Finite Group Databasebeta

← #37550321

Subgroups — #37550321

Order 23168.

800 conjugacy classes of subgroups; only the 198 normal subgroups are shown.

normal subgroups only

This group has 800 conjugacy classes of subgroups — too many to draw. Only its 198 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#26042346C212yesmaximal, complemented
2#26042260C212yesmaximal, complemented
3#26042258C212yesmaximal, complemented
4C2×#19153875C214yesmaximal, complemented
5D181⋊#99C214yesmaximal, complemented
6C2×#19153862C211yesmaximal, characteristic
7D4×#10624382—28no—
8#19153895C2214yescomplemented
9C2×C22⋊#10624382C2218yescomplemented
10#19153875C2218yescomplemented
11#19153875C2214yescomplemented
12#19153874C2214yescomplemented
13#19153862—28no—
14C2×#15668450C411yescharacteristic, complemented
15C2×#15668450C2211yescharacteristic
16C2×#15668449C412yescomplemented
17C2×#15668449C2211yescharacteristic
18C22×D724C411yescharacteristic, complemented
19C22×D724C2212yes—
20C2×D4×D181C411yescharacteristic, complemented
21C2×D4×D181C412yescomplemented
22C2×D4×D181C2211yescharacteristic
23C2×D4×D181C2212yes—
24C2×#41×C181C411yescharacteristic, complemented
25C22×#10624382—28no—
26C4⋊#10624382—24no—
27C22⋊#10624382—28no—
28C22⋊#10624382C2318yes—
29C4×#10624382—24no—
30C23×D181C2×C412yescomplemented
31C23×D181C2×C412yescomplemented
32C23×D181C2312yes—
33#15668450—24no—
34#15668450—24no—
35#15668449—28no—
36#15668449—24no—
37C2×Dic362C2×C411yescharacteristic, complemented
38C2×D724—28no—
39C2×D724—24no—
40C2×D724C2×C411yescharacteristic
41C2×D724C2×C411yescharacteristic, complemented
42C2×D724C2×C412yescomplemented
43C2×D724C2×C412yescomplemented
44C2×D724C2312yes—
45C2×D724C2311yescharacteristic
46C2×C22⋊D181C2×C412yes—
47C2×C22⋊D181C2×C411yescharacteristic, complemented
48C2×C22⋊D181C2×C412yescomplemented
49C2×C22⋊D181C2311yescharacteristic
50D4×D181—28no—
51D4×D181—28no—
52D4×D181—24no—
53D4×D181D418yes—
54C2×C4×D181C2×C412yescomplemented
55C2×C4×D181C2×C411yescharacteristic, complemented
56C2×C4×D181C2×C412yescomplemented
57C2×C4×D181C2311yescharacteristic
58#41×C181—24no—
59C2×Q8×C181C2×C411yescharacteristic, complemented
60C2×D4×C181C2×C411yescharacteristic
61C2×D4×C181C2×C412yescomplemented
62C22×C4×C181C2×C411yescharacteristic, complemented
63C22×C4×C181C2×C412yescomplemented
64C2×#10624382—28no—
65C2×#10624382—88no—
66C22×D181—24no—
67C22×D181—24no—
68C22×D181—82no—
69C22×D181—82no—
70C22×D181—82no—
71C22×D181C22×C412yes—
72C22×D181C22×C411yescharacteristic
73C22×D181C22×C412yes—
74C22×D181C2×D418yes—
75C22×D181C2411yescharacteristic
76Dic362—22no—
77D724—24no—
78D724—24no—
79D724—24no—
80D724—22no—
81D724—22no—
82D724C22⋊C414yes—
83C22⋊D181—24no—
84C22⋊D181—22no—
85C22⋊D181—22no—
86C22⋊D181C22⋊C418yes—
87C2×Dic181C22×C411yescharacteristic
88C2×Dic181C22×C411yescharacteristic
89C4×D181—24no—
90C4×D181—22no—
91C4×D181—24no—
92C4×D181C2×D414yes—
93C23×C181C22×C411yescharacteristic
94C23×C181C22×C412yes—
95Q8×C181—22no—
96D4×C181—24no—
97D4×C181C22⋊C414yes—
98C2×C4×C181—24no—
99C2×C4×C181—22no—
100C2×C4×C181C22×C411yescharacteristic
101C2×C4×C181C22×C412yes—
102C2×C4×C181C22×C411yescharacteristic
103C22×C181—21no—
104C22×C181—42no—
105C22×C181—41no—
106C22×C181C2×C22⋊C414yes—
107C22×C181C23×C411yescharacteristic
108C2×D181—21no—
109C2×D181—22no—
110C2×D181—42no—
111C2×D181—41no—
112C2×D181—48no—
113C2×D181—42no—
114C2×D181—41no—
115C2×D181—84no—
116C2×D181—84no—
117C2×D181C2×C22⋊C414yes—
118C2×D181C22×D412yes—
119#10624382—48no—
120C4×C181—21no—
121C4×C181—22no—
122C4×C181C2×C22⋊C412yes—
123Dic181—21no—
124Dic181C2×C22⋊C412yes—
125C2×C181—21no—
126C2×C181—24no—
127C2×C181—41no—
128C2×C181C22×C22⋊C411yescharacteristic
129D181—22no—
130D181—24no—
131D181—42no—
132D181—41no—
133C181#288311yescharacteristic, complemented, Sylow
134#2883—1811nomaximal, Sylow
135C2×H(5,𝔽2)—1811no—
136#518—1812no—
137#410—1812no—
138#410—1812no—
139C2×#99—1814no—
140C2×#99—1814no—
141H(5,𝔽2)—3628no—
142C2×#41—1811no—
143C2×#41—1811no—
144C2×#41—1811no—
145C2×#41—1812no—
146C2×#41#1062438211yescharacteristic, complemented
147C22×D4—1811no—
148C22×D4—1812no—
149C22×D4—1812no—
150C22×D4—1811no—
151C22×D4—1812no—
152C22×D4—1811no—
153C4×D4—3628no—
154#117—1814no—
155C2×C22⋊C4—1818no—
156#99—1814no—
157#99—1818no—
158#99—1814no—
159C24—1812no—
160C24—1812no—
161C24—1812no—
162#41—24no—
163#41—3628no—
164#41—3624no—
165#41—3624no—
166#41—3624no—
167C2×Q8—1811no—
168C2×Q8C2×#1062438211yescharacteristic, complemented
169C2×D4—3628no—
170C2×D4—3624no—
171C2×D4—3628no—
172C2×D4—3628no—
173C2×D4—3624no—
174C2×D4—1811no—
175C2×D4—1812no—
176C2×D4—1811no—
177C2×D4—1812no—
178C2×D4—1818no—
179C2×D4—1812no—
180C2×D4—1811no—
181C2×D4—1812no—
182C2×D4—1812no—
183C2×D4—1811no—
184C2×D4—1811no—
185C2×D4C2×#1062438211yescharacteristic
186C2×D4C2×#1062438212yescomplemented
187C22×C4—3628no—
188C22×C4—1812no—
189C22×C4—1812no—
190C22×C4—1811no—
191C22×C4—1811no—
192C22×C4C2×#1062438212yescomplemented
193C22×C4C2×#1062438211yescharacteristic, complemented
194C4⋊C4—3624no—
195C22⋊C4—3628no—
196C22⋊C4—1818no—
197C42—3624no—
198C23—3624no—
199C23—3624no—
200C23—14482no—
201C23—14482no—
202C23—14482no—
203C23—1812no—
204C23—1811no—
205C23—1818no—
206C23—1811no—
207C23—1812no—
208C23C22×#1062438211yescharacteristic
209C23C22×#1062438212yes—
210Q8—22no—
211Q8—3622no—
212D4—24no—
213D4—3622no—
214D4—3624no—
215D4—3624no—
216D4—3624no—
217D4—3622no—
218D4—3624no—
219D4—3622no—
220D4—3622no—
221D4—1818no—
222D4—1814no—
223D4C22⋊#1062438214yes—
224C2×C4—24no—
225C2×C4—22no—
226C2×C4—3624no—
227C2×C4—3624no—
228C2×C4—3622no—
229C2×C4—3628no—
230C2×C4—14488no—
231C2×C4—1811no—
232C2×C4—1811no—
233C2×C4—1814no—
234C2×C4C22×#1062438212yes—
235C2×C4C22×#1062438211yescharacteristic
236C2×C4C22×#1062438211yescharacteristic
237C22—21no—
238C22—42no—
239C22—41no—
240C22—3622no—
241C22—3621no—
242C22—14484no—
243C22—14484no—
244C22—1812no—
245C22—1814no—
246C22—7248no—
247C22—7241no—
248C22—7242no—
249C22—7241no—
250C22—7242no—
251C22C2×C22⋊#1062438214yes—
252C22C23×#1062438211yescharacteristic
253C4—21no—
254C4—22no—
255C4—3621no—
256C4—1812no—
257C4—7248no—
258C4C2×C22⋊#1062438212yes—
259C2—21no—
260C2—24no—
261C2—41no—
262C2—3622no—
263C2—3624no—
264C2—7241no—
265C2—7242no—
266C2C22×C22⋊#1062438211yescharacteristic
subgroupG/H
centerC2C22×C22⋊#10624382
commutatorC22×C181C23×C4
FrattiniC22C23×#10624382
FittingC2×#41×C181C4
socleC2×C181C22×C22⋊C4
prime radicalC2×#19153862C2
cosocleC22×D181C24
hypercenterC2×#41#10624382
hypocenterC181#2883
LCSG ▷ C22×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C2 ◁ C2×D4 ◁ C2×#41

Complements

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

NH ≅ G/Ntype
1. #26042346C2semidirect
2. #26042260C2semidirect
3. #26042258C2semidirect
4. C2×#19153875C2semidirect
5. D181⋊#99C2semidirect
8. #19153895C22semidirect
9. C2×C22⋊#10624382C22semidirect
10. #19153875C22semidirect
11. #19153875C22semidirect
12. #19153874C22semidirect
14. C2×#15668450C4semidirect
16. C2×#15668449C4semidirect
18. C22×D724C4semidirect
20. C2×D4×D181C4semidirect
21. C2×D4×D181C4semidirect
24. C2×#41×C181C4semidirect
30. C23×D181C2×C4semidirect
31. C23×D181C2×C4semidirect
37. C2×Dic362C2×C4semidirect
41. C2×D724C2×C4semidirect
42. C2×D724C2×C4semidirect
43. C2×D724C2×C4semidirect
47. C2×C22⋊D181C2×C4semidirect
48. C2×C22⋊D181C2×C4semidirect
54. C2×C4×D181C2×C4semidirect
55. C2×C4×D181C2×C4semidirect
56. C2×C4×D181C2×C4semidirect
59. C2×Q8×C181C2×C4semidirect
61. C2×D4×C181C2×C4semidirect
62. C22×C4×C181C2×C4semidirect
63. C22×C4×C181C2×C4semidirect
133. C181#2883semidirect
146. C2×#41#10624382semidirect
168. C2×Q8C2×#10624382semidirect
186. C2×D4C2×#10624382semidirect
192. C22×C4C2×#10624382semidirect
193. C22×C4C2×#10624382semidirect

Maximal subgroups

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

subgroupindex
#260423462
#260422602
#260422582
C2×#191538752
D181⋊#992
C2×#191538622
#2883181