Finite Group Databasebeta

← #37549048

Subgroups — #37549048

Order 23168.

312 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 74 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#26042296C211yesmaximal, characteristic, complemented
2#26042282C211yesmaximal, characteristic, complemented
3#26042127C214yesmaximal, complemented
4C2×#19153778C211yesmaximal, characteristic
5C2×#15668458—21no—
6#19153883—21no—
7#19153883—21no—
8C4⋊#13089685C2212yescomplemented
9C22⋊#13089685—21no—
10C4×#13089685C2212yescomplemented
11C22×D724C411yescharacteristic
12C22×C4×D181—21no—
13#19153778—21no—
14#19153778—21no—
15#19153778C412yes—
16#19153778C2212yes—
17C2×#15668426—21no—
18C2×C4×Dic181C2211yescharacteristic
19C2×C4⋊C4×C181C411yescharacteristic
20#15668458—42no—
21C2×#13089685—22no—
22C2×#13089685—22no—
23C23×D181—21no—
24C2×D724—24no—
25C2×D724—21no—
26C2×D724C2×C412yes—
27C22×Dic181D411yescharacteristic
28C22×Dic181D411yescharacteristic
29C2×C4×D181—22no—
30C2×C4×D181—22no—
31C2×C4×D181—42no—
32C2×C4×D181—42no—
33#15668426—22no—
34#15668426—22no—
35C4×Dic181C2×C411yescharacteristic
36C4×Dic181D411yescharacteristic
37C4×Dic181D411yescharacteristic
38C4×Dic181C2311yescharacteristic
39C22×C4×C181—21no—
40C22×C4×C181C2×C411yescharacteristic
41C4⋊C4×C181—21no—
42C4⋊C4×C181C2×C412yes—
43#13089685—42no—
44#13089685—42no—
45C22×D181—24no—
46C22×D181—22no—
47C22×D181—42no—
48C22×D181—42no—
49D724—24no—
50D724—22no—
51D724M1614yescomplemented
52C2×Dic181—41no—
53C2×Dic181C22⋊C411yescharacteristic
54C2×Dic181C22⋊C411yescharacteristic
55C2×Dic181D812yes—
56C2×Dic181SD1612yes—
57C2×Dic181C2×D411yescharacteristic
58C2×Dic181C2×D411yescharacteristic
59C4×D181—44no—
60C4×D181—42no—
61C4×D181—42no—
62C23×C181C22⋊C411yescharacteristic
63C2×C4×C181—22no—
64C2×C4×C181—22no—
65C2×C4×C181—42no—
66C2×C4×C181C22⋊C411yescharacteristic
67C2×C4×C181C22×C411yescharacteristic
68C22×C181—21no—
69C22×C181—21no—
70C22×C181#10212yes—
71C22×C181C2×C22⋊C411yescharacteristic
72C2×D181—24no—
73C2×D181—48no—
74C2×D181—42no—
75C2×D181—42no—
76C4×C181—21no—
77C4×C181—42no—
78C4×C181C2×M1612yes—
79Dic181—21no—
80Dic181—21no—
81Dic181#10212yes—
82Dic181C2×D811yescharacteristic
83Dic181C2×SD1611yescharacteristic
84C2×C181—22no—
85C2×C181#40711yescharacteristic
86C2×C181C2×#10211yescharacteristic
87C2×C181#42111yescharacteristic
88D181—44no—
89C181#148611yescharacteristic, complemented, Sylow
90#1486—1811nomaximal, Sylow
91C2×C4×D4—1811no—
92#431—1811no—
93#423—1811no—
94D4⋊C8—1814no—
95C22×D4—1811no—
96C23×C4—3621no—
97C2×M16—3621no—
98C4×D4—3621no—
99C4×D4—3621no—
100C4×D4—1812no—
101C4×D4—1812no—
102C2×C4⋊C4#1062438211yescharacteristic
103C2×C22⋊C4—3621no—
104C2×C42—1811no—
105C4⋊C8—3621no—
106C4⋊C8—3621no—
107C4⋊C8—1812no—
108C22⋊C8—3621no—
109C4×C8—1812no—
110C24—3621no—
111C2×D4—3624no—
112C2×D4—3621no—
113C2×D4—1812no—
114C22×C4—21no—
115C22×C4—3622no—
116C22×C4—3622no—
117C22×C4—1811no—
118C22×C4—1811no—
119C22×C4—7242no—
120C22×C4—7242no—
121C22×C4C2×#1062438211yescharacteristic
122M16—7242no—
123C2×C8—3622no—
124C2×C8—3622no—
125C4⋊C4—21no—
126C4⋊C4C2×#1062438212yes—
127C22⋊C4—3622no—
128C22⋊C4—3622no—
129C42—1811no—
130C42—1811no—
131C42—1811no—
132C42—1811no—
133C23—3624no—
134C23—3622no—
135C23—7242no—
136C23—7242no—
137C23C22⋊#1062438211yescharacteristic
138D4—3624no—
139D4—3622no—
140D4—1814no—
141C2×C4—22no—
142C2×C4—22no—
143C2×C4—42no—
144C2×C4—1811no—
145C2×C4—1811no—
146C2×C4—1812no—
147C2×C4—1812no—
148C2×C4—1811no—
149C2×C4—1811no—
150C2×C4—7242no—
151C2×C4—7244no—
152C2×C4—7242no—
153C2×C4—7241no—
154C2×C4C22⋊#1062438211yescharacteristic
155C2×C4C22×#1062438211yescharacteristic
156C8—7242no—
157C8—7242no—
158C22—21no—
159C22—21no—
160C22—3624no—
161C22—7248no—
162C22—7242no—
163C22—7242no—
164C22#1915387812yes—
165C22C2×C22⋊#1062438211yescharacteristic
166C4—21no—
167C4—42no—
168C4—3621no—
169C4—3621no—
170C4—1811no—
171C4—1812no—
172C4—1811no—
173C4D181⋊M1612yes—
174C2—22no—
175C2—7244no—
176C2#2604225111yescharacteristic
177C2C2×#1915387811yescharacteristic
178C2#2604227811yescharacteristic
subgroupG/H
centerC22C2×C22⋊#10624382
commutatorC2×C4×C181C22×C4
FrattiniC2×C4C22×#10624382
FittingC2×C4⋊C4×C181C4
socleC22×C181C2×C22⋊C4
prime radicalC22×D724C4
cosocleC4×Dic181C23
hypercenterC2×C4⋊C4#10624382
hypocenterC181#1486
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. #26042296C2semidirect
2. #26042282C2semidirect
3. #26042127C2semidirect
8. C4⋊#13089685C22semidirect
10. C4×#13089685C22semidirect
51. D724M16semidirect
89. C181#1486semidirect

Maximal subgroups

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

subgroupindex
#260422962
#260422822
#260421272
C2×#191537782
#1486181