Finite Group Databasebeta

← #37541864

Subgroups — #37541864

Order 23168.

408 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 408 conjugacy classes of subgroups — too many to draw. Only its 117 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#26041209C211yesmaximal, characteristic, complemented
2#26041207C212yesmaximal, complemented
3#26041207C212yesmaximal, complemented
4#26041202C212yesmaximal, complemented
5#26041199C211yesmaximal, characteristic, complemented
6#26041193C212yesmaximal
7#26041191C212yesmaximal, complemented
8#26041163C212yesmaximal, complemented
9#390×C181C211yesmaximal, characteristic
10C22×Dic362C2211yescharacteristic
11C2×C4⋊Dic181C2211yescharacteristic
12C2×#15668428C2212yes—
13C2×#15668428C2212yes—
14C2×#15668428C2212yes—
15C2×C22⋊Dic181C2212yes—
16C2×C22⋊Dic181C2212yes—
17C2×C22⋊Dic181C2212yes—
18C2×C22⋊Dic181C2212yes—
19#19153723C2212yes—
20#19153723C2212yes—
21#19153723C2212yes—
22#19153723C2212yes—
23#19153723C2212yes—
24#19153723C2211yescharacteristic
25#19153723C2211yescharacteristic
26C22×D4×C181C2211yescharacteristic
27C2×C4⋊C4×C181C2211yescharacteristic
28C2×C22⋊C4×C181C2212yes—
29C2×C22⋊C4×C181C2212yes—
30C2×C42×C181C2211yescharacteristic
31C2×Dic362—42no—
32C2×Dic362—41no—
33C22×Dic181C2312yes—
34C22×Dic181C2312yes—
35C22×Dic181C2312yes—
36C22×Dic181C2312yes—
37C4⋊Dic181—41no—
38#15668428—24no—
39#15668428—42no—
40#15668428—42no—
41C22⋊Dic181—42no—
42C22⋊Dic181—42no—
43C22⋊Dic181—42no—
44C22⋊Dic181—42no—
45C24×C181C2312yes—
46C2×D4×C181—22no—
47C2×D4×C181—41no—
48C2×D4×C181—41no—
49C22×C4×C181C2312yes—
50C22×C4×C181C2311yescharacteristic
51C22×C4×C181C2311yescharacteristic
52C22×C4×C181C2311yescharacteristic
53C4⋊C4×C181—41no—
54C22⋊C4×C181—42no—
55C22⋊C4×C181—42no—
56C42×C181—41no—
57Dic362—44no—
58C2×Dic181—22no—
59C2×Dic181—22no—
60C2×Dic181—22no—
61C2×Dic181—22no—
62C2×Dic181—22no—
63C2×Dic181—22no—
64C2×Dic181—22no—
65C2×Dic181—22no—
66C2×Dic181—22no—
67C2×Dic181—22no—
68C2×Dic181—22no—
69C2×Dic181—22no—
70C23×C181—22no—
71C23×C181—22no—
72C23×C181—22no—
73C23×C181—22no—
74C23×C181—22no—
75C23×C181—22no—
76C23×C181—22no—
77C23×C181C2411yescharacteristic
78D4×C181—44no—
79C2×C4×C181—21no—
80C2×C4×C181—21no—
81C2×C4×C181—22no—
82C2×C4×C181—22no—
83C2×C4×C181—22no—
84C2×C4×C181—22no—
85C2×C4×C181—21no—
86C2×C4×C181—21no—
87C2×C4×C181—22no—
88C2×C4×C181#4112yes—
89C22×C181—42no—
90C22×C181—42no—
91C22×C181—42no—
92C22×C181—42no—
93C22×C181—42no—
94C22×C181—42no—
95C22×C181—42no—
96C22×C181C2×#4111yescharacteristic
97C22×C181H(5,𝔽2)11yescharacteristic
98C22×C181H(5,𝔽2)11yescharacteristic
99C22×C181H(5,𝔽2)12yes—
100C22×C1812−1+412yes—
101C4×C181—22no—
102C4×C181—41no—
103C4×C181—41no—
104C4×C181—42no—
105Dic181—42no—
106Dic181—42no—
107Dic181—42no—
108Dic181—42no—
109C2×C181—82no—
110C2×C181#53811yescharacteristic
111C2×C181#54212yes—
112C2×C181#56011yescharacteristic
113C2×C181#56212yes—
114C2×C181#56311yescharacteristic
115C181#283011yescharacteristic, complemented, Sylow
116#2830—1811nomaximal, Sylow
117#398—1811no—
118#397—1812no—
119#397—1812no—
120#396—1812no—
121#395—1811no—
122#394—1812no—
123#393—1812no—
124#390D18111yescharacteristic
125#387—1812no—
126C22×Q8—1811no—
127C22×D4C2×D18111yescharacteristic
128C2×C4⋊C4—1812no—
129C2×C4⋊C4—1812no—
130C2×C4⋊C4—1812no—
131C2×C4⋊C4—1811no—
132C2×C4⋊C4C2×D18111yescharacteristic
133C2×C22⋊C4—1812no—
134C2×C22⋊C4—1812no—
135C2×C22⋊C4—1812no—
136C2×C22⋊C4—1812no—
137C2×C22⋊C4C2×D18112yes—
138C2×C22⋊C4C2×D18112yes—
139C2×C42C2×D18111yescharacteristic
140#95—1812no—
141#95—1812no—
142#95—1812no—
143#95—1812no—
144#95—1812no—
145#95—1811no—
146#95—1811no—
147C24C22×D18112yes—
148C2×Q8—7242no—
149C2×Q8—7241no—
150C2×D4—22no—
151C2×D4—41no—
152C2×D4—41no—
153C22×C4—1812no—
154C22×C4—1812no—
155C22×C4—1812no—
156C22×C4—1812no—
157C22×C4C22×D18112yes—
158C22×C4C22×D18111yescharacteristic
159C22×C4C22×D18111yescharacteristic
160C22×C4C22×D18111yescharacteristic
161C4⋊C4—41no—
162C4⋊C4—3624no—
163C4⋊C4—7242no—
164C4⋊C4—7242no—
165C4⋊C4—7241no—
166C22⋊C4—42no—
167C22⋊C4—42no—
168C22⋊C4—7242no—
169C22⋊C4—7242no—
170C22⋊C4—7242no—
171C22⋊C4—7242no—
172C42—41no—
173C23—22no—
174C23—22no—
175C23—22no—
176C23—22no—
177C23—22no—
178C23—22no—
179C23—22no—
180C23C23×D18111yescharacteristic
181Q8—7244no—
182D4—44no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—22no—
192C2×C4—3622no—
193C2×C4—3622no—
194C2×C4—3622no—
195C2×C4—3622no—
196C2×C4—3622no—
197C2×C4—3622no—
198C2×C4—3622no—
199C2×C4—3622no—
200C2×C4—3622no—
201C2×C4—3622no—
202C2×C4—3622no—
203C2×C4—3622no—
204C2×C4#1566844912yes—
205C22—42no—
206C22—42no—
207C22—42no—
208C22—42no—
209C22—42no—
210C22—42no—
211C22—42no—
212C22C2×#1566844911yescharacteristic
213C22#1915386111yescharacteristic
214C22#1915386111yescharacteristic
215C22#1915386112yes—
216C22#1915386312yes—
217C4—22no—
218C4—41no—
219C4—41no—
220C4—42no—
221C4—7242no—
222C4—7242no—
223C4—7242no—
224C4—7242no—
225C2—82no—
226C2#2604186711yescharacteristic
227C2#2604189412yes—
228C2#2604201611yescharacteristic
229C2#2604202412yes—
230C2#2604202911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D181
commutator, socle, cosocleC23×C181C24
Fitting#390×C181C2
prime radicalC22×D4×C181C22
hypercenter#390D181
hypocenterC181#2830
LCSG ▷ C23×C181 ▷ C181
UCSC1 ◁ C23 ◁ #390

Complements

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

NH ≅ G/Ntype
1. #26041209C2semidirect
2. #26041207C2semidirect
3. #26041207C2semidirect
4. #26041202C2semidirect
5. #26041199C2semidirect
7. #26041191C2semidirect
8. #26041163C2semidirect
115. C181#2830semidirect

Maximal subgroups

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

subgroupindex
#260412092
#260412072
#260412022
#260411992
#260411932
#260411912
#260411632
#390×C1812
#2830181