Finite Group Databasebeta

← #37532739

Subgroups — #37532739

Order 23168.

288 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 288 conjugacy classes of subgroups — too many to draw. Only its 51 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#26041837C211yesmaximal, characteristic
2#26041597C211yesmaximal, characteristic, complemented
3#26041214C211yesmaximal, characteristic, complemented
4#26041014C211yesmaximal, characteristic, complemented
5#26041013C211yesmaximal, characteristic, complemented
6#26041002C211yesmaximal, characteristic, complemented
7#481×C181C211yesmaximal, characteristic, complemented
8C2×D4×D181—21no—
9#19153801—21no—
10#19153798—21no—
11#19153790C2211yescharacteristic
12#19153788—21no—
13#19153786—21no—
14#19153784—21no—
15C2×#15668428—21no—
16C2×#15668426—21no—
17C2×#15668426C2211yescharacteristic
18#19153747—21no—
19#19153736—21no—
20C22⋊#13089678C2211yescharacteristic, complemented
21#19153723C2211yescharacteristic, complemented
22#121×C181C2211yescharacteristic
23C2×C4⋊C4×C181C2211yescharacteristic, complemented
24#106×C181—21no—
25#102×C181—21no—
26C22⋊C8×C181C2211yescharacteristic, complemented
27C23×D181—21no—
28C23×D181D411yescharacteristic
29C2×D724—21no—
30C2×D724D411yescharacteristic
31C2×C22⋊D181—21no—
32C2×C22⋊D181—21no—
33C2×C22⋊D181—21no—
34D4×D181—42no—
35D4×D181—42no—
36C22×Dic181—21no—
37C22×Dic181D411yescharacteristic
38C2×C4×D181—21no—
39C2×#13089678—21no—
40C4⋊Dic181D411yescharacteristic
41#15668428—21no—
42#15668428—42no—
43C22⋊Dic181—21no—
44#15668426—21no—
45#15668426—21no—
46#15668426—21no—
47#15668426—21no—
48#15668426—42no—
49C2×D4×C181—21no—
50C2×D4×C181D411yescharacteristic
51C22×C4×C181—21no—
52C22×C4×C181C2311yescharacteristic
53C2×C8×C181—21no—
54C4⋊C4×C181—21no—
55C4⋊C4×C181—21no—
56C4⋊C4×C181D411yescharacteristic
57C22⋊C4×C181—21no—
58C22×D181—21no—
59C22×D181—21no—
60C22×D181—21no—
61C22×D181—42no—
62C22×D181—41no—
63C22×D181—41no—
64C22×D181—81no—
65C22×D181—81no—
66C22×D181SD1612yes—
67D724—41no—
68D724—42no—
69C22⋊D181—42no—
70C22⋊D181—41no—
71C22⋊D181—41no—
72C22⋊D181—81no—
73C2×Dic181—22no—
74C2×Dic181—21no—
75C2×Dic181—21no—
76C2×Dic181—41no—
77C2×Dic181—41no—
78C2×Dic181—41no—
79C4×D181—42no—
80#13089678—41no—
81C23×C181—21no—
82C23×C181C2×D411yescharacteristic
83D4×C181—41no—
84D4×C181—41no—
85D4×C181—41no—
86C2×C4×C181—21no—
87C2×C4×C181—41no—
88C2×C4×C181—41no—
89C2×C4×C181—41no—
90C2×C4×C181—41no—
91C2×C4×C181C2×D411yescharacteristic
92C2×C4×C181C2×D411yescharacteristic
93C8×C181—41no—
94C22×C181—21no—
95C22×C181—21no—
96C22×C181—41no—
97C22×C181—41no—
98C22×C181—41no—
99C22×C181#12011yescharacteristic
100C22×C181C2×SD1611yescharacteristic
101C22×C181AGL(1,ℤ/8)11yescharacteristic
102C2×D181—22no—
103C2×D181—41no—
104C2×D181—41no—
105C2×D181—42no—
106C2×D181—41no—
107C2×D181—41no—
108C2×D181—41no—
109C2×D181—81no—
110C2×D181—84no—
111C4×C181—21no—
112C4×C181—21no—
113C4×C181—41no—
114C4×C181—81no—
115Dic181—42no—
116Dic181—41no—
117Dic181—41no—
118C2×C181—21no—
119C2×C181—21no—
120C2×C181—81no—
121C2×C181#45011yescharacteristic
122C2×C181#45311yescharacteristic
123C2×C181#45811yescharacteristic
124D181—42no—
125D181—81no—
126C181#161311yescharacteristic, complemented, Sylow
127#1613—1811nomaximal, Sylow
128#534—1811no—
129#481—1811no—
130#481D18111yescharacteristic, complemented
131#399—1811no—
132#327—1811no—
133#327—1811no—
134#323—1811no—
135C22×D4—3621no—
136#123—3621no—
137#123—3621no—
138#121—3621no—
139#121—1811no—
140#121C2×D18111yescharacteristic
141#120—3621no—
142#120—3621no—
143C2×C4⋊C4—3621no—
144C2×C4⋊C4C2×D18111yescharacteristic, complemented
145C2×C22⋊C4—3621no—
146C2×C22⋊C4—1811no—
147#106—21no—
148#106—3621no—
149#102—21no—
150#102—3621no—
151C22⋊C8—1811no—
152C22⋊C8C2×D18111yescharacteristic, complemented
153#95—1811no—
154C24—3621no—
155C24—1811no—
156C2×D4—21no—
157C2×D4—3621no—
158C2×D4—3621no—
159C2×D4—3621no—
160C2×D4—3621no—
161C2×D4—1811no—
162C2×D4—7242no—
163C2×D4—7242no—
164C2×D4C22⋊D18111yescharacteristic
165C22×C4—21no—
166C22×C4—3621no—
167C22×C4—3621no—
168C22×C4—1811no—
169C22×C4C22×D18111yescharacteristic
170C2×C8—21no—
171C2×C8—3621no—
172C4⋊C4—21no—
173C4⋊C4—21no—
174C4⋊C4—3621no—
175C4⋊C4—1811no—
176C4⋊C4—7242no—
177C4⋊C4C22⋊D18111yescharacteristic
178C22⋊C4—21no—
179C22⋊C4—3621no—
180C22⋊C4—3621no—
181C22⋊C4—3621no—
182C22⋊C4—3621no—
183C22⋊C4—3621no—
184C22⋊C4—7242no—
185C23—21no—
186C23—3621no—
187C23—3621no—
188C23—3621no—
189C23—14481no—
190C23—14481no—
191C23—1812no—
192C23—7241no—
193C23—7241no—
194C23—7242no—
195C23D4×D18111yescharacteristic
196D4—41no—
197D4—41no—
198D4—41no—
199D4—14481no—
200D4—7241no—
201D4—7242no—
202D4—7242no—
203D4—7241no—
204D4—7241no—
205C2×C4—21no—
206C2×C4—41no—
207C2×C4—41no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—3622no—
211C2×C4—3621no—
212C2×C4—3621no—
213C2×C4—7242no—
214C2×C4—7241no—
215C2×C4—7241no—
216C2×C4—7241no—
217C2×C4D4×D18111yescharacteristic
218C2×C4C2×C22⋊D18111yescharacteristic
219C8—41no—
220C8—7241no—
221C22—21no—
222C22—21no—
223C22—41no—
224C22—41no—
225C22—41no—
226C22—3622no—
227C22—14481no—
228C22—14484no—
229C22—7242no—
230C22—7241no—
231C22—7241no—
232C22—7241no—
233C22—7241no—
234C22—7241no—
235C22#1915378411yescharacteristic
236C22SD16×D18111yescharacteristic
237C22#1915384311yescharacteristic
238C4—21no—
239C4—21no—
240C4—41no—
241C4—81no—
242C4—7242no—
243C4—7241no—
244C4—7241no—
245C2—21no—
246C2—21no—
247C2—81no—
248C2—14481no—
249C2—7242no—
250C2#2604142811yescharacteristic
251C2#2604144511yescharacteristic
252C2#2604146811yescharacteristic
subgroupG/H
centerC22#19153784
commutator, cosocleC22×C4×C181C23
FrattiniC22×C4C22×D181
Fitting#481×C181C2
socleC22×C181#120
prime radical#26041837C2
hypercenter#481D181
hypocenterC181#1613
LCSG ▷ C22×C4×C181 ▷ C22×C181 ▷ C181
UCSC1 ◁ C22 ◁ C22×C4 ◁ #481

Complements

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

NH ≅ G/Ntype
2. #26041597C2semidirect
3. #26041214C2semidirect
4. #26041014C2semidirect
5. #26041013C2semidirect
6. #26041002C2semidirect
7. #481×C181C2semidirect
20. C22⋊#13089678C22semidirect
21. #19153723C22semidirect
23. C2×C4⋊C4×C181C22semidirect
26. C22⋊C8×C181C22semidirect
126. C181#1613semidirect
130. #481D181semidirect
144. C2×C4⋊C4C2×D181semidirect
152. C22⋊C8C2×D181semidirect

Maximal subgroups

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

subgroupindex
#260418372
#260415972
#260412142
#260410142
#260410132
#260410022
#481×C1812
#1613181