Finite Group Databasebeta

← #37534750

Subgroups — #37534750

Order 23168.

394 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 75 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×#19153847C211yesmaximal, characteristic, complemented
2#26041846C211yesmaximal, characteristic
3C2×#19153744C211yesmaximal, characteristic, complemented
4#26041288C211yesmaximal, characteristic, complemented
5C2×#19153733C211yesmaximal, characteristic, complemented
6#26041039C211yesmaximal, characteristic, complemented
7C2×#124×C181C211yesmaximal, characteristic, complemented
8C2×#15668449C2211yescharacteristic
9#19153847—22no—
10#19153847—22no—
11C2×#15668439—21no—
12C2×#15668436—21no—
13C2×#15668433C2211yescharacteristic
14C2×#15668433C2211yescharacteristic
15#19153812—21no—
16#19153803—21no—
17#19153787—22no—
18#19153785—22no—
19#19153775C2211yescharacteristic
20#19153744—22no—
21#19153740—22no—
22#19153733—22no—
23C22×Q8×C181C2211yescharacteristic
24C22×D4×C181C2211yescharacteristic
25#124×C181—22no—
26#124×C181—22no—
27C2×C22⋊C4×C181—22no—
28C2×C42×C181C2211yescharacteristic
29#15668449—21no—
30#15668449—21no—
31#15668449—21no—
32#15668449—21no—
33C2×Dic362D411yescharacteristic
34C2×D724D411yescharacteristic
35C2×C22⋊D181—24no—
36C2×C22⋊D181—21no—
37C2×C4×D181—21no—
38#15668439—24no—
39#15668436—24no—
40#15668433—21no—
41#15668433—21no—
42#15668433—21no—
43#15668433D412yescomplemented
44C2×#13089678—21no—
45C2×#13089678—21no—
46C4⋊Dic181D412yes—
47C22⋊Dic181—24no—
48C22⋊Dic181—21no—
49C4×Dic181—21no—
50C24×C181D412yes—
51C2×Q8×C181—21no—
52C2×Q8×C181—21no—
53C2×Q8×C181—21no—
54C2×Q8×C181—21no—
55C2×Q8×C181—21no—
56C2×Q8×C181D412yescomplemented
57C2×D4×C181—21no—
58C2×D4×C181—21no—
59C2×D4×C181—21no—
60C2×D4×C181—21no—
61C2×D4×C181—21no—
62C2×D4×C181—21no—
63C22×C4×C181—21no—
64C22×C4×C181—21no—
65C22×C4×C181C2311yescharacteristic
66C22⋊C4×C181—28no—
67C42×C181—21no—
68C42×C181D412yescomplemented
69C22×D181—21no—
70Dic362—21no—
71Dic362—21no—
72D724—21no—
73D724—21no—
74C22⋊D181—41no—
75C22⋊D181—41no—
76C22⋊D181—84no—
77C2×Dic181—22no—
78C2×Dic181—21no—
79C4×D181—41no—
80C4×D181—41no—
81#13089678—22no—
82#13089678—41no—
83C23×C181—22no—
84C23×C181—22no—
85C23×C181—22no—
86C23×C181—44no—
87C23×C181C2×D411yescharacteristic
88Q8×C181—22no—
89Q8×C181—22no—
90Q8×C181—41no—
91Q8×C181—41no—
92D4×C181—41no—
93D4×C181—41no—
94D4×C181—41no—
95D4×C181—41no—
96C2×C4×C181—22no—
97C2×C4×C181—22no—
98C2×C4×C181—42no—
99C2×C4×C181—41no—
100C2×C4×C181—41no—
101C2×C4×C181C2×D411yescharacteristic
102C2×C4×C181C2×D411yescharacteristic
103C2×C4×C181C2×D411yescharacteristic
104C2×C4×C181C2×D411yescharacteristic
105C2×C4×C181C2×D411yescharacteristic
106C2×C4×C181#4111yescharacteristic
107C22×C181—22no—
108C22×C181—44no—
109C22×C181—42no—
110C22×C181—44no—
111C22×C181—82no—
112C22×C181#12011yescharacteristic
113C22×C181#12011yescharacteristic
114C22×C181#12111yescharacteristic
115C2×D181—41no—
116C2×D181—42no—
117C4×C181—42no—
118C4×C181—42no—
119C4×C181#12011yescharacteristic
120C4×C181#12111yescharacteristic
121C4×C181#12111yescharacteristic
122C4×C181#12711yescharacteristic
123Dic181—41no—
124Dic181—42no—
125C2×C181—21no—
126C2×C181—21no—
127C2×C181—82no—
128C2×C181#39211yescharacteristic
129C2×C181#45512yes—
130D181—81no—
131C181#200411yescharacteristic, complemented, Sylow
132#2004—1811nomaximal, Sylow
133C2×#137—1811no—
134#535—1811no—
135C2×#124D18111yescharacteristic, complemented
136C2×#104—1811no—
137#419—1811no—
138C2×#100—1811no—
139#339—1811no—
140C2×#41—1811no—
141C22×Q8C2×D18111yescharacteristic
142C22×D4C2×D18111yescharacteristic
143#137—3622no—
144#137—3622no—
145C2×Q16—3621no—
146C2×SD16—3621no—
147C2×M16—1811no—
148C2×M16—1811no—
149#127—3621no—
150#124—22no—
151#124—22no—
152#124—3621no—
153#121—3622no—
154#120—3622no—
155#117—1811no—
156C2×C22⋊C4—22no—
157C2×C42C2×D18111yescharacteristic
158#104—3622no—
159#103—3622no—
160#100—3622no—
161C24C22⋊D18112yes—
162#41—3621no—
163#41—3621no—
164#41—3621no—
165#41—3621no—
166C2×Q8—21no—
167C2×Q8—21no—
168C2×Q8—21no—
169C2×Q8—21no—
170C2×Q8—21no—
171C2×Q8—1811no—
172C2×Q8C22⋊D18112yescomplemented
173C2×D4—21no—
174C2×D4—21no—
175C2×D4—21no—
176C2×D4—21no—
177C2×D4—21no—
178C2×D4—21no—
179C2×D4—3624no—
180C2×D4—3621no—
181C2×D4—1811no—
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—3621no—
185C22×C4C22×D18111yescharacteristic
186Q16—3624no—
187SD16—3624no—
188M16—3621no—
189M16—3621no—
190M16—3621no—
191M16—1812no—
192C2×C8—3621no—
193C2×C8—3621no—
194C4⋊C4—1812no—
195C22⋊C4—28no—
196C22⋊C4—3624no—
197C22⋊C4—3621no—
198C42—21no—
199C42—3621no—
200C42C22⋊D18112yescomplemented
201C23—22no—
202C23—22no—
203C23—22no—
204C23—44no—
205C23—3621no—
206C23C2×C22⋊D18111yescharacteristic
207Q8—22no—
208Q8—22no—
209Q8—41no—
210Q8—41no—
211Q8—3621no—
212Q8—3621no—
213D4—41no—
214D4—41no—
215D4—41no—
216D4—41no—
217D4—3621no—
218D4—3621no—
219D4—14484no—
220D4—7241no—
221D4—7241no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—42no—
225C2×C4—41no—
226C2×C4—41no—
227C2×C4—3622no—
228C2×C4—3621no—
229C2×C4—7241no—
230C2×C4—7241no—
231C2×C4D4×D18111yescharacteristic
232C2×C4D4×D18111yescharacteristic
233C2×C4D4×D18111yescharacteristic
234C2×C4C2×C22⋊D18111yescharacteristic
235C2×C4C2×C22⋊D18111yescharacteristic
236C2×C4#1566844811yescharacteristic
237C8—3622no—
238C8—7241no—
239C22—22no—
240C22—44no—
241C22—42no—
242C22—44no—
243C22—82no—
244C22—7241no—
245C22—7242no—
246C22#1915378411yescharacteristic
247C22#1915378511yescharacteristic
248C22#1915378811yescharacteristic
249C4—42no—
250C4—42no—
251C4—7241no—
252C4—7242no—
253C4#1915378411yescharacteristic
254C4#1915378711yescharacteristic
255C4#1915378811yescharacteristic
256C4#1915381211yescharacteristic
257C2—21no—
258C2—21no—
259C2—82no—
260C2—14481no—
261C2#2604118711yescharacteristic
262C2#2604145612yes—
subgroupG/H
centerC22#19153784
commutator, cosocleC22×C4×C181C23
FrattiniC22×C4C22×D181
FittingC2×#124×C181C2
socleC22×C181#120
prime radical#26041846C2
hypercenterC2×#124D181
hypocenterC181#2004
LCSG ▷ C22×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#124

Complements

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

NH ≅ G/Ntype
1. C2×#19153847C2semidirect
3. C2×#19153744C2semidirect
4. #26041288C2semidirect
5. C2×#19153733C2semidirect
6. #26041039C2semidirect
7. C2×#124×C181C2semidirect
43. #15668433D4semidirect
56. C2×Q8×C181D4semidirect
68. C42×C181D4semidirect
131. C181#2004semidirect
135. C2×#124D181semidirect
172. C2×Q8C22⋊D181semidirect
200. C42C22⋊D181semidirect

Maximal subgroups

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

subgroupindex
C2×#191538472
#260418462
C2×#191537442
#260412882
C2×#191537332
#260410392
C2×#124×C1812
#2004181