Finite Group Databasebeta

← #37549976

Subgroups — #37549976

Order 23168.

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

normal subgroups only

This group has 280 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
1C2×#19153876C211yesmaximal, characteristic, complemented
2C2×#19153874C211yesmaximal, characteristic, complemented
3#26042181C212yesmaximal, complemented
4#26042179C212yesmaximal, complemented
5#26041838C211yesmaximal, characteristic
6C2×#15668458—21no—
7C2×C22⋊#10624382—21no—
8#19153876—21no—
9#19153876C2212yes—
10#19153874—21no—
11#19153874C2212yes—
12C22×C22⋊D181C2211yescharacteristic
13#19153799C411yescharacteristic, complemented
14#19153799C2211yescharacteristic
15#19153797—21no—
16#19153788—21no—
17#19153788—21no—
18#19153785C411yescharacteristic, complemented
19#19153785C2211yescharacteristic
20#19153784—21no—
21C2×C22⋊Dic181C411yescharacteristic, complemented
22C22×D4×C181C411yescharacteristic, complemented
23C22×#10624382—41no—
24#15668458—22no—
25#15668458—41no—
26C2×#13089685—41no—
27C22⋊#10624382—22no—
28C22⋊#10624382—41no—
29C23×D181D411yescharacteristic
30C2×C22⋊D181—21no—
31C2×C22⋊D181—21no—
32C2×C22⋊D181—41no—
33C2×C22⋊D181—41no—
34C2×C22⋊D181C2×C411yescharacteristic
35C2×C22⋊D181D411yescharacteristic
36C2×C22⋊D181D411yescharacteristic
37C2×C22⋊D181C2311yescharacteristic
38C22×Dic181—21no—
39C22×Dic181D411yescharacteristic
40#15668428—21no—
41#15668428—21no—
42C22⋊Dic181—21no—
43C22⋊Dic181—21no—
44C22⋊Dic181—21no—
45C22⋊Dic181C2×C411yescharacteristic, complemented
46C22⋊Dic181C2×C411yescharacteristic, complemented
47#15668426—21no—
48#15668426—21no—
49C24×C181C2×C411yescharacteristic
50C24×C181C2×C411yescharacteristic, complemented
51C2×D4×C181—21no—
52C2×D4×C181—21no—
53C2×D4×C181—41no—
54C2×D4×C181—41no—
55C22×C4×C181C2×C411yescharacteristic, complemented
56C2×#10624382—42no—
57C2×#10624382—82no—
58#13089685—42no—
59C22×D181—21no—
60C22×D181—21no—
61C22×D181—81no—
62C22×D181C22⋊C411yescharacteristic
63C22×D181C2×D411yescharacteristic
64C22⋊D181—41no—
65C22⋊D181—41no—
66C22⋊D181—41no—
67C22⋊D181—41no—
68C22⋊D181—81no—
69C22⋊D181—81no—
70C2×Dic181—21no—
71C2×Dic181—21no—
72C2×Dic181—21no—
73C2×Dic181—21no—
74C2×Dic181—81no—
75C2×Dic181C22⋊C411yescharacteristic
76C2×Dic181C2×D411yescharacteristic
77C23×C181—21no—
78C23×C181—21no—
79C23×C181—21no—
80C23×C181—41no—
81C23×C181—81no—
82C23×C181—81no—
83C23×C181C22⋊C411yescharacteristic
84C23×C181C22⋊C411yescharacteristic
85C23×C181C22×C411yescharacteristic
86D4×C181—42no—
87D4×C181—41no—
88D4×C181—41no—
89C2×C4×C181—21no—
90C2×C4×C181—41no—
91C22×C181—21no—
92C22×C181—21no—
93C22×C181—41no—
94C22×C181—41no—
95C22×C181—41no—
96C22×C181—41no—
97C22×C181—41no—
98C22×C181—41no—
99C22×C181—81no—
100C22×C181—81no—
101C22×C181—82no—
102C22×C181#9911yescharacteristic
103C22×C181#9911yescharacteristic
104C22×C181C2×C22⋊C411yescharacteristic
105C2×D181—21no—
106C2×D181—21no—
107C2×D181—42no—
108C2×D181—41no—
109C2×D181—41no—
110C2×D181—81no—
111#10624382—82no—
112C4×C181—41no—
113Dic181—21no—
114Dic181—21no—
115Dic181—41no—
116Dic181—41no—
117C2×C181—21no—
118C2×C181—21no—
119C2×C181—21no—
120C2×C181—41no—
121C2×C181—41no—
122C2×C181—81no—
123C2×C181C2×#9911yescharacteristic
124D181—41no—
125D181—41no—
126C181#212111yescharacteristic, complemented, Sylow
127#2121—1811nomaximal, Sylow
128#534—1811no—
129C2×#100—1811no—
130C2×#99—1811no—
131#352—1812no—
132AΣL(1,𝔽16)—1812no—
133C22×D4—1811no—
134C22×D4#1062438211yescharacteristic, complemented
135C2×M16—3621no—
136#123—3621no—
137#123—1811no—
138#123—1811no—
139#121—3621no—
140#121—3621no—
141#120—3621no—
142#120—1811no—
143#120—1811no—
144C2×C22⋊C4—3621no—
145C2×C22⋊C4—1811no—
146#100—3621no—
147#100—1812no—
148#99—3621no—
149#99—1812no—
150C24—1811no—
151C24C2×#1062438211yescharacteristic
152C24C2×#1062438211yescharacteristic, complemented
153C2×D4—21no—
154C2×D4—21no—
155C2×D4—41no—
156C2×D4—41no—
157C2×D4—3621no—
158C2×D4—3621no—
159C2×D4—1811no—
160C2×D4—1811no—
161C2×D4—1811no—
162C2×D4—1811no—
163C2×D4—7241no—
164C2×D4—7241no—
165C22×C4—3621no—
166C22×C4—1811no—
167C22×C4—7241no—
168C22×C4C2×#1062438211yescharacteristic, complemented
169M16—3622no—
170M16—7241no—
171C2×C8—7241no—
172C4⋊C4—3621no—
173C4⋊C4—3621no—
174C22⋊C4—3621no—
175C22⋊C4—3621no—
176C22⋊C4—3621no—
177C22⋊C4—3621no—
178C22⋊C4—3621no—
179C22⋊C4—3622no—
180C22⋊C4—1811no—
181C22⋊C4—1811no—
182C22⋊C4—7241no—
183C23—21no—
184C23—21no—
185C23—21no—
186C23—41no—
187C23—81no—
188C23—81no—
189C23—3621no—
190C23—3621no—
191C23—14481no—
192C23—1811no—
193C23—1811no—
194C23C22⋊#1062438211yescharacteristic
195C23C22⋊#1062438211yescharacteristic
196C23C22×#1062438211yescharacteristic
197D4—42no—
198D4—41no—
199D4—41no—
200D4—14481no—
201D4—14481no—
202D4—7241no—
203D4—7241no—
204D4—7241no—
205D4—7241no—
206C2×C4—21no—
207C2×C4—41no—
208C2×C4—3621no—
209C2×C4—3621no—
210C2×C4—3621no—
211C2×C4—3621no—
212C2×C4—14482no—
213C2×C4—14481no—
214C2×C4—1811no—
215C2×C4—1811no—
216C2×C4—7242no—
217C8—7242no—
218C22—21no—
219C22—21no—
220C22—41no—
221C22—41no—
222C22—41no—
223C22—41no—
224C22—41no—
225C22—41no—
226C22—81no—
227C22—81no—
228C22—82no—
229C22—3621no—
230C22—3621no—
231C22—14481no—
232C22—7241no—
233C22—7241no—
234C22—7242no—
235C22#1915387411yescharacteristic
236C22#1915387411yescharacteristic
237C22C2×C22⋊#1062438211yescharacteristic
238C4—41no—
239C4—3621no—
240C4—3621no—
241C4—14482no—
242C4—7241no—
243C4—7241no—
244C2—21no—
245C2—21no—
246C2—21no—
247C2—41no—
248C2—41no—
249C2—81no—
250C2—7241no—
251C2—7241no—
252C2C2×#1915387411yescharacteristic
subgroupG/H
centerC2C2×#19153874
commutatorC23×C181C22×C4
FrattiniC23C22×#10624382
FittingC22×D4×C181C4
socleC2×C181C2×#99
prime radical#26041838C2
cosocleC2×C22⋊D181C23
hypercenterC22×D4#10624382
hypocenterC181#2121
LCSG ▷ C23×C181 ▷ C22×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C2 ◁ C23 ◁ C24 ◁ C22×D4

Complements

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

NH ≅ G/Ntype
1. C2×#19153876C2semidirect
2. C2×#19153874C2semidirect
3. #26042181C2semidirect
4. #26042179C2semidirect
13. #19153799C4semidirect
18. #19153785C4semidirect
21. C2×C22⋊Dic181C4semidirect
22. C22×D4×C181C4semidirect
45. C22⋊Dic181C2×C4semidirect
46. C22⋊Dic181C2×C4semidirect
50. C24×C181C2×C4semidirect
55. C22×C4×C181C2×C4semidirect
126. C181#2121semidirect
134. C22×D4#10624382semidirect
152. C24C2×#10624382semidirect
168. C22×C4C2×#10624382semidirect

Maximal subgroups

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

subgroupindex
C2×#191538762
C2×#191538742
#260421812
#260421792
#260418382
#2121181