Finite Group Databasebeta

← #37549748

Subgroups — #37549748

Order 23168.

338 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 338 conjugacy classes of subgroups — too many to draw. Only its 74 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
2C2×#19153877C212yesmaximal, complemented
3#26042249C211yesmaximal, characteristic
4#26042159C212yesmaximal, complemented
5C2×#19153793C211yesmaximal, characteristic
6C2×#15668458—24no—
7C2×#15668458—21no—
8C2×#15668458C2212yes—
9C2×#15668458C2212yes—
10C22×#13089685—21no—
11#19153877—24no—
12C22⋊#13089685—21no—
13C22⋊#13089685—21no—
14C22×Dic362C2211yescharacteristic
15C23×Dic181C2211yescharacteristic
16#19153793—21no—
17#19153793—21no—
18#19153793—21no—
19#19153793—21no—
20C2×C4⋊Dic181C411yescharacteristic
21C2×#15668428C411yescharacteristic
22C2×#15668428C2211yescharacteristic
23C2×C22⋊Dic181C411yescharacteristic
24C2×C22⋊C4×C181C411yescharacteristic
25#15668458—24no—
26#15668458—22no—
27#15668458—22no—
28#15668458—22no—
29#15668458D414yes—
30C2×#13089685—24no—
31C2×#13089685—22no—
32C2×#13089685—22no—
33C2×Dic362—21no—
34C2×Dic362—21no—
35C2×Dic362—22no—
36C2×Dic362—22no—
37C22×Dic181—22no—
38C22×Dic181—22no—
39C22×Dic181C2×C411yescharacteristic
40C22×Dic181C2×C411yescharacteristic
41C22×Dic181C2×C411yescharacteristic
42C22×Dic181D411yescharacteristic
43C22×Dic181D411yescharacteristic
44C22×Dic181D411yescharacteristic
45C22×Dic181D411yescharacteristic
46C22×Dic181C2311yescharacteristic
47C4⋊Dic181—21no—
48C4⋊Dic181—21no—
49#15668428—21no—
50#15668428—21no—
51#15668428—21no—
52#15668428—21no—
53C22⋊Dic181—21no—
54C22⋊Dic181—21no—
55C24×C181C2×C411yescharacteristic
56C22×C4×C181C2×C411yescharacteristic
57C22×C4×C181C2×C411yescharacteristic
58C22⋊C4×C181—21no—
59C22⋊C4×C181—21no—
60#13089685—24no—
61#13089685—42no—
62Dic362—41no—
63Dic362—42no—
64Dic362—41no—
65C2×Dic181—22no—
66C2×Dic181—21no—
67C2×Dic181—21no—
68C2×Dic181—22no—
69C2×Dic181—22no—
70C2×Dic181—21no—
71C2×Dic181—21no—
72C2×Dic181—21no—
73C2×Dic181—21no—
74C2×Dic181—22no—
75C2×Dic181—41no—
76C2×Dic181C22⋊C411yescharacteristic
77C2×Dic181C22⋊C411yescharacteristic
78C2×Dic181C2×D412yes—
79C2×Dic181C2×D411yescharacteristic
80C2×Dic181C2×D411yescharacteristic
81C2×Dic181#4112yes—
82C23×C181—22no—
83C23×C181—22no—
84C23×C181—41no—
85C23×C181C22⋊C411yescharacteristic
86C23×C181C22⋊C411yescharacteristic
87C23×C181C22×C411yescharacteristic
88C2×C4×C181—21no—
89C2×C4×C181—21no—
90C2×C4×C181—22no—
91C2×C4×C181—41no—
92C22×C181—21no—
93C22×C181—21no—
94C22×C181—21no—
95C22×C181—21no—
96C22×C181—21no—
97C22×C181—21no—
98C22×C181—21no—
99C22×C181—44no—
100C22×C181#10112yescomplemented
101C22×C181C2×C22⋊C411yescharacteristic
102C22×C181C4×D412yes—
103C4×C181—41no—
104C4×C181—41no—
105Dic181—21no—
106Dic181—21no—
107Dic181—41no—
108Dic181—41no—
109Dic181#12011yescharacteristic
110Dic181#12112yes—
111Dic181#12311yescharacteristic
112C2×C181—22no—
113C2×C181—22no—
114C2×C181—41no—
115C2×C181#38611yescharacteristic
116C2×C181C2×#10111yescharacteristic
117C2×C181#41311yescharacteristic
118C181#190111yescharacteristic, complemented, Sylow
119#1901—1811nomaximal, Sylow
120C22×M16—1811no—
121C2×C22⋊Q8—1811no—
122C2×#101—1812no—
123#407—1811no—
124#343—1812no—
125C22×Q8—1811no—
126C23×C4—1811no—
127C2×M16—3621no—
128C2×M16—3624no—
129C2×M16—1812no—
130C2×M16—1812no—
131C22×C8—3621no—
132C22⋊Q8—3621no—
133C22⋊Q8—3621no—
134C22⋊Q8—3621no—
135C22⋊Q8—3621no—
136C2×C4⋊C4—1811no—
137C2×C4⋊C4—1811no—
138C2×C4⋊C4—1811no—
139C2×C22⋊C4—1811no—
140C2×C22⋊C4#1062438211yescharacteristic
141#101—3624no—
142C22⋊C8—3621no—
143C22⋊C8—3621no—
144C24C2×#1062438211yescharacteristic
145C2×Q8—3622no—
146C2×Q8—3621no—
147C2×Q8—3621no—
148C2×Q8—3622no—
149C22×C4—3622no—
150C22×C4—3622no—
151C22×C4—1811no—
152C22×C4—1811no—
153C22×C4—1811no—
154C22×C4—1811no—
155C22×C4—1811no—
156C22×C4—1811no—
157C22×C4—1811no—
158C22×C4—1811no—
159C22×C4C2×#1062438211yescharacteristic
160C22×C4C2×#1062438211yescharacteristic
161M16—3624no—
162M16—3622no—
163M16—3622no—
164M16—3622no—
165M16—1814no—
166C2×C8—3624no—
167C2×C8—3622no—
168C2×C8—3622no—
169C4⋊C4—3621no—
170C4⋊C4—3621no—
171C4⋊C4—3621no—
172C4⋊C4—3621no—
173C4⋊C4—3621no—
174C4⋊C4—3621no—
175C22⋊C4—21no—
176C22⋊C4—21no—
177C22⋊C4—3621no—
178C22⋊C4—3621no—
179C23—22no—
180C23—22no—
181C23—41no—
182C23C22⋊#1062438211yescharacteristic
183C23C22⋊#1062438211yescharacteristic
184C23C22×#1062438211yescharacteristic
185Q8—7241no—
186Q8—7242no—
187Q8—7241no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—22no—
191C2×C4—41no—
192C2×C4—3622no—
193C2×C4—3621no—
194C2×C4—3621no—
195C2×C4—3622no—
196C2×C4—3622no—
197C2×C4—3621no—
198C2×C4—3621no—
199C2×C4—3621no—
200C2×C4—3621no—
201C2×C4—3622no—
202C2×C4—1811no—
203C2×C4—1811no—
204C2×C4—1811no—
205C2×C4—1811no—
206C2×C4—1812no—
207C2×C4—1812no—
208C2×C4—7241no—
209C8—3624no—
210C8—7242no—
211C22—21no—
212C22—21no—
213C22—21no—
214C22—21no—
215C22—21no—
216C22—21no—
217C22—21no—
218C22—44no—
219C22#1915387712yescomplemented
220C22C2×C22⋊#1062438211yescharacteristic
221C22D4×#1062438212yes—
222C4—41no—
223C4—41no—
224C4—3621no—
225C4—3621no—
226C4—1811no—
227C4—1812no—
228C4—1811no—
229C4—7241no—
230C4—7241no—
231C2—22no—
232C2—22no—
233C2—41no—
234C2#2604222711yescharacteristic
235C2C2×#1915387711yescharacteristic
236C2#2604226611yescharacteristic
subgroupG/H
centerC22C2×C22⋊#10624382
commutatorC23×C181C22×C4
FrattiniC23C22×#10624382
FittingC2×C22⋊C4×C181C4
socleC22×C181C2×C22⋊C4
prime radicalC24×C181C2×C4
cosocleC22×Dic181C23
hypercenterC2×C22⋊C4#10624382
hypocenterC181#1901
LCSG ▷ C23×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C24 ◁ C2×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×#19153877C2semidirect
4. #26042159C2semidirect
100. C22×C181#101semidirect
118. C181#1901semidirect
219. C22#19153877semidirect

Maximal subgroups

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

subgroupindex
C22×#156684582
C2×#191538772
#260422492
#260421592
C2×#191537932
#1901181