Finite Group Databasebeta

← C2×#26041526 #37543310

Subgroups — C2×#26041526 #37543310

Order 23168.

544 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 544 conjugacy classes of subgroups — too many to draw. Only its 157 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
1C22×#15668438C211yesmaximal, characteristic, complemented
2C22×#15668432C211yesmaximal, characteristic, complemented
3C2×#19153796C211yesmaximal, characteristic, complemented
4C2×#19153791C211yesmaximal, characteristic
5#26041526C218yesmaximal, direct product factor
6C2×#19153739C211yesmaximal, characteristic, complemented
7C2×#19153736C211yesmaximal, characteristic, complemented
8C2×#107×C181C211yesmaximal, characteristic, complemented
9C22×Dic362C2211yescharacteristic
10C22×D724—21no—
11C22×D724C2211yescharacteristic
12C22×C4×D181C2211yescharacteristic
13C2×#15668438—24no—
14C2×#15668438C2214yescomplemented
15C2×#15668432—24no—
16C2×#15668432C2214yescomplemented
17C22×#13089678C2211yescharacteristic
18#19153796—22no—
19#19153796C2214yescomplemented
20#19153791—22no—
21#19153791C2214yes—
22C2×C4⋊Dic181—21no—
23C2×#15668426—21no—
24C2×#15668426—21no—
25#19153739C2214yescomplemented
26#19153736C2214yescomplemented
27C22×C8×C181C2211yescharacteristic
28C2×C4⋊C4×C181C2211yescharacteristic
29C2×C4⋊C4×C181C2211yescharacteristic
30#107×C181C2214yescomplemented
31C23×D181—21no—
32C23×D181D411yescharacteristic
33C2×Dic362—21no—
34C2×Dic362—22no—
35C2×Dic362—22no—
36C2×Dic362C2312yes—
37C2×D724—24no—
38C2×D724—22no—
39C2×D724—21no—
40C2×D724—22no—
41C2×D724—44no—
42C2×D724C2312yes—
43C22×Dic181—21no—
44C22×Dic181D411yescharacteristic
45C2×C4×D181—22no—
46C2×C4×D181—22no—
47C2×C4×D181D412yes—
48C2×C4×D181C2312yes—
49#15668438—28no—
50#15668432—28no—
51C2×#13089678—22no—
52C2×#13089678C2312yes—
53C4⋊Dic181—24no—
54#15668426—24no—
55#15668426—24no—
56C22×C4×C181—21no—
57C22×C4×C181—21no—
58C22×C4×C181C2311yescharacteristic
59C2×C8×C181D414yescomplemented
60C2×C8×C181C2312yes—
61C4⋊C4×C181—21no—
62C4⋊C4×C181—21no—
63C4⋊C4×C181C2312yes—
64C4⋊C4×C181C2312yes—
65C22×D181—22no—
66C22×D181—22no—
67C22×D181—22no—
68C22×D181—22no—
69C22×D181—42no—
70C22×D181—42no—
71C22×D181—42no—
72C22×D181C2×D412yes—
73Dic362—22no—
74Dic362—22no—
75Dic362—42no—
76D724—22no—
77D724—22no—
78D724—48no—
79D724—42no—
80C2×Dic181—22no—
81C2×Dic181—22no—
82C2×Dic181—42no—
83C2×Dic181C2×D412yes—
84C4×D181—24no—
85C4×D181—22no—
86C4×D181—22no—
87#13089678—22no—
88C23×C181C2×D411yescharacteristic
89C2×C4×C181—22no—
90C2×C4×C181—22no—
91C2×C4×C181—42no—
92C2×C4×C181—42no—
93C2×C4×C181C2×D411yescharacteristic
94C2×C4×C181C2×D412yes—
95C2×C4×C181#4112yes—
96C2×C4×C181C2411yescharacteristic
97C8×C181C2×D414yescomplemented
98C22×C181#12112yes—
99C22×C181AGL(1,ℤ/8)12yes—
100C22×C181#13712yes—
101C22×C181C22×D411yescharacteristic
102C2×D181—22no—
103C2×D181—22no—
104C2×D181—22no—
105C2×D181—44no—
106C2×D181—42no—
107C2×D181—42no—
108C2×D181—42no—
109C2×D181—84no—
110C4×C181—42no—
111C4×C181—42no—
112C4×C181#12112yes—
113C4×C181C22×D411yescharacteristic
114C4×C181C2×#4111yescharacteristic
115Dic181—22no—
116Dic181—42no—
117C2×C181#46814yescomplemented
118C2×C181C2×#12111yescharacteristic
119C2×C181C2×AGL(1,ℤ/8)11yescharacteristic
120C2×C181C2×#13711yescharacteristic
121D181—42no—
122D181—82no—
123C181C2×#46811yescharacteristic, complemented, Sylow
124C2×#468—1811nomaximal, Sylow
125C22×SD16—1811no—
126C22×M16—1811no—
127C2×C22⋊Q8—1811no—
128C2×#121—1811no—
129#468—1818no—
130C2×#107D18111yescharacteristic, complemented
131C2×#103—1811no—
132C2×#102—1811no—
133C22×Q8—1811no—
134C22×D4—3621no—
135C22×D4—1811no—
136C23×C4—1811no—
137C2×SD16—3624no—
138C2×SD16—1814no—
139C2×M16—3624no—
140C2×M16—1814no—
141C22×C8—1811no—
142C22×C8C2×D18111yescharacteristic
143C22⋊Q8—3622no—
144C22⋊Q8—1814no—
145#121—3622no—
146#121—1814no—
147C2×C4⋊C4—3621no—
148C2×C4⋊C4C2×D18111yescharacteristic
149C2×C4⋊C4C2×D18111yescharacteristic
150C2×C22⋊C4—3621no—
151C2×C22⋊C4—3621no—
152#107C2×D18114yescomplemented
153#103—1814no—
154#102—1814no—
155C24—3621no—
156C24—1811no—
157C2×Q8—3621no—
158C2×Q8—3622no—
159C2×Q8—3622no—
160C2×Q8—1812no—
161C2×D4—3624no—
162C2×D4—3621no—
163C2×D4—3622no—
164C2×D4—3622no—
165C2×D4—1812no—
166C2×D4—7244no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—3622no—
170C22×C4—3622no—
171C22×C4—3621no—
172C22×C4—1811no—
173C22×C4—1812no—
174C22×C4—1812no—
175C22×C4C22×D18111yescharacteristic
176SD16—3628no—
177M16—3628no—
178C2×C8—3622no—
179C2×C8—1812no—
180C2×C8D72414yescomplemented
181C2×C8C22×D18112yes—
182C4⋊C4—21no—
183C4⋊C4—21no—
184C4⋊C4—3624no—
185C4⋊C4C22×D18112yes—
186C4⋊C4C22×D18112yes—
187C22⋊C4—3624no—
188C22⋊C4—3624no—
189C23—3622no—
190C23—3622no—
191C23—3622no—
192C23—3622no—
193C23—1812no—
194C23—7242no—
195C23—7242no—
196C23—7242no—
197C23D4×D18111yescharacteristic
198Q8—3622no—
199Q8—3622no—
200Q8—7242no—
201D4—3622no—
202D4—3622no—
203D4—7248no—
204D4—7242no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—42no—
208C2×C4—42no—
209C2×C4—3622no—
210C2×C4—3624no—
211C2×C4—3622no—
212C2×C4—3622no—
213C2×C4—3622no—
214C2×C4—1812no—
215C2×C4—7242no—
216C2×C4D4×D18111yescharacteristic
217C2×C4C2×D72412yes—
218C2×C4#1566845012yes—
219C2×C4C23×D18111yescharacteristic
220C8—3622no—
221C8C2×D72414yescomplemented
222C22—3622no—
223C22—3622no—
224C22—3622no—
225C22—14484no—
226C22—7242no—
227C22—7242no—
228C22—7242no—
229C22—7244no—
230C22#1915379112yes—
231C22#1915384012yes—
232C22#1915384812yes—
233C22C2×D4×D18111yescharacteristic
234C4—42no—
235C4—42no—
236C4—3622no—
237C4—7242no—
238C4#1915379112yes—
239C4C22×D72411yescharacteristic
240C4C2×#1566845011yescharacteristic
241C2—14482no—
242C2—7242no—
243C2#2604152614yesdirect product factor
244C2C2×#1915379111yescharacteristic
245C2C2×#1915384011yescharacteristic
246C2C2×#1915384811yescharacteristic
subgroupG/H
centerC23D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
FittingC2×#107×C181C2
socleC23×C181C2×D4
prime radicalC2×#19153791C2
hypercenterC2×#107D181
hypocenterC181C2×#468
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C23 ◁ C22×C4 ◁ C2×#107

Complements

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

NH ≅ G/Ntype
1. C22×#15668438C2semidirect
2. C22×#15668432C2semidirect
3. C2×#19153796C2semidirect
5. #26041526C2direct (N × H)
6. C2×#19153739C2semidirect
7. C2×#19153736C2semidirect
8. C2×#107×C181C2semidirect
14. C2×#15668438C22semidirect
16. C2×#15668432C22semidirect
19. #19153796C22semidirect
25. #19153739C22semidirect
26. #19153736C22semidirect
30. #107×C181C22semidirect
59. C2×C8×C181D4semidirect
97. C8×C181C2×D4semidirect
117. C2×C181#468semidirect
123. C181C2×#468semidirect
130. C2×#107D181semidirect
152. #107C2×D181semidirect
180. C2×C8D724semidirect
221. C8C2×D724semidirect
243. C2#26041526direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#156684382
C22×#156684322
C2×#191537962
C2×#191537912
#260415262
C2×#191537392
C2×#191537362
C2×#107×C1812
C2×#468181