Finite Group Databasebeta

← #37534969

Subgroups — #37534969

Order 23168.

382 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 382 conjugacy classes of subgroups — too many to draw. Only its 71 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×#19153731C212yesmaximal, complemented
2#26041159C212yesmaximal, complemented
3#26041151C211yesmaximal, characteristic, complemented
4#26041046C211yesmaximal, characteristic, complemented
5#534×C181C211yesmaximal, characteristic
6C23×Dic181C2211yescharacteristic
7C2×#15668428—21no—
8C2×C22⋊Dic181—22no—
9C2×C22⋊Dic181C2212yes—
10C2×C22⋊Dic181C2211yescharacteristic
11#19153731—24no—
12#19153723—21no—
13#19153723—22no—
14C22×D4×C181C2212yes—
15#123×C181—22no—
16#121×C181—21no—
17#121×C181—21no—
18#120×C181—22no—
19C2×C22⋊C4×C181C2211yescharacteristic
20C22×Dic181—22no—
21C22×Dic181—21no—
22C22×Dic181—21no—
23C22×Dic181—22no—
24C22×Dic181—42no—
25#15668428—42no—
26C22⋊Dic181—22no—
27C22⋊Dic181—21no—
28C22⋊Dic181—22no—
29C22⋊Dic181—44no—
30C22⋊Dic181D412yescomplemented
31C24×C181D412yes—
32C24×C181C2311yescharacteristic
33C2×D4×C181—22no—
34C2×D4×C181—22no—
35C2×D4×C181—22no—
36C2×D4×C181—22no—
37C2×D4×C181—22no—
38C2×D4×C181—22no—
39C2×D4×C181—21no—
40C2×D4×C181—21no—
41C22×C4×C181D412yes—
42C22×C4×C181D411yescharacteristic
43C22×C4×C181D411yescharacteristic
44C4⋊C4×C181—21no—
45C4⋊C4×C181—21no—
46C22⋊C4×C181—22no—
47C22⋊C4×C181—22no—
48C22⋊C4×C181—22no—
49C2×Dic181—24no—
50C2×Dic181—22no—
51C2×Dic181—22no—
52C2×Dic181—41no—
53C2×Dic181—41no—
54C2×Dic181—41no—
55C2×Dic181—42no—
56C2×Dic181—41no—
57C2×Dic181—42no—
58C2×Dic181—42no—
59C2×Dic181—42no—
60C23×C181—22no—
61C23×C181—22no—
62C23×C181—22no—
63C23×C181—41no—
64C23×C181—41no—
65C23×C181—42no—
66C23×C181—42no—
67C23×C181C2×D411yescharacteristic
68C23×C181C2×D412yes—
69C23×C181C2×D411yescharacteristic
70C23×C181#4111yescharacteristic
71C23×C181#4112yes—
72D4×C181—42no—
73D4×C181—42no—
74D4×C181—42no—
75D4×C181—42no—
76D4×C181—81no—
77D4×C181—81no—
78C2×C4×C181—21no—
79C2×C4×C181—21no—
80C2×C4×C181—22no—
81C2×C4×C181—22no—
82C2×C4×C181—22no—
83C2×C4×C181—41no—
84C2×C4×C181—41no—
85C22×C181—21no—
86C22×C181—22no—
87C22×C181—21no—
88C22×C181—22no—
89C22×C181—41no—
90C22×C181—41no—
91C22×C181—42no—
92C22×C181—44no—
93C22×C181—42no—
94C22×C181—42no—
95C22×C181—42no—
96C22×C181—82no—
97C22×C181#12011yescharacteristic
98C22×C181#12111yescharacteristic
99C22×C181#12112yes—
100C22×C181#12312yes—
101C22×C181#12411yescharacteristic
102C4×C181—42no—
103C4×C181—41no—
104C4×C181—41no—
105Dic181—42no—
106Dic181—42no—
107Dic181—82no—
108C2×C181—21no—
109C2×C181—22no—
110C2×C181—22no—
111C2×C181—21no—
112C2×C181—82no—
113C2×C181#39411yescharacteristic
114C2×C181B(4,𝔽2)11yescharacteristic
115C2×C181#45811yescharacteristic
116C181#204411yescharacteristic, complemented, Sylow
117#2044—1811nomaximal, Sylow
118#534D18111yescharacteristic
119C2×#99—1812no—
120#386—1812no—
121#385—1811no—
122#342—1811no—
123C22×D4C2×D18112yes—
124C23×C4—1811no—
125#123—22no—
126#121—21no—
127#121—21no—
128#120—22no—
129C2×C4⋊C4—3621no—
130C2×C22⋊C4—3622no—
131C2×C22⋊C4—1812no—
132C2×C22⋊C4—1811no—
133C2×C22⋊C4C2×D18111yescharacteristic
134#99—3624no—
135#95—3621no—
136#95—3622no—
137C24C22⋊D18112yes—
138C24C22×D18111yescharacteristic
139C2×D4—22no—
140C2×D4—22no—
141C2×D4—22no—
142C2×D4—22no—
143C2×D4—22no—
144C2×D4—22no—
145C2×D4—21no—
146C2×D4—21no—
147C22×C4—3622no—
148C22×C4—3621no—
149C22×C4—3621no—
150C22×C4—3622no—
151C22×C4—7242no—
152C22×C4C22⋊D18112yes—
153C22×C4C22⋊D18111yescharacteristic
154C22×C4C22⋊D18111yescharacteristic
155C4⋊C4—21no—
156C4⋊C4—21no—
157C4⋊C4—7242no—
158C22⋊C4—22no—
159C22⋊C4—22no—
160C22⋊C4—22no—
161C22⋊C4—3621no—
162C22⋊C4—3622no—
163C22⋊C4—3622no—
164C22⋊C4—1812no—
165C22⋊C4—7244no—
166C23—22no—
167C23—22no—
168C23—22no—
169C23—41no—
170C23—41no—
171C23—42no—
172C23—42no—
173C23D4×D18111yescharacteristic
174C23C2×C22⋊D18111yescharacteristic
175C23C2×C22⋊D18112yes—
176C23#1566844811yescharacteristic
177C23#1566844812yes—
178D4—42no—
179D4—42no—
180D4—42no—
181D4—42no—
182D4—81no—
183D4—81no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—41no—
190C2×C4—41no—
191C2×C4—3624no—
192C2×C4—3622no—
193C2×C4—3622no—
194C2×C4—7242no—
195C2×C4—7242no—
196C2×C4—7242no—
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—44no—
210C22—42no—
211C22—42no—
212C22—42no—
213C22—82no—
214C22#1915378511yescharacteristic
215C22#1915378711yescharacteristic
216C22#1915378812yes—
217C22#1915379712yes—
218C22#1915380311yescharacteristic
219C4—42no—
220C4—41no—
221C4—41no—
222C4—14482no—
223C4—7242no—
224C4—7242no—
225C2—22no—
226C2—21no—
227C2—22no—
228C2—21no—
229C2—82no—
230C2#2604119311yescharacteristic
231C2#2604146611yescharacteristic
232C2#2604146911yescharacteristic
subgroupG/H
centerC22#19153785
commutator, cosocleC24×C181C23
FrattiniC24C22×D181
Fitting, prime radical#534×C181C2
socleC22×C181#120
hypercenter#534D181
hypocenterC181#2044
LCSG ▷ C24×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C24 ◁ #534

Complements

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

NH ≅ G/Ntype
1. C2×#19153731C2semidirect
2. #26041159C2semidirect
3. #26041151C2semidirect
4. #26041046C2semidirect
30. C22⋊Dic181D4semidirect
116. C181#2044semidirect

Maximal subgroups

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

subgroupindex
C2×#191537312
#260411592
#260411512
#260410462
#534×C1812
#2044181