Finite Group Databasebeta

← #37542183

Subgroups — #37542183

Order 23168.

716 conjugacy classes of subgroups; only the 209 normal subgroups are shown.

normal subgroups only

This group has 716 conjugacy classes of subgroups — too many to draw. Only its 209 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
1C2×#19153863C211yesmaximal, characteristic
2#26041756C212yesmaximal, complemented
3#26041253C211yesmaximal, characteristic, complemented
4#26041252C212yesmaximal, complemented
5#26041251C218yesmaximal, complemented
6#410×C181C211yesmaximal, characteristic, complemented
7#19153863—24no—
8#19153863—24no—
9C2×#15668449C412yescomplemented
10C2×#15668449C2211yescharacteristic
11C2×#15668448C412yescomplemented
12C2×#15668448C2214yes—
13C22×Dic362C412yescomplemented
14C22×Dic362C2211yescharacteristic
15C2×Q8×D181C412yescomplemented
16C4×Dic362—24no—
17#19153781—24no—
18#19153775C2212yes—
19#19153772C2218yes—
20#19153731C2214yes—
21#19153730C2218yes—
22C2×#41×C181C2211yescharacteristic
23#117×C181C2212yes—
24#99×C181C2214yes—
25#15668449—24no—
26#15668449—24no—
27#15668449—22no—
28#15668449—22no—
29#15668448—28no—
30#15668448—216no—
31C2×Dic362—24no—
32C2×Dic362—24no—
33C2×Dic362C2×C414yes—
34C2×Dic362C2×C411yescharacteristic
35C2×Dic362C2×C412yes—
36C2×Dic362C2×C412yes—
37C2×Dic362D414yes—
38C2×Dic362D414yes—
39Q8×D181—28no—
40C2×D724C2×C411yescharacteristic
41C2×C22⋊D181C2×C414yes—
42C2×C22⋊D181C2312yes—
43C22×Dic181C2×C414yes—
44C22×Dic181C2312yes—
45C2×C4×D181C2×C412yes—
46C2×C4×D181C2×C414yes—
47C4⋊Dic181—22no—
48#15668428—28no—
49C22⋊Dic181C2314yes—
50C4×Dic181—28no—
51C4×Dic181—22no—
52#41×C181—22no—
53#41×C181—22no—
54C2×Q8×C181C2×C411yescharacteristic
55C2×D4×C181C2×C411yescharacteristic
56C2×D4×C181C2312yes—
57C22×C4×C181C2×C412yes—
58C22×C4×C181C2311yescharacteristic
59C4⋊C4×C181—22no—
60C22⋊C4×C181C2314yes—
61C42×C181—22no—
62C22×D181C22×C412yes—
63Dic362—21no—
64Dic362—21no—
65Dic362—28no—
66Dic362C22⋊C418yes—
67Dic362C22⋊C414yes—
68Dic362C22⋊C414yes—
69D724—21no—
70D724—21no—
71C22⋊D181—24no—
72C22⋊D181—28no—
73C2×Dic181—28no—
74C2×Dic181—24no—
75C2×Dic181C22×C412yes—
76C2×Dic181C22×C414yes—
77C2×Dic181C2×D418yes—
78C4×D181—24no—
79C4×D181—28no—
80C23×C181C22×C412yes—
81C23×C181C2411yescharacteristic
82Q8×C181—22no—
83D4×C181—24no—
84D4×C181—22no—
85C2×C4×C181—22no—
86C2×C4×C181—22no—
87C2×C4×C181—24no—
88C2×C4×C181C22×C412yes—
89C2×C4×C181C22×C411yescharacteristic
90C2×C4×C181C22×C411yescharacteristic
91C2×C4×C181C2×D412yes—
92C2×C4×C181C2×D412yes—
93C22×C181—22no—
94C22×C181—41no—
95C22×C181—42no—
96C22×C181C23×C411yescharacteristic
97C22×C181C22×D412yes—
98C2×D181—22no—
99C2×D181—42no—
100C4×C181—22no—
101C4×C181—44no—
102C4×C181C2×C22⋊C411yescharacteristic
103C4×C181C2×C22⋊C411yescharacteristic
104C4×C181C2×C22⋊C412yes—
105Dic181—22no—
106Dic181—44no—
107Dic181C2×C22⋊C418yes—
108C2×C181—22no—
109C2×C181—21no—
110C2×C181—42no—
111C2×C181C22×C22⋊C411yescharacteristic
112D181—42no—
113C181#288411yescharacteristic, complemented, Sylow
114#2884—1811nomaximal, Sylow
115C2×2−1+4—1811no—
116#519—1812no—
117#410—1811no—
118#410—1812no—
119#410—1818no—
120#410D18111yescharacteristic, complemented
1212−1+4—3624no—
1222−1+4—3624no—
123C2×#41—1812no—
124C2×#41—1814no—
125C2×#41—1812no—
126C2×#41—1811no—
127C2×#41C2×D18111yescharacteristic
128C22×Q8—1812no—
129C22×Q8—1812no—
130C22×Q8—1811no—
131C4×Q8—3624no—
132C4×Q8—3624no—
133#117—1812no—
134#117—1818no—
135#117C2×D18112yes—
136#99—1814no—
137#99—1818no—
138#99C2×D18114yes—
139#41—22no—
140#41—22no—
141#41—3624no—
142#41—3624no—
143#41—3628no—
144#41—36216no—
145#41—3622no—
146#41—3622no—
147C2×Q8—3624no—
148C2×Q8—3624no—
149C2×Q8—3628no—
150C2×Q8—1811no—
151C2×Q8—1814no—
152C2×Q8—1812no—
153C2×Q8—1812no—
154C2×Q8—1814no—
155C2×Q8—1814no—
156C2×Q8C4×D18111yescharacteristic
157C2×D4—1814no—
158C2×D4—1812no—
159C2×D4—1811no—
160C2×D4C4×D18111yescharacteristic
161C2×D4C22×D18112yes—
162C22×C4—1814no—
163C22×C4—1814no—
164C22×C4—1812no—
165C22×C4—1812no—
166C22×C4C4×D18112yes—
167C22×C4C22×D18111yescharacteristic
168C4⋊C4—22no—
169C4⋊C4—3628no—
170C4⋊C4—3622no—
171C22⋊C4—1814no—
172C22⋊C4C22×D18114yes—
173C42—22no—
174C42—3628no—
175C42—3622no—
176C23—1812no—
177C23C2×C4×D18112yes—
178C23C23×D18111yescharacteristic
179Q8—22no—
180Q8—3628no—
181Q8—3621no—
182Q8—3621no—
183Q8—1814no—
184Q8—1814no—
185Q8—1818no—
186D4—24no—
187D4—22no—
188D4—3624no—
189D4—3628no—
190D4—3621no—
191D4—3621no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—24no—
195C2×C4—3628no—
196C2×C4—3628no—
197C2×C4—3624no—
198C2×C4—3624no—
199C2×C4—1812no—
200C2×C4—1814no—
201C2×C4—1818no—
202C2×C4C2×C4×D18112yes—
203C2×C4C2×C4×D18111yescharacteristic
204C2×C4C2×C4×D18111yescharacteristic
205C2×C4D4×D18112yes—
206C2×C4D4×D18112yes—
207C22—22no—
208C22—41no—
209C22—42no—
210C22—3622no—
211C22—7242no—
212C22C22×C4×D18111yescharacteristic
213C22C2×D4×D18112yes—
214C4—22no—
215C4—44no—
216C4—3622no—
217C4—1818no—
218C4—7244no—
219C4C22⋊C4×D18111yescharacteristic
220C4C22⋊C4×D18111yescharacteristic
221C4C22⋊C4×D18112yes—
222C2—22no—
223C2—21no—
224C2—42no—
225C2—7242no—
226C2C2×C22⋊C4×D18111yescharacteristic
subgroupG/H
centerC2C2×C22⋊C4×D181
commutatorC22×C181C23×C4
FrattiniC23C23×D181
Fitting#410×C181C2
socleC2×C181C22×C22⋊C4
prime radicalC2×#19153863C2
cosocleC23×C181C24
hypercenter#410D181
hypocenterC181#2884
LCSG ▷ C22×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C2 ◁ C22×C4 ◁ #410

Complements

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

NH ≅ G/Ntype
2. #26041756C2semidirect
3. #26041253C2semidirect
4. #26041252C2semidirect
5. #26041251C2semidirect
6. #410×C181C2semidirect
9. C2×#15668449C4semidirect
11. C2×#15668448C4semidirect
13. C22×Dic362C4semidirect
15. C2×Q8×D181C4semidirect
113. C181#2884semidirect
120. #410D181semidirect

Maximal subgroups

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

subgroupindex
C2×#191538632
#260417562
#260412532
#260412522
#260412512
#410×C1812
#2884181