×C373C2312yes—
59D4×C373—22no—
60D4×C373—42no—
61C2×C4×C373—22no—
62C2×C4×C373C2311yescharacteristic
63C2×C4×C373C2312yes—
64C2×C4×C373C2311yescharacteristic
65C22×C373—21no—
66C22×C373—22no—
67C22×C373—21no—
68C22×C373—22no—
69C22×C373—21no—
70C22×C373—22no—
71C22×C373C2411yescharacteristic
72C2×D373—21no—
73C2×D373—21no—
74C2×D373—21no—
75C2×D373—22no—
76C2×D373—21no—
77C2×D373—22no—
78C2×D373—21no—
79C2×D373—22no—
80C2×D373—48no—
81C2×D373C2×D412yes—
82C4×C373—21no—
83C4×C373—22no—
84C4×C373C2×D412yes—
85Dic373—21no—
86Dic373—22no—
87Dic373C2×D412yes—
88C2×C373—42no—
89C2×C373—41no—
90C2×C373C22×D411yescharacteristic
91C2×C373H(5,𝔽2)11yescharacteristic
92C2×C373H(5,𝔽2)11yescharacteristic
93D373—22no—
94D373—41no—
95D373—42no—
96C373#53511yescharacteristic, complemented, Sylow
97#535—3731nomaximal, Sylow
98C2×#41—3731no—
99C22×D4—3731no—
100#127—3732no—
101#124—3732no—
102#121—3731no—
103#121—3731no—
104#121—3731no—
105#121D37311yescharacteristic, complemented
106#120—3732no—
107#120—3732no—
108#117—3731no—
109C24—3732no—
110#41—7464no—
111C2×Q8—3731no—
112C2×D4—3732no—
113C2×D4—3732no—
114C2×D4—3731no—
115C2×D4—3732no—
116C2×D4—3731no—
117C2×D4—3732no—
118C2×D4—3731no—
119C2×D4—3731no—
120C2×D4—7464no—
121C2×D4C2×D37311yescharacteristic
122C2×D4C2×D37312yescomplemented
123C22×C4—3731no—
124C22×C4—3731no—
125C22×C4C2×D37311yescharacteristic, complemented
126C4⋊C4—3731no—
127C4⋊C4C2×D37311yescharacteristic, complemented
128C22⋊C4—3732no—
129C22⋊C4—3732no—
130C22⋊C4—3732no—
131C22⋊C4—3732no—
132C22⋊C4C2×D37312yescomplemented
133C42—3732no—
134C23—3731no—
135C23—3732no—
136C23—3731no—
137C23—14922no—
138C23—14922no—
139C23—7464no—
140C23C22×D37312yes—
141C23C22×D37311yescharacteristic
142Q8—7462no—
143D4—22no—
144D4—42no—
145D4—14922no—
146D4—14922no—
147D4—14922no—
148D4—7462no—
149D4—7464no—
150D4—7462no—
151D4—7462no—
152D4—7462no—
153C2×C4—22no—
154C2×C4—3731no—
155C2×C4—3734no—
156C2×C4—3732no—
157C2×C4—3731no—
158C2×C4—7462no—
159C2×C4C22×D37312yes—
160C2×C4C22×D37311yescharacteristic
161C2×C4C22×D37311yescharacteristic
162C22—21no—
163C22—22no—
164C22—21no—
165C22—22no—
166C22—21no—
167C22—22no—
168C22—3732no—
169C22—14928no—
170C22—7461no—
171C22—7462no—
172C22—7461no—
173C22—7462no—
174C22—7461no—
175C22—7462no—
176C22—7461no—
177C22—7461no—
178C22C23×D37311yescharacteristic
179C4—21no—
180C4—22no—
181C4—3732no—
182C4—7461no—
183C4—7462no—
184C4D4×D37312yes—
185C2—42no—
186C2—41no—
187C2—14922no—
188C2—14921no—
189C2—7462no—
190C2C2×D4×D37311yescharacteristic
191C2#2604583711yescharacteristic
192C2#2604583811yescharacteristic
subgroupG/H
center, FrattiniC22C23×D373
commutator, socle, cosocleC22×C373C24
Fitting#121×C373C2
hypercenter#121D373
hypocenterC373#535
LCSG ▷ C22×C373 ▷ C373
UCSC1 ◁ C22 ◁ #121

Complements

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

NH ≅ G/Ntype
1. C2×#20238610C2semidirect
2. C2×D4×D373C2semidirect
3. #26045788C2semidirect
4. #26045778C2semidirect
5. #26045767C2semidirect
6. #26045766C2semidirect
7. #26045763C2semidirect
8. #26045762C2semidirect
9. #26045760C2semidirect
10. #26045750C2semidirect
11. #121×C373C2semidirect
14. C2×Dic746C22semidirect
17. C2×D1492C22semidirect
20. C2×C22⋊D373C22semidirect
21. C2×C22⋊D373C22semidirect
22. C2×C22⋊D373C22semidirect
25. C2×C4×D373C22semidirect
26. C4⋊Dic373C22semidirect
27. C22⋊Dic373C22semidirect
29. #20238587C22semidirect
30. #20238587C22semidirect
31. C4×Dic373C22semidirect
33. C2×D4×C373C22semidirect
34. C22×C4×C373C22semidirect
35. C4⋊C4×C373C22semidirect
36. C22⋊C4×C373C22semidirect
56. C4×D373D4semidirect
96. C373#535semidirect
105. #121D373semidirect
122. C2×D4C2×D373semidirect
125. C22×C4C2×D373semidirect
127. C4⋊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
C2×#202386102
C2×D4×D3732
#260457882
#260457782
#260457672
#260457662
#260457632
#260457622
#260457602
#260457502
#121×C3732
#535373