Finite Group Databasebeta

← #37549119

Subgroups — #37549119

Order 23168.

294 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 294 conjugacy classes of subgroups — too many to draw. Only its 60 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
2C4×C22⋊#10624382C211yesmaximal, characteristic
3#26042133C212yesmaximal, complemented
4#26042132C212yesmaximal, complemented
5C22⋊Q8×D181C211yesmaximal, characteristic
6C2×#15668458—21no—
7C2×C22⋊#10624382—21no—
8C2×C4×#10624382—21no—
9#19153873—21no—
10#19153873C2212yes—
11(C2×#10624382)⋊C4C2212yes—
12C2×Q8×D181C411yescharacteristic, complemented
13C22×C4×D181C2211yescharacteristic
14#19153796C2211yescharacteristic
15#19153795C411yescharacteristic, complemented
16#19153794—21no—
17#19153793—21no—
18#19153792C2211yescharacteristic
19C4⋊C4×D181—21no—
20C4⋊C4×D181C411yescharacteristic, complemented
21C22⋊C4×D181—21no—
22C22⋊Q8×C181C411yescharacteristic, complemented
23C22×#10624382—22no—
24#15668458—42no—
25C2×#13089685—22no—
26C22⋊#10624382—24no—
27C4×#10624382—42no—
28C23×D181D411yescharacteristic
29C2×Dic362—21no—
30C2×Dic362C2×C411yescharacteristic
31Q8×D181—42no—
32C22×Dic181D411yescharacteristic
33C2×C4×D181—21no—
34C2×C4×D181—21no—
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—
43#15668428—21no—
44#15668428—21no—
45C22⋊Dic181—21no—
46#15668426—21no—
47#15668426—21no—
48C2×Q8×C181C2×C411yescharacteristic
49C22×C4×C181C2×C411yescharacteristic
50C4⋊C4×C181—21no—
51C4⋊C4×C181C2×C411yescharacteristic
52C22⋊C4×C181—21no—
53C2×#10624382—24no—
54C2×#10624382—42no—
55C2×#10624382—42no—
56#13089685—42no—
57C22×D181—22no—
58C22×D181—22no—
59C22×D181—41no—
60C22×D181C22⋊C411yescharacteristic
61C22×D181C2×D411yescharacteristic
62Dic362—42no—
63Dic362—41no—
64C2×Dic181—22no—
65C2×Dic181—21no—
66C2×Dic181—21no—
67C2×Dic181C22⋊C411yescharacteristic
68C2×Dic181C2×D411yescharacteristic
69C4×D181—22no—
70C4×D181—22no—
71C4×D181—41no—
72C4×D181—41no—
73C4×D181—42no—
74C4×D181—42no—
75C23×C181C22⋊C411yescharacteristic
76Q8×C181—41no—
77C2×C4×C181—21no—
78C2×C4×C181—21no—
79C2×C4×C181—41no—
80C2×C4×C181C22⋊C411yescharacteristic
81C2×C4×C181C22×C411yescharacteristic
82C22×C181—21no—
83C22×C181—21no—
84C22×C181—21no—
85C22×C181C2×C22⋊C411yescharacteristic
86C2×D181—21no—
87C2×D181—21no—
88C2×D181—21no—
89C2×D181—21no—
90C2×D181—44no—
91C2×D181#10412yes—
92C2×D181#13712yes—
93#10624382—44no—
94C4×C181—21no—
95C4×C181—21no—
96C4×C181—41no—
97C4×C181—41no—
98Dic181—21no—
99Dic181—41no—
100Dic181—41no—
101Dic181#9912yes—
102C2×C181—41no—
103C2×C181C2×#9911yescharacteristic
104C2×C181#41911yescharacteristic
105C2×C181C2×#10411yescharacteristic
106D181—22no—
107D181—22no—
108C181#150811yescharacteristic, complemented, Sylow
109#1508—1811nomaximal, Sylow
110C2×C22⋊Q8—1811no—
111#407—1811no—
112C4×C22⋊C4—1811no—
113#328—1812no—
114#328—1812no—
115C22×Q8—1811no—
116C23×C4—1811no—
117C2×M16—3621no—
118C22⋊Q8—3621no—
119C22⋊Q8—3621no—
120C22⋊Q8—1811no—
121C22⋊Q8—1811no—
122C22⋊Q8—1811no—
123C22⋊Q8#1062438211yescharacteristic, complemented
124C2×C4⋊C4—3621no—
125C2×C4⋊C4—1811no—
126C2×C22⋊C4—3621no—
127C2×C22⋊C4—3621no—
128C2×C42—3621no—
129C22⋊C8—3621no—
130C22⋊C8—1812no—
131#95—1812no—
132C24—1811no—
133C2×Q8—3621no—
134C2×Q8—1811no—
135C2×Q8—7242no—
136C2×Q8C2×#1062438211yescharacteristic
137C22×C4—3621no—
138C22×C4—3621no—
139C22×C4—3622no—
140C22×C4—1811no—
141C22×C4—1811no—
142C22×C4—1811no—
143C22×C4—1811no—
144C22×C4—1811no—
145C22×C4—7242no—
146C22×C4C2×#1062438211yescharacteristic
147M16—7242no—
148C2×C8—3622no—
149C4⋊C4—21no—
150C4⋊C4—3621no—
151C4⋊C4—3621no—
152C4⋊C4—3621no—
153C4⋊C4—3621no—
154C4⋊C4—1811no—
155C4⋊C4C2×#1062438211yescharacteristic
156C22⋊C4—21no—
157C22⋊C4—3621no—
158C22⋊C4—3621no—
159C22⋊C4—3624no—
160C22⋊C4—3621no—
161C42—7242no—
162C23—3622no—
163C23—3622no—
164C23—1811no—
165C23—1811no—
166C23—7241no—
167C23C22⋊#1062438211yescharacteristic
168Q8—41no—
169Q8—7241no—
170Q8—7242no—
171C2×C4—21no—
172C2×C4—21no—
173C2×C4—41no—
174C2×C4—3622no—
175C2×C4—3622no—
176C2×C4—3622no—
177C2×C4—3624no—
178C2×C4—3621no—
179C2×C4—3621no—
180C2×C4—1811no—
181C2×C4—1811no—
182C2×C4—7241no—
183C2×C4—7241no—
184C2×C4—7242no—
185C2×C4—7242no—
186C2×C4—7242no—
187C2×C4—7242no—
188C2×C4C22⋊#1062438211yescharacteristic
189C2×C4C22×#1062438211yescharacteristic
190C8—7242no—
191C22—21no—
192C22—21no—
193C22—21no—
194C22—3621no—
195C22—3621no—
196C22—3621no—
197C22—3621no—
198C22—1812no—
199C22—1812no—
200C22—7244no—
201C22C2×C22⋊#1062438211yescharacteristic
202C4—21no—
203C4—21no—
204C4—41no—
205C4—41no—
206C4—3621no—
207C4—1812no—
208C4—7244no—
209C4—7241no—
210C4—7241no—
211C2—41no—
212C2—3622no—
213C2—3622no—
214C2D181⋊#9911yescharacteristic
215C2#2604227211yescharacteristic
216C2D181⋊#10411yescharacteristic
subgroupG/H
centerC22C2×C22⋊#10624382
commutatorC2×C4×C181C22×C4
FrattiniC2×C4C22×#10624382
FittingC22⋊Q8×C181C4
socleC22×C181C2×C22⋊C4
prime radicalC23×D181D4
cosocleC2×C4×D181C23
hypercenterC22⋊Q8#10624382
hypocenterC181#1508
LCSG ▷ C2×C4×C181 ▷ C22×C181 ▷ C181
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
3. #26042133C2semidirect
4. #26042132C2semidirect
12. C2×Q8×D181C4semidirect
15. #19153795C4semidirect
20. C4⋊C4×D181C4semidirect
22. C22⋊Q8×C181C4semidirect
108. C181#1508semidirect
123. C22⋊Q8#10624382semidirect

Maximal subgroups

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

subgroupindex
#260422512
C4×C22⋊#106243822
#260421332
#260421322
C22⋊Q8×D1812
#1508181