Finite Group Databasebeta

← #37549984

Subgroups — #37549984

Order 23168.

234 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 52 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#26042266C211yesmaximal, characteristic
2#26042259C211yesmaximal, characteristic
3#26042182C212yesmaximal, complemented
4#26042181C212yesmaximal, complemented
5#26041851C211yesmaximal, characteristic
6C2×#15668458—21no—
7#19153895—21no—
8#19153877—21no—
9#19153876C2212yes—
10#19153874—21no—
11#19153874C2212yes—
12C2×#15668448C2211yescharacteristic
13C22×Dic362C411yescharacteristic, complemented
14#19153815—21no—
15#19153802—21no—
16#19153800C2211yescharacteristic
17#19153799C2211yescharacteristic
18#19153797C411yescharacteristic, complemented
19#19153794—21no—
20#19153793—21no—
21#19153772C411yescharacteristic, complemented
22#123×C181C411yescharacteristic, complemented
23#15668458—22no—
24#15668458—41no—
25C2×#13089685—41no—
26C4⋊#10624382—41no—
27C22⋊#10624382—22no—
28C4×#10624382—41no—
29#15668448—41no—
30#15668448—41no—
31C2×Dic362—21no—
32C2×Dic362—21no—
33C2×Dic362—41no—
34C2×Dic362—41no—
35C2×Dic362D411yescharacteristic
36C2×C22⋊D181D411yescharacteristic
37C2×C22⋊D181C2311yescharacteristic
38C22×Dic181C2×C411yescharacteristic
39C22×Dic181C2×C411yescharacteristic
40C22×Dic181D411yescharacteristic
41C2×C4×D181D411yescharacteristic
42C4⋊Dic181—21no—
43#15668428—21no—
44#15668428—21no—
45#15668428—21no—
46C22⋊Dic181—21no—
47C22⋊Dic181C2×C411yescharacteristic
48#15668426—21no—
49#15668426—21no—
50C4×Dic181—21no—
51C2×D4×C181C2×C411yescharacteristic
52C22×C4×C181C2×C411yescharacteristic
53C4⋊C4×C181—21no—
54C22⋊C4×C181—21no—
55C22⋊C4×C181C2×C411yescharacteristic
56C2×#10624382—42no—
57#13089685—42no—
58C22×D181C2×D411yescharacteristic
59Dic362—41no—
60Dic362—42no—
61Dic362—41no—
62Dic362—41no—
63C22⋊D181—41no—
64C22⋊D181—41no—
65C2×Dic181—21no—
66C2×Dic181—21no—
67C2×Dic181—21no—
68C2×Dic181—21no—
69C2×Dic181—21no—
70C2×Dic181—21no—
71C2×Dic181—41no—
72C2×Dic181C22⋊C411yescharacteristic
73C2×Dic181C22⋊C411yescharacteristic
74C2×Dic181C2×D411yescharacteristic
75C4×D181—21no—
76C4×D181—21no—
77C23×C181C22⋊C411yescharacteristic
78C23×C181C22×C411yescharacteristic
79D4×C181—41no—
80C2×C4×C181—21no—
81C2×C4×C181—21no—
82C2×C4×C181—41no—
83C2×C4×C181C22⋊C411yescharacteristic
84C22×C181—21no—
85C22×C181—21no—
86C22×C181—41no—
87C22×C181—41no—
88C22×C181C2×C22⋊C411yescharacteristic
89C2×D181—21no—
90C2×D181—41no—
91#10624382—82no—
92C4×C181—21no—
93C4×C181—41no—
94C4×C181—41no—
95Dic181—21no—
96Dic181—21no—
97Dic181—41no—
98Dic181—41no—
99Dic181#9911yescharacteristic
100Dic181#9911yescharacteristic
101C2×C181—21no—
102C2×C181—41no—
103C2×C181—41no—
104C2×C181C2×#9911yescharacteristic
105D181—41no—
106C181#212311yescharacteristic, complemented, Sylow
107#2123—1811nomaximal, Sylow
108#536—1811no—
109#413—1811no—
110#410—1811no—
111#352—1812no—
112#352—1812no—
113C2×#41—1811no—
114C22×Q8—1811no—
115C2×M16—3621no—
116C4⋊Q8—3621no—
117#124—3621no—
118#123—1811no—
119#123—1811no—
120#123—1811no—
121#123#1062438211yescharacteristic, complemented
122C22⋊Q8—3621no—
123C22⋊Q8—3621no—
124#117—3621no—
125#117—1811no—
126#101—3621no—
127#100—1812no—
128#99—3621no—
129#99—1812no—
130#41—7241no—
131#41—7241no—
132C2×Q8—3621no—
133C2×Q8—3621no—
134C2×Q8—1811no—
135C2×Q8—7241no—
136C2×Q8—7241no—
137C2×D4—1811no—
138C2×D4—1811no—
139C2×D4C2×#1062438211yescharacteristic
140C22×C4—1811no—
141C22×C4—1811no—
142C22×C4—1811no—
143C22×C4—1811no—
144C22×C4C2×#1062438211yescharacteristic
145M16—3622no—
146M16—7241no—
147C2×C8—7241no—
148C4⋊C4—21no—
149C4⋊C4—3621no—
150C4⋊C4—3621no—
151C4⋊C4—3621no—
152C4⋊C4—3621no—
153C4⋊C4—7241no—
154C22⋊C4—21no—
155C22⋊C4—3621no—
156C22⋊C4—3621no—
157C22⋊C4—3621no—
158C22⋊C4—3622no—
159C22⋊C4—1811no—
160C22⋊C4C2×#1062438211yescharacteristic
161C42—3621no—
162C42—7241no—
163C23—1811no—
164C23C22⋊#1062438211yescharacteristic
165C23C22×#1062438211yescharacteristic
166Q8—7241no—
167Q8—7242no—
168Q8—7241no—
169Q8—7241no—
170D4—41no—
171D4—7241no—
172D4—7241no—
173C2×C4—21no—
174C2×C4—21no—
175C2×C4—41no—
176C2×C4—3621no—
177C2×C4—3621no—
178C2×C4—3621no—
179C2×C4—3621no—
180C2×C4—3621no—
181C2×C4—3621no—
182C2×C4—3621no—
183C2×C4—3621no—
184C2×C4—1811no—
185C2×C4—1811no—
186C2×C4—1811no—
187C2×C4—7242no—
188C2×C4—7241no—
189C2×C4C22⋊#1062438211yescharacteristic
190C8—7242no—
191C22—21no—
192C22—21no—
193C22—41no—
194C22—41no—
195C22—3621no—
196C22—7241no—
197C22C2×C22⋊#1062438211yescharacteristic
198C4—21no—
199C4—41no—
200C4—41no—
201C4—3621no—
202C4—3621no—
203C4—14482no—
204C4—1811no—
205C4—1811no—
206C4—7241no—
207C4—7241no—
208C2—21no—
209C2—41no—
210C2—41no—
211C2—7241no—
212C2D181⋊#9911yescharacteristic
subgroupG/H
centerC2D181⋊#99
commutatorC23×C181C22×C4
FrattiniC23C22×#10624382
Fitting#123×C181C4
socleC2×C181C2×#99
prime radicalC2×#15668448C22
cosocleC2×C22⋊D181C23
hypercenter#123#10624382
hypocenterC181#2123
LCSG ▷ C23×C181 ▷ C22×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #123

Complements

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

NH ≅ G/Ntype
3. #26042182C2semidirect
4. #26042181C2semidirect
13. C22×Dic362C4semidirect
18. #19153797C4semidirect
21. #19153772C4semidirect
22. #123×C181C4semidirect
106. C181#2123semidirect
121. #123#10624382semidirect

Maximal subgroups

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

subgroupindex
#260422662
#260422592
#260421822
#260421812
#260418512
#2123181