Finite Group Databasebeta

← #37542065

Subgroups — #37542065

Order 23168.

924 conjugacy classes of subgroups; only the 189 normal subgroups are shown.

normal subgroups only

This group has 924 conjugacy classes of subgroups — too many to draw. Only its 189 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×#19153812C211yesmaximal, characteristic, complemented
2C2×#19153791C211yesmaximal, characteristic, complemented
3C2×#19153789C214yesmaximal, complemented
4C2×#19153787C211yesmaximal, characteristic, complemented
5C2×C42×D181C211yesmaximal, characteristic, complemented
6#26041181C211yesmaximal, characteristic, complemented
7#26041179C214yesmaximal, complemented
8#26041148C211yesmaximal, characteristic, complemented
9#390×C181C211yesmaximal, characteristic, complemented
10C22×D724C2211yescharacteristic
11C22×D724C2212yes—
12C22×C22⋊D181C2214yes—
13C22×C22⋊D181C2212yes—
14C22×C4×D181C2211yescharacteristic
15C22×C4×D181C2212yes—
16#19153812—24no—
17#19153791—24no—
18#19153789—216no—
19#19153787—24no—
20C2×C4⋊Dic181C2211yescharacteristic
21C2×#15668428C2214yes—
22C2×C22⋊Dic181C2212yes—
23C2×#15668426C2212yes—
24C2×#15668426C2214yes—
25C2×C4×Dic181C2212yes—
26C2×C4×Dic181C2211yescharacteristic
27C42×D181—24no—
28C22×D4×C181C2211yescharacteristic
29C2×C4⋊C4×C181C2211yescharacteristic
30C2×C22⋊C4×C181C2214yes—
31C2×C42×C181C2211yescharacteristic
32C23×D181C2312yes—
33C23×D181C2311yescharacteristic
34C2×D724—28no—
35C2×D724—24no—
36C2×D724—22no—
37C2×D724—22no—
38C2×D724—22no—
39C2×C22⋊D181—24no—
40C2×C22⋊D181—28no—
41C2×C22⋊D181—28no—
42C2×C22⋊D181—28no—
43C2×C22⋊D181—28no—
44C22×Dic181C2312yes—
45C22×Dic181C2311yescharacteristic
46C22×Dic181C2312yes—
47C2×C4×D181—24no—
48C2×C4×D181—24no—
49C2×C4×D181—22no—
50C2×C4×D181—22no—
51C2×C4×D181D414yescomplemented
52C2×C4×D181D418yescomplemented
53C4⋊Dic181—22no—
54#15668428—28no—
55C22⋊Dic181—24no—
56#15668426—24no—
57#15668426—28no—
58C4×Dic181—24no—
59C4×Dic181D414yescomplemented
60C24×C181C2312yes—
61C2×D4×C181—22no—
62C2×D4×C181—22no—
63C2×D4×C181—22no—
64C22×C4×C181C2312yes—
65C22×C4×C181C2312yes—
66C22×C4×C181C2311yescharacteristic
67C4⋊C4×C181—22no—
68C22⋊C4×C181—28no—
69C42×C181—22no—
70C22×D181—21no—
71C22×D181—22no—
72C22×D181—21no—
73C22×D181—24no—
74C22×D181—22no—
75C22×D181—22no—
76C22×D181—24no—
77C22×D181—22no—
78C22×D181C2×D412yes—
79C22×D181C2×D414yes—
80D724—48no—
81D724—44no—
82C22⋊D181—48no—
83C22⋊D181—416no—
84C2×Dic181—21no—
85C2×Dic181—21no—
86C2×Dic181—24no—
87C2×Dic181—22no—
88C2×Dic181C2×D412yes—
89C2×Dic181C2×D414yes—
90C2×Dic181C2×D414yes—
91C2×Dic181#4114yes—
92C4×D181—28no—
93C4×D181—216no—
94C23×C181—24no—
95C23×C181—22no—
96C23×C181—22no—
97C23×C181—22no—
98C23×C181—24no—
99C23×C181C2411yescharacteristic
100D4×C181—44no—
101C2×C4×C181—22no—
102C2×C4×C181—22no—
103C2×C4×C181—24no—
104C2×C4×C181—22no—
105C2×C4×C181C2×D414yes—
106C2×C4×C181C2×D412yes—
107C2×C4×C181C2×D412yes—
108C2×C4×C181#4112yes—
109C22×C181—42no—
110C22×C181—44no—
111C22×C181—42no—
112C22×C181—44no—
113C22×C181—42no—
114C22×C181C22×D411yescharacteristic
115C22×C181C22×D412yes—
116C22×C181C22×D411yescharacteristic
117C22×C181C2×#4112yes—
118C22×C181C2×#4111yescharacteristic
119C2×D181—24no—
120C2×D181—22no—
121C2×D181—24no—
122C2×D181—22no—
123C2×D181—44no—
124C2×D181—42no—
125C2×D181—42no—
126C2×D181—44no—
127C2×D181—42no—
128C2×D181#12714yescomplemented
129C4×C181—24no—
130C4×C181—42no—
131C4×C181#12114yescomplemented
132Dic181—22no—
133Dic181—42no—
134Dic181#12118yescomplemented
135C2×C181—82no—
136C2×C181C2×#12112yes—
137C2×C181C2×#12111yescharacteristic
138C2×C181C2×#12711yescharacteristic
139C2×C181#53212yes—
140C2×C181#53211yescharacteristic
141D181—24no—
142D181—82no—
143C181#285611yescharacteristic, complemented, Sylow
144#2856—1811nomaximal, Sylow
145C2×#127—1811no—
146C2×#121—1811no—
147C2×#121—1811no—
148C2×#121—1814no—
149C22×C42—1811no—
150#390—1814no—
151#390—1811no—
152#390D18111yescharacteristic, complemented
153#384—1811no—
154C22×D4—1812no—
155C22×D4—1814no—
156C22×D4—1812no—
157C22×D4—1811no—
158C22×D4C2×D18111yescharacteristic
159C23×C4—1812no—
160C23×C4—1811no—
161#127—3624no—
162#121—36216no—
163#121—3624no—
164#121—3624no—
165C2×C4⋊C4—1814no—
166C2×C4⋊C4—1811no—
167C2×C4⋊C4C2×D18111yescharacteristic
168C2×C22⋊C4—1812no—
169C2×C22⋊C4—1812no—
170C2×C22⋊C4—1814no—
171C2×C22⋊C4C2×D18114yes—
172C2×C42—3624no—
173C2×C42—1812no—
174C2×C42—1811no—
175C2×C42C2×D18111yescharacteristic
176C24—1812no—
177C24—1811no—
178C24C22×D18112yes—
179C2×D4—22no—
180C2×D4—22no—
181C2×D4—22no—
182C2×D4—3628no—
183C2×D4—3624no—
184C2×D4—3628no—
185C2×D4—3624no—
186C2×D4—3628no—
187C2×D4—3628no—
188C2×D4—3628no—
189C2×D4—3622no—
190C2×D4—3622no—
191C2×D4—3622no—
192C22×C4—3624no—
193C22×C4—3624no—
194C22×C4—3622no—
195C22×C4—3622no—
196C22×C4—1812no—
197C22×C4—1811no—
198C22×C4—1818no—
199C22×C4—1814no—
200C22×C4—1812no—
201C22×C4C22×D18112yes—
202C22×C4C22×D18112yes—
203C22×C4C22×D18111yescharacteristic
204C4⋊C4—22no—
205C4⋊C4—3628no—
206C4⋊C4—3622no—
207C22⋊C4—28no—
208C22⋊C4—3624no—
209C22⋊C4—3624no—
210C22⋊C4—3628no—
211C42—22no—
212C42—3624no—
213C42—1814no—
214C23—24no—
215C23—22no—
216C23—22no—
217C23—24no—
218C23—22no—
219C23—3621no—
220C23—3622no—
221C23—3621no—
222C23—3624no—
223C23—3622no—
224C23—3622no—
225C23—3624no—
226C23—3622no—
227C23—1814no—
228C23—1812no—
229C23C23×D18111yescharacteristic
230D4—44no—
231D4—7248no—
232D4—72416no—
233D4—7248no—
234D4—7244no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—24no—
238C2×C4—22no—
239C2×C4—36216no—
240C2×C4—3621no—
241C2×C4—3621no—
242C2×C4—3628no—
243C2×C4—3624no—
244C2×C4—3622no—
245C2×C4—1812no—
246C2×C4—1814no—
247C2×C4—1814no—
248C2×C4—1814no—
249C2×C4D4×D18114yes—
250C2×C4D4×D18112yes—
251C2×C4D4×D18112yes—
252C2×C4#1566844912yes—
253C22—44no—
254C22—42no—
255C22—42no—
256C22—44no—
257C22—42no—
258C22—3624no—
259C22—3622no—
260C22—3624no—
261C22—3622no—
262C22—1814no—
263C22—7244no—
264C22—7242no—
265C22—7242no—
266C22—7244no—
267C22—7242no—
268C22C2×D4×D18111yescharacteristic
269C22C2×D4×D18112yes—
270C22C2×D4×D18111yescharacteristic
271C22#41×D18112yes—
272C22C2×#1566844911yescharacteristic
273C4—24no—
274C4—42no—
275C4—3622no—
276C4—1818no—
277C4—7242no—
278C4#1915378914yescomplemented
279C2—82no—
280C2—3624no—
281C2—14482no—
282C2#121×D18112yes—
283C2C2×#1915378911yescharacteristic
284C2#127×D18111yescharacteristic
285C2#2604182612yes—
286C2#2604182811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D181
commutator, socle, cosocleC23×C181C24
Fitting#390×C181C2
hypercenter#390D181
hypocenterC181#2856
LCSG ▷ C23×C181 ▷ C181
UCSC1 ◁ C23 ◁ #390

Complements

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

NH ≅ G/Ntype
1. C2×#19153812C2semidirect
2. C2×#19153791C2semidirect
3. C2×#19153789C2semidirect
4. C2×#19153787C2semidirect
5. C2×C42×D181C2semidirect
6. #26041181C2semidirect
7. #26041179C2semidirect
8. #26041148C2semidirect
9. #390×C181C2semidirect
51. C2×C4×D181D4semidirect
52. C2×C4×D181D4semidirect
59. C4×Dic181D4semidirect
128. C2×D181#127semidirect
131. C4×C181#121semidirect
134. Dic181#121semidirect
143. C181#2856semidirect
152. #390D181semidirect
278. C4#19153789semidirect

Maximal subgroups

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

subgroupindex
C2×#191538122
C2×#191537912
C2×#191537892
C2×#191537872
C2×C42×D1812
#260411812
#260411792
#260411482
#390×C1812
#2856181