Finite Group Databasebeta

← #37549137

Subgroups — #37549137

Order 23168.

286 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 286 conjugacy classes of subgroups — too many to draw. Only its 58 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#26042251C211yesmaximal, characteristic, complemented
2#26042221C211yesmaximal, characteristic, complemented
3#26042132C212yesmaximal, complemented
4#26042130C212yesmaximal, complemented
5#26041789C211yesmaximal, characteristic
6C2×#15668458—21no—
7C2×C4⋊#10624382—21no—
8C2×C22⋊#10624382—21no—
9#19153873—21no—
10#19153873C2212yes—
11(C2×#10624382)⋊C4C2212yes—
12C2×#15668448C411yescharacteristic, complemented
13C22×C4×D181C2211yescharacteristic
14#19153798—21no—
15#19153796C2211yescharacteristic
16#19153795C411yescharacteristic, complemented
17#19153787C2211yescharacteristic
18#19153784—21no—
19D4×Dic181—21no—
20#19153776—21no—
21#19153774C411yescharacteristic, complemented
22#121×C181C411yescharacteristic, complemented
23C22×#10624382—22no—
24#15668458—42no—
25C2×#13089685—22no—
26C4⋊#10624382—42no—
27C22⋊#10624382—42no—
28C23×D181D411yescharacteristic
29#15668448—42no—
30C2×Dic362C2×C411yescharacteristic
31C2×C22⋊D181—21no—
32C2×C22⋊D181—21no—
33C22×Dic181—21no—
34C22×Dic181D411yescharacteristic
35C2×C4×D181—42no—
36C2×C4×D181C2×C411yescharacteristic
37C2×C4×D181D411yescharacteristic
38C2×C4×D181D411yescharacteristic
39C2×C4×D181C2311yescharacteristic
40C4⋊Dic181—21no—
41C4⋊Dic181C2×C411yescharacteristic
42#15668428—21no—
43C22⋊Dic181—21no—
44C22⋊Dic181—21no—
45#15668426—21no—
46#15668426—21no—
47C4×Dic181—21no—
48C2×D4×C181—21no—
49C2×D4×C181C2×C411yescharacteristic
50C22×C4×C181C2×C411yescharacteristic
51C4⋊C4×C181C2×C411yescharacteristic
52C22⋊C4×C181—21no—
53C2×#10624382—42no—
54C2×#10624382—42no—
55C2×#10624382—42no—
56#13089685—42no—
57C22×D181—21no—
58C22×D181—21no—
59C22×D181—41no—
60C22×D181—41no—
61C22×D181C22⋊C411yescharacteristic
62C22×D181C2×D411yescharacteristic
63Dic362—41no—
64C22⋊D181—42no—
65C22⋊D181—41no—
66C22⋊D181—41no—
67C2×Dic181—22no—
68C2×Dic181—21no—
69C2×Dic181—21no—
70C2×Dic181—42no—
71C2×Dic181C22⋊C411yescharacteristic
72C2×Dic181C2×D411yescharacteristic
73C4×D181—22no—
74C4×D181—22no—
75C4×D181—41no—
76C4×D181—41no—
77C23×C181—21no—
78C23×C181C22⋊C411yescharacteristic
79D4×C181—41no—
80D4×C181—42no—
81C2×C4×C181—21no—
82C2×C4×C181—41no—
83C2×C4×C181C22⋊C411yescharacteristic
84C2×C4×C181C22×C411yescharacteristic
85C22×C181—21no—
86C22×C181—21no—
87C22×C181—21no—
88C22×C181—41no—
89C22×C181—41no—
90C22×C181—41no—
91C22×C181C2×C22⋊C411yescharacteristic
92C2×D181—21no—
93C2×D181—21no—
94C2×D181—21no—
95C2×D181—21no—
96C2×D181—21no—
97C2×D181—42no—
98C2×D181—42no—
99C2×D181AGL(1,ℤ/8)11yescharacteristic
100C2×D181#13711yescharacteristic
101#10624382—82no—
102C4×C181—21no—
103C4×C181—21no—
104C4×C181—41no—
105Dic181—21no—
106Dic181—41no—
107Dic181—41no—
108Dic181#9912yes—
109C2×C181—41no—
110C2×C181—81no—
111C2×C181C2×#9911yescharacteristic
112C2×C181#41711yescharacteristic
113C2×C181#42111yescharacteristic
114D181—21no—
115D181—21no—
116D181—41no—
117C181#151311yescharacteristic, complemented, Sylow
118#1513—1811nomaximal, Sylow
119#525—1811no—
120#407—1811no—
121#380—1811no—
122#328—1812no—
123#327—1812no—
124C2×#41—1811no—
125C23×C4—1811no—
126C2×M16—3621no—
127#123—3621no—
128C22⋊Q8—1811no—
129C22⋊Q8—1811no—
130#121—1811no—
131#121#1062438211yescharacteristic, complemented
132#120—3621no—
133C4×D4—3621no—
134C4×D4—3621no—
135#117—1811no—
136C2×C4⋊C4—3621no—
137C2×C22⋊C4—3621no—
138C22⋊C8—3621no—
139C22⋊C8—1812no—
140#95—1812no—
141C24—1811no—
142#41—7242no—
143C2×Q8—1811no—
144C2×D4—21no—
145C2×D4—3621no—
146C2×D4—3621no—
147C2×D4C2×#1062438211yescharacteristic
148C22×C4—3621no—
149C22×C4—3622no—
150C22×C4—1811no—
151C22×C4—1811no—
152C22×C4—1811no—
153C22×C4—1811no—
154C22×C4—1811no—
155C22×C4—7242no—
156C22×C4C2×#1062438211yescharacteristic
157M16—7242no—
158C2×C8—3622no—
159C4⋊C4—3621no—
160C4⋊C4—3621no—
161C4⋊C4—1811no—
162C4⋊C4—7242no—
163C4⋊C4C2×#1062438211yescharacteristic
164C22⋊C4—21no—
165C22⋊C4—3621no—
166C22⋊C4—3621no—
167C22⋊C4—3621no—
168C22⋊C4—3621no—
169C22⋊C4—7242no—
170C42—3621no—
171C23—21no—
172C23—3621no—
173C23—3621no—
174C23—1811no—
175C23—1811no—
176C23—7241no—
177C23—7241no—
178C23C22⋊#1062438211yescharacteristic
179Q8—7241no—
180D4—41no—
181D4—42no—
182D4—7242no—
183D4—7241no—
184D4—7241no—
185C2×C4—21no—
186C2×C4—41no—
187C2×C4—3622no—
188C2×C4—3622no—
189C2×C4—3622no—
190C2×C4—3621no—
191C2×C4—3621no—
192C2×C4—1811no—
193C2×C4—1811no—
194C2×C4—7241no—
195C2×C4—7241no—
196C2×C4—7242no—
197C2×C4—7242no—
198C2×C4—7242no—
199C2×C4—7242no—
200C2×C4C22⋊#1062438211yescharacteristic
201C2×C4C22×#1062438211yescharacteristic
202C8—7242no—
203C22—21no—
204C22—21no—
205C22—21no—
206C22—41no—
207C22—41no—
208C22—41no—
209C22—3621no—
210C22—3621no—
211C22—3621no—
212C22—3621no—
213C22—3621no—
214C22—1811no—
215C22—1811no—
216C22—7242no—
217C22—7242no—
218C22C2×C22⋊#1062438211yescharacteristic
219C4—21no—
220C4—21no—
221C4—41no—
222C4—3621no—
223C4—14482no—
224C4—1812no—
225C4—7241no—
226C4—7241no—
227C2—41no—
228C2—81no—
229C2—3621no—
230C2—3621no—
231C2—7241no—
232C2D181⋊#9911yescharacteristic
233C2#2604227011yescharacteristic
234C2#2604227711yescharacteristic
subgroupG/H
centerC22C2×C22⋊#10624382
commutatorC2×C4×C181C22×C4
FrattiniC2×C4C22×#10624382
Fitting#121×C181C4
socleC22×C181C2×C22⋊C4
prime radical#26041789C2
cosocleC2×C4×D181C23
hypercenter#121#10624382
hypocenterC181#1513
LCSG ▷ C2×C4×C181 ▷ C22×C181 ▷ C181
UCSC1 ◁ C22 ◁ C22×C4 ◁ #121

Complements

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

NH ≅ G/Ntype
1. #26042251C2semidirect
2. #26042221C2semidirect
3. #26042132C2semidirect
4. #26042130C2semidirect
12. C2×#15668448C4semidirect
16. #19153795C4semidirect
21. #19153774C4semidirect
22. #121×C181C4semidirect
117. C181#1513semidirect
131. #121#10624382semidirect

Maximal subgroups

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

subgroupindex
#260422512
#260422212
#260421322
#260421302
#260417892
#1513181