Finite Group Databasebeta

← #37549707

Subgroups — #37549707

Order 23168.

360 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 78 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
1C2×#19153895C211yesmaximal, characteristic
2C2×#19153879C212yesmaximal, complemented
3D181⋊(C22⋊C8)C211yesmaximal, characteristic
4#26042152C212yesmaximal, complemented
5C22⋊Q8×D181C211yesmaximal, characteristic
6C2×D181⋊C8C2212yes—
7#19153895—24no—
8C2×C4⋊#10624382C2212yes—
9C2×C22⋊#10624382—21no—
10C2×C4×#10624382—21no—
11#19153879—24no—
12#19153873—21no—
13C22⋊#13089685—21no—
14C2×Q8×D181C2211yescharacteristic
15C22×C4×D181C2211yescharacteristic
16#19153796C411yescharacteristic, complemented
17#19153795C411yescharacteristic, complemented
18#19153794—21no—
19#19153793—21no—
20#19153792C411yescharacteristic, complemented
21C4⋊C4×D181—21no—
22C4⋊C4×D181C2211yescharacteristic
23C22⋊C4×D181—21no—
24C22⋊Q8×C181C411yescharacteristic, complemented
25C22×#10624382—22no—
26D181⋊C8—22no—
27D181⋊C8—22no—
28C2×#13089685—22no—
29C4⋊#10624382—22no—
30C4⋊#10624382D414yes—
31C22⋊#10624382—24no—
32C4×#10624382—24no—
33C23×D181D411yescharacteristic
34C2×Dic362—21no—
35C2×Dic362C2×C411yescharacteristic
36Q8×D181—22no—
37Q8×D181—22no—
38C22×Dic181D411yescharacteristic
39C2×C4×D181—22no—
40C2×C4×D181—22no—
41C2×C4×D181—21no—
42C2×C4×D181—21no—
43C2×C4×D181C2×C411yescharacteristic
44C2×C4×D181D411yescharacteristic
45C2×C4×D181D411yescharacteristic
46C2×C4×D181C2311yescharacteristic
47C4⋊Dic181—21no—
48C4⋊Dic181C2×C411yescharacteristic
49#15668428—21no—
50#15668428—21no—
51#15668428—21no—
52C22⋊Dic181—21no—
53#15668426—21no—
54#15668426—21no—
55C2×Q8×C181C2×C411yescharacteristic
56C22×C4×C181C2×C411yescharacteristic
57C4⋊C4×C181—21no—
58C4⋊C4×C181C2×C411yescharacteristic
59C22⋊C4×C181—21no—
60C2×#10624382—24no—
61C2×#10624382—42no—
62C2×#10624382—42no—
63#13089685—42no—
64C22×D181—21no—
65C22×D181—21no—
66C22×D181—22no—
67C22×D181—22no—
68C22×D181C22⋊C411yescharacteristic
69C22×D181C2×D411yescharacteristic
70Dic362—22no—
71Dic362—42no—
72C2×Dic181—22no—
73C2×Dic181—21no—
74C2×Dic181—21no—
75C2×Dic181C22⋊C411yescharacteristic
76C2×Dic181C2×D411yescharacteristic
77C4×D181—21no—
78C4×D181—21no—
79C4×D181—22no—
80C4×D181—22no—
81C4×D181—41no—
82C4×D181—41no—
83C4×D181—42no—
84C4×D181C2×D412yes—
85C4×D181#4112yes—
86C23×C181C22⋊C411yescharacteristic
87Q8×C181—22no—
88C2×C4×C181—22no—
89C2×C4×C181—21no—
90C2×C4×C181—21no—
91C2×C4×C181C22⋊C411yescharacteristic
92C2×C4×C181C22×C411yescharacteristic
93C22×C181—21no—
94C22×C181—21no—
95C22×C181—21no—
96C22×C181C2×C22⋊C411yescharacteristic
97C2×D181—21no—
98C2×D181—21no—
99C2×D181—21no—
100C2×D181—42no—
101C2×D181—42no—
102C2×D181#12011yescharacteristic
103C2×D181#12311yescharacteristic
104C2×D181#13512yes—
105C2×D181#13712yes—
106#10624382—44no—
107C4×C181—21no—
108C4×C181—41no—
109C4×C181—41no—
110C4×C181C4×D412yes—
111Dic181—21no—
112Dic181—41no—
113Dic181—41no—
114Dic181#12112yes—
115C2×C181—41no—
116C2×C181#38611yescharacteristic
117C2×C181#41611yescharacteristic
118C2×C181#41911yescharacteristic
119D181—41no—
120D181#45212yes—
121D181#48512yes—
122C181#187311yescharacteristic, complemented, Sylow
123#1873—1811nomaximal, Sylow
124C2×C22⋊Q8—1811no—
125C2×#117—1811no—
126C2×#103—1812no—
127C2×C22⋊C8—1811no—
128#340—1812no—
129C22×Q8—1811no—
130C23×C4—1811no—
131C22×C8—1812no—
132C22⋊Q8—3621no—
133C22⋊Q8—3621no—
134C22⋊Q8—1811no—
135C22⋊Q8—1811no—
136C22⋊Q8—1811no—
137C22⋊Q8#1062438211yescharacteristic, complemented
138#117—3624no—
139C2×C4⋊C4—3621no—
140C2×C4⋊C4—1812no—
141C2×C4⋊C4—1811no—
142C2×C22⋊C4—3621no—
143C2×C22⋊C4—3621no—
144C2×C42—3621no—
145#103—3624no—
146C22⋊C8—3621no—
147C22⋊C8—3621no—
148C24—1811no—
149C2×Q8—3621no—
150C2×Q8—3622no—
151C2×Q8—3622no—
152C2×Q8—1811no—
153C2×Q8C2×#1062438211yescharacteristic
154C22×C4—3622no—
155C22×C4—3622no—
156C22×C4—3621no—
157C22×C4—3621no—
158C22×C4—3622no—
159C22×C4—1811no—
160C22×C4—1811no—
161C22×C4—1811no—
162C22×C4—1811no—
163C22×C4—1811no—
164C22×C4C2×#1062438211yescharacteristic
165C2×C8—3622no—
166C2×C8—3622no—
167C2×C8—3622no—
168C4⋊C4—21no—
169C4⋊C4—3621no—
170C4⋊C4—3621no—
171C4⋊C4—3621no—
172C4⋊C4—3622no—
173C4⋊C4—3621no—
174C4⋊C4—1814no—
175C4⋊C4—1811no—
176C4⋊C4C2×#1062438211yescharacteristic
177C22⋊C4—21no—
178C22⋊C4—3621no—
179C22⋊C4—3621no—
180C22⋊C4—3624no—
181C22⋊C4—3621no—
182C42—3624no—
183C23—3621no—
184C23—3621no—
185C23—3622no—
186C23—3622no—
187C23—1811no—
188C23—1811no—
189C23C22⋊#1062438211yescharacteristic
190Q8—22no—
191Q8—3622no—
192Q8—7242no—
193C2×C4—22no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—3621no—
197C2×C4—3621no—
198C2×C4—3622no—
199C2×C4—3622no—
200C2×C4—3622no—
201C2×C4—3624no—
202C2×C4—3621no—
203C2×C4—3621no—
204C2×C4—1811no—
205C2×C4—1811no—
206C2×C4—1812no—
207C2×C4—1812no—
208C2×C4—7241no—
209C2×C4—7241no—
210C2×C4—7242no—
211C2×C4—7242no—
212C2×C4—7242no—
213C2×C4C22⋊#1062438211yescharacteristic
214C2×C4C22×#1062438211yescharacteristic
215C8—7242no—
216C22—21no—
217C22—21no—
218C22—21no—
219C22—3621no—
220C22—3621no—
221C22—3621no—
222C22—1811no—
223C22—1811no—
224C22—1812no—
225C22—1812no—
226C22—7242no—
227C22—7242no—
228C22C2×C22⋊#1062438211yescharacteristic
229C4—21no—
230C4—41no—
231C4—41no—
232C4—3621no—
233C4—1812no—
234C4—7244no—
235C4—7241no—
236C4—7241no—
237C4D4×#1062438212yes—
238C2—41no—
239C2—1812no—
240C2—1812no—
241C2—7241no—
242C2#2604222711yescharacteristic
243C2#2604226911yescharacteristic
244C2#2604227211yescharacteristic
subgroupG/H
centerC22C2×C22⋊#10624382
commutatorC2×C4×C181C22×C4
FrattiniC2×C4C22×#10624382
FittingC22⋊Q8×C181C4
socleC22×C181C2×C22⋊C4
prime radicalC23×D181D4
cosocleC2×C4×D181C23
hypercenterC22⋊Q8#10624382
hypocenterC181#1873
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. C2×#19153879C2semidirect
4. #26042152C2semidirect
16. #19153796C4semidirect
17. #19153795C4semidirect
20. #19153792C4semidirect
24. C22⋊Q8×C181C4semidirect
122. C181#1873semidirect
137. C22⋊Q8#10624382semidirect

Maximal subgroups

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

subgroupindex
C2×#191538952
C2×#191538792
D181⋊(C22⋊C8)2
#260421522
C22⋊Q8×D1812
#1873181