Finite Group Databasebeta

← #37542981

Subgroups — #37542981

Order 23168.

754 conjugacy classes of subgroups; only the 169 normal subgroups are shown.

normal subgroups only

This group has 754 conjugacy classes of subgroups — too many to draw. Only its 169 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×#19153846C212yesmaximal, complemented
2C2×#19153795C211yesmaximal, characteristic, complemented
3#26041441C214yesmaximal, complemented
4#26041435C212yesmaximal, complemented
5#26041430C212yesmaximal, complemented
6#26041280C212yesmaximal, complemented
7#26041243C211yesmaximal, characteristic, complemented
8C22×#41×C181C211yesmaximal, characteristic
9C22×Dic362C2211yescharacteristic, complemented
10#19153846—28no—
11C2×#15668439C2214yescomplemented
12C2×#15668437C2214yescomplemented
13C2×#15668433C2212yescomplemented
14#19153795—21no—
15#19153795—21no—
16#19153795C2212yes—
17#19153795C2212yescomplemented
18C2×C4⋊Dic181C2211yescharacteristic, complemented
19C2×#15668428—21no—
20C2×C22⋊Dic181—21no—
21#19153740C2214yescomplemented
22#19153737C2214yescomplemented
23C22⋊#13089678C2212yes—
24C22⋊#13089678C2212yescomplemented
25C2×#41×C181—28no—
26C2×#41×C181—21no—
27C2×#41×C181—21no—
28C2×#41×C181C2212yes—
29C2×#41×C181C2212yes—
30C22×Q8×C181C2211yescharacteristic
31C22×D4×C181—21no—
32C22×D4×C181C2211yescharacteristic
33C23×C4×C181—21no—
34C23×C4×C181C2211yescharacteristic
35C2×Dic362—21no—
36C2×Dic362—24no—
37C2×Dic362C2312yes—
38C22×Dic181—21no—
39C22×Dic181—21no—
40#15668439—28no—
41#15668437—28no—
42#15668433—24no—
43C2×#13089678C2314yes—
44C4⋊Dic181—21no—
45C4⋊Dic181C2312yes—
46#15668428—22no—
47#15668428—22no—
48C22⋊Dic181—22no—
49C22⋊Dic181—22no—
50C24×C181—21no—
51C24×C181D411yescharacteristic
52#41×C181—28no—
53#41×C181—28no—
54#41×C181—24no—
55#41×C181—24no—
56#41×C181—24no—
57#41×C181D418yes—
58C2×Q8×C181—24no—
59C2×Q8×C181—21no—
60C2×Q8×C181C2312yes—
61C2×D4×C181—28no—
62C2×D4×C181—22no—
63C2×D4×C181—22no—
64C2×D4×C181—24no—
65C2×D4×C181—21no—
66C2×D4×C181C2312yes—
67C22×C4×C181—28no—
68C22×C4×C181—21no—
69C22×C4×C181—22no—
70C22×C4×C181—22no—
71C22×C4×C181—24no—
72C22×C4×C181D411yescharacteristic
73C22×C4×C181D411yescharacteristic
74C22×C4×C181D411yescharacteristic
75C22×C4×C181C2311yescharacteristic
76C22×C4×C181C2311yescharacteristic
77C22×C4×C181C2311yescharacteristic
78Dic362—24no—
79Dic362—42no—
80C2×Dic181—22no—
81C2×Dic181—22no—
82C2×Dic181—42no—
83C2×Dic181—42no—
84#13089678—24no—
85C23×C181—24no—
86C23×C181—22no—
87C23×C181—22no—
88C23×C181—22no—
89C23×C181—41no—
90C23×C181—42no—
91C23×C181—42no—
92C23×C181C2×D411yescharacteristic
93C23×C181C2×D411yescharacteristic
94C23×C181C2×D411yescharacteristic
95Q8×C181—24no—
96Q8×C181—22no—
97Q8×C181C2×D414yes—
98D4×C181—28no—
99D4×C181—24no—
100D4×C181—28no—
101D4×C181—22no—
102D4×C181C2×D414yes—
103C2×C4×C181—24no—
104C2×C4×C181—28no—
105C2×C4×C181—22no—
106C2×C4×C181—22no—
107C2×C4×C181—22no—
108C2×C4×C181—28no—
109C2×C4×C181—22no—
110C2×C4×C181—22no—
111C2×C4×C181—22no—
112C2×C4×C181C2×D411yescharacteristic
113C2×C4×C181C2×D411yescharacteristic
114C2×C4×C181C2×D411yescharacteristic
115C2×C4×C181C2×D414yes—
116C2×C4×C181C2411yescharacteristic
117C22×C181—21no—
118C22×C181—21no—
119C22×C181—21no—
120C22×C181—21no—
121C22×C181—21no—
122C22×C181—24no—
123C22×C181—21no—
124C22×C181—21no—
125C22×C181—44no—
126C22×C181—48no—
127C22×C181—42no—
128C22×C181—42no—
129C22×C181#12012yes—
130C22×C181#13712yescomplemented
131C22×C181C22×D411yescharacteristic
132C4×C181—21no—
133C4×C181—24no—
134C4×C181—21no—
135C4×C181#12012yes—
136C4×C181C22×D412yes—
137Dic181—42no—
138Dic181—42no—
139C2×C181—22no—
140C2×C181—22no—
141C2×C181—41no—
142C2×C181—44no—
143C2×C181C2×#12011yescharacteristic
144C2×C181C2×#13711yescharacteristic
145C2×C181#57511yescharacteristic
146C181#300711yescharacteristic, complemented, Sylow
147#3007—1811nomaximal, Sylow
148C22×#41D18111yescharacteristic
149C2×#137—1812no—
150C2×C22⋊Q8—1811no—
151#452—1814no—
152#451—1812no—
153#450—1812no—
154#417—1812no—
155#407—1811no—
156C2×#41—28no—
157C2×#41—21no—
158C2×#41—21no—
159C2×#41C2×D18112yes—
160C2×#41C2×D18112yes—
161C22×Q8—1811no—
162C22×Q8C2×D18111yescharacteristic
163C22×D4—21no—
164C22×D4C2×D18111yescharacteristic
165C23×C4—21no—
166C23×C4C2×D18111yescharacteristic
167#137—3628no—
168C2×Q16—1814no—
169C2×SD16—1814no—
170C2×M16—1812no—
171C22⋊Q8—3621no—
172C22⋊Q8—3621no—
173C22⋊Q8—1812no—
174C22⋊Q8—1812no—
175C2×C4⋊C4—3621no—
176C2×C4⋊C4—1811no—
177C2×C22⋊C4—3621no—
178#103—1814no—
179#102—1814no—
180C22⋊C8—1812no—
181C22⋊C8—1812no—
182C24—21no—
183C24C22⋊D18111yescharacteristic
184#41—28no—
185#41—28no—
186#41—24no—
187#41—24no—
188#41—24no—
189#41C22⋊D18118yes—
190C2×Q8—24no—
191C2×Q8—21no—
192C2×Q8—3621no—
193C2×Q8—3624no—
194C2×Q8—1812no—
195C2×Q8C22×D18112yes—
196C2×D4—28no—
197C2×D4—22no—
198C2×D4—22no—
199C2×D4—24no—
200C2×D4—21no—
201C2×D4C22×D18112yes—
202C22×C4—28no—
203C22×C4—21no—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—24no—
207C22×C4—3621no—
208C22×C4—3621no—
209C22×C4C22⋊D18111yescharacteristic
210C22×C4C22⋊D18111yescharacteristic
211C22×C4C22⋊D18111yescharacteristic
212C22×C4C22×D18111yescharacteristic
213C22×C4C22×D18111yescharacteristic
214C22×C4C22×D18111yescharacteristic
215Q16—3628no—
216SD16—3628no—
217M16—3624no—
218C2×C8—1814no—
219C4⋊C4—3622no—
220C4⋊C4—3622no—
221C4⋊C4—3621no—
222C4⋊C4—1812no—
223C22⋊C4—3622no—
224C22⋊C4—3622no—
225C23—24no—
226C23—22no—
227C23—22no—
228C23—22no—
229C23—41no—
230C23—42no—
231C23—42no—
232C23C2×C22⋊D18111yescharacteristic
233C23C2×C22⋊D18111yescharacteristic
234C23C2×C22⋊D18111yescharacteristic
235Q8—24no—
236Q8—22no—
237Q8—3624no—
238Q8—7242no—
239Q8C2×C22⋊D18114yes—
240D4—28no—
241D4—24no—
242D4—28no—
243D4—22no—
244D4C2×C22⋊D18114yes—
245C2×C4—24no—
246C2×C4—28no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—28no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—3622no—
255C2×C4—3622no—
256C2×C4—7242no—
257C2×C4—7242no—
258C2×C4C2×C22⋊D18111yescharacteristic
259C2×C4C2×C22⋊D18111yescharacteristic
260C2×C4C2×C22⋊D18111yescharacteristic
261C2×C4C2×C22⋊D18114yes—
262C2×C4C23×D18111yescharacteristic
263C8—3624no—
264C22—21no—
265C22—21no—
266C22—21no—
267C22—21no—
268C22—21no—
269C22—24no—
270C22—21no—
271C22—21no—
272C22—44no—
273C22—48no—
274C22—42no—
275C22—42no—
276C22#1915378512yes—
277C22#1915384612yescomplemented
278C22C22×C22⋊D18111yescharacteristic
279C4—21no—
280C4—24no—
281C4—21no—
282C4—7242no—
283C4—7242no—
284C4#1915378512yes—
285C4C22×C22⋊D18112yes—
286C2—22no—
287C2—22no—
288C2—41no—
289C2—44no—
290C2C2×#1915378511yescharacteristic
291C2C2×#1915384611yescharacteristic
292C2#2604207711yescharacteristic
subgroupG/H
centerC22C22×C22⋊D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting, prime radicalC22×#41×C181C2
socleC22×C181C22×D4
hypercenterC22×#41D181
hypocenterC181#3007
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×#41

Complements

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

NH ≅ G/Ntype
1. C2×#19153846C2semidirect
2. C2×#19153795C2semidirect
3. #26041441C2semidirect
4. #26041435C2semidirect
5. #26041430C2semidirect
6. #26041280C2semidirect
7. #26041243C2semidirect
9. C22×Dic362C22semidirect
11. C2×#15668439C22semidirect
12. C2×#15668437C22semidirect
13. C2×#15668433C22semidirect
17. #19153795C22semidirect
18. C2×C4⋊Dic181C22semidirect
21. #19153740C22semidirect
22. #19153737C22semidirect
24. C22⋊#13089678C22semidirect
130. C22×C181#137semidirect
146. C181#3007semidirect
277. C22#19153846semidirect

Maximal subgroups

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

subgroupindex
C2×#191538462
C2×#191537952
#260414412
#260414352
#260414302
#260412802
#260412432
C22×#41×C1812
#3007181