Finite Group Databasebeta

← #37541888

Subgroups — #37541888

Order 23168.

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

normal subgroups only

This group has 416 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#26041220C212yesmaximal, complemented
2#26041208C212yesmaximal, complemented
3#26041199C211yesmaximal, characteristic, complemented
4#26041197C212yesmaximal
5#26041196C211yesmaximal, characteristic
6#26041192C212yesmaximal, complemented
7#26041190C212yesmaximal, complemented
8#26041170C212yesmaximal, complemented
9#391×C181C211yesmaximal, characteristic, complemented
10C22×Dic362C2211yescharacteristic
11C22×D724C2211yescharacteristic
12C2×#15668428C2212yes—
13C2×#15668428C2212yes—
14C2×#15668426C2212yes—
15C2×#15668426C2212yes—
16C2×#15668426C2212yes—
17C2×#15668426C2212yes—
18C2×#15668426C2212yes—
19C2×#15668426C2212yes—
20#19153723C2212yes—
21#19153723C2212yes—
22#19153723C2212yes—
23#19153723C2212yes—
24#19153723C2211yescharacteristic
25#19153723C2211yescharacteristic
26C22×Q8×C181C2211yescharacteristic
27C2×C4⋊C4×C181C2211yescharacteristic
28C2×C42×C181C2211yescharacteristic
29#95×C181C2212yes—
30#95×C181C2212yes—
31C23×D181C2312yes—
32C2×Dic362—41no—
33C2×Dic362—42no—
34C2×D724—41no—
35C2×D724—42no—
36C22×Dic181C2312yes—
37C22×Dic181C2312yes—
38C22×Dic181C2312yes—
39#15668428—42no—
40#15668428—42no—
41#15668426—24no—
42#15668426—42no—
43#15668426—42no—
44#15668426—42no—
45#15668426—42no—
46#15668426—42no—
47C2×Q8×C181—22no—
48C2×Q8×C181—41no—
49C2×Q8×C181—41no—
50C22×C4×C181C2312yes—
51C22×C4×C181C2311yescharacteristic
52C22×C4×C181C2311yescharacteristic
53C22×C4×C181C2312yes—
54C22×C4×C181C2311yescharacteristic
55C4⋊C4×C181—41no—
56C42×C181—41no—
57C22×D181—22no—
58C22×D181—22no—
59C22×D181—22no—
60C22×D181—22no—
61C22×D181—22no—
62C22×D181—22no—
63C22×D181—22no—
64Dic362—44no—
65D724—44no—
66C2×Dic181—22no—
67C2×Dic181—22no—
68C2×Dic181—22no—
69C2×Dic181—22no—
70C2×Dic181—22no—
71C2×Dic181—22no—
72C2×Dic181—22no—
73C2×Dic181—22no—
74C2×Dic181—22no—
75C23×C181C2411yescharacteristic
76Q8×C181—44no—
77C2×C4×C181—21no—
78C2×C4×C181—21no—
79C2×C4×C181—22no—
80C2×C4×C181—22no—
81C2×C4×C181—21no—
82C2×C4×C181—21no—
83C2×C4×C181—22no—
84C2×C4×C181—22no—
85C2×C4×C181—22no—
86C2×C4×C181—22no—
87C2×C4×C181—22no—
88C2×C4×C181—22no—
89C2×C4×C181#4112yes—
90C22×C181C2×#4111yescharacteristic
91C22×C181H(5,𝔽2)11yescharacteristic
92C22×C181H(5,𝔽2)11yescharacteristic
93C22×C181H(5,𝔽2)12yes—
94C22×C1812−1+412yes—
95C2×D181—42no—
96C2×D181—42no—
97C2×D181—42no—
98C2×D181—42no—
99C2×D181—42no—
100C2×D181—42no—
101C2×D181—42no—
102C4×C181—22no—
103C4×C181—42no—
104C4×C181—41no—
105C4×C181—41no—
106C4×C181—42no—
107Dic181—42no—
108Dic181—42no—
109Dic181—42no—
110C2×C181#53811yescharacteristic
111C2×C181#54212yes—
112C2×C181H(3,𝔽4)11yescharacteristic
113C2×C181#56212yes—
114C2×C181#56311yescharacteristic
115D181—82no—
116C181#283211yescharacteristic, complemented, Sylow
117#2832—1811nomaximal, Sylow
118#400—1812no—
119#397—1812no—
120#395—1811no—
121#394—1811no—
122#394—1812no—
123#393—1812no—
124#393—1812no—
125#391D18111yescharacteristic, complemented
126#388—1812no—
127C22×Q8—1811no—
128C22×Q8C2×D18111yescharacteristic
129C22×D4—1811no—
130C2×C4⋊C4—1812no—
131C2×C4⋊C4—1812no—
132C2×C4⋊C4C2×D18111yescharacteristic
133C2×C22⋊C4—1812no—
134C2×C22⋊C4—1812no—
135C2×C22⋊C4—1812no—
136C2×C22⋊C4—1812no—
137C2×C22⋊C4—1812no—
138C2×C22⋊C4—1812no—
139C2×C42C2×D18111yescharacteristic
140#95—1812no—
141#95—1812no—
142#95—1812no—
143#95—1812no—
144#95—1811no—
145#95—1811no—
146#95C2×D18112yes—
147#95C2×D18112yes—
148C24—1812no—
149C2×Q8—22no—
150C2×Q8—41no—
151C2×Q8—41no—
152C2×Q8—7241no—
153C2×Q8—7242no—
154C2×D4—7241no—
155C2×D4—7242no—
156C22×C4—1812no—
157C22×C4—1812no—
158C22×C4—1812no—
159C22×C4C22×D18112yes—
160C22×C4C22×D18112yes—
161C22×C4C22×D18111yescharacteristic
162C22×C4C22×D18111yescharacteristic
163C22×C4C22×D18111yescharacteristic
164C4⋊C4—41no—
165C4⋊C4—7242no—
166C4⋊C4—7242no—
167C22⋊C4—3624no—
168C22⋊C4—7242no—
169C22⋊C4—7242no—
170C22⋊C4—7242no—
171C22⋊C4—7242no—
172C22⋊C4—7242no—
173C42—41no—
174C23—3622no—
175C23—3622no—
176C23—3622no—
177C23—3622no—
178C23—3622no—
179C23—3622no—
180C23—3622no—
181C23C23×D18111yescharacteristic
182Q8—44no—
183Q8—7244no—
184D4—7244no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—3622no—
198C2×C4—3622no—
199C2×C4—3622no—
200C2×C4—3622no—
201C2×C4—3622no—
202C2×C4—3622no—
203C2×C4—3622no—
204C2×C4—3622no—
205C2×C4—3622no—
206C2×C4#1566844912yes—
207C22—7242no—
208C22—7242no—
209C22—7242no—
210C22—7242no—
211C22—7242no—
212C22—7242no—
213C22—7242no—
214C22C2×#1566844911yescharacteristic
215C22#1915386111yescharacteristic
216C22#1915386111yescharacteristic
217C22#1915386212yes—
218C22#1915386412yes—
219C4—22no—
220C4—42no—
221C4—41no—
222C4—41no—
223C4—42no—
224C4—7242no—
225C4—7242no—
226C4—7242no—
227C2—14482no—
228C2#2604186711yescharacteristic
229C2#2604190112yes—
230C2#2604202111yescharacteristic
231C2#2604202512yes—
232C2#2604202811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D181
commutator, socle, cosocleC23×C181C24
Fitting#391×C181C2
prime radicalC22×D724C22
hypercenter#391D181
hypocenterC181#2832
LCSG ▷ C23×C181 ▷ C181
UCSC1 ◁ C23 ◁ #391

Complements

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

NH ≅ G/Ntype
1. #26041220C2semidirect
2. #26041208C2semidirect
3. #26041199C2semidirect
6. #26041192C2semidirect
7. #26041190C2semidirect
8. #26041170C2semidirect
9. #391×C181C2semidirect
116. C181#2832semidirect
125. #391D181semidirect

Maximal subgroups

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

subgroupindex
#260412202
#260412082
#260411992
#260411972
#260411962
#260411922
#260411902
#260411702
#391×C1812
#2832181