Finite Group Databasebeta

← #37536985

Subgroups — #37536985

Order 23168.

560 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 560 conjugacy classes of subgroups — too many to draw. Only its 147 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×#19153778C211yesmaximal, characteristic
2C2×#19153774C211yesmaximal, characteristic, complemented
3#26041218C211yesmaximal, characteristic, complemented
4#26041216C211yesmaximal, characteristic, complemented
5#26041206C212yesmaximal, complemented
6#26041205C211yesmaximal, characteristic, complemented
7#26041196C212yesmaximal
8C4⋊C4×Dic181C212yesmaximal, complemented
9#26041126C212yesmaximal, complemented
10#26041126C211yesmaximal, characteristic, complemented
11#398×C181C211yesmaximal, characteristic, complemented
12C22×D724C2211yescharacteristic
13C22×C4×D181C2211yescharacteristic
14C22×C4×D181C2211yescharacteristic
15#19153778—24no—
16#19153774—24no—
17C2×C4⋊Dic181C2212yes—
18C2×C4⋊Dic181C2211yescharacteristic
19C2×#15668426C2212yes—
20C2×#15668426C2212yes—
21C2×#15668426C2212yes—
22C2×#15668426C2212yes—
23C2×#15668426C2211yescharacteristic
24C2×#15668426C2211yescharacteristic
25C2×C4×Dic181C2212yes—
26C2×C4×Dic181C2212yes—
27C2×C4×Dic181C2211yescharacteristic
28#19153723C2211yescharacteristic
29#19153723C2212yes—
30#19153723C2212yes—
31#19153723C2212yes—
32C2×C4⋊C4×C181C2212yes—
33C2×C4⋊C4×C181C2211yescharacteristic
34C2×C4⋊C4×C181C2211yescharacteristic
35#95×C181C2212yes—
36#95×C181C2211yescharacteristic
37C23×D181C2311yescharacteristic
38C23×D181C2311yescharacteristic
39C2×D724—22no—
40C2×D724—22no—
41C2×D724—22no—
42C22×Dic181C2312yes—
43C22×Dic181C2311yescharacteristic
44C22×Dic181C2312yes—
45C22×Dic181C2311yescharacteristic
46C2×C4×D181—24no—
47C2×C4×D181—22no—
48C2×C4×D181—24no—
49C2×C4×D181D414yes—
50C4⋊Dic181—22no—
51C4⋊Dic181—24no—
52#15668426—24no—
53#15668426—22no—
54#15668426—24no—
55#15668426—22no—
56#15668426—24no—
57#15668426—24no—
58C4×Dic181—24no—
59C4×Dic181—22no—
60C4×Dic181—24no—
61C22×C4×C181C2311yescharacteristic
62C22×C4×C181C2312yes—
63C22×C4×C181C2311yescharacteristic
64C22×C4×C181C2312yes—
65C22×C4×C181C2311yescharacteristic
66C4⋊C4×C181—22no—
67C4⋊C4×C181—22no—
68C4⋊C4×C181—24no—
69C22×D181—21no—
70C22×D181—22no—
71C22×D181—22no—
72C22×D181—22no—
73C22×D181—22no—
74C22×D181—21no—
75C22×D181—21no—
76C22×D181—21no—
77C22×D181C2×D412yes—
78C22×D181#4112yes—
79D724—28no—
80C2×Dic181—21no—
81C2×Dic181—22no—
82C2×Dic181—22no—
83C2×Dic181—22no—
84C2×Dic181—22no—
85C2×Dic181—22no—
86C2×Dic181—22no—
87C2×Dic181—22no—
88C2×Dic181C2×D412yes—
89C2×Dic181#4114yes—
90C4×D181—28no—
91C4×D181—44no—
92C23×C181C2411yescharacteristic
93C2×C4×C181—21no—
94C2×C4×C181—22no—
95C2×C4×C181—22no—
96C2×C4×C181—22no—
97C2×C4×C181—21no—
98C2×C4×C181—22no—
99C2×C4×C181—21no—
100C2×C4×C181—22no—
101C2×C4×C181—22no—
102C2×C4×C181—22no—
103C2×C4×C181C2×D412yes—
104C2×C4×C181#4112yes—
105C2×C4×C181#4114yes—
106C22×C181C22×D411yescharacteristic
107C22×C181C2×#4112yes—
108C22×C181C2×#4111yescharacteristic
109C22×C181C2×#4111yescharacteristic
110C22×C181C2×#4112yes—
111C2×D181—22no—
112C2×D181—22no—
113C2×D181—22no—
114C2×D181—24no—
115C2×D181—22no—
116C2×D181—41no—
117C2×D181—42no—
118C2×D181—42no—
119C2×D181—42no—
120C2×D181—41no—
121C4×C181—22no—
122C4×C181—22no—
123C4×C181—24no—
124C4×C181—41no—
125C4×C181—42no—
126Dic181—42no—
127Dic181—41no—
128Dic181—42no—
129Dic181#12314yes—
130C2×C181C2×#12311yescharacteristic
131C2×C181#52511yescharacteristic
132C2×C181#52912yes—
133C2×C181#52912yes—
134C2×C181#53211yescharacteristic
135D181—42no—
136D181—42no—
137C181#239511yescharacteristic, complemented, Sylow
138#2395—1811nomaximal, Sylow
139C2×C4×D4—1811no—
140C2×#117—1811no—
141#400—1811no—
142#399—1811no—
143#398D18111yescharacteristic, complemented
144#397—1811no—
145#397—1812no—
146#394—1812no—
147C4×C4⋊C4—1812no—
148C4×C22⋊C4—1812no—
149C4×C22⋊C4—1811no—
150C22×D4—1811no—
151C23×C4—1811no—
152C23×C4—1811no—
153C4×D4—3624no—
154#117—3624no—
155C2×C4⋊C4—1811no—
156C2×C4⋊C4—1812no—
157C2×C4⋊C4C2×D18111yescharacteristic
158C2×C4⋊C4C2×D18111yescharacteristic
159C2×C4⋊C4C2×D18112yes—
160C2×C22⋊C4—1812no—
161C2×C22⋊C4—1811no—
162C2×C22⋊C4—1811no—
163C2×C22⋊C4—1812no—
164C2×C22⋊C4—1812no—
165C2×C22⋊C4—1812no—
166C2×C42—1812no—
167C2×C42—1811no—
168C2×C42—1812no—
169#95—1811no—
170#95—1812no—
171#95—1812no—
172#95—1812no—
173#95C2×D18112yes—
174#95C2×D18111yescharacteristic
175C24—1811no—
176C24—1811no—
177C2×D4—3622no—
178C2×D4—3622no—
179C2×D4—3622no—
180C22×C4—3624no—
181C22×C4—3622no—
182C22×C4—3624no—
183C22×C4—1812no—
184C22×C4—1811no—
185C22×C4—1812no—
186C22×C4—1814no—
187C22×C4—1811no—
188C22×C4C22×D18111yescharacteristic
189C22×C4C22×D18112yes—
190C22×C4C22×D18111yescharacteristic
191C22×C4C22×D18111yescharacteristic
192C22×C4C22×D18112yes—
193C4⋊C4—22no—
194C4⋊C4—22no—
195C4⋊C4—24no—
196C4⋊C4—3622no—
197C4⋊C4—3624no—
198C22⋊C4—3624no—
199C22⋊C4—3622no—
200C22⋊C4—3624no—
201C22⋊C4—3622no—
202C22⋊C4—3624no—
203C22⋊C4—3624no—
204C42—3624no—
205C42—3622no—
206C42—3624no—
207C23—3621no—
208C23—3622no—
209C23—3622no—
210C23—3621no—
211C23—3621no—
212C23—3621no—
213C23—3622no—
214C23—3622no—
215C23—1812no—
216C23—1812no—
217C23C23×D18111yescharacteristic
218D4—3628no—
219C2×C4—21no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—21no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—21no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—3628no—
230C2×C4—3621no—
231C2×C4—3622no—
232C2×C4—3622no—
233C2×C4—3622no—
234C2×C4—3622no—
235C2×C4—3622no—
236C2×C4—3622no—
237C2×C4—3622no—
238C2×C4—1812no—
239C2×C4—1814no—
240C2×C4—7244no—
241C2×C4D4×D18112yes—
242C2×C4#1566844812yes—
243C2×C4#1566845014yes—
244C22—3622no—
245C22—3622no—
246C22—3622no—
247C22—3624no—
248C22—3622no—
249C22—7241no—
250C22—7242no—
251C22—7242no—
252C22—7242no—
253C22—7241no—
254C22C2×D4×D18111yescharacteristic
255C22#41×D18112yes—
256C22#41×D18111yescharacteristic
257C22C2×#1566844811yescharacteristic
258C22C2×#1566845012yes—
259C4—22no—
260C4—22no—
261C4—24no—
262C4—41no—
263C4—42no—
264C4—1814no—
265C4—7242no—
266C4—7241no—
267C4—7242no—
268C2—7242no—
269C2—7242no—
270C2#123×D18111yescharacteristic
271C2#2604178911yescharacteristic
272C2#2604181412yes—
273C2#2604181512yes—
274C2#2604183011yescharacteristic
subgroupG/H
center, FrattiniC23C23×D181
commutator, socle, cosocleC23×C181C24
Fitting#398×C181C2
prime radicalC22×D724C22
hypercenter#398D181
hypocenterC181#2395
LCSG ▷ C23×C181 ▷ C181
UCSC1 ◁ C23 ◁ #398

Complements

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

NH ≅ G/Ntype
2. C2×#19153774C2semidirect
3. #26041218C2semidirect
4. #26041216C2semidirect
5. #26041206C2semidirect
6. #26041205C2semidirect
8. C4⋊C4×Dic181C2semidirect
9. #26041126C2semidirect
10. #26041126C2semidirect
11. #398×C181C2semidirect
137. C181#2395semidirect
143. #398D181semidirect

Maximal subgroups

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

subgroupindex
C2×#191537782
C2×#191537742
#260412182
#260412162
#260412062
#260412052
#260411962
C4⋊C4×Dic1812
#260411262
#398×C1812
#2395181