Finite Group Databasebeta

← #37542955

Subgroups — #37542955

Order 23168.

738 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 738 conjugacy classes of subgroups — too many to draw. Only its 173 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×#19153847C212yesmaximal, complemented
2#26041790C211yesmaximal, characteristic
3#26041435C214yesmaximal, complemented
4#26041418C214yesmaximal, complemented
5#26041288C212yesmaximal, complemented
6#26041243C211yesmaximal, characteristic, complemented
7C23×Q8×C181C211yesmaximal, characteristic, complemented
8C2×#15668449C2211yescharacteristic
9#19153847—28no—
10C2×#15668439C2214yescomplemented
11C2×#15668436C2214yescomplemented
12C2×#15668433C2212yes—
13#19153799—21no—
14#19153795C2212yes—
15#19153790C2212yes—
16#19153785—21no—
17C4×C22⋊D181—22no—
18#19153775C2211yescharacteristic
19#19153740C2218yescomplemented
20C22⋊#13089678C2214yes—
21C22×Q8×C181—28no—
22C22×Q8×C181—22no—
23C22×Q8×C181—21no—
24C22×Q8×C181C2214yes—
25C22×Q8×C181C2212yes—
26C23×C4×C181—21no—
27C23×C4×C181C2211yescharacteristic
28#15668449—24no—
29C2×Dic362C2311yescharacteristic
30C2×D724C2311yescharacteristic
31C2×C22⋊D181—22no—
32C2×C22⋊D181—21no—
33C2×C4×D181—21no—
34#15668439—28no—
35#15668436—28no—
36#15668433—24no—
37C2×#13089678C2314yes—
38C4⋊Dic181C2312yes—
39#15668428—22no—
40C22⋊Dic181—21no—
41C22⋊Dic181—22no—
42#15668426—22no—
43C4×Dic181—21no—
44C24×C181D411yescharacteristic
45C2×Q8×C181—216no—
46C2×Q8×C181—28no—
47C2×Q8×C181—216no—
48C2×Q8×C181—24no—
49C2×Q8×C181—24no—
50C2×Q8×C181—22no—
51C2×Q8×C181D418yescomplemented
52C2×Q8×C181C2314yes—
53C22×C4×C181—28no—
54C22×C4×C181—24no—
55C22×C4×C181—22no—
56C22×C4×C181—24no—
57C22×C4×C181D412yes—
58C22×C4×C181D411yescharacteristic
59C22×C4×C181C2312yes—
60C22×C4×C181C2311yescharacteristic
61C22×D181—21no—
62Dic362—22no—
63D724—22no—
64C22⋊D181—42no—
65C22⋊D181—44no—
66C2×Dic181—22no—
67C2×Dic181—21no—
68C4×D181—42no—
69#13089678—24no—
70C23×C181—22no—
71C23×C181—24no—
72C23×C181C2×D411yescharacteristic
73C23×C181C2×D412yes—
74Q8×C181—216no—
75Q8×C181—28no—
76Q8×C181—24no—
77Q8×C181C2×D418yes—
78C2×C4×C181—216no—
79C2×C4×C181—28no—
80C2×C4×C181—22no—
81C2×C4×C181—24no—
82C2×C4×C181—24no—
83C2×C4×C181—24no—
84C2×C4×C181C2×D412yes—
85C2×C4×C181C2×D411yescharacteristic
86C2×C4×C181C2×D414yes—
87C2×C4×C181C2411yescharacteristic
88C22×C181—28no—
89C22×C181—22no—
90C22×C181—22no—
91C22×C181—22no—
92C22×C181#12012yes—
93C22×C181#13714yescomplemented
94C22×C181C22×D411yescharacteristic
95C2×D181—41no—
96C2×D181—42no—
97C4×C181—28no—
98C4×C181—22no—
99C4×C181#12012yes—
100C4×C181C22×D412yes—
101Dic181—41no—
102Dic181—42no—
103C2×C181—22no—
104C2×C181—24no—
105C2×C181C2×#12011yescharacteristic
106C2×C181C2×#13712yes—
107D181—81no—
108C181#300411yescharacteristic, complemented, Sylow
109#3004—1811nomaximal, Sylow
110C23×Q8D18111yescharacteristic, complemented
111C2×#137—1812no—
112#525—1811no—
113#451—1814no—
114#448—1814no—
115#419—1812no—
116#407—1811no—
117C2×#41—1811no—
118C22×Q8—28no—
119C22×Q8—22no—
120C22×Q8—21no—
121C22×Q8C2×D18114yes—
122C22×Q8C2×D18112yes—
123C23×C4—21no—
124C23×C4C2×D18111yescharacteristic
125#137—3628no—
126C2×Q16—1814no—
127C2×SD16—1814no—
128C2×M16—1812no—
129#123—3621no—
130C22⋊Q8—1812no—
131#121—1812no—
132#120—3621no—
133C4×D4—3622no—
134#117—1811no—
135#103—1818no—
136C22⋊C8—1814no—
137C24C22⋊D18111yescharacteristic
138#41—3624no—
139C2×Q8—216no—
140C2×Q8—28no—
141C2×Q8—216no—
142C2×Q8—24no—
143C2×Q8—24no—
144C2×Q8—22no—
145C2×Q8—1811no—
146C2×Q8C22⋊D18118yescomplemented
147C2×Q8C22×D18114yes—
148C2×D4—3622no—
149C2×D4—3621no—
150C2×D4—1811no—
151C22×C4—28no—
152C22×C4—24no—
153C22×C4—22no—
154C22×C4—24no—
155C22×C4—3621no—
156C22×C4C22⋊D18112yes—
157C22×C4C22⋊D18111yescharacteristic
158C22×C4C22×D18112yes—
159C22×C4C22×D18111yescharacteristic
160Q16—3628no—
161SD16—3628no—
162M16—3624no—
163C2×C8—1814no—
164C4⋊C4—3622no—
165C4⋊C4—1812no—
166C22⋊C4—3621no—
167C22⋊C4—3622no—
168C22⋊C4—3622no—
169C42—3621no—
170C23—22no—
171C23—24no—
172C23—3621no—
173C23C2×C22⋊D18111yescharacteristic
174C23C2×C22⋊D18112yes—
175Q8—216no—
176Q8—28no—
177Q8—24no—
178Q8—3622no—
179Q8C2×C22⋊D18118yes—
180D4—3622no—
181D4—7242no—
182D4—7244no—
183C2×C4—216no—
184C2×C4—28no—
185C2×C4—22no—
186C2×C4—24no—
187C2×C4—24no—
188C2×C4—24no—
189C2×C4—3622no—
190C2×C4—3621no—
191C2×C4—7242no—
192C2×C4C2×C22⋊D18112yes—
193C2×C4C2×C22⋊D18111yescharacteristic
194C2×C4C2×C22⋊D18114yes—
195C2×C4C23×D18111yescharacteristic
196C8—3624no—
197C22—28no—
198C22—22no—
199C22—22no—
200C22—22no—
201C22—7241no—
202C22—7242no—
203C22#1915378512yes—
204C22#1915384714yescomplemented
205C22C22×C22⋊D18111yescharacteristic
206C4—28no—
207C4—22no—
208C4—7241no—
209C4—7242no—
210C4#1915378512yes—
211C4C22×C22⋊D18112yes—
212C2—22no—
213C2—24no—
214C2—14481no—
215C2C2×#1915378511yescharacteristic
216C2C2×#1915384712yes—
subgroupG/H
centerC22C22×C22⋊D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
FittingC23×Q8×C181C2
socleC22×C181C22×D4
prime radical#26041790C2
hypercenterC23×Q8D181
hypocenterC181#3004
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C23×C4 ◁ C23×Q8

Complements

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

NH ≅ G/Ntype
1. C2×#19153847C2semidirect
3. #26041435C2semidirect
4. #26041418C2semidirect
5. #26041288C2semidirect
6. #26041243C2semidirect
7. C23×Q8×C181C2semidirect
10. C2×#15668439C22semidirect
11. C2×#15668436C22semidirect
19. #19153740C22semidirect
51. C2×Q8×C181D4semidirect
93. C22×C181#137semidirect
108. C181#3004semidirect
110. C23×Q8D181semidirect
146. C2×Q8C22⋊D181semidirect
204. C22#19153847semidirect

Maximal subgroups

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

subgroupindex
C2×#191538472
#260417902
#260414352
#260414182
#260412882
#260412432
C23×Q8×C1812
#3004181