Finite Group Databasebeta

← #37536302

Subgroups — #37536302

Order 23168.

1104 conjugacy classes of subgroups; only the 229 normal subgroups are shown.

normal subgroups only

This group has 1104 conjugacy classes of subgroups — too many to draw. Only its 229 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
1C23×C22⋊D181C211yesmaximal, characteristic
2C2×#19153772C212yesmaximal, complemented
3#26041179C218yesmaximal, complemented
4#26041135C211yesmaximal, characteristic, complemented
5#26041134C212yesmaximal, complemented
6#379×C181C211yesmaximal, characteristic, complemented
7C24×D181C2211yescharacteristic
8C22×C22⋊D181—28no—
9C22×C22⋊D181C418yescomplemented
10C22×C22⋊D181C2214yes—
11C23×Dic181C2211yescharacteristic
12#19153772—28no—
13C2×#15668428C2214yescomplemented
14C2×C22⋊Dic181C2212yes—
15C2×C22⋊Dic181C2214yescomplemented
16C2×#15668426C2218yescomplemented
17C2×C4×Dic181C2214yescomplemented
18C25×C181C2211yescharacteristic
19C2×C22⋊C4×C181C2212yes—
20C2×C22⋊C4×C181C2214yescomplemented
21C23×D181—24no—
22C23×D181—44no—
23C23×D181C2×C414yes—
24C23×D181C2312yes—
25C2×C22⋊D181—232no—
26C2×C22⋊D181—28no—
27C2×C22⋊D181—28no—
28C2×C22⋊D181C2×C4116yes—
29C22×Dic181C2×C414yes—
30C22×Dic181D418yes—
31C22×Dic181C2314yes—
32C22×Dic181C2312yes—
33#15668428—28no—
34C22⋊Dic181—24no—
35C22⋊Dic181—44no—
36#15668426—48no—
37C4×Dic181—28no—
38C24×C181—24no—
39C24×C181—44no—
40C24×C181C2×C414yes—
41C24×C181C2312yes—
42C24×C181C2311yescharacteristic
43C22×C4×C181C2314yes—
44C22⋊C4×C181—24no—
45C22⋊C4×C181—44no—
46C22×D181—28no—
47C22×D181—24no—
48C22×D181—28no—
49C22×D181—48no—
50C22×D181—416no—
51C22×D181C22×C414yes—
52C22⋊D181—232no—
53C2×Dic181—24no—
54C2×Dic181—28no—
55C2×Dic181C22⋊C4116yescomplemented
56C2×Dic181C22×C414yes—
57C2×Dic181C2×D414yes—
58C2×Dic181C2×D414yes—
59C23×C181—24no—
60C23×C181—28no—
61C23×C181—24no—
62C23×C181—24no—
63C23×C181—24no—
64C23×C181—48no—
65C23×C181—416no—
66C23×C181C22×C414yes—
67C23×C181C22×C412yes—
68C23×C181C2×D414yes—
69C23×C181C2411yescharacteristic
70C2×C4×C181—24no—
71C2×C4×C181—28no—
72C22×C181—24no—
73C22×C181—28no—
74C22×C181—24no—
75C22×C181—24no—
76C22×C181—24no—
77C22×C181—416no—
78C22×C181—48no—
79C22×C181C2×C22⋊C414yes—
80C22×C181C23×C411yescharacteristic
81C22×C181C22×D412yes—
82C22×C181H(5,𝔽2)14yes—
83C2×D181—24no—
84C2×D181—28no—
85C2×D181—416no—
86C2×D181—48no—
87C4×C181—44no—
88Dic181—44no—
89Dic181C2×C22⋊C418yes—
90C2×C181—24no—
91C2×C181—44no—
92C2×C181C22×C22⋊C411yescharacteristic
93C2×C181#51812yes—
94C2×C181#53514yes—
95D181—44no—
96C181#230911yescharacteristic, complemented, Sylow
97#2309—1811nomaximal, Sylow
98C23×D4—1811no—
99C2×#117—1812no—
100#390—1818no—
101#379—1811no—
102#379—1812no—
103#379D18111yescharacteristic, complemented
104C25—1811no—
105C25C2×D18111yescharacteristic
106C22×D4—3628no—
107C22×D4—1818no—
108C22×D4—1814no—
109C23×C4—1811no—
110#117—3628no—
111C2×C4⋊C4—1814no—
112C2×C22⋊C4—1812no—
113C2×C22⋊C4—1814no—
114C2×C22⋊C4—1818no—
115C2×C22⋊C4C2×D18112yes—
116C2×C22⋊C4C2×D18114yescomplemented
117C2×C42—1814no—
118C24—24no—
119C24—44no—
120C24—3624no—
121C24—1812no—
122C24—1814no—
123C24—7244no—
124C24C4×D18114yes—
125C24C22×D18112yes—
126C24C22×D18111yescharacteristic
127C2×D4—36232no—
128C2×D4—3628no—
129C2×D4—3628no—
130C2×D4—18116no—
131C22×C4—1814no—
132C22×C4—1814no—
133C22×C4—1818no—
134C22×C4—1812no—
135C22×C4C22×D18114yes—
136C4⋊C4—3628no—
137C22⋊C4—24no—
138C22⋊C4—44no—
139C22⋊C4—3624no—
140C22⋊C4—7248no—
141C22⋊C4—7244no—
142C42—3628no—
143C23—24no—
144C23—28no—
145C23—24no—
146C23—24no—
147C23—24no—
148C23—48no—
149C23—416no—
150C23—3628no—
151C23—3624no—
152C23—3628no—
153C23—1814no—
154C23—7248no—
155C23—72416no—
156C23C2×C4×D18114yes—
157C23C2×C4×D18112yes—
158C23D4×D18114yes—
159C23C23×D18111yescharacteristic
160D4—36232no—
161C2×C4—24no—
162C2×C4—28no—
163C2×C4—3624no—
164C2×C4—3628no—
165C2×C4—18116no—
166C2×C4—1814no—
167C2×C4—1814no—
168C2×C4—1814no—
169C22—24no—
170C22—28no—
171C22—24no—
172C22—24no—
173C22—24no—
174C22—416no—
175C22—48no—
176C22—3624no—
177C22—3628no—
178C22—72416no—
179C22—7248no—
180C22C22⋊C4×D18114yes—
181C22C22×C4×D18111yescharacteristic
182C22C2×D4×D18112yes—
183C22#1915386114yes—
184C4—44no—
185C4—1818no—
186C4—7244no—
187C2—24no—
188C2—44no—
189C2—7244no—
190C2C2×C22⋊C4×D18111yescharacteristic
191C2#2604175212yes—
192C2#2604184414yes—
subgroupG/H
center, FrattiniC23C23×D181
commutatorC22×C181C23×C4
Fitting#379×C181C2
socle, cosocleC23×C181C24
prime radicalC23×C22⋊D181C2
hypercenter#379D181
hypocenterC181#2309
LCSG ▷ C22×C181 ▷ C181
UCSC1 ◁ C23 ◁ #379

Complements

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

NH ≅ G/Ntype
2. C2×#19153772C2semidirect
3. #26041179C2semidirect
4. #26041135C2semidirect
5. #26041134C2semidirect
6. #379×C181C2semidirect
9. C22×C22⋊D181C4semidirect
13. C2×#15668428C22semidirect
15. C2×C22⋊Dic181C22semidirect
16. C2×#15668426C22semidirect
17. C2×C4×Dic181C22semidirect
20. C2×C22⋊C4×C181C22semidirect
55. C2×Dic181C22⋊C4semidirect
96. C181#2309semidirect
103. #379D181semidirect
116. C2×C22⋊C4C2×D181semidirect

Maximal subgroups

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

subgroupindex
C23×C22⋊D1812
C2×#191537722
#260411792
#260411352
#260411342
#379×C1812
#2309181