Finite Group Databasebeta

← #37544563

Subgroups — #37544563

Order 23168.

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

normal subgroups only

This group has 418 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#26041790C212yesmaximal, complemented
2#26041621C212yesmaximal, complemented
3#26041604C212yesmaximal, complemented
4#26041537C211yesmaximal, characteristic, complemented
5#26041532C211yesmaximal, characteristic, complemented
6#26041525C212yesmaximal, complemented
7#26041443C212yesmaximal, complemented
8#26041426C212yesmaximal, complemented
9#407×C181C211yesmaximal, characteristic, complemented
10C2×#15668449C2212yes—
11C2×Dic724C2211yescharacteristic, complemented
12C2×#15668438C2212yescomplemented
13C2×D1448C2211yescharacteristic, complemented
14#19153799—22no—
15#19153795C2212yes—
16#19153795C2212yes—
17#19153790C2212yes—
18#19153790C2212yes—
19#19153785—22no—
20C4×C22⋊D181—22no—
21C4×C22⋊D181—22no—
22#19153775C2212yes—
23#19153750C2212yescomplemented
24#19153748C2211yescharacteristic, complemented
25#19153748C2211yescharacteristic, complemented
26#19153741C2212yescomplemented
27#19153741C2212yescomplemented
28#19153738C2212yescomplemented
29#19153738C2212yescomplemented
30C23×C4×C181C2211yescharacteristic
31C2×M16×C181C2211yescharacteristic
32C2×M16×C181C2211yescharacteristic
33C22⋊C8×C181C2212yes—
34C22⋊C8×C181C2212yes—
35#15668449—42no—
36#15668449—42no—
37C2×Dic362C2312yes—
38C2×D724C2312yes—
39C2×C22⋊D181—22no—
40C2×C22⋊D181—22no—
41C2×C22⋊D181—22no—
42C2×C4×D181—22no—
43Dic724—41no—
44#15668438—42no—
45D1448—41no—
46C4⋊Dic181C2312yes—
47C4⋊Dic181C2312yes—
48#15668428—22no—
49#15668428—22no—
50C22⋊Dic181—22no—
51C22⋊Dic181—22no—
52C22⋊Dic181—22no—
53#15668426—22no—
54#15668426—22no—
55C4×Dic181—22no—
56C24×C181D411yescharacteristic
57C22×C4×C181—41no—
58C22×C4×C181—41no—
59C22×C4×C181D411yescharacteristic
60C22×C4×C181D412yes—
61C22×C4×C181C2312yes—
62C22×C4×C181C2311yescharacteristic
63M16×C181—41no—
64M16×C181—41no—
65C2×C8×C181C2311yescharacteristic
66C2×C8×C181C2312yes—
67C2×C8×C181C2311yescharacteristic
68C22×D181—22no—
69Dic362—22no—
70Dic362—22no—
71D724—22no—
72D724—22no—
73C22⋊D181—44no—
74C22⋊D181—44no—
75C22⋊D181—44no—
76C2×Dic181—22no—
77C2×Dic181—22no—
78C2×Dic181—22no—
79C4×D181—44no—
80C23×C181—41no—
81C23×C181—42no—
82C23×C181C2×D411yescharacteristic
83C23×C181C2×D412yes—
84C2×C4×C181—21no—
85C2×C4×C181—21no—
86C2×C4×C181—22no—
87C2×C4×C181—22no—
88C2×C4×C181—42no—
89C2×C4×C181—41no—
90C2×C4×C181C2×D412yes—
91C2×C4×C181C2×D411yescharacteristic
92C2×C4×C181C2411yescharacteristic
93C8×C181—21no—
94C8×C181—22no—
95C8×C181—21no—
96C22×C181—21no—
97C22×C181—22no—
98C22×C181—22no—
99C22×C181—22no—
100C22×C181—22no—
101C22×C181—44no—
102C22×C181C22×D411yescharacteristic
103C2×D181—42no—
104C2×D181—42no—
105C2×D181—42no—
106C4×C181—21no—
107C4×C181—22no—
108C4×C181H(5,𝔽2)11yescharacteristic
109C4×C181H(5,𝔽2)11yescharacteristic
110Dic181—42no—
111Dic181—42no—
112Dic181—42no—
113C2×C181—41no—
114C2×C181—42no—
115C2×C181#53411yescharacteristic
116C2×C181#57512yes—
117D181—82no—
118C181#319711yescharacteristic, complemented, Sylow
119#3197—1811nomaximal, Sylow
120#525—1812no—
121#485—1812no—
122#482—1812no—
123#470—1811no—
124#469—1811no—
125#468—1812no—
126#452—1812no—
127#449—1812no—
128#407D18111yescharacteristic, complemented
129C2×#41—1812no—
130C23×C4C2×D18111yescharacteristic
131C2×Q16—1811no—
132C2×SD16—1812no—
133C2×D8—1811no—
134C2×M16C2×D18111yescharacteristic
135C2×M16C2×D18111yescharacteristic
136#123—3622no—
137C22⋊Q8—1812no—
138C22⋊Q8—1812no—
139#121—1812no—
140#121—1812no—
141#120—3622no—
142C4×D4—3622no—
143C4×D4—3622no—
144#117—1812no—
145#107—1812no—
146#106—1811no—
147#106—1811no—
148#103—1812no—
149#103—1812no—
150#102—1812no—
151#102—1812no—
152C22⋊C8C2×D18112yes—
153C22⋊C8C2×D18112yes—
154C24D72411yescharacteristic
155#41—7242no—
156#41—7242no—
157C2×Q8—1812no—
158C2×D4—3622no—
159C2×D4—3622no—
160C2×D4—3622no—
161C2×D4—1812no—
162C22×C4—41no—
163C22×C4—41no—
164C22×C4—3622no—
165C22×C4D72411yescharacteristic
166C22×C4D72412yes—
167C22×C4C22×D18112yes—
168C22×C4C22×D18111yescharacteristic
169Q16—7241no—
170SD16—7242no—
171D8—7241no—
172M16—41no—
173M16—41no—
174C2×C8C22×D18111yescharacteristic
175C2×C8C22×D18112yes—
176C2×C8C22×D18111yescharacteristic
177C4⋊C4—3622no—
178C4⋊C4—3622no—
179C4⋊C4—1812no—
180C4⋊C4—1812no—
181C22⋊C4—3622no—
182C22⋊C4—3622no—
183C22⋊C4—3622no—
184C22⋊C4—3622no—
185C22⋊C4—3622no—
186C42—3622no—
187C23—41no—
188C23—42no—
189C23—3622no—
190C23C2×D72411yescharacteristic
191C23C2×D72412yes—
192Q8—3622no—
193Q8—3622no—
194D4—3622no—
195D4—3622no—
196D4—7244no—
197D4—7244no—
198D4—7244no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—41no—
204C2×C4—42no—
205C2×C4—3622no—
206C2×C4—3622no—
207C2×C4—3622no—
208C2×C4—7244no—
209C2×C4C2×D72412yes—
210C2×C4C2×D72411yescharacteristic
211C2×C4C23×D18111yescharacteristic
212C8—21no—
213C8—22no—
214C8—21no—
215C22—22no—
216C22—22no—
217C22—22no—
218C22—22no—
219C22—21no—
220C22—44no—
221C22—7242no—
222C22—7242no—
223C22—7242no—
224C22C22×D72411yescharacteristic
225C4—22no—
226C4—21no—
227C4—7242no—
228C4—7242no—
229C4—7242no—
230C4#1915386111yescharacteristic
231C4#1915386111yescharacteristic
232C2—41no—
233C2—42no—
234C2—14482no—
235C2#2604184111yescharacteristic
236C2#2604207912yes—
subgroupG/H
centerC22C22×D724
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#407×C181C2
socleC22×C181C22×D4
hypercenter#407D181
hypocenterC181#3197
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C23×C4 ◁ #407

Complements

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

NH ≅ G/Ntype
1. #26041790C2semidirect
2. #26041621C2semidirect
3. #26041604C2semidirect
4. #26041537C2semidirect
5. #26041532C2semidirect
6. #26041525C2semidirect
7. #26041443C2semidirect
8. #26041426C2semidirect
9. #407×C181C2semidirect
11. C2×Dic724C22semidirect
12. C2×#15668438C22semidirect
13. C2×D1448C22semidirect
23. #19153750C22semidirect
24. #19153748C22semidirect
25. #19153748C22semidirect
26. #19153741C22semidirect
27. #19153741C22semidirect
28. #19153738C22semidirect
29. #19153738C22semidirect
118. C181#3197semidirect
128. #407D181semidirect

Maximal subgroups

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

subgroupindex
#260417902
#260416212
#260416042
#260415372
#260415322
#260415252
#260414432
#260414262
#407×C1812
#3197181