Finite Group Databasebeta

← #37544588

Subgroups — #37544588

Order 23168.

410 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 117 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#26041843C211yesmaximal, characteristic
2#26041617C212yesmaximal, complemented
3#26041603C212yesmaximal, complemented
4#26041527C212yesmaximal, complemented
5#26041510C212yesmaximal, complemented
6#26041429C212yesmaximal, complemented
7#26041418C212yesmaximal, complemented
8#26041246C211yesmaximal, characteristic, complemented
9#536×C181C211yesmaximal, characteristic, complemented
10C22×D724C2211yescharacteristic
11C2×#15668436C2212yes—
12C2×#15668436C2212yes—
13C2×#15668433C2211yescharacteristic, complemented
14C22×#13089678C2211yescharacteristic, complemented
15#19153812—21no—
16#19153805—21no—
17#19153790C2212yes—
18#19153790C2212yes—
19#19153784—21no—
20#19153784—21no—
21#19153775C2211yescharacteristic
22#19153749C2212yescomplemented
23#19153747C2212yescomplemented
24#19153740C2212yescomplemented
25#19153740C2212yescomplemented
26#19153736C2212yes—
27#19153736C2212yes—
28C22⋊#13089678C2212yes—
29C22⋊#13089678C2212yescomplemented
30C2×#41×C181C2211yescharacteristic
31C22×Q8×C181C2211yescharacteristic, complemented
32C4⋊Q8×C181—21no—
33#124×C181—21no—
34#123×C181—21no—
35#123×C181—21no—
36C22⋊Q8×C181C2212yes—
37C22⋊Q8×C181C2212yescomplemented
38#117×C181C2211yescharacteristic, complemented
39C23×D181—21no—
40C2×D724—21no—
41C2×D724—41no—
42C2×D724—41no—
43C2×D724C2312yes—
44C2×C22⋊D181—22no—
45C2×C22⋊D181—22no—
46#15668436—42no—
47#15668436—42no—
48#15668433—41no—
49C2×#13089678—41no—
50C2×#13089678C2312yes—
51C2×#13089678C2312yes—
52C4⋊Dic181C2312yes—
53C22⋊Dic181—21no—
54#15668426—22no—
55#15668426—22no—
56C4×Dic181—21no—
57#41×C181—41no—
58#41×C181—41no—
59C2×Q8×C181—21no—
60C2×Q8×C181—41no—
61C2×Q8×C181—41no—
62C2×Q8×C181D411yescharacteristic
63C2×Q8×C181C2312yes—
64C2×D4×C181D411yescharacteristic
65C2×D4×C181D411yescharacteristic
66C2×D4×C181D411yescharacteristic
67C22×C4×C181—21no—
68C22×C4×C181C2311yescharacteristic
69C22×C4×C181C2311yescharacteristic
70C22×C4×C181C2311yescharacteristic
71C4⋊C4×C181—22no—
72C4⋊C4×C181—22no—
73C4⋊C4×C181C2312yes—
74C22⋊C4×C181—21no—
75C22⋊C4×C181—22no—
76C22⋊C4×C181—22no—
77C42×C181—21no—
78C22×D181—22no—
79C22×D181—42no—
80C22×D181—82no—
81D724—22no—
82D724—22no—
83D724—42no—
84C22⋊D181—82no—
85C22⋊D181—82no—
86C2×Dic181—22no—
87#13089678—22no—
88#13089678—22no—
89C23×C181C2×D411yescharacteristic
90C23×C181C2×D411yescharacteristic
91C23×C181C2×D411yescharacteristic
92Q8×C181—22no—
93Q8×C181—22no—
94Q8×C181—41no—
95Q8×C181—42no—
96D4×C181—41no—
97D4×C181—41no—
98D4×C181—41no—
99C2×C4×C181—21no—
100C2×C4×C181—21no—
101C2×C4×C181—21no—
102C2×C4×C181—21no—
103C2×C4×C181—21no—
104C2×C4×C181—21no—
105C2×C4×C181—22no—
106C2×C4×C181—22no—
107C2×C4×C181—42no—
108C2×C4×C181C2×D411yescharacteristic
109C2×C4×C181C2×D411yescharacteristic
110C2×C4×C181C2×D411yescharacteristic
111C2×C4×C181C2411yescharacteristic
112C22×C181—21no—
113C22×C181—21no—
114C22×C181—21no—
115C22×C181—22no—
116C22×C181—22no—
117C22×C181—22no—
118C22×C181C22×D411yescharacteristic
119C2×D181—42no—
120C2×D181—42no—
121C2×D181—42no—
122C2×D181—84no—
123C4×C181—21no—
124C4×C181—21no—
125C4×C181—21no—
126C4×C181—42no—
127C4×C181—42no—
128C4×C181H(5,𝔽2)11yescharacteristic
129C4×C181H(5,𝔽2)11yescharacteristic
130Dic181—42no—
131C2×C181—41no—
132C2×C181—41no—
133C2×C181—41no—
134C2×C181#53411yescharacteristic
135C2×C181#57712yes—
136D181—82no—
137C181#320011yescharacteristic, complemented, Sylow
138#3200—1811nomaximal, Sylow
139#536D18111yescharacteristic, complemented
140#535—1811no—
141#485—1812no—
142#482—1812no—
143#468—1812no—
144#465—1812no—
145#450—1812no—
146#448—1812no—
147#408—1811no—
148C2×#41C2×D18111yescharacteristic
149C22×Q8C2×D18111yescharacteristic, complemented
150C22×D4—1811no—
151C2×SD16—1812no—
152C2×SD16—1812no—
153C2×M16—1811no—
154C22×C8—1811no—
155C4⋊Q8—21no—
156#127—3621no—
157#124—21no—
158#124—3621no—
159#123—21no—
160#123—21no—
161C22⋊Q8C2×D18112yes—
162C22⋊Q8C2×D18112yescomplemented
163#121—1812no—
164#121—1812no—
165#120—3621no—
166#120—3621no—
167#117—1811no—
168#117C2×D18111yescharacteristic, complemented
169#107—1812no—
170#106—1812no—
171#103—1812no—
172#103—1812no—
173#102—1812no—
174#102—1812no—
175C22⋊C8—1812no—
176C22⋊C8—1812no—
177C24—3621no—
178#41—41no—
179#41—41no—
180C2×Q8—21no—
181C2×Q8—41no—
182C2×Q8—41no—
183C2×Q8C22⋊D18111yescharacteristic
184C2×Q8C22×D18112yes—
185C2×D4—3622no—
186C2×D4—3622no—
187C2×D4—3621no—
188C2×D4—1812no—
189C2×D4—7241no—
190C2×D4—7241no—
191C2×D4C22⋊D18111yescharacteristic
192C2×D4C22⋊D18111yescharacteristic
193C2×D4C22⋊D18111yescharacteristic
194C22×C4—21no—
195C22×C4C22×D18111yescharacteristic
196C22×C4C22×D18111yescharacteristic
197C22×C4C22×D18111yescharacteristic
198SD16—7242no—
199SD16—7242no—
200M16—7241no—
201C2×C8—1812no—
202C2×C8—1812no—
203C2×C8—7241no—
204C4⋊C4—22no—
205C4⋊C4—22no—
206C4⋊C4—1812no—
207C4⋊C4C22×D18112yes—
208C22⋊C4—21no—
209C22⋊C4—22no—
210C22⋊C4—22no—
211C22⋊C4—3621no—
212C22⋊C4—3622no—
213C22⋊C4—3622no—
214C42—21no—
215C42—3621no—
216C23—3622no—
217C23—14482no—
218C23—7242no—
219C23C2×C22⋊D18111yescharacteristic
220C23C2×C22⋊D18111yescharacteristic
221C23C2×C22⋊D18111yescharacteristic
222Q8—22no—
223Q8—22no—
224Q8—41no—
225Q8—42no—
226D4—41no—
227D4—41no—
228D4—41no—
229D4—3622no—
230D4—3622no—
231D4—14482no—
232D4—14482no—
233D4—7242no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—42no—
243C2×C4—3622no—
244C2×C4C2×C22⋊D18111yescharacteristic
245C2×C4C2×C22⋊D18111yescharacteristic
246C2×C4C2×C22⋊D18111yescharacteristic
247C2×C4C23×D18111yescharacteristic
248C8—3622no—
249C8—3622no—
250C22—22no—
251C22—22no—
252C22—22no—
253C22—21no—
254C22—21no—
255C22—21no—
256C22—14484no—
257C22—7242no—
258C22—7242no—
259C22—7242no—
260C22C22×C22⋊D18111yescharacteristic
261C4—21no—
262C4—21no—
263C4—21no—
264C4—42no—
265C4—42no—
266C4—7242no—
267C4#1915386111yescharacteristic
268C4#1915386111yescharacteristic
269C2—41no—
270C2—41no—
271C2—41no—
272C2—14482no—
273C2#2604183811yescharacteristic
274C2#2604208712yes—
subgroupG/H
centerC22C22×C22⋊D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#536×C181C2
socleC22×C181C22×D4
prime radical#26041843C2
hypercenter#536D181
hypocenterC181#3200
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C2×#41 ◁ #536

Complements

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

NH ≅ G/Ntype
2. #26041617C2semidirect
3. #26041603C2semidirect
4. #26041527C2semidirect
5. #26041510C2semidirect
6. #26041429C2semidirect
7. #26041418C2semidirect
8. #26041246C2semidirect
9. #536×C181C2semidirect
13. C2×#15668433C22semidirect
14. C22×#13089678C22semidirect
22. #19153749C22semidirect
23. #19153747C22semidirect
24. #19153740C22semidirect
25. #19153740C22semidirect
29. C22⋊#13089678C22semidirect
31. C22×Q8×C181C22semidirect
37. C22⋊Q8×C181C22semidirect
38. #117×C181C22semidirect
137. C181#3200semidirect
139. #536D181semidirect
149. C22×Q8C2×D181semidirect
162. C22⋊Q8C2×D181semidirect
168. #117C2×D181semidirect

Maximal subgroups

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

subgroupindex
#260418432
#260416172
#260416032
#260415272
#260415102
#260414292
#260414182
#260412462
#536×C1812
#3200181