Finite Group Databasebeta

← #37544673

Subgroups — #37544673

Order 23168.

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

normal subgroups only

This group has 394 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#26041829C211yesmaximal, characteristic
2#26041658C211yesmaximal, characteristic, complemented
3#26041653C211yesmaximal, characteristic, complemented
4#26041631C211yesmaximal, characteristic, complemented
5#26041627C211yesmaximal, characteristic, complemented
6#26041501C214yesmaximal, complemented
7#26041439C214yesmaximal, complemented
8#26041332C211yesmaximal, characteristic, complemented
9#533×C181C211yesmaximal, characteristic, complemented
10C2×#15668449C2212yes—
11C2×#15668439C2214yes—
12C2×#15668436C2214yes—
13#19153816C2211yescharacteristic
14#19153813C2211yescharacteristic
15#19153804C2212yes—
16#19153790—21no—
17#19153790—21no—
18C4×D724—21no—
19C4×D724—21no—
20C4⋊#13089678C2212yes—
21#19153739C2214yes—
22#19153736C2214yes—
23C22⋊#13089678C2212yes—
24C4×#13089678C2211yescharacteristic
25C4×#13089678C2211yescharacteristic
26C4⋊Q8×C181C2211yescharacteristic
27C4⋊Q8×C181C2211yescharacteristic
28#125×C181—21no—
29C22⋊Q8×C181—21no—
30C22⋊Q8×C181C2212yes—
31C4×Q8×C181—21no—
32C4×Q8×C181C2212yes—
33#117×C181—21no—
34C2×C42×C181C2211yescharacteristic
35#15668449—44no—
36C2×Dic362C2312yes—
37C2×D724—21no—
38C2×D724—21no—
39C2×D724C2312yes—
40C2×C22⋊D181—22no—
41C2×C4×D181—22no—
42#15668439—44no—
43#15668436—44no—
44C2×#13089678C2312yes—
45C2×#13089678C2312yes—
46C4⋊Dic181—21no—
47C4⋊Dic181—21no—
48#15668426—22no—
49#15668426—22no—
50C2×Q8×C181—21no—
51C2×Q8×C181C2312yes—
52C22×C4×C181D411yescharacteristic
53C22×C4×C181D411yescharacteristic
54C22×C4×C181C2311yescharacteristic
55C4⋊C4×C181—21no—
56C4⋊C4×C181—21no—
57C4⋊C4×C181—21no—
58C4⋊C4×C181—22no—
59C4⋊C4×C181—22no—
60C4⋊C4×C181C2312yes—
61C22⋊C4×C181—22no—
62C42×C181—22no—
63C42×C181D411yescharacteristic
64C42×C181D411yescharacteristic
65C42×C181C2311yescharacteristic
66C42×C181C2311yescharacteristic
67C22×D181—22no—
68Dic362—24no—
69D724—24no—
70D724—42no—
71D724—42no—
72C22⋊D181—44no—
73C2×Dic181—22no—
74C4×D181—44no—
75#13089678—22no—
76#13089678—22no—
77C23×C181C2×D411yescharacteristic
78Q8×C181—24no—
79Q8×C181—42no—
80C2×C4×C181—22no—
81C2×C4×C181—22no—
82C2×C4×C181—22no—
83C2×C4×C181—22no—
84C2×C4×C181—22no—
85C2×C4×C181C2×D411yescharacteristic
86C2×C4×C181C2×D411yescharacteristic
87C2×C4×C181C2×D411yescharacteristic
88C2×C4×C181C2×D411yescharacteristic
89C2×C4×C181C2×D411yescharacteristic
90C2×C4×C181C2411yescharacteristic
91C22×C181—21no—
92C22×C181—21no—
93C22×C181—21no—
94C22×C181C22×D411yescharacteristic
95C2×D181—42no—
96C2×D181—42no—
97C2×D181—42no—
98C4×C181—21no—
99C4×C181—21no—
100C4×C181—21no—
101C4×C181—42no—
102C4×C181—42no—
103C4×C181#13512yes—
104C4×C181#13712yes—
105C4×C181H(5,𝔽2)12yes—
106Dic181—42no—
107C2×C181—41no—
108C2×C181#53511yescharacteristic
109C2×C181C2×#13511yescharacteristic
110C2×C181C2×#13711yescharacteristic
111D181—82no—
112C181#321011yescharacteristic, complemented, Sylow
113#3210—1811nomaximal, Sylow
114#533D18111yescharacteristic, complemented
115#532—1811no—
116#494—1811no—
117#492—1811no—
118#487—1811no—
119#486—1811no—
120#464—1814no—
121#452—1814no—
122#431—1811no—
123C2×#41—1812no—
124C2×Q16—1814no—
125C2×SD16—1814no—
126C4⋊Q8—1811no—
127C4⋊Q8C2×D18111yescharacteristic
128C4⋊Q8C2×D18111yescharacteristic
129#127—1811no—
130#125—21no—
131#124—1812no—
132C22⋊Q8—21no—
133C22⋊Q8C2×D18112yes—
134#121—3621no—
135#121—3621no—
136C4×Q8—21no—
137C4×Q8C2×D18112yes—
138C4×D4—3621no—
139C4×D4—3621no—
140#117—21no—
141C2×C42C2×D18111yescharacteristic
142C4⋊C8—1812no—
143#103—1814no—
144#102—1814no—
145C22⋊C8—1812no—
146C4×C8—1811no—
147C4×C8—1811no—
148#41—7244no—
149C2×Q8—21no—
150C2×Q8—1812no—
151C2×Q8C22×D18112yes—
152C2×D4—3622no—
153C2×D4—3621no—
154C2×D4—3621no—
155C2×D4—1812no—
156C22×C4—3622no—
157C22×C4C22⋊D18111yescharacteristic
158C22×C4C22⋊D18111yescharacteristic
159C22×C4C22×D18111yescharacteristic
160Q16—7244no—
161SD16—7244no—
162C2×C8—1812no—
163C2×C8—1812no—
164C4⋊C4—21no—
165C4⋊C4—21no—
166C4⋊C4—21no—
167C4⋊C4—22no—
168C4⋊C4—22no—
169C4⋊C4—3621no—
170C4⋊C4—3621no—
171C4⋊C4C22×D18112yes—
172C22⋊C4—22no—
173C22⋊C4—3622no—
174C22⋊C4—3622no—
175C42—22no—
176C42C22⋊D18111yescharacteristic
177C42C22⋊D18111yescharacteristic
178C42C22×D18111yescharacteristic
179C42C22×D18111yescharacteristic
180C23—3622no—
181C23C2×C22⋊D18111yescharacteristic
182Q8—24no—
183Q8—42no—
184Q8—3624no—
185D4—3624no—
186D4—7244no—
187D4—7242no—
188D4—7242no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—3622no—
195C2×C4—7244no—
196C2×C4C2×C22⋊D18111yescharacteristic
197C2×C4C2×C22⋊D18111yescharacteristic
198C2×C4C2×C22⋊D18111yescharacteristic
199C2×C4C2×C22⋊D18111yescharacteristic
200C2×C4C2×C22⋊D18111yescharacteristic
201C2×C4C23×D18111yescharacteristic
202C8—3622no—
203C8—3622no—
204C22—21no—
205C22—21no—
206C22—21no—
207C22—7242no—
208C22—7242no—
209C22—7242no—
210C22C22×C22⋊D18111yescharacteristic
211C4—21no—
212C4—21no—
213C4—21no—
214C4—42no—
215C4—42no—
216C4—7242no—
217C4#1915383612yes—
218C4#1915384712yes—
219C4#1915386212yes—
220C2—41no—
221C2—14482no—
222C2#2604184311yescharacteristic
223C2C2×#1915383611yescharacteristic
224C2C2×#1915384711yescharacteristic
subgroupG/H
centerC22C22×C22⋊D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#533×C181C2
socleC22×C181C22×D4
prime radical#26041829C2
hypercenter#533D181
hypocenterC181#3210
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C2×C42 ◁ #533

Complements

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

NH ≅ G/Ntype
2. #26041658C2semidirect
3. #26041653C2semidirect
4. #26041631C2semidirect
5. #26041627C2semidirect
6. #26041501C2semidirect
7. #26041439C2semidirect
8. #26041332C2semidirect
9. #533×C181C2semidirect
112. C181#3210semidirect
114. #533D181semidirect

Maximal subgroups

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

subgroupindex
#260418292
#260416582
#260416532
#260416312
#260416272
#260415012
#260414392
#260413322
#533×C1812
#3210181