Finite Group Databasebeta

← #37549773

Subgroups — #37549773

Order 23168.

308 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 308 conjugacy classes of subgroups — too many to draw. Only its 66 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#26042357C211yesmaximal, characteristic
2#26042266C211yesmaximal, characteristic
3C2×#19153877C212yesmaximal, complemented
4#26042160C212yesmaximal, complemented
5#26041851C211yesmaximal, characteristic
6#19153903—24no—
7D181⋊M16—21no—
8C2×#15668458—21no—
9C2×#15668458C2212yes—
10C2×#15668458C2212yes—
11#19153877—24no—
12#19153877—21no—
13#19153876—21no—
14C2×#15668448C2211yescharacteristic
15C22×Dic362C2211yescharacteristic
16#19153815—21no—
17#19153802—21no—
18#19153800C411yescharacteristic
19#19153799C411yescharacteristic
20#19153797C411yescharacteristic
21#19153794—21no—
22#19153793—21no—
23#19153772C2211yescharacteristic
24#123×C181C411yescharacteristic
25#15668459—22no—
26#15668458—22no—
27#15668458—22no—
28#15668458—22no—
29#15668458—22no—
30#15668458D414yes—
31D181⋊C8—22no—
32C2×#13089685—22no—
33C2×#13089685—22no—
34C2×#13089685—22no—
35#15668448—22no—
36#15668448—22no—
37C2×Dic362—21no—
38C2×Dic362—21no—
39C2×Dic362—22no—
40C2×Dic362—22no—
41C2×Dic362D411yescharacteristic
42C2×C22⋊D181C2×C411yescharacteristic
43C2×C22⋊D181D411yescharacteristic
44C22×Dic181C2×C411yescharacteristic
45C22×Dic181D411yescharacteristic
46C22×Dic181C2311yescharacteristic
47C2×C4×D181D411yescharacteristic
48C4⋊Dic181—21no—
49#15668428—21no—
50#15668428—21no—
51#15668428—21no—
52C22⋊Dic181—21no—
53C22⋊Dic181C2×C411yescharacteristic
54#15668426—21no—
55#15668426—21no—
56C4×Dic181—21no—
57C2×D4×C181C2×C411yescharacteristic
58C22×C4×C181C2×C411yescharacteristic
59C4⋊C4×C181—21no—
60C22⋊C4×C181—21no—
61C22⋊C4×C181C2×C411yescharacteristic
62#13089685—24no—
63#13089685—42no—
64C22×D181C22⋊C411yescharacteristic
65Dic362—22no—
66Dic362—41no—
67Dic362—42no—
68Dic362—41no—
69C22⋊D181—22no—
70C22⋊D181—22no—
71C2×Dic181—21no—
72C2×Dic181—21no—
73C2×Dic181—21no—
74C2×Dic181—21no—
75C2×Dic181—22no—
76C2×Dic181C22⋊C411yescharacteristic
77C2×Dic181C2×D412yes—
78C2×Dic181C2×D411yescharacteristic
79C2×Dic181C2×D411yescharacteristic
80C2×Dic181#4112yes—
81C4×D181—21no—
82C4×D181—21no—
83C23×C181C22⋊C411yescharacteristic
84C23×C181C22×C411yescharacteristic
85D4×C181—22no—
86C2×C4×C181—21no—
87C2×C4×C181—21no—
88C2×C4×C181—22no—
89C2×C4×C181C22⋊C411yescharacteristic
90C22×C181—21no—
91C22×C181—41no—
92C22×C181—41no—
93C22×C181C2×C22⋊C411yescharacteristic
94C22×C181C4×D412yes—
95C2×D181—21no—
96C2×D181—41no—
97C4×C181—21no—
98C4×C181—41no—
99C4×C181—41no—
100Dic181—21no—
101Dic181—41no—
102Dic181—41no—
103Dic181#12011yescharacteristic
104Dic181#12112yes—
105Dic181#12311yescharacteristic
106C2×C181—21no—
107C2×C181—22no—
108C2×C181—41no—
109C2×C181#38611yescharacteristic
110D181—41no—
111C181#191511yescharacteristic, complemented, Sylow
112#1915—1811nomaximal, Sylow
113#568—1811no—
114#536—1811no—
115#413—1811no—
116C2×#101—1812no—
117#344—1812no—
118C2×#41—1811no—
119C22×Q8—1811no—
120#131—3624no—
121C2×M16—3621no—
122C2×M16—3621no—
123C2×M16—1812no—
124C2×M16—1812no—
125C4⋊Q8—3621no—
126#124—3621no—
127#123—1811no—
128#123—1811no—
129#123—1811no—
130#123#1062438211yescharacteristic
131C22⋊Q8—3621no—
132C22⋊Q8—3621no—
133#117—1811no—
134#101—3621no—
135#101—3624no—
136#100—3621no—
137#41—3622no—
138#41—3622no—
139C2×Q8—3622no—
140C2×Q8—3621no—
141C2×Q8—3621no—
142C2×Q8—3622no—
143C2×Q8—1811no—
144C2×D4—1811no—
145C2×D4—1811no—
146C2×D4C2×#1062438211yescharacteristic
147C22×C4—1811no—
148C22×C4—1811no—
149C22×C4—1811no—
150C22×C4—1811no—
151C22×C4C2×#1062438211yescharacteristic
152M16—3622no—
153M16—3622no—
154M16—3622no—
155M16—3622no—
156M16—3622no—
157M16—1814no—
158C2×C8—3622no—
159C2×C8—3622no—
160C2×C8—3622no—
161C2×C8—3622no—
162C4⋊C4—21no—
163C4⋊C4—3621no—
164C4⋊C4—3621no—
165C4⋊C4—3621no—
166C4⋊C4—3621no—
167C22⋊C4—21no—
168C22⋊C4—3621no—
169C22⋊C4—3621no—
170C22⋊C4—3621no—
171C22⋊C4—1811no—
172C22⋊C4C2×#1062438211yescharacteristic
173C42—3621no—
174C23—1811no—
175C23C22⋊#1062438211yescharacteristic
176C23C22×#1062438211yescharacteristic
177Q8—3622no—
178Q8—7241no—
179Q8—7242no—
180Q8—7241no—
181D4—22no—
182D4—3622no—
183D4—3622no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—22no—
187C2×C4—3621no—
188C2×C4—3621no—
189C2×C4—3621no—
190C2×C4—3621no—
191C2×C4—3621no—
192C2×C4—3621no—
193C2×C4—3622no—
194C2×C4—1811no—
195C2×C4—1811no—
196C2×C4—1811no—
197C2×C4—1812no—
198C2×C4—1812no—
199C2×C4C22⋊#1062438211yescharacteristic
200C8—3624no—
201C8—7242no—
202C22—21no—
203C22—41no—
204C22—41no—
205C22—3621no—
206C22—7241no—
207C22C2×C22⋊#1062438211yescharacteristic
208C22D4×#1062438212yes—
209C4—21no—
210C4—41no—
211C4—41no—
212C4—3621no—
213C4—1811no—
214C4—1812no—
215C4—1811no—
216C4—7241no—
217C4—7241no—
218C2—21no—
219C2—22no—
220C2—41no—
221C2—7241no—
222C2#2604222711yescharacteristic
subgroupG/H
centerC2#26042227
commutatorC23×C181C22×C4
FrattiniC23C22×#10624382
Fitting#123×C181C4
socleC2×C181#386
prime radicalC2×#15668448C22
cosocleC22×Dic181C23
hypercenter#123#10624382
hypocenterC181#1915
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. C2×#19153877C2semidirect
4. #26042160C2semidirect
111. C181#1915semidirect

Maximal subgroups

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

subgroupindex
#260423572
#260422662
C2×#191538772
#260421602
#260418512
#1915181