Finite Group Databasebeta

← #37534852

Subgroups — #37534852

Order 23168.

430 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 430 conjugacy classes of subgroups — too many to draw. Only its 73 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#26041841C211yesmaximal, characteristic
2C2×#19153799C212yesmaximal, complemented
3C2×#19153730C212yesmaximal, complemented
4#26041046C211yesmaximal, characteristic, complemented
5#381×C181C211yesmaximal, characteristic, complemented
6C22×C22⋊D181C2212yes—
7#19153800—22no—
8#19153799—24no—
9#19153799—24no—
10#19153790—22no—
11#19153786—22no—
12C2×#15668428—22no—
13C2×C22⋊Dic181C2212yes—
14C2×#15668426—22no—
15#19153730—24no—
16C23×C4×C181C2211yescharacteristic
17C2×C22⋊C4×C181C2211yescharacteristic
18C2×C22⋊C4×C181C2211yescharacteristic
19#95×C181—22no—
20C23×D181D412yes—
21C2×D724—22no—
22C2×C22⋊D181—22no—
23C2×C22⋊D181—22no—
24C2×C22⋊D181—22no—
25C2×C22⋊D181—22no—
26C2×C22⋊D181—22no—
27C2×C22⋊D181—22no—
28C22×Dic181—22no—
29C22×Dic181D412yes—
30C4⋊Dic181—22no—
31#15668428—28no—
32C22⋊Dic181—22no—
33C22⋊Dic181D414yescomplemented
34#15668426—28no—
35#15668426—24no—
36C24×C181C2311yescharacteristic
37C22×C4×C181—21no—
38C22×C4×C181—22no—
39C22×C4×C181—22no—
40C22×C4×C181—21no—
41C22×C4×C181—21no—
42C22×C4×C181D412yes—
43C22×C4×C181D412yescomplemented
44C22⋊C4×C181—22no—
45C22⋊C4×C181—22no—
46C22×D181—22no—
47C22×D181—22no—
48C22×D181—22no—
49C22×D181—44no—
50D724—82no—
51C22⋊D181—42no—
52C22⋊D181—42no—
53C22⋊D181—42no—
54C22⋊D181—42no—
55C2×Dic181—24no—
56C2×Dic181—22no—
57C2×Dic181—22no—
58C2×Dic181—22no—
59C2×Dic181—44no—
60C23×C181—42no—
61C23×C181C2×D412yes—
62C23×C181C2×D412yes—
63C23×C181C2×D411yescharacteristic
64C23×C181C2×D411yescharacteristic
65C23×C181#4111yescharacteristic
66C2×C4×C181—22no—
67C2×C4×C181—24no—
68C2×C4×C181—22no—
69C2×C4×C181—41no—
70C2×C4×C181—42no—
71C2×C4×C181—41no—
72C2×C4×C181—42no—
73C2×C4×C181—42no—
74C2×C4×C181—41no—
75C22×C181—22no—
76C22×C181—24no—
77C22×C181—42no—
78C22×C181—41no—
79C22×C181—41no—
80C22×C181#12011yescharacteristic
81C22×C181#12012yes—
82C22×C181#12112yes—
83C22×C181#12111yescharacteristic
84C22×C181#12711yescharacteristic
85C2×D181—44no—
86C2×D181—42no—
87C2×D181—44no—
88C2×D181—82no—
89C4×C181—42no—
90C4×C181—42no—
91C4×C181—81no—
92Dic181—44no—
93Dic181—42no—
94C2×C181—22no—
95C2×C181—21no—
96C2×C181—22no—
97C2×C181—21no—
98C2×C181#39211yescharacteristic
99C2×C181#45812yes—
100D181—82no—
101C181#202211yescharacteristic, complemented, Sylow
102#2022—1811nomaximal, Sylow
103#534—1811no—
104C2×#123—1812no—
105C2×#99—1812no—
106#381D18111yescharacteristic, complemented
107#342—1811no—
108C22×D4—1812no—
109C23×C4C2×D18111yescharacteristic
110#123—3622no—
111#123—3624no—
112#123—3624no—
113#121—3622no—
114#120—3622no—
115C2×C4⋊C4—3622no—
116C2×C22⋊C4—3622no—
117C2×C22⋊C4—1812no—
118C2×C22⋊C4C2×D18111yescharacteristic
119C2×C22⋊C4C2×D18111yescharacteristic
120#99—3624no—
121#95—22no—
122C24—1812no—
123C24C22×D18111yescharacteristic
124C2×D4—3622no—
125C2×D4—3622no—
126C2×D4—3622no—
127C2×D4—3622no—
128C2×D4—3622no—
129C2×D4—3622no—
130C2×D4—3622no—
131C22×C4—22no—
132C22×C4—22no—
133C22×C4—21no—
134C22×C4—21no—
135C22×C4—21no—
136C22×C4—3622no—
137C22×C4—1812no—
138C22×C4C22⋊D18112yes—
139C22×C4D72412yescomplemented
140C4⋊C4—3628no—
141C4⋊C4—3622no—
142C22⋊C4—22no—
143C22⋊C4—22no—
144C22⋊C4—3628no—
145C22⋊C4—3624no—
146C22⋊C4—3622no—
147C22⋊C4—1814no—
148C23—42no—
149C23—3622no—
150C23—3622no—
151C23—3622no—
152C23—7244no—
153C23D4×D18112yes—
154C23D4×D18112yes—
155C23C2×C22⋊D18111yescharacteristic
156C23C2×D72411yescharacteristic
157C23#1566844911yescharacteristic
158D4—14482no—
159D4—7242no—
160D4—7242no—
161D4—7242no—
162D4—7242no—
163C2×C4—22no—
164C2×C4—24no—
165C2×C4—22no—
166C2×C4—41no—
167C2×C4—42no—
168C2×C4—41no—
169C2×C4—42no—
170C2×C4—42no—
171C2×C4—41no—
172C2×C4—3622no—
173C2×C4—3624no—
174C2×C4—3622no—
175C2×C4—3622no—
176C2×C4—7244no—
177C22—22no—
178C22—24no—
179C22—42no—
180C22—41no—
181C22—41no—
182C22—14482no—
183C22—7244no—
184C22—7242no—
185C22—7244no—
186C22#1915378411yescharacteristic
187C22#1915378612yes—
188C22#1915378912yes—
189C22#1915379011yescharacteristic
190C22#1915381211yescharacteristic
191C4—42no—
192C4—42no—
193C4—81no—
194C4—7244no—
195C4—7242no—
196C2—21no—
197C2—22no—
198C2—22no—
199C2—21no—
200C2—14482no—
201C2#2604118811yescharacteristic
202C2#2604146812yes—
subgroupG/H
centerC22#19153784
commutator, cosocleC24×C181C23
FrattiniC24C22×D181
Fitting#381×C181C2
socleC22×C181#120
prime radical#26041841C2
hypercenter#381D181
hypocenterC181#2022
LCSG ▷ C24×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C24 ◁ #381

Complements

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

NH ≅ G/Ntype
2. C2×#19153799C2semidirect
3. C2×#19153730C2semidirect
4. #26041046C2semidirect
5. #381×C181C2semidirect
33. C22⋊Dic181D4semidirect
43. C22×C4×C181D4semidirect
101. C181#2022semidirect
106. #381D181semidirect
139. C22×C4D724semidirect

Maximal subgroups

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

subgroupindex
#260418412
C2×#191537992
C2×#191537302
#260410462
#381×C1812
#2022181