Finite Group Databasebeta

← #37545092

Subgroups — #37545092

Order 23168.

348 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 113 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#26041834C211yesmaximal, characteristic, complemented
2#26041816C211yesmaximal, characteristic
3#26041584C212yesmaximal, complemented
4#26041574C211yesmaximal, characteristic, complemented
5#26041570C211yesmaximal, characteristic, complemented
6#26041534C212yesmaximal, complemented
7#26041528C212yesmaximal
8#26041503C212yesmaximal, complemented
9#26041481C211yesmaximal, characteristic, complemented
10#26041301C211yesmaximal, characteristic, complemented
11#462×C181C211yesmaximal, characteristic, complemented
12C2×#15668436C2212yes—
13C2×#15668432C2212yes—
14#19153816C2211yescharacteristic
15#19153811—21no—
16#19153807—21no—
17#19153807C2211yescharacteristic
18#19153805C2211yescharacteristic
19#19153801—21no—
20#19153796—21no—
21#19153792C2212yes—
22#19153792C2211yescharacteristic
23#19153791C2211yescharacteristic
24Q8×Dic181C2212yes—
25#19153781—21no—
26C4×D724—21no—
27C4×D724C2211yescharacteristic
28#19153774—21no—
29#19153773—21no—
30C42×D181C2211yescharacteristic
31#19153749C2212yes—
32#19153748C2212yes—
33C4⋊#13089678C2211yescharacteristic
34#19153745C2212yes—
35#19153740C2212yes—
36#19153740C2212yes—
37#19153738C2212yes—
38C2×Q16×C181C2212yes—
39C4⋊Q8×C181C2211yescharacteristic
40C4×Q8×C181C2211yescharacteristic
41C4⋊C8×C181C2211yescharacteristic
42#103×C181C2212yes—
43C2×Dic362—21no—
44C2×D724—21no—
45C2×D724C2311yescharacteristic
46C2×C4×D181—21no—
47C2×C4×D181D411yescharacteristic
48C2×C4×D181D411yescharacteristic
49C2×C4×D181C2311yescharacteristic
50#15668436—42no—
51#15668432—42no—
52C2×#13089678C2312yes—
53C4⋊Dic181—21no—
54C4⋊Dic181—22no—
55C4⋊Dic181—21no—
56C4⋊Dic181C2312yes—
57C4⋊Dic181C2311yescharacteristic
58#15668428—21no—
59#15668428—21no—
60#15668428—22no—
61#15668426—22no—
62#15668426—21no—
63#15668426—21no—
64#15668426—21no—
65#15668426—21no—
66#15668426—21no—
67C4×Dic181—22no—
68C4×Dic181D411yescharacteristic
69C4×Dic181D411yescharacteristic
70C4×Dic181C2311yescharacteristic
71C2×Q8×C181C2312yes—
72C2×Q8×C181C2311yescharacteristic
73Q16×C181—42no—
74C2×C8×C181C2312yes—
75C4⋊C4×C181—21no—
76C4⋊C4×C181—21no—
77C4⋊C4×C181—21no—
78C4⋊C4×C181C2311yescharacteristic
79C42×C181—21no—
80C42×C181C2311yescharacteristic
81C22×D181—21no—
82C22×D181C2×D411yescharacteristic
83Dic362—42no—
84D724—21no—
85D724—21no—
86D724—42no—
87C2×Dic181—22no—
88C2×Dic181—21no—
89C2×Dic181C2×D411yescharacteristic
90C2×Dic181C2×D411yescharacteristic
91C2×Dic181C2×D411yescharacteristic
92C4×D181—22no—
93C4×D181—21no—
94C4×D181—21no—
95C4×D181—41no—
96C4×D181—42no—
97#13089678—22no—
98Q8×C181—21no—
99Q8×C181—22no—
100Q8×C181—22no—
101Q8×C181—21no—
102C2×C4×C181—21no—
103C2×C4×C181—21no—
104C2×C4×C181—22no—
105C2×C4×C181C2×D411yescharacteristic
106C2×C4×C181C2×D411yescharacteristic
107C2×C4×C181C2411yescharacteristic
108C8×C181—22no—
109C22×C181C22×D411yescharacteristic
110C2×D181—21no—
111C2×D181—21no—
112C2×D181—21no—
113C2×D181—41no—
114C2×D181—41no—
115C2×D181—41no—
116C4×C181—21no—
117C4×C181—41no—
118C4×C181—42no—
119C4×C181—41no—
120C4×C181#13712yes—
121C4×C181H(5,𝔽2)11yescharacteristic
122C4×C1812−1+411yescharacteristic
123Dic181—21no—
124Dic181—21no—
125Dic181—21no—
126Dic181—41no—
127Dic181—42no—
128C2×C181#53711yescharacteristic
129C2×C181C2×#13711yescharacteristic
130C2×C181#57511yescharacteristic
131D181—41no—
132D181—81no—
133C181#325211yescharacteristic, complemented, Sylow
134#3252—1811nomaximal, Sylow
135#533—1811no—
136#529—1811no—
137#479—1812no—
138#477—1811no—
139#476—1811no—
140#470—1812no—
141#468—1812no—
142#464—1812no—
143#462D18111yescharacteristic, complemented
144#461—1811no—
145#423—1811no—
146C2×Q16C2×D18112yes—
147C2×SD16—1812no—
148C2×M16—1812no—
149C4⋊Q8—1811no—
150C4⋊Q8C2×D18111yescharacteristic
151#126—3621no—
152#125—3621no—
153#125—1811no—
154#124—1811no—
155#123—3621no—
156C22⋊Q8—3621no—
157C22⋊Q8—1811no—
158C22⋊Q8—1812no—
159#121—1811no—
160C4×Q8—3621no—
161C4×Q8—1812no—
162C4×Q8C2×D18111yescharacteristic
163C4×D4—3621no—
164C4×D4—1811no—
165#117—3621no—
166#117—3621no—
167C2×C42—1811no—
168#107—1812no—
169#106—1812no—
170C4⋊C8—1812no—
171C4⋊C8—1811no—
172C4⋊C8C2×D18111yescharacteristic
173#103—1812no—
174#103—1812no—
175#103C2×D18112yes—
176#102—1812no—
177C2×Q8—3621no—
178C2×Q8C22×D18111yescharacteristic
179C2×Q8C22×D18112yes—
180C2×D4—3621no—
181C2×D4—1811no—
182C22×C4—3621no—
183C22×C4—1811no—
184C22×C4—1811no—
185C22×C4—1811no—
186Q16—42no—
187SD16—7242no—
188M16—7242no—
189C2×C8—1812no—
190C2×C8C22×D18112yes—
191C4⋊C4—21no—
192C4⋊C4—21no—
193C4⋊C4—21no—
194C4⋊C4—3621no—
195C4⋊C4—3621no—
196C4⋊C4—3621no—
197C4⋊C4—3622no—
198C4⋊C4—3622no—
199C4⋊C4—3621no—
200C4⋊C4—1812no—
201C4⋊C4—1811no—
202C4⋊C4C22×D18111yescharacteristic
203C22⋊C4—3622no—
204C22⋊C4—3621no—
205C22⋊C4—3621no—
206C22⋊C4—3621no—
207C22⋊C4—3621no—
208C22⋊C4—3621no—
209C42—21no—
210C42—3622no—
211C42—1811no—
212C42—1811no—
213C42—1811no—
214C42C22×D18111yescharacteristic
215C23—3621no—
216C23—1811no—
217Q8—22no—
218Q8—21no—
219Q8—21no—
220Q8—22no—
221Q8—7242no—
222D4—3621no—
223D4—3621no—
224D4—7242no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—22no—
228C2×C4—3622no—
229C2×C4—3621no—
230C2×C4—3621no—
231C2×C4—3622no—
232C2×C4—3621no—
233C2×C4—1811no—
234C2×C4—1811no—
235C2×C4—1811no—
236C2×C4—7241no—
237C2×C4—7242no—
238C2×C4D4×D18111yescharacteristic
239C2×C4D4×D18111yescharacteristic
240C2×C4C23×D18111yescharacteristic
241C8—22no—
242C8—3622no—
243C22—3621no—
244C22—3621no—
245C22—3621no—
246C22—7241no—
247C22—7241no—
248C22—7241no—
249C22C2×D4×D18111yescharacteristic
250C4—21no—
251C4—41no—
252C4—42no—
253C4—41no—
254C4—3621no—
255C4—3621no—
256C4—3621no—
257C4—7241no—
258C4—7242no—
259C4#1915384812yes—
260C4#1915386111yescharacteristic
261C4#1915386311yescharacteristic
262C2—14481no—
263C2—7241no—
264C2#2604185811yescharacteristic
265C2C2×#1915384811yescharacteristic
266C2#2604207811yescharacteristic
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#462×C181C2
socleC22×C181C22×D4
prime radical#19153791C22
hypercenter#462D181
hypocenterC181#3252
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C42 ◁ #462

Complements

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

NH ≅ G/Ntype
1. #26041834C2semidirect
3. #26041584C2semidirect
4. #26041574C2semidirect
5. #26041570C2semidirect
6. #26041534C2semidirect
8. #26041503C2semidirect
9. #26041481C2semidirect
10. #26041301C2semidirect
11. #462×C181C2semidirect
133. C181#3252semidirect
143. #462D181semidirect

Maximal subgroups

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

subgroupindex
#260418342
#260418162
#260415842
#260415742
#260415702
#260415342
#260415282
#260415032
#260414812
#260413012
#462×C1812
#3252181