Finite Group Databasebeta

← #37541868

Subgroups — #37541868

Order 23168.

408 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 408 conjugacy classes of subgroups — too many to draw. Only its 117 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
1#26041208C212yesmaximal, complemented
2#26041206C212yesmaximal, complemented
3#26041201C212yesmaximal, complemented
4#26041199C211yesmaximal, characteristic, complemented
5#26041196C212yesmaximal
6#26041192C212yesmaximal, complemented
7#26041180C211yesmaximal, characteristic
8#26041165C212yesmaximal, complemented
9#398×C181C211yesmaximal, characteristic, complemented
10C22×Dic362C2211yescharacteristic
11C22×D724C2211yescharacteristic
12C2×C4⋊Dic181C2212yes—
13C2×#15668428C2212yes—
14C2×#15668426C2212yes—
15C2×#15668426C2212yes—
16C2×#15668426C2212yes—
17C2×#15668426C2212yes—
18C2×#15668426C2212yes—
19C2×#15668426C2212yes—
20C2×C4×Dic181C2211yescharacteristic
21#19153723C2212yes—
22#19153723C2212yes—
23#19153723C2212yes—
24#19153723C2212yes—
25#19153723C2211yescharacteristic
26C2×C4⋊C4×C181C2212yes—
27C2×C4⋊C4×C181C2211yescharacteristic
28C2×C4⋊C4×C181C2211yescharacteristic
29#95×C181C2212yes—
30#95×C181C2211yescharacteristic
31C23×D181C2312yes—
32C2×Dic362—41no—
33C2×Dic362—42no—
34C2×D724—22no—
35C2×D724—41no—
36C2×D724—41no—
37C22×Dic181C2312yes—
38C22×Dic181C2312yes—
39C22×Dic181C2312yes—
40C4⋊Dic181—42no—
41#15668428—42no—
42#15668426—42no—
43#15668426—42no—
44#15668426—42no—
45#15668426—42no—
46#15668426—42no—
47#15668426—42no—
48C4×Dic181—41no—
49C22×C4×C181C2312yes—
50C22×C4×C181C2311yescharacteristic
51C22×C4×C181C2312yes—
52C22×C4×C181C2311yescharacteristic
53C22×C4×C181C2311yescharacteristic
54C4⋊C4×C181—24no—
55C4⋊C4×C181—41no—
56C4⋊C4×C181—41no—
57C22×D181—22no—
58C22×D181—22no—
59C22×D181—22no—
60C22×D181—22no—
61C22×D181—22no—
62C22×D181—22no—
63C22×D181—22no—
64Dic362—44no—
65D724—44no—
66C2×Dic181—22no—
67C2×Dic181—22no—
68C2×Dic181—22no—
69C2×Dic181—22no—
70C2×Dic181—22no—
71C2×Dic181—22no—
72C2×Dic181—22no—
73C2×Dic181—22no—
74C2×Dic181—22no—
75C23×C181C2411yescharacteristic
76C2×C4×C181—21no—
77C2×C4×C181—21no—
78C2×C4×C181—22no—
79C2×C4×C181—22no—
80C2×C4×C181—21no—
81C2×C4×C181—22no—
82C2×C4×C181—21no—
83C2×C4×C181—22no—
84C2×C4×C181—22no—
85C2×C4×C181—22no—
86C2×C4×C181—22no—
87C2×C4×C181—22no—
88C2×C4×C181#4112yes—
89C22×C181C2×#4111yescharacteristic
90C22×C181H(5,𝔽2)11yescharacteristic
91C22×C181H(5,𝔽2)11yescharacteristic
92C22×C181H(5,𝔽2)12yes—
93C22×C1812−1+412yes—
94C2×D181—42no—
95C2×D181—42no—
96C2×D181—42no—
97C2×D181—42no—
98C2×D181—42no—
99C2×D181—42no—
100C2×D181—42no—
101C4×C181—22no—
102C4×C181—42no—
103C4×C181—42no—
104C4×C181—41no—
105C4×C181—41no—
106Dic181—42no—
107Dic181—42no—
108Dic181—42no—
109C2×C181#53811yescharacteristic
110C2×C181#54212yes—
111C2×C181#56011yescharacteristic
112C2×C181#56212yes—
113C2×C181#56311yescharacteristic
114D181—82no—
115C181#283011yescharacteristic, complemented, Sylow
116#2830—1811nomaximal, Sylow
117#398D18111yescharacteristic, complemented
118#397—1812no—
119#397—1812no—
120#396—1812no—
121#395—1811no—
122#394—1812no—
123#393—1812no—
124#390—1811no—
125#387—1812no—
126C22×Q8—1811no—
127C22×D4—1811no—
128C2×C4⋊C4—1812no—
129C2×C4⋊C4—1812no—
130C2×C4⋊C4C2×D18112yes—
131C2×C4⋊C4C2×D18111yescharacteristic
132C2×C4⋊C4C2×D18111yescharacteristic
133C2×C22⋊C4—1812no—
134C2×C22⋊C4—1812no—
135C2×C22⋊C4—1812no—
136C2×C22⋊C4—1812no—
137C2×C22⋊C4—1812no—
138C2×C22⋊C4—1812no—
139C2×C42—1811no—
140#95—1812no—
141#95—1812no—
142#95—1812no—
143#95—1812no—
144#95—1811no—
145#95C2×D18112yes—
146#95C2×D18111yescharacteristic
147C24—1812no—
148C2×Q8—7241no—
149C2×Q8—7242no—
150C2×D4—3622no—
151C2×D4—7241no—
152C2×D4—7241no—
153C22×C4—1812no—
154C22×C4—1812no—
155C22×C4—1812no—
156C22×C4C22×D18112yes—
157C22×C4C22×D18111yescharacteristic
158C22×C4C22×D18112yes—
159C22×C4C22×D18111yescharacteristic
160C22×C4C22×D18111yescharacteristic
161C4⋊C4—24no—
162C4⋊C4—41no—
163C4⋊C4—41no—
164C4⋊C4—7242no—
165C4⋊C4—7242no—
166C22⋊C4—7242no—
167C22⋊C4—7242no—
168C22⋊C4—7242no—
169C22⋊C4—7242no—
170C22⋊C4—7242no—
171C22⋊C4—7242no—
172C42—7241no—
173C23—3622no—
174C23—3622no—
175C23—3622no—
176C23—3622no—
177C23—3622no—
178C23—3622no—
179C23—3622no—
180C23C23×D18111yescharacteristic
181Q8—7244no—
182D4—7244no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—3622no—
196C2×C4—3622no—
197C2×C4—3622no—
198C2×C4—3622no—
199C2×C4—3622no—
200C2×C4—3622no—
201C2×C4—3622no—
202C2×C4—3622no—
203C2×C4—3622no—
204C2×C4#1566844812yes—
205C22—7242no—
206C22—7242no—
207C22—7242no—
208C22—7242no—
209C22—7242no—
210C22—7242no—
211C22—7242no—
212C22C2×#1566844811yescharacteristic
213C22#1915386111yescharacteristic
214C22#1915386211yescharacteristic
215C22#1915386212yes—
216C22#1915386412yes—
217C4—22no—
218C4—42no—
219C4—42no—
220C4—41no—
221C4—41no—
222C4—7242no—
223C4—7242no—
224C4—7242no—
225C2—14482no—
226C2#2604186611yescharacteristic
227C2#2604190212yes—
228C2#2604201911yescharacteristic
229C2#2604202712yes—
230C2#2604202811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D181
commutator, socle, cosocleC23×C181C24
Fitting#398×C181C2
prime radicalC22×D724C22
hypercenter#398D181
hypocenterC181#2830
LCSG ▷ C23×C181 ▷ C181
UCSC1 ◁ C23 ◁ #398

Complements

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

NH ≅ G/Ntype
1. #26041208C2semidirect
2. #26041206C2semidirect
3. #26041201C2semidirect
4. #26041199C2semidirect
6. #26041192C2semidirect
8. #26041165C2semidirect
9. #398×C181C2semidirect
115. C181#2830semidirect
117. #398D181semidirect

Maximal subgroups

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

subgroupindex
#260412082
#260412062
#260412012
#260411992
#260411962
#260411922
#260411802
#260411652
#398×C1812
#2830181