Finite Group Databasebeta

← #37543332

Subgroups — #37543332

Order 23168.

500 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 127 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×#19153839C211yesmaximal, characteristic, complemented
2C2×M16×D181C211yesmaximal, characteristic, complemented
3#26041792C212yesmaximal, complemented
4#26041534C211yesmaximal, characteristic, complemented
5#26041533C211yesmaximal, characteristic, complemented
6#26041532C211yesmaximal, characteristic, complemented
7#26041528C212yesmaximal, complemented
8#26041527C212yesmaximal, complemented
9#26041314C211yesmaximal, characteristic, complemented
10#26041274C212yesmaximal, complemented
11#470×C181C211yesmaximal, characteristic, complemented
12C2×#15668450C2212yes—
13C22×C4×D181C2211yescharacteristic
14#19153839—24no—
15C2×#15668436C2212yescomplemented
16C2×D1448C2211yescharacteristic, complemented
17C2×#15668433C2211yescharacteristic
18C2×#15668432C2211yescharacteristic
19C2×#15668432C2211yescharacteristic
20M16×D181—24no—
21C2×C8×D181C2211yescharacteristic
22C2×C8×D181C2211yescharacteristic
23#19153801—22no—
24#19153792C2212yes—
25#19153791C2212yes—
26#19153790C2212yes—
27#19153786—22no—
28#19153778—22no—
29#19153776—22no—
30#19153774C2212yes—
31#19153749C2212yescomplemented
32#19153748C2211yescharacteristic, complemented
33#19153740C2212yescomplemented
34#19153738C2212yescomplemented
35#19153736C2212yescomplemented
36C2×Q16×C181C2211yescharacteristic, complemented
37C2×M16×C181C2211yescharacteristic
38C22⋊Q8×C181C2212yes—
39#106×C181C2211yescharacteristic, complemented
40#103×C181C2212yescomplemented
41C23×D181D411yescharacteristic
42#15668450—24no—
43#15668450—24no—
44C2×D724—22no—
45C2×D724—22no—
46C2×D724C2312yes—
47C2×C22⋊D181—22no—
48C22×Dic181D411yescharacteristic
49C2×C4×D181—22no—
50C2×C4×D181—22no—
51C2×C4×D181—22no—
52C2×C4×D181D411yescharacteristic
53C2×C4×D181D411yescharacteristic
54C2×C4×D181C2311yescharacteristic
55C2×C4×D181C2311yescharacteristic
56#15668436—24no—
57D1448—22no—
58#15668433—22no—
59#15668432—22no—
60#15668432—22no—
61C2×#13089678C2311yescharacteristic
62C2×#13089678C2311yescharacteristic
63C8×D181—22no—
64C8×D181D414yescomplemented
65C4⋊Dic181C2312yes—
66#15668428—22no—
67#15668426—22no—
68#15668426—22no—
69#15668426—22no—
70#15668426—22no—
71C4×Dic181—22no—
72C2×Q8×C181C2312yes—
73C22×C4×C181C2311yescharacteristic
74Q16×C181—22no—
75M16×C181—22no—
76C2×C8×C181C2311yescharacteristic
77C2×C8×C181C2311yescharacteristic
78C4⋊C4×C181—22no—
79C4⋊C4×C181C2312yes—
80C22⋊C4×C181—22no—
81C22×D181—22no—
82C22×D181—22no—
83C22×D181—42no—
84C22×D181C2×D411yescharacteristic
85C22×D181C2×D411yescharacteristic
86D724—22no—
87D724—22no—
88D724—44no—
89D724—44no—
90C22⋊D181—44no—
91C2×Dic181—22no—
92C2×Dic181—22no—
93C2×Dic181C2×D411yescharacteristic
94C2×Dic181C2×D411yescharacteristic
95C4×D181—22no—
96C4×D181—21no—
97C4×D181—21no—
98C4×D181—22no—
99C4×D181—44no—
100C4×D181C2×D412yes—
101C4×D181#4112yes—
102#13089678—21no—
103#13089678C2×D412yes—
104C23×C181C2×D411yescharacteristic
105Q8×C181—22no—
106Q8×C181—22no—
107C2×C4×C181—21no—
108C2×C4×C181—21no—
109C2×C4×C181—22no—
110C2×C4×C181—22no—
111C2×C4×C181C2×D411yescharacteristic
112C2×C4×C181C2411yescharacteristic
113C8×C181—21no—
114C8×C181C2×D412yes—
115C22×C181—21no—
116C22×C181—22no—
117C22×C181C22×D411yescharacteristic
118C2×D181—22no—
119C2×D181—21no—
120C2×D181—22no—
121C2×D181—44no—
122C2×D181—42no—
123C2×D181—42no—
124C2×D181—42no—
125C2×D181#12112yes—
126C4×C181—21no—
127C4×C181—42no—
128C4×C181—42no—
129C4×C181C22×D411yescharacteristic
130C4×C181C2×#4111yescharacteristic
131Dic181—21no—
132Dic181—42no—
133Dic181#12112yes—
134C2×C181—41no—
135C2×C181C2×#12111yescharacteristic
136C2×C181#57512yes—
137D181—22no—
138D181—41no—
139D181—82no—
140C181#305411yescharacteristic, complemented, Sylow
141#3054—1811nomaximal, Sylow
142C2×#135—1811no—
143C22×M16—1811no—
144#525—1812no—
145#470—1811no—
146#470D18111yescharacteristic, complemented
147#469—1811no—
148#469—1811no—
149#468—1812no—
150#468—1812no—
151#427—1811no—
152#416—1812no—
153C2×#41—1812no—
154C23×C4—1811no—
155#135—3624no—
156C2×Q16C2×D18111yescharacteristic, complemented
157C2×SD16—1812no—
158C2×D8—1811no—
159C2×M16—3624no—
160C2×M16—1811no—
161C2×M16—1811no—
162C2×M16—1811no—
163C2×M16C2×D18111yescharacteristic
164C22×C8—1811no—
165C22×C8—1811no—
166#123—3622no—
167C22⋊Q8—1812no—
168C22⋊Q8C2×D18112yes—
169#121—1812no—
170#121—1812no—
171#120—3622no—
172C4×D4—3622no—
173C4×D4—3622no—
174#117—1812no—
175#107—1812no—
176#106—1811no—
177#106C2×D18111yescharacteristic, complemented
178#103—1812no—
179#103C2×D18112yescomplemented
180#102—1812no—
181#102—1812no—
182C24—1811no—
183#41—3624no—
184#41—3624no—
185C2×Q8C22×D18112yes—
186C2×D4—3622no—
187C2×D4—3622no—
188C2×D4—3622no—
189C2×D4—1812no—
190C22×C4—3622no—
191C22×C4—3622no—
192C22×C4—3622no—
193C22×C4—1811no—
194C22×C4—1811no—
195C22×C4—1811no—
196C22×C4—1811no—
197C22×C4—1811no—
198C22×C4C22×D18111yescharacteristic
199Q16—22no—
200SD16—3624no—
201D8—3622no—
202M16—22no—
203M16—3622no—
204M16—3622no—
205M16—3622no—
206C2×C8—3622no—
207C2×C8—1811no—
208C2×C8—1811no—
209C2×C8—1814no—
210C2×C8C22×D18111yescharacteristic
211C2×C8C22×D18111yescharacteristic
212C4⋊C4—22no—
213C4⋊C4—3622no—
214C4⋊C4—1812no—
215C4⋊C4C22×D18112yes—
216C22⋊C4—22no—
217C22⋊C4—3622no—
218C22⋊C4—3622no—
219C22⋊C4—3622no—
220C22⋊C4—3622no—
221C42—3622no—
222C23—3622no—
223C23—3622no—
224C23—1811no—
225C23—1811no—
226C23—7242no—
227C23D4×D18111yescharacteristic
228Q8—22no—
229Q8—22no—
230D4—3622no—
231D4—3622no—
232D4—7244no—
233D4—7244no—
234D4—7244no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—3622no—
240C2×C4—3622no—
241C2×C4—3622no—
242C2×C4—3621no—
243C2×C4—3621no—
244C2×C4—3622no—
245C2×C4—1811no—
246C2×C4—1811no—
247C2×C4—1812no—
248C2×C4—1812no—
249C2×C4—7244no—
250C2×C4D4×D18111yescharacteristic
251C2×C4C23×D18111yescharacteristic
252C8—21no—
253C8—3621no—
254C8—1812no—
255C8D4×D18112yes—
256C22—21no—
257C22—22no—
258C22—3622no—
259C22—3621no—
260C22—3622no—
261C22—1812no—
262C22—7244no—
263C22—7242no—
264C22—7242no—
265C22—7242no—
266C22C2×D4×D18111yescharacteristic
267C4—21no—
268C4—42no—
269C4—42no—
270C4—3621no—
271C4—1812no—
272C4—7242no—
273C4C2×D4×D18111yescharacteristic
274C4#41×D18111yescharacteristic
275C2—41no—
276C2—3622no—
277C2—14482no—
278C2—7241no—
279C2#121×D18111yescharacteristic
280C2#2604207812yes—
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#470×C181C2
socleC22×C181C22×D4
hypercenter#470D181
hypocenterC181#3054
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C22×C4 ◁ #470

Complements

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

NH ≅ G/Ntype
1. C2×#19153839C2semidirect
2. C2×M16×D181C2semidirect
3. #26041792C2semidirect
4. #26041534C2semidirect
5. #26041533C2semidirect
6. #26041532C2semidirect
7. #26041528C2semidirect
8. #26041527C2semidirect
9. #26041314C2semidirect
10. #26041274C2semidirect
11. #470×C181C2semidirect
15. C2×#15668436C22semidirect
16. C2×D1448C22semidirect
31. #19153749C22semidirect
32. #19153748C22semidirect
33. #19153740C22semidirect
34. #19153738C22semidirect
35. #19153736C22semidirect
36. C2×Q16×C181C22semidirect
39. #106×C181C22semidirect
40. #103×C181C22semidirect
64. C8×D181D4semidirect
140. C181#3054semidirect
146. #470D181semidirect
156. C2×Q16C2×D181semidirect
177. #106C2×D181semidirect
179. #103C2×D181semidirect

Maximal subgroups

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

subgroupindex
C2×#191538392
C2×M16×D1812
#260417922
#260415342
#260415332
#260415322
#260415282
#260415272
#260413142
#260412742
#470×C1812
#3054181