Finite Group Databasebeta

← C2×#26041423 #37542921

Subgroups — C2×#26041423 #37542921

Order 23168.

812 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 812 conjugacy classes of subgroups — too many to draw. Only its 165 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×#15668450C211yesmaximal, characteristic, complemented
2C22×#15668434C211yesmaximal, characteristic, complemented
3C2×#19153787C211yesmaximal, characteristic, complemented
4#26041423C218yesmaximal, direct product factor
5C2×#19153740C211yesmaximal, characteristic, complemented
6C2×#19153738C211yesmaximal, characteristic, complemented
7C2×#19153728C211yesmaximal, characteristic, complemented
8C22×SD16×C181C211yesmaximal, characteristic, complemented
9C2×#15668450—24no—
10C2×#15668450—22no—
11C2×#15668450—24no—
12C2×#15668450C2214yescomplemented
13C22×D724—21no—
14C22×D724C2211yescharacteristic
15C22×C22⋊D181—21no—
16C22×C4×D181—21no—
17C22×C4×D181C2211yescharacteristic
18C2×#15668434—24no—
19C2×#15668434C2214yescomplemented
20C22×#13089678C2211yescharacteristic
21#19153787—22no—
22#19153787C2214yescomplemented
23C2×C4⋊Dic181C2211yescharacteristic
24C2×C22⋊Dic181—21no—
25#19153740C2214yescomplemented
26#19153738C2214yescomplemented
27#19153728C2214yescomplemented
28C22×Q8×C181C2211yescharacteristic
29C22×D4×C181C2211yescharacteristic
30C2×SD16×C181—24no—
31C2×SD16×C181C2214yescomplemented
32C22×C8×C181C2211yescharacteristic
33C23×D181—21no—
34C23×D181D411yescharacteristic
35#15668450—216no—
36#15668450—28no—
37#15668450—28no—
38C2×D724—28no—
39C2×D724—24no—
40C2×D724—21no—
41C2×D724—22no—
42C2×D724D414yescomplemented
43C2×D724C2312yes—
44C2×C22⋊D181—24no—
45C2×C22⋊D181—44no—
46C22×Dic181—21no—
47C22×Dic181D411yescharacteristic
48C2×C4×D181—28no—
49C2×C4×D181—24no—
50C2×C4×D181—22no—
51C2×C4×D181—22no—
52C2×C4×D181D412yes—
53C2×C4×D181C2312yes—
54#15668434—28no—
55C2×#13089678—22no—
56C2×#13089678C2312yes—
57C4⋊Dic181—21no—
58C4⋊Dic181C2312yes—
59C22⋊Dic181—24no—
60C24×C181—21no—
61C2×Q8×C181—22no—
62C2×Q8×C181—21no—
63C2×Q8×C181D414yescomplemented
64C2×Q8×C181C2312yes—
65C2×D4×C181—22no—
66C2×D4×C181—22no—
67C2×D4×C181—21no—
68C2×D4×C181C2312yes—
69C22×C4×C181—21no—
70C22×C4×C181C2311yescharacteristic
71SD16×C181—28no—
72C2×C8×C181—22no—
73C2×C8×C181C2312yes—
74C22×D181—22no—
75C22×D181—24no—
76C22×D181—22no—
77C22×D181—22no—
78C22×D181—22no—
79C22×D181—42no—
80C22×D181—42no—
81C22×D181C2×D412yes—
82D724—216no—
83D724—24no—
84D724—22no—
85D724C2×D414yescomplemented
86C22⋊D181—48no—
87C2×Dic181—22no—
88C2×Dic181—22no—
89C2×Dic181—42no—
90C2×Dic181C2×D412yes—
91C4×D181—216no—
92C4×D181—24no—
93C4×D181—22no—
94C4×D181—22no—
95#13089678—22no—
96C23×C181—22no—
97C23×C181—42no—
98C23×C181—42no—
99C23×C181—42no—
100C23×C181C2×D411yescharacteristic
101Q8×C181—24no—
102Q8×C181—22no—
103Q8×C181C2×D414yescomplemented
104D4×C181—22no—
105D4×C181—22no—
106D4×C181—42no—
107C2×C4×C181—24no—
108C2×C4×C181—22no—
109C2×C4×C181C2×D411yescharacteristic
110C2×C4×C181C2×D412yes—
111C2×C4×C181C2×D412yes—
112C2×C4×C181C2411yescharacteristic
113C8×C181—22no—
114C22×C181—42no—
115C22×C181—42no—
116C22×C181—42no—
117C22×C181—84no—
118C22×C181#12012yes—
119C22×C181#13512yes—
120C22×C181AGL(1,ℤ/8)12yes—
121C22×C181C22×D411yescharacteristic
122C2×D181—22no—
123C2×D181—24no—
124C2×D181—22no—
125C2×D181—22no—
126C2×D181—42no—
127C2×D181—42no—
128C2×D181—44no—
129C2×D181—48no—
130C4×C181—24no—
131C4×C181#12012yes—
132C4×C181C22×D411yescharacteristic
133C4×C181C22×D411yescharacteristic
134Dic181—22no—
135Dic181—42no—
136C2×C181—82no—
137C2×C181#44914yescomplemented
138C2×C181C2×#12011yescharacteristic
139C2×C181C2×#13511yescharacteristic
140C2×C181C2×AGL(1,ℤ/8)11yescharacteristic
141D181—42no—
142D181—44no—
143C181C2×#44911yescharacteristic, complemented, Sylow
144C2×#449—1811nomaximal, Sylow
145C22×#41—1811no—
146C22×SD16D18111yescharacteristic, complemented
147C22×D8—1811no—
148C2×#121—1811no—
149#449—1818no—
150C2×#103—1811no—
151C2×#102—1811no—
152C2×C22⋊C8—1811no—
153C2×#41—3624no—
154C2×#41—3624no—
155C2×#41—3622no—
156C2×#41—1814no—
157C22×Q8C2×D18111yescharacteristic
158C22×D4—3621no—
159C22×D4—3621no—
160C22×D4—1811no—
161C22×D4C2×D18111yescharacteristic
162C23×C4—3621no—
163C23×C4—1811no—
164C2×SD16—24no—
165C2×SD16C2×D18114yescomplemented
166C2×D8—3624no—
167C2×D8—1814no—
168C22×C8—1811no—
169C22×C8C2×D18111yescharacteristic
170#121—3622no—
171#121—1814no—
172C2×C4⋊C4—1811no—
173C2×C22⋊C4—3621no—
174#103—1814no—
175#102—1814no—
176C22⋊C8—1814no—
177C24—21no—
178C24—3621no—
179C24—1811no—
180#41—36216no—
181#41—3628no—
182#41—3628no—
183C2×Q8—22no—
184C2×Q8—21no—
185C2×Q8C22⋊D18114yescomplemented
186C2×Q8C22×D18112yes—
187C2×D4—22no—
188C2×D4—21no—
189C2×D4—22no—
190C2×D4—3628no—
191C2×D4—3624no—
192C2×D4—3624no—
193C2×D4—3621no—
194C2×D4—3622no—
195C2×D4—1814no—
196C2×D4—1812no—
197C2×D4—7244no—
198C2×D4C22×D18112yes—
199C22×C4—21no—
200C22×C4—3628no—
201C22×C4—3624no—
202C22×C4—3622no—
203C22×C4—3622no—
204C22×C4—3621no—
205C22×C4—1811no—
206C22×C4—1812no—
207C22×C4—1812no—
208C22×C4C22×D18111yescharacteristic
209SD16—28no—
210D8—3628no—
211C2×C8—22no—
212C2×C8—3622no—
213C2×C8—1812no—
214C2×C8C22×D18112yes—
215C4⋊C4—3621no—
216C4⋊C4—1812no—
217C22⋊C4—3624no—
218C23—22no—
219C23—42no—
220C23—42no—
221C23—42no—
222C23—3622no—
223C23—3624no—
224C23—3622no—
225C23—3622no—
226C23—3622no—
227C23—1812no—
228C23—7242no—
229C23—7242no—
230C23D4×D18111yescharacteristic
231Q8—24no—
232Q8—22no—
233Q8C2×C22⋊D18114yescomplemented
234D4—22no—
235D4—22no—
236D4—42no—
237D4—36216no—
238D4—3624no—
239D4—3622no—
240D4—1814no—
241D4—7248no—
242C2×C4—24no—
243C2×C4—22no—
244C2×C4—36216no—
245C2×C4—3622no—
246C2×C4—3624no—
247C2×C4—3622no—
248C2×C4—3622no—
249C2×C4—3622no—
250C2×C4—1812no—
251C2×C4—7242no—
252C2×C4D4×D18111yescharacteristic
253C2×C4D4×D18112yes—
254C2×C4C2×C22⋊D18112yes—
255C2×C4C23×D18111yescharacteristic
256C8—22no—
257C8—3622no—
258C22—42no—
259C22—42no—
260C22—42no—
261C22—84no—
262C22—3622no—
263C22—3624no—
264C22—3622no—
265C22—3622no—
266C22—7244no—
267C22—7248no—
268C22—7242no—
269C22—7242no—
270C22#1915378412yes—
271C22#1915383712yes—
272C22#1915384312yes—
273C22C2×D4×D18111yescharacteristic
274C4—24no—
275C4—3622no—
276C4—7242no—
277C4#1915378412yes—
278C4C2×D4×D18111yescharacteristic
279C4C22×C22⋊D18111yescharacteristic
280C2—82no—
281C2—7242no—
282C2—7244no—
283C2#2604142314yesdirect product factor
284C2C2×#1915378411yescharacteristic
285C2C2×#1915383711yescharacteristic
286C2C2×#1915384311yescharacteristic
subgroupG/H
centerC23D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
FittingC22×SD16×C181C2
socleC23×C181C2×D4
hypercenterC22×SD16D181
hypocenterC181C2×#449
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×SD16

Complements

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

NH ≅ G/Ntype
1. C22×#15668450C2semidirect
2. C22×#15668434C2semidirect
3. C2×#19153787C2semidirect
4. #26041423C2direct (N × H)
5. C2×#19153740C2semidirect
6. C2×#19153738C2semidirect
7. C2×#19153728C2semidirect
8. C22×SD16×C181C2semidirect
12. C2×#15668450C22semidirect
19. C2×#15668434C22semidirect
22. #19153787C22semidirect
25. #19153740C22semidirect
26. #19153738C22semidirect
27. #19153728C22semidirect
31. C2×SD16×C181C22semidirect
42. C2×D724D4semidirect
63. C2×Q8×C181D4semidirect
85. D724C2×D4semidirect
103. Q8×C181C2×D4semidirect
137. C2×C181#449semidirect
143. C181C2×#449semidirect
146. C22×SD16D181semidirect
165. C2×SD16C2×D181semidirect
185. C2×Q8C22⋊D181semidirect
233. Q8C2×C22⋊D181semidirect
283. C2#26041423direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#156684502
C22×#156684342
C2×#191537872
#260414232
C2×#191537402
C2×#191537382
C2×#191537282
C22×SD16×C1812
C2×#449181