Finite Group Databasebeta

← #37536960

Subgroups — #37536960

Order 23168.

696 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 696 conjugacy classes of subgroups — too many to draw. Only its 155 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×D4×Dic181C211yesmaximal, characteristic, complemented
2C2×#19153776C212yesmaximal, complemented
3C2×#19153774C211yesmaximal, characteristic, complemented
4#26041209C211yesmaximal, characteristic, complemented
5#26041206C211yesmaximal, characteristic, complemented
6#26041202C212yesmaximal, complemented
7#26041194C212yesmaximal, complemented
8#26041187C211yesmaximal, characteristic
9#26041130C211yesmaximal, characteristic, complemented
10C22⋊C4×Dic181C212yesmaximal, complemented
11#394×C181C211yesmaximal, characteristic, complemented
12C22×C22⋊D181C2212yes—
13C23×Dic181C2212yes—
14C22×C4×D181C2211yescharacteristic
15D4×Dic181—24no—
16#19153776—28no—
17#19153774—24no—
18C2×C4⋊Dic181C2211yescharacteristic
19C2×C4⋊Dic181C2211yescharacteristic
20C2×#15668428C2212yes—
21C2×#15668428C2212yes—
22C2×C22⋊Dic181C2212yes—
23C2×C22⋊Dic181C2212yes—
24C2×#15668426C2211yescharacteristic
25C2×#15668426C2211yescharacteristic
26C2×#15668426C2212yes—
27C2×C4×Dic181C2211yescharacteristic
28C2×C4×Dic181C2212yes—
29C2×C4×Dic181C2211yescharacteristic
30#19153723C2212yes—
31#19153723C2212yes—
32#19153723C2211yescharacteristic
33C22×D4×C181C2211yescharacteristic
34C2×C4⋊C4×C181C2211yescharacteristic
35C2×C22⋊C4×C181C2212yes—
36C2×C22⋊C4×C181C2212yes—
37#95×C181C2211yescharacteristic
38C23×D181C2311yescharacteristic
39C2×C22⋊D181—24no—
40C2×C22⋊D181—24no—
41C2×C22⋊D181—24no—
42C22×Dic181—24no—
43C22×Dic181—24no—
44C22×Dic181—24no—
45C22×Dic181C2311yescharacteristic
46C22×Dic181C2312yes—
47C22×Dic181C2311yescharacteristic
48C22×Dic181C2312yes—
49C22×Dic181C2311yescharacteristic
50C2×C4×D181—22no—
51C2×C4×D181—22no—
52C2×C4×D181D414yescomplemented
53C4⋊Dic181—22no—
54C4⋊Dic181—22no—
55#15668428—24no—
56#15668428—24no—
57C22⋊Dic181—24no—
58C22⋊Dic181—24no—
59#15668426—24no—
60#15668426—22no—
61#15668426—22no—
62C4×Dic181—22no—
63C4×Dic181—24no—
64C4×Dic181D414yescomplemented
65C24×C181C2312yes—
66C2×D4×C181—24no—
67C2×D4×C181—22no—
68C22×C4×C181C2312yes—
69C22×C4×C181C2311yescharacteristic
70C22×C4×C181C2311yescharacteristic
71C22×C4×C181C2311yescharacteristic
72C4⋊C4×C181—22no—
73C22⋊C4×C181—24no—
74C22⋊C4×C181—24no—
75C22×D181—21no—
76C22×D181—21no—
77C22×D181—22no—
78C22×D181—22no—
79C22×D181C2×D412yes—
80C22⋊D181—216no—
81C2×Dic181—22no—
82C2×Dic181—22no—
83C2×Dic181—22no—
84C2×Dic181—21no—
85C2×Dic181—22no—
86C2×Dic181—22no—
87C2×Dic181—22no—
88C2×Dic181—22no—
89C2×Dic181—48no—
90C2×Dic181C2×D412yes—
91C2×Dic181C2×D412yes—
92C2×Dic181C2×D412yes—
93C2×Dic181#4114yes—
94C2×Dic181#4112yes—
95C4×D181—28no—
96C23×C181—22no—
97C23×C181—22no—
98C23×C181—22no—
99C23×C181—22no—
100C23×C181—22no—
101C23×C181—22no—
102C23×C181#4114yes—
103C23×C181C2411yescharacteristic
104D4×C181—28no—
105C2×C4×C181—21no—
106C2×C4×C181—22no—
107C2×C4×C181—21no—
108C2×C4×C181—22no—
109C2×C4×C181—21no—
110C2×C4×C181—22no—
111C2×C4×C181—22no—
112C2×C4×C181—22no—
113C2×C4×C181C2×D412yes—
114C2×C4×C181C2×D412yes—
115C22×C181—24no—
116C22×C181—24no—
117C22×C181—24no—
118C22×C181—42no—
119C22×C181—42no—
120C22×C181—42no—
121C22×C181—42no—
122C22×C181C22×D411yescharacteristic
123C22×C181C22×D411yescharacteristic
124C22×C181C2×#4111yescharacteristic
125C22×C181C2×#4112yes—
126C22×C181C2×#4112yes—
127C2×D181—22no—
128C2×D181—24no—
129C2×D181—41no—
130C2×D181—42no—
131C2×D181—41no—
132C4×C181—22no—
133C4×C181—22no—
134C4×C181—41no—
135C4×C181—42no—
136Dic181—24no—
137Dic181—22no—
138Dic181—42no—
139Dic181—41no—
140Dic181#12114yes—
141C2×C181—44no—
142C2×C181C2×#12111yescharacteristic
143C2×C181#52511yescharacteristic
144C2×C181#52512yes—
145C2×C181#52912yes—
146C2×C181#53211yescharacteristic
147D181—42no—
148C181#239211yescharacteristic, complemented, Sylow
149#2392—1811nomaximal, Sylow
150C2×C4×D4—1811no—
151C2×C4×D4—1812no—
152C2×#117—1811no—
153#398—1811no—
154#397—1811no—
155#396—1812no—
156#394—1812no—
157#394D18111yescharacteristic, complemented
158#392—1811no—
159C4×C4⋊C4—1811no—
160C4×C22⋊C4—1812no—
161C22×D4—1812no—
162C22×D4C2×D18111yescharacteristic
163C23×C4—1812no—
164C23×C4—1811no—
165C4×D4—3628no—
166C4×D4—3624no—
167#117—3624no—
168C2×C4⋊C4—1811no—
169C2×C4⋊C4—1812no—
170C2×C4⋊C4—1811no—
171C2×C4⋊C4—1812no—
172C2×C4⋊C4C2×D18111yescharacteristic
173C2×C22⋊C4—1812no—
174C2×C22⋊C4—1812no—
175C2×C22⋊C4—1811no—
176C2×C22⋊C4—1811no—
177C2×C22⋊C4—1812no—
178C2×C22⋊C4C2×D18112yes—
179C2×C22⋊C4C2×D18112yes—
180C2×C42—1811no—
181C2×C42—1812no—
182C2×C42—1811no—
183#95—1812no—
184#95—1812no—
185#95—1811no—
186#95C2×D18111yescharacteristic
187C24—1811no—
188C24C22×D18112yes—
189C2×D4—24no—
190C2×D4—22no—
191C2×D4—3624no—
192C2×D4—3624no—
193C2×D4—3624no—
194C22×C4—3622no—
195C22×C4—3622no—
196C22×C4—3624no—
197C22×C4—3624no—
198C22×C4—3624no—
199C22×C4—1811no—
200C22×C4—1812no—
201C22×C4—1812no—
202C22×C4—1811no—
203C22×C4—1814no—
204C22×C4—1811no—
205C22×C4C22×D18111yescharacteristic
206C22×C4C22×D18112yes—
207C22×C4C22×D18111yescharacteristic
208C22×C4C22×D18111yescharacteristic
209C4⋊C4—22no—
210C4⋊C4—3622no—
211C4⋊C4—3624no—
212C4⋊C4—3624no—
213C4⋊C4—3622no—
214C22⋊C4—24no—
215C22⋊C4—24no—
216C22⋊C4—3624no—
217C22⋊C4—3622no—
218C22⋊C4—3624no—
219C22⋊C4—3624no—
220C22⋊C4—3622no—
221C42—3622no—
222C42—3624no—
223C42—1814no—
224C23—22no—
225C23—22no—
226C23—22no—
227C23—22no—
228C23—22no—
229C23—22no—
230C23—3621no—
231C23—3621no—
232C23—3622no—
233C23—3622no—
234C23—1812no—
235C23#1566844814yes—
236C23C23×D18111yescharacteristic
237D4—28no—
238D4—36216no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—21no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—3628no—
248C2×C4—3622no—
249C2×C4—3622no—
250C2×C4—3622no—
251C2×C4—3621no—
252C2×C4—3622no—
253C2×C4—3622no—
254C2×C4—3622no—
255C2×C4—3622no—
256C2×C4—1812no—
257C2×C4—1814no—
258C2×C4—1812no—
259C2×C4—1812no—
260C2×C4—1812no—
261C2×C4—7248no—
262C2×C4D4×D18112yes—
263C2×C4D4×D18112yes—
264C22—24no—
265C22—24no—
266C22—24no—
267C22—42no—
268C22—42no—
269C22—42no—
270C22—42no—
271C22—3622no—
272C22—3624no—
273C22—7241no—
274C22—7242no—
275C22—7241no—
276C22C2×D4×D18111yescharacteristic
277C22C2×D4×D18111yescharacteristic
278C22#41×D18111yescharacteristic
279C22#41×D18112yes—
280C22C2×#1566844812yes—
281C4—22no—
282C4—22no—
283C4—41no—
284C4—42no—
285C4—3624no—
286C4—3622no—
287C4—1814no—
288C4—7242no—
289C4—7241no—
290C2—44no—
291C2—7242no—
292C2#121×D18111yescharacteristic
293C2#2604178511yescharacteristic
294C2#2604178912yes—
295C2#2604180612yes—
296C2#2604182811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D181
commutator, socle, cosocleC23×C181C24
Fitting#394×C181C2
prime radical#26041187C2
hypercenter#394D181
hypocenterC181#2392
LCSG ▷ C23×C181 ▷ C181
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
1. C2×D4×Dic181C2semidirect
2. C2×#19153776C2semidirect
3. C2×#19153774C2semidirect
4. #26041209C2semidirect
5. #26041206C2semidirect
6. #26041202C2semidirect
7. #26041194C2semidirect
9. #26041130C2semidirect
10. C22⋊C4×Dic181C2semidirect
11. #394×C181C2semidirect
52. C2×C4×D181D4semidirect
64. C4×Dic181D4semidirect
148. C181#2392semidirect
157. #394D181semidirect

Maximal subgroups

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

subgroupindex
C2×D4×Dic1812
C2×#191537762
C2×#191537742
#260412092
#260412062
#260412022
#260411942
#260411872
#260411302
C22⋊C4×Dic1812
#394×C1812
#2392181