Finite Group Databasebeta

← #37549733

Subgroups — #37549733

Order 23168.

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

normal subgroups only

This group has 342 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, complemented
2#26042266C211yesmaximal, characteristic, complemented
3#26042251C212yesmaximal, complemented
4#26042160C212yesmaximal, complemented
5#26041787C211yesmaximal, characteristic
6#19153903—24no—
7D181⋊M16—21no—
8C2×#15668458—21no—
9C2×#15668458C2212yes—
10C2×#15668458C2212yes—
11#19153877—21no—
12#19153876—21no—
13#19153873—24no—
14C2×#15668448C2211yescharacteristic
15C22×C4×D181C2211yescharacteristic
16#19153799C411yescharacteristic
17#19153794—21no—
18#19153789—21no—
19#19153786C411yescharacteristic
20#19153784C411yescharacteristic
21#19153778—21no—
22#19153776—21no—
23#19153772C2211yescharacteristic
24#123×C181C411yescharacteristic
25#15668459—22no—
26#15668458—22no—
27#15668458—22no—
28#15668458—22no—
29#15668458—22no—
30#15668458—22no—
31D181⋊C8—22no—
32C2×#13089685—22no—
33C2×#13089685—22no—
34C2×#13089685D414yescomplemented
35C23×D181C2×C411yescharacteristic
36#15668448—22no—
37#15668448—22no—
38C2×Dic362D411yescharacteristic
39C2×D724—21no—
40C2×C22⋊D181—21no—
41C2×C22⋊D181C2×C411yescharacteristic
42C2×C22⋊D181D411yescharacteristic
43C22×Dic181D411yescharacteristic
44C22×Dic181C2311yescharacteristic
45C2×C4×D181—21no—
46C2×C4×D181—21no—
47C2×C4×D181—22no—
48C2×C4×D181—22no—
49C2×C4×D181D411yescharacteristic
50#15668428—21no—
51#15668428—21no—
52C22⋊Dic181C2×C411yescharacteristic
53#15668426—21no—
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×D181—21no—
65C22×D181—22no—
66C22×D181—42no—
67C22×D181C22⋊C411yescharacteristic
68Dic362—22no—
69D724—42no—
70C22⋊D181—22no—
71C22⋊D181—22no—
72C22⋊D181—42no—
73C2×Dic181—21no—
74C2×Dic181—21no—
75C2×Dic181—21no—
76C2×Dic181C22⋊C411yescharacteristic
77C2×Dic181C2×D412yes—
78C2×Dic181C2×D411yescharacteristic
79C2×Dic181C2×D411yescharacteristic
80C2×Dic181#4112yes—
81C4×D181—21no—
82C4×D181—21no—
83C4×D181—41no—
84C4×D181—42no—
85C4×D181—41no—
86C23×C181C22⋊C411yescharacteristic
87C23×C181C22×C411yescharacteristic
88D4×C181—22no—
89C2×C4×C181—21no—
90C2×C4×C181—21no—
91C2×C4×C181—22no—
92C2×C4×C181C22⋊C411yescharacteristic
93C22×C181—21no—
94C22×C181—41no—
95C22×C181—41no—
96C22×C181C2×C22⋊C411yescharacteristic
97C22×C181C4×D412yes—
98C2×D181—22no—
99C2×D181—24no—
100C2×D181—21no—
101C2×D181—41no—
102C2×D181—42no—
103C2×D181—41no—
104C2×D181—41no—
105C4×C181—21no—
106C4×C181—41no—
107C4×C181—41no—
108Dic181—21no—
109Dic181—41no—
110Dic181#12011yescharacteristic
111Dic181#12112yes—
112Dic181#12311yescharacteristic
113C2×C181—21no—
114C2×C181—22no—
115C2×C181—41no—
116C2×C181#38611yescharacteristic
117D181—42no—
118D181—41no—
119C181#189111yescharacteristic, complemented, Sylow
120#1891—1811nomaximal, Sylow
121#568—1811no—
122#525—1811no—
123#413—1811no—
124#407—1812no—
125#344—1812no—
126C2×#41—1811no—
127C23×C4—1811no—
128#131—3624no—
129C2×M16—3621no—
130C2×M16—3621no—
131C2×M16—1812no—
132C2×M16—1812no—
133#123—1811no—
134#123#1062438211yescharacteristic
135C22⋊Q8—3621no—
136#121—3621no—
137#120—1811no—
138#120—1811no—
139C4×D4—3621no—
140C4×D4—3621no—
141#117—1811no—
142#101—3621no—
143#100—3621no—
144C22⋊C8—3624no—
145C24—1811no—
146#41—3622no—
147#41—3622no—
148C2×Q8—1811no—
149C2×D4—3621no—
150C2×D4—3621no—
151C2×D4—1811no—
152C2×D4—1811no—
153C2×D4C2×#1062438211yescharacteristic
154C22×C4—3622no—
155C22×C4—3621no—
156C22×C4—3621no—
157C22×C4—3622no—
158C22×C4—1811no—
159C22×C4—1811no—
160C22×C4—1811no—
161C22×C4C2×#1062438211yescharacteristic
162M16—3622no—
163M16—3622no—
164M16—3622no—
165M16—3622no—
166M16—3622no—
167M16—3622no—
168C2×C8—3622no—
169C2×C8—3622no—
170C2×C8—3622no—
171C2×C8—1814no—
172C4⋊C4—21no—
173C4⋊C4—3621no—
174C4⋊C4—3621no—
175C22⋊C4—21no—
176C22⋊C4—3621no—
177C22⋊C4—3621no—
178C22⋊C4—3621no—
179C22⋊C4—1811no—
180C22⋊C4C2×#1062438211yescharacteristic
181C42—3621no—
182C23—3621no—
183C23—3622no—
184C23—1811no—
185C23—7242no—
186C23C22⋊#1062438211yescharacteristic
187C23C22×#1062438211yescharacteristic
188Q8—3622no—
189D4—22no—
190D4—3622no—
191D4—3622no—
192D4—7242no—
193D4—7242no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—22no—
197C2×C4—3621no—
198C2×C4—3621no—
199C2×C4—3621no—
200C2×C4—3621no—
201C2×C4—3621no—
202C2×C4—1811no—
203C2×C4—1811no—
204C2×C4—1811no—
205C2×C4—1812no—
206C2×C4—1812no—
207C2×C4—7241no—
208C2×C4—7242no—
209C2×C4—7241no—
210C2×C4C22⋊#1062438211yescharacteristic
211C8—3624no—
212C8—7242no—
213C22—21no—
214C22—41no—
215C22—41no—
216C22—3622no—
217C22—3624no—
218C22—3621no—
219C22—7241no—
220C22—7241no—
221C22—7242no—
222C22—7241no—
223C22C2×C22⋊#1062438211yescharacteristic
224C22D4×#1062438212yes—
225C4—21no—
226C4—41no—
227C4—41no—
228C4—3621no—
229C4—1811no—
230C4—1812no—
231C4—1811no—
232C4—7241no—
233C2—21no—
234C2—22no—
235C2—41no—
236C2—7242no—
237C2—7241no—
238C2#2604222711yescharacteristic
subgroupG/H
centerC2#26042227
commutatorC23×C181C22×C4
FrattiniC23C22×#10624382
Fitting#123×C181C4
socleC2×C181#386
prime radical#26041787C2
cosocleC22×Dic181C23
hypercenter#123#10624382
hypocenterC181#1891
LCSG ▷ C23×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C2 ◁ C23 ◁ #123

Complements

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

NH ≅ G/Ntype
1. #26042357C2semidirect
2. #26042266C2semidirect
3. #26042251C2semidirect
4. #26042160C2semidirect
34. C2×#13089685D4semidirect
119. C181#1891semidirect

Maximal subgroups

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

subgroupindex
#260423572
#260422662
#260422512
#260421602
#260417872
#1891181