Finite Group Databasebeta

← #37532248

Subgroups — #37532248

Order 23168.

326 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 326 conjugacy classes of subgroups — too many to draw. Only its 63 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#121×D181C211yesmaximal, characteristic
2#26041241C211yesmaximal, characteristic, complemented
3#26041141C211yesmaximal, characteristic, complemented
4#26041014C211yesmaximal, characteristic, complemented
5#26041013C211yesmaximal, characteristic, complemented
6#26041012C211yesmaximal, characteristic, complemented
7#327×C181C211yesmaximal, characteristic, complemented
8C2×D4×D181—21no—
9C2×D4×D181C411yescharacteristic, complemented
10C22×C4×D181C2211yescharacteristic
11C2×#15668432—21no—
12#19153791C411yescharacteristic, complemented
13#19153790C2211yescharacteristic
14#19153789—21no—
15#19153788—21no—
16#19153787C411yescharacteristic, complemented
17C4⋊C4×D181C411yescharacteristic, complemented
18C2×#15668426—21no—
19C22⋊C4×D181—21no—
20C22⋊#13089678C2211yescharacteristic, complemented
21#19153728—21no—
22#19153723—21no—
23#19153723C2211yescharacteristic, complemented
24#121×C181C2211yescharacteristic
25C22⋊C8×C181C2211yescharacteristic, complemented
26#95×C181C2211yescharacteristic, complemented
27C23×D181—21no—
28C23×D181D411yescharacteristic
29C2×D724—21no—
30C2×D724C2×C411yescharacteristic
31C2×C22⋊D181—21no—
32C2×C22⋊D181—21no—
33C2×C22⋊D181—21no—
34D4×D181—42no—
35D4×D181—42no—
36D4×D181—42no—
37C22×Dic181—21no—
38C22×Dic181D411yescharacteristic
39C2×C4×D181—21no—
40C2×C4×D181—42no—
41C2×C4×D181C2×C411yescharacteristic
42C2×C4×D181C2×C411yescharacteristic
43C2×C4×D181D411yescharacteristic
44C2×C4×D181D411yescharacteristic
45#15668432—42no—
46C2×#13089678—21no—
47C4⋊Dic181C2×C411yescharacteristic
48#15668428—21no—
49C22⋊Dic181—21no—
50#15668426—21no—
51#15668426—21no—
52#15668426—42no—
53C2×D4×C181—21no—
54C2×D4×C181C2×C411yescharacteristic
55C22×C4×C181—21no—
56C22×C4×C181C2311yescharacteristic
57C2×C8×C181—21no—
58C4⋊C4×C181C2×C411yescharacteristic
59C22⋊C4×C181—21no—
60C22×D181—21no—
61C22×D181—21no—
62C22×D181—22no—
63C22×D181—21no—
64C22×D181—41no—
65C22×D181—42no—
66C22×D181—42no—
67C22×D181—42no—
68C22×D181C22⋊C411yescharacteristic
69C22×D181C22⋊C411yescharacteristic
70D724—41no—
71D724—42no—
72C22⋊D181—42no—
73C22⋊D181—42no—
74C22⋊D181—41no—
75C22⋊D181—41no—
76C2×Dic181—21no—
77C2×Dic181—41no—
78C2×Dic181—41no—
79C2×Dic181—41no—
80C2×Dic181—41no—
81C2×Dic181C22⋊C411yescharacteristic
82C2×Dic181C22⋊C411yescharacteristic
83C4×D181—22no—
84C4×D181—22no—
85C4×D181—41no—
86C4×D181—41no—
87C4×D181—42no—
88#13089678—41no—
89C23×C181—21no—
90C23×C181C2×D411yescharacteristic
91D4×C181—41no—
92D4×C181—41no—
93D4×C181—41no—
94C2×C4×C181—21no—
95C2×C4×C181—41no—
96C2×C4×C181—41no—
97C2×C4×C181—41no—
98C2×C4×C181—41no—
99C2×C4×C181C22×C411yescharacteristic
100C2×C4×C181C2×D411yescharacteristic
101C8×C181—41no—
102C22×C181—21no—
103C22×C181—21no—
104C22×C181—41no—
105C22×C181—41no—
106C22×C181—41no—
107C22×C181C2×C22⋊C411yescharacteristic
108C22×C181AGL(1,ℤ/8)11yescharacteristic
109C22×C181AGL(1,ℤ/8)11yescharacteristic
110C2×D181—21no—
111C2×D181—21no—
112C2×D181—21no—
113C2×D181—21no—
114C2×D181—21no—
115C2×D181—42no—
116C2×D181—42no—
117C2×D181—41no—
118C2×D181—41no—
119C2×D181—41no—
120C2×D181—84no—
121C2×D181#9912yes—
122C4×C181—21no—
123C4×C181—21no—
124C4×C181—41no—
125C4×C181—81no—
126Dic181—21no—
127Dic181—21no—
128Dic181—41no—
129Dic181—81no—
130C2×C181—21no—
131C2×C181—21no—
132C2×C181—81no—
133C2×C181C2×#9911yescharacteristic
134C2×C181#41811yescharacteristic
135C2×C181#42111yescharacteristic
136D181—22no—
137D181—41no—
138D181—81no—
139C181#151011yescharacteristic, complemented, Sylow
140#1510—1811nomaximal, Sylow
141C2×#121—1811no—
142#407—1811no—
143#381—1811no—
144#327—1811no—
145#327—1811no—
146#327—1811no—
147#327D18111yescharacteristic, complemented
148C22×D4—3621no—
149C22×D4—1811no—
150C23×C4—1811no—
151C2×M16—3621no—
152#121—3621no—
153#121—3621no—
154#121—1811no—
155#121—1811no—
156#121—1811no—
157#121C2×D18111yescharacteristic
158C2×C4⋊C4—1811no—
159C2×C22⋊C4—3621no—
160C2×C22⋊C4—3621no—
161C22⋊C8—3621no—
162C22⋊C8—1811no—
163C22⋊C8C2×D18111yescharacteristic, complemented
164#95—3621no—
165#95—1811no—
166#95C2×D18111yescharacteristic, complemented
167C24—3621no—
168C24—1811no—
169C2×D4—21no—
170C2×D4—3621no—
171C2×D4—3621no—
172C2×D4—3621no—
173C2×D4—3621no—
174C2×D4—1811no—
175C2×D4—7242no—
176C2×D4—7242no—
177C2×D4—7242no—
178C2×D4C4×D18111yescharacteristic
179C22×C4—21no—
180C22×C4—3621no—
181C22×C4—3621no—
182C22×C4—1811no—
183C22×C4—1811no—
184C22×C4—1811no—
185C22×C4—1811no—
186C22×C4—1811no—
187C22×C4—7242no—
188C22×C4C22×D18111yescharacteristic
189M16—7242no—
190C2×C8—21no—
191C2×C8—3621no—
192C4⋊C4—3621no—
193C4⋊C4—1811no—
194C4⋊C4C4×D18111yescharacteristic
195C22⋊C4—21no—
196C22⋊C4—3621no—
197C22⋊C4—3621no—
198C22⋊C4—3621no—
199C22⋊C4—7242no—
200C23—21no—
201C23—3622no—
202C23—3621no—
203C23—3621no—
204C23—3621no—
205C23—1811no—
206C23—1811no—
207C23—7241no—
208C23—7242no—
209C23—7242no—
210C23—7242no—
211C23D4×D18111yescharacteristic
212D4—41no—
213D4—41no—
214D4—41no—
215D4—7241no—
216D4—7242no—
217D4—7242no—
218D4—7242no—
219D4—7241no—
220D4—7241no—
221C2×C4—21no—
222C2×C4—41no—
223C2×C4—41no—
224C2×C4—41no—
225C2×C4—41no—
226C2×C4—3622no—
227C2×C4—3622no—
228C2×C4—3621no—
229C2×C4—1811no—
230C2×C4—1811no—
231C2×C4—7241no—
232C2×C4—7241no—
233C2×C4—7242no—
234C2×C4—7241no—
235C2×C4—7241no—
236C2×C4—7241no—
237C2×C4—7241no—
238C2×C4C2×C4×D18111yescharacteristic
239C2×C4D4×D18111yescharacteristic
240C8—41no—
241C8—7241no—
242C22—21no—
243C22—21no—
244C22—41no—
245C22—41no—
246C22—41no—
247C22—3621no—
248C22—3621no—
249C22—3621no—
250C22—3621no—
251C22—3621no—
252C22—14484no—
253C22—1812no—
254C22—7242no—
255C22—7242no—
256C22—7241no—
257C22—7241no—
258C22—7241no—
259C22C22⋊C4×D18111yescharacteristic
260C22#1915384011yescharacteristic
261C22#1915384311yescharacteristic
262C4—21no—
263C4—21no—
264C4—41no—
265C4—81no—
266C4—3621no—
267C4—3621no—
268C4—14481no—
269C4—7241no—
270C2—21no—
271C2—21no—
272C2—81no—
273C2—3622no—
274C2—14481no—
275C2—7241no—
276C2#99×D18111yescharacteristic
277C2#2604128211yescharacteristic
278C2#2604129411yescharacteristic
subgroupG/H
centerC22C22⋊C4×D181
commutatorC2×C4×C181C22×C4
FrattiniC22×C4C22×D181
Fitting#327×C181C2
socleC22×C181C2×C22⋊C4
prime radical#121×D181C2
cosocleC22×C4×C181C23
hypercenter#327D181
hypocenterC181#1510
LCSG ▷ C2×C4×C181 ▷ C22×C181 ▷ C181
UCSC1 ◁ C22 ◁ C22×C4 ◁ #327

Complements

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

NH ≅ G/Ntype
2. #26041241C2semidirect
3. #26041141C2semidirect
4. #26041014C2semidirect
5. #26041013C2semidirect
6. #26041012C2semidirect
7. #327×C181C2semidirect
9. C2×D4×D181C4semidirect
12. #19153791C4semidirect
16. #19153787C4semidirect
17. C4⋊C4×D181C4semidirect
20. C22⋊#13089678C22semidirect
23. #19153723C22semidirect
25. C22⋊C8×C181C22semidirect
26. #95×C181C22semidirect
139. C181#1510semidirect
147. #327D181semidirect
163. C22⋊C8C2×D181semidirect
166. #95C2×D181semidirect

Maximal subgroups

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

subgroupindex
#121×D1812
#260412412
#260411412
#260410142
#260410132
#260410122
#327×C1812
#1510181