Finite Group Databasebeta

← #37549982

Subgroups — #37549982

Order 23168.

266 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 266 conjugacy classes of subgroups — too many to draw. Only its 54 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#26042265C211yesmaximal, characteristic, complemented
2#26042258C211yesmaximal, characteristic, complemented
3#26042180C214yesmaximal, complemented
4#26041848C211yesmaximal, characteristic
5D181⋊M16—21no—
6#19153895—21no—
7#19153877—21no—
8#19153876C2212yes—
9#19153875—21no—
10#19153874C2212yes—
11C2×#15668449C2211yescharacteristic
12C22×D724C411yescharacteristic, complemented
13#19153813—21no—
14#19153804—21no—
15#19153791—21no—
16#19153791—21no—
17#19153786C412yescomplemented
18#19153786C2212yes—
19#117×C181C411yescharacteristic, complemented
20#15668459—41no—
21#15668458—22no—
22D181⋊C8—41no—
23C4⋊#10624382—41no—
24C22⋊#10624382—22no—
25C4×#10624382—41no—
26C23×D181C2×C412yes—
27#15668449—41no—
28#15668449—41no—
29C2×Dic362D411yescharacteristic
30C2×D724—22no—
31C2×D724—22no—
32C2×D724—21no—
33C2×D724—21no—
34C2×D724—41no—
35C2×D724—41no—
36C2×D724D411yescharacteristic
37C2×C22⋊D181C2×C411yescharacteristic
38C2×C22⋊D181C2311yescharacteristic
39C2×C4×D181D411yescharacteristic
40C2×C4×D181D411yescharacteristic
41#15668426—22no—
42#15668426—22no—
43C22×C4×C181C2×C411yescharacteristic
44C4⋊C4×C181—21no—
45C22⋊C4×C181C2×C412yes—
46C42×C181—21no—
47C2×#10624382—42no—
48#13089685—42no—
49C22×D181—22no—
50C22×D181—42no—
51C22×D181—82no—
52C22×D181C22⋊C411yescharacteristic
53C22×D181C2×D411yescharacteristic
54Dic362—41no—
55D724—41no—
56D724—42no—
57D724—41no—
58D724—41no—
59D724—82no—
60D724—82no—
61C22⋊D181—41no—
62C22⋊D181—41no—
63C2×Dic181C22⋊C411yescharacteristic
64C2×Dic181C2×D411yescharacteristic
65C4×D181—21no—
66C4×D181—21no—
67C4×D181—21no—
68C4×D181—21no—
69C23×C181C22×C411yescharacteristic
70C2×C4×C181—21no—
71C2×C4×C181—21no—
72C2×C4×C181—22no—
73C2×C4×C181C22⋊C411yescharacteristic
74C2×C4×C181C22⋊C411yescharacteristic
75C22×C181—21no—
76C22×C181—41no—
77C22×C181C2×C22⋊C411yescharacteristic
78C2×D181—21no—
79C2×D181—21no—
80C2×D181—41no—
81C2×D181—41no—
82C2×D181—42no—
83C2×D181—42no—
84C2×D181—42no—
85C2×D181—84no—
86#10624382—82no—
87C4×C181—21no—
88C4×C181—42no—
89C4×C181#9911yescharacteristic
90C4×C181#9911yescharacteristic
91Dic181—21no—
92Dic181—21no—
93C2×C181—21no—
94C2×C181—41no—
95C2×C181C2×#9911yescharacteristic
96D181—41no—
97D181—41no—
98D181—82no—
99C181#212211yescharacteristic, complemented, Sylow
100#2122—1811nomaximal, Sylow
101#535—1811no—
102#413—1811no—
103#410—1811no—
104AΣL(1,𝔽16)—1814no—
105C2×#41—1811no—
106C22×D4—1811no—
107C2×M16—3621no—
108#127—3621no—
109#124—3621no—
110#121—3621no—
111#121—3621no—
112#120—1812no—
113#120—1812no—
114#117—3621no—
115#117#1062438211yescharacteristic, complemented
116#101—3621no—
117#100—1812no—
118#99—3621no—
119#99—1812no—
120C24—1812no—
121#41—7241no—
122#41—7241no—
123C2×Q8—1811no—
124C2×D4—3622no—
125C2×D4—3622no—
126C2×D4—3621no—
127C2×D4—3621no—
128C2×D4—1811no—
129C2×D4—1811no—
130C2×D4—1811no—
131C2×D4—7241no—
132C2×D4—7241no—
133C22×C4—1811no—
134C22×C4—1811no—
135C22×C4C2×#1062438211yescharacteristic
136M16—3622no—
137M16—7241no—
138C2×C8—7241no—
139C4⋊C4—21no—
140C4⋊C4—7241no—
141C22⋊C4—3622no—
142C22⋊C4—3622no—
143C22⋊C4—3622no—
144C22⋊C4C2×#1062438212yes—
145C42—21no—
146C42—7241no—
147C23—3622no—
148C23—14482no—
149C23—1811no—
150C23—1811no—
151C23—7242no—
152C23C22×#1062438211yescharacteristic
153Q8—7241no—
154D4—14482no—
155D4—14482no—
156D4—7241no—
157D4—7242no—
158D4—7241no—
159D4—7241no—
160D4—7241no—
161D4—7241no—
162C2×C4—21no—
163C2×C4—21no—
164C2×C4—22no—
165C2×C4—3621no—
166C2×C4—3621no—
167C2×C4—3621no—
168C2×C4—3621no—
169C2×C4—1811no—
170C2×C4—1811no—
171C2×C4—7242no—
172C2×C4C22⋊#1062438211yescharacteristic
173C2×C4C22⋊#1062438211yescharacteristic
174C8—7242no—
175C22—21no—
176C22—41no—
177C22—3621no—
178C22—3621no—
179C22—14484no—
180C22—7242no—
181C22—7242no—
182C22—7241no—
183C22—7241no—
184C22—7242no—
185C22C2×C22⋊#1062438211yescharacteristic
186C4—21no—
187C4—42no—
188C4—3621no—
189C4—3621no—
190C4—14482no—
191C4#1915387511yescharacteristic
192C4#1915387511yescharacteristic
193C2—21no—
194C2—41no—
195C2—14482no—
196C2—7241no—
197C2—7241no—
198C2C2×#1915387511yescharacteristic
subgroupG/H
centerC2C2×#19153875
commutatorC23×C181C22×C4
FrattiniC23C22×#10624382
Fitting#117×C181C4
socleC2×C181C2×#99
prime radical#26041848C2
cosocleC2×C22⋊D181C23
hypercenter#117#10624382
hypocenterC181#2122
LCSG ▷ C23×C181 ▷ C22×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C2 ◁ C2×C4 ◁ C22×C4 ◁ #117

Complements

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

NH ≅ G/Ntype
1. #26042265C2semidirect
2. #26042258C2semidirect
3. #26042180C2semidirect
12. C22×D724C4semidirect
17. #19153786C4semidirect
19. #117×C181C4semidirect
99. C181#2122semidirect
115. #117#10624382semidirect

Maximal subgroups

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

subgroupindex
#260422652
#260422582
#260421802
#260418482
#2122181