Finite Group Databasebeta

← #37544796

Subgroups — #37544796

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#26041892C211yesmaximal, characteristic, complemented
2#26041847C211yesmaximal, characteristic
3#26041674C211yesmaximal, characteristic, complemented
4#26041652C211yesmaximal, characteristic, complemented
5#26041645C211yesmaximal, characteristic, complemented
6#26041504C212yesmaximal, complemented
7#26041478C212yesmaximal, complemented
8#26041444C212yesmaximal, complemented
9#26041431C212yesmaximal, complemented
10#26041338C211yesmaximal, characteristic, complemented
11#488×C181C211yesmaximal, characteristic, complemented
12C2×#15668450C2211yescharacteristic
13C2×D4×D181C2211yescharacteristic
14C2×#15668439C2212yescomplemented
15C2×#15668438C2212yes—
16C2×#15668438C2212yescomplemented
17C2×#15668437C2212yes—
18#19153816C2211yescharacteristic, complemented
19#19153813C2211yescharacteristic
20#19153812C2211yescharacteristic
21#19153811—22no—
22#19153807—22no—
23#19153805C2211yescharacteristic
24#19153796C2212yes—
25#19153789—22no—
26#19153786—22no—
27#19153781C2212yescomplemented
28#19153773C2211yescharacteristic
29#19153745C2212yescomplemented
30#19153736C2212yes—
31#19153736C2212yescomplemented
32#19153728C2212yes—
33#19153727C2211yescharacteristic, complemented
34#125×C181C2211yescharacteristic, complemented
35#124×C181C2211yescharacteristic
36#103×C181C2212yescomplemented
37#102×C181C2212yes—
38C4×C8×C181C2211yescharacteristic, complemented
39C23×D181—21no—
40#15668450—42no—
41C2×Dic362C2312yes—
42C2×D724—21no—
43C2×D724—22no—
44C2×D724C2311yescharacteristic
45C2×D724C2311yescharacteristic
46C2×C22⋊D181—21no—
47C2×C22⋊D181—22no—
48D4×D181—42no—
49C2×C4×D181—21no—
50C2×C4×D181C2311yescharacteristic
51#15668439—42no—
52#15668438—42no—
53#15668438—42no—
54#15668437—42no—
55C2×#13089678C2312yes—
56C4⋊Dic181—22no—
57C4⋊Dic181—22no—
58#15668428—22no—
59#15668428D412yes—
60#15668426—22no—
61#15668426—22no—
62#15668426D412yes—
63C4×Dic181—22no—
64C4×Dic181C2311yescharacteristic
65C2×Q8×C181C2311yescharacteristic
66C2×D4×C181C2311yescharacteristic
67C2×C8×C181C2312yes—
68C4⋊C4×C181—22no—
69C4⋊C4×C181C2312yes—
70C22⋊C4×C181—22no—
71C42×C181C2311yescharacteristic
72C22×D181—21no—
73C22×D181—21no—
74C22×D181—42no—
75C22×D181—82no—
76C22×D181C2×D411yescharacteristic
77Dic362—22no—
78Dic362—22no—
79D724—22no—
80D724—22no—
81D724—42no—
82D724—82no—
83C22⋊D181—42no—
84C22⋊D181—82no—
85C2×Dic181—22no—
86C2×Dic181C2×D412yes—
87C2×Dic181C2×D411yescharacteristic
88C4×D181—22no—
89C4×D181—42no—
90#13089678—22no—
91C23×C181—21no—
92Q8×C181—22no—
93D4×C181—22no—
94C2×C4×C181—21no—
95C2×C4×C181—22no—
96C2×C4×C181C2×D412yes—
97C2×C4×C181C2411yescharacteristic
98C8×C181—22no—
99C22×C181—41no—
100C22×C181—42no—
101C22×C181C22×D411yescharacteristic
102C2×D181—21no—
103C2×D181—22no—
104C2×D181—41no—
105C2×D181—41no—
106C2×D181—42no—
107C2×D181—42no—
108C2×D181—84no—
109C4×C181—22no—
110C4×C181—41no—
111C4×C181—42no—
112C4×C181H(5,𝔽2)12yes—
113Dic181—21no—
114Dic181—22no—
115Dic181—42no—
116C2×C181—81no—
117C2×C181#53511yescharacteristic
118C2×C181#57712yes—
119D181—41no—
120D181—81no—
121D181—81no—
122C181#322211yescharacteristic, complemented, Sylow
123#3222—1811nomaximal, Sylow
124#541—1811no—
125#535—1811no—
126#497—1811no—
127#492—1811no—
128#490—1811no—
129#488D18111yescharacteristic, complemented
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×SD16—1812no—
140C2×SD16—1812no—
141C4⋊Q8—1811no—
142#127—1811no—
143#127—1811no—
144#126—3622no—
145#125—3622no—
146#125C2×D18111yescharacteristic, complemented
147#124—1811no—
148#124C2×D18111yescharacteristic
149C22⋊Q8—1812no—
150#121—3622no—
151#120—3622no—
152C4×Q8—1812no—
153#117—1811no—
154C4⋊C8—1812no—
155#103C2×D18112yescomplemented
156#102—1812no—
157#102—1812no—
158#102C2×D18112yes—
159C22⋊C8—1812no—
160#97—1811no—
161C4×C8C2×D18111yescharacteristic, complemented
162C24—3621no—
163#41—7242no—
164C2×Q8—1812no—
165C2×Q8C22×D18111yescharacteristic
166C2×D4—3621no—
167C2×D4—3622no—
168C2×D4—3621no—
169C2×D4—3622no—
170C2×D4—1811no—
171C2×D4—1811no—
172C2×D4—7242no—
173C2×D4C22×D18111yescharacteristic
174C22×C4—3621no—
175C22×C4—1811no—
176Q16—7242no—
177SD16—7242no—
178SD16—7242no—
179SD16—7242no—
180C2×C8—1812no—
181C2×C8C22×D18112yes—
182C4⋊C4—22no—
183C4⋊C4—3622no—
184C4⋊C4—3622no—
185C4⋊C4—3622no—
186C4⋊C4—1812no—
187C4⋊C4C22×D18112yes—
188C22⋊C4—22no—
189C22⋊C4—3622no—
190C22⋊C4—3622no—
191C22⋊C4—1812no—
192C42—3622no—
193C42—1811no—
194C42C22×D18111yescharacteristic
195C23—21no—
196C23—3621no—
197C23—3621no—
198C23—14482no—
199C23—1811no—
200C23—7242no—
201Q8—22no—
202Q8—3622no—
203Q8—3622no—
204D4—22no—
205D4—3622no—
206D4—3622no—
207D4—14482no—
208D4—14482no—
209D4—7242no—
210D4—7242no—
211C2×C4—22no—
212C2×C4—21no—
213C2×C4—3622no—
214C2×C4—3622no—
215C2×C4—1812no—
216C2×C4—1811no—
217C2×C4—7242no—
218C2×C4D4×D18112yes—
219C2×C4C23×D18111yescharacteristic
220C8—22no—
221C8—3622no—
222C22—42no—
223C22—41no—
224C22—3622no—
225C22—3621no—
226C22—14484no—
227C22—7242no—
228C22—7242no—
229C22—7241no—
230C22—7241no—
231C22C2×D4×D18111yescharacteristic
232C4—22no—
233C4—42no—
234C4—41no—
235C4—3622no—
236C4—3621no—
237C4—7242no—
238C4#1915386212yes—
239C2—81no—
240C2—14481no—
241C2—14481no—
242C2—7241no—
243C2#2604184711yescharacteristic
244C2#2604208612yes—
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#488×C181C2
socleC22×C181C22×D4
prime radical#26041847C2
hypercenter#488D181
hypocenterC181#3222
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C42 ◁ #488

Complements

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

NH ≅ G/Ntype
1. #26041892C2semidirect
3. #26041674C2semidirect
4. #26041652C2semidirect
5. #26041645C2semidirect
6. #26041504C2semidirect
7. #26041478C2semidirect
8. #26041444C2semidirect
9. #26041431C2semidirect
10. #26041338C2semidirect
11. #488×C181C2semidirect
14. C2×#15668439C22semidirect
16. C2×#15668438C22semidirect
18. #19153816C22semidirect
27. #19153781C22semidirect
29. #19153745C22semidirect
31. #19153736C22semidirect
33. #19153727C22semidirect
34. #125×C181C22semidirect
36. #103×C181C22semidirect
38. C4×C8×C181C22semidirect
122. C181#3222semidirect
129. #488D181semidirect
146. #125C2×D181semidirect
155. #103C2×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
#260418922
#260418472
#260416742
#260416522
#260416452
#260415042
#260414782
#260414442
#260414312
#260413382
#488×C1812
#3222181