Finite Group Databasebeta

← #37549972

Subgroups — #37549972

Order 23168.

264 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 264 conjugacy classes of subgroups — too many to draw. Only its 54 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#26042265C211yesmaximal, characteristic, complemented
2#26042258C211yesmaximal, characteristic, complemented
3#26042181C212yesmaximal, complemented
4#26042179C212yesmaximal, complemented
5#26041790C211yesmaximal, characteristic
6D181⋊M16—21no—
7#19153895—21no—
8#19153877—21no—
9#19153876C2212yes—
10#19153875—21no—
11#19153874C2212yes—
12C2×#15668449C2211yescharacteristic
13#19153799C411yescharacteristic, complemented
14#19153799C2211yescharacteristic
15#19153795—21no—
16#19153790—21no—
17#19153785C411yescharacteristic, complemented
18#19153785C2211yescharacteristic
19C4×C22⋊D181—21no—
20C4×C22⋊D181—21no—
21#19153775C411yescharacteristic, complemented
22C23×C4×C181C411yescharacteristic, complemented
23#15668459—41no—
24#15668458—22no—
25D181⋊C8—41no—
26C4⋊#10624382—41no—
27C22⋊#10624382—22no—
28C4×#10624382—41no—
29#15668449—41no—
30#15668449—41no—
31C2×Dic362D411yescharacteristic
32C2×D724D411yescharacteristic
33C2×C22⋊D181—21no—
34C2×C22⋊D181—21no—
35C2×C22⋊D181C2×C411yescharacteristic
36C2×C22⋊D181C2311yescharacteristic
37C2×C4×D181D411yescharacteristic
38C2×C4×D181D411yescharacteristic
39C4⋊Dic181—21no—
40#15668428—21no—
41#15668428—21no—
42C22⋊Dic181—21no—
43C22⋊Dic181—21no—
44C22⋊Dic181C2×C411yescharacteristic
45C22⋊Dic181C2×C411yescharacteristic
46#15668426—21no—
47#15668426—21no—
48C4×Dic181—21no—
49C24×C181C2×C411yescharacteristic
50C22×C4×C181—21no—
51C22×C4×C181—21no—
52C22×C4×C181—41no—
53C22×C4×C181—41no—
54C22×C4×C181C2×C411yescharacteristic
55C22×C4×C181C2×C411yescharacteristic
56C2×#10624382—42no—
57#13089685—42no—
58C22×D181C22⋊C411yescharacteristic
59C22×D181C2×D411yescharacteristic
60Dic362—41no—
61D724—41no—
62C22⋊D181—41no—
63C22⋊D181—41no—
64C22⋊D181—42no—
65C22⋊D181—42no—
66C2×Dic181—21no—
67C2×Dic181—21no—
68C2×Dic181C22⋊C411yescharacteristic
69C2×Dic181C2×D411yescharacteristic
70C4×D181—21no—
71C4×D181—21no—
72C4×D181—21no—
73C4×D181—21no—
74C23×C181—21no—
75C23×C181—41no—
76C23×C181—42no—
77C23×C181C22×C411yescharacteristic
78C2×C4×C181—21no—
79C2×C4×C181—21no—
80C2×C4×C181—21no—
81C2×C4×C181—42no—
82C2×C4×C181—41no—
83C2×C4×C181—41no—
84C2×C4×C181—41no—
85C2×C4×C181C22⋊C411yescharacteristic
86C2×C4×C181C22⋊C411yescharacteristic
87C22×C181—22no—
88C22×C181—21no—
89C22×C181—41no—
90C22×C181—41no—
91C22×C181—42no—
92C22×C181—42no—
93C22×C181—41no—
94C22×C181C2×C22⋊C411yescharacteristic
95C2×D181—21no—
96C2×D181—21no—
97C2×D181—41no—
98C2×D181—41no—
99#10624382—82no—
100C4×C181—21no—
101C4×C181—41no—
102C4×C181#9911yescharacteristic
103C4×C181#9911yescharacteristic
104Dic181—21no—
105Dic181—21no—
106Dic181—41no—
107Dic181—41no—
108C2×C181—21no—
109C2×C181—42no—
110C2×C181—41no—
111C2×C181C2×#9911yescharacteristic
112D181—41no—
113D181—41no—
114C181#212011yescharacteristic, complemented, Sylow
115#2120—1811nomaximal, Sylow
116#525—1811no—
117#413—1811no—
118#410—1811no—
119#352—1812no—
120AΣL(1,𝔽16)—1812no—
121C2×#41—1811no—
122C23×C4#1062438211yescharacteristic, complemented
123C2×M16—3621no—
124#123—1811no—
125#123—1811no—
126C22⋊Q8—3621no—
127#121—3621no—
128#120—1811no—
129#120—1811no—
130C4×D4—3621no—
131C4×D4—3621no—
132#117—3621no—
133#117—1811no—
134#101—3621no—
135#100—1812no—
136#99—3621no—
137#99—1812no—
138C24C2×#1062438211yescharacteristic
139#41—7241no—
140#41—7241no—
141C2×Q8—1811no—
142C2×D4—3621no—
143C2×D4—3621no—
144C2×D4—1811no—
145C2×D4—1811no—
146C2×D4—1811no—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—41no—
150C22×C4—41no—
151C22×C4—1811no—
152C22×C4—1811no—
153C22×C4C2×#1062438211yescharacteristic
154C22×C4C2×#1062438211yescharacteristic
155M16—3622no—
156M16—7241no—
157C2×C8—7241no—
158C4⋊C4—3621no—
159C4⋊C4—3621no—
160C4⋊C4—3621no—
161C4⋊C4—7241no—
162C22⋊C4—3621no—
163C22⋊C4—3621no—
164C22⋊C4—3621no—
165C22⋊C4—3621no—
166C22⋊C4—3622no—
167C22⋊C4—1811no—
168C22⋊C4—1811no—
169C42—3621no—
170C42—7241no—
171C23—21no—
172C23—41no—
173C23—42no—
174C23—1811no—
175C23—1811no—
176C23C22×#1062438211yescharacteristic
177Q8—7241no—
178D4—7241no—
179D4—7241no—
180D4—7241no—
181D4—7242no—
182D4—7242no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—42no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—41no—
190C2×C4—3621no—
191C2×C4—3621no—
192C2×C4—3621no—
193C2×C4—3621no—
194C2×C4—3621no—
195C2×C4—3621no—
196C2×C4—1811no—
197C2×C4—1811no—
198C2×C4—7242no—
199C2×C4C22⋊#1062438211yescharacteristic
200C2×C4C22⋊#1062438211yescharacteristic
201C8—7242no—
202C22—22no—
203C22—21no—
204C22—41no—
205C22—41no—
206C22—42no—
207C22—42no—
208C22—41no—
209C22—3621no—
210C22—3621no—
211C22—7241no—
212C22—7241no—
213C22C2×C22⋊#1062438211yescharacteristic
214C4—21no—
215C4—41no—
216C4—3621no—
217C4—3621no—
218C4—14482no—
219C4—7241no—
220C4—7241no—
221C4#1915387411yescharacteristic
222C4#1915387411yescharacteristic
223C2—21no—
224C2—42no—
225C2—41no—
226C2—7241no—
227C2—7241no—
228C2C2×#1915387411yescharacteristic
subgroupG/H
centerC4#19153874
commutatorC23×C181C22×C4
FrattiniC23C22×#10624382
FittingC23×C4×C181C4
socleC2×C181C2×#99
prime radical#26041790C2
cosocleC2×C22⋊D181C23
hypercenterC23×C4#10624382
hypocenterC181#2120
LCSG ▷ C23×C181 ▷ C22×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C4 ◁ C2×C4 ◁ C22×C4 ◁ C23×C4

Complements

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

NH ≅ G/Ntype
1. #26042265C2semidirect
2. #26042258C2semidirect
3. #26042181C2semidirect
4. #26042179C2semidirect
13. #19153799C4semidirect
17. #19153785C4semidirect
21. #19153775C4semidirect
22. C23×C4×C181C4semidirect
114. C181#2120semidirect
122. C23×C4#10624382semidirect

Maximal subgroups

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

subgroupindex
#260422652
#260422582
#260421812
#260421792
#260417902
#2120181