Finite Group Databasebeta

← #37543085

Subgroups — #37543085

Order 23168.

698 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 698 conjugacy classes of subgroups — too many to draw. Only its 161 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#26042077C212yesmaximal, complemented
2#26041853C211yesmaximal, characteristic, complemented
3#26041462C214yesmaximal, complemented
4#26041452C214yesmaximal, complemented
5#26041296C212yesmaximal, complemented
6C2×#19153735C211yesmaximal, characteristic, complemented
7C2×2−1+4×C181C211yesmaximal, characteristic, complemented
8C2×#15668449C2211yescharacteristic
9#19153847—22no—
10#19153846C2214yescomplemented
11#19153842—22no—
12#19153841C2214yescomplemented
13#19153836—24no—
14C2×#15668433C2212yes—
15#19153814C2212yescomplemented
16#19153805C2212yescomplemented
17#19153799—22no—
18#19153792—22no—
19#19153775C2211yescharacteristic
20#19153743C2218yescomplemented
21#19153735C2214yescomplemented
222−1+4×C181—24no—
232−1+4×C181—22no—
242−1+4×C181C2214yes—
25C2×#41×C181—22no—
26C2×#41×C181—22no—
27C2×#41×C181C2212yes—
28C22×Q8×C181—22no—
29C22×Q8×C181C2211yescharacteristic
30#15668449—22no—
31#15668449—22no—
32C2×Dic362C2311yescharacteristic
33C2×D724C2311yescharacteristic
34C2×C22⋊D181—21no—
35C2×C4×D181—21no—
36#15668439—24no—
37#15668437—24no—
38#15668436—24no—
39#15668434—24no—
40#15668433—22no—
41#15668433C2314yes—
42C2×#13089678—22no—
43C4⋊Dic181—21no—
44#15668428—24no—
45C22⋊Dic181—21no—
46#15668426—24no—
47C4×Dic181C2312yes—
48#41×C181—28no—
49#41×C181—28no—
50#41×C181—28no—
51#41×C181—28no—
52#41×C181—22no—
53#41×C181—22no—
54#41×C181D414yes—
55#41×C181C2314yes—
56C2×Q8×C181—22no—
57C2×Q8×C181—28no—
58C2×Q8×C181—28no—
59C2×Q8×C181—22no—
60C2×Q8×C181—22no—
61C2×Q8×C181D412yes—
62C2×Q8×C181D412yes—
63C2×Q8×C181C2312yes—
64C2×D4×C181—22no—
65C2×D4×C181—22no—
66C2×D4×C181D412yes—
67C22×C4×C181—24no—
68C22×C4×C181—22no—
69C22×C4×C181D412yes—
70C22×C4×C181C2311yescharacteristic
71C22×D181—21no—
72Dic362—22no—
73D724—22no—
74C22⋊D181—41no—
75C22⋊D181—41no—
76C2×Dic181—22no—
77C2×Dic181—21no—
78C4×D181—42no—
79#13089678—24no—
80C23×C181—22no—
81C23×C181C2×D411yescharacteristic
82Q8×C181—28no—
83Q8×C181—22no—
84Q8×C181—24no—
85Q8×C181—24no—
86Q8×C181C2×D414yes—
87D4×C181—28no—
88D4×C181—24no—
89D4×C181—24no—
90D4×C181—22no—
91D4×C181C2×D414yes—
92C2×C4×C181—216no—
93C2×C4×C181—22no—
94C2×C4×C181—22no—
95C2×C4×C181—22no—
96C2×C4×C181—24no—
97C2×C4×C181—24no—
98C2×C4×C181C2×D414yes—
99C2×C4×C181C2×D411yescharacteristic
100C2×C4×C181C2×D412yes—
101C2×C4×C181C2×D412yes—
102C2×C4×C181C2411yescharacteristic
103C22×C181—24no—
104C22×C181—41no—
105C22×C181—44no—
106C22×C181#12011yescharacteristic
107C22×C181#12011yescharacteristic
108C22×C181C22×D411yescharacteristic
109C2×D181—42no—
110C2×D181—41no—
111C4×C181—24no—
112C4×C181—24no—
113C4×C181#12012yes—
114C4×C181C22×D412yes—
115Dic181—42no—
116Dic181—41no—
117C2×C181—21no—
118C2×C181—21no—
119C2×C181—21no—
120C2×C181—44no—
121C2×C181C2×#12011yescharacteristic
122D181—81no—
123C181#302011yescharacteristic, complemented, Sylow
124#3020—1811nomaximal, Sylow
125C2×2−1+4D18111yescharacteristic, complemented
126#575—1812no—
127#536—1811no—
128#456—1814no—
129#454—1814no—
130#421—1812no—
131C2×#101—1811no—
1322−1+4—24no—
1332−1+4—22no—
1342−1+4C2×D18114yes—
135C2×#41—22no—
136C2×#41—22no—
137C2×#41—1811no—
138C2×#41C2×D18112yes—
139C22×Q8—22no—
140C22×Q8C2×D18111yescharacteristic
141#137—3622no—
142#137—1814no—
143AGL(1,ℤ/8)—3622no—
144AGL(1,ℤ/8)—1814no—
145#135—3624no—
146C2×M16—1812no—
147C4⋊Q8—1812no—
148#124—1812no—
149#123—3622no—
150C22⋊Q8—3622no—
151#117—1811no—
152#104—1818no—
153#101—1814no—
154#41—28no—
155#41—28no—
156#41—28no—
157#41—28no—
158#41—22no—
159#41—22no—
160#41—3622no—
161#41—3622no—
162#41C22⋊D18114yes—
163#41C22×D18114yes—
164C2×Q8—22no—
165C2×Q8—28no—
166C2×Q8—28no—
167C2×Q8—22no—
168C2×Q8—22no—
169C2×Q8—1811no—
170C2×Q8C22⋊D18112yes—
171C2×Q8C22⋊D18112yes—
172C2×Q8C22×D18112yes—
173C2×D4—22no—
174C2×D4—22no—
175C2×D4—3621no—
176C2×D4—1811no—
177C2×D4C22⋊D18112yes—
178C22×C4—24no—
179C22×C4—22no—
180C22×C4—3621no—
181C22×C4C22⋊D18112yes—
182C22×C4C22×D18111yescharacteristic
183Q16—3624no—
184SD16—3624no—
185SD16—3624no—
186D8—3624no—
187M16—3622no—
188M16—1814no—
189C2×C8—3622no—
190C4⋊C4—3621no—
191C4⋊C4—3624no—
192C22⋊C4—3621no—
193C22⋊C4—3624no—
194C42—1812no—
195C23—22no—
196C23—3621no—
197C23C2×C22⋊D18111yescharacteristic
198Q8—28no—
199Q8—22no—
200Q8—24no—
201Q8—24no—
202Q8—3622no—
203Q8C2×C22⋊D18114yes—
204D4—28no—
205D4—24no—
206D4—24no—
207D4—22no—
208D4—3622no—
209D4—7241no—
210D4—7241no—
211D4C2×C22⋊D18114yes—
212C2×C4—216no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—24no—
217C2×C4—24no—
218C2×C4—3622no—
219C2×C4—3621no—
220C2×C4—7242no—
221C2×C4C2×C22⋊D18114yes—
222C2×C4C2×C22⋊D18111yescharacteristic
223C2×C4C2×C22⋊D18112yes—
224C2×C4C2×C22⋊D18112yes—
225C2×C4C23×D18111yescharacteristic
226C8—3624no—
227C22—24no—
228C22—41no—
229C22—44no—
230C22—7242no—
231C22—7241no—
232C22#1915378511yescharacteristic
233C22#1915378511yescharacteristic
234C22C22×C22⋊D18111yescharacteristic
235C4—24no—
236C4—24no—
237C4—7242no—
238C4—7241no—
239C4#1915378512yes—
240C4C22×C22⋊D18112yes—
241C2—21no—
242C2—21no—
243C2—21no—
244C2—44no—
245C2—14481no—
246C2C2×#1915378511yescharacteristic
subgroupG/H
centerC2C2×#19153785
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
FittingC2×2−1+4×C181C2
socleC2×C181C2×#120
hypercenterC2×2−1+4D181
hypocenterC181#3020
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×2−1+4

Complements

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

NH ≅ G/Ntype
1. #26042077C2semidirect
2. #26041853C2semidirect
3. #26041462C2semidirect
4. #26041452C2semidirect
5. #26041296C2semidirect
6. C2×#19153735C2semidirect
7. C2×2−1+4×C181C2semidirect
10. #19153846C22semidirect
12. #19153841C22semidirect
15. #19153814C22semidirect
16. #19153805C22semidirect
20. #19153743C22semidirect
21. #19153735C22semidirect
123. C181#3020semidirect
125. C2×2−1+4D181semidirect

Maximal subgroups

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

subgroupindex
#260420772
#260418532
#260414622
#260414522
#260412962
C2×#191537352
C2×2−1+4×C1812
#3020181