Finite Group Databasebeta

← #37549131

Subgroups — #37549131

Order 23168.

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

normal subgroups only

This group has 274 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
2#26042222C211yesmaximal, characteristic
3#26042133C214yesmaximal, complemented
4C2×#19153792C211yesmaximal, characteristic
5C2×#15668458—21no—
6C2×C22⋊#10624382—21no—
7#19153873—21no—
8#19153873C2212yes—
9(C2×#10624382)⋊C4—21no—
10(C2×#10624382)⋊C4C2212yes—
11C22×C4×D181C2211yescharacteristic
12#19153792—21no—
13#19153792—21no—
14#19153792C412yescomplemented
15#19153792C2212yes—
16C2×C4⋊Dic181C411yescharacteristic, complemented
17C2×#15668428—21no—
18C2×#15668426—21no—
19C22×Q8×C181C411yescharacteristic, complemented
20C22×#10624382—22no—
21#15668458—42no—
22C2×#13089685—22no—
23C22⋊#10624382—42no—
24C23×D181D411yescharacteristic
25C22×Dic181—21no—
26C22×Dic181D411yescharacteristic
27C2×C4×D181—42no—
28C2×C4×D181C2×C411yescharacteristic
29C2×C4×D181D411yescharacteristic
30C2×C4×D181D411yescharacteristic
31C2×C4×D181C2311yescharacteristic
32C4⋊Dic181—21no—
33C4⋊Dic181C2×C412yes—
34#15668428—22no—
35#15668428—22no—
36#15668426—22no—
37#15668426—22no—
38C2×Q8×C181—21no—
39C2×Q8×C181—42no—
40C2×Q8×C181C2×C412yes—
41C22×C4×C181—21no—
42C22×C4×C181C2×C411yescharacteristic
43C2×#10624382—42no—
44C2×#10624382—42no—
45C2×#10624382—42no—
46#13089685—42no—
47C22×D181—22no—
48C22×D181—22no—
49C22×D181—41no—
50C22×D181C22⋊C411yescharacteristic
51C22×D181C2×D411yescharacteristic
52C2×Dic181—22no—
53C2×Dic181—22no—
54C2×Dic181—42no—
55C2×Dic181C22⋊C411yescharacteristic
56C2×Dic181C2×D411yescharacteristic
57C4×D181—41no—
58C4×D181—41no—
59C4×D181—41no—
60C4×D181—41no—
61C23×C181C22⋊C411yescharacteristic
62Q8×C181—42no—
63Q8×C181—42no—
64C2×C4×C181—22no—
65C2×C4×C181—22no—
66C2×C4×C181—42no—
67C2×C4×C181C22⋊C411yescharacteristic
68C2×C4×C181C22×C411yescharacteristic
69C22×C181—21no—
70C22×C181—21no—
71C22×C181#9912yes—
72C22×C181C2×C22⋊C411yescharacteristic
73C2×D181—21no—
74C2×D181—21no—
75C2×D181—21no—
76C2×D181—21no—
77C2×D181—21no—
78C2×D181—44no—
79C2×D181#13712yes—
80#10624382—82no—
81C4×C181—21no—
82C4×C181—21no—
83C4×C181—42no—
84Dic181—21no—
85Dic181—21no—
86Dic181—42no—
87C2×C181—22no—
88C2×C181C2×#9911yescharacteristic
89C2×C181#41911yescharacteristic
90C2×C181#42111yescharacteristic
91D181—22no—
92D181—41no—
93C181#151111yescharacteristic, complemented, Sylow
94#1511—1811nomaximal, Sylow
95C2×C22⋊Q8—1811no—
96#407—1811no—
97#381—1811no—
98#328—1814no—
99C22×Q8#1062438211yescharacteristic, complemented
100C23×C4—1811no—
101C2×M16—3621no—
102C22⋊Q8—3621no—
103C22⋊Q8—3621no—
104C22⋊Q8—1812no—
105C22⋊Q8—1812no—
106C2×C4⋊C4—3621no—
107C2×C4⋊C4—1811no—
108C2×C22⋊C4—3621no—
109C2×C22⋊C4—3621no—
110C22⋊C8—3621no—
111C22⋊C8—1812no—
112#95—3621no—
113#95—1812no—
114C24—1811no—
115C2×Q8—21no—
116C2×Q8—42no—
117C2×Q8C2×#1062438212yes—
118C22×C4—21no—
119C22×C4—3621no—
120C22×C4—3622no—
121C22×C4—1811no—
122C22×C4—1811no—
123C22×C4—1811no—
124C22×C4—1811no—
125C22×C4—1811no—
126C22×C4—7242no—
127C22×C4C2×#1062438211yescharacteristic
128M16—7242no—
129C2×C8—3622no—
130C4⋊C4—3622no—
131C4⋊C4—3622no—
132C4⋊C4—3621no—
133C4⋊C4—1812no—
134C22⋊C4—3622no—
135C22⋊C4—3622no—
136C22⋊C4—7242no—
137C23—3622no—
138C23—3622no—
139C23—1811no—
140C23—1811no—
141C23—7241no—
142C23C22⋊#1062438211yescharacteristic
143Q8—42no—
144Q8—42no—
145C2×C4—22no—
146C2×C4—22no—
147C2×C4—42no—
148C2×C4—3622no—
149C2×C4—3622no—
150C2×C4—1811no—
151C2×C4—1811no—
152C2×C4—7241no—
153C2×C4—7241no—
154C2×C4—7241no—
155C2×C4—7241no—
156C2×C4—7242no—
157C2×C4—7242no—
158C2×C4—7242no—
159C2×C4—7242no—
160C2×C4C22⋊#1062438211yescharacteristic
161C2×C4C22×#1062438211yescharacteristic
162C8—7242no—
163C22—21no—
164C22—21no—
165C22—3621no—
166C22—3621no—
167C22—3621no—
168C22—3621no—
169C22—3621no—
170C22—1812no—
171C22—7244no—
172C22#1915387412yes—
173C22C2×C22⋊#1062438211yescharacteristic
174C4—21no—
175C4—21no—
176C4—42no—
177C4—3621no—
178C4—3621no—
179C4—14482no—
180C4—7242no—
181C2—22no—
182C2—3622no—
183C2—7241no—
184C2C2×#1915387411yescharacteristic
185C2#2604227211yescharacteristic
186C2#2604227811yescharacteristic
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#1511
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
14. #19153792C4semidirect
16. C2×C4⋊Dic181C4semidirect
19. C22×Q8×C181C4semidirect
93. C181#1511semidirect
99. C22×Q8#10624382semidirect

Maximal subgroups

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

subgroupindex
#260422512
#260422222
#260421332
C2×#191537922
#1511181