Finite Group Databasebeta

← #37549865

Subgroups — #37549865

Order 23168.

312 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 70 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#26042282C211yesmaximal, characteristic, complemented
2C2×#19153877C212yesmaximal, complemented
3C2×#19153876C212yesmaximal
4C4×#15668458C211yesmaximal, characteristic, complemented
5C2×#19153802C211yesmaximal, characteristic
6C2×#15668458C2212yes—
7C2×#15668458C2212yes—
8#19153883—21no—
9C4⋊#13089685—21no—
10#19153877—24no—
11#19153876—24no—
12#19153871—21no—
13C4×#13089685—21no—
14C22×Dic362C2211yescharacteristic
15C22×C22⋊D181C2211yescharacteristic
16#19153802—21no—
17#19153802—21no—
18#19153802—21no—
19#19153802—21no—
20C2×C22⋊Dic181C411yescharacteristic
21C2×#15668426C411yescharacteristic
22C2×#15668426C411yescharacteristic
23C2×C4×Dic181C2211yescharacteristic
24C2×C22⋊C4×C181C411yescharacteristic
25#15668458—22no—
26#15668458—22no—
27#15668458—22no—
28#15668458D414yes—
29C2×#13089685—22no—
30C2×#13089685—22no—
31C23×D181C2×C411yescharacteristic
32C2×Dic362—21no—
33C2×Dic362—21no—
34C2×Dic362—22no—
35C2×Dic362—22no—
36C2×C22⋊D181—21no—
37C2×C22⋊D181—21no—
38C2×C22⋊D181—22no—
39C2×C22⋊D181—22no—
40C22×Dic181C2×C411yescharacteristic
41C22×Dic181C2×C411yescharacteristic
42C22×Dic181C2311yescharacteristic
43C22⋊Dic181—21no—
44C22⋊Dic181—21no—
45#15668426—21no—
46#15668426—21no—
47#15668426—21no—
48#15668426—21no—
49C4×Dic181D411yescharacteristic
50C4×Dic181D411yescharacteristic
51C4×Dic181D411yescharacteristic
52C4×Dic181D411yescharacteristic
53C24×C181C2×C411yescharacteristic
54C22×C4×C181C2×C411yescharacteristic
55C22×C4×C181C2×C411yescharacteristic
56C22⋊C4×C181—21no—
57C22⋊C4×C181—21no—
58#13089685—24no—
59#13089685—42no—
60C22×D181—21no—
61C22×D181—22no—
62C22×D181—41no—
63C22×D181—41no—
64Dic362—42no—
65Dic362—41no—
66Dic362—41no—
67C22⋊D181—42no—
68C22⋊D181—41no—
69C22⋊D181—41no—
70C2×Dic181—21no—
71C2×Dic181—22no—
72C2×Dic181—41no—
73C2×Dic181C22⋊C411yescharacteristic
74C2×Dic181C22⋊C411yescharacteristic
75C2×Dic181C2×D411yescharacteristic
76C2×Dic181C2×D412yes—
77C2×Dic181C2×D411yescharacteristic
78C2×Dic181#4112yes—
79C23×C181—21no—
80C23×C181—22no—
81C23×C181—41no—
82C23×C181—41no—
83C23×C181C22×C411yescharacteristic
84C2×C4×C181—21no—
85C2×C4×C181—22no—
86C2×C4×C181—41no—
87C2×C4×C181C22⋊C411yescharacteristic
88C2×C4×C181C22⋊C411yescharacteristic
89C22×C181—21no—
90C22×C181—21no—
91C22×C181—41no—
92C22×C181—41no—
93C22×C181—41no—
94C22×C181—82no—
95C22×C181C2×C22⋊C411yescharacteristic
96C22×C181C4×D412yes—
97C2×D181—41no—
98C2×D181—41no—
99C2×D181—41no—
100C2×D181—82no—
101C4×C181—21no—
102C4×C181—21no—
103C4×C181—41no—
104Dic181—21no—
105Dic181—21no—
106Dic181—41no—
107Dic181#12112yes—
108Dic181#12411yescharacteristic
109Dic181#12711yescharacteristic
110C2×C181—22no—
111C2×C181—81no—
112C2×C181#39011yescharacteristic
113C2×C181#41311yescharacteristic
114C2×C181#41311yescharacteristic
115D181—81no—
116C181#196711yescharacteristic, complemented, Sylow
117#1967—1811nomaximal, Sylow
118C2×#124—1811no—
119#423—1811no—
120C2×#101—1812no—
121C2×#100—1812no—
122C4×M16—1811no—
123C22×Q8—1811no—
124C22×D4—1811no—
125C2×M16—1812no—
126C2×M16—1812no—
127#124—3621no—
128#124—3621no—
129#124—3621no—
130#124—3621no—
131C2×C22⋊C4—1811no—
132C2×C22⋊C4—1811no—
133C2×C22⋊C4—1811no—
134C2×C22⋊C4#1062438211yescharacteristic
135C2×C42—1811no—
136C4⋊C8—3621no—
137C4⋊C8—3621no—
138#101—3624no—
139#100—3624no—
140#97—3621no—
141C4×C8—3621no—
142C24—1811no—
143C24C2×#1062438211yescharacteristic
144C2×Q8—3622no—
145C2×Q8—3621no—
146C2×Q8—3621no—
147C2×Q8—3622no—
148C2×D4—3622no—
149C2×D4—3621no—
150C2×D4—3621no—
151C2×D4—3622no—
152C22×C4—1811no—
153C22×C4—1811no—
154C22×C4—1811no—
155C22×C4C2×#1062438211yescharacteristic
156C22×C4C2×#1062438211yescharacteristic
157M16—3622no—
158M16—3622no—
159M16—3622no—
160M16—1814no—
161C2×C8—3622no—
162C2×C8—3622no—
163C22⋊C4—21no—
164C22⋊C4—21no—
165C22⋊C4—3621no—
166C22⋊C4—3621no—
167C22⋊C4—3621no—
168C22⋊C4—3621no—
169C22⋊C4—3621no—
170C22⋊C4—3621no—
171C42—1811no—
172C42—1811no—
173C42—1811no—
174C42—1811no—
175C23—21no—
176C23—22no—
177C23—41no—
178C23—41no—
179C23—3622no—
180C23—3621no—
181C23—7241no—
182C23—7241no—
183C23C22×#1062438211yescharacteristic
184Q8—7242no—
185Q8—7241no—
186Q8—7241no—
187D4—7242no—
188D4—7241no—
189D4—7241no—
190C2×C4—21no—
191C2×C4—22no—
192C2×C4—41no—
193C2×C4—3621no—
194C2×C4—3622no—
195C2×C4—1811no—
196C2×C4—1811no—
197C2×C4—1811no—
198C2×C4—1811no—
199C2×C4—1812no—
200C2×C4—1812no—
201C2×C4—7241no—
202C2×C4C22⋊#1062438211yescharacteristic
203C2×C4C22⋊#1062438211yescharacteristic
204C8—3624no—
205C8—7242no—
206C22—21no—
207C22—21no—
208C22—41no—
209C22—41no—
210C22—41no—
211C22—82no—
212C22—14482no—
213C22—7241no—
214C22—7241no—
215C22—7241no—
216C22C2×C22⋊#1062438211yescharacteristic
217C22D4×#1062438212yes—
218C4—21no—
219C4—21no—
220C4—41no—
221C4—3621no—
222C4—3621no—
223C4—1812no—
224C4—1811no—
225C4—1811no—
226C4—7241no—
227C2—22no—
228C2—81no—
229C2—14481no—
230C2#2604223111yescharacteristic
231C2#2604226511yescharacteristic
232C2#2604226611yescharacteristic
subgroupG/H
centerC22C2×C22⋊#10624382
commutatorC23×C181C22×C4
FrattiniC23C22×#10624382
FittingC2×C22⋊C4×C181C4
socleC22×C181C2×C22⋊C4
prime radicalC22×C22⋊D181C22
cosocleC22×Dic181C23
hypercenterC2×C22⋊C4#10624382
hypocenterC181#1967
LCSG ▷ C23×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. #26042282C2semidirect
2. C2×#19153877C2semidirect
4. C4×#15668458C2semidirect
116. C181#1967semidirect

Maximal subgroups

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

subgroupindex
#260422822
C2×#191538772
C2×#191538762
C4×#156684582
C2×#191538022
#1967181