Finite Group Databasebeta

← #37544896

Subgroups — #37544896

Order 23168.

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

normal subgroups only

This group has 332 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#26041836C211yesmaximal, characteristic, complemented
2#26041678C212yesmaximal, complemented
3#26041633C212yesmaximal, complemented
4#26041617C212yesmaximal, complemented
5#26041600C212yesmaximal
6#26041574C211yesmaximal, characteristic, complemented
7#26041569C211yesmaximal, characteristic, complemented
8#26041564C211yesmaximal, characteristic, complemented
9#26041559C211yesmaximal, characteristic, complemented
10#26041332C211yesmaximal, characteristic, complemented
11#529×C181C211yesmaximal, characteristic
12#19153816C2211yescharacteristic
13#19153816C2211yescharacteristic
14#19153808C2212yes—
15#19153795—21no—
16#19153795—21no—
17C4×Dic362—21no—
18C4×Dic362—21no—
19#19153775C2212yes—
20#19153749C2212yes—
21#19153749C2212yes—
22#19153747C2212yes—
23#19153747C2212yes—
24C4⋊#13089678C2211yescharacteristic
25C4⋊#13089678C2211yescharacteristic
26#19153740C2212yes—
27#19153740C2212yes—
28#19153737C2212yes—
29#19153737C2212yes—
30C22⋊#13089678C2212yes—
31C4×#13089678C2212yes—
32#126×C181—21no—
33#125×C181C2211yescharacteristic
34#124×C181C2211yescharacteristic
35#123×C181—21no—
36C22⋊Q8×C181C2211yescharacteristic
37#121×C181C2211yescharacteristic
38C4×Q8×C181C2211yescharacteristic
39C4×D4×C181—21no—
40C4×D4×C181C2211yescharacteristic
41#117×C181—21no—
42C2×C42×C181C2211yescharacteristic
43C2×Dic362—21no—
44C2×Dic362—21no—
45C2×#13089678C2312yes—
46C2×#13089678C2312yes—
47C4⋊Dic181—21no—
48C4⋊Dic181—21no—
49C4⋊Dic181C2312yes—
50C4⋊Dic181C2312yes—
51#15668428—22no—
52#15668428—22no—
53C22⋊Dic181—22no—
54C4×Dic181—22no—
55C2×Q8×C181C2311yescharacteristic
56C2×D4×C181—21no—
57C2×D4×C181C2311yescharacteristic
58C22×C4×C181—21no—
59C22×C4×C181D411yescharacteristic
60C22×C4×C181D411yescharacteristic
61C22×C4×C181C2311yescharacteristic
62C4⋊C4×C181—21no—
63C4⋊C4×C181—21no—
64C4⋊C4×C181—21no—
65C4⋊C4×C181—21no—
66C4⋊C4×C181C2311yescharacteristic
67C4⋊C4×C181C2311yescharacteristic
68C22⋊C4×C181—21no—
69C22⋊C4×C181—21no—
70C22⋊C4×C181—21no—
71C22⋊C4×C181—21no—
72C22⋊C4×C181—21no—
73C42×C181—21no—
74C42×C181D411yescharacteristic
75C42×C181D411yescharacteristic
76C42×C181C2311yescharacteristic
77C42×C181C2311yescharacteristic
78Dic362—42no—
79Dic362—42no—
80C2×Dic181—22no—
81C2×Dic181—22no—
82#13089678—22no—
83#13089678—22no—
84C23×C181—21no—
85C23×C181C2×D411yescharacteristic
86Q8×C181—21no—
87Q8×C181—21no—
88D4×C181—21no—
89D4×C181—21no—
90D4×C181—42no—
91C2×C4×C181—22no—
92C2×C4×C181—21no—
93C2×C4×C181—21no—
94C2×C4×C181—21no—
95C2×C4×C181—21no—
96C2×C4×C181—21no—
97C2×C4×C181—41no—
98C2×C4×C181—42no—
99C2×C4×C181C2×D411yescharacteristic
100C2×C4×C181C2×D411yescharacteristic
101C2×C4×C181C2×D411yescharacteristic
102C2×C4×C181C2×D411yescharacteristic
103C2×C4×C181C2×D411yescharacteristic
104C2×C4×C181C2411yescharacteristic
105C22×C181—21no—
106C22×C181—21no—
107C22×C181—21no—
108C22×C181—41no—
109C22×C181—41no—
110C22×C181—41no—
111C22×C181C22×D411yescharacteristic
112C4×C181—21no—
113C4×C181—21no—
114C4×C181—21no—
115C4×C181—21no—
116C4×C181—41no—
117C4×C181—41no—
118C4×C181—41no—
119C4×C181#13512yes—
120C4×C1812−1+411yescharacteristic
121C4×C1812−1+411yescharacteristic
122Dic181—42no—
123Dic181—42no—
124C2×C181—41no—
125C2×C181—81no—
126C2×C181#53611yescharacteristic
127C2×C181C2×#13511yescharacteristic
128C2×C181#57511yescharacteristic
129C181#323311yescharacteristic, complemented, Sylow
130#3233—1811nomaximal, Sylow
131#533—1811no—
132#529D18111yescharacteristic
133#499—1812no—
134#488—1812no—
135#485—1812no—
136#482—1812no—
137#477—1811no—
138#476—1811no—
139#475—1811no—
140#474—1811no—
141#431—1811no—
142C4⋊Q8—1811no—
143C4⋊Q8—1811no—
144#126—21no—
145#125—1812no—
146#125C2×D18111yescharacteristic
147#124C2×D18111yescharacteristic
148#123—21no—
149C22⋊Q8—3621no—
150C22⋊Q8—3621no—
151C22⋊Q8C2×D18111yescharacteristic
152#121C2×D18111yescharacteristic
153C4×Q8—3621no—
154C4×Q8—3621no—
155C4×Q8C2×D18111yescharacteristic
156C4×D4—21no—
157C4×D4C2×D18111yescharacteristic
158#117—21no—
159#117—1812no—
160C2×C42C2×D18111yescharacteristic
161#107—1812no—
162#107—1812no—
163#106—1812no—
164#106—1812no—
165C4⋊C8—1811no—
166C4⋊C8—1811no—
167#103—1812no—
168#103—1812no—
169#102—1812no—
170#102—1812no—
171C22⋊C8—1812no—
172C4×C8—1812no—
173C2×Q8—3621no—
174C2×Q8—3621no—
175C2×Q8C22×D18111yescharacteristic
176C2×D4—21no—
177C2×D4C22×D18111yescharacteristic
178C22×C4—21no—
179C22×C4C22⋊D18111yescharacteristic
180C22×C4C22⋊D18111yescharacteristic
181C22×C4C22×D18111yescharacteristic
182C2×C8—1812no—
183C2×C8—1812no—
184C4⋊C4—21no—
185C4⋊C4—21no—
186C4⋊C4—21no—
187C4⋊C4—21no—
188C4⋊C4—3622no—
189C4⋊C4—3622no—
190C4⋊C4—3621no—
191C4⋊C4—3621no—
192C4⋊C4—1812no—
193C4⋊C4—1812no—
194C4⋊C4C22×D18111yescharacteristic
195C4⋊C4C22×D18111yescharacteristic
196C22⋊C4—21no—
197C22⋊C4—21no—
198C22⋊C4—21no—
199C22⋊C4—21no—
200C22⋊C4—21no—
201C22⋊C4—3622no—
202C42—21no—
203C42—3622no—
204C42C22⋊D18111yescharacteristic
205C42C22⋊D18111yescharacteristic
206C42C22×D18111yescharacteristic
207C42C22×D18111yescharacteristic
208C23—21no—
209C23C2×C22⋊D18111yescharacteristic
210Q8—21no—
211Q8—21no—
212Q8—7242no—
213Q8—7242no—
214D4—21no—
215D4—21no—
216D4—42no—
217C2×C4—22no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—41no—
224C2×C4—42no—
225C2×C4—3622no—
226C2×C4—3622no—
227C2×C4C2×C22⋊D18111yescharacteristic
228C2×C4C2×C22⋊D18111yescharacteristic
229C2×C4C2×C22⋊D18111yescharacteristic
230C2×C4C2×C22⋊D18111yescharacteristic
231C2×C4C2×C22⋊D18111yescharacteristic
232C2×C4C23×D18111yescharacteristic
233C8—3622no—
234C8—3622no—
235C22—21no—
236C22—21no—
237C22—21no—
238C22—41no—
239C22—41no—
240C22—41no—
241C22C22×C22⋊D18111yescharacteristic
242C4—21no—
243C4—21no—
244C4—21no—
245C4—21no—
246C4—41no—
247C4—41no—
248C4—41no—
249C4—7242no—
250C4—7242no—
251C4#1915383612yes—
252C4#1915386311yescharacteristic
253C4#1915386311yescharacteristic
254C2—41no—
255C2—81no—
256C2#2604185011yescharacteristic
257C2C2×#1915383611yescharacteristic
258C2#2604207711yescharacteristic
subgroupG/H
centerC22C22×C22⋊D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#529×C181C2
socleC22×C181C22×D4
prime radical#121×C181C22
hypercenter#529D181
hypocenterC181#3233
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C2×C42 ◁ #529

Complements

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

NH ≅ G/Ntype
1. #26041836C2semidirect
2. #26041678C2semidirect
3. #26041633C2semidirect
4. #26041617C2semidirect
6. #26041574C2semidirect
7. #26041569C2semidirect
8. #26041564C2semidirect
9. #26041559C2semidirect
10. #26041332C2semidirect
129. C181#3233semidirect

Maximal subgroups

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

subgroupindex
#260418362
#260416782
#260416332
#260416172
#260416002
#260415742
#260415692
#260415642
#260415592
#260413322
#529×C1812
#3233181