Finite Group Databasebeta

← #37532069

Subgroups — #37532069

Order 23168.

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

normal subgroups only

This group has 280 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
1#26041243C211yesmaximal, characteristic, complemented
2C2×C22⋊#13089678C211yesmaximal, characteristic, complemented
3#26041005C212yesmaximal, complemented
4#26041003C212yesmaximal, complemented
5C2×#123×C181C211yesmaximal, characteristic
6C2×#15668433—21no—
7C22×#13089678—21no—
8C22⋊#13089678—21no—
9C22⋊#13089678—21no—
10C22⋊#13089678C2212yes—
11C22⋊#13089678C2212yes—
12C22×D4×C181C411yescharacteristic
13C23×C4×C181C2211yescharacteristic
14#123×C181—21no—
15#123×C181—21no—
16#123×C181—21no—
17#123×C181—21no—
18C2×C4⋊C4×C181C411yescharacteristic
19C2×C4⋊C4×C181C2211yescharacteristic
20C2×C22⋊C4×C181C411yescharacteristic
21C2×C22⋊C4×C181C411yescharacteristic
22C2×C22⋊C4×C181C2211yescharacteristic
23#15668433—42no—
24C2×#13089678—22no—
25C2×#13089678—22no—
26C2×#13089678—42no—
27C24×C181C2×C411yescharacteristic
28C24×C181C2×C411yescharacteristic
29C2×D4×C181—21no—
30C2×D4×C181—21no—
31C2×D4×C181—42no—
32C22×C4×C181—42no—
33C22×C4×C181C2×C411yescharacteristic
34C22×C4×C181C2×C411yescharacteristic
35C22×C4×C181C2×C411yescharacteristic
36C22×C4×C181C2×C411yescharacteristic
37C22×C4×C181D411yescharacteristic
38C22×C4×C181D411yescharacteristic
39C22×C4×C181D411yescharacteristic
40C22×C4×C181D411yescharacteristic
41C22×C4×C181C2311yescharacteristic
42C4⋊C4×C181—21no—
43C4⋊C4×C181—21no—
44C4⋊C4×C181—21no—
45C4⋊C4×C181—21no—
46C22⋊C4×C181—21no—
47C22⋊C4×C181—21no—
48C22⋊C4×C181—21no—
49C22⋊C4×C181—21no—
50C22⋊C4×C181—21no—
51C22⋊C4×C181—21no—
52#13089678—42no—
53#13089678—42no—
54C23×C181—22no—
55C23×C181—22no—
56C23×C181—21no—
57C23×C181—22no—
58C23×C181—41no—
59C23×C181—41no—
60C23×C181—41no—
61C23×C181C22⋊C411yescharacteristic
62C23×C181C22⋊C411yescharacteristic
63C23×C181C22×C411yescharacteristic
64D4×C181—42no—
65D4×C181—42no—
66C2×C4×C181—22no—
67C2×C4×C181—22no—
68C2×C4×C181—21no—
69C2×C4×C181—21no—
70C2×C4×C181—21no—
71C2×C4×C181—22no—
72C2×C4×C181—41no—
73C2×C4×C181—41no—
74C2×C4×C181—41no—
75C2×C4×C181—41no—
76C2×C4×C181—41no—
77C2×C4×C181—41no—
78C2×C4×C181C22⋊C411yescharacteristic
79C2×C4×C181C22⋊C411yescharacteristic
80C2×C4×C181C2×D411yescharacteristic
81C2×C4×C181C2×D411yescharacteristic
82C22×C181—21no—
83C22×C181—21no—
84C22×C181—21no—
85C22×C181—21no—
86C22×C181—21no—
87C22×C181—21no—
88C22×C181—21no—
89C22×C181—44no—
90C22×C181—41no—
91C22×C181—41no—
92C22×C181—41no—
93C22×C181—82no—
94C22×C181#9912yes—
95C22×C181C2×C22⋊C411yescharacteristic
96C22×C181#13112yes—
97C4×C181—21no—
98C4×C181—21no—
99C4×C181—21no—
100C4×C181—21no—
101C4×C181—41no—
102C4×C181—41no—
103C4×C181—41no—
104C2×C181—22no—
105C2×C181—22no—
106C2×C181—41no—
107C2×C181—81no—
108C2×C181#40811yescharacteristic
109C2×C181C2×#9911yescharacteristic
110C2×C181#41311yescharacteristic
111C181#146811yescharacteristic, complemented, Sylow
112#1468—1811nomaximal, Sylow
113C2×#123D18111yescharacteristic
114#407—1811no—
115C2×C22⋊C8—1811no—
116#324—1812no—
117#323—1812no—
118C22×D4Dic18111yescharacteristic
119C23×C4C2×D18111yescharacteristic
120C2×M16—3621no—
121C22×C8—3621no—
122#123—21no—
123#123—21no—
124#123—21no—
125#123—21no—
126C2×C4⋊C4Dic18111yescharacteristic
127C2×C4⋊C4C2×D18111yescharacteristic
128C2×C22⋊C4Dic18111yescharacteristic
129C2×C22⋊C4Dic18111yescharacteristic
130C2×C22⋊C4C2×D18111yescharacteristic
131C22⋊C8—3621no—
132C22⋊C8—3621no—
133C22⋊C8—1812no—
134C22⋊C8—1812no—
135C24C2×Dic18111yescharacteristic
136C24C2×Dic18111yescharacteristic
137C2×D4—21no—
138C2×D4—21no—
139C2×D4—42no—
140C22×C4—42no—
141C22×C4C2×Dic18111yescharacteristic
142C22×C4C2×Dic18111yescharacteristic
143C22×C4C2×Dic18111yescharacteristic
144C22×C4C2×Dic18111yescharacteristic
145C22×C4C22⋊D18111yescharacteristic
146C22×C4C22⋊D18111yescharacteristic
147C22×C4C22⋊D18111yescharacteristic
148C22×C4C22⋊D18111yescharacteristic
149C22×C4C22×D18111yescharacteristic
150M16—7242no—
151C2×C8—3622no—
152C2×C8—3622no—
153C2×C8—7242no—
154C4⋊C4—21no—
155C4⋊C4—21no—
156C4⋊C4—21no—
157C4⋊C4—21no—
158C22⋊C4—21no—
159C22⋊C4—21no—
160C22⋊C4—21no—
161C22⋊C4—21no—
162C22⋊C4—21no—
163C22⋊C4—21no—
164C23—22no—
165C23—22no—
166C23—21no—
167C23—22no—
168C23—41no—
169C23—41no—
170C23—41no—
171C23C22⋊Dic18111yescharacteristic
172C23C22⋊Dic18111yescharacteristic
173C23C22×Dic18111yescharacteristic
174D4—42no—
175D4—42no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—22no—
182C2×C4—41no—
183C2×C4—41no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4C22⋊Dic18111yescharacteristic
189C2×C4C22⋊Dic18111yescharacteristic
190C2×C4C2×C22⋊D18111yescharacteristic
191C2×C4C2×C22⋊D18111yescharacteristic
192C8—7242no—
193C8—7242no—
194C22—21no—
195C22—21no—
196C22—21no—
197C22—21no—
198C22—21no—
199C22—21no—
200C22—21no—
201C22—44no—
202C22—41no—
203C22—41no—
204C22—41no—
205C22—82no—
206C22#1915373112yes—
207C22C2×C22⋊Dic18111yescharacteristic
208C22#1915382312yes—
209C4—21no—
210C4—21no—
211C4—21no—
212C4—21no—
213C4—41no—
214C4—41no—
215C4—41no—
216C2—22no—
217C2—22no—
218C2—41no—
219C2—81no—
220C2#2604124611yescharacteristic
221C2C2×#1915373111yescharacteristic
222C2#2604126311yescharacteristic
subgroupG/H
centerC22C2×C22⋊Dic181
commutatorC23×C181C22×C4
FrattiniC22×C4C22×D181
FittingC2×#123×C181C2
socleC22×C181C2×C22⋊C4
prime radicalC22×D4×C181C4
cosocleC22×C4×C181C23
hypercenterC2×#123D181
hypocenterC181#1468
LCSG ▷ C23×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#123

Complements

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

NH ≅ G/Ntype
1. #26041243C2semidirect
2. C2×C22⋊#13089678C2semidirect
3. #26041005C2semidirect
4. #26041003C2semidirect
111. C181#1468semidirect

Maximal subgroups

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

subgroupindex
#260412432
C2×C22⋊#130896782
#260410052
#260410032
C2×#123×C1812
#1468181