Finite Group Databasebeta

← #37549975

Subgroups — #37549975

Order 23168.

264 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 264 conjugacy classes of subgroups — too many to draw. Only its 52 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#26042266C211yesmaximal, characteristic, complemented
2#26042259C211yesmaximal, characteristic, complemented
3#26042181C212yesmaximal, complemented
4#26042180C212yesmaximal, complemented
5#26041787C211yesmaximal, characteristic
6C2×#15668458—21no—
7#19153895—21no—
8#19153877—21no—
9#19153876C2212yes—
10#19153874—21no—
11#19153874C2212yes—
12C2×#15668448C2211yescharacteristic
13C22×C4×D181C411yescharacteristic, complemented
14#19153799C2211yescharacteristic
15#19153794—21no—
16#19153789—21no—
17#19153786C2211yescharacteristic
18#19153784C411yescharacteristic, complemented
19#19153778—21no—
20#19153776—21no—
21#19153772C411yescharacteristic, complemented
22#123×C181C411yescharacteristic, complemented
23#15668458—22no—
24#15668458—41no—
25C2×#13089685—41no—
26C4⋊#10624382—41no—
27C22⋊#10624382—22no—
28C4×#10624382—41no—
29C23×D181C2×C411yescharacteristic
30#15668448—41no—
31#15668448—41no—
32C2×Dic362D411yescharacteristic
33C2×D724—21no—
34C2×C22⋊D181—21no—
35C2×C22⋊D181D411yescharacteristic
36C2×C22⋊D181C2311yescharacteristic
37C22×Dic181C2×C411yescharacteristic
38C22×Dic181D411yescharacteristic
39C2×C4×D181—21no—
40C2×C4×D181—21no—
41C2×C4×D181—41no—
42C2×C4×D181—41no—
43C2×C4×D181D411yescharacteristic
44#15668428—21no—
45#15668428—21no—
46C22⋊Dic181C2×C411yescharacteristic
47#15668426—21no—
48#15668426—21no—
49#15668426—21no—
50C4×Dic181—21no—
51C2×D4×C181C2×C411yescharacteristic
52C22×C4×C181C2×C411yescharacteristic
53C4⋊C4×C181—21no—
54C22⋊C4×C181—21no—
55C22⋊C4×C181C2×C411yescharacteristic
56C2×#10624382—42no—
57#13089685—42no—
58C22×D181—21no—
59C22×D181—42no—
60C22×D181—41no—
61C22×D181C2×D411yescharacteristic
62Dic362—41no—
63D724—42no—
64C22⋊D181—41no—
65C22⋊D181—41no—
66C22⋊D181—42no—
67C2×Dic181—21no—
68C2×Dic181—21no—
69C2×Dic181—21no—
70C2×Dic181—21no—
71C2×Dic181—21no—
72C2×Dic181C22⋊C411yescharacteristic
73C2×Dic181C22⋊C411yescharacteristic
74C2×Dic181C2×D411yescharacteristic
75C4×D181—21no—
76C4×D181—21no—
77C4×D181—42no—
78C4×D181—41no—
79C4×D181—41no—
80C23×C181C22⋊C411yescharacteristic
81C23×C181C22×C411yescharacteristic
82D4×C181—41no—
83C2×C4×C181—21no—
84C2×C4×C181—21no—
85C2×C4×C181—41no—
86C2×C4×C181C22⋊C411yescharacteristic
87C22×C181—21no—
88C22×C181—21no—
89C22×C181—41no—
90C22×C181—41no—
91C22×C181C2×C22⋊C411yescharacteristic
92C2×D181—22no—
93C2×D181—21no—
94C2×D181—41no—
95C2×D181—42no—
96C2×D181—41no—
97C2×D181—41no—
98C2×D181—42no—
99#10624382—82no—
100C4×C181—21no—
101C4×C181—41no—
102C4×C181—41no—
103Dic181—21no—
104Dic181—21no—
105Dic181—41no—
106Dic181#9911yescharacteristic
107Dic181#9911yescharacteristic
108C2×C181—21no—
109C2×C181—41no—
110C2×C181—41no—
111C2×C181C2×#9911yescharacteristic
112D181—41no—
113D181—42no—
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—1811no—
123C2×M16—3621no—
124#123—1811no—
125#123#1062438211yescharacteristic, complemented
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—
138C24—1811no—
139#41—7241no—
140#41—7241no—
141C2×Q8—1811no—
142C2×D4—3621no—
143C2×D4—3621no—
144C2×D4—1811no—
145C2×D4—1811no—
146C2×D4C2×#1062438211yescharacteristic
147C22×C4—3621no—
148C22×C4—3621no—
149C22×C4—1811no—
150C22×C4—1811no—
151C22×C4—1811no—
152C22×C4—7241no—
153C22×C4—7241no—
154C22×C4C2×#1062438211yescharacteristic
155M16—3622no—
156M16—7241no—
157C2×C8—7241no—
158C4⋊C4—21no—
159C4⋊C4—3621no—
160C4⋊C4—3621no—
161C4⋊C4—7241no—
162C22⋊C4—21no—
163C22⋊C4—3621no—
164C22⋊C4—3621no—
165C22⋊C4—3621no—
166C22⋊C4—3622no—
167C22⋊C4—1811no—
168C22⋊C4C2×#1062438211yescharacteristic
169C42—3621no—
170C42—7241no—
171C23—3621no—
172C23—1811no—
173C23—7241no—
174C23—7242no—
175C23C22⋊#1062438211yescharacteristic
176C23C22×#1062438211yescharacteristic
177Q8—7241no—
178D4—41no—
179D4—7242no—
180D4—7241no—
181D4—7241no—
182D4—7242no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—41no—
186C2×C4—3621no—
187C2×C4—3621no—
188C2×C4—3621no—
189C2×C4—3621no—
190C2×C4—3621no—
191C2×C4—3621no—
192C2×C4—3621no—
193C2×C4—1811no—
194C2×C4—1811no—
195C2×C4—1811no—
196C2×C4—7242no—
197C2×C4—7241no—
198C2×C4—7241no—
199C2×C4—7242no—
200C2×C4C22⋊#1062438211yescharacteristic
201C8—7242no—
202C22—21no—
203C22—21no—
204C22—41no—
205C22—41no—
206C22—3622no—
207C22—3621no—
208C22—7241no—
209C22—7241no—
210C22—7242no—
211C22—7242no—
212C22—7241no—
213C22C2×C22⋊#1062438211yescharacteristic
214C4—21no—
215C4—41no—
216C4—41no—
217C4—3621no—
218C4—3621no—
219C4—14482no—
220C4—1811no—
221C4—1811no—
222C4—7241no—
223C2—21no—
224C2—41no—
225C2—41no—
226C2—7241no—
227C2—7242no—
228C2D181⋊#9911yescharacteristic
subgroupG/H
centerC2D181⋊#99
commutatorC23×C181C22×C4
FrattiniC23C22×#10624382
Fitting#123×C181C4
socleC2×C181C2×#99
prime radical#26041787C2
cosocleC2×C22⋊D181C23
hypercenter#123#10624382
hypocenterC181#2120
LCSG ▷ C23×C181 ▷ C22×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #123

Complements

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

NH ≅ G/Ntype
1. #26042266C2semidirect
2. #26042259C2semidirect
3. #26042181C2semidirect
4. #26042180C2semidirect
13. C22×C4×D181C4semidirect
18. #19153784C4semidirect
21. #19153772C4semidirect
22. #123×C181C4semidirect
114. C181#2120semidirect
125. #123#10624382semidirect

Maximal subgroups

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

subgroupindex
#260422662
#260422592
#260421812
#260421802
#260417872
#2120181