Finite Group Databasebeta

← #37534973

Subgroups — #37534973

Order 23168.

382 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 382 conjugacy classes of subgroups — too many to draw. Only its 67 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×#19153730C212yesmaximal, complemented
3#26041156C212yesmaximal, complemented
4#26041151C211yesmaximal, characteristic, complemented
5#342×C181C211yesmaximal, characteristic, complemented
6C22×C22⋊D181C2212yes—
7C23×Dic181C2211yescharacteristic
8#19153800—22no—
9#19153790—21no—
10#19153790—21no—
11#19153786—22no—
12C2×#15668428—21no—
13C2×C22⋊Dic181C2211yescharacteristic
14C2×#15668426—22no—
15#19153730—24no—
16#19153723—22no—
17#19153723—21no—
18C2×C22⋊C4×C181C2212yes—
19C2×C22⋊C4×C181C2211yescharacteristic
20C23×D181D412yes—
21C2×D724—21no—
22C2×D724—21no—
23C2×C22⋊D181—22no—
24C2×C22⋊D181—22no—
25C2×C22⋊D181—22no—
26C2×C22⋊D181—22no—
27C2×C22⋊D181—22no—
28C2×C22⋊D181—22no—
29C22×Dic181—22no—
30C22×Dic181—21no—
31C22×Dic181—21no—
32C22×Dic181—42no—
33C22×Dic181D412yes—
34C4⋊Dic181—21no—
35C4⋊Dic181—21no—
36#15668428—42no—
37C22⋊Dic181—21no—
38C22⋊Dic181D412yescomplemented
39#15668426—22no—
40#15668426—22no—
41#15668426—44no—
42C24×C181C2311yescharacteristic
43C22×C4×C181—22no—
44C22×C4×C181D411yescharacteristic
45C22×C4×C181D411yescharacteristic
46C22⋊C4×C181—22no—
47C22⋊C4×C181—22no—
48C22⋊C4×C181—22no—
49C22×D181—22no—
50C22×D181—22no—
51C22×D181—22no—
52C22×D181—42no—
53C22×D181—42no—
54D724—81no—
55D724—81no—
56C22⋊D181—42no—
57C22⋊D181—42no—
58C22⋊D181—42no—
59C22⋊D181—42no—
60C2×Dic181—24no—
61C2×Dic181—22no—
62C2×Dic181—22no—
63C2×Dic181—22no—
64C2×Dic181—22no—
65C2×Dic181—22no—
66C2×Dic181—41no—
67C2×Dic181—41no—
68C2×Dic181—41no—
69C2×Dic181—42no—
70C2×Dic181—41no—
71C23×C181—41no—
72C23×C181—41no—
73C23×C181C2×D412yes—
74C23×C181C2×D411yescharacteristic
75C23×C181C2×D411yescharacteristic
76C23×C181#4112yes—
77C23×C181#4111yescharacteristic
78C2×C4×C181—21no—
79C2×C4×C181—21no—
80C2×C4×C181—41no—
81C2×C4×C181—41no—
82C2×C4×C181—42no—
83C2×C4×C181—42no—
84C2×C4×C181—42no—
85C22×C181—21no—
86C22×C181—22no—
87C22×C181—21no—
88C22×C181—22no—
89C22×C181—41no—
90C22×C181—42no—
91C22×C181—41no—
92C22×C181#12011yescharacteristic
93C22×C181#12112yes—
94C22×C181#12111yescharacteristic
95C22×C181#12312yes—
96C22×C181#12411yescharacteristic
97C2×D181—44no—
98C2×D181—42no—
99C2×D181—42no—
100C2×D181—42no—
101C2×D181—82no—
102C4×C181—41no—
103C4×C181—41no—
104C4×C181—82no—
105Dic181—42no—
106Dic181—42no—
107Dic181—42no—
108C2×C181—21no—
109C2×C181—22no—
110C2×C181—22no—
111C2×C181—21no—
112C2×C181#39411yescharacteristic
113C2×C181B(4,𝔽2)11yescharacteristic
114C2×C181#45811yescharacteristic
115D181—82no—
116C181#204411yescharacteristic, complemented, Sylow
117#2044—1811nomaximal, Sylow
118#534—1811no—
119C2×#99—1812no—
120#386—1812no—
121#385—1811no—
122#342D18111yescharacteristic, complemented
123C22×D4—1812no—
124C23×C4—1811no—
125#123—3622no—
126#121—3621no—
127#121—3621no—
128#120—3622no—
129C2×C4⋊C4—3621no—
130C2×C22⋊C4—3622no—
131C2×C22⋊C4—1811no—
132C2×C22⋊C4C2×D18112yes—
133C2×C22⋊C4C2×D18111yescharacteristic
134#99—3624no—
135#95—3622no—
136#95—3621no—
137C24—1812no—
138C24C22×D18111yescharacteristic
139C2×D4—3621no—
140C2×D4—3621no—
141C2×D4—3622no—
142C2×D4—3622no—
143C2×D4—3622no—
144C2×D4—3622no—
145C2×D4—3622no—
146C2×D4—3622no—
147C22×C4—22no—
148C22×C4—3622no—
149C22×C4—3621no—
150C22×C4—3621no—
151C22×C4—1812no—
152C22×C4—7242no—
153C22×C4D72411yescharacteristic
154C22×C4D72411yescharacteristic
155C4⋊C4—3621no—
156C4⋊C4—3621no—
157C4⋊C4—7242no—
158C22⋊C4—22no—
159C22⋊C4—22no—
160C22⋊C4—22no—
161C22⋊C4—3622no—
162C22⋊C4—3622no—
163C22⋊C4—3621no—
164C22⋊C4—1812no—
165C22⋊C4—7244no—
166C23—41no—
167C23—41no—
168C23—3622no—
169C23—3622no—
170C23—3622no—
171C23—7242no—
172C23—7242no—
173C23D4×D18112yes—
174C23D4×D18111yescharacteristic
175C23C2×D72411yescharacteristic
176C23#1566844912yes—
177C23#1566845011yescharacteristic
178D4—14481no—
179D4—14481no—
180D4—7242no—
181D4—7242no—
182D4—7242no—
183D4—7242no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4—42no—
189C2×C4—42no—
190C2×C4—42no—
191C2×C4—3624no—
192C2×C4—3622no—
193C2×C4—3622no—
194C2×C4—3622no—
195C2×C4—3622no—
196C2×C4—3622no—
197C2×C4—7241no—
198C2×C4—7241no—
199C2×C4—7242no—
200C2×C4—7241no—
201C2×C4—7241no—
202C22—21no—
203C22—22no—
204C22—21no—
205C22—22no—
206C22—41no—
207C22—42no—
208C22—41no—
209C22—14482no—
210C22—7244no—
211C22—7242no—
212C22—7242no—
213C22—7242no—
214C22#1915378611yescharacteristic
215C22#1915378912yes—
216C22#1915379111yescharacteristic
217C22#1915380112yes—
218C22#1915380411yescharacteristic
219C4—41no—
220C4—41no—
221C4—82no—
222C4—7242no—
223C4—7242no—
224C4—7242no—
225C2—22no—
226C2—21no—
227C2—22no—
228C2—21no—
229C2—14482no—
230C2#2604119711yescharacteristic
231C2#2604146711yescharacteristic
232C2#2604147011yescharacteristic
subgroupG/H
centerC22#19153786
commutator, cosocleC24×C181C23
FrattiniC24C22×D181
Fitting#342×C181C2
socleC22×C181#120
prime radical#26041841C2
hypercenter#342D181
hypocenterC181#2044
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×#19153730C2semidirect
3. #26041156C2semidirect
4. #26041151C2semidirect
5. #342×C181C2semidirect
38. C22⋊Dic181D4semidirect
116. C181#2044semidirect
122. #342D181semidirect

Maximal subgroups

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

subgroupindex
#260418412
C2×#191537302
#260411562
#260411512
#342×C1812
#2044181