Finite Group Databasebeta

← #37543552

Subgroups — #37543552

Order 23168.

398 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 398 conjugacy classes of subgroups — too many to draw. Only its 149 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#26041907C211yesmaximal, characteristic, complemented
2#26041579C212yesmaximal, complemented
3#26041579C212yesmaximal, complemented
4#26041569C212yesmaximal, complemented
5#26041559C212yesmaximal, complemented
6#26041322C211yesmaximal, characteristic, complemented
7#26041315C211yesmaximal, characteristic, complemented
8#26041306C211yesmaximal, characteristic, complemented
9#26041284C211yesmaximal, characteristic
10C2×#19153737C211yesmaximal, characteristic
11#518×C181C211yesmaximal, characteristic
12C2×#15668433C2211yescharacteristic
13C22×#13089678C2211yescharacteristic
14#19153816C2212yescomplemented
15#19153815—21no—
16#19153808C2212yescomplemented
17#19153807—21no—
18#19153793—22no—
19#19153775C2211yescharacteristic
20C2×C4⋊Dic181C2211yescharacteristic
21#19153749C2211yescharacteristic, complemented
22#19153749C2212yescomplemented
23#19153749C2211yescharacteristic, complemented
24#19153747C2211yescharacteristic, complemented
25#19153747C2212yescomplemented
26#19153747C2211yescharacteristic, complemented
27C4⋊#13089678C2212yescomplemented
28C4⋊#13089678C2212yescomplemented
29#19153737C2212yes—
30#19153737C2212yes—
31#19153737C2212yes—
32#19153737C2212yes—
33C22×D4×C181C2211yescharacteristic
34C4×D4×C181—21no—
35C4×D4×C181—21no—
36C4×D4×C181C2212yes—
37C4×D4×C181C2212yes—
38#117×C181C2211yescharacteristic
39#117×C181C2211yescharacteristic
40C2×C22⋊C4×C181—22no—
41C2×Dic362—22no—
42C22×Dic181—21no—
43#15668433—22no—
44C2×#13089678—22no—
45C2×#13089678C2311yescharacteristic
46C2×#13089678C2312yes—
47C2×#13089678C2311yescharacteristic
48C4⋊Dic181—21no—
49C4⋊Dic181—22no—
50C4⋊Dic181C2312yes—
51C4⋊Dic181C2311yescharacteristic
52C4⋊Dic181C2311yescharacteristic
53#15668428—22no—
54#15668428—22no—
55C22⋊Dic181—21no—
56C4×Dic181—21no—
57C24×C181—21no—
58C2×D4×C181—21no—
59C2×D4×C181—22no—
60C2×D4×C181Q814yescomplemented
61C2×D4×C181C2312yes—
62C22×C4×C181—22no—
63C22×C4×C181—22no—
64C22×C4×C181C2311yescharacteristic
65C4⋊C4×C181D411yescharacteristic
66C4⋊C4×C181D411yescharacteristic
67C4⋊C4×C181C2311yescharacteristic
68C4⋊C4×C181C2311yescharacteristic
69C22⋊C4×C181—21no—
70C22⋊C4×C181—22no—
71C22⋊C4×C181—22no—
72C22⋊C4×C181D412yes—
73C42×C181—21no—
74C42×C181C2312yes—
75Dic362—82no—
76C2×Dic181—21no—
77C2×Dic181—21no—
78C2×Dic181—22no—
79C2×Dic181—81no—
80#13089678—21no—
81#13089678—21no—
82#13089678—22no—
83C23×C181—22no—
84C23×C181—24no—
85C23×C181—81no—
86C23×C181—81no—
87C23×C181C2×D411yescharacteristic
88D4×C181—24no—
89D4×C181—22no—
90D4×C181C2×Q814yes—
91C2×C4×C181—21no—
92C2×C4×C181—21no—
93C2×C4×C181—82no—
94C2×C4×C181—82no—
95C2×C4×C181C2×D412yes—
96C2×C4×C181C2×D412yes—
97C2×C4×C181C2×D411yescharacteristic
98C2×C4×C181C2×Q812yes—
99C2×C4×C181#4112yes—
100C2×C4×C181C2411yescharacteristic
101C22×C181—21no—
102C22×C181—21no—
103C22×C181—24no—
104C22×C181—42no—
105C22×C181—42no—
106C22×C181—84no—
107C22×C181C22⋊Q812yes—
108C22×C181C22×D411yescharacteristic
109C4×C181—22no—
110C4×C181—22no—
111C4×C181—41no—
112C4×C181—41no—
113C4×C181C22⋊Q812yes—
114C4×C181C22×Q811yescharacteristic
115C4×C181C2×#4111yescharacteristic
116Dic181—42no—
117Dic181—41no—
118Dic181—41no—
119C2×C181—22no—
120C2×C181—44no—
121C2×C181C2×C22⋊Q811yescharacteristic
122C2×C181#57611yescharacteristic
123C2×C181#57711yescharacteristic
124C181#308111yescharacteristic, complemented, Sylow
125#3081—1811nomaximal, Sylow
126#544—1811no—
127#518D18111yescharacteristic
128#478—1812no—
129#478—1812no—
130#476—1812no—
131#474—1812no—
132#428—1811no—
133#427—1811no—
134#424—1811no—
135#418—1811no—
136C2×#102—1811no—
137C22×D4C2×D18111yescharacteristic
138C2×M16—1811no—
139C22×C8—1811no—
140C4⋊Q8—3621no—
141C4⋊Q8—1812no—
142#125—3621no—
143#125—1812no—
144C22⋊Q8—3622no—
145C4×D4—21no—
146C4×D4—21no—
147C4×D4C2×D18112yes—
148C4×D4C2×D18112yes—
149#117—1811no—
150#117C2×D18111yescharacteristic
151#117C2×D18111yescharacteristic
152C2×C4⋊C4—1811no—
153C2×C22⋊C4—22no—
154#107—1811no—
155#107—1811no—
156#107—1812no—
157#106—1811no—
158#106—1811no—
159#106—1812no—
160C4⋊C8—1812no—
161C4⋊C8—1812no—
162#102—1812no—
163#102—1812no—
164#102—1812no—
165#102—1812no—
166C24—21no—
167C2×Q8—3622no—
168C2×D4—21no—
169C2×D4—22no—
170C2×D4Dic36214yescomplemented
171C2×D4C22×D18112yes—
172C22×C4—22no—
173C22×C4—22no—
174C22×C4—3621no—
175C22×C4C22×D18111yescharacteristic
176M16—3622no—
177C2×C8—3622no—
178C2×C8—1811no—
179C2×C8—1812no—
180C2×C8—1811no—
181C4⋊C4—3621no—
182C4⋊C4—3622no—
183C4⋊C4—3622no—
184C4⋊C4—3622no—
185C4⋊C4—1812no—
186C4⋊C4—1811no—
187C4⋊C4—1811no—
188C4⋊C4C22⋊D18111yescharacteristic
189C4⋊C4C22⋊D18111yescharacteristic
190C4⋊C4C22×D18111yescharacteristic
191C4⋊C4C22×D18111yescharacteristic
192C22⋊C4—21no—
193C22⋊C4—22no—
194C22⋊C4—22no—
195C22⋊C4—3621no—
196C22⋊C4C22⋊D18112yes—
197C42—21no—
198C42—3621no—
199C42C22×D18112yes—
200C23—22no—
201C23—24no—
202C23—81no—
203C23—81no—
204C23C2×C22⋊D18111yescharacteristic
205Q8—14482no—
206D4—24no—
207D4—22no—
208D4C2×Dic36214yes—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—82no—
212C2×C4—82no—
213C2×C4—3621no—
214C2×C4—3621no—
215C2×C4—3622no—
216C2×C4—14481no—
217C2×C4C2×C22⋊D18112yes—
218C2×C4C2×C22⋊D18112yes—
219C2×C4C2×C22⋊D18111yescharacteristic
220C2×C4C2×Dic36212yes—
221C2×C4#1566844912yes—
222C2×C4C23×D18111yescharacteristic
223C8—3621no—
224C8—3621no—
225C8—3622no—
226C22—21no—
227C22—21no—
228C22—24no—
229C22—42no—
230C22—42no—
231C22—84no—
232C22#1915379512yes—
233C22C22×C22⋊D18111yescharacteristic
234C4—22no—
235C4—22no—
236C4—41no—
237C4—41no—
238C4—7242no—
239C4—7241no—
240C4—7241no—
241C4#1915379512yes—
242C4C22×Dic36211yescharacteristic
243C4C2×#1566844911yescharacteristic
244C2—22no—
245C2—44no—
246C2C2×#1915379511yescharacteristic
247C2#2604208211yescharacteristic
248C2#2604209011yescharacteristic
subgroupG/H
centerC22C22×C22⋊D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#518×C181C2
socleC22×C181C22×D4
prime radicalC22×D4×C181C22
hypercenter#518D181
hypocenterC181#3081
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ #117 ◁ #518

Complements

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

NH ≅ G/Ntype
1. #26041907C2semidirect
2. #26041579C2semidirect
3. #26041579C2semidirect
4. #26041569C2semidirect
5. #26041559C2semidirect
6. #26041322C2semidirect
7. #26041315C2semidirect
8. #26041306C2semidirect
14. #19153816C22semidirect
16. #19153808C22semidirect
21. #19153749C22semidirect
22. #19153749C22semidirect
23. #19153749C22semidirect
24. #19153747C22semidirect
25. #19153747C22semidirect
26. #19153747C22semidirect
27. C4⋊#13089678C22semidirect
28. C4⋊#13089678C22semidirect
60. C2×D4×C181Q8semidirect
124. C181#3081semidirect
170. C2×D4Dic362semidirect

Maximal subgroups

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

subgroupindex
#260419072
#260415792
#260415692
#260415592
#260413222
#260413152
#260413062
#260412842
C2×#191537372
#518×C1812
#3081181