Finite Group Databasebeta

← #37542020

Subgroups — #37542020

Order 23168.

696 conjugacy classes of subgroups; only the 177 normal subgroups are shown.

normal subgroups only

This group has 696 conjugacy classes of subgroups — too many to draw. Only its 177 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×#19153803C211yesmaximal, characteristic, complemented
2C2×#19153795C211yesmaximal, characteristic, complemented
3C22×C4×Dic181C211yesmaximal, characteristic, complemented
4#26041184C211yesmaximal, characteristic, complemented
5#26041182C211yesmaximal, characteristic, complemented
6#26041173C214yesmaximal, complemented
7#26041159C214yesmaximal, complemented
8#26041144C211yesmaximal, characteristic, complemented
9C2×#121×C181C211yesmaximal, characteristic
10C22×Dic362C2211yescharacteristic
11C23×Dic181C2212yes—
12#19153803—24no—
13#19153795—24no—
14C2×C4⋊Dic181C2211yescharacteristic
15C2×#15668428C2212yes—
16C2×C22⋊Dic181C2212yes—
17C2×C22⋊Dic181C2214yes—
18C2×C22⋊Dic181C2214yes—
19C2×C4×Dic181—24no—
20C2×C4×Dic181C2212yes—
21C2×C4×Dic181C2211yescharacteristic
22C2×C4×Dic181C2211yescharacteristic
23#19153723C2214yes—
24#19153723C2214yes—
25C22×D4×C181C2212yes—
26C22×D4×C181C2211yescharacteristic
27C23×C4×C181C2211yescharacteristic
28#121×C181—24no—
29C2×C4⋊C4×C181C2211yescharacteristic
30C2×C22⋊C4×C181C2212yes—
31C2×Dic362—22no—
32C2×Dic362—22no—
33C2×Dic362—22no—
34C22×Dic181—24no—
35C22×Dic181—24no—
36C22×Dic181—24no—
37C22×Dic181C2312yes—
38C22×Dic181C2312yes—
39C22×Dic181C2312yes—
40C22×Dic181C2312yes—
41C4⋊Dic181—22no—
42#15668428—24no—
43C22⋊Dic181—28no—
44C22⋊Dic181—28no—
45C22⋊Dic181—24no—
46C4×Dic181—24no—
47C4×Dic181—22no—
48C4×Dic181D414yescomplemented
49C24×C181C2312yes—
50C24×C181C2311yescharacteristic
51C2×D4×C181—24no—
52C2×D4×C181—28no—
53C2×D4×C181—22no—
54C2×D4×C181—22no—
55C2×D4×C181—22no—
56C22×C4×C181—22no—
57C22×C4×C181—22no—
58C22×C4×C181D414yes—
59C22×C4×C181C2312yes—
60C22×C4×C181C2311yescharacteristic
61C22×C4×C181C2311yescharacteristic
62C4⋊C4×C181—22no—
63C22⋊C4×C181—24no—
64Dic362—44no—
65C2×Dic181—216no—
66C2×Dic181—22no—
67C2×Dic181—22no—
68C2×Dic181—22no—
69C2×Dic181—22no—
70C2×Dic181—22no—
71C2×Dic181—22no—
72C2×Dic181—22no—
73C2×Dic181—24no—
74C2×Dic181—22no—
75C2×Dic181C2×D414yes—
76C2×Dic181#4114yes—
77C23×C181—21no—
78C23×C181—21no—
79C23×C181—22no—
80C23×C181—22no—
81C23×C181—22no—
82C23×C181—22no—
83C23×C181—24no—
84C23×C181—24no—
85C23×C181C2×D412yes—
86C23×C181#4114yes—
87C23×C181C2411yescharacteristic
88D4×C181—48no—
89D4×C181—44no—
90C2×C4×C181—21no—
91C2×C4×C181—21no—
92C2×C4×C181—28no—
93C2×C4×C181—22no—
94C2×C4×C181—24no—
95C2×C4×C181C2×D412yes—
96C2×C4×C181C2×D412yes—
97C2×C4×C181C2×D412yes—
98C2×C4×C181#4112yes—
99C22×C181—22no—
100C22×C181—24no—
101C22×C181—24no—
102C22×C181—22no—
103C22×C181—42no—
104C22×C181—44no—
105C22×C181—42no—
106C22×C181—44no—
107C22×C181—42no—
108C22×C181#12414yescomplemented
109C22×C181C22×D411yescharacteristic
110C22×C181C22×D411yescharacteristic
111C22×C181C2×#4112yes—
112C22×C181C2×#4111yescharacteristic
113C22×C181C2×#4112yes—
114C4×C181—22no—
115C4×C181—42no—
116C4×C181#12114yes—
117Dic181—24no—
118Dic181—24no—
119Dic181—42no—
120Dic181—42no—
121C2×C181—24no—
122C2×C181—82no—
123C2×C181C2×#12111yescharacteristic
124C2×C181#52512yes—
125C2×C181C2×#12411yescharacteristic
126C2×C181#52912yes—
127C2×C181#53211yescharacteristic
128C181#285011yescharacteristic, complemented, Sylow
129#2850—1811nomaximal, Sylow
130C2×#124—1811no—
131C2×C22⋊Q8—1811no—
132C2×#121D18111yescharacteristic
133C22×C42—1811no—
134#391—1811no—
135#390—1811no—
136#388—1814no—
137#386—1814no—
138#382—1811no—
139C22×Q8—1811no—
140C22×D4C2×D18111yescharacteristic
141C22×D4C2×D18112yes—
142C23×C4—1812no—
143C23×C4C2×D18111yescharacteristic
144#124—3624no—
145C22⋊Q8—3624no—
146#121—24no—
147C2×C4⋊C4—1812no—
148C2×C4⋊C4—1811no—
149C2×C4⋊C4C2×D18111yescharacteristic
150C2×C22⋊C4—1812no—
151C2×C22⋊C4—1814no—
152C2×C22⋊C4—1814no—
153C2×C22⋊C4C2×D18112yes—
154C2×C42—3624no—
155C2×C42—1812no—
156C2×C42—1811no—
157C2×C42—1811no—
158#95—1814no—
159#95—1814no—
160C24C22×D18112yes—
161C24C22×D18111yescharacteristic
162C2×Q8—3622no—
163C2×Q8—3622no—
164C2×Q8—3622no—
165C2×D4—24no—
166C2×D4—28no—
167C2×D4—22no—
168C2×D4—22no—
169C2×D4—22no—
170C22×C4—22no—
171C22×C4—22no—
172C22×C4—3624no—
173C22×C4—3624no—
174C22×C4—3624no—
175C22×C4—1812no—
176C22×C4—1812no—
177C22×C4—1812no—
178C22×C4—1812no—
179C22×C4C22⋊D18114yes—
180C22×C4C22×D18112yes—
181C22×C4C22×D18111yescharacteristic
182C22×C4C22×D18111yescharacteristic
183C4⋊C4—22no—
184C4⋊C4—3624no—
185C4⋊C4—3622no—
186C22⋊C4—24no—
187C22⋊C4—3624no—
188C22⋊C4—3628no—
189C22⋊C4—3628no—
190C42—3624no—
191C42—3622no—
192C42—1814no—
193C23—21no—
194C23—22no—
195C23—21no—
196C23—22no—
197C23—22no—
198C23—24no—
199C23—22no—
200C23—24no—
201C23C2×C22⋊D18112yes—
202C23#1566844814yes—
203C23C23×D18111yescharacteristic
204Q8—7244no—
205D4—48no—
206D4—44no—
207C2×C4—21no—
208C2×C4—21no—
209C2×C4—28no—
210C2×C4—22no—
211C2×C4—24no—
212C2×C4—36216no—
213C2×C4—3622no—
214C2×C4—3622no—
215C2×C4—3622no—
216C2×C4—3622no—
217C2×C4—3622no—
218C2×C4—3622no—
219C2×C4—3622no—
220C2×C4—3624no—
221C2×C4—3622no—
222C2×C4—1814no—
223C2×C4—1814no—
224C2×C4D4×D18112yes—
225C2×C4C2×C22⋊D18112yes—
226C2×C4C2×C22⋊D18112yes—
227C2×C4#1566844812yes—
228C22—22no—
229C22—24no—
230C22—24no—
231C22—22no—
232C22—42no—
233C22—44no—
234C22—42no—
235C22—44no—
236C22—42no—
237C22#1915380314yescomplemented
238C22C2×D4×D18111yescharacteristic
239C22C22×C22⋊D18111yescharacteristic
240C22#41×D18112yes—
241C22C2×#1566844811yescharacteristic
242C22C2×#1566844812yes—
243C4—22no—
244C4—42no—
245C4—3624no—
246C4—3624no—
247C4—7242no—
248C4—7242no—
249C4#1915378814yes—
250C2—24no—
251C2—82no—
252C2C2×#1915378811yescharacteristic
253C2#2604178912yes—
254C2C2×#1915380311yescharacteristic
255C2#2604181012yes—
256C2#2604182511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D181
commutator, socle, cosocleC23×C181C24
Fitting, prime radicalC2×#121×C181C2
hypercenterC2×#121D181
hypocenterC181#2850
LCSG ▷ C23×C181 ▷ C181
UCSC1 ◁ C23 ◁ C2×#121

Complements

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

NH ≅ G/Ntype
1. C2×#19153803C2semidirect
2. C2×#19153795C2semidirect
3. C22×C4×Dic181C2semidirect
4. #26041184C2semidirect
5. #26041182C2semidirect
6. #26041173C2semidirect
7. #26041159C2semidirect
8. #26041144C2semidirect
48. C4×Dic181D4semidirect
108. C22×C181#124semidirect
128. C181#2850semidirect
237. C22#19153803semidirect

Maximal subgroups

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

subgroupindex
C2×#191538032
C2×#191537952
C22×C4×Dic1812
#260411842
#260411822
#260411732
#260411592
#260411442
C2×#121×C1812
#2850181