Finite Group Databasebeta

← #37543643

Subgroups — #37543643

Order 23168.

458 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 123 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
1#26041789C211yesmaximal, characteristic
2C2×C4⋊C4×D181C211yesmaximal, characteristic, complemented
3#26041615C211yesmaximal, characteristic, complemented
4#26041614C211yesmaximal, characteristic, complemented
5#26041611C212yesmaximal, complemented
6#26041595C211yesmaximal, characteristic, complemented
7#26041591C212yesmaximal, complemented
8#26041318C212yesmaximal, complemented
9#26041277C212yesmaximal, complemented
10#26041241C211yesmaximal, characteristic, complemented
11#481×C181C211yesmaximal, characteristic, complemented
12C2×#15668448C2211yescharacteristic
13C22×C4×D181—21no—
14C22×C4×D181C2211yescharacteristic
15C2×#15668432C2212yes—
16#19153798—21no—
17#19153796C2211yescharacteristic
18#19153795C2211yescharacteristic
19#19153787C2211yescharacteristic
20#19153784—21no—
21D4×Dic181—21no—
22#19153776—21no—
23#19153774C2211yescharacteristic
24C2×C4⋊Dic181C2211yescharacteristic
25C2×#15668428—21no—
26C4⋊C4×D181—21no—
27C4⋊C4×D181—21no—
28C4⋊C4×D181C2212yes—
29C4⋊C4×D181C2212yes—
30#19153750C2212yescomplemented
31#19153749C2212yescomplemented
32#19153747C2212yescomplemented
33#19153741C2212yescomplemented
34#19153739C2212yescomplemented
35#19153737C2212yescomplemented
36C22⋊#13089678C2211yescharacteristic
37#19153728C2212yes—
38#121×C181C2211yescharacteristic
39C2×C4⋊C4×C181C2211yescharacteristic
40#106×C181C2212yescomplemented
41#102×C181C2212yescomplemented
42C22⋊C8×C181C2211yescharacteristic
43C23×D181D411yescharacteristic
44#15668448—24no—
45C2×Dic362C2311yescharacteristic
46C2×C22⋊D181—21no—
47C2×C22⋊D181—21no—
48C22×Dic181—21no—
49C22×Dic181—21no—
50C22×Dic181D411yescharacteristic
51C2×C4×D181—24no—
52C2×C4×D181—22no—
53C2×C4×D181—22no—
54C2×C4×D181—44no—
55C2×C4×D181D411yescharacteristic
56C2×C4×D181D411yescharacteristic
57C2×C4×D181C2311yescharacteristic
58C2×C4×D181C2311yescharacteristic
59#15668432—24no—
60C2×#13089678C2312yes—
61C4⋊Dic181—21no—
62C4⋊Dic181—21no—
63C4⋊Dic181C2312yes—
64C4⋊Dic181C2311yescharacteristic
65#15668428—22no—
66#15668428—22no—
67#15668428—21no—
68C22⋊Dic181—21no—
69C22⋊Dic181—21no—
70#15668426—21no—
71#15668426—21no—
72C4×Dic181—21no—
73C2×D4×C181—21no—
74C2×D4×C181C2311yescharacteristic
75C22×C4×C181—21no—
76C22×C4×C181C2311yescharacteristic
77C2×C8×C181C2312yes—
78C4⋊C4×C181—21no—
79C4⋊C4×C181C2312yes—
80C4⋊C4×C181C2311yescharacteristic
81C22⋊C4×C181—21no—
82C22×D181—22no—
83C22×D181—22no—
84C22×D181—22no—
85C22×D181C2×D411yescharacteristic
86C22×D181C2×D411yescharacteristic
87Dic362—22no—
88C22⋊D181—42no—
89C22⋊D181—42no—
90C2×Dic181—22no—
91C2×Dic181—21no—
92C2×Dic181—21no—
93C2×Dic181—22no—
94C2×Dic181—42no—
95C2×Dic181—42no—
96C2×Dic181C2×D411yescharacteristic
97C2×Dic181C2×D411yescharacteristic
98C4×D181—22no—
99C4×D181—22no—
100C4×D181—22no—
101C4×D181—44no—
102C4×D181—44no—
103C4×D181#4114yes—
104#13089678—22no—
105C23×C181—21no—
106C23×C181C2×D411yescharacteristic
107D4×C181—22no—
108D4×C181—42no—
109C2×C4×C181—22no—
110C2×C4×C181—21no—
111C2×C4×C181—22no—
112C2×C4×C181—42no—
113C2×C4×C181C2×D411yescharacteristic
114C2×C4×C181C2411yescharacteristic
115C8×C181—22no—
116C22×C181—21no—
117C22×C181—21no—
118C22×C181—41no—
119C22×C181—41no—
120C22×C181—41no—
121C22×C181#13712yes—
122C22×C181C22×D411yescharacteristic
123C2×D181—28no—
124C2×D181—21no—
125C2×D181—21no—
126C2×D181—21no—
127C2×D181—21no—
128C2×D181#12312yes—
129C2×D181AGL(1,ℤ/8)12yes—
130C4×C181—21no—
131C4×C181—42no—
132C4×C181—41no—
133C4×C181C2×#4112yes—
134Dic181—21no—
135Dic181—41no—
136Dic181—42no—
137Dic181—41no—
138Dic181#12312yes—
139C2×C181—22no—
140C2×C181—81no—
141C2×C181C2×#12311yescharacteristic
142C2×C181C2×AGL(1,ℤ/8)11yescharacteristic
143C2×C181C2×#13711yescharacteristic
144D181—22no—
145D181—22no—
146C181#309211yescharacteristic, complemented, Sylow
147#3092—1811nomaximal, Sylow
148#525—1811no—
149C22×C4⋊C4—1811no—
150#484—1811no—
151#484—1811no—
152#483—1812no—
153#481—1811no—
154#481D18111yescharacteristic, complemented
155#480—1812no—
156#428—1812no—
157#417—1812no—
158#407—1811no—
159C2×#41—1811no—
160C23×C4—3621no—
161C23×C4—1811no—
162C2×M16—1812no—
163#123—3621no—
164C22⋊Q8—1811no—
165C22⋊Q8—1811no—
166#121—1811no—
167#121C2×D18111yescharacteristic
168#120—3621no—
169C4×D4—3621no—
170C4×D4—3621no—
171#117—1811no—
172C2×C4⋊C4—3621no—
173C2×C4⋊C4—3621no—
174C2×C4⋊C4—3621no—
175C2×C4⋊C4—1811no—
176C2×C4⋊C4—1812no—
177C2×C4⋊C4—1812no—
178C2×C4⋊C4C2×D18111yescharacteristic
179#107—1812no—
180#107—1812no—
181#106—1812no—
182#106C2×D18112yescomplemented
183#103—1812no—
184#103—1812no—
185#102—1812no—
186#102C2×D18112yescomplemented
187C22⋊C8—1811no—
188C22⋊C8—1812no—
189C22⋊C8C2×D18111yescharacteristic
190C24—1811no—
191#41—3624no—
192C2×Q8—1811no—
193C2×D4—21no—
194C2×D4—3621no—
195C2×D4—3621no—
196C2×D4C22×D18111yescharacteristic
197C22×C4—21no—
198C22×C4—3624no—
199C22×C4—3622no—
200C22×C4—3622no—
201C22×C4—3621no—
202C22×C4—3621no—
203C22×C4—1811no—
204C22×C4—1811no—
205C22×C4—1811no—
206C22×C4—1811no—
207C22×C4—1811no—
208C22×C4—7244no—
209C22×C4C22×D18111yescharacteristic
210M16—3624no—
211C2×C8—1812no—
212C2×C8C22×D18112yes—
213C4⋊C4—21no—
214C4⋊C4—3622no—
215C4⋊C4—3622no—
216C4⋊C4—3621no—
217C4⋊C4—3621no—
218C4⋊C4—3621no—
219C4⋊C4—1812no—
220C4⋊C4—1811no—
221C4⋊C4C22×D18112yes—
222C4⋊C4C22×D18111yescharacteristic
223C22⋊C4—21no—
224C22⋊C4—3621no—
225C22⋊C4—3621no—
226C22⋊C4—3621no—
227C22⋊C4—3621no—
228C42—3621no—
229C23—21no—
230C23—3622no—
231C23—3622no—
232C23—3622no—
233C23—1811no—
234C23—1811no—
235C23D4×D18111yescharacteristic
236Q8—3622no—
237D4—22no—
238D4—42no—
239D4—7242no—
240D4—7242no—
241C2×C4—22no—
242C2×C4—21no—
243C2×C4—22no—
244C2×C4—42no—
245C2×C4—3622no—
246C2×C4—3622no—
247C2×C4—3622no—
248C2×C4—3622no—
249C2×C4—3621no—
250C2×C4—3621no—
251C2×C4—3622no—
252C2×C4—1811no—
253C2×C4—1811no—
254C2×C4—1814no—
255C2×C4—7244no—
256C2×C4—7244no—
257C2×C4—7242no—
258C2×C4—7242no—
259C2×C4D4×D18111yescharacteristic
260C2×C4C23×D18111yescharacteristic
261C8—22no—
262C8—3622no—
263C22—21no—
264C22—21no—
265C22—41no—
266C22—41no—
267C22—41no—
268C22—3628no—
269C22—3621no—
270C22—3621no—
271C22—3621no—
272C22—3621no—
273C22—1812no—
274C22—1812no—
275C22#1915384512yes—
276C22C2×D4×D18111yescharacteristic
277C4—21no—
278C4—42no—
279C4—41no—
280C4—3621no—
281C4—1812no—
282C4—7241no—
283C4—7242no—
284C4—7241no—
285C4#41×D18112yes—
286C2—22no—
287C2—81no—
288C2—3622no—
289C2—3622no—
290C2#123×D18111yescharacteristic
291C2AGL(1,ℤ/8)×D18111yescharacteristic
292C2C2×#1915384511yescharacteristic
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#481×C181C2
socleC22×C181C22×D4
prime radical#26041789C2
hypercenter#481D181
hypocenterC181#3092
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C22×C4 ◁ #481

Complements

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

NH ≅ G/Ntype
2. C2×C4⋊C4×D181C2semidirect
3. #26041615C2semidirect
4. #26041614C2semidirect
5. #26041611C2semidirect
6. #26041595C2semidirect
7. #26041591C2semidirect
8. #26041318C2semidirect
9. #26041277C2semidirect
10. #26041241C2semidirect
11. #481×C181C2semidirect
30. #19153750C22semidirect
31. #19153749C22semidirect
32. #19153747C22semidirect
33. #19153741C22semidirect
34. #19153739C22semidirect
35. #19153737C22semidirect
40. #106×C181C22semidirect
41. #102×C181C22semidirect
146. C181#3092semidirect
154. #481D181semidirect
182. #106C2×D181semidirect
186. #102C2×D181semidirect

Maximal subgroups

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

subgroupindex
#260417892
C2×C4⋊C4×D1812
#260416152
#260416142
#260416112
#260415952
#260415912
#260413182
#260412772
#260412412
#481×C1812
#3092181