Finite Group Databasebeta

← C2×#26041530 #37543315

Subgroups — C2×#26041530 #37543315

Order 23168.

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

normal subgroups only

This group has 596 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×#15668434C211yesmaximal, characteristic, complemented
2C22×#15668433C211yesmaximal, characteristic, complemented
3C2×#19153790C211yesmaximal, characteristic, complemented
4#26041530C218yesmaximal, direct product factor
5C2×#19153747C211yesmaximal, characteristic, complemented
6C2×#19153737C211yesmaximal, characteristic, complemented
7C2×#19153736C211yesmaximal, characteristic, complemented
8C2×#121×C181C211yesmaximal, characteristic, complemented
9C22×D724C2211yescharacteristic
10C22×C22⋊D181—21no—
11C2×#15668434—24no—
12C2×#15668434C2214yescomplemented
13C2×#15668433—24no—
14C2×#15668433C2214yescomplemented
15C22×#13089678C2211yescharacteristic
16C22×#13089678C2211yescharacteristic
17#19153790—22no—
18#19153790C2214yescomplemented
19C2×C4⋊Dic181C2211yescharacteristic
20C2×#15668426—21no—
21#19153747C2214yescomplemented
22#19153737C2214yescomplemented
23#19153736C2214yescomplemented
24C22×D4×C181—21no—
25C22×D4×C181C2211yescharacteristic
26C23×C4×C181C2211yescharacteristic
27#121×C181—22no—
28#121×C181C2214yescomplemented
29C2×C4⋊C4×C181C2211yescharacteristic
30C2×C22⋊C4×C181—21no—
31C23×D181—21no—
32C2×D724—22no—
33C2×D724—21no—
34C2×D724—22no—
35C2×D724C2312yes—
36C2×C22⋊D181—24no—
37C2×C22⋊D181—44no—
38C22×Dic181—21no—
39#15668434—28no—
40#15668433—28no—
41C2×#13089678—22no—
42C2×#13089678D414yescomplemented
43C2×#13089678C2312yes—
44C2×#13089678C2312yes—
45C4⋊Dic181—21no—
46C4⋊Dic181C2312yes—
47#15668426—24no—
48C24×C181—21no—
49C24×C181D411yescharacteristic
50C2×D4×C181—24no—
51C2×D4×C181—22no—
52C2×D4×C181—21no—
53C2×D4×C181—22no—
54C2×D4×C181—44no—
55C2×D4×C181C2312yes—
56C22×C4×C181—22no—
57C22×C4×C181—22no—
58C22×C4×C181—21no—
59C22×C4×C181D411yescharacteristic
60C22×C4×C181D412yes—
61C22×C4×C181C2312yes—
62C22×C4×C181C2311yescharacteristic
63C4⋊C4×C181—21no—
64C4⋊C4×C181C2312yes—
65C22⋊C4×C181—24no—
66C22×D181—22no—
67C22×D181—42no—
68C22×D181—42no—
69C22×D181—42no—
70D724—22no—
71D724—22no—
72D724—42no—
73C22⋊D181—48no—
74C2×Dic181—22no—
75C2×Dic181—42no—
76#13089678—22no—
77#13089678C2×D414yescomplemented
78C23×C181—22no—
79C23×C181—22no—
80C23×C181—22no—
81C23×C181—22no—
82C23×C181—42no—
83C23×C181—42no—
84C23×C181—42no—
85C23×C181C2×D412yes—
86C23×C181C2×D411yescharacteristic
87D4×C181—22no—
88D4×C181—22no—
89D4×C181—48no—
90D4×C181—42no—
91C2×C4×C181—22no—
92C2×C4×C181—24no—
93C2×C4×C181—22no—
94C2×C4×C181—22no—
95C2×C4×C181—22no—
96C2×C4×C181—42no—
97C2×C4×C181C2×D412yes—
98C2×C4×C181C2×D412yes—
99C2×C4×C181C2×D411yescharacteristic
100C2×C4×C181#4112yes—
101C2×C4×C181C2411yescharacteristic
102C22×C181—22no—
103C22×C181—22no—
104C22×C181—22no—
105C22×C181—44no—
106C22×C181—42no—
107C22×C181—42no—
108C22×C181—42no—
109C22×C181—84no—
110C22×C181#12112yes—
111C22×C181AGL(1,ℤ/8)12yes—
112C22×C181AGL(1,ℤ/8)12yes—
113C22×C181C22×D411yescharacteristic
114C2×D181—42no—
115C2×D181—42no—
116C2×D181—42no—
117C2×D181—84no—
118C4×C181—22no—
119C4×C181—42no—
120C4×C181#12112yes—
121C4×C181C22×D411yescharacteristic
122C4×C181C2×#4111yescharacteristic
123Dic181—42no—
124C2×C181—42no—
125C2×C181—82no—
126C2×C181#46914yescomplemented
127C2×C181C2×#12111yescharacteristic
128C2×C181C2×AGL(1,ℤ/8)11yescharacteristic
129C2×C181C2×AGL(1,ℤ/8)11yescharacteristic
130D181—82no—
131C181C2×#46911yescharacteristic, complemented, Sylow
132C2×#469—1811nomaximal, Sylow
133C22×D8—1811no—
134C22×M16—1811no—
135C2×#121—1811no—
136C2×#121D18111yescharacteristic, complemented
137#469—1818no—
138C2×#106—1811no—
139C2×#102—1811no—
140C2×#102—1811no—
141C22×D4—21no—
142C22×D4—3621no—
143C22×D4—1811no—
144C22×D4C2×D18111yescharacteristic
145C23×C4C2×D18111yescharacteristic
146C2×D8—3624no—
147C2×D8—1814no—
148C2×M16—3624no—
149C2×M16—1814no—
150C22×C8—1811no—
151C22×C8—1811no—
152#121—22no—
153#121—3622no—
154#121—1814no—
155#121C2×D18114yescomplemented
156C2×C4⋊C4—1811no—
157C2×C4⋊C4C2×D18111yescharacteristic
158C2×C22⋊C4—21no—
159C2×C22⋊C4—3621no—
160#106—1814no—
161#102—1814no—
162#102—1814no—
163C24—21no—
164C24—3621no—
165C24C22⋊D18111yescharacteristic
166C2×D4—24no—
167C2×D4—22no—
168C2×D4—21no—
169C2×D4—22no—
170C2×D4—44no—
171C2×D4—3624no—
172C2×D4—3622no—
173C2×D4—3621no—
174C2×D4—3622no—
175C2×D4—1812no—
176C2×D4—7244no—
177C2×D4C22×D18112yes—
178C22×C4—22no—
179C22×C4—22no—
180C22×C4—21no—
181C22×C4—3621no—
182C22×C4C22⋊D18112yes—
183C22×C4C22⋊D18111yescharacteristic
184C22×C4C22×D18112yes—
185C22×C4C22×D18111yescharacteristic
186D8—3628no—
187M16—3628no—
188C2×C8—3622no—
189C2×C8—1812no—
190C2×C8—1814no—
191C2×C8—1812no—
192C4⋊C4—21no—
193C4⋊C4—3621no—
194C4⋊C4—1812no—
195C4⋊C4C22×D18112yes—
196C22⋊C4—24no—
197C22⋊C4—3624no—
198C23—22no—
199C23—22no—
200C23—22no—
201C23—22no—
202C23—42no—
203C23—42no—
204C23—42no—
205C23—3622no—
206C23—7242no—
207C23—7242no—
208C23—7242no—
209C23C2×C22⋊D18112yes—
210C23C2×C22⋊D18111yescharacteristic
211D4—22no—
212D4—22no—
213D4—48no—
214D4—42no—
215D4—3622no—
216D4—3622no—
217D4—7248no—
218D4—7242no—
219C2×C4—22no—
220C2×C4—24no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—42no—
225C2×C4—3622no—
226C2×C4—7242no—
227C2×C4D4×D18112yes—
228C2×C4C2×C22⋊D18112yes—
229C2×C4C2×C22⋊D18111yescharacteristic
230C2×C4#1566844812yes—
231C2×C4C23×D18111yescharacteristic
232C8—3622no—
233C8—1814no—
234C22—22no—
235C22—22no—
236C22—22no—
237C22—44no—
238C22—42no—
239C22—42no—
240C22—42no—
241C22—84no—
242C22—14484no—
243C22—7242no—
244C22—7242no—
245C22—7242no—
246C22#1915378812yes—
247C22#1915384112yes—
248C22#1915384212yes—
249C22C22×C22⋊D18111yescharacteristic
250C4—22no—
251C4—42no—
252C4—7242no—
253C4#1915378812yes—
254C4C2×D4×D18111yescharacteristic
255C4C2×#1566844811yescharacteristic
256C2—42no—
257C2—82no—
258C2—14482no—
259C2#2604153014yesdirect product factor
260C2C2×#1915378811yescharacteristic
261C2C2×#1915384111yescharacteristic
262C2C2×#1915384211yescharacteristic
subgroupG/H
centerC23C2×C22⋊D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
FittingC2×#121×C181C2
socleC23×C181C2×D4
hypercenterC2×#121D181
hypocenterC181C2×#469
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#121

Complements

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

NH ≅ G/Ntype
1. C22×#15668434C2semidirect
2. C22×#15668433C2semidirect
3. C2×#19153790C2semidirect
4. #26041530C2direct (N × H)
5. C2×#19153747C2semidirect
6. C2×#19153737C2semidirect
7. C2×#19153736C2semidirect
8. C2×#121×C181C2semidirect
12. C2×#15668434C22semidirect
14. C2×#15668433C22semidirect
18. #19153790C22semidirect
21. #19153747C22semidirect
22. #19153737C22semidirect
23. #19153736C22semidirect
28. #121×C181C22semidirect
42. C2×#13089678D4semidirect
77. #13089678C2×D4semidirect
126. C2×C181#469semidirect
131. C181C2×#469semidirect
136. C2×#121D181semidirect
155. #121C2×D181semidirect
259. C2#26041530direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#156684342
C22×#156684332
C2×#191537902
#260415302
C2×#191537472
C2×#191537372
C2×#191537362
C2×#121×C1812
C2×#469181