Finite Group Databasebeta

← #37544898

Subgroups — #37544898

Order 23168.

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

normal subgroups only

This group has 332 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#26041680C212yesmaximal, complemented
4#26041632C212yesmaximal, complemented
5#26041622C212yesmaximal, complemented
6#26041601C212yesmaximal
7#26041575C211yesmaximal, characteristic, complemented
8#26041569C211yesmaximal, characteristic, complemented
9#26041561C211yesmaximal, characteristic, complemented
10#26041331C211yesmaximal, characteristic, complemented
11#474×C181C211yesmaximal, characteristic, complemented
12#19153816C2211yescharacteristic
13#19153809—21no—
14#19153807C2212yes—
15#19153807C2211yescharacteristic
16#19153803C2211yescharacteristic
17#19153798—21no—
18#19153796—21no—
19#19153796C2211yescharacteristic
20#19153792—21no—
21#19153787C2211yescharacteristic
22Q8×Dic181—21no—
23C4×Dic362C2211yescharacteristic
24#19153781—21no—
25C4×C22⋊D181—21no—
26#19153775—21no—
27#19153774C2212yes—
28C42×D181C2211yescharacteristic
29#19153750C2212yes—
30#19153747C2212yes—
31#19153747C2212yes—
32C4⋊#13089678C2211yescharacteristic
33#19153741C2212yes—
34#19153739C2212yes—
35#19153737C2212yes—
36#19153728C2212yes—
37C8×Dic181C2212yes—
38C4⋊Q8×C181C2211yescharacteristic
39C4×D4×C181C2211yescharacteristic
40#107×C181C2212yes—
41C4⋊C8×C181C2211yescharacteristic
42#102×C181C2212yes—
43C2×Dic362—21no—
44C2×Dic362C2311yescharacteristic
45C2×C22⋊D181—21no—
46C2×C4×D181D411yescharacteristic
47C2×C4×D181D411yescharacteristic
48C2×C4×D181C2311yescharacteristic
49C2×#13089678C2312yes—
50C4⋊Dic181—21no—
51C4⋊Dic181—21no—
52C4⋊Dic181—22no—
53C4⋊Dic181—21no—
54C4⋊Dic181C2312yes—
55C4⋊Dic181C2311yescharacteristic
56#15668428—21no—
57#15668428—22no—
58#15668428—21no—
59C22⋊Dic181—21no—
60C22⋊Dic181—21no—
61C22⋊Dic181—21no—
62#15668426—22no—
63#15668426—21no—
64C4×Dic181—22no—
65C4×Dic181—21no—
66C4×Dic181D411yescharacteristic
67C4×Dic181D411yescharacteristic
68C4×Dic181C2311yescharacteristic
69C2×Q8×C181—21no—
70C2×D4×C181C2311yescharacteristic
71C22×C4×C181—21no—
72C2×C8×C181C2312yes—
73C4⋊C4×C181—21no—
74C4⋊C4×C181C2312yes—
75C4⋊C4×C181C2311yescharacteristic
76C22⋊C4×C181—21no—
77C42×C181C2311yescharacteristic
78C22×D181C2×D411yescharacteristic
79Dic362—21no—
80Dic362—21no—
81Dic362—42no—
82C22⋊D181—42no—
83C2×Dic181—21no—
84C2×Dic181—22no—
85C2×Dic181—21no—
86C2×Dic181C2×D411yescharacteristic
87C2×Dic181C2×D411yescharacteristic
88C2×Dic181C2×D411yescharacteristic
89C4×D181—22no—
90C4×D181—21no—
91C4×D181—21no—
92C4×D181—41no—
93#13089678—22no—
94C23×C181—21no—
95Q8×C181—42no—
96D4×C181—21no—
97D4×C181—21no—
98C2×C4×C181—21no—
99C2×C4×C181—22no—
100C2×C4×C181—42no—
101C2×C4×C181C2×D411yescharacteristic
102C2×C4×C181C2×D411yescharacteristic
103C2×C4×C181C2411yescharacteristic
104C8×C181—22no—
105C22×C181—41no—
106C22×C181—41no—
107C22×C181—41no—
108C22×C181C22×D411yescharacteristic
109C2×D181—21no—
110C2×D181—21no—
111C2×D181—21no—
112C4×C181—21no—
113C4×C181—41no—
114C4×C181—42no—
115C4×C181#13512yes—
116C4×C1812−1+411yescharacteristic
117C4×C1812−1+411yescharacteristic
118Dic181—21no—
119Dic181—21no—
120Dic181—21no—
121Dic181—41no—
122Dic181—41no—
123Dic181—42no—
124C2×C181—81no—
125C2×C181#53611yescharacteristic
126C2×C181C2×#13511yescharacteristic
127C2×C181#57511yescharacteristic
128D181—41no—
129C181#323311yescharacteristic, complemented, Sylow
130#3233—1811nomaximal, Sylow
131#533—1811no—
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#474D18111yescharacteristic, complemented
141#431—1811no—
142C4⋊Q8—1811no—
143C4⋊Q8C2×D18111yescharacteristic
144#126—3621no—
145#125—1811no—
146#125—1812no—
147#124—1811no—
148#123—3621no—
149C22⋊Q8—3621no—
150C22⋊Q8—3621no—
151C22⋊Q8—1811no—
152#121—1811no—
153C4×Q8—3621no—
154C4×Q8—3621no—
155C4×Q8—1811no—
156C4×D4—3621no—
157C4×D4C2×D18111yescharacteristic
158#117—3621no—
159#117—1812no—
160C2×C42—1811no—
161#107—1812no—
162#107C2×D18112yes—
163#106—1812no—
164#106—1812no—
165C4⋊C8—1811no—
166C4⋊C8C2×D18111yescharacteristic
167#103—1812no—
168#103—1812no—
169#102—1812no—
170#102C2×D18112yes—
171C22⋊C8—1812no—
172C4×C8—1812no—
173C2×Q8—21no—
174C2×Q8—3621no—
175C2×Q8—1811no—
176C2×D4—3621no—
177C2×D4C22×D18111yescharacteristic
178C22×C4—21no—
179C22×C4—1811no—
180C22×C4—1811no—
181C22×C4—1811no—
182C2×C8—1812no—
183C2×C8C22×D18112yes—
184C4⋊C4—21no—
185C4⋊C4—3621no—
186C4⋊C4—3621no—
187C4⋊C4—3621no—
188C4⋊C4—3622no—
189C4⋊C4—3622no—
190C4⋊C4—3621no—
191C4⋊C4—3621no—
192C4⋊C4—1812no—
193C4⋊C4—1811no—
194C4⋊C4C22×D18112yes—
195C4⋊C4C22×D18111yescharacteristic
196C22⋊C4—21no—
197C22⋊C4—3622no—
198C22⋊C4—3621no—
199C22⋊C4—3621no—
200C22⋊C4—3621no—
201C22⋊C4—3621no—
202C42—3622no—
203C42—3621no—
204C42—1811no—
205C42—1811no—
206C42—1811no—
207C42C22×D18111yescharacteristic
208C23—21no—
209C23—1811no—
210Q8—42no—
211Q8—3621no—
212Q8—3621no—
213Q8—7242no—
214D4—21no—
215D4—21no—
216D4—7242no—
217C2×C4—22no—
218C2×C4—21no—
219C2×C4—42no—
220C2×C4—3622no—
221C2×C4—3621no—
222C2×C4—3621no—
223C2×C4—3621no—
224C2×C4—3621no—
225C2×C4—3622no—
226C2×C4—1811no—
227C2×C4—1811no—
228C2×C4—1811no—
229C2×C4—7241no—
230C2×C4D4×D18111yescharacteristic
231C2×C4D4×D18111yescharacteristic
232C2×C4C23×D18111yescharacteristic
233C8—22no—
234C8—3622no—
235C22—41no—
236C22—41no—
237C22—41no—
238C22—3621no—
239C22—3621no—
240C22—3621no—
241C22C2×D4×D18111yescharacteristic
242C4—21no—
243C4—41no—
244C4—42no—
245C4—3621no—
246C4—3621no—
247C4—3621no—
248C4—7241no—
249C4—7242no—
250C4—7241no—
251C4#1915383512yes—
252C4#1915386311yescharacteristic
253C4#1915386411yescharacteristic
254C2—81no—
255C2—7241no—
256C2#2604185211yescharacteristic
257C2C2×#1915383511yescharacteristic
258C2#2604207811yescharacteristic
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#474×C181C2
socleC22×C181C22×D4
prime radical#19153787C22
hypercenter#474D181
hypocenterC181#3233
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C42 ◁ #474

Complements

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

NH ≅ G/Ntype
1. #26041834C2semidirect
3. #26041680C2semidirect
4. #26041632C2semidirect
5. #26041622C2semidirect
7. #26041575C2semidirect
8. #26041569C2semidirect
9. #26041561C2semidirect
10. #26041331C2semidirect
11. #474×C181C2semidirect
129. C181#3233semidirect
140. #474D181semidirect

Maximal subgroups

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

subgroupindex
#260418342
#260418112
#260416802
#260416322
#260416222
#260416012
#260415752
#260415692
#260415612
#260413312
#474×C1812
#3233181