Finite Group Databasebeta

← #37544642

Subgroups — #37544642

Order 23168.

428 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 428 conjugacy classes of subgroups — too many to draw. Only its 125 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#26041827C211yesmaximal, characteristic
2C4⋊Q8×D181C211yesmaximal, characteristic, complemented
3#26041651C211yesmaximal, characteristic, complemented
4#26041630C211yesmaximal, characteristic, complemented
5#26041629C211yesmaximal, characteristic, complemented
6#26041482C214yesmaximal, complemented
7#26041416C214yesmaximal, complemented
8#26041330C211yesmaximal, characteristic, complemented
9#492×C181C211yesmaximal, characteristic, complemented
10C2×#15668448—21no—
11C2×Q8×D181C2212yes—
12C2×#15668437C2214yes—
13#19153816C2211yescharacteristic
14#19153816C2211yescharacteristic
15#19153815—21no—
16#19153815—21no—
17#19153814C2212yes—
18#19153803—21no—
19#19153787—21no—
20#19153787C2212yes—
21D4×Dic181—21no—
22D4×Dic181C2212yes—
23C4⋊C4×D181—21no—
24C4⋊C4×D181—21no—
25C42×D181C2211yescharacteristic
26#19153745C2212yes—
27#19153741C2214yes—
28#19153740C2214yes—
29#19153728C2212yes—
30C4×#13089678C2211yescharacteristic
31C2×SD16×C181C2214yes—
32C4⋊Q8×C181C2211yescharacteristic
33#127×C181C2211yescharacteristic
34C4×C8×C181C2211yescharacteristic
35#15668448—44no—
36C2×Dic362—22no—
37C2×Dic362—21no—
38C2×Dic362C2312yes—
39Q8×D181—44no—
40C2×C22⋊D181—22no—
41C22×Dic181—22no—
42C2×C4×D181—22no—
43C2×C4×D181D411yescharacteristic
44C2×C4×D181D411yescharacteristic
45C2×C4×D181C2311yescharacteristic
46#15668437—44no—
47C2×#13089678C2312yes—
48C4⋊Dic181—21no—
49C4⋊Dic181—21no—
50C4⋊Dic181—21no—
51C4⋊Dic181C2312yes—
52#15668428—22no—
53#15668428—22no—
54C22⋊Dic181—22no—
55C22⋊Dic181—22no—
56C4×Dic181D411yescharacteristic
57C4×Dic181D411yescharacteristic
58C4×Dic181C2311yescharacteristic
59C2×Q8×C181C2312yes—
60C2×D4×C181—21no—
61C2×D4×C181—21no—
62C2×D4×C181C2312yes—
63SD16×C181—44no—
64C2×C8×C181C2312yes—
65C4⋊C4×C181—21no—
66C4⋊C4×C181—21no—
67C42×C181C2311yescharacteristic
68C22×D181C2×D411yescharacteristic
69Dic362—24no—
70Dic362—44no—
71Dic362—42no—
72C22⋊D181—44no—
73C2×Dic181—22no—
74C2×Dic181—22no—
75C2×Dic181—44no—
76C2×Dic181C2×D411yescharacteristic
77C2×Dic181C2×D411yescharacteristic
78C2×Dic181C2×D411yescharacteristic
79C4×D181—22no—
80C4×D181—22no—
81C4×D181—44no—
82C4×D181SD1614yescomplemented
83#13089678—22no—
84C23×C181—22no—
85Q8×C181—24no—
86D4×C181—24no—
87D4×C181—42no—
88D4×C181—42no—
89C2×C4×C181—22no—
90C2×C4×C181C2×D411yescharacteristic
91C2×C4×C181C2×D411yescharacteristic
92C2×C4×C181C2411yescharacteristic
93C8×C181—22no—
94C22×C181—42no—
95C22×C181—42no—
96C22×C181—42no—
97C22×C181C22×D411yescharacteristic
98C2×D181—21no—
99C2×D181—21no—
100C2×D181C2×SD1612yes—
101C4×C181—42no—
102C4×C181C2×SD1612yes—
103C4×C181#13712yes—
104C4×C181H(5,𝔽2)12yes—
105Dic181—21no—
106Dic181—21no—
107Dic181—42no—
108Dic181—42no—
109Dic181C2×SD1612yes—
110C2×C181—82no—
111C2×C181#53511yescharacteristic
112C2×C181C22×SD1611yescharacteristic
113C2×C181C2×#13711yescharacteristic
114D181—22no—
115C181#320611yescharacteristic, complemented, Sylow
116#3206—1811nomaximal, Sylow
117#532—1811no—
118C2×C4⋊Q8—1811no—
119#492—1811no—
120#492D18111yescharacteristic, complemented
121#487—1811no—
122#487—1811no—
123#461—1814no—
124#448—1814no—
125#431—1811no—
126C2×#41—3621no—
127C22×Q8—1812no—
128C2×SD16—1814no—
129C2×SD16C2×D18114yes—
130C4⋊Q8—3621no—
131C4⋊Q8—3621no—
132C4⋊Q8—1812no—
133C4⋊Q8—1811no—
134C4⋊Q8—1811no—
135C4⋊Q8C2×D18111yescharacteristic
136#127C2×D18111yescharacteristic
137#124—3621no—
138#121—3621no—
139#121—1812no—
140C4×D4—3621no—
141C4×D4—1812no—
142C2×C4⋊C4—3621no—
143C2×C4⋊C4—3621no—
144C2×C42—1811no—
145C4⋊C8—1812no—
146#103—1814no—
147#103—1814no—
148C22⋊C8—1812no—
149C4×C8—1811no—
150C4×C8C2×D18111yescharacteristic
151#41—7244no—
152C2×Q8—3622no—
153C2×Q8—3621no—
154C2×Q8—1812no—
155C2×Q8—7244no—
156C2×Q8C22×D18112yes—
157C2×D4—21no—
158C2×D4—21no—
159C2×D4—3622no—
160C2×D4C22×D18112yes—
161C22×C4—3622no—
162C22×C4—3622no—
163C22×C4—1811no—
164C22×C4—1811no—
165C22×C4—1811no—
166SD16—44no—
167SD16—7244no—
168C2×C8—1812no—
169C2×C8C22×D18112yes—
170C4⋊C4—21no—
171C4⋊C4—21no—
172C4⋊C4—3622no—
173C4⋊C4—3622no—
174C4⋊C4—3621no—
175C4⋊C4—3621no—
176C4⋊C4—3621no—
177C4⋊C4—1812no—
178C22⋊C4—3622no—
179C22⋊C4—3622no—
180C42—1811no—
181C42—1811no—
182C42—1811no—
183C42C22×D18111yescharacteristic
184C23—22no—
185C23—1811no—
186Q8—24no—
187Q8—3624no—
188Q8—7244no—
189Q8—7242no—
190D4—24no—
191D4—42no—
192D4—42no—
193D4—7244no—
194C2×C4—22no—
195C2×C4—3622no—
196C2×C4—3622no—
197C2×C4—3622no—
198C2×C4—3622no—
199C2×C4—1811no—
200C2×C4—1811no—
201C2×C4—1811no—
202C2×C4—1814no—
203C2×C4—7244no—
204C2×C4—7244no—
205C2×C4D4×D18111yescharacteristic
206C2×C4D4×D18111yescharacteristic
207C2×C4C23×D18111yescharacteristic
208C8—22no—
209C8—3622no—
210C22—42no—
211C22—42no—
212C22—42no—
213C22—3621no—
214C22—3621no—
215C22—1812no—
216C22C2×D4×D18111yescharacteristic
217C4—42no—
218C4—3621no—
219C4—3621no—
220C4—1812no—
221C4—7242no—
222C4—7242no—
223C4SD16×D18112yes—
224C4#1915384512yes—
225C4#1915386112yes—
226C2—82no—
227C2—3622no—
228C2#2604184211yescharacteristic
229C2C2×SD16×D18111yescharacteristic
230C2C2×#1915384511yescharacteristic
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#492×C181C2
socleC22×C181C22×D4
prime radical#26041827C2
hypercenter#492D181
hypocenterC181#3206
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C42 ◁ #492

Complements

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

NH ≅ G/Ntype
2. C4⋊Q8×D181C2semidirect
3. #26041651C2semidirect
4. #26041630C2semidirect
5. #26041629C2semidirect
6. #26041482C2semidirect
7. #26041416C2semidirect
8. #26041330C2semidirect
9. #492×C181C2semidirect
82. C4×D181SD16semidirect
115. C181#3206semidirect
120. #492D181semidirect

Maximal subgroups

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

subgroupindex
#260418272
C4⋊Q8×D1812
#260416512
#260416302
#260416292
#260414822
#260414162
#260413302
#492×C1812
#3206181