Finite Group Databasebeta

← #37549159

Subgroups — #37549159

Order 23168.

288 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 288 conjugacy classes of subgroups — too many to draw. Only its 68 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
1C2×#19153873C211yesmaximal, characteristic
2#26042221C211yesmaximal, characteristic
3#26042133C212yesmaximal, complemented
4#26042132C212yesmaximal, complemented
5C22⋊Q8×D181C211yesmaximal, characteristic
6C22×#13089685—21no—
7C2×C4⋊#10624382—21no—
8C2×C22⋊#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—
24C2×#13089685—22no—
25C2×#13089685—42no—
26C4⋊#10624382—42no—
27C22⋊#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—42no—
54C2×#10624382—42no—
55C2×#10624382—42no—
56#13089685—42no—
57C22×D181—41no—
58C22×D181—41no—
59C22×D181C22⋊C411yescharacteristic
60C22×D181SD1612yes—
61C22×D181Q1612yes—
62C22×D181C2×D411yescharacteristic
63Dic362—42no—
64Dic362—41no—
65C2×Dic181—22no—
66C2×Dic181—21no—
67C2×Dic181—21no—
68C2×Dic181C22⋊C411yescharacteristic
69C2×Dic181C2×D411yescharacteristic
70C4×D181—22no—
71C4×D181—22no—
72C4×D181—41no—
73C4×D181—41no—
74C4×D181—42no—
75C4×D181—42no—
76C23×C181C22⋊C411yescharacteristic
77Q8×C181—41no—
78C2×C4×C181—21no—
79C2×C4×C181—21no—
80C2×C4×C181—41no—
81C2×C4×C181C22⋊C411yescharacteristic
82C2×C4×C181C22×C411yescharacteristic
83C22×C181—21no—
84C22×C181—21no—
85C22×C181#10312yes—
86C22×C181C2×C22⋊C411yescharacteristic
87C2×D181—21no—
88C2×D181—21no—
89C2×D181—21no—
90C2×D181—21no—
91C2×D181—42no—
92C2×D181—42no—
93C2×D181#10312yes—
94C2×D181C2×SD1611yescharacteristic
95C2×D181C2×Q1611yescharacteristic
96#10624382—82no—
97C4×C181—21no—
98C4×C181—21no—
99C4×C181—41no—
100C4×C181—41no—
101Dic181—21no—
102Dic181—41no—
103Dic181—41no—
104Dic181#9912yes—
105C2×C181—22no—
106C2×C181C2×#9911yescharacteristic
107C2×C181C2×#10311yescharacteristic
108C2×C181#42111yescharacteristic
109D181—21no—
110D181—22no—
111D181—21no—
112C181#152011yescharacteristic, complemented, Sylow
113#1520—1811nomaximal, Sylow
114C2×C22⋊Q8—1811no—
115C2×C22⋊C8—1811no—
116#380—1811no—
117#328—1812no—
118#328—1812no—
119C22×Q8—1811no—
120C23×C4—1811no—
121C22×C8—3621no—
122C22⋊Q8—3621no—
123C22⋊Q8—3621no—
124C22⋊Q8—1811no—
125C22⋊Q8—1811no—
126C22⋊Q8—1811no—
127C22⋊Q8#1062438211yescharacteristic, complemented
128C2×C4⋊C4—3621no—
129C2×C4⋊C4—3621no—
130C2×C4⋊C4—1811no—
131C2×C22⋊C4—3621no—
132C2×C22⋊C4—3621no—
133C22⋊C8—3621no—
134C22⋊C8—1812no—
135#95—1812no—
136C24—1811no—
137C2×Q8—3621no—
138C2×Q8—1811no—
139C2×Q8—7242no—
140C2×Q8C2×#1062438211yescharacteristic
141C22×C4—3621no—
142C22×C4—3621no—
143C22×C4—3622no—
144C22×C4—1811no—
145C22×C4—1811no—
146C22×C4—1811no—
147C22×C4—1811no—
148C22×C4—1811no—
149C22×C4—7242no—
150C22×C4C2×#1062438211yescharacteristic
151C2×C8—3622no—
152C2×C8—7242no—
153C4⋊C4—21no—
154C4⋊C4—3621no—
155C4⋊C4—3621no—
156C4⋊C4—3621no—
157C4⋊C4—3621no—
158C4⋊C4—1811no—
159C4⋊C4—7242no—
160C4⋊C4C2×#1062438211yescharacteristic
161C22⋊C4—21no—
162C22⋊C4—3621no—
163C22⋊C4—3621no—
164C22⋊C4—3621no—
165C22⋊C4—7242no—
166C23—1811no—
167C23—1811no—
168C23—1812no—
169C23—1812no—
170C23—7241no—
171C23—7241no—
172C23C22⋊#1062438211yescharacteristic
173Q8—41no—
174Q8—7241no—
175Q8—7242no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—41no—
179C2×C4—3622no—
180C2×C4—3622no—
181C2×C4—3622no—
182C2×C4—3621no—
183C2×C4—3621no—
184C2×C4—1811no—
185C2×C4—1811no—
186C2×C4—7241no—
187C2×C4—7241no—
188C2×C4—7242no—
189C2×C4—7242no—
190C2×C4—7242no—
191C2×C4—7242no—
192C2×C4—7242no—
193C2×C4C22⋊#1062438211yescharacteristic
194C2×C4C22×#1062438211yescharacteristic
195C8—7242no—
196C22—21no—
197C22—21no—
198C22—3621no—
199C22—3621no—
200C22—3621no—
201C22—3621no—
202C22—1811no—
203C22—1812no—
204C22—1811no—
205C22—7242no—
206C22—7242no—
207C22#1915387912yes—
208C22C2×C22⋊#1062438211yescharacteristic
209C4—21no—
210C4—21no—
211C4—41no—
212C4—41no—
213C4—3621no—
214C4—14482no—
215C4—1812no—
216C4—7241no—
217C4—7241no—
218C2—22no—
219C2—3621no—
220C2—3622no—
221C2—3621no—
222C2D181⋊#9911yescharacteristic
223C2C2×#1915387911yescharacteristic
224C2#2604227611yescharacteristic
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#1520
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
112. C181#1520semidirect
127. C22⋊Q8#10624382semidirect

Maximal subgroups

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

subgroupindex
C2×#191538732
#260422212
#260421332
#260421322
C22⋊Q8×D1812
#1520181