Finite Group Databasebeta

← #37532691

Subgroups — #37532691

Order 23168.

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

normal subgroups only

This group has 282 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#26041856C211yesmaximal, characteristic
2#26041420C211yesmaximal, characteristic, complemented
3#26041417C211yesmaximal, characteristic, complemented
4#26041200C211yesmaximal, characteristic, complemented
5#26041012C211yesmaximal, characteristic, complemented
6#26041004C211yesmaximal, characteristic, complemented
7#327×C181C211yesmaximal, characteristic, complemented
8C2×#15668448—21no—
9C22×Dic362C2211yescharacteristic
10C2×#15668438—21no—
11C2×#15668437—21no—
12#19153794—21no—
13#19153793—21no—
14#19153790C2211yescharacteristic
15#19153789—21no—
16#19153788—21no—
17#19153788—21no—
18C2×C4⋊Dic181—21no—
19C2×#15668428C2211yescharacteristic
20#19153741—21no—
21#19153739—21no—
22C22⋊#13089678C2211yescharacteristic
23#19153723—21no—
24#121×C181C2211yescharacteristic
25C22⋊C8×C181C2211yescharacteristic
26#95×C181C2211yescharacteristic
27#15668448—42no—
28#15668448—42no—
29C2×Dic362—21no—
30C2×Dic362—21no—
31C2×Dic362—21no—
32C2×Dic362—42no—
33C2×D724D411yescharacteristic
34C2×C22⋊D181—21no—
35C2×C22⋊D181—21no—
36C2×C22⋊D181—21no—
37C22×Dic181—21no—
38C22×Dic181—21no—
39C22×Dic181D411yescharacteristic
40C22×Dic181D411yescharacteristic
41C2×C4×D181—21no—
42#15668438—42no—
43#15668437—42no—
44C2×#13089678—21no—
45C4⋊Dic181—42no—
46C4⋊Dic181D411yescharacteristic
47#15668428—21no—
48#15668428—21no—
49#15668428—21no—
50C22⋊Dic181—21no—
51#15668426—21no—
52#15668426—21no—
53C2×D4×C181—21no—
54C2×D4×C181D411yescharacteristic
55C22×C4×C181—21no—
56C22×C4×C181C2311yescharacteristic
57C2×C8×C181—21no—
58C4⋊C4×C181D411yescharacteristic
59C22⋊C4×C181—21no—
60C22×D181—21no—
61Dic362—22no—
62Dic362—22no—
63Dic362—42no—
64Dic362—41no—
65Dic362—41no—
66D724—41no—
67C22⋊D181—42no—
68C22⋊D181—41no—
69C22⋊D181—41no—
70C22⋊D181—81no—
71C2×Dic181—21no—
72C2×Dic181—21no—
73C2×Dic181—22no—
74C2×Dic181—21no—
75C2×Dic181—21no—
76C2×Dic181—42no—
77C2×Dic181—41no—
78C2×Dic181—41no—
79C2×Dic181—41no—
80C2×Dic181SD1612yescomplemented
81C4×D181—42no—
82#13089678—41no—
83C23×C181—21no—
84C23×C181C2×D411yescharacteristic
85D4×C181—41no—
86D4×C181—41no—
87D4×C181—41no—
88C2×C4×C181—22no—
89C2×C4×C181—21no—
90C2×C4×C181—41no—
91C2×C4×C181—41no—
92C2×C4×C181—41no—
93C2×C4×C181C2×D411yescharacteristic
94C2×C4×C181C2×D411yescharacteristic
95C8×C181—41no—
96C22×C181—21no—
97C22×C181—21no—
98C22×C181—41no—
99C22×C181—41no—
100C22×C181—41no—
101C22×C181#12011yescharacteristic
102C22×C181C2×SD1611yescharacteristic
103C22×C181AGL(1,ℤ/8)11yescharacteristic
104C2×D181—41no—
105C2×D181—41no—
106C2×D181—41no—
107C4×C181—21no—
108C4×C181—21no—
109C4×C181—42no—
110C4×C181—41no—
111Dic181—22no—
112Dic181—41no—
113Dic181—42no—
114Dic181—41no—
115C2×C181—21no—
116C2×C181—21no—
117C2×C181—81no—
118C2×C181#45011yescharacteristic
119C2×C181#45611yescharacteristic
120C2×C181B(4,𝔽2)11yescharacteristic
121D181—81no—
122C181#160411yescharacteristic, complemented, Sylow
123#1604—1811nomaximal, Sylow
124#537—1811no—
125#448—1811no—
126#448—1811no—
127#395—1811no—
128#327—1811no—
129#327D18111yescharacteristic, complemented
130#324—1811no—
131C2×#41—3621no—
132C22×Q8—1811no—
133C2×SD16—3621no—
134C2×SD16—3621no—
135C22⋊Q8—3621no—
136C22⋊Q8—3621no—
137#121—3621no—
138#121—3621no—
139#121—3621no—
140#121—1811no—
141#121C2×D18111yescharacteristic
142C2×C4⋊C4—3621no—
143C2×C4⋊C4—1811no—
144#103—3621no—
145#103—3621no—
146C22⋊C8—1811no—
147C22⋊C8C2×D18111yescharacteristic
148#95—3621no—
149#95C2×D18111yescharacteristic
150#41—7242no—
151#41—7242no—
152C2×Q8—3621no—
153C2×Q8—3621no—
154C2×Q8—3621no—
155C2×Q8—7242no—
156C2×D4—21no—
157C2×D4—3621no—
158C2×D4—3621no—
159C2×D4—3621no—
160C2×D4—1811no—
161C2×D4D72411yescharacteristic
162C22×C4—21no—
163C22×C4—3621no—
164C22×C4—3621no—
165C22×C4—3621no—
166C22×C4—1811no—
167C22×C4—1811no—
168C22×C4C22×D18111yescharacteristic
169SD16—7242no—
170SD16—7242no—
171C2×C8—21no—
172C2×C8—3621no—
173C4⋊C4—3621no—
174C4⋊C4—3621no—
175C4⋊C4—3621no—
176C4⋊C4—1811no—
177C4⋊C4—7242no—
178C4⋊C4D72411yescharacteristic
179C22⋊C4—21no—
180C22⋊C4—3621no—
181C22⋊C4—3621no—
182C22⋊C4—3621no—
183C23—21no—
184C23—3621no—
185C23D4×D18111yescharacteristic
186Q8—3622no—
187Q8—3622no—
188Q8—7241no—
189Q8—7241no—
190Q8—7242no—
191D4—41no—
192D4—41no—
193D4—41no—
194D4—14481no—
195D4—7241no—
196D4—7242no—
197D4—7241no—
198D4—7241no—
199C2×C4—22no—
200C2×C4—21no—
201C2×C4—41no—
202C2×C4—41no—
203C2×C4—41no—
204C2×C4—3621no—
205C2×C4—3621no—
206C2×C4—3622no—
207C2×C4—3621no—
208C2×C4—3621no—
209C2×C4—1812no—
210C2×C4—7242no—
211C2×C4—7242no—
212C2×C4—7241no—
213C2×C4—7241no—
214C2×C4—7241no—
215C2×C4D4×D18111yescharacteristic
216C2×C4C2×D72411yescharacteristic
217C8—41no—
218C8—7241no—
219C22—21no—
220C22—21no—
221C22—41no—
222C22—41no—
223C22—41no—
224C22—7241no—
225C22—7241no—
226C22—7241no—
227C22#1915378611yescharacteristic
228C22SD16×D18111yescharacteristic
229C22#1915384011yescharacteristic
230C4—21no—
231C4—21no—
232C4—42no—
233C4—41no—
234C4—3622no—
235C4—7241no—
236C4—7242no—
237C4—7241no—
238C2—21no—
239C2—21no—
240C2—81no—
241C2—14481no—
242C2#2604143111yescharacteristic
243C2#2604146311yescharacteristic
244C2#2604146711yescharacteristic
subgroupG/H
centerC22#19153786
commutator, cosocleC22×C4×C181C23
FrattiniC22×C4C22×D181
Fitting#327×C181C2
socleC22×C181#120
prime radical#26041856C2
hypercenter#327D181
hypocenterC181#1604
LCSG ▷ C22×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. #26041420C2semidirect
3. #26041417C2semidirect
4. #26041200C2semidirect
5. #26041012C2semidirect
6. #26041004C2semidirect
7. #327×C181C2semidirect
80. C2×Dic181SD16semidirect
122. C181#1604semidirect
129. #327D181semidirect

Maximal subgroups

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

subgroupindex
#260418562
#260414202
#260414172
#260412002
#260410122
#260410042
#327×C1812
#1604181