Finite Group Databasebeta

← #37549629

Subgroups — #37549629

Order 23168.

306 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 306 conjugacy classes of subgroups — too many to draw. Only its 68 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#26042250C211yesmaximal, characteristic
2C4×C22⋊#10624382C211yesmaximal, characteristic
3#26042149C212yesmaximal, complemented
4#26042149C212yesmaximal, complemented
5#117×D181C211yesmaximal, characteristic
6C2×#15668459C2212yes—
7C2×C22⋊#10624382—21no—
8C2×C4×#10624382C2212yes—
9#19153873—21no—
10C22⋊#13089685—21no—
11(C2×#10624382)⋊C4—21no—
12C22×C4×D181C2211yescharacteristic
13#19153775C411yescharacteristic, complemented
14#19153774C2211yescharacteristic
15#19153774C2211yescharacteristic
16#19153773—21no—
17#19153772—21no—
18C4⋊C4×D181C411yescharacteristic, complemented
19C4⋊C4×D181C411yescharacteristic, complemented
20C22⋊C4×D181—21no—
21C42×D181—21no—
22#117×C181C411yescharacteristic, complemented
23C22×#10624382—22no—
24#15668459—24no—
25C2×#13089685—22no—
26C22⋊#10624382—24no—
27C4×#10624382—24no—
28C23×D181D411yescharacteristic
29C22×Dic181D411yescharacteristic
30C2×C4×D181—24no—
31C2×C4×D181—21no—
32C2×C4×D181—21no—
33C2×C4×D181C2×C411yescharacteristic
34C2×C4×D181D411yescharacteristic
35C2×C4×D181D411yescharacteristic
36C2×C4×D181C2311yescharacteristic
37C4⋊Dic181C2×C411yescharacteristic
38C4⋊Dic181C2×C411yescharacteristic
39#15668428—21no—
40#15668428—21no—
41C22⋊Dic181—21no—
42#15668426—21no—
43#15668426—21no—
44C4×Dic181—21no—
45C4×Dic181—21no—
46C4×Dic181—21no—
47C22×C4×C181C2×C411yescharacteristic
48C4⋊C4×C181C2×C411yescharacteristic
49C4⋊C4×C181C2×C411yescharacteristic
50C22⋊C4×C181—21no—
51C42×C181—21no—
52C2×#10624382—24no—
53C2×#10624382—42no—
54C2×#10624382—42no—
55#13089685—42no—
56C22×D181—22no—
57C22×D181—22no—
58C22×D181—41no—
59C22×D181C22⋊C411yescharacteristic
60C22×D181C2×D411yescharacteristic
61C2×Dic181—22no—
62C2×Dic181—21no—
63C2×Dic181—21no—
64C2×Dic181C22⋊C411yescharacteristic
65C2×Dic181C2×D411yescharacteristic
66C4×D181—22no—
67C4×D181—22no—
68C4×D181—22no—
69C4×D181—42no—
70C4×D181—42no—
71C4×D181#4114yes—
72C23×C181C22⋊C411yescharacteristic
73C2×C4×C181—22no—
74C2×C4×C181—21no—
75C2×C4×C181—21no—
76C2×C4×C181C22⋊C411yescharacteristic
77C2×C4×C181C22×C411yescharacteristic
78C22×C181—21no—
79C22×C181—21no—
80C22×C181—21no—
81C22×C181C2×C22⋊C411yescharacteristic
82C2×D181—21no—
83C2×D181—21no—
84C2×D181—21no—
85C2×D181—21no—
86C2×D181—44no—
87C2×D181#10412yes—
88C2×D181#12312yes—
89#10624382—44no—
90C4×C181—21no—
91C4×C181—41no—
92C4×C181—41no—
93C4×C181#11712yes—
94Dic181—21no—
95Dic181—41no—
96Dic181—41no—
97Dic181#12312yes—
98C2×C181—41no—
99C2×C181#38111yescharacteristic
100C2×C181C2×#10411yescharacteristic
101C2×C181#42111yescharacteristic
102D181—22no—
103D181—22no—
104C181#182611yescharacteristic, complemented, Sylow
105#1826—1811nomaximal, Sylow
106C2×#117—1811no—
107#407—1811no—
108C4×C22⋊C4—1811no—
109#339—1812no—
110#339—1812no—
111C23×C4—1811no—
112C2×M16—1812no—
113#117—3621no—
114#117—3621no—
115#117—1811no—
116#117—1811no—
117#117—1811no—
118#117#1062438211yescharacteristic, complemented
119C2×C4⋊C4—1811no—
120C2×C4⋊C4—1811no—
121C2×C22⋊C4—3621no—
122C2×C22⋊C4—3621no—
123C2×C42—3621no—
124C2×C42—1812no—
125C22⋊C8—3621no—
126C22⋊C8—3621no—
127#95—3621no—
128C24—1811no—
129C22×C4—3624no—
130C22×C4—3621no—
131C22×C4—3621no—
132C22×C4—3622no—
133C22×C4—1811no—
134C22×C4—1811no—
135C22×C4—1811no—
136C22×C4—1811no—
137C22×C4—1811no—
138C22×C4C2×#1062438211yescharacteristic
139M16—3624no—
140C2×C8—3622no—
141C4⋊C4—3621no—
142C4⋊C4—3621no—
143C4⋊C4—1811no—
144C4⋊C4—1811no—
145C4⋊C4C2×#1062438211yescharacteristic
146C4⋊C4C2×#1062438211yescharacteristic
147C22⋊C4—21no—
148C22⋊C4—3621no—
149C22⋊C4—3621no—
150C22⋊C4—3624no—
151C22⋊C4—3621no—
152C42—21no—
153C42—3621no—
154C42—3621no—
155C42—3621no—
156C42—3624no—
157C23—3622no—
158C23—3622no—
159C23—1811no—
160C23—1811no—
161C23—7241no—
162C23C22⋊#1062438211yescharacteristic
163C2×C4—22no—
164C2×C4—21no—
165C2×C4—21no—
166C2×C4—3622no—
167C2×C4—3622no—
168C2×C4—3622no—
169C2×C4—3622no—
170C2×C4—3624no—
171C2×C4—3621no—
172C2×C4—3621no—
173C2×C4—1811no—
174C2×C4—1811no—
175C2×C4—1814no—
176C2×C4—7242no—
177C2×C4—7242no—
178C2×C4—7242no—
179C2×C4—7242no—
180C2×C4C22⋊#1062438211yescharacteristic
181C2×C4C22×#1062438211yescharacteristic
182C8—7242no—
183C22—21no—
184C22—21no—
185C22—21no—
186C22—3621no—
187C22—3621no—
188C22—3621no—
189C22—3621no—
190C22—1812no—
191C22—1812no—
192C22—7244no—
193C22C2×C22⋊#1062438211yescharacteristic
194C4—21no—
195C4—41no—
196C4—41no—
197C4—3621no—
198C4—1812no—
199C4—7244no—
200C4—7241no—
201C4—7241no—
202C4#1915389512yes—
203C2—41no—
204C2—3622no—
205C2—3622no—
206C2#2604222211yescharacteristic
207C2D181⋊#10411yescharacteristic
208C2#2604227611yescharacteristic
subgroupG/H
centerC22C2×C22⋊#10624382
commutatorC2×C4×C181C22×C4
FrattiniC2×C4C22×#10624382
Fitting#117×C181C4
socleC22×C181C2×C22⋊C4
prime radicalC23×D181D4
cosocleC2×C4×D181C23
hypercenter#117#10624382
hypocenterC181#1826
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C22×C4 ◁ #117

Complements

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

NH ≅ G/Ntype
3. #26042149C2semidirect
4. #26042149C2semidirect
13. #19153775C4semidirect
18. C4⋊C4×D181C4semidirect
19. C4⋊C4×D181C4semidirect
22. #117×C181C4semidirect
104. C181#1826semidirect
118. #117#10624382semidirect

Maximal subgroups

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

subgroupindex
#260422502
C4×C22⋊#106243822
#260421492
#117×D1812
#1826181