Finite Group Databasebeta

← #37559987

Subgroups — #37559987

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×#20238609C211yesmaximal, characteristic, complemented
2C22×C22⋊D373C211yesmaximal, characteristic, complemented
3#26045788C212yesmaximal, complemented
4#26045778C212yesmaximal, complemented
5#26045765C212yesmaximal, complemented
6#26045764C212yesmaximal, complemented
7#26045762C211yesmaximal, characteristic, complemented
8#26045761C211yesmaximal, characteristic, complemented
9#26045760C211yesmaximal, characteristic, complemented
10#26045748C211yesmaximal, characteristic, complemented
11#120×C373C211yesmaximal, characteristic, complemented
12C23×D373C2211yescharacteristic
13#20238609—24no—
14C2×Dic746C2211yescharacteristic, complemented
15C2×D1492C2212yescomplemented
16C2×C22⋊D373—24no—
17C2×C22⋊D373C2211yescharacteristic
18C2×C22⋊D373C2212yes—
19C2×C22⋊D373C2212yes—
20C2×C22⋊D373C2211yescharacteristic, complemented
21C2×C22⋊D373C2212yescomplemented
22C2×C22⋊D373C2212yescomplemented
23C22×Dic373C2211yescharacteristic
24C22×Dic373C2211yescharacteristic, complemented
25C2×C4×D373C2211yescharacteristic, complemented
26#20238589C2212yescomplemented
27C22⋊Dic373C2211yescharacteristic
28C22⋊Dic373C2212yescomplemented
29#20238587C2212yescomplemented
30#20238587C2212yescomplemented
31C4×Dic373C2212yescomplemented
32C24×C373C2211yescharacteristic
33C2×D4×C373C2211yescharacteristic
34C2×D4×C373C2212yescomplemented
35C22⋊C4×C373C2211yescharacteristic
36C22⋊C4×C373C2212yescomplemented
37C22×D373—22no—
38C22×D373—42no—
39C22×D373C2311yescharacteristic
40C22×D373C2312yes—
41Dic746—22no—
42D1492—42no—
43C22⋊D373—22no—
44C22⋊D373—24no—
45C22⋊D373—24no—
46C22⋊D373—22no—
47C22⋊D373—42no—
48C22⋊D373—42no—
49C2×Dic373—22no—
50C2×Dic373D414yescomplemented
51C2×Dic373C2311yescharacteristic
52C2×Dic373C2312yes—
53C2×Dic373C2311yescharacteristic
54C2×Dic373C2311yescharacteristic
55C4×D373—22no—
56C23×C373—22no—
57C23×C373—42no—
58C23×C373C2311yescharacteristic
59C23×C373C2312yes—
60C23×C373C2311yescharacteristic
61D4×C373—22no—
62D4×C373—42no—
63C2×C4×C373C2311yescharacteristic
64C2×C4×C373C2312yes—
65C22×C373—22no—
66C22×C373—21no—
67C22×C373—22no—
68C22×C373—22no—
69C22×C373—22no—
70C22×C373—22no—
71C22×C373—44no—
72C22×C373C2×D412yes—
73C22×C373C2411yescharacteristic
74C2×D373—21no—
75C2×D373—22no—
76C2×D373—22no—
77C2×D373—22no—
78C2×D373—22no—
79C2×D373—44no—
80C4×C373—21no—
81C4×C373—22no—
82Dic373—21no—
83Dic373—22no—
84Dic373C2×D412yes—
85Dic373C2×D412yes—
86C2×C373—22no—
87C2×C373—41no—
88C2×C373—42no—
89C2×C373C22×D411yescharacteristic
90C2×C373H(5,𝔽2)12yes—
91D373—41no—
92D373—42no—
93C373#53511yescharacteristic, complemented, Sylow
94#535—3731nomaximal, Sylow
95C2×#41—3731no—
96C22×D4—3731no—
97#127—3732no—
98#124—3732no—
99#121—3732no—
100#121—3732no—
101#120—3731no—
102#120—3731no—
103#120—3731no—
104#120D37311yescharacteristic, complemented
105#117—3731no—
106C24—3731no—
107C24C2×D37311yescharacteristic
108#41—7464no—
109C2×Q8—3731no—
110C2×D4—3732no—
111C2×D4—3732no—
112C2×D4—3731no—
113C2×D4—3732no—
114C2×D4—3732no—
115C2×D4—3731no—
116C2×D4—3732no—
117C2×D4—7464no—
118C2×D4C2×D37311yescharacteristic
119C2×D4C2×D37312yescomplemented
120C22×C4—3731no—
121C22×C4—3731no—
122C22×C4—3731no—
123C4⋊C4—3732no—
124C22⋊C4—3731no—
125C22⋊C4—3732no—
126C22⋊C4—3732no—
127C22⋊C4—3732no—
128C22⋊C4C2×D37311yescharacteristic
129C22⋊C4C2×D37312yescomplemented
130C42—3732no—
131C23—22no—
132C23—42no—
133C23—3732no—
134C23—3731no—
135C23—14922no—
136C23—7462no—
137C23C22×D37312yes—
138C23C22×D37311yescharacteristic
139C23C22×D37311yescharacteristic
140Q8—7462no—
141D4—22no—
142D4—42no—
143D4—14922no—
144D4—14922no—
145D4—14922no—
146D4—7464no—
147D4—7464no—
148D4—7462no—
149D4—7462no—
150C2×C4—3734no—
151C2×C4—3731no—
152C2×C4—3731no—
153C2×C4—3732no—
154C2×C4—3731no—
155C2×C4—7462no—
156C2×C4—7462no—
157C2×C4C22×D37312yes—
158C2×C4C22×D37311yescharacteristic
159C22—22no—
160C22—21no—
161C22—22no—
162C22—22no—
163C22—22no—
164C22—22no—
165C22—44no—
166C22—14924no—
167C22—7461no—
168C22—7462no—
169C22—7462no—
170C22—7462no—
171C22—7462no—
172C22D4×D37312yes—
173C22C23×D37311yescharacteristic
174C4—21no—
175C4—22no—
176C4—3732no—
177C4—3732no—
178C4—7461no—
179C4—7462no—
180C2—22no—
181C2—42no—
182C2—41no—
183C2—14922no—
184C2—14921no—
185C2C2×D4×D37311yescharacteristic
186C2#2604583712yes—
subgroupG/H
center, FrattiniC22C23×D373
commutator, socle, cosocleC22×C373C24
Fitting#120×C373C2
hypercenter#120D373
hypocenterC373#535
LCSG ▷ C22×C373 ▷ C373
UCSC1 ◁ C22 ◁ #120

Complements

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

NH ≅ G/Ntype
1. C2×#20238609C2semidirect
2. C22×C22⋊D373C2semidirect
3. #26045788C2semidirect
4. #26045778C2semidirect
5. #26045765C2semidirect
6. #26045764C2semidirect
7. #26045762C2semidirect
8. #26045761C2semidirect
9. #26045760C2semidirect
10. #26045748C2semidirect
11. #120×C373C2semidirect
14. C2×Dic746C22semidirect
15. C2×D1492C22semidirect
20. C2×C22⋊D373C22semidirect
21. C2×C22⋊D373C22semidirect
22. C2×C22⋊D373C22semidirect
24. C22×Dic373C22semidirect
25. C2×C4×D373C22semidirect
26. #20238589C22semidirect
28. C22⋊Dic373C22semidirect
29. #20238587C22semidirect
30. #20238587C22semidirect
31. C4×Dic373C22semidirect
34. C2×D4×C373C22semidirect
36. C22⋊C4×C373C22semidirect
50. C2×Dic373D4semidirect
93. C373#535semidirect
104. #120D373semidirect
119. C2×D4C2×D373semidirect
129. C22⋊C4C2×D373semidirect

Maximal subgroups

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

subgroupindex
C2×#202386092
C22×C22⋊D3732
#260457882
#260457782
#260457652
#260457642
#260457622
#260457612
#260457602
#260457482
#120×C3732
#535373