Finite Group Databasebeta

← #37534021

Subgroups — #37534021

Order 23168.

376 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 376 conjugacy classes of subgroups — too many to draw. Only its 93 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×#15668433C211yesmaximal, characteristic, complemented
2#26041790C211yesmaximal, characteristic
3C2×#19153744C212yesmaximal, complemented
4#26041039C212yesmaximal, complemented
5C4×C22⋊C4×C181C211yesmaximal, characteristic, complemented
6C2×#15668449C2211yescharacteristic
7C2×#15668433—24no—
8C2×#15668433—21no—
9C2×#15668433C2212yes—
10C22×#13089678—21no—
11#19153799—21no—
12#19153795C411yescharacteristic, complemented
13#19153795C411yescharacteristic, complemented
14#19153790C411yescharacteristic, complemented
15#19153790C411yescharacteristic, complemented
16#19153785—21no—
17C4×C22⋊D181—21no—
18C4×C22⋊D181—21no—
19#19153775C2211yescharacteristic
20#19153744—24no—
21C23×C4×C181C2211yescharacteristic
22C2×C22⋊C4×C181—21no—
23C2×C42×C181C2212yes—
24#95×C181—21no—
25#15668449—22no—
26#15668449—22no—
27C2×Dic362C2×C411yescharacteristic
28C2×D724C2×C411yescharacteristic
29C2×C22⋊D181—21no—
30C2×C22⋊D181—21no—
31C2×C22⋊D181—21no—
32C2×C4×D181—21no—
33#15668433—24no—
34#15668433—22no—
35#15668433D414yescomplemented
36C2×#13089678—24no—
37C2×#13089678—22no—
38C4⋊Dic181C2×C411yescharacteristic
39C4⋊Dic181C2×C411yescharacteristic
40#15668428—21no—
41#15668428—21no—
42C22⋊Dic181—21no—
43C22⋊Dic181—21no—
44C22⋊Dic181—21no—
45#15668426—21no—
46#15668426—21no—
47C4×Dic181—21no—
48C24×C181D411yescharacteristic
49C22×C4×C181—22no—
50C22×C4×C181—22no—
51C22×C4×C181—22no—
52C22×C4×C181C2×C411yescharacteristic
53C22×C4×C181C2×C411yescharacteristic
54C22×C4×C181D411yescharacteristic
55C22×C4×C181D411yescharacteristic
56C22×C4×C181D411yescharacteristic
57C22×C4×C181C2311yescharacteristic
58C22⋊C4×C181—24no—
59C42×C181—22no—
60C42×C181—22no—
61C22×D181—21no—
62Dic362—22no—
63D724—22no—
64C22⋊D181—42no—
65C22⋊D181—42no—
66C22⋊D181—42no—
67C2×Dic181—21no—
68C2×Dic181—21no—
69C2×Dic181—21no—
70C4×D181—41no—
71C4×D181—41no—
72#13089678—24no—
73C23×C181—22no—
74C23×C181—22no—
75C23×C181—41no—
76C23×C181C22⋊C411yescharacteristic
77C23×C181C22⋊C411yescharacteristic
78C23×C181C2×D411yescharacteristic
79C2×C4×C181—22no—
80C2×C4×C181—22no—
81C2×C4×C181—22no—
82C2×C4×C181—21no—
83C2×C4×C181—21no—
84C2×C4×C181—21no—
85C2×C4×C181—21no—
86C2×C4×C181—24no—
87C2×C4×C181—42no—
88C2×C4×C181—42no—
89C2×C4×C181C22⋊C411yescharacteristic
90C2×C4×C181C22⋊C411yescharacteristic
91C2×C4×C181C22×C411yescharacteristic
92C2×C4×C181C2×D412yes—
93C2×C4×C181C2×D411yescharacteristic
94C2×C4×C181#4112yes—
95C22×C181—21no—
96C22×C181—21no—
97C22×C181—21no—
98C22×C181—21no—
99C22×C181—21no—
100C22×C181—21no—
101C22×C181—21no—
102C22×C181—44no—
103C22×C181#10412yescomplemented
104C22×C181C2×C22⋊C411yescharacteristic
105C22×C181#12112yes—
106C2×D181—41no—
107C2×D181—41no—
108C2×D181—41no—
109C4×C181—21no—
110C4×C181—21no—
111C4×C181—44no—
112C4×C181C4×D412yes—
113C4×C181#12011yescharacteristic
114C4×C181#12311yescharacteristic
115Dic181—41no—
116Dic181—41no—
117Dic181—41no—
118C2×C181—22no—
119C2×C181—22no—
120C2×C181—41no—
121C2×C181#38611yescharacteristic
122C2×C181C2×#10411yescharacteristic
123C2×C181#42111yescharacteristic
124D181—81no—
125C181#187111yescharacteristic, complemented, Sylow
126#1871—1811nomaximal, Sylow
127C22×M16—1811no—
128#525—1811no—
129C2×#104—1812no—
130C4×C22⋊C4D18111yescharacteristic, complemented
131#339—1812no—
132C2×#41—1811no—
133C23×C4C2×D18111yescharacteristic
134C2×M16—3621no—
135C2×M16—3624no—
136C2×M16—1812no—
137C22×C8—3621no—
138#123—3621no—
139C22⋊Q8—1811no—
140C22⋊Q8—1811no—
141#121—1811no—
142#121—1811no—
143#120—3621no—
144C4×D4—3621no—
145C4×D4—3621no—
146#117—1811no—
147C2×C22⋊C4—21no—
148C2×C42C2×D18112yes—
149#104—3624no—
150#95—21no—
151C24C22⋊D18111yescharacteristic
152#41—3622no—
153#41—3622no—
154C2×Q8—1811no—
155C2×D4—3621no—
156C2×D4—3621no—
157C2×D4—3621no—
158C2×D4—1811no—
159C22×C4—22no—
160C22×C4—22no—
161C22×C4—22no—
162C22×C4—3621no—
163C22×C4C4×D18111yescharacteristic
164C22×C4C4×D18111yescharacteristic
165C22×C4C22⋊D18111yescharacteristic
166C22×C4D72411yescharacteristic
167C22×C4D72411yescharacteristic
168C22×C4C22×D18111yescharacteristic
169M16—3624no—
170M16—3622no—
171M16—1814no—
172C2×C8—3624no—
173C2×C8—3622no—
174C4⋊C4—3621no—
175C4⋊C4—3621no—
176C4⋊C4—1811no—
177C4⋊C4—1811no—
178C22⋊C4—24no—
179C22⋊C4—3621no—
180C22⋊C4—3621no—
181C22⋊C4—3621no—
182C22⋊C4—3621no—
183C22⋊C4—3621no—
184C42—22no—
185C42—22no—
186C42—3621no—
187C23—22no—
188C23—22no—
189C23—41no—
190C23—3621no—
191C23#1566842611yescharacteristic
192C23#1566842611yescharacteristic
193C23C2×C22⋊D18111yescharacteristic
194Q8—3622no—
195D4—3622no—
196D4—7242no—
197D4—7242no—
198D4—7242no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—24no—
207C2×C4—42no—
208C2×C4—42no—
209C2×C4—3621no—
210C2×C4—3621no—
211C2×C4—3621no—
212C2×C4—7241no—
213C2×C4—7241no—
214C2×C4#1566842611yescharacteristic
215C2×C4#1566842611yescharacteristic
216C2×C4C2×C4×D18111yescharacteristic
217C2×C4D4×D18112yes—
218C2×C4C2×D72411yescharacteristic
219C2×C4#1566844812yes—
220C8—3624no—
221C22—21no—
222C22—21no—
223C22—21no—
224C22—21no—
225C22—21no—
226C22—21no—
227C22—21no—
228C22—44no—
229C22—7241no—
230C22—7241no—
231C22—7241no—
232C22#1915374412yescomplemented
233C22C2×#1566842611yescharacteristic
234C22#1915378812yes—
235C4—21no—
236C4—21no—
237C4—44no—
238C4—7241no—
239C4—7241no—
240C4—7241no—
241C4#1915377612yes—
242C4#1915378611yescharacteristic
243C4#1915380011yescharacteristic
244C2—22no—
245C2—22no—
246C2—41no—
247C2—14481no—
248C2#2604115611yescharacteristic
249C2C2×#1915374411yescharacteristic
250C2#2604129711yescharacteristic
subgroupG/H
centerC2×C4C2×D724
commutatorC2×C4×C181C22×C4
FrattiniC22×C4C22×D181
FittingC4×C22⋊C4×C181C2
socleC22×C181C2×C22⋊C4
prime radical#26041790C2
cosocleC22×C4×C181C23
hypercenterC4×C22⋊C4D181
hypocenterC181#1871
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C22×#15668433C2semidirect
3. C2×#19153744C2semidirect
4. #26041039C2semidirect
5. C4×C22⋊C4×C181C2semidirect
12. #19153795C4semidirect
13. #19153795C4semidirect
14. #19153790C4semidirect
15. #19153790C4semidirect
35. #15668433D4semidirect
103. C22×C181#104semidirect
125. C181#1871semidirect
130. C4×C22⋊C4D181semidirect
232. C22#19153744semidirect

Maximal subgroups

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

subgroupindex
C22×#156684332
#260417902
C2×#191537442
#260410392
C4×C22⋊C4×C1812
#1871181