Finite Group Databasebeta

← #37544786

Subgroups — #37544786

Order 23168.

396 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 396 conjugacy classes of subgroups — too many to draw. Only its 109 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#26041890C211yesmaximal, characteristic, complemented
2#26041842C211yesmaximal, characteristic
3#26041671C211yesmaximal, characteristic, complemented
4#26041643C211yesmaximal, characteristic, complemented
5#26041634C211yesmaximal, characteristic, complemented
6#26041499C212yesmaximal, complemented
7#26041477C212yesmaximal, complemented
8#26041438C212yesmaximal, complemented
9#26041428C212yesmaximal, complemented
10#26041338C211yesmaximal, characteristic, complemented
11#492×C181C211yesmaximal, characteristic, complemented
12C2×#15668448C2211yescharacteristic
13C2×D4×D181C2211yescharacteristic
14C2×#15668439C2212yescomplemented
15C2×#15668436C2212yes—
16#19153812C2211yescharacteristic
17#19153811—22no—
18#19153808C2211yescharacteristic, complemented
19#19153806—22no—
20#19153804C2211yescharacteristic
21#19153803C2211yescharacteristic
22#19153792C2212yes—
23#19153788—22no—
24#19153784—22no—
25Q8×Dic181C2212yescomplemented
26#19153773C2211yescharacteristic
27#19153745C2212yescomplemented
28#19153741C2212yescomplemented
29#19153738C2212yes—
30#19153737C2212yes—
31#19153737C2212yescomplemented
32#19153728C2212yes—
33#19153727C2211yescharacteristic, complemented
34C2×SD16×C181C2212yes—
35C2×SD16×C181C2212yescomplemented
36C4⋊Q8×C181C2211yescharacteristic, complemented
37#127×C181C2211yescharacteristic
38C4×C8×C181C2211yescharacteristic, complemented
39C23×D181—21no—
40#15668448—42no—
41C2×Dic362C2311yescharacteristic
42C2×D724C2311yescharacteristic
43C2×C22⋊D181—21no—
44C2×C22⋊D181—21no—
45C2×C22⋊D181—22no—
46D4×D181—42no—
47C22×Dic181—21no—
48C2×C4×D181C2311yescharacteristic
49#15668439—42no—
50#15668436—42no—
51C2×#13089678C2312yes—
52C4⋊Dic181—22no—
53C4⋊Dic181C2312yes—
54#15668428—22no—
55#15668428—22no—
56#15668428D412yes—
57C22⋊Dic181—22no—
58#15668426—22no—
59#15668426—22no—
60#15668426D412yes—
61C4×Dic181—22no—
62C4×Dic181C2311yescharacteristic
63C2×Q8×C181C2312yes—
64C2×D4×C181—22no—
65C2×D4×C181C2311yescharacteristic
66C2×D4×C181C2311yescharacteristic
67SD16×C181—42no—
68SD16×C181—42no—
69C2×C8×C181C2312yes—
70C4⋊C4×C181—22no—
71C42×C181C2311yescharacteristic
72C22×D181—21no—
73C22×D181—42no—
74C22×D181—82no—
75C22×D181C2×D411yescharacteristic
76Dic362—22no—
77D724—22no—
78C22⋊D181—42no—
79C22⋊D181—42no—
80C22⋊D181—82no—
81C2×Dic181—22no—
82C2×Dic181—21no—
83C2×Dic181—42no—
84C2×Dic181C2×D412yes—
85C2×Dic181C2×D411yescharacteristic
86C4×D181—22no—
87#13089678—22no—
88C23×C181—21no—
89C23×C181—21no—
90Q8×C181—22no—
91Q8×C181—22no—
92D4×C181—22no—
93D4×C181—22no—
94D4×C181—82no—
95C2×C4×C181—22no—
96C2×C4×C181C2×D412yes—
97C2×C4×C181C2411yescharacteristic
98C8×C181—22no—
99C22×C181—41no—
100C22×C181—41no—
101C22×C181—42no—
102C22×C181—42no—
103C22×C181C22×D411yescharacteristic
104C2×D181—21no—
105C2×D181—22no—
106C2×D181—41no—
107C2×D181—42no—
108C2×D181—84no—
109C4×C181—22no—
110C4×C181—42no—
111C4×C181H(5,𝔽2)12yes—
112Dic181—21no—
113Dic181—22no—
114Dic181—41no—
115Dic181—42no—
116C2×C181—81no—
117C2×C181—81no—
118C2×C181#53511yescharacteristic
119C2×C181#57712yes—
120D181—41no—
121D181—81no—
122C181#322211yescharacteristic, complemented, Sylow
123#3222—1811nomaximal, Sylow
124#541—1811no—
125#535—1811no—
126#497—1811no—
127#492D18111yescharacteristic, complemented
128#490—1811no—
129#488—1811no—
130#464—1812no—
131#460—1812no—
132#452—1812no—
133#450—1812no—
134#433—1811no—
135C2×#41—1811no—
136C22×D4—1811no—
137C2×Q16—1812no—
138C2×SD16—1812no—
139C2×SD16C2×D18112yes—
140C2×SD16C2×D18112yescomplemented
141C4⋊Q8C2×D18111yescharacteristic, complemented
142#127—1811no—
143#127C2×D18111yescharacteristic
144#126—3622no—
145#125—3622no—
146#125—1811no—
147#124—1811no—
148#124—1811no—
149C22⋊Q8—1812no—
150#121—3622no—
151#120—3622no—
152C4×Q8—1812no—
153#117—1811no—
154C4⋊C8—1812no—
155#103—1812no—
156#102—1812no—
157#102—1812no—
158#102—1812no—
159C22⋊C8—1812no—
160#97—1811no—
161C4×C8C2×D18111yescharacteristic, complemented
162C24—3621no—
163#41—7242no—
164C2×Q8—1811no—
165C2×Q8C22×D18112yes—
166C2×D4—22no—
167C2×D4—3621no—
168C2×D4—3622no—
169C2×D4—3621no—
170C2×D4—1811no—
171C2×D4—7242no—
172C2×D4C22×D18111yescharacteristic
173C2×D4C22×D18111yescharacteristic
174C22×C4—3621no—
175C22×C4—1811no—
176Q16—7242no—
177SD16—42no—
178SD16—42no—
179SD16—7242no—
180C2×C8—1812no—
181C2×C8C22×D18112yes—
182C4⋊C4—22no—
183C4⋊C4—3622no—
184C4⋊C4—3622no—
185C4⋊C4—3622no—
186C4⋊C4—1812no—
187C4⋊C4—1812no—
188C22⋊C4—3622no—
189C22⋊C4—3622no—
190C22⋊C4—3622no—
191C22⋊C4—1812no—
192C42—3622no—
193C42—1811no—
194C42C22×D18111yescharacteristic
195C23—21no—
196C23—21no—
197C23—3621no—
198C23—14482no—
199C23—1811no—
200C23—7242no—
201Q8—22no—
202Q8—22no—
203Q8—3622no—
204D4—22no—
205D4—22no—
206D4—82no—
207D4—3622no—
208D4—14482no—
209D4—7242no—
210D4—7242no—
211C2×C4—22no—
212C2×C4—3622no—
213C2×C4—3622no—
214C2×C4—3621no—
215C2×C4—1812no—
216C2×C4—1811no—
217C2×C4—7242no—
218C2×C4D4×D18112yes—
219C2×C4C23×D18111yescharacteristic
220C8—22no—
221C8—3622no—
222C22—42no—
223C22—42no—
224C22—41no—
225C22—41no—
226C22—3622no—
227C22—3621no—
228C22—14484no—
229C22—7242no—
230C22—7241no—
231C22C2×D4×D18111yescharacteristic
232C4—22no—
233C4—42no—
234C4—3622no—
235C4—3621no—
236C4—7242no—
237C4—7241no—
238C4#1915386112yes—
239C2—81no—
240C2—81no—
241C2—14481no—
242C2—7241no—
243C2#2604184211yescharacteristic
244C2#2604208512yes—
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#492×C181C2
socleC22×C181C22×D4
prime radical#26041842C2
hypercenter#492D181
hypocenterC181#3222
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C42 ◁ #492

Complements

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

NH ≅ G/Ntype
1. #26041890C2semidirect
3. #26041671C2semidirect
4. #26041643C2semidirect
5. #26041634C2semidirect
6. #26041499C2semidirect
7. #26041477C2semidirect
8. #26041438C2semidirect
9. #26041428C2semidirect
10. #26041338C2semidirect
11. #492×C181C2semidirect
14. C2×#15668439C22semidirect
18. #19153808C22semidirect
25. Q8×Dic181C22semidirect
27. #19153745C22semidirect
28. #19153741C22semidirect
31. #19153737C22semidirect
33. #19153727C22semidirect
35. C2×SD16×C181C22semidirect
36. C4⋊Q8×C181C22semidirect
38. C4×C8×C181C22semidirect
122. C181#3222semidirect
127. #492D181semidirect
140. C2×SD16C2×D181semidirect
141. C4⋊Q8C2×D181semidirect
161. C4×C8C2×D181semidirect

Maximal subgroups

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

subgroupindex
#260418902
#260418422
#260416712
#260416432
#260416342
#260414992
#260414772
#260414382
#260414282
#260413382
#492×C1812
#3222181