Finite Group Databasebeta

← #37545088

Subgroups — #37545088

Order 23168.

348 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 121 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
1#26041813C211yesmaximal, characteristic
2#26041586C212yesmaximal, complemented
3#26041571C211yesmaximal, characteristic, complemented
4#26041566C211yesmaximal, characteristic, complemented
5#26041535C212yesmaximal, complemented
6#26041527C212yesmaximal
7#26041501C212yesmaximal, complemented
8#26041486C211yesmaximal, characteristic, complemented
9#26041481C211yesmaximal, characteristic, complemented
10#26041303C211yesmaximal, characteristic, complemented
11#533×C181C211yesmaximal, characteristic, complemented
12C2×#15668439C2212yes—
13C2×#15668436C2212yes—
14C2×#15668433C2212yes—
15#19153810—21no—
16#19153808C2211yescharacteristic
17#19153804C2211yescharacteristic
18#19153799—21no—
19#19153795C2211yescharacteristic
20#19153790C2211yescharacteristic
21C4×Dic362C2211yescharacteristic
22C4×D724C2211yescharacteristic
23C4×C22⋊D181—21no—
24#19153773—21no—
25#19153749C2212yes—
26#19153747C2212yes—
27C4⋊#13089678C2211yescharacteristic
28C4⋊#13089678C2211yescharacteristic
29C4⋊#13089678C2212yes—
30#19153740C2212yes—
31#19153740C2212yes—
32#19153739C2212yes—
33#19153736C2212yes—
34C4⋊Q8×C181C2211yescharacteristic
35C4⋊Q8×C181C2211yescharacteristic
36#125×C181—21no—
37C22⋊Q8×C181—21no—
38C22⋊Q8×C181C2212yes—
39C4×Q8×C181—21no—
40C4×Q8×C181C2212yes—
41#117×C181—21no—
42C2×C42×C181C2211yescharacteristic
43C2×Dic362C2311yescharacteristic
44C2×D724C2311yescharacteristic
45C2×C22⋊D181—21no—
46C2×C4×D181—21no—
47#15668439—42no—
48#15668436—42no—
49#15668433—42no—
50C2×#13089678C2312yes—
51C2×#13089678C2312yes—
52C4⋊Dic181C2311yescharacteristic
53C4⋊Dic181C2311yescharacteristic
54#15668428—21no—
55#15668428—21no—
56#15668428—21no—
57#15668428—21no—
58C22⋊Dic181—21no—
59#15668426—21no—
60#15668426—21no—
61#15668426—21no—
62#15668426—21no—
63C4×Dic181—21no—
64C2×Q8×C181—21no—
65C2×Q8×C181C2312yes—
66C22×C4×C181D411yescharacteristic
67C22×C4×C181D411yescharacteristic
68C22×C4×C181C2311yescharacteristic
69C4⋊C4×C181—21no—
70C4⋊C4×C181—21no—
71C4⋊C4×C181—21no—
72C4⋊C4×C181—22no—
73C4⋊C4×C181—22no—
74C4⋊C4×C181C2312yes—
75C22⋊C4×C181—22no—
76C42×C181—22no—
77C42×C181D411yescharacteristic
78C42×C181D411yescharacteristic
79C42×C181C2311yescharacteristic
80C42×C181C2311yescharacteristic
81C22×D181—21no—
82Dic362—21no—
83Dic362—21no—
84D724—21no—
85D724—21no—
86C22⋊D181—42no—
87C2×Dic181—21no—
88C2×Dic181—21no—
89C2×Dic181—21no—
90C4×D181—42no—
91#13089678—22no—
92#13089678—22no—
93C23×C181C2×D411yescharacteristic
94Q8×C181—22no—
95Q8×C181—22no—
96Q8×C181—42no—
97C2×C4×C181—22no—
98C2×C4×C181—21no—
99C2×C4×C181—21no—
100C2×C4×C181—22no—
101C2×C4×C181—22no—
102C2×C4×C181—41no—
103C2×C4×C181C2×D411yescharacteristic
104C2×C4×C181C2×D411yescharacteristic
105C2×C4×C181C2×D411yescharacteristic
106C2×C4×C181C2×D411yescharacteristic
107C2×C4×C181C2×D411yescharacteristic
108C2×C4×C181C2411yescharacteristic
109C22×C181—21no—
110C22×C181—21no—
111C22×C181—21no—
112C22×C181C22×D411yescharacteristic
113C2×D181—41no—
114C2×D181—41no—
115C2×D181—41no—
116C4×C181—21no—
117C4×C181—21no—
118C4×C181—21no—
119C4×C181—21no—
120C4×C181—42no—
121C4×C181—42no—
122C4×C181#13712yes—
123C4×C181H(5,𝔽2)11yescharacteristic
124C4×C1812−1+411yescharacteristic
125Dic181—41no—
126Dic181—41no—
127Dic181—41no—
128C2×C181—41no—
129C2×C181#53711yescharacteristic
130C2×C181C2×#13711yescharacteristic
131C2×C181#57511yescharacteristic
132D181—81no—
133C181#325211yescharacteristic, complemented, Sylow
134#3252—1811nomaximal, Sylow
135#533D18111yescharacteristic, complemented
136#529—1811no—
137#479—1812no—
138#477—1811no—
139#476—1811no—
140#470—1812no—
141#468—1812no—
142#464—1812no—
143#462—1811no—
144#461—1811no—
145#423—1811no—
146C2×Q16—1812no—
147C2×SD16—1812no—
148C2×M16—1812no—
149C4⋊Q8C2×D18111yescharacteristic
150C4⋊Q8C2×D18111yescharacteristic
151#126—3621no—
152#125—21no—
153#125—1811no—
154#124—1811no—
155#123—3621no—
156C22⋊Q8—21no—
157C22⋊Q8—1811no—
158C22⋊Q8C2×D18112yes—
159#121—1811no—
160C4×Q8—21no—
161C4×Q8—1811no—
162C4×Q8C2×D18112yes—
163C4×D4—3621no—
164C4×D4—1811no—
165#117—21no—
166#117—3621no—
167C2×C42C2×D18111yescharacteristic
168#107—1812no—
169#106—1812no—
170C4⋊C8—1811no—
171C4⋊C8—1811no—
172C4⋊C8—1812no—
173#103—1812no—
174#103—1812no—
175#103—1812no—
176#102—1812no—
177C2×Q8—21no—
178C2×Q8—1811no—
179C2×Q8C22×D18112yes—
180C2×D4—3621no—
181C2×D4—1811no—
182C22×C4—3621no—
183C22×C4C22⋊D18111yescharacteristic
184C22×C4C22⋊D18111yescharacteristic
185C22×C4C22×D18111yescharacteristic
186Q16—7242no—
187SD16—7242no—
188M16—7242no—
189C2×C8—1812no—
190C2×C8—1812no—
191C4⋊C4—21no—
192C4⋊C4—21no—
193C4⋊C4—21no—
194C4⋊C4—22no—
195C4⋊C4—22no—
196C4⋊C4—3621no—
197C4⋊C4—3621no—
198C4⋊C4—3621no—
199C4⋊C4—3621no—
200C4⋊C4—1811no—
201C4⋊C4—1811no—
202C4⋊C4C22×D18112yes—
203C22⋊C4—22no—
204C22⋊C4—3621no—
205C22⋊C4—3621no—
206C22⋊C4—3621no—
207C22⋊C4—3621no—
208C22⋊C4—3621no—
209C42—22no—
210C42—3621no—
211C42C22⋊D18111yescharacteristic
212C42C22⋊D18111yescharacteristic
213C42C22×D18111yescharacteristic
214C42C22×D18111yescharacteristic
215C23—3621no—
216C23C2×C22⋊D18111yescharacteristic
217Q8—22no—
218Q8—22no—
219Q8—42no—
220Q8—3621no—
221Q8—3621no—
222D4—3621no—
223D4—3621no—
224D4—7242no—
225C2×C4—22no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—41no—
231C2×C4—3621no—
232C2×C4—3621no—
233C2×C4—3621no—
234C2×C4—7242no—
235C2×C4C2×C22⋊D18111yescharacteristic
236C2×C4C2×C22⋊D18111yescharacteristic
237C2×C4C2×C22⋊D18111yescharacteristic
238C2×C4C2×C22⋊D18111yescharacteristic
239C2×C4C2×C22⋊D18111yescharacteristic
240C2×C4C23×D18111yescharacteristic
241C8—3622no—
242C8—3622no—
243C22—21no—
244C22—21no—
245C22—21no—
246C22—7241no—
247C22—7241no—
248C22—7241no—
249C22C22×C22⋊D18111yescharacteristic
250C4—21no—
251C4—21no—
252C4—21no—
253C4—21no—
254C4—42no—
255C4—42no—
256C4—7241no—
257C4—7241no—
258C4—7241no—
259C4#1915384712yes—
260C4#1915386211yescharacteristic
261C4#1915386311yescharacteristic
262C2—41no—
263C2—14481no—
264C2#2604185711yescharacteristic
265C2C2×#1915384711yescharacteristic
266C2#2604207711yescharacteristic
subgroupG/H
centerC22C22×C22⋊D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#533×C181C2
socleC22×C181C22×D4
prime radical#19153790C22
hypercenter#533D181
hypocenterC181#3252
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C2×C42 ◁ #533

Complements

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

NH ≅ G/Ntype
2. #26041586C2semidirect
3. #26041571C2semidirect
4. #26041566C2semidirect
5. #26041535C2semidirect
7. #26041501C2semidirect
8. #26041486C2semidirect
9. #26041481C2semidirect
10. #26041303C2semidirect
11. #533×C181C2semidirect
133. C181#3252semidirect
135. #533D181semidirect

Maximal subgroups

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

subgroupindex
#260418132
#260415862
#260415712
#260415662
#260415352
#260415272
#260415012
#260414862
#260414812
#260413032
#533×C1812
#3252181