Finite Group Databasebeta

← #37543200

Subgroups — #37543200

Order 23168.

584 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 584 conjugacy classes of subgroups — too many to draw. Only its 137 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×#19153844C211yesmaximal, characteristic, complemented
2C2×#19153842C211yesmaximal, characteristic, complemented
3C2×#19153812C211yesmaximal, characteristic, complemented
4#26041747C211yesmaximal, characteristic, complemented
5#26041478C214yesmaximal, complemented
6#26041473C214yesmaximal, complemented
7#26041302C211yesmaximal, characteristic, complemented
8#26041283C211yesmaximal, characteristic, complemented
9#418×C181C211yesmaximal, characteristic, complemented
10C2×#15668449C2211yescharacteristic
11C22×D724C2211yescharacteristic
12C22×C22⋊D181—22no—
13#19153844—24no—
14#19153842—24no—
15C2×#15668438C2212yescomplemented
16C2×#15668437C2212yescomplemented
17C2×D1448C2212yescomplemented
18C2×#15668434C2212yescomplemented
19C2×#15668433C2211yescharacteristic
20#19153812—21no—
21#19153812—21no—
22#19153812C2214yes—
23#19153781C2212yescomplemented
24#19153778C2212yescomplemented
25#19153776—22no—
26#19153773—21no—
27C2×C4×Dic181C2211yescharacteristic
28C42×D181—21no—
29#19153745C2214yescomplemented
30#19153736C2214yescomplemented
31C22×D4×C181C2211yescharacteristic
32C2×M16×C181C2211yescharacteristic
33#117×C181C2211yescharacteristic
34#102×C181C2214yescomplemented
35C23×D181—21no—
36#15668449—22no—
37#15668449D414yescomplemented
38C2×Dic362C2311yescharacteristic
39C2×D724—22no—
40C2×D724—21no—
41C2×D724—22no—
42C2×D724C2312yes—
43C2×D724C2311yescharacteristic
44C2×C22⋊D181—24no—
45C2×C22⋊D181—24no—
46C2×C22⋊D181—21no—
47C2×C22⋊D181—48no—
48C22×Dic181D412yes—
49C2×C4×D181—22no—
50C2×C4×D181—21no—
51#15668438—24no—
52#15668437—24no—
53D1448—24no—
54#15668434—24no—
55#15668433—22no—
56C2×#13089678C2312yes—
57#15668428—22no—
58#15668426—22no—
59C4×Dic181—22no—
60C4×Dic181D411yescharacteristic
61C4×Dic181D411yescharacteristic
62C4×Dic181C2312yes—
63C24×C181—21no—
64C2×D4×C181—22no—
65C2×D4×C181—21no—
66C2×D4×C181—22no—
67C2×D4×C181C2312yes—
68C22×C4×C181C2311yescharacteristic
69M16×C181—22no—
70C2×C8×C181C2312yes—
71C4⋊C4×C181C2312yes—
72C22⋊C4×C181—21no—
73C42×C181—21no—
74C22×D181—21no—
75C22×D181—22no—
76C22×D181—44no—
77C22×D181—42no—
78Dic362—21no—
79Dic362C2×D412yescomplemented
80D724—22no—
81D724—21no—
82D724—22no—
83D724—42no—
84D724C2×D412yescomplemented
85C22⋊D181—24no—
86C22⋊D181—48no—
87C22⋊D181—48no—
88C2×Dic181—21no—
89C2×Dic181—24no—
90C2×Dic181C2×D412yes—
91C2×Dic181C2×D412yes—
92C4×D181—24no—
93C4×D181—44no—
94#13089678—22no—
95C23×C181—22no—
96C23×C181—44no—
97C23×C181—42no—
98C23×C181C2×D411yescharacteristic
99D4×C181—22no—
100D4×C181—22no—
101D4×C181—42no—
102C2×C4×C181—22no—
103C2×C4×C181C2×D411yescharacteristic
104C2×C4×C181C2×D412yes—
105C2×C4×C181#4112yes—
106C2×C4×C181C2411yescharacteristic
107C8×C181—22no—
108C22×C181—22no—
109C22×C181—42no—
110C22×C181—44no—
111C22×C181—84no—
112C22×C181#12112yes—
113C22×C181C22×D411yescharacteristic
114C2×D181—22no—
115C2×D181—42no—
116C2×D181—42no—
117C2×D181—44no—
118C2×D181—84no—
119C4×C181—42no—
120C4×C181#12112yes—
121C4×C181C22×D411yescharacteristic
122C4×C181C2×#4111yescharacteristic
123Dic181—22no—
124Dic181—22no—
125Dic181AGL(1,ℤ/8)14yescomplemented
126C2×C181—22no—
127C2×C181—82no—
128C2×C181C2×#12111yescharacteristic
129C2×C181C2×AGL(1,ℤ/8)12yes—
130D181—42no—
131D181—82no—
132C181#303811yescharacteristic, complemented, Sylow
133#3038—1811nomaximal, Sylow
134C2×AGL(1,ℤ/8)—1811no—
135C2×AGL(1,ℤ/8)—1811no—
136C2×#127—1811no—
137C4×#41—1811no—
138#460—1814no—
139#459—1814no—
140#423—1811no—
141#418—1811no—
142#418D18111yescharacteristic, complemented
143C2×#41—1811no—
144C22×D4—3622no—
145C22×D4—1811no—
146C22×D4C2×D18111yescharacteristic
147AGL(1,ℤ/8)—3624no—
148AGL(1,ℤ/8)—3624no—
149C2×SD16—1812no—
150C2×SD16—1812no—
151C2×D8—1812no—
152C2×D8—1812no—
153C2×M16—1811no—
154C2×M16C2×D18111yescharacteristic
155#127—3621no—
156#127—3621no—
157#127—1814no—
158C4×Q8—1812no—
159C4×D4—3622no—
160C4×D4—1812no—
161#117—3621no—
162#117C2×D18111yescharacteristic
163C2×C42—3621no—
164C2×C42—1811no—
165C4⋊C8—1814no—
166#102—1814no—
167#102C2×D18114yescomplemented
168C24—21no—
169C24—3621no—
170#41—3622no—
171#41—1814no—
172C2×Q8—1811no—
173C2×D4—22no—
174C2×D4—21no—
175C2×D4—22no—
176C2×D4—3621no—
177C2×D4—3624no—
178C2×D4—3624no—
179C2×D4—3622no—
180C2×D4—3621no—
181C2×D4—3622no—
182C2×D4—1812no—
183C2×D4—1811no—
184C2×D4—7248no—
185C2×D4C22×D18112yes—
186C22×C4—3622no—
187C22×C4—3621no—
188C22×C4—1812no—
189C22×C4C22×D18111yescharacteristic
190SD16—3624no—
191SD16—3624no—
192D8—3624no—
193D8—3624no—
194M16—22no—
195M16—3622no—
196C2×C8—1812no—
197C2×C8C22×D18112yes—
198C4⋊C4—3622no—
199C4⋊C4C22×D18112yes—
200C22⋊C4—21no—
201C22⋊C4—3622no—
202C42—21no—
203C42—3622no—
204C42—1811no—
205C42—1811no—
206C42—1812no—
207C23—22no—
208C23—44no—
209C23—42no—
210C23—3621no—
211C23—3622no—
212C23—7244no—
213C23—7242no—
214C23D4×D18111yescharacteristic
215Q8—3621no—
216Q8—1812no—
217D4—22no—
218D4—22no—
219D4—42no—
220D4—3624no—
221D4—3622no—
222D4—3621no—
223D4—3622no—
224D4—1812no—
225D4—7248no—
226D4—7248no—
227D4—7242no—
228C2×C4—22no—
229C2×C4—3621no—
230C2×C4—3624no—
231C2×C4—3624no—
232C2×C4—1812no—
233C2×C4—1812no—
234C2×C4—7244no—
235C2×C4D4×D18111yescharacteristic
236C2×C4D4×D18112yes—
237C2×C4#1566844912yes—
238C2×C4C23×D18111yescharacteristic
239C8—22no—
240C8—3622no—
241C22—22no—
242C22—42no—
243C22—44no—
244C22—84no—
245C22—3622no—
246C22—14484no—
247C22—7242no—
248C22—7244no—
249C22—7242no—
250C22#1915378912yes—
251C22C2×D4×D18111yescharacteristic
252C4—42no—
253C4—3622no—
254C4—3622no—
255C4—1814no—
256C4#1915378912yes—
257C4C2×D4×D18111yescharacteristic
258C4C2×#1566844911yescharacteristic
259C2—22no—
260C2—82no—
261C2—14482no—
262C2—7242no—
263C2C2×#1915378911yescharacteristic
264C2AGL(1,ℤ/8)×D18112yes—
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#418×C181C2
socleC22×C181C22×D4
hypercenter#418D181
hypocenterC181#3038
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C22×C4 ◁ #418

Complements

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

NH ≅ G/Ntype
1. C2×#19153844C2semidirect
2. C2×#19153842C2semidirect
3. C2×#19153812C2semidirect
4. #26041747C2semidirect
5. #26041478C2semidirect
6. #26041473C2semidirect
7. #26041302C2semidirect
8. #26041283C2semidirect
9. #418×C181C2semidirect
15. C2×#15668438C22semidirect
16. C2×#15668437C22semidirect
17. C2×D1448C22semidirect
18. C2×#15668434C22semidirect
23. #19153781C22semidirect
24. #19153778C22semidirect
29. #19153745C22semidirect
30. #19153736C22semidirect
34. #102×C181C22semidirect
37. #15668449D4semidirect
79. Dic362C2×D4semidirect
84. D724C2×D4semidirect
125. Dic181AGL(1,ℤ/8)semidirect
132. C181#3038semidirect
142. #418D181semidirect
167. #102C2×D181semidirect

Maximal subgroups

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

subgroupindex
C2×#191538442
C2×#191538422
C2×#191538122
#260417472
#260414782
#260414732
#260413022
#260412832
#418×C1812
#3038181