Finite Group Databasebeta

← #37544900

Subgroups — #37544900

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#26041813C211yesmaximal, characteristic
2#26041678C212yesmaximal, complemented
3#26041636C212yesmaximal, complemented
4#26041620C212yesmaximal, complemented
5#26041603C212yesmaximal
6#26041571C211yesmaximal, characteristic, complemented
7#26041566C211yesmaximal, characteristic, complemented
8#26041561C211yesmaximal, characteristic, complemented
9#26041556C211yesmaximal, characteristic, complemented
10#26041332C211yesmaximal, characteristic, complemented
11#533×C181C211yesmaximal, characteristic, complemented
12#19153810—21no—
13#19153808C2211yescharacteristic
14#19153804C2211yescharacteristic
15#19153799—21no—
16#19153795C2211yescharacteristic
17#19153790C2211yescharacteristic
18C4×Dic362C2211yescharacteristic
19C4×D724C2211yescharacteristic
20C4×C22⋊D181—21no—
21#19153773—21no—
22#19153749C2212yes—
23#19153749C2212yes—
24#19153747C2212yes—
25#19153747C2212yes—
26C4⋊#13089678C2211yescharacteristic
27C4⋊#13089678C2211yescharacteristic
28#19153739C2212yes—
29#19153739C2212yes—
30#19153736C2212yes—
31#19153736C2212yes—
32C22⋊#13089678C2212yes—
33C4×#13089678C2212yes—
34C4⋊Q8×C181C2211yescharacteristic
35C4⋊Q8×C181C2211yescharacteristic
36#125×C181C2212yes—
37C22⋊Q8×C181—21no—
38C22⋊Q8×C181—21no—
39C4×Q8×C181—21no—
40C4×Q8×C181—21no—
41#117×C181C2212yes—
42C2×C42×C181C2211yescharacteristic
43C2×Dic362C2311yescharacteristic
44C2×D724C2311yescharacteristic
45C2×C22⋊D181—21no—
46C2×C4×D181—21no—
47C2×#13089678C2312yes—
48C2×#13089678C2312yes—
49C4⋊Dic181C2311yescharacteristic
50C4⋊Dic181C2311yescharacteristic
51#15668428—21no—
52#15668428—21no—
53#15668428—21no—
54#15668428—21no—
55C22⋊Dic181—21no—
56#15668426—21no—
57#15668426—21no—
58#15668426—21no—
59#15668426—21no—
60C4×Dic181—21no—
61C2×Q8×C181—21no—
62C2×Q8×C181—21no—
63C22×C4×C181D411yescharacteristic
64C22×C4×C181D411yescharacteristic
65C22×C4×C181C2311yescharacteristic
66C4⋊C4×C181—22no—
67C4⋊C4×C181—22no—
68C4⋊C4×C181—21no—
69C4⋊C4×C181—21no—
70C4⋊C4×C181C2312yes—
71C4⋊C4×C181C2312yes—
72C22⋊C4×C181—22no—
73C42×C181—22no—
74C42×C181D411yescharacteristic
75C42×C181D411yescharacteristic
76C42×C181C2311yescharacteristic
77C42×C181C2311yescharacteristic
78C22×D181—21no—
79Dic362—21no—
80Dic362—21no—
81D724—21no—
82D724—21no—
83C22⋊D181—42no—
84C2×Dic181—21no—
85C2×Dic181—21no—
86C2×Dic181—21no—
87C4×D181—42no—
88#13089678—22no—
89#13089678—22no—
90C23×C181C2×D411yescharacteristic
91Q8×C181—42no—
92Q8×C181—42no—
93C2×C4×C181—22no—
94C2×C4×C181—21no—
95C2×C4×C181—21no—
96C2×C4×C181—22no—
97C2×C4×C181—22no—
98C2×C4×C181—41no—
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×C181C22×D411yescharacteristic
109C2×D181—41no—
110C2×D181—41no—
111C2×D181—41no—
112C4×C181—21no—
113C4×C181—21no—
114C4×C181—21no—
115C4×C181—21no—
116C4×C181—42no—
117C4×C181—42no—
118C4×C181#13512yes—
119C4×C1812−1+411yescharacteristic
120C4×C1812−1+411yescharacteristic
121Dic181—41no—
122Dic181—41no—
123Dic181—41no—
124C2×C181—41no—
125C2×C181#53611yescharacteristic
126C2×C181C2×#13511yescharacteristic
127C2×C181#57511yescharacteristic
128D181—81no—
129C181#323311yescharacteristic, complemented, Sylow
130#3233—1811nomaximal, Sylow
131#533D18111yescharacteristic, complemented
132#529—1811no—
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⋊Q8C2×D18111yescharacteristic
143C4⋊Q8C2×D18111yescharacteristic
144#126—3621no—
145#125—1811no—
146#125C2×D18112yes—
147#124—1811no—
148#123—3621no—
149C22⋊Q8—21no—
150C22⋊Q8—21no—
151C22⋊Q8—1811no—
152#121—1811no—
153C4×Q8—21no—
154C4×Q8—21no—
155C4×Q8—1811no—
156C4×D4—3621no—
157C4×D4—1811no—
158#117—3621no—
159#117C2×D18112yes—
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—21no—
174C2×Q8—21no—
175C2×Q8—1811no—
176C2×D4—3621no—
177C2×D4—1811no—
178C22×C4—3621no—
179C22×C4C22⋊D18111yescharacteristic
180C22×C4C22⋊D18111yescharacteristic
181C22×C4C22×D18111yescharacteristic
182C2×C8—1812no—
183C2×C8—1812no—
184C4⋊C4—22no—
185C4⋊C4—22no—
186C4⋊C4—21no—
187C4⋊C4—21no—
188C4⋊C4—3621no—
189C4⋊C4—3621no—
190C4⋊C4—3621no—
191C4⋊C4—3621no—
192C4⋊C4—1811no—
193C4⋊C4—1811no—
194C4⋊C4C22×D18112yes—
195C4⋊C4C22×D18112yes—
196C22⋊C4—22no—
197C22⋊C4—3621no—
198C22⋊C4—3621no—
199C22⋊C4—3621no—
200C22⋊C4—3621no—
201C22⋊C4—3621no—
202C42—22no—
203C42—3621no—
204C42C22⋊D18111yescharacteristic
205C42C22⋊D18111yescharacteristic
206C42C22×D18111yescharacteristic
207C42C22×D18111yescharacteristic
208C23—3621no—
209C23C2×C22⋊D18111yescharacteristic
210Q8—42no—
211Q8—42no—
212Q8—3621no—
213Q8—3621no—
214D4—3621no—
215D4—3621no—
216D4—7242no—
217C2×C4—22no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—41no—
223C2×C4—3621no—
224C2×C4—3621no—
225C2×C4—3621no—
226C2×C4—7242no—
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—7241no—
239C22—7241no—
240C22—7241no—
241C22C22×C22⋊D18111yescharacteristic
242C4—21no—
243C4—21no—
244C4—21no—
245C4—21no—
246C4—42no—
247C4—42no—
248C4—7241no—
249C4—7241no—
250C4—7241no—
251C4#1915383612yes—
252C4#1915386411yescharacteristic
253C4#1915386411yescharacteristic
254C2—41no—
255C2—14481no—
256C2#2604185311yescharacteristic
257C2C2×#1915383611yescharacteristic
258C2#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#3233
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. #26041678C2semidirect
3. #26041636C2semidirect
4. #26041620C2semidirect
6. #26041571C2semidirect
7. #26041566C2semidirect
8. #26041561C2semidirect
9. #26041556C2semidirect
10. #26041332C2semidirect
11. #533×C181C2semidirect
129. C181#3233semidirect
131. #533D181semidirect

Maximal subgroups

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

subgroupindex
#260418132
#260416782
#260416362
#260416202
#260416032
#260415712
#260415662
#260415612
#260415562
#260413322
#533×C1812
#3233181