Finite Group Databasebeta

← #37542180

Subgroups — #37542180

Order 23168.

800 conjugacy classes of subgroups; only the 209 normal subgroups are shown.

normal subgroups only

This group has 800 conjugacy classes of subgroups — too many to draw. Only its 209 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
1C2×#19153862C211yesmaximal, characteristic
2#26041753C212yesmaximal, complemented
3#26041253C211yesmaximal, characteristic, complemented
4#26041252C212yesmaximal, complemented
5#99×D181C218yesmaximal, complemented
6#410×C181C211yesmaximal, characteristic, complemented
7#19153862—24no—
8#19153862—24no—
9C2×#15668450C412yescomplemented
10C2×#15668449C412yescomplemented
11C2×#15668449C2211yescharacteristic
12C22×D724C412yescomplemented
13C22×D724C2211yescharacteristic
14C2×D4×D181C412yescomplemented
15C2×D4×D181C2214yes—
16C4×D724—24no—
17#19153778—24no—
18#19153775C2212yescomplemented
19C22⋊C4×D181C2218yescomplemented
20#19153731C2214yescomplemented
21#19153730C2218yescomplemented
22C2×#41×C181C2211yescharacteristic
23#117×C181C2212yescomplemented
24#99×C181C2214yescomplemented
25C23×D181C2×C414yescomplemented
26C23×D181C2312yes—
27#15668450—28no—
28#15668449—24no—
29#15668449—24no—
30#15668449—22no—
31#15668449—22no—
32C2×Dic362C2×C411yescharacteristic, complemented
33C2×D724—24no—
34C2×D724—24no—
35C2×D724C2×C412yes—
36C2×D724C2×C412yes—
37C2×D724C2×C414yescomplemented
38C2×D724C2×C411yescharacteristic, complemented
39C2×D724D414yes—
40C2×D724D414yes—
41C2×C22⋊D181C2×C414yescomplemented
42C2×C22⋊D181C2312yes—
43D4×D181—216no—
44D4×D181—28no—
45C2×C4×D181—28no—
46C2×C4×D181C2×C412yescomplemented
47C2×C4×D181C2×C414yescomplemented
48C4⋊Dic181—22no—
49C22⋊Dic181C2314yes—
50#15668426—28no—
51C4×Dic181—22no—
52#41×C181—22no—
53#41×C181—22no—
54C2×Q8×C181C2×C411yescharacteristic, complemented
55C2×D4×C181C2×C411yescharacteristic, complemented
56C2×D4×C181C2312yes—
57C22×C4×C181C2×C412yescomplemented
58C22×C4×C181C2311yescharacteristic
59C4⋊C4×C181—22no—
60C22⋊C4×C181C2314yes—
61C42×C181—22no—
62C22×D181—28no—
63C22×D181—82no—
64C22×D181—84no—
65C22×D181C22×C412yes—
66C22×D181C22×C414yes—
67C22×D181C2×D418yes—
68Dic362—21no—
69Dic362—21no—
70D724—28no—
71D724—21no—
72D724—21no—
73D724C22⋊C418yes—
74D724C22⋊C414yes—
75D724C22⋊C414yes—
76C22⋊D181—24no—
77C22⋊D181—28no—
78C2×Dic181—24no—
79C2×Dic181C22×C412yes—
80C4×D181—28no—
81C4×D181—24no—
82C4×D181—88no—
83C23×C181C22×C412yes—
84C23×C181C2411yescharacteristic
85Q8×C181—22no—
86D4×C181—24no—
87D4×C181—22no—
88C2×C4×C181—22no—
89C2×C4×C181—22no—
90C2×C4×C181—24no—
91C2×C4×C181C22×C412yes—
92C2×C4×C181C22×C411yescharacteristic
93C2×C4×C181C22×C411yescharacteristic
94C2×C4×C181C2×D412yes—
95C2×C4×C181C2×D412yes—
96C22×C181—22no—
97C22×C181—41no—
98C22×C181—42no—
99C22×C181C23×C411yescharacteristic
100C22×C181C22×D412yes—
101C2×D181—22no—
102C2×D181—48no—
103C2×D181—44no—
104C2×D181—42no—
105C2×D181—88no—
106C2×D181C2×C22⋊C418yes—
107C4×C181—22no—
108C4×C181—44no—
109C4×C181C2×C22⋊C411yescharacteristic
110C4×C181C2×C22⋊C411yescharacteristic
111C4×C181C2×C22⋊C412yes—
112Dic181—22no—
113Dic181—44no—
114C2×C181—22no—
115C2×C181—21no—
116C2×C181—42no—
117C2×C181C22×C22⋊C411yescharacteristic
118D181—28no—
119D181—42no—
120C181#288311yescharacteristic, complemented, Sylow
121#2883—1811nomaximal, Sylow
122C2×H(5,𝔽2)—1811no—
123#518—1812no—
124#410—1811no—
125#410—1812no—
126#410D18111yescharacteristic, complemented
127C2×#99—1818no—
128H(5,𝔽2)—3624no—
129H(5,𝔽2)—3624no—
130C2×#41—1812no—
131C2×#41—1812no—
132C2×#41—1811no—
133C2×#41C2×D18111yescharacteristic
134C22×D4—1812no—
135C22×D4—1814no—
136C22×D4—1812no—
137C22×D4—1811no—
138C4×D4—3624no—
139C4×D4—3624no—
140#117—1812no—
141#117C2×D18112yescomplemented
142C2×C22⋊C4—1818no—
143#99—1814no—
144#99—1818no—
145#99C2×D18114yescomplemented
146C24—1814no—
147C24—1812no—
148#41—22no—
149#41—22no—
150#41—3624no—
151#41—3624no—
152#41—3628no—
153#41—3622no—
154#41—3622no—
155C2×Q8—1811no—
156C2×Q8C4×D18111yescharacteristic, complemented
157C2×D4—3624no—
158C2×D4—3624no—
159C2×D4—36216no—
160C2×D4—3628no—
161C2×D4—1814no—
162C2×D4—1812no—
163C2×D4—1814no—
164C2×D4—1811no—
165C2×D4—1812no—
166C2×D4—1812no—
167C2×D4—1814no—
168C2×D4—1814no—
169C2×D4C4×D18111yescharacteristic, complemented
170C2×D4C22×D18112yes—
171C22×C4—3628no—
172C22×C4—1814no—
173C22×C4—1812no—
174C22×C4C4×D18112yescomplemented
175C22×C4C22×D18111yescharacteristic
176C4⋊C4—22no—
177C4⋊C4—3622no—
178C22⋊C4—3628no—
179C22⋊C4—1814no—
180C22⋊C4C22×D18114yes—
181C42—22no—
182C42—3622no—
183C23—3628no—
184C23—14484no—
185C23—14482no—
186C23—1812no—
187C23—1814no—
188C23—1818no—
189C23C2×C4×D18112yes—
190C23C23×D18111yescharacteristic
191Q8—22no—
192Q8—3621no—
193Q8—3621no—
194D4—24no—
195D4—22no—
196D4—3628no—
197D4—3628no—
198D4—3624no—
199D4—3621no—
200D4—3621no—
201D4—1814no—
202D4—1814no—
203D4—1818no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—24no—
207C2×C4—3628no—
208C2×C4—3624no—
209C2×C4—3624no—
210C2×C4—14488no—
211C2×C4—1812no—
212C2×C4C2×C4×D18112yes—
213C2×C4C2×C4×D18111yescharacteristic
214C2×C4C2×C4×D18111yescharacteristic
215C2×C4D4×D18112yes—
216C2×C4D4×D18112yes—
217C22—22no—
218C22—41no—
219C22—42no—
220C22—3622no—
221C22—14488no—
222C22—1818no—
223C22—7248no—
224C22—7244no—
225C22—7242no—
226C22C22×C4×D18111yescharacteristic
227C22C2×D4×D18112yes—
228C4—22no—
229C4—44no—
230C4—3622no—
231C4—7244no—
232C4C22⋊C4×D18111yescharacteristic
233C4C22⋊C4×D18111yescharacteristic
234C4C22⋊C4×D18112yes—
235C2—22no—
236C2—21no—
237C2—42no—
238C2—3628no—
239C2—7242no—
240C2C2×C22⋊C4×D18111yescharacteristic
subgroupG/H
centerC2C2×C22⋊C4×D181
commutatorC22×C181C23×C4
FrattiniC23C23×D181
Fitting#410×C181C2
socleC2×C181C22×C22⋊C4
prime radicalC2×#19153862C2
cosocleC23×C181C24
hypercenter#410D181
hypocenterC181#2883
LCSG ▷ C22×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C2 ◁ C22×C4 ◁ #410

Complements

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

NH ≅ G/Ntype
2. #26041753C2semidirect
3. #26041253C2semidirect
4. #26041252C2semidirect
5. #99×D181C2semidirect
6. #410×C181C2semidirect
9. C2×#15668450C4semidirect
10. C2×#15668449C4semidirect
12. C22×D724C4semidirect
14. C2×D4×D181C4semidirect
18. #19153775C22semidirect
19. C22⋊C4×D181C22semidirect
20. #19153731C22semidirect
21. #19153730C22semidirect
23. #117×C181C22semidirect
24. #99×C181C22semidirect
25. C23×D181C2×C4semidirect
32. C2×Dic362C2×C4semidirect
37. C2×D724C2×C4semidirect
38. C2×D724C2×C4semidirect
41. C2×C22⋊D181C2×C4semidirect
46. C2×C4×D181C2×C4semidirect
47. C2×C4×D181C2×C4semidirect
54. C2×Q8×C181C2×C4semidirect
55. C2×D4×C181C2×C4semidirect
57. C22×C4×C181C2×C4semidirect
120. C181#2883semidirect
126. #410D181semidirect
141. #117C2×D181semidirect
145. #99C2×D181semidirect
156. C2×Q8C4×D181semidirect
169. C2×D4C4×D181semidirect
174. C22×C4C4×D181semidirect

Maximal subgroups

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

subgroupindex
C2×#191538622
#260417532
#260412532
#260412522
#99×D1812
#410×C1812
#2883181