Finite Group Databasebeta

← #37544799

Subgroups — #37544799

Order 23168.

364 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 109 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#26041890C211yesmaximal, characteristic, complemented
2#26041849C211yesmaximal, characteristic
3#26041658C211yesmaximal, characteristic, complemented
4#26041648C211yesmaximal, characteristic, complemented
5#26041633C211yesmaximal, characteristic, complemented
6#26041499C212yesmaximal, complemented
7#26041487C212yesmaximal, complemented
8#26041438C212yesmaximal
9#26041433C212yesmaximal, complemented
10#26041341C211yesmaximal, characteristic, complemented
11#497×C181C211yesmaximal, characteristic, complemented
12C2×#15668448C2211yescharacteristic
13C2×Q8×D181C2211yescharacteristic
14C2×#15668439C2212yes—
15C2×#15668439C2212yes—
16#19153816C2211yescharacteristic
17#19153814C2211yescharacteristic
18#19153811—22no—
19#19153808C2211yescharacteristic
20#19153806—22no—
21#19153803C2211yescharacteristic
22#19153798—22no—
23#19153793—22no—
24#19153792C2212yes—
25Q8×Dic181C2212yes—
26#19153773C2211yescharacteristic
27#19153745C2212yes—
28#19153741C2212yes—
29#19153741C2212yes—
30#19153740C2212yes—
31#19153737C2212yes—
32#19153728C2212yes—
33C4×#13089678C2211yescharacteristic
34C2×Q16×C181C2212yes—
35C2×SD16×C181C2212yes—
36C4⋊Q8×C181C2211yescharacteristic
37#124×C181C2211yescharacteristic
38#97×C181C2211yescharacteristic
39#15668448—42no—
40C2×Dic362—21no—
41C2×Dic362C2311yescharacteristic
42C2×Dic362C2311yescharacteristic
43Q8×D181—42no—
44C2×C22⋊D181—21no—
45C22×Dic181—21no—
46C2×C4×D181—21no—
47C2×C4×D181C2311yescharacteristic
48#15668439—42no—
49#15668439—42no—
50C2×#13089678C2312yes—
51C4⋊Dic181—22no—
52C4⋊Dic181—22no—
53C4⋊Dic181C2312yes—
54#15668428—22no—
55#15668428—22no—
56#15668428—22no—
57#15668428D412yes—
58C22⋊Dic181—22no—
59#15668426—22no—
60#15668426D412yes—
61C4×Dic181—22no—
62C4×Dic181C2311yescharacteristic
63C2×Q8×C181C2311yescharacteristic
64C2×Q8×C181C2312yes—
65C2×D4×C181C2311yescharacteristic
66Q16×C181—42no—
67SD16×C181—42no—
68C2×C8×C181C2312yes—
69C4⋊C4×C181—22no—
70C22⋊C4×C181—22no—
71C42×C181C2311yescharacteristic
72C22×D181C2×D411yescharacteristic
73Dic362—22no—
74Dic362—22no—
75Dic362—42no—
76C22⋊D181—42no—
77C2×Dic181—22no—
78C2×Dic181—21no—
79C2×Dic181—21no—
80C2×Dic181—42no—
81C2×Dic181C2×D412yes—
82C2×Dic181C2×D411yescharacteristic
83C4×D181—22no—
84C4×D181—42no—
85#13089678—22no—
86C23×C181—21no—
87Q8×C181—22no—
88Q8×C181—22no—
89Q8×C181—22no—
90D4×C181—22no—
91C2×C4×C181—21no—
92C2×C4×C181—22no—
93C2×C4×C181C2×D412yes—
94C2×C4×C181C2411yescharacteristic
95C8×C181—22no—
96C22×C181—41no—
97C22×C181—42no—
98C22×C181C22×D411yescharacteristic
99C2×D181—21no—
100C2×D181—22no—
101C4×C181—22no—
102C4×C181—42no—
103C4×C181—41no—
104C4×C181H(5,𝔽2)12yes—
105Dic181—21no—
106Dic181—22no—
107Dic181—41no—
108Dic181—41no—
109Dic181—42no—
110C2×C181—81no—
111C2×C181#53511yescharacteristic
112C2×C181#57812yes—
113D181—41no—
114C181#322311yescharacteristic, complemented, Sylow
115#3223—1811nomaximal, Sylow
116#541—1811no—
117#536—1811no—
118#497D18111yescharacteristic, complemented
119#494—1811no—
120#491—1811no—
121#488—1811no—
122#464—1812no—
123#462—1812no—
124#452—1812no—
125#451—1812no—
126#433—1811no—
127C2×#41—1811no—
128C22×Q8—1811no—
129C2×Q16—1812no—
130C2×Q16—1812no—
131C2×Q16C2×D18112yes—
132C2×SD16C2×D18112yes—
133C4⋊Q8—1811no—
134C4⋊Q8—1811no—
135C4⋊Q8C2×D18111yescharacteristic
136#126—3622no—
137#125—3622no—
138#125—1811no—
139#124—1811no—
140#124C2×D18111yescharacteristic
141#123—3622no—
142C22⋊Q8—3622no—
143C22⋊Q8—1812no—
144C4×Q8—1812no—
145#117—1811no—
146C4⋊C8—1812no—
147#103—1812no—
148#103—1812no—
149#103—1812no—
150#102—1812no—
151C22⋊C8—1812no—
152#97C2×D18111yescharacteristic
153C4×C8—1811no—
154#41—7242no—
155C2×Q8—3621no—
156C2×Q8—1811no—
157C2×Q8—1811no—
158C2×Q8—7242no—
159C2×Q8C22×D18111yescharacteristic
160C2×Q8C22×D18112yes—
161C2×D4—3621no—
162C2×D4C22×D18111yescharacteristic
163C22×C4—3621no—
164C22×C4—3621no—
165C22×C4—1811no—
166Q16—42no—
167Q16—7242no—
168Q16—7242no—
169SD16—42no—
170C2×C8—1812no—
171C2×C8C22×D18112yes—
172C4⋊C4—22no—
173C4⋊C4—3622no—
174C4⋊C4—3622no—
175C4⋊C4—3622no—
176C4⋊C4—3622no—
177C4⋊C4—3622no—
178C4⋊C4—1812no—
179C4⋊C4—1812no—
180C22⋊C4—22no—
181C22⋊C4—3622no—
182C22⋊C4—3622no—
183C22⋊C4—1812no—
184C42—3622no—
185C42—1811no—
186C42C22×D18111yescharacteristic
187C23—21no—
188C23—1811no—
189Q8—22no—
190Q8—22no—
191Q8—22no—
192Q8—3622no—
193Q8—3622no—
194Q8—7242no—
195D4—22no—
196D4—7242no—
197C2×C4—22no—
198C2×C4—21no—
199C2×C4—3622no—
200C2×C4—3622no—
201C2×C4—3621no—
202C2×C4—3621no—
203C2×C4—1812no—
204C2×C4—1811no—
205C2×C4—7242no—
206C2×C4—7242no—
207C2×C4D4×D18112yes—
208C2×C4C23×D18111yescharacteristic
209C8—22no—
210C8—3622no—
211C22—42no—
212C22—41no—
213C22—3622no—
214C22—3621no—
215C22C2×D4×D18111yescharacteristic
216C4—22no—
217C4—41no—
218C4—42no—
219C4—3622no—
220C4—3621no—
221C4—7242no—
222C4—7241no—
223C4—7241no—
224C4#1915386112yes—
225C2—81no—
226C2—7241no—
227C2#2604184211yescharacteristic
228C2#2604209512yes—
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#497×C181C2
socleC22×C181C22×D4
prime radicalC2×#15668448C22
hypercenter#497D181
hypocenterC181#3223
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C42 ◁ #497

Complements

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

NH ≅ G/Ntype
1. #26041890C2semidirect
3. #26041658C2semidirect
4. #26041648C2semidirect
5. #26041633C2semidirect
6. #26041499C2semidirect
7. #26041487C2semidirect
9. #26041433C2semidirect
10. #26041341C2semidirect
11. #497×C181C2semidirect
114. C181#3223semidirect
118. #497D181semidirect

Maximal subgroups

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

subgroupindex
#260418902
#260418492
#260416582
#260416482
#260416332
#260414992
#260414872
#260414382
#260414332
#260413412
#497×C1812
#3223181