Finite Group Databasebeta

← #37549731

Subgroups — #37549731

Order 23168.

386 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 68 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#26042346C211yesmaximal, characteristic
2D181⋊#99C211yesmaximal, characteristic
3#26042222C212yesmaximal, complemented
4#26042156C212yesmaximal, complemented
5#123×D181C211yesmaximal, characteristic
6D4×#10624382—24no—
7#19153895—21no—
8C2×C22⋊#10624382—21no—
9C2×C22⋊#10624382C2212yes—
10C2×C22⋊#10624382C2212yes—
11#19153875—21no—
12#19153874—21no—
13(C2×#10624382)⋊C4—24no—
14C2×D4×D181C2211yescharacteristic
15C22×C4×D181C2211yescharacteristic
16#19153801—21no—
17#19153800C411yescharacteristic, complemented
18#19153799C411yescharacteristic, complemented
19#19153798—21no—
20#19153797C411yescharacteristic, complemented
21C4⋊C4×D181—21no—
22C22⋊C4×D181—21no—
23C22⋊C4×D181C2211yescharacteristic
24#123×C181C411yescharacteristic, complemented
25C22×#10624382—22no—
26C22×#10624382—22no—
27C22×#10624382D414yes—
28C4⋊#10624382—22no—
29C22⋊#10624382—22no—
30C22⋊#10624382—22no—
31C22⋊#10624382—22no—
32C22⋊#10624382—22no—
33C22⋊#10624382—22no—
34C4×#10624382—22no—
35C23×D181D411yescharacteristic
36C23×D181C2311yescharacteristic
37C2×D724D411yescharacteristic
38C2×C22⋊D181C2×C411yescharacteristic
39C2×C22⋊D181D411yescharacteristic
40D4×D181—22no—
41D4×D181—22no—
42C22×Dic181C2×C411yescharacteristic, complemented
43C2×C4×D181—21no—
44C2×C4×D181—21no—
45C2×C4×D181—21no—
46C2×C4×D181—22no—
47C2×C4×D181—22no—
48C2×C4×D181D411yescharacteristic
49C4⋊Dic181—21no—
50#15668428—21no—
51#15668428—21no—
52C22⋊Dic181—21no—
53C22⋊Dic181C2×C411yescharacteristic, complemented
54#15668426—21no—
55#15668426—21no—
56#15668426—21no—
57C2×D4×C181C2×C411yescharacteristic
58C22×C4×C181C2×C411yescharacteristic, complemented
59C4⋊C4×C181—21no—
60C22⋊C4×C181—21no—
61C22⋊C4×C181C2×C411yescharacteristic, complemented
62C2×#10624382—24no—
63C2×#10624382—44no—
64C2×#10624382—44no—
65C2×#10624382—42no—
66C2×#10624382—82no—
67C22×D181—21no—
68C22×D181—21no—
69C22×D181—42no—
70C22×D181—42no—
71C22×D181C22⋊C411yescharacteristic
72C22×D181C2×D412yes—
73C22×D181C2×D411yescharacteristic
74C22×D181C2×D411yescharacteristic
75C22×D181#4112yes—
76D724—22no—
77C22⋊D181—22no—
78C22⋊D181—22no—
79C2×Dic181—21no—
80C2×Dic181—21no—
81C2×Dic181—22no—
82C2×Dic181C22⋊C411yescharacteristic
83C4×D181—21no—
84C4×D181—21no—
85C4×D181—42no—
86C4×D181—41no—
87C4×D181—42no—
88C4×D181—41no—
89C23×C181C22⋊C411yescharacteristic
90C23×C181C22×C411yescharacteristic
91D4×C181—22no—
92C2×C4×C181—21no—
93C2×C4×C181—21no—
94C2×C4×C181—22no—
95C2×C4×C181C22⋊C411yescharacteristic
96C22×C181—21no—
97C22×C181—41no—
98C22×C181—41no—
99C22×C181C2×C22⋊C411yescharacteristic
100C22×C181C4×D412yes—
101C2×D181—22no—
102C2×D181—24no—
103C2×D181—21no—
104C2×D181—41no—
105C2×D181—42no—
106C2×D181—42no—
107C2×D181—41no—
108C2×D181—81no—
109C2×D181#12011yescharacteristic
110C2×D181#12112yes—
111C2×D181#12311yescharacteristic
112#10624382—44no—
113#10624382—82no—
114C4×C181—21no—
115C4×C181—41no—
116C4×C181—41no—
117Dic181—21no—
118Dic181—41no—
119Dic181—41no—
120C2×C181—21no—
121C2×C181—22no—
122C2×C181—41no—
123C2×C181#38611yescharacteristic
124D181—22no—
125D181—21no—
126D181—41no—
127D181#45812yes—
128C181#189011yescharacteristic, complemented, Sylow
129#1890—1811nomaximal, Sylow
130C2×#123—1811no—
131#518—1811no—
132C2×#99—1811no—
133#381—1812no—
134#342—1812no—
135C22×D4—1811no—
136C23×C4—1811no—
137#123—3621no—
138#123—3621no—
139#123—1811no—
140#123—1811no—
141#123—1811no—
142#123#1062438211yescharacteristic, complemented
143C4×D4—3624no—
144#117—3621no—
145C2×C4⋊C4—3621no—
146C2×C22⋊C4—3621no—
147C2×C22⋊C4—3621no—
148C2×C22⋊C4—1812no—
149C2×C22⋊C4—1812no—
150C2×C22⋊C4—1811no—
151#99—3621no—
152#99—3621no—
153#95—3624no—
154C24—1811no—
155C24—1811no—
156C2×D4—3622no—
157C2×D4—3622no—
158C2×D4—1811no—
159C2×D4—1811no—
160C2×D4—1811no—
161C2×D4C2×#1062438211yescharacteristic
162C22×C4—3622no—
163C22×C4—3621no—
164C22×C4—3621no—
165C22×C4—3621no—
166C22×C4—3622no—
167C22×C4—3622no—
168C22×C4—3622no—
169C22×C4—1811no—
170C22×C4—1811no—
171C22×C4—1814no—
172C22×C4C2×#1062438211yescharacteristic, complemented
173C4⋊C4—21no—
174C4⋊C4—3621no—
175C4⋊C4—3621no—
176C4⋊C4—3622no—
177C4⋊C4—3621no—
178C22⋊C4—21no—
179C22⋊C4—3621no—
180C22⋊C4—3621no—
181C22⋊C4—3621no—
182C22⋊C4—3622no—
183C22⋊C4—3622no—
184C22⋊C4—3622no—
185C22⋊C4—3621no—
186C22⋊C4—3622no—
187C22⋊C4—3622no—
188C22⋊C4—1811no—
189C22⋊C4C2×#1062438211yescharacteristic, complemented
190C42—3622no—
191C23—3621no—
192C23—3621no—
193C23—1811no—
194C23—1812no—
195C23—1811no—
196C23—1811no—
197C23—1812no—
198C23—7242no—
199C23—7242no—
200C23C22⋊#1062438211yescharacteristic
201C23C22×#1062438211yescharacteristic
202D4—22no—
203D4—3622no—
204D4—3622no—
205D4—3622no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—22no—
209C2×C4—3621no—
210C2×C4—3621no—
211C2×C4—3624no—
212C2×C4—3621no—
213C2×C4—3621no—
214C2×C4—3622no—
215C2×C4—14482no—
216C2×C4—1811no—
217C2×C4—7242no—
218C2×C4—7241no—
219C2×C4—7242no—
220C2×C4—7241no—
221C2×C4—7244no—
222C2×C4—7242no—
223C2×C4—7244no—
224C2×C4C22⋊#1062438211yescharacteristic
225C22—21no—
226C22—41no—
227C22—41no—
228C22—3624no—
229C22—3622no—
230C22—3621no—
231C22—14481no—
232C22—1811no—
233C22—1812no—
234C22—1811no—
235C22—7241no—
236C22—7242no—
237C22—7241no—
238C22—7242no—
239C22C2×C22⋊#1062438211yescharacteristic
240C22D4×#1062438212yes—
241C4—21no—
242C4—41no—
243C4—41no—
244C4—3621no—
245C4—14482no—
246C4—7241no—
247C4—7244no—
248C4—7241no—
249C2—21no—
250C2—22no—
251C2—41no—
252C2—3622no—
253C2—3621no—
254C2—1812no—
255C2—7241no—
256C2#2604222711yescharacteristic
subgroupG/H
centerC2#26042227
commutatorC23×C181C22×C4
FrattiniC23C22×#10624382
Fitting#123×C181C4
socleC2×C181#386
prime radicalC2×D4×D181C22
cosocleC23×D181C23
hypercenter#123#10624382
hypocenterC181#1890
LCSG ▷ C23×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C2 ◁ C23 ◁ #123

Complements

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

NH ≅ G/Ntype
3. #26042222C2semidirect
4. #26042156C2semidirect
17. #19153800C4semidirect
18. #19153799C4semidirect
20. #19153797C4semidirect
24. #123×C181C4semidirect
42. C22×Dic181C2×C4semidirect
53. C22⋊Dic181C2×C4semidirect
58. C22×C4×C181C2×C4semidirect
61. C22⋊C4×C181C2×C4semidirect
128. C181#1890semidirect
142. #123#10624382semidirect
172. C22×C4C2×#10624382semidirect
189. C22⋊C4C2×#10624382semidirect

Maximal subgroups

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

subgroupindex
#260423462
D181⋊#992
#260422222
#260421562
#123×D1812
#1890181