Finite Group Databasebeta

← #37545090

Subgroups — #37545090

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#26041811C211yesmaximal, characteristic
3#26041588C212yesmaximal, complemented
4#26041575C211yesmaximal, characteristic, complemented
5#26041569C211yesmaximal, characteristic, complemented
6#26041536C212yesmaximal, complemented
7#26041522C212yesmaximal
8#26041499C212yesmaximal, complemented
9#26041486C211yesmaximal, characteristic, complemented
10#26041301C211yesmaximal, characteristic, complemented
11#461×C181C211yesmaximal, characteristic, complemented
12C2×#15668439C2212yes—
13C2×#15668432C2212yes—
14#19153816C2211yescharacteristic
15#19153809—21no—
16#19153807—21no—
17#19153807C2211yescharacteristic
18#19153803C2211yescharacteristic
19#19153798—21no—
20#19153796—21no—
21#19153796C2211yescharacteristic
22#19153792C2212yes—
23#19153787C2211yescharacteristic
24Q8×Dic181C2212yes—
25C4×Dic362C2211yescharacteristic
26#19153781—21no—
27C4×C22⋊D181—21no—
28#19153775—21no—
29#19153774—21no—
30C42×D181C2211yescharacteristic
31#19153750C2212yes—
32#19153747C2212yes—
33C4⋊#13089678C2211yescharacteristic
34#19153745C2212yes—
35#19153741C2212yes—
36#19153740C2212yes—
37#19153737C2212yes—
38C2×SD16×C181C2212yes—
39C4⋊Q8×C181C2211yescharacteristic
40C4×D4×C181C2211yescharacteristic
41C4⋊C8×C181C2211yescharacteristic
42#103×C181C2212yes—
43C2×Dic362—21no—
44C2×Dic362C2311yescharacteristic
45C2×C22⋊D181—21no—
46C2×C4×D181D411yescharacteristic
47C2×C4×D181D411yescharacteristic
48C2×C4×D181C2311yescharacteristic
49#15668439—42no—
50#15668432—42no—
51C2×#13089678C2312yes—
52C4⋊Dic181—21no—
53C4⋊Dic181—21no—
54C4⋊Dic181—22no—
55C4⋊Dic181—21no—
56C4⋊Dic181C2312yes—
57C4⋊Dic181C2311yescharacteristic
58#15668428—21no—
59#15668428—22no—
60#15668428—21no—
61C22⋊Dic181—21no—
62C22⋊Dic181—21no—
63C22⋊Dic181—21no—
64#15668426—22no—
65#15668426—21no—
66C4×Dic181—22no—
67C4×Dic181—21no—
68C4×Dic181D411yescharacteristic
69C4×Dic181D411yescharacteristic
70C4×Dic181C2311yescharacteristic
71C2×Q8×C181C2312yes—
72C2×D4×C181C2311yescharacteristic
73C22×C4×C181—21no—
74SD16×C181—42no—
75C2×C8×C181C2312yes—
76C4⋊C4×C181—21no—
77C4⋊C4×C181—21no—
78C4⋊C4×C181C2311yescharacteristic
79C22⋊C4×C181—21no—
80C42×C181C2311yescharacteristic
81C22×D181C2×D411yescharacteristic
82Dic362—21no—
83Dic362—21no—
84Dic362—42no—
85C22⋊D181—42no—
86C2×Dic181—21no—
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—
96#13089678—22no—
97C23×C181—21no—
98Q8×C181—22no—
99Q8×C181—22no—
100D4×C181—21no—
101D4×C181—21no—
102C2×C4×C181—21no—
103C2×C4×C181—22no—
104C2×C4×C181—42no—
105C2×C4×C181C2×D411yescharacteristic
106C2×C4×C181C2×D411yescharacteristic
107C2×C4×C181C2411yescharacteristic
108C8×C181—22no—
109C22×C181—41no—
110C22×C181—41no—
111C22×C181—41no—
112C22×C181C22×D411yescharacteristic
113C2×D181—21no—
114C2×D181—21no—
115C2×D181—21no—
116C4×C181—21no—
117C4×C181—42no—
118C4×C181—41no—
119C4×C181#13712yes—
120C4×C181H(5,𝔽2)11yescharacteristic
121C4×C1812−1+411yescharacteristic
122Dic181—21no—
123Dic181—21no—
124Dic181—21no—
125Dic181—41no—
126Dic181—41no—
127Dic181—42no—
128C2×C181—81no—
129C2×C181#53711yescharacteristic
130C2×C181C2×#13711yescharacteristic
131C2×C181#57511yescharacteristic
132D181—41no—
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#462—1811no—
144#461D18111yescharacteristic, complemented
145#423—1811no—
146C2×Q16—1812no—
147C2×SD16C2×D18112yes—
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—1812no—
158C22⋊Q8—1811no—
159#121—1811no—
160C4×Q8—3621no—
161C4×Q8—1812no—
162C4×Q8—1811no—
163C4×D4—3621no—
164C4×D4C2×D18111yescharacteristic
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×Q8—1811no—
179C2×Q8C22×D18112yes—
180C2×D4—3621no—
181C2×D4C22×D18111yescharacteristic
182C22×C4—21no—
183C22×C4—1811no—
184C22×C4—1811no—
185C22×C4—1811no—
186Q16—7242no—
187SD16—42no—
188M16—7242no—
189C2×C8—1812no—
190C2×C8C22×D18112yes—
191C4⋊C4—21no—
192C4⋊C4—21no—
193C4⋊C4—3621no—
194C4⋊C4—3621no—
195C4⋊C4—3621no—
196C4⋊C4—3622no—
197C4⋊C4—3622no—
198C4⋊C4—3621no—
199C4⋊C4—3621no—
200C4⋊C4—1812no—
201C4⋊C4—1811no—
202C4⋊C4C22×D18111yescharacteristic
203C22⋊C4—21no—
204C22⋊C4—3622no—
205C22⋊C4—3621no—
206C22⋊C4—3621no—
207C22⋊C4—3621no—
208C22⋊C4—3621no—
209C42—3622no—
210C42—3621no—
211C42—1811no—
212C42—1811no—
213C42—1811no—
214C42C22×D18111yescharacteristic
215C23—21no—
216C23—1811no—
217Q8—22no—
218Q8—22no—
219Q8—3621no—
220Q8—3621no—
221Q8—7242no—
222D4—21no—
223D4—21no—
224D4—7242no—
225C2×C4—22no—
226C2×C4—21no—
227C2×C4—42no—
228C2×C4—3622no—
229C2×C4—3621no—
230C2×C4—3621no—
231C2×C4—3621no—
232C2×C4—3621no—
233C2×C4—3622no—
234C2×C4—1811no—
235C2×C4—1811no—
236C2×C4—1811no—
237C2×C4—7241no—
238C2×C4D4×D18111yescharacteristic
239C2×C4D4×D18111yescharacteristic
240C2×C4C23×D18111yescharacteristic
241C8—22no—
242C8—3622no—
243C22—41no—
244C22—41no—
245C22—41no—
246C22—3621no—
247C22—3621no—
248C22—3621no—
249C22C2×D4×D18111yescharacteristic
250C4—21no—
251C4—42no—
252C4—41no—
253C4—3621no—
254C4—3621no—
255C4—3621no—
256C4—7241no—
257C4—7241no—
258C4—7242no—
259C4#1915384512yes—
260C4#1915386111yescharacteristic
261C4#1915386311yescharacteristic
262C2—81no—
263C2—7241no—
264C2#2604185811yescharacteristic
265C2C2×#1915384511yescharacteristic
266C2#2604207811yescharacteristic
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#461×C181C2
socleC22×C181C22×D4
prime radical#19153787C22
hypercenter#461D181
hypocenterC181#3252
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C42 ◁ #461

Complements

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

NH ≅ G/Ntype
1. #26041834C2semidirect
3. #26041588C2semidirect
4. #26041575C2semidirect
5. #26041569C2semidirect
6. #26041536C2semidirect
8. #26041499C2semidirect
9. #26041486C2semidirect
10. #26041301C2semidirect
11. #461×C181C2semidirect
133. C181#3252semidirect
144. #461D181semidirect

Maximal subgroups

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

subgroupindex
#260418342
#260418112
#260415882
#260415752
#260415692
#260415362
#260415222
#260414992
#260414862
#260413012
#461×C1812
#3252181