Finite Group Databasebeta

← #37544674

Subgroups — #37544674

Order 23168.

394 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 117 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#26041834C211yesmaximal, characteristic, complemented
2#26041830C211yesmaximal, characteristic
3#26041653C211yesmaximal, characteristic, complemented
4#26041629C211yesmaximal, characteristic, complemented
5#26041626C211yesmaximal, characteristic, complemented
6#26041503C214yesmaximal, complemented
7#26041440C214yesmaximal, complemented
8#26041330C211yesmaximal, characteristic, complemented
9#494×C181C211yesmaximal, characteristic, complemented
10C2×#15668450C2212yes—
11C2×#15668436C2214yes—
12#19153816C2211yescharacteristic
13#19153813C2211yescharacteristic
14#19153807—21no—
15#19153805C2212yes—
16#19153796—21no—
17#19153792C2212yes—
18#19153791—21no—
19#19153791—21no—
20Q8×Dic181C2212yes—
21#19153781—21no—
22#19153778—21no—
23#19153778—21no—
24#19153774—21no—
25C42×D181C2211yescharacteristic
26#19153745C2212yes—
27#19153740C2214yes—
28#19153738C2214yes—
29#19153728C2212yes—
30C4×#13089678C2211yescharacteristic
31C2×Q16×C181C2214yes—
32C4⋊Q8×C181C2211yescharacteristic
33C4⋊Q8×C181C2211yescharacteristic
34C4×C8×C181C2211yescharacteristic
35#15668450—44no—
36C2×Dic362—21no—
37C2×D724—22no—
38C2×D724—21no—
39C2×D724—21no—
40C2×D724C2312yes—
41C2×C4×D181—22no—
42C2×C4×D181D411yescharacteristic
43C2×C4×D181D411yescharacteristic
44C2×C4×D181C2311yescharacteristic
45#15668436—44no—
46C2×#13089678C2312yes—
47C4⋊Dic181—22no—
48C4⋊Dic181—21no—
49C4⋊Dic181C2312yes—
50#15668428—22no—
51#15668426—22no—
52#15668426—22no—
53#15668426—22no—
54C4×Dic181—22no—
55C4×Dic181D411yescharacteristic
56C4×Dic181D411yescharacteristic
57C4×Dic181C2311yescharacteristic
58C2×Q8×C181C2312yes—
59C2×Q8×C181C2312yes—
60Q16×C181—44no—
61C2×C8×C181C2312yes—
62C4⋊C4×C181—21no—
63C4⋊C4×C181—21no—
64C4⋊C4×C181—21no—
65C4⋊C4×C181—21no—
66C42×C181C2311yescharacteristic
67C22×D181—22no—
68C22×D181C2×D411yescharacteristic
69Dic362—42no—
70D724—24no—
71D724—44no—
72D724—42no—
73D724—42no—
74C2×Dic181—22no—
75C2×Dic181C2×D411yescharacteristic
76C2×Dic181C2×D411yescharacteristic
77C2×Dic181C2×D411yescharacteristic
78C4×D181—22no—
79C4×D181—22no—
80C4×D181—22no—
81C4×D181—44no—
82#13089678—22no—
83Q8×C181—24no—
84Q8×C181—24no—
85C2×C4×C181—22no—
86C2×C4×C181—22no—
87C2×C4×C181C2×D411yescharacteristic
88C2×C4×C181C2×D411yescharacteristic
89C2×C4×C181C2411yescharacteristic
90C8×C181—22no—
91C22×C181C22×D411yescharacteristic
92C2×D181—21no—
93C2×D181—21no—
94C2×D181—21no—
95C2×D181—42no—
96C2×D181—42no—
97C2×D181—42no—
98C4×C181—42no—
99C4×C181—42no—
100C4×C181#13512yes—
101C4×C181#13712yes—
102C4×C181H(5,𝔽2)12yes—
103Dic181—21no—
104Dic181—21no—
105Dic181—21no—
106Dic181—42no—
107C2×C181#53511yescharacteristic
108C2×C181C2×#13511yescharacteristic
109C2×C181C2×#13711yescharacteristic
110D181—41no—
111D181—82no—
112C181#321011yescharacteristic, complemented, Sylow
113#3210—1811nomaximal, Sylow
114#533—1811no—
115#532—1811no—
116#494D18111yescharacteristic, complemented
117#492—1811no—
118#487—1811no—
119#486—1811no—
120#464—1814no—
121#452—1814no—
122#431—1811no—
123C2×#41—1812no—
124C2×Q16C2×D18114yes—
125C2×SD16—1814no—
126C4⋊Q8—1811no—
127C4⋊Q8C2×D18111yescharacteristic
128C4⋊Q8C2×D18111yescharacteristic
129#127—1811no—
130#125—3621no—
131#124—1812no—
132C22⋊Q8—3621no—
133C22⋊Q8—1812no—
134#121—3621no—
135#121—3621no—
136C4×Q8—3621no—
137C4×Q8—1812no—
138C4×D4—3621no—
139C4×D4—3621no—
140#117—3621no—
141C2×C42—1811no—
142C4⋊C8—1812no—
143#103—1814no—
144#102—1814no—
145C22⋊C8—1812no—
146C4×C8—1811no—
147C4×C8C2×D18111yescharacteristic
148#41—7244no—
149C2×Q8—3621no—
150C2×Q8C22×D18112yes—
151C2×Q8C22×D18112yes—
152C2×D4—3622no—
153C2×D4—3621no—
154C2×D4—3621no—
155C2×D4—1812no—
156C22×C4—3622no—
157C22×C4—1811no—
158C22×C4—1811no—
159C22×C4—1811no—
160Q16—44no—
161SD16—7244no—
162C2×C8—1812no—
163C2×C8C22×D18112yes—
164C4⋊C4—21no—
165C4⋊C4—21no—
166C4⋊C4—21no—
167C4⋊C4—21no—
168C4⋊C4—3622no—
169C4⋊C4—3622no—
170C4⋊C4—3621no—
171C4⋊C4—1812no—
172C22⋊C4—3622no—
173C22⋊C4—3622no—
174C22⋊C4—3622no—
175C42—3622no—
176C42—1811no—
177C42—1811no—
178C42—1811no—
179C42C22×D18111yescharacteristic
180C23—3622no—
181C23—1811no—
182Q8—24no—
183Q8—24no—
184Q8—7242no—
185D4—3624no—
186D4—7244no—
187D4—7242no—
188D4—7242no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—3622no—
192C2×C4—3622no—
193C2×C4—3622no—
194C2×C4—3622no—
195C2×C4—1811no—
196C2×C4—1811no—
197C2×C4—1811no—
198C2×C4—7244no—
199C2×C4D4×D18111yescharacteristic
200C2×C4D4×D18111yescharacteristic
201C2×C4C23×D18111yescharacteristic
202C8—22no—
203C8—3622no—
204C22—3621no—
205C22—3621no—
206C22—3621no—
207C22—7242no—
208C22—7242no—
209C22—7242no—
210C22C2×D4×D18111yescharacteristic
211C4—42no—
212C4—42no—
213C4—3621no—
214C4—3621no—
215C4—3621no—
216C4—7242no—
217C4#1915383912yes—
218C4#1915384812yes—
219C4#1915386112yes—
220C2—14482no—
221C2—7241no—
222C2#2604184211yescharacteristic
223C2C2×#1915383911yescharacteristic
224C2C2×#1915384811yescharacteristic
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#494×C181C2
socleC22×C181C22×D4
prime radical#26041830C2
hypercenter#494D181
hypocenterC181#3210
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C42 ◁ #494

Complements

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

NH ≅ G/Ntype
1. #26041834C2semidirect
3. #26041653C2semidirect
4. #26041629C2semidirect
5. #26041626C2semidirect
6. #26041503C2semidirect
7. #26041440C2semidirect
8. #26041330C2semidirect
9. #494×C181C2semidirect
112. C181#3210semidirect
116. #494D181semidirect

Maximal subgroups

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

subgroupindex
#260418342
#260418302
#260416532
#260416292
#260416262
#260415032
#260414402
#260413302
#494×C1812
#3210181