Finite Group Databasebeta

← #37549728

Subgroups — #37549728

Order 23168.

444 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 68 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#26042346C211yesmaximal, characteristic, complemented
2#26042271C212yesmaximal, complemented
3D181⋊#100C211yesmaximal, characteristic, complemented
4#26042150C212yesmaximal, complemented
5#127×D181C211yesmaximal, characteristic
6D181⋊M16C2212yes—
7D4×#10624382—24no—
8#19153895C2212yes—
9C2×C22⋊#10624382—22no—
10#19153878—24no—
11#19153876—22no—
12C2×D4×D181—22no—
13C2×D4×D181C2211yescharacteristic
14C2×D4×D181C2211yescharacteristic
15#19153813C411yescharacteristic, complemented
16#19153812—22no—
17#19153812C411yescharacteristic, complemented
18#19153812C411yescharacteristic, complemented
19C42×D181C2211yescharacteristic
20#127×C181C411yescharacteristic, complemented
21C22×#10624382—24no—
22#15668459—22no—
23#15668458—22no—
24D181⋊C8—22no—
25C4⋊#10624382D414yescomplemented
26C22⋊#10624382—24no—
27C22⋊#10624382—22no—
28C4×#10624382—22no—
29C23×D181—21no—
30C23×D181D412yes—
31C2×D724—22no—
32C2×D724C2×C411yescharacteristic
33C2×D724C2×C411yescharacteristic, complemented
34C2×C22⋊D181—22no—
35C2×C22⋊D181—21no—
36C2×C22⋊D181—22no—
37C2×C22⋊D181D412yes—
38D4×D181—22no—
39D4×D181—22no—
40D4×D181—22no—
41D4×D181—22no—
42D4×D181—48no—
43C2×C4×D181—21no—
44C2×C4×D181C2311yescharacteristic
45C4×Dic181—21no—
46C4×Dic181C2×C411yescharacteristic, complemented
47C2×D4×C181—22no—
48C2×D4×C181C2×C411yescharacteristic
49C2×D4×C181C2×C411yescharacteristic, complemented
50C42×C181C2×C411yescharacteristic, complemented
51C2×#10624382—24no—
52C2×#10624382—84no—
53#13089685—42no—
54C22×D181—22no—
55C22×D181—22no—
56C22×D181—21no—
57C22×D181—44no—
58C22×D181—44no—
59C22×D181—41no—
60C22×D181—41no—
61C22×D181C22⋊C412yes—
62C22×D181C2×D411yescharacteristic
63D724—22no—
64D724—22no—
65D724—44no—
66C22⋊D181—24no—
67C22⋊D181—44no—
68C22⋊D181—44no—
69C22⋊D181—42no—
70C2×Dic181—21no—
71C2×Dic181C2×D411yescharacteristic
72C4×D181—24no—
73C4×D181C2×D412yes—
74C4×D181#4112yes—
75C23×C181—21no—
76C23×C181C22⋊C412yes—
77D4×C181—22no—
78D4×C181—22no—
79D4×C181—44no—
80C2×C4×C181—21no—
81C2×C4×C181C22×C411yescharacteristic
82C22×C181—22no—
83C22×C181—42no—
84C22×C181—42no—
85C22×C181—41no—
86C22×C181C2×C22⋊C411yescharacteristic
87C2×D181—22no—
88C2×D181—22no—
89C2×D181—44no—
90C2×D181—42no—
91C2×D181—42no—
92C2×D181—41no—
93C2×D181—82no—
94C2×D181—82no—
95C2×D181—82no—
96C2×D181#12011yescharacteristic
97C2×D181#12311yescharacteristic
98#10624382—44no—
99C4×C181—22no—
100C4×C181C4×D412yes—
101Dic181—22no—
102Dic181#12112yes—
103C2×C181—21no—
104C2×C181—42no—
105C2×C181—81no—
106C2×C181#38611yescharacteristic
107D181—21no—
108D181—42no—
109D181—81no—
110D181#45312yes—
111C181#188711yescharacteristic, complemented, Sylow
112#1887—1811nomaximal, Sylow
113C2×#127—1811no—
114#518—1811no—
115#418—1812no—
116C2×#100—1811no—
117#339—1812no—
118C22×D4—3622no—
119C22×D4—1811no—
120C22×D4—1811no—
121C2×M16—1812no—
122#127—3622no—
123#127—1811no—
124#127—1811no—
125#127—1811no—
126#127#1062438211yescharacteristic, complemented
127C4×D4—3624no—
128#117—1812no—
129C2×C22⋊C4—3622no—
130C2×C42—1811no—
131#102—3624no—
132#100—3622no—
133C24—3621no—
134C24—1812no—
135C2×D4—22no—
136C2×D4—3622no—
137C2×D4—3622no—
138C2×D4—3622no—
139C2×D4—3622no—
140C2×D4—3622no—
141C2×D4—3622no—
142C2×D4—3622no—
143C2×D4—3621no—
144C2×D4—1812no—
145C2×D4—1811no—
146C2×D4—1811no—
147C2×D4—7248no—
148C2×D4C2×#1062438211yescharacteristic
149C2×D4C2×#1062438211yescharacteristic, complemented
150C22×C4—3621no—
151C22×C4—3624no—
152C22×C4—1811no—
153M16—3622no—
154M16—3622no—
155C2×C8—3622no—
156C4⋊C4—1814no—
157C22⋊C4—3624no—
158C22⋊C4—3622no—
159C42—3621no—
160C42—3622no—
161C42—1811no—
162C42C2×#1062438211yescharacteristic, complemented
163C23—21no—
164C23—3622no—
165C23—3622no—
166C23—3621no—
167C23—1811no—
168C23—1812no—
169C23—7244no—
170C23—7244no—
171C23—7241no—
172C23—7241no—
173C23C22⋊#1062438212yes—
174D4—22no—
175D4—22no—
176D4—44no—
177D4—3624no—
178D4—3622no—
179D4—3622no—
180D4—7244no—
181D4—7244no—
182D4—7244no—
183D4—7242no—
184C2×C4—21no—
185C2×C4—3624no—
186C2×C4—3621no—
187C2×C4—3624no—
188C2×C4—14484no—
189C2×C4—1811no—
190C2×C4—1812no—
191C2×C4—1812no—
192C2×C4C22×#1062438211yescharacteristic
193C8—7242no—
194C22—22no—
195C22—42no—
196C22—42no—
197C22—41no—
198C22—3622no—
199C22—3622no—
200C22—14482no—
201C22—14482no—
202C22—14482no—
203C22—1811no—
204C22—1811no—
205C22—7242no—
206C22—7242no—
207C22—7244no—
208C22—7241no—
209C22C2×C22⋊#1062438211yescharacteristic
210C4—22no—
211C4—3622no—
212C4—1812no—
213C4—7244no—
214C4D4×#1062438212yes—
215C2—21no—
216C2—42no—
217C2—81no—
218C2—3621no—
219C2—14481no—
220C2—1812no—
221C2—7242no—
222C2#2604222711yescharacteristic
subgroupG/H
centerC2#26042227
commutatorC2×C4×C181C22×C4
FrattiniC2×C4C22×#10624382
Fitting#127×C181C4
socleC2×C181#386
prime radical#127×D181C2
cosocleC2×C4×D181C23
hypercenter#127#10624382
hypocenterC181#1887
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C2 ◁ C2×C4 ◁ #127

Complements

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

NH ≅ G/Ntype
1. #26042346C2semidirect
2. #26042271C2semidirect
3. D181⋊#100C2semidirect
4. #26042150C2semidirect
15. #19153813C4semidirect
17. #19153812C4semidirect
18. #19153812C4semidirect
20. #127×C181C4semidirect
25. C4⋊#10624382D4semidirect
33. C2×D724C2×C4semidirect
46. C4×Dic181C2×C4semidirect
49. C2×D4×C181C2×C4semidirect
50. C42×C181C2×C4semidirect
111. C181#1887semidirect
126. #127#10624382semidirect
149. C2×D4C2×#10624382semidirect
162. C42C2×#10624382semidirect

Maximal subgroups

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

subgroupindex
#260423462
#260422712
D181⋊#1002
#260421502
#127×D1812
#1887181