Finite Group Databasebeta

← #37549981

Subgroups — #37549981

Order 23168.

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

normal subgroups only

This group has 266 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, complemented
2#26042259C211yesmaximal, characteristic, complemented
3#26042180C212yesmaximal, complemented
4#26042179C212yesmaximal, complemented
5#26041844C211yesmaximal, characteristic
6C2×#15668458—21no—
7#19153895—21no—
8#19153877—21no—
9#19153876C2212yes—
10#19153874—21no—
11#19153874C2212yes—
12C2×#15668448C2211yescharacteristic
13C22×C22⋊D181C411yescharacteristic, complemented
14#19153812—21no—
15#19153802—21no—
16#19153789—21no—
17#19153788—21no—
18#19153786C2211yescharacteristic
19#19153785C2211yescharacteristic
20#19153784C411yescharacteristic, complemented
21#19153772C411yescharacteristic, complemented
22#120×C181C411yescharacteristic, complemented
23#15668458—22no—
24#15668458—41no—
25C2×#13089685—41no—
26C4⋊#10624382—41no—
27C22⋊#10624382—22no—
28C4×#10624382—41no—
29C23×D181C2×C411yescharacteristic
30#15668448—41no—
31#15668448—41no—
32C2×Dic362D411yescharacteristic
33C2×D724—21no—
34C2×C22⋊D181—21no—
35C2×C22⋊D181—21no—
36C2×C22⋊D181—21no—
37C2×C22⋊D181—21no—
38C2×C22⋊D181—41no—
39C2×C22⋊D181—41no—
40C2×C22⋊D181D411yescharacteristic
41C2×C22⋊D181C2311yescharacteristic
42C22×Dic181C2×C411yescharacteristic
43C22×Dic181D411yescharacteristic
44C2×C4×D181D411yescharacteristic
45#15668428—21no—
46C22⋊Dic181—21no—
47C22⋊Dic181C2×C411yescharacteristic
48#15668426—21no—
49#15668426—21no—
50C4×Dic181—21no—
51C24×C181C2×C411yescharacteristic
52C2×D4×C181—21no—
53C2×D4×C181C2×C411yescharacteristic
54C22⋊C4×C181—21no—
55C22⋊C4×C181C2×C411yescharacteristic
56C2×#10624382—42no—
57#13089685—42no—
58C22×D181—21no—
59C22×D181—41no—
60C22×D181—81no—
61C22×D181C2×D411yescharacteristic
62Dic362—41no—
63D724—81no—
64C22⋊D181—41no—
65C22⋊D181—41no—
66C22⋊D181—42no—
67C22⋊D181—41no—
68C22⋊D181—41no—
69C22⋊D181—81no—
70C22⋊D181—81no—
71C2×Dic181—21no—
72C2×Dic181—21no—
73C2×Dic181—21no—
74C2×Dic181—21no—
75C2×Dic181—21no—
76C2×Dic181C22⋊C411yescharacteristic
77C2×Dic181C22⋊C411yescharacteristic
78C2×Dic181C2×D411yescharacteristic
79C4×D181—21no—
80C4×D181—21no—
81C23×C181—21no—
82C23×C181—41no—
83C23×C181—81no—
84C23×C181C22⋊C411yescharacteristic
85C23×C181C22×C411yescharacteristic
86D4×C181—41no—
87D4×C181—81no—
88C2×C4×C181—21no—
89C2×C4×C181C22⋊C411yescharacteristic
90C22×C181—21no—
91C22×C181—21no—
92C22×C181—41no—
93C22×C181—41no—
94C22×C181—41no—
95C22×C181—41no—
96C22×C181—41no—
97C22×C181—82no—
98C22×C181C2×C22⋊C411yescharacteristic
99C2×D181—21no—
100C2×D181—41no—
101C2×D181—41no—
102C2×D181—41no—
103C2×D181—41no—
104C2×D181—82no—
105#10624382—82no—
106C4×C181—21no—
107C4×C181—41no—
108Dic181—21no—
109Dic181—21no—
110Dic181—41no—
111Dic181#9911yescharacteristic
112Dic181#9911yescharacteristic
113C2×C181—21no—
114C2×C181—41no—
115C2×C181—41no—
116C2×C181—81no—
117C2×C181C2×#9911yescharacteristic
118D181—41no—
119D181—81no—
120C181#212211yescharacteristic, complemented, Sylow
121#2122—1811nomaximal, Sylow
122#535—1811no—
123#413—1811no—
124#410—1811no—
125AΣL(1,𝔽16)—1812no—
126AΣL(1,𝔽16)—1812no—
127C2×#41—1811no—
128C22×D4—1811no—
129C2×M16—3621no—
130#127—3621no—
131#124—3621no—
132#121—3621no—
133#121—3621no—
134#120—1811no—
135#120—1811no—
136#120—1811no—
137#120#1062438211yescharacteristic, complemented
138#117—3621no—
139#117—1811no—
140#101—3621no—
141#100—1812no—
142#99—3621no—
143#99—1812no—
144C24—1811no—
145C24C2×#1062438211yescharacteristic
146#41—7241no—
147#41—7241no—
148C2×Q8—1811no—
149C2×D4—21no—
150C2×D4—3621no—
151C2×D4—3621no—
152C2×D4—3621no—
153C2×D4—3621no—
154C2×D4—3621no—
155C2×D4—1811no—
156C2×D4—1811no—
157C2×D4—7241no—
158C2×D4—7241no—
159C2×D4C2×#1062438211yescharacteristic
160C22×C4—1811no—
161C22×C4—1811no—
162C22×C4—1811no—
163M16—3622no—
164M16—7241no—
165C2×C8—7241no—
166C4⋊C4—3621no—
167C4⋊C4—7241no—
168C22⋊C4—21no—
169C22⋊C4—3621no—
170C22⋊C4—3621no—
171C22⋊C4—3621no—
172C22⋊C4—3622no—
173C22⋊C4—1811no—
174C22⋊C4C2×#1062438211yescharacteristic
175C42—3621no—
176C42—7241no—
177C23—21no—
178C23—41no—
179C23—81no—
180C23—3621no—
181C23—14481no—
182C23—1811no—
183C23—7241no—
184C23C22⋊#1062438211yescharacteristic
185C23C22×#1062438211yescharacteristic
186Q8—7241no—
187D4—41no—
188D4—81no—
189D4—14481no—
190D4—14481no—
191D4—14481no—
192D4—7241no—
193D4—7241no—
194D4—7242no—
195D4—7241no—
196D4—7241no—
197C2×C4—21no—
198C2×C4—3621no—
199C2×C4—3621no—
200C2×C4—3621no—
201C2×C4—3621no—
202C2×C4—3621no—
203C2×C4—3621no—
204C2×C4—3621no—
205C2×C4—1811no—
206C2×C4—1811no—
207C2×C4—1811no—
208C2×C4—7242no—
209C2×C4C22⋊#1062438211yescharacteristic
210C8—7242no—
211C22—21no—
212C22—21no—
213C22—41no—
214C22—41no—
215C22—41no—
216C22—41no—
217C22—41no—
218C22—82no—
219C22—3621no—
220C22—14482no—
221C22—7241no—
222C22—7241no—
223C22—7241no—
224C22—7241no—
225C22C2×C22⋊#1062438211yescharacteristic
226C4—21no—
227C4—41no—
228C4—3621no—
229C4—3621no—
230C4—14482no—
231C4—1811no—
232C4—1811no—
233C4—7241no—
234C2—21no—
235C2—41no—
236C2—41no—
237C2—81no—
238C2—14481no—
239C2—7241no—
240C2D181⋊#9911yescharacteristic
subgroupG/H
centerC2D181⋊#99
commutatorC23×C181C22×C4
FrattiniC23C22×#10624382
Fitting#120×C181C4
socleC2×C181C2×#99
prime radical#26041844C2
cosocleC2×C22⋊D181C23
hypercenter#120#10624382
hypocenterC181#2122
LCSG ▷ C23×C181 ▷ C22×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #120

Complements

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

NH ≅ G/Ntype
1. #26042266C2semidirect
2. #26042259C2semidirect
3. #26042180C2semidirect
4. #26042179C2semidirect
13. C22×C22⋊D181C4semidirect
20. #19153784C4semidirect
21. #19153772C4semidirect
22. #120×C181C4semidirect
120. C181#2122semidirect
137. #120#10624382semidirect

Maximal subgroups

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

subgroupindex
#260422662
#260422592
#260421802
#260421792
#260418442
#2122181