Finite Group Databasebeta

← #37532038

Subgroups — #37532038

Order 23168.

332 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 332 conjugacy classes of subgroups — too many to draw. Only its 81 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×#19153778C211yesmaximal, characteristic
2C22⋊C8×D181C212yesmaximal, complemented
3#26041005C211yesmaximal, characteristic, complemented
4#26041002C212yesmaximal, complemented
5#324×C181C211yesmaximal, characteristic, complemented
6C22×D724C411yescharacteristic
7C22×C4×D181C411yescharacteristic
8C22×C4×D181C2211yescharacteristic
9C2×C8×D181—22no—
10#19153778—22no—
11#19153778—22no—
12C2×#15668426C411yescharacteristic
13C2×#15668426C2211yescharacteristic
14C2×C4×Dic181C411yescharacteristic
15C22⋊#13089678C2212yes—
16#19153728—22no—
17C2×C4⋊C4×C181C2211yescharacteristic
18C22⋊C8×C181C2212yes—
19C23×D181C2×C411yescharacteristic
20C23×D181C2×C411yescharacteristic
21C2×D724—22no—
22C2×D724—22no—
23C2×D724C814yescomplemented
24C22×Dic181C2×C411yescharacteristic
25C22×Dic181C2×C411yescharacteristic
26C2×C4×D181—21no—
27C2×C4×D181—21no—
28C2×C4×D181—42no—
29C2×C4×D181—42no—
30C2×C4×D181D412yes—
31C2×C4×D181D412yes—
32C2×#13089678—22no—
33C8×D181—44no—
34#15668426—22no—
35#15668426—22no—
36C4×Dic181—21no—
37C4×Dic181—21no—
38C22×C4×C181C2×C411yescharacteristic
39C22×C4×C181C2×C411yescharacteristic
40C22×C4×C181C2311yescharacteristic
41C2×C8×C181—22no—
42C4⋊C4×C181—22no—
43C22×D181—21no—
44C22×D181—21no—
45C22×D181—41no—
46C22×D181—41no—
47C22×D181—41no—
48C22×D181—41no—
49C22×D181C22⋊C412yes—
50C22×D181C2×C812yes—
51C22×D181C2×C812yescomplemented
52C22×D181M1612yes—
53D724—28no—
54C2×Dic181—22no—
55C2×Dic181—21no—
56C2×Dic181—41no—
57C2×Dic181C22⋊C411yescharacteristic
58C2×Dic181C22⋊C411yescharacteristic
59C4×D181—24no—
60C4×D181—42no—
61C4×D181—42no—
62C4×D181—42no—
63#13089678—42no—
64C23×C181C22×C411yescharacteristic
65C2×C4×C181—21no—
66C2×C4×C181—21no—
67C2×C4×C181—21no—
68C2×C4×C181—41no—
69C2×C4×C181—41no—
70C2×C4×C181C2×C812yescomplemented
71C2×C4×C181C2×D412yes—
72C8×C181—42no—
73C22×C181—21no—
74C22×C181—21no—
75C22×C181C2×C22⋊C411yescharacteristic
76C22×C181C22×C811yescharacteristic
77C22×C181C2×M1611yescharacteristic
78C2×D181—22no—
79C2×D181—22no—
80C2×D181—22no—
81C2×D181—42no—
82C2×D181—42no—
83C2×D181—42no—
84C2×D181—41no—
85C2×D181—41no—
86C2×D181—42no—
87C2×D181C22⋊C814yes—
88C4×C181—22no—
89C4×C181—22no—
90C4×C181—41no—
91Dic181—21no—
92Dic181—41no—
93Dic181#9912yes—
94C2×C181—21no—
95C2×C181—21no—
96C2×C181C2×C22⋊C811yescharacteristic
97C2×C181C2×#9911yescharacteristic
98C2×C181#41311yescharacteristic
99D181—24no—
100D181—42no—
101C181#145911yescharacteristic, complemented, Sylow
102#1459—1811nomaximal, Sylow
103C2×C4×D4—1811no—
104C2×C22⋊C8—1812no—
105#324—1811no—
106#324D18111yescharacteristic, complemented
107#323—1812no—
108C22×D4—1811no—
109C23×C4—1811no—
110C23×C4—1811no—
111C22×C8—3622no—
112C4×D4—3622no—
113C4×D4—3622no—
114C2×C4⋊C4C2×D18111yescharacteristic
115C2×C22⋊C4—1811no—
116C2×C22⋊C4—1811no—
117C2×C42—1811no—
118C22⋊C8—3622no—
119C22⋊C8—1812no—
120C22⋊C8C2×D18112yes—
121C24—1811no—
122C24—1811no—
123C2×D4—3622no—
124C2×D4—3622no—
125C2×D4—1814no—
126C22×C4—3621no—
127C22×C4—3621no—
128C22×C4—1811no—
129C22×C4—1812no—
130C22×C4—1811no—
131C22×C4—1812no—
132C22×C4—7242no—
133C22×C4—7242no—
134C22×C4C4×D18111yescharacteristic
135C22×C4C4×D18111yescharacteristic
136C22×C4C22×D18111yescharacteristic
137C2×C8—22no—
138C2×C8—3622no—
139C2×C8—7244no—
140C4⋊C4—22no—
141C22⋊C4—3622no—
142C22⋊C4—3622no—
143C42—3621no—
144C42—3621no—
145C23—3621no—
146C23—3621no—
147C23—1812no—
148C23—1812no—
149C23—1812no—
150C23—1812no—
151C23—7241no—
152C23—7241no—
153C23—7241no—
154C23—7241no—
155C23C2×C4×D18111yescharacteristic
156D4—3628no—
157C2×C4—21no—
158C2×C4—21no—
159C2×C4—21no—
160C2×C4—41no—
161C2×C4—41no—
162C2×C4—3624no—
163C2×C4—3622no—
164C2×C4—3621no—
165C2×C4—1811no—
166C2×C4—1811no—
167C2×C4—7242no—
168C2×C4—7242no—
169C2×C4—7242no—
170C2×C4—7241no—
171C2×C4C8×D18112yescomplemented
172C2×C4D4×D18112yes—
173C8—42no—
174C8—7242no—
175C22—21no—
176C22—21no—
177C22—3622no—
178C22—3622no—
179C22—3622no—
180C22—1814no—
181C22—7242no—
182C22—7242no—
183C22—7242no—
184C22—7242no—
185C22—7241no—
186C22—7241no—
187C22C22⋊C4×D18111yescharacteristic
188C22C2×C8×D18111yescharacteristic
189C22M16×D18111yescharacteristic
190C4—22no—
191C4—22no—
192C4—41no—
193C4—3621no—
194C4—1812no—
195C4—7241no—
196C2—21no—
197C2—21no—
198C2—3624no—
199C2—7242no—
200C2C22⋊C8×D18111yescharacteristic
201C2#99×D18111yescharacteristic
202C2#2604126211yescharacteristic
subgroupG/H
centerC22C22⋊C4×D181
commutatorC22×C181C22×C8
FrattiniC22×C4C22×D181
Fitting#324×C181C2
socleC22×C181C2×C22⋊C4
prime radicalC22×D724C4
cosocleC22×C4×C181C23
hypercenter#324D181
hypocenterC181#1459
LCSG ▷ C22×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C22×C4 ◁ #324

Complements

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

NH ≅ G/Ntype
2. C22⋊C8×D181C2semidirect
3. #26041005C2semidirect
4. #26041002C2semidirect
5. #324×C181C2semidirect
23. C2×D724C8semidirect
51. C22×D181C2×C8semidirect
70. C2×C4×C181C2×C8semidirect
101. C181#1459semidirect
106. #324D181semidirect
171. C2×C4C8×D181semidirect

Maximal subgroups

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

subgroupindex
C2×#191537782
C22⋊C8×D1812
#260410052
#260410022
#324×C1812
#1459181