Finite Group Databasebeta

← #37537010

Subgroups — #37537010

Order 23168.

568 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 568 conjugacy classes of subgroups — too many to draw. Only its 143 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×#19153781C211yesmaximal, characteristic, complemented
2C2×#19153778C211yesmaximal, characteristic
3#26041217C211yesmaximal, characteristic, complemented
4#26041208C212yesmaximal, complemented
5#26041197C211yesmaximal, characteristic
6#26041196C211yesmaximal, characteristic
7#26041192C212yesmaximal, complemented
8#26041126C212yesmaximal, complemented
9#26041126C212yesmaximal, complemented
10#26041124C211yesmaximal, characteristic, complemented
11#400×C181C211yesmaximal, characteristic, complemented
12C22×Dic362C2211yescharacteristic
13C22×D724C2211yescharacteristic
14C22×C4×D181C2212yes—
15#19153781—24no—
16#19153778—24no—
17C2×#15668428C2211yescharacteristic
18C2×#15668428C2211yescharacteristic
19C2×#15668426C2212yes—
20C2×#15668426C2212yes—
21C2×#15668426C2212yes—
22C2×#15668426C2212yes—
23C2×#15668426C2212yes—
24C2×C4×Dic181C2212yes—
25C2×C4×Dic181C2211yescharacteristic
26C2×C4×Dic181C2211yescharacteristic
27C2×C4×Dic181C2211yescharacteristic
28#19153723C2212yes—
29#19153723C2211yescharacteristic
30#19153723C2212yes—
31#19153723C2212yes—
32C2×C4⋊C4×C181C2211yescharacteristic
33C2×C4⋊C4×C181C2211yescharacteristic
34#95×C181C2211yescharacteristic
35#95×C181C2212yes—
36#95×C181C2212yes—
37C23×D181C2312yes—
38C2×Dic362—22no—
39C2×Dic362—22no—
40C2×Dic362—22no—
41C2×D724—24no—
42C2×D724—22no—
43C22×Dic181C2312yes—
44C22×Dic181C2311yescharacteristic
45C22×Dic181C2311yescharacteristic
46C22×Dic181C2311yescharacteristic
47C22×Dic181C2311yescharacteristic
48C2×C4×D181—24no—
49C2×C4×D181—24no—
50C2×C4×D181—24no—
51#15668428—22no—
52#15668428—22no—
53#15668426—24no—
54#15668426—24no—
55#15668426—24no—
56#15668426—24no—
57#15668426—24no—
58C4×Dic181—22no—
59C4×Dic181—22no—
60C4×Dic181—24no—
61C4×Dic181D414yes—
62C22×C4×C181C2312yes—
63C22×C4×C181C2312yes—
64C22×C4×C181C2311yescharacteristic
65C22×C4×C181C2311yescharacteristic
66C22×C4×C181C2311yescharacteristic
67C4⋊C4×C181—22no—
68C4⋊C4×C181—22no—
69C22×D181—22no—
70C22×D181—22no—
71C22×D181—22no—
72C22×D181—22no—
73C22×D181—22no—
74C22×D181—22no—
75C22×D181#4114yes—
76Dic362—28no—
77D724—28no—
78C2×Dic181—21no—
79C2×Dic181—21no—
80C2×Dic181—21no—
81C2×Dic181—21no—
82C2×Dic181—21no—
83C2×Dic181—22no—
84C2×Dic181—22no—
85C2×Dic181—22no—
86C2×Dic181—22no—
87C2×Dic181C2×D412yes—
88C2×Dic181C2×D412yes—
89C2×Dic181#4112yes—
90C2×Dic181#4114yes—
91C4×D181—48no—
92C23×C181C2411yescharacteristic
93C2×C4×C181—21no—
94C2×C4×C181—22no—
95C2×C4×C181—21no—
96C2×C4×C181—21no—
97C2×C4×C181—22no—
98C2×C4×C181—22no—
99C2×C4×C181—22no—
100C2×C4×C181—22no—
101C2×C4×C181—22no—
102C2×C4×C181—22no—
103C2×C4×C181—22no—
104C2×C4×C181C2×D412yes—
105C2×C4×C181#4112yes—
106C22×C181C22×D411yescharacteristic
107C22×C181C2×#4112yes—
108C22×C181C2×#4111yescharacteristic
109C22×C181C2×#4112yes—
110C22×C181C2×#4111yescharacteristic
111C2×D181—24no—
112C2×D181—24no—
113C2×D181—24no—
114C2×D181—42no—
115C2×D181—42no—
116C2×D181—42no—
117C2×D181—42no—
118C4×C181—22no—
119C4×C181—22no—
120C4×C181—41no—
121C4×C181—42no—
122C4×C181—42no—
123Dic181—22no—
124Dic181—22no—
125Dic181—41no—
126Dic181—42no—
127Dic181#12414yes—
128C2×C181#52511yescharacteristic
129C2×C181#52511yescharacteristic
130C2×C181C2×#12411yescharacteristic
131C2×C181#52912yes—
132C2×C181#52912yes—
133D181—44no—
134C181#239811yescharacteristic, complemented, Sylow
135#2398—1811nomaximal, Sylow
136C2×C4×Q8—1811no—
137C2×C4×D4—1811no—
138#400—1811no—
139#400D18111yescharacteristic, complemented
140#397—1812no—
141#394—1811no—
142#394—1811no—
143#393—1812no—
144C4×C22⋊C4—1812no—
145C4×C22⋊C4—1812no—
146#376—1811no—
147C22×Q8—1811no—
148C22×D4—1811no—
149C23×C4—1812no—
150C4×Q8—3624no—
151C4×D4—3624no—
152C2×C4⋊C4—1811no—
153C2×C4⋊C4—1811no—
154C2×C4⋊C4C2×D18111yescharacteristic
155C2×C4⋊C4C2×D18111yescharacteristic
156C2×C22⋊C4—1812no—
157C2×C22⋊C4—1812no—
158C2×C22⋊C4—1812no—
159C2×C22⋊C4—1812no—
160C2×C22⋊C4—1812no—
161C2×C42—1811no—
162C2×C42—1811no—
163C2×C42—1812no—
164C2×C42—1811no—
165#95—1812no—
166#95—1811no—
167#95—1812no—
168#95—1812no—
169#95C2×D18111yescharacteristic
170#95C2×D18112yes—
171#95C2×D18112yes—
172C24—1812no—
173C2×Q8—3622no—
174C2×Q8—3622no—
175C2×Q8—3622no—
176C2×D4—3624no—
177C2×D4—3622no—
178C22×C4—3624no—
179C22×C4—3624no—
180C22×C4—3624no—
181C22×C4—1812no—
182C22×C4—1811no—
183C22×C4—1811no—
184C22×C4—1811no—
185C22×C4—1811no—
186C22×C4C22×D18112yes—
187C22×C4C22×D18112yes—
188C22×C4C22×D18111yescharacteristic
189C22×C4C22×D18111yescharacteristic
190C22×C4C22×D18111yescharacteristic
191C4⋊C4—22no—
192C4⋊C4—22no—
193C4⋊C4—3622no—
194C4⋊C4—3622no—
195C22⋊C4—3624no—
196C22⋊C4—3624no—
197C22⋊C4—3624no—
198C22⋊C4—3624no—
199C22⋊C4—3624no—
200C42—3622no—
201C42—3622no—
202C42—3624no—
203C42—1814no—
204C23—3622no—
205C23—3622no—
206C23—3622no—
207C23—3622no—
208C23—3622no—
209C23—3622no—
210C23—1814no—
211C23C23×D18111yescharacteristic
212Q8—3628no—
213D4—3628no—
214C2×C4—21no—
215C2×C4—22no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—3621no—
226C2×C4—3621no—
227C2×C4—3621no—
228C2×C4—3621no—
229C2×C4—3621no—
230C2×C4—3622no—
231C2×C4—3622no—
232C2×C4—3622no—
233C2×C4—3622no—
234C2×C4—1812no—
235C2×C4—1812no—
236C2×C4—1812no—
237C2×C4—1814no—
238C2×C4—7248no—
239C2×C4D4×D18112yes—
240C2×C4#1566845012yes—
241C22—3624no—
242C22—3624no—
243C22—3624no—
244C22—7242no—
245C22—7242no—
246C22—7242no—
247C22—7242no—
248C22C2×D4×D18111yescharacteristic
249C22#41×D18112yes—
250C22#41×D18111yescharacteristic
251C22#41×D18112yes—
252C22C2×#1566845011yescharacteristic
253C4—22no—
254C4—22no—
255C4—41no—
256C4—42no—
257C4—42no—
258C4—3622no—
259C4—3622no—
260C4—1814no—
261C4—7241no—
262C4—7242no—
263C2—7244no—
264C2#2604178711yescharacteristic
265C2#2604179211yescharacteristic
266C2#124×D18111yescharacteristic
267C2#2604181212yes—
268C2#2604181412yes—
subgroupG/H
center, FrattiniC23C23×D181
commutator, socle, cosocleC23×C181C24
Fitting#400×C181C2
prime radicalC22×D724C22
hypercenter#400D181
hypocenterC181#2398
LCSG ▷ C23×C181 ▷ C181
UCSC1 ◁ C23 ◁ #400

Complements

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

NH ≅ G/Ntype
1. C2×#19153781C2semidirect
3. #26041217C2semidirect
4. #26041208C2semidirect
7. #26041192C2semidirect
8. #26041126C2semidirect
9. #26041126C2semidirect
10. #26041124C2semidirect
11. #400×C181C2semidirect
134. C181#2398semidirect
139. #400D181semidirect

Maximal subgroups

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

subgroupindex
C2×#191537812
C2×#191537782
#260412172
#260412082
#260411972
#260411962
#260411922
#260411262
#260411242
#400×C1812
#2398181