Finite Group Databasebeta

← #37549700

Subgroups — #37549700

Order 23168.

408 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 408 conjugacy classes of subgroups — too many to draw. Only its 90 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
1C22×#15668458C211yesmaximal, characteristic, complemented
2C2×#19153873C212yesmaximal, complemented
3C2×C22⋊#13089685C211yesmaximal, characteristic, complemented
4#26042145C212yesmaximal, complemented
5C2×#19153776C211yesmaximal, characteristic
6C2×#15668458—24no—
7C2×#15668458—22no—
8C22×#13089685C2212yes—
9C22×#13089685C2212yes—
10#19153873—24no—
11C22⋊#13089685—22no—
12C22×C22⋊D181C411yescharacteristic
13C23×Dic181C2211yescharacteristic
14C22×C4×D181C2211yescharacteristic
15#19153776—22no—
16#19153776—22no—
17C2×#15668428C411yescharacteristic
18C2×#15668426C411yescharacteristic
19C2×C4×Dic181C2211yescharacteristic
20C2×C22⋊C4×C181C411yescharacteristic
21#15668458—28no—
22C2×#13089685—22no—
23C2×#13089685—22no—
24C2×#13089685—22no—
25C2×#13089685—22no—
26C2×#13089685—22no—
27C2×#13089685D414yescomplemented
28C23×D181C2×C411yescharacteristic
29C2×C22⋊D181—22no—
30C2×C22⋊D181—22no—
31C2×C22⋊D181—22no—
32C22×Dic181—22no—
33C22×Dic181—22no—
34C22×Dic181C2×C411yescharacteristic
35C22×Dic181C2×C411yescharacteristic
36C22×Dic181D412yes—
37C22×Dic181D412yes—
38C22×Dic181C2311yescharacteristic
39C2×C4×D181—21no—
40C2×C4×D181—21no—
41C2×C4×D181—22no—
42C2×C4×D181—22no—
43#15668428—22no—
44#15668426—22no—
45C4×Dic181—21no—
46C4×Dic181—21no—
47C24×C181C2×C411yescharacteristic
48C22×C4×C181C2×C411yescharacteristic
49C22×C4×C181C2×C411yescharacteristic
50C22⋊C4×C181—22no—
51#13089685—24no—
52#13089685—42no—
53C22×D181—21no—
54C22×D181—22no—
55C22×D181—21no—
56C22×D181—22no—
57C22×D181M1612yescomplemented
58C22⋊D181—28no—
59C2×Dic181—21no—
60C2×Dic181—22no—
61C2×Dic181—21no—
62C2×Dic181—22no—
63C2×Dic181—24no—
64C2×Dic181—22no—
65C2×Dic181C22⋊C412yes—
66C2×Dic181M1612yescomplemented
67C2×Dic181C2×D412yes—
68C2×Dic181C2×D411yescharacteristic
69C2×Dic181C2×D411yescharacteristic
70C2×Dic181#4112yes—
71C4×D181—41no—
72C4×D181—42no—
73C4×D181—41no—
74C23×C181—22no—
75C23×C181—21no—
76C23×C181—21no—
77C23×C181—22no—
78C23×C181C22⋊C412yes—
79C23×C181C22×C411yescharacteristic
80C2×C4×C181—21no—
81C2×C4×C181—21no—
82C2×C4×C181—22no—
83C2×C4×C181—22no—
84C22×C181—22no—
85C22×C181—22no—
86C22×C181—22no—
87C22×C181—42no—
88C22×C181—42no—
89C22×C181C2×C22⋊C411yescharacteristic
90C22×C181C4×D412yes—
91C22×C181C2×M1611yescharacteristic
92C22×C181C2×M1611yescharacteristic
93C22×C181#13112yes—
94C2×D181—22no—
95C2×D181—24no—
96C2×D181—41no—
97C2×D181—41no—
98C2×D181—42no—
99C4×C181—41no—
100C4×C181—41no—
101Dic181—22no—
102Dic181—22no—
103Dic181#12011yescharacteristic
104Dic181#12112yes—
105Dic181#12311yescharacteristic
106C2×C181—42no—
107C2×C181#38611yescharacteristic
108C2×C181#40711yescharacteristic
109C2×C181#40811yescharacteristic
110C2×C181#43512yes—
111C2×C181#43612yes—
112D181—42no—
113C181#187011yescharacteristic, complemented, Sylow
114#1870—1811nomaximal, Sylow
115C22×M16—1811no—
116C2×C4×D4—1811no—
117C2×C22⋊C8—1811no—
118C2×C22⋊C8—1812no—
119#336—1812no—
120C22×D4—1811no—
121C23×C4—1811no—
122C23×C4—1811no—
123C2×M16—3622no—
124C2×M16—3624no—
125C22×C8—1812no—
126C22×C8—1812no—
127C4×D4—3622no—
128C4×D4—3622no—
129C2×C4⋊C4—1811no—
130C2×C22⋊C4—1811no—
131C2×C22⋊C4#1062438211yescharacteristic
132C2×C42—1811no—
133C22⋊C8—3622no—
134C22⋊C8—3624no—
135C24—1811no—
136C24C2×#1062438211yescharacteristic
137C2×D4—3622no—
138C2×D4—3622no—
139C2×D4—3622no—
140C22×C4—3622no—
141C22×C4—3621no—
142C22×C4—3621no—
143C22×C4—3622no—
144C22×C4—3622no—
145C22×C4—3622no—
146C22×C4—1812no—
147C22×C4—1812no—
148C22×C4—1811no—
149C22×C4—1811no—
150C22×C4—1811no—
151C22×C4C2×#1062438211yescharacteristic
152C22×C4C2×#1062438211yescharacteristic
153M16—3628no—
154C2×C8—3622no—
155C2×C8—3622no—
156C2×C8—3622no—
157C2×C8—3622no—
158C2×C8—3622no—
159C2×C8—1814no—
160C4⋊C4—3622no—
161C22⋊C4—22no—
162C22⋊C4—3622no—
163C42—3621no—
164C42—3621no—
165C23—22no—
166C23—21no—
167C23—21no—
168C23—22no—
169C23—3621no—
170C23—3622no—
171C23—3621no—
172C23—3622no—
173C23—1812no—
174C23C22⋊#1062438212yes—
175C23C22×#1062438211yescharacteristic
176D4—3628no—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—3621no—
182C2×C4—3622no—
183C2×C4—3621no—
184C2×C4—3622no—
185C2×C4—3624no—
186C2×C4—3622no—
187C2×C4—1812no—
188C2×C4—1812no—
189C2×C4—1811no—
190C2×C4—1811no—
191C2×C4—1812no—
192C2×C4—1812no—
193C2×C4—7241no—
194C2×C4—7242no—
195C2×C4—7241no—
196C8—3624no—
197C8—7242no—
198C22—22no—
199C22—22no—
200C22—22no—
201C22—42no—
202C22—42no—
203C22—3622no—
204C22—3624no—
205C22—7241no—
206C22—7241no—
207C22—7242no—
208C22C2×C22⋊#1062438211yescharacteristic
209C22D4×#1062438212yes—
210C22D181⋊M1611yescharacteristic
211C22D181⋊M1611yescharacteristic
212C22#1915390312yes—
213C4—41no—
214C4—41no—
215C4—3622no—
216C4—3622no—
217C4—1811no—
218C4—1812no—
219C4—1811no—
220C2—42no—
221C2—7242no—
222C2#2604222711yescharacteristic
223C2#2604225111yescharacteristic
224C2#2604225211yescharacteristic
225C2#2604231112yes—
226C2#2604231212yes—
subgroupG/H
center, FrattiniC23C22×#10624382
commutator, socleC23×C181C22×C4
FittingC2×C22⋊C4×C181C4
prime radicalC22×C22⋊D181C4
cosocleC22×Dic181C23
hypercenterC2×C22⋊C4#10624382
hypocenterC181#1870
LCSG ▷ C23×C181 ▷ C181
UCSC1 ◁ C23 ◁ C2×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C22×#15668458C2semidirect
2. C2×#19153873C2semidirect
3. C2×C22⋊#13089685C2semidirect
4. #26042145C2semidirect
27. C2×#13089685D4semidirect
57. C22×D181M16semidirect
66. C2×Dic181M16semidirect
113. C181#1870semidirect

Maximal subgroups

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

subgroupindex
C22×#156684582
C2×#191538732
C2×C22⋊#130896852
#260421452
C2×#191537762
#1870181