Finite Group Databasebeta

← #37544805

Subgroups — #37544805

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#26041892C211yesmaximal, characteristic, complemented
2#26041854C211yesmaximal, characteristic
3#26041673C211yesmaximal, characteristic, complemented
4#26041658C211yesmaximal, characteristic, complemented
5#26041636C211yesmaximal, characteristic, complemented
6#26041504C212yesmaximal, complemented
7#26041488C212yesmaximal, complemented
8#26041444C212yesmaximal
9#26041436C212yesmaximal, complemented
10#26041341C211yesmaximal, characteristic, complemented
11#491×C181C211yesmaximal, characteristic, complemented
12C2×#15668450C2211yescharacteristic
13C2×Q8×D181C2211yescharacteristic
14C2×Dic724C2212yes—
15C2×#15668439C2212yes—
16C2×#15668439C2212yes—
17C2×#15668438C2212yes—
18#19153816C2211yescharacteristic
19#19153814C2211yescharacteristic
20#19153811—22no—
21#19153807—22no—
22#19153805C2211yescharacteristic
23#19153804C2211yescharacteristic
24#19153801—22no—
25#19153796C2212yes—
26#19153794—22no—
27#19153781C2212yes—
28#19153773C2211yescharacteristic
29#19153745C2212yes—
30#19153739C2212yes—
31#19153736C2212yes—
32#19153728C2212yes—
33C4×#13089678C2211yescharacteristic
34C4⋊Q8×C181C2211yescharacteristic
35#125×C181C2211yescharacteristic
36#103×C181C2212yes—
37#103×C181C2212yes—
38#97×C181C2211yescharacteristic
39#15668450—42no—
40C2×Dic362—21no—
41C2×Dic362C2311yescharacteristic
42C2×Dic362C2312yes—
43Q8×D181—42no—
44C2×D724—21no—
45C2×D724C2311yescharacteristic
46C2×C4×D181—21no—
47C2×C4×D181—21no—
48C2×C4×D181C2311yescharacteristic
49Dic724—42no—
50#15668439—42no—
51#15668439—42no—
52#15668438—42no—
53C2×#13089678C2312yes—
54C4⋊Dic181—22no—
55C4⋊Dic181—22no—
56#15668428—22no—
57#15668428—22no—
58#15668428D412yes—
59#15668426—22no—
60#15668426—22no—
61#15668426—22no—
62#15668426D412yes—
63C4×Dic181—22no—
64C4×Dic181C2311yescharacteristic
65C2×Q8×C181C2311yescharacteristic
66C2×Q8×C181C2311yescharacteristic
67C2×C8×C181C2312yes—
68C4⋊C4×C181—22no—
69C4⋊C4×C181—22no—
70C4⋊C4×C181C2312yes—
71C42×C181C2311yescharacteristic
72C22×D181—21no—
73C22×D181C2×D411yescharacteristic
74Dic362—22no—
75Dic362—22no—
76Dic362—22no—
77Dic362—42no—
78D724—22no—
79D724—42no—
80C2×Dic181—22no—
81C2×Dic181—21no—
82C2×Dic181C2×D412yes—
83C2×Dic181C2×D411yescharacteristic
84C4×D181—22no—
85C4×D181—42no—
86C4×D181—42no—
87#13089678—22no—
88Q8×C181—22no—
89Q8×C181—22no—
90C2×C4×C181—22no—
91C2×C4×C181—21no—
92C2×C4×C181—21no—
93C2×C4×C181C2×D412yes—
94C2×C4×C181C2411yescharacteristic
95C8×C181—22no—
96C22×C181C22×D411yescharacteristic
97C2×D181—21no—
98C2×D181—22no—
99C2×D181—41no—
100C2×D181—42no—
101C4×C181—22no—
102C4×C181—41no—
103C4×C181—41no—
104C4×C181—42no—
105C4×C181H(5,𝔽2)12yes—
106Dic181—21no—
107Dic181—22no—
108Dic181—41no—
109Dic181—42no—
110C2×C181#53511yescharacteristic
111C2×C181#57812yes—
112D181—41no—
113D181—81no—
114C181#322311yescharacteristic, complemented, Sylow
115#3223—1811nomaximal, Sylow
116#541—1811no—
117#536—1811no—
118#497—1811no—
119#494—1811no—
120#491D18111yescharacteristic, complemented
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×Q16—1812no—
132C2×SD16—1812no—
133C4⋊Q8—1811no—
134C4⋊Q8—1811no—
135C4⋊Q8C2×D18111yescharacteristic
136#126—3622no—
137#125—3622no—
138#125C2×D18111yescharacteristic
139#124—1811no—
140#124—1811no—
141#123—3622no—
142C22⋊Q8—3622no—
143C22⋊Q8—1812no—
144C4×Q8—1812no—
145#117—1811no—
146C4⋊C8—1812no—
147#103—1812no—
148#103C2×D18112yes—
149#103C2×D18112yes—
150#102—1812no—
151C22⋊C8—1812no—
152#97C2×D18111yescharacteristic
153C4×C8—1811no—
154#41—7242no—
155C2×Q8—3621no—
156C2×Q8—1811no—
157C2×Q8—1812no—
158C2×Q8—7242no—
159C2×Q8C22×D18111yescharacteristic
160C2×Q8C22×D18111yescharacteristic
161C2×D4—3621no—
162C2×D4—1811no—
163C22×C4—3621no—
164C22×C4—3621no—
165C22×C4—1811no—
166Q16—7242no—
167Q16—7242no—
168Q16—7242no—
169SD16—7242no—
170C2×C8—1812no—
171C2×C8C22×D18112yes—
172C4⋊C4—22no—
173C4⋊C4—22no—
174C4⋊C4—3622no—
175C4⋊C4—3622no—
176C4⋊C4—3622no—
177C4⋊C4—3622no—
178C4⋊C4—1812no—
179C4⋊C4C22×D18112yes—
180C22⋊C4—3622no—
181C22⋊C4—3622no—
182C22⋊C4—3622no—
183C22⋊C4—1812no—
184C42—3622no—
185C42—1811no—
186C42C22×D18111yescharacteristic
187C23—3621no—
188C23—1811no—
189Q8—22no—
190Q8—22no—
191Q8—3622no—
192Q8—3622no—
193Q8—3622no—
194Q8—7242no—
195D4—3622no—
196D4—7242no—
197C2×C4—22no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—3622no—
201C2×C4—3622no—
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—3622no—
212C22—3621no—
213C22—7242no—
214C22—7241no—
215C22C2×D4×D18111yescharacteristic
216C4—22no—
217C4—42no—
218C4—41no—
219C4—41no—
220C4—3622no—
221C4—3621no—
222C4—7241no—
223C4—7242no—
224C4#1915386212yes—
225C2—14481no—
226C2—7241no—
227C2#2604184711yescharacteristic
228C2#2604209512yes—
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#491×C181C2
socleC22×C181C22×D4
prime radicalC2×#15668450C22
hypercenter#491D181
hypocenterC181#3223
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C42 ◁ #491

Complements

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

NH ≅ G/Ntype
1. #26041892C2semidirect
3. #26041673C2semidirect
4. #26041658C2semidirect
5. #26041636C2semidirect
6. #26041504C2semidirect
7. #26041488C2semidirect
9. #26041436C2semidirect
10. #26041341C2semidirect
11. #491×C181C2semidirect
114. C181#3223semidirect
120. #491D181semidirect

Maximal subgroups

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

subgroupindex
#260418922
#260418542
#260416732
#260416582
#260416362
#260415042
#260414882
#260414442
#260414362
#260413412
#491×C1812
#3223181