Finite Group Databasebeta

← #37534853

Subgroups — #37534853

Order 23168.

430 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 430 conjugacy classes of subgroups — too many to draw. Only its 75 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×#19153800C212yesmaximal, complemented
3C2×#19153730C212yesmaximal, complemented
4#26041140C211yesmaximal, characteristic, complemented
5#342×C181C211yesmaximal, characteristic, complemented
6C22×C22⋊D181C2212yes—
7C23×Dic181C2211yescharacteristic
8#19153800—22no—
9#19153800—24no—
10#19153800—24no—
11#19153790—22no—
12#19153786—22no—
13C2×C4⋊Dic181—22no—
14C2×C22⋊Dic181C2211yescharacteristic
15C2×#15668426—22no—
16#19153730—24no—
17#19153723—22no—
18C2×C22⋊C4×C181C2211yescharacteristic
19C2×C22⋊C4×C181C2212yes—
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—21no—
29C22×Dic181—22no—
30C22×Dic181—22no—
31C22×Dic181—21no—
32C22×Dic181—21no—
33C22×Dic181D412yes—
34C22×Dic181D412yescomplemented
35C4⋊Dic181—28no—
36C4⋊Dic181—22no—
37C22⋊Dic181—22no—
38#15668426—28no—
39#15668426—24no—
40C24×C181C2311yescharacteristic
41C22×C4×C181—22no—
42C22×C4×C181D412yes—
43C22⋊C4×C181—22no—
44C22⋊C4×C181—22no—
45C22⋊C4×C181D414yescomplemented
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—22no—
60C2×Dic181—41no—
61C2×Dic181—42no—
62C2×Dic181—41no—
63C2×Dic181—42no—
64C2×Dic181—41no—
65C23×C181—42no—
66C23×C181C2×D412yes—
67C23×C181C2×D411yescharacteristic
68C23×C181C2×D411yescharacteristic
69C23×C181C2×D412yes—
70C23×C181#4111yescharacteristic
71C2×C4×C181—22no—
72C2×C4×C181—24no—
73C2×C4×C181—44no—
74C2×C4×C181—42no—
75C22×C181—22no—
76C22×C181—24no—
77C22×C181—41no—
78C22×C181—42no—
79C22×C181—41no—
80C22×C181#12012yes—
81C22×C181#12011yescharacteristic
82C22×C181#12111yescharacteristic
83C22×C181#12112yes—
84C22×C181#12711yescharacteristic
85C2×D181—44no—
86C2×D181—42no—
87C2×D181—44no—
88C2×D181—82no—
89C4×C181—44no—
90C4×C181—42no—
91Dic181—42no—
92Dic181—42no—
93Dic181—81no—
94C2×C181—21no—
95C2×C181—22no—
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#381—1811no—
107#342D18111yescharacteristic, complemented
108C22×D4—1812no—
109C23×C4—1811no—
110#123—3624no—
111#123—3622no—
112#123—3624no—
113#121—3622no—
114#120—3622no—
115C2×C4⋊C4—3622no—
116C2×C22⋊C4—3622no—
117C2×C22⋊C4—1811no—
118C2×C22⋊C4C2×D18111yescharacteristic
119C2×C22⋊C4C2×D18112yes—
120#99—3624no—
121#95—3622no—
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—3622no—
133C22×C4—3621no—
134C22×C4—3622no—
135C22×C4—3621no—
136C22×C4—3621no—
137C22×C4—1812no—
138C22×C4—1812no—
139C22×C4D72412yes—
140C4⋊C4—3628no—
141C4⋊C4—3622no—
142C22⋊C4—22no—
143C22⋊C4—22no—
144C22⋊C4—3628no—
145C22⋊C4—3624no—
146C22⋊C4—3622no—
147C22⋊C4D72414yescomplemented
148C23—42no—
149C23—3622no—
150C23—3622no—
151C23—3622no—
152C23—7244no—
153C23D4×D18112yes—
154C23D4×D18111yescharacteristic
155C23C2×D72411yescharacteristic
156C23C2×D72412yes—
157C23#1566845011yescharacteristic
158D4—14482no—
159D4—7242no—
160D4—7242no—
161D4—7242no—
162D4—7242no—
163C2×C4—24no—
164C2×C4—22no—
165C2×C4—44no—
166C2×C4—42no—
167C2×C4—3624no—
168C2×C4—3622no—
169C2×C4—3622no—
170C2×C4—3622no—
171C2×C4—3622no—
172C2×C4—7242no—
173C2×C4—7241no—
174C2×C4—7241no—
175C2×C4—7242no—
176C2×C4—7241no—
177C22—22no—
178C22—24no—
179C22—41no—
180C22—42no—
181C22—41no—
182C22—14482no—
183C22—7244no—
184C22—7242no—
185C22—7244no—
186C22#1915378612yes—
187C22#1915378611yescharacteristic
188C22#1915379111yescharacteristic
189C22#1915379112yes—
190C22#1915381311yescharacteristic
191C4—44no—
192C4—42no—
193C4—14481no—
194C4—7242no—
195C4—7242no—
196C2—22no—
197C2—21no—
198C2—22no—
199C2—21no—
200C2—14482no—
201C2#2604118911yescharacteristic
202C2#2604147012yes—
subgroupG/H
centerC22#19153786
commutator, cosocleC24×C181C23
FrattiniC24C22×D181
Fitting#342×C181C2
socleC22×C181#120
prime radical#26041841C2
hypercenter#342D181
hypocenterC181#2022
LCSG ▷ C24×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C24 ◁ #342

Complements

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

NH ≅ G/Ntype
2. C2×#19153800C2semidirect
3. C2×#19153730C2semidirect
4. #26041140C2semidirect
5. #342×C181C2semidirect
34. C22×Dic181D4semidirect
45. C22⋊C4×C181D4semidirect
101. C181#2022semidirect
107. #342D181semidirect
147. C22⋊C4D724semidirect

Maximal subgroups

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

subgroupindex
#260418412
C2×#191538002
C2×#191537302
#260411402
#342×C1812
#2022181