Finite Group Databasebeta

← #37537003

Subgroups — #37537003

Order 23168.

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

normal subgroups only

This group has 568 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×Q8×Dic181C211yesmaximal, characteristic, complemented
2C2×#19153776C211yesmaximal, characteristic
3#26041221C212yesmaximal, complemented
4#26041206C211yesmaximal, characteristic, complemented
5#26041204C211yesmaximal, characteristic, complemented
6#26041195C212yesmaximal
7#26041190C211yesmaximal, characteristic, complemented
8C22⋊C4×Dic181C212yesmaximal, complemented
9#26041126C212yesmaximal, complemented
10#26041124C211yesmaximal, characteristic, complemented
11#393×C181C211yesmaximal, characteristic, complemented
12C22×C22⋊D181C2211yescharacteristic
13C23×Dic181C2211yescharacteristic
14C22×C4×D181C2211yescharacteristic
15Q8×Dic181—24no—
16#19153776—24no—
17C2×C4⋊Dic181C2212yes—
18C2×C4⋊Dic181C2211yescharacteristic
19C2×#15668428C2211yescharacteristic
20C2×C22⋊Dic181C2212yes—
21C2×#15668426C2212yes—
22C2×#15668426C2212yes—
23C2×#15668426C2211yescharacteristic
24C2×C4×Dic181C2212yes—
25C2×C4×Dic181C2212yes—
26C2×C4×Dic181C2211yescharacteristic
27#19153723C2211yescharacteristic
28#19153723C2212yes—
29#19153723C2212yes—
30#19153723C2212yes—
31#19153723C2212yes—
32C22×Q8×C181C2211yescharacteristic
33C2×C22⋊C4×C181C2212yes—
34C2×C22⋊C4×C181C2211yescharacteristic
35#95×C181C2211yescharacteristic
36#95×C181C2212yes—
37C23×D181C2311yescharacteristic
38C2×C22⋊D181—22no—
39C2×C22⋊D181—22no—
40C2×C22⋊D181—22no—
41C22×Dic181—22no—
42C22×Dic181—24no—
43C22×Dic181C2312yes—
44C22×Dic181C2311yescharacteristic
45C22×Dic181C2312yes—
46C22×Dic181C2311yescharacteristic
47C22×Dic181C2311yescharacteristic
48C2×C4×D181—22no—
49C2×C4×D181—24no—
50C4⋊Dic181—24no—
51C4⋊Dic181—22no—
52#15668428—22no—
53C22⋊Dic181—24no—
54#15668426—24no—
55#15668426—22no—
56#15668426—24no—
57C4×Dic181—24no—
58C4×Dic181—24no—
59C4×Dic181D414yes—
60C24×C181C2311yescharacteristic
61C2×Q8×C181—22no—
62C2×Q8×C181—24no—
63C22×C4×C181C2312yes—
64C22×C4×C181C2311yescharacteristic
65C22×C4×C181C2311yescharacteristic
66C22×C4×C181C2312yes—
67C22⋊C4×C181—22no—
68C22⋊C4×C181—24no—
69C22×D181—21no—
70C22×D181—22no—
71C22×D181—22no—
72C22×D181—21no—
73C22×D181#4112yes—
74C22⋊D181—28no—
75C2×Dic181—21no—
76C2×Dic181—21no—
77C2×Dic181—22no—
78C2×Dic181—21no—
79C2×Dic181—22no—
80C2×Dic181—22no—
81C2×Dic181—22no—
82C2×Dic181—22no—
83C2×Dic181—22no—
84C2×Dic181—22no—
85C2×Dic181—44no—
86C2×Dic181C2×D412yes—
87C2×Dic181C2×D412yes—
88C2×Dic181#4114yes—
89C4×D181—44no—
90C23×C181—21no—
91C23×C181—22no—
92C23×C181—22no—
93C23×C181—21no—
94C23×C181#4112yes—
95C23×C181C2411yescharacteristic
96Q8×C181—28no—
97C2×C4×C181—22no—
98C2×C4×C181—22no—
99C2×C4×C181—22no—
100C2×C4×C181—21no—
101C2×C4×C181—22no—
102C2×C4×C181—22no—
103C2×C4×C181—22no—
104C2×C4×C181—22no—
105C2×C4×C181C2×D412yes—
106C2×C4×C181#4114yes—
107C22×C181—22no—
108C22×C181—24no—
109C22×C181—41no—
110C22×C181—41no—
111C22×C181—42no—
112C22×C181C22×D411yescharacteristic
113C22×C181C2×#4111yescharacteristic
114C22×C181C2×#4112yes—
115C22×C181C2×#4112yes—
116C22×C181C2×#4111yescharacteristic
117C2×D181—22no—
118C2×D181—24no—
119C2×D181—41no—
120C2×D181—41no—
121C2×D181—42no—
122C4×C181—22no—
123C4×C181—24no—
124C4×C181—41no—
125C4×C181—42no—
126Dic181—22no—
127Dic181—42no—
128Dic181—41no—
129Dic181—42no—
130Dic181#12414yes—
131C2×C181—42no—
132C2×C181#52511yescharacteristic
133C2×C181#52511yescharacteristic
134C2×C181C2×#12411yescharacteristic
135C2×C181#52912yes—
136C2×C181#52912yes—
137D181—42no—
138C181#239811yescharacteristic, complemented, Sylow
139#2398—1811nomaximal, Sylow
140C2×C4×Q8—1811no—
141C2×C4×D4—1811no—
142#400—1812no—
143#397—1811no—
144#397—1811no—
145#394—1812no—
146#393—1811no—
147#393D18111yescharacteristic, complemented
148C4×C22⋊C4—1812no—
149C4×C22⋊C4—1812no—
150#376—1811no—
151C22×Q8C2×D18111yescharacteristic
152C22×D4—1811no—
153C23×C4—1811no—
154C23×C4—1811no—
155C4×Q8—3624no—
156C4×D4—3624no—
157C2×C4⋊C4—1811no—
158C2×C4⋊C4—1812no—
159C2×C4⋊C4—1811no—
160C2×C22⋊C4—1812no—
161C2×C22⋊C4—1811no—
162C2×C22⋊C4—1812no—
163C2×C22⋊C4—1812no—
164C2×C22⋊C4C2×D18111yescharacteristic
165C2×C22⋊C4C2×D18112yes—
166C2×C42—1812no—
167C2×C42—1812no—
168C2×C42—1811no—
169#95—1811no—
170#95—1812no—
171#95—1812no—
172#95—1812no—
173#95—1812no—
174#95C2×D18111yescharacteristic
175#95C2×D18112yes—
176C24—1811no—
177C24C22×D18111yescharacteristic
178C2×Q8—22no—
179C2×Q8—24no—
180C2×D4—3622no—
181C2×D4—3622no—
182C2×D4—3622no—
183C22×C4—3622no—
184C22×C4—3624no—
185C22×C4—3622no—
186C22×C4—3624no—
187C22×C4—1812no—
188C22×C4—1811no—
189C22×C4—1812no—
190C22×C4—1811no—
191C22×C4—1811no—
192C22×C4C22×D18112yes—
193C22×C4C22×D18111yescharacteristic
194C22×C4C22×D18111yescharacteristic
195C22×C4C22×D18112yes—
196C4⋊C4—3622no—
197C4⋊C4—3624no—
198C4⋊C4—3622no—
199C22⋊C4—22no—
200C22⋊C4—24no—
201C22⋊C4—3624no—
202C22⋊C4—3624no—
203C22⋊C4—3622no—
204C22⋊C4—3624no—
205C42—3624no—
206C42—3624no—
207C42—1814no—
208C23—21no—
209C23—22no—
210C23—21no—
211C23—22no—
212C23—3621no—
213C23—3622no—
214C23—3622no—
215C23—3621no—
216C23—1812no—
217C23#1566845012yes—
218C23C23×D18111yescharacteristic
219Q8—28no—
220D4—3628no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—21no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—3621no—
230C2×C4—3621no—
231C2×C4—3622no—
232C2×C4—3621no—
233C2×C4—3622no—
234C2×C4—3622no—
235C2×C4—3622no—
236C2×C4—3622no—
237C2×C4—3622no—
238C2×C4—3622no—
239C2×C4—1812no—
240C2×C4—1812no—
241C2×C4—1814no—
242C2×C4—7244no—
243C2×C4—7244no—
244C2×C4D4×D18112yes—
245C2×C4#1566844814yes—
246C22—22no—
247C22—24no—
248C22—41no—
249C22—41no—
250C22—42no—
251C22—3622no—
252C22—3624no—
253C22—7241no—
254C22—7241no—
255C22—7242no—
256C22C2×D4×D18111yescharacteristic
257C22#41×D18111yescharacteristic
258C22#41×D18112yes—
259C22C2×#1566844812yes—
260C22C2×#1566845011yescharacteristic
261C4—22no—
262C4—24no—
263C4—41no—
264C4—42no—
265C4—3622no—
266C4—1814no—
267C4—7242no—
268C4—7241no—
269C4—7242no—
270C2—42no—
271C2—7242no—
272C2#2604178511yescharacteristic
273C2#2604179211yescharacteristic
274C2#124×D18111yescharacteristic
275C2#2604180812yes—
276C2#2604181512yes—
subgroupG/H
center, FrattiniC23C23×D181
commutator, socle, cosocleC23×C181C24
Fitting#393×C181C2
prime radicalC22×C22⋊D181C22
hypercenter#393D181
hypocenterC181#2398
LCSG ▷ C23×C181 ▷ C181
UCSC1 ◁ C23 ◁ #393

Complements

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

NH ≅ G/Ntype
1. C2×Q8×Dic181C2semidirect
3. #26041221C2semidirect
4. #26041206C2semidirect
5. #26041204C2semidirect
7. #26041190C2semidirect
8. C22⋊C4×Dic181C2semidirect
9. #26041126C2semidirect
10. #26041124C2semidirect
11. #393×C181C2semidirect
138. C181#2398semidirect
147. #393D181semidirect

Maximal subgroups

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

subgroupindex
C2×Q8×Dic1812
C2×#191537762
#260412212
#260412062
#260412042
#260411952
#260411902
C22⋊C4×Dic1812
#260411262
#260411242
#393×C1812
#2398181