Finite Group Databasebeta

← #37549097

Subgroups — #37549097

Order 23168.

304 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 304 conjugacy classes of subgroups — too many to draw. Only its 62 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×#19153873C211yesmaximal, characteristic, complemented
2C4×C22⋊#10624382C211yesmaximal, characteristic, complemented
3#26042132C212yesmaximal, complemented
4#26042130C212yesmaximal, complemented
5#26041789C211yesmaximal, characteristic
6C22×#13089685—21no—
7C2×C22⋊#10624382—21no—
8C2×C4×#10624382—21no—
9#19153873—21no—
10#19153873C2212yes—
11(C2×#10624382)⋊C4C2212yes—
12C2×#15668448C411yescharacteristic, complemented
13C22×C4×D181C2211yescharacteristic
14#19153798—21no—
15#19153796C2211yescharacteristic
16#19153795C411yescharacteristic, complemented
17#19153787C2211yescharacteristic
18#19153784—21no—
19D4×Dic181—21no—
20#19153776—21no—
21#19153774C411yescharacteristic, complemented
22#121×C181C411yescharacteristic, complemented
23C22×#10624382—22no—
24C2×#13089685—22no—
25C2×#13089685—42no—
26C22⋊#10624382—24no—
27C4×#10624382—42no—
28C23×D181D411yescharacteristic
29#15668448—42no—
30C2×Dic362C2×C411yescharacteristic
31C2×C22⋊D181—21no—
32C2×C22⋊D181—21no—
33C22×Dic181—21no—
34C22×Dic181D411yescharacteristic
35C2×C4×D181—42no—
36C2×C4×D181C2×C411yescharacteristic
37C2×C4×D181D411yescharacteristic
38C2×C4×D181D411yescharacteristic
39C2×C4×D181C2311yescharacteristic
40C4⋊Dic181—21no—
41C4⋊Dic181C2×C411yescharacteristic
42#15668428—21no—
43C22⋊Dic181—21no—
44C22⋊Dic181—21no—
45#15668426—21no—
46#15668426—21no—
47C4×Dic181—21no—
48C2×D4×C181—21no—
49C2×D4×C181C2×C411yescharacteristic
50C22×C4×C181C2×C411yescharacteristic
51C4⋊C4×C181C2×C411yescharacteristic
52C22⋊C4×C181—21no—
53C2×#10624382—24no—
54C2×#10624382—42no—
55C2×#10624382—42no—
56#13089685—42no—
57C22×D181—22no—
58C22×D181—22no—
59C22×D181—41no—
60C22×D181C22⋊C411yescharacteristic
61C22×D181C2×D411yescharacteristic
62Dic362—41no—
63C22⋊D181—42no—
64C22⋊D181—42no—
65C2×Dic181—22no—
66C2×Dic181—21no—
67C2×Dic181—21no—
68C2×Dic181—42no—
69C2×Dic181C22⋊C411yescharacteristic
70C2×Dic181C2×D411yescharacteristic
71C4×D181—22no—
72C4×D181—22no—
73C4×D181—41no—
74C4×D181—41no—
75C23×C181—21no—
76C23×C181C22⋊C411yescharacteristic
77D4×C181—41no—
78D4×C181—42no—
79C2×C4×C181—21no—
80C2×C4×C181—41no—
81C2×C4×C181C22⋊C411yescharacteristic
82C2×C4×C181C22×C411yescharacteristic
83C22×C181—21no—
84C22×C181—21no—
85C22×C181—41no—
86C22×C181—41no—
87C22×C181—41no—
88C22×C181#10412yes—
89C22×C181C2×C22⋊C411yescharacteristic
90C2×D181—21no—
91C2×D181—21no—
92C2×D181—21no—
93C2×D181—21no—
94C2×D181—21no—
95C2×D181—44no—
96C2×D181#13512yes—
97#10624382—44no—
98C4×C181—21no—
99C4×C181—21no—
100C4×C181—41no—
101Dic181—21no—
102Dic181—41no—
103Dic181—41no—
104Dic181#9912yes—
105C2×C181—22no—
106C2×C181—81no—
107C2×C181C2×#9911yescharacteristic
108C2×C181#41611yescharacteristic
109C2×C181C2×#10411yescharacteristic
110D181—22no—
111D181—41no—
112C181#150111yescharacteristic, complemented, Sylow
113#1501—1811nomaximal, Sylow
114#525—1811no—
115C2×C22⋊C8—1811no—
116C4×C22⋊C4—1811no—
117#328—1812no—
118#327—1812no—
119C2×#41—1811no—
120C23×C4—1811no—
121C22×C8—3621no—
122#123—3621no—
123C22⋊Q8—1811no—
124C22⋊Q8—1811no—
125#121—1811no—
126#121#1062438211yescharacteristic, complemented
127#120—3621no—
128C4×D4—3621no—
129C4×D4—3621no—
130#117—1811no—
131C2×C22⋊C4—3621no—
132C2×C42—3621no—
133C22⋊C8—3621no—
134C22⋊C8—1812no—
135#95—1812no—
136C24—1811no—
137#41—7242no—
138C2×Q8—1811no—
139C2×D4—21no—
140C2×D4—3621no—
141C2×D4—3621no—
142C2×D4C2×#1062438211yescharacteristic
143C22×C4—3622no—
144C22×C4—3621no—
145C22×C4—1811no—
146C22×C4—1811no—
147C22×C4—1811no—
148C22×C4—1811no—
149C22×C4—1811no—
150C22×C4—7242no—
151C22×C4C2×#1062438211yescharacteristic
152C2×C8—3622no—
153C2×C8—7242no—
154C4⋊C4—3621no—
155C4⋊C4—3621no—
156C4⋊C4—1811no—
157C4⋊C4C2×#1062438211yescharacteristic
158C22⋊C4—21no—
159C22⋊C4—3621no—
160C22⋊C4—3621no—
161C22⋊C4—3624no—
162C22⋊C4—3621no—
163C22⋊C4—3621no—
164C42—3621no—
165C42—7242no—
166C23—21no—
167C23—3622no—
168C23—3622no—
169C23—1811no—
170C23—1811no—
171C23—7241no—
172C23C22⋊#1062438211yescharacteristic
173Q8—7241no—
174D4—41no—
175D4—42no—
176D4—7242no—
177D4—7242no—
178C2×C4—21no—
179C2×C4—41no—
180C2×C4—3622no—
181C2×C4—3622no—
182C2×C4—3622no—
183C2×C4—3624no—
184C2×C4—3621no—
185C2×C4—3621no—
186C2×C4—1811no—
187C2×C4—1811no—
188C2×C4—7241no—
189C2×C4—7241no—
190C2×C4—7242no—
191C2×C4—7242no—
192C2×C4—7242no—
193C2×C4C22⋊#1062438211yescharacteristic
194C2×C4C22×#1062438211yescharacteristic
195C8—7242no—
196C22—21no—
197C22—21no—
198C22—41no—
199C22—41no—
200C22—41no—
201C22—3621no—
202C22—3621no—
203C22—3621no—
204C22—3621no—
205C22—3621no—
206C22—1812no—
207C22—7244no—
208C22#1915388012yes—
209C22C2×C22⋊#1062438211yescharacteristic
210C4—21no—
211C4—21no—
212C4—41no—
213C4—3621no—
214C4—1812no—
215C4—7244no—
216C4—7241no—
217C4—7241no—
218C2—22no—
219C2—81no—
220C2—3622no—
221C2—7241no—
222C2D181⋊#9911yescharacteristic
223C2#2604226911yescharacteristic
224C2C2×#1915388011yescharacteristic
subgroupG/H
centerC22C2×C22⋊#10624382
commutatorC2×C4×C181C22×C4
FrattiniC2×C4C22×#10624382
Fitting#121×C181C4
socleC22×C181C2×C22⋊C4
prime radical#26041789C2
cosocleC2×C4×D181C23
hypercenter#121#10624382
hypocenterC181#1501
LCSG ▷ C2×C4×C181 ▷ C22×C181 ▷ C181
UCSC1 ◁ C22 ◁ C22×C4 ◁ #121

Complements

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

NH ≅ G/Ntype
1. C2×#19153873C2semidirect
2. C4×C22⋊#10624382C2semidirect
3. #26042132C2semidirect
4. #26042130C2semidirect
12. C2×#15668448C4semidirect
16. #19153795C4semidirect
21. #19153774C4semidirect
22. #121×C181C4semidirect
112. C181#1501semidirect
126. #121#10624382semidirect

Maximal subgroups

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

subgroupindex
C2×#191538732
C4×C22⋊#106243822
#260421322
#260421302
#260417892
#1501181