Finite Group Databasebeta

← #37544647

Subgroups — #37544647

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#26041830C211yesmaximal, characteristic
2C4⋊Q8×D181C211yesmaximal, characteristic, complemented
3#26041653C211yesmaximal, characteristic, complemented
4#26041652C211yesmaximal, characteristic, complemented
5#26041631C211yesmaximal, characteristic, complemented
6#26041483C214yesmaximal, complemented
7#26041419C214yesmaximal, complemented
8#26041330C211yesmaximal, characteristic, complemented
9#487×C181C211yesmaximal, characteristic, complemented
10C2×#15668450—21no—
11C2×Q8×D181C2212yes—
12C2×#15668438C2214yes—
13C2×#15668436C2214yes—
14#19153816C2211yescharacteristic
15#19153815—21no—
16#19153815—21no—
17#19153814C2212yes—
18#19153813C2211yescharacteristic
19#19153805—21no—
20#19153791—21no—
21#19153791C2212yes—
22#19153778—21no—
23#19153778C2212yes—
24C4⋊C4×D181—21no—
25C4⋊C4×D181—21no—
26C42×D181C2211yescharacteristic
27#19153745C2212yes—
28#19153739C2214yes—
29#19153728C2212yes—
30C4×#13089678C2211yescharacteristic
31C4⋊Q8×C181C2211yescharacteristic
32C4⋊Q8×C181C2211yescharacteristic
33#103×C181C2214yes—
34C4×C8×C181C2211yescharacteristic
35#15668450—44no—
36C2×Dic362—22no—
37C2×Dic362C2312yes—
38Q8×D181—44no—
39C2×D724—22no—
40C2×D724—21no—
41C2×D724—21no—
42C2×D724C2312yes—
43C2×C4×D181—22no—
44C2×C4×D181—22no—
45C2×C4×D181D411yescharacteristic
46C2×C4×D181D411yescharacteristic
47C2×C4×D181C2311yescharacteristic
48#15668438—44no—
49#15668436—44no—
50C2×#13089678C2312yes—
51C4⋊Dic181—21no—
52C4⋊Dic181—21no—
53#15668428—22no—
54#15668428—22no—
55#15668426—22no—
56#15668426—22no—
57C4×Dic181D411yescharacteristic
58C4×Dic181D411yescharacteristic
59C4×Dic181C2311yescharacteristic
60C2×Q8×C181—21no—
61C2×Q8×C181C2312yes—
62C2×C8×C181C2312yes—
63C4⋊C4×C181—21no—
64C4⋊C4×C181—21no—
65C4⋊C4×C181—21no—
66C4⋊C4×C181C2312yes—
67C42×C181C2311yescharacteristic
68C22×D181—22no—
69C22×D181C2×D411yescharacteristic
70Dic362—24no—
71Dic362—44no—
72D724—24no—
73D724—44no—
74D724—42no—
75D724—42no—
76C2×Dic181—22no—
77C2×Dic181C2×D411yescharacteristic
78C2×Dic181C2×D411yescharacteristic
79C2×Dic181C2×D411yescharacteristic
80C4×D181—22no—
81C4×D181—22no—
82C4×D181—44no—
83C4×D181—44no—
84C4×D181SD1614yescomplemented
85#13089678—22no—
86Q8×C181—24no—
87Q8×C181—42no—
88C2×C4×C181—22no—
89C2×C4×C181—22no—
90C2×C4×C181C2×D411yescharacteristic
91C2×C4×C181C2×D411yescharacteristic
92C2×C4×C181C2411yescharacteristic
93C8×C181—22no—
94C22×C181C22×D411yescharacteristic
95C2×D181—21no—
96C2×D181—21no—
97C2×D181—42no—
98C2×D181—42no—
99C2×D181—42no—
100C2×D181C2×SD1612yes—
101C4×C181—42no—
102C4×C181—42no—
103C4×C181C2×SD1612yes—
104C4×C181#13712yes—
105C4×C181H(5,𝔽2)12yes—
106Dic181—21no—
107Dic181—21no—
108Dic181—42no—
109Dic181C2×SD1612yes—
110C2×C181#53511yescharacteristic
111C2×C181C22×SD1611yescharacteristic
112C2×C181C2×#13711yescharacteristic
113D181—22no—
114D181—82no—
115C181#320611yescharacteristic, complemented, Sylow
116#3206—1811nomaximal, Sylow
117#532—1811no—
118C2×C4⋊Q8—1811no—
119#492—1811no—
120#492—1811no—
121#487—1811no—
122#487D18111yescharacteristic, complemented
123#461—1814no—
124#448—1814no—
125#431—1811no—
126C2×#41—3621no—
127C22×Q8—1812no—
128C2×SD16—1814no—
129C2×SD16—1814no—
130C4⋊Q8—3621no—
131C4⋊Q8—3621no—
132C4⋊Q8—1812no—
133C4⋊Q8—1811no—
134C4⋊Q8C2×D18111yescharacteristic
135C4⋊Q8C2×D18111yescharacteristic
136#127—1811no—
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#103C2×D18114yes—
148C22⋊C8—1812no—
149C4×C8—1811no—
150C4×C8C2×D18111yescharacteristic
151#41—7244no—
152C2×Q8—21no—
153C2×Q8—3622no—
154C2×Q8—1812no—
155C2×Q8—7244no—
156C2×Q8C22×D18112yes—
157C2×D4—3622no—
158C2×D4—3621no—
159C2×D4—3621no—
160C2×D4—1812no—
161C22×C4—3622no—
162C22×C4—3622no—
163C22×C4—1811no—
164C22×C4—1811no—
165C22×C4—1811no—
166SD16—7244no—
167SD16—7244no—
168C2×C8—1812no—
169C2×C8C22×D18112yes—
170C4⋊C4—21no—
171C4⋊C4—21no—
172C4⋊C4—21no—
173C4⋊C4—3622no—
174C4⋊C4—3622no—
175C4⋊C4—3621no—
176C4⋊C4—3621no—
177C4⋊C4C22×D18112yes—
178C22⋊C4—3622no—
179C22⋊C4—3622no—
180C42—1811no—
181C42—1811no—
182C42—1811no—
183C42C22×D18111yescharacteristic
184C23—3622no—
185C23—1811no—
186Q8—24no—
187Q8—42no—
188Q8—3624no—
189Q8—7244no—
190D4—3624no—
191D4—7244no—
192D4—7242no—
193D4—7242no—
194C2×C4—22no—
195C2×C4—22no—
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—3621no—
211C22—3621no—
212C22—1812no—
213C22—7242no—
214C22—7242no—
215C22—7242no—
216C22C2×D4×D18111yescharacteristic
217C4—42no—
218C4—42no—
219C4—3621no—
220C4—3621no—
221C4—1812no—
222C4—7242no—
223C4SD16×D18112yes—
224C4#1915384812yes—
225C4#1915386212yes—
226C2—3622no—
227C2—14482no—
228C2#2604184711yescharacteristic
229C2C2×SD16×D18111yescharacteristic
230C2C2×#1915384811yescharacteristic
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#487×C181C2
socleC22×C181C22×D4
prime radical#26041830C2
hypercenter#487D181
hypocenterC181#3206
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C42 ◁ #487

Complements

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

NH ≅ G/Ntype
2. C4⋊Q8×D181C2semidirect
3. #26041653C2semidirect
4. #26041652C2semidirect
5. #26041631C2semidirect
6. #26041483C2semidirect
7. #26041419C2semidirect
8. #26041330C2semidirect
9. #487×C181C2semidirect
84. C4×D181SD16semidirect
115. C181#3206semidirect
122. #487D181semidirect

Maximal subgroups

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

subgroupindex
#260418302
C4⋊Q8×D1812
#260416532
#260416522
#260416312
#260414832
#260414192
#260413302
#487×C1812
#3206181