Finite Group Databasebeta

← #37542979

Subgroups — #37542979

Order 23168.

754 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 754 conjugacy classes of subgroups — too many to draw. Only its 153 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
1C22×#15668449C211yesmaximal, characteristic
2C2×#19153847C212yesmaximal, complemented
3#26041439C214yesmaximal, complemented
4#26041434C212yesmaximal, complemented
5#26041429C212yesmaximal, complemented
6#26041278C212yesmaximal, complemented
7#26041243C211yesmaximal, characteristic, complemented
8C2×C22⋊Q8×C181C211yesmaximal, characteristic, complemented
9C2×#15668449—28no—
10C2×#15668449—21no—
11C2×#15668449—21no—
12C2×#15668449C2212yes—
13C2×#15668449C2212yes—
14C22×Dic362C2211yescharacteristic
15C22×D724C2211yescharacteristic
16C22×C22⋊D181—21no—
17C22×C4×D181—21no—
18#19153847—28no—
19C2×#15668439C2214yescomplemented
20C2×#15668436C2214yescomplemented
21C2×#15668433C2212yescomplemented
22#19153739C2214yescomplemented
23#19153736C2214yescomplemented
24C22⋊#13089678C2212yes—
25C22⋊#13089678C2212yescomplemented
26C22×Q8×C181C2211yescharacteristic, complemented
27C23×C4×C181C2211yescharacteristic
28C22⋊Q8×C181—21no—
29C22⋊Q8×C181—21no—
30C22⋊Q8×C181C2212yes—
31C22⋊Q8×C181C2212yescomplemented
32C2×C4⋊C4×C181—21no—
33C2×C4⋊C4×C181C2211yescharacteristic, complemented
34C2×C22⋊C4×C181—21no—
35C23×D181—21no—
36#15668449—28no—
37#15668449—28no—
38#15668449—24no—
39#15668449—24no—
40#15668449—24no—
41#15668449D418yes—
42C2×Dic362—24no—
43C2×Dic362—21no—
44C2×Dic362C2312yes—
45C2×D724—24no—
46C2×D724—21no—
47C2×D724C2312yes—
48C2×C22⋊D181—28no—
49C2×C22⋊D181—22no—
50C2×C22⋊D181—22no—
51C22×Dic181—21no—
52C2×C4×D181—28no—
53C2×C4×D181—22no—
54C2×C4×D181—22no—
55#15668439—28no—
56#15668436—28no—
57#15668433—24no—
58C2×#13089678C2314yes—
59C24×C181D411yescharacteristic
60C2×Q8×C181—21no—
61C2×Q8×C181—24no—
62C2×Q8×C181C2312yes—
63C22×C4×C181—24no—
64C22×C4×C181—21no—
65C22×C4×C181—21no—
66C22×C4×C181D411yescharacteristic
67C22×C4×C181D411yescharacteristic
68C22×C4×C181D411yescharacteristic
69C22×C4×C181C2311yescharacteristic
70C22×C4×C181C2311yescharacteristic
71C22×C4×C181C2311yescharacteristic
72C4⋊C4×C181—22no—
73C4⋊C4×C181—22no—
74C4⋊C4×C181—21no—
75C4⋊C4×C181C2312yes—
76C22⋊C4×C181—22no—
77C22⋊C4×C181—22no—
78C22×D181—24no—
79C22×D181—22no—
80C22×D181—42no—
81C22×D181—42no—
82Dic362—24no—
83Dic362—22no—
84Dic362C2×D414yes—
85D724—24no—
86D724—22no—
87D724C2×D414yes—
88C22⋊D181—28no—
89C22⋊D181—28no—
90C2×Dic181—24no—
91C2×Dic181—22no—
92C4×D181—28no—
93C4×D181—28no—
94#13089678—24no—
95C23×C181—22no—
96C23×C181—22no—
97C23×C181—41no—
98C23×C181C2×D411yescharacteristic
99C23×C181C2×D411yescharacteristic
100C23×C181C2×D411yescharacteristic
101Q8×C181—24no—
102Q8×C181—42no—
103C2×C4×C181—22no—
104C2×C4×C181—22no—
105C2×C4×C181—22no—
106C2×C4×C181—22no—
107C2×C4×C181—22no—
108C2×C4×C181—22no—
109C2×C4×C181—22no—
110C2×C4×C181—42no—
111C2×C4×C181—42no—
112C2×C4×C181C2×D414yes—
113C2×C4×C181C2×D411yescharacteristic
114C2×C4×C181C2×D411yescharacteristic
115C2×C4×C181C2×D411yescharacteristic
116C2×C4×C181C2411yescharacteristic
117C22×C181—21no—
118C22×C181—21no—
119C22×C181—21no—
120C22×C181—21no—
121C22×C181—21no—
122C22×C181—21no—
123C22×C181—21no—
124C22×C181—44no—
125C22×C181#12012yes—
126C22×C181#13712yescomplemented
127C22×C181C22×D411yescharacteristic
128C2×D181—24no—
129C2×D181—48no—
130C2×D181—42no—
131C2×D181—42no—
132C4×C181—21no—
133C4×C181—21no—
134C4×C181—42no—
135C4×C181—42no—
136C4×C181#12012yes—
137C4×C181C22×D412yes—
138Dic181—24no—
139C2×C181—22no—
140C2×C181—22no—
141C2×C181—41no—
142C2×C181C2×#12011yescharacteristic
143C2×C181C2×#13711yescharacteristic
144C2×C181#57511yescharacteristic
145D181—44no—
146C181#300711yescharacteristic, complemented, Sylow
147#3007—1811nomaximal, Sylow
148C22×#41—1811no—
149C2×#137—1812no—
150C2×C22⋊Q8D18111yescharacteristic, complemented
151#452—1814no—
152#451—1812no—
153#450—1812no—
154#417—1812no—
155#407—1811no—
156C2×#41—3628no—
157C2×#41—3621no—
158C2×#41—3621no—
159C2×#41—1812no—
160C2×#41—1812no—
161C22×Q8—1811no—
162C22×Q8C2×D18111yescharacteristic, complemented
163C22×D4—3621no—
164C22×D4—1811no—
165C23×C4—3621no—
166C23×C4C2×D18111yescharacteristic
167#137—3628no—
168C2×Q16—1814no—
169C2×SD16—1814no—
170C2×M16—1812no—
171C22⋊Q8—21no—
172C22⋊Q8—21no—
173C22⋊Q8C2×D18112yes—
174C22⋊Q8C2×D18112yescomplemented
175C2×C4⋊C4—21no—
176C2×C4⋊C4C2×D18111yescharacteristic, complemented
177C2×C22⋊C4—21no—
178#103—1814no—
179#102—1814no—
180C22⋊C8—1812no—
181C22⋊C8—1812no—
182C24—3621no—
183C24C22⋊D18111yescharacteristic
184#41—3628no—
185#41—3628no—
186#41—3624no—
187#41—3624no—
188#41—3624no—
189#41—1818no—
190C2×Q8—21no—
191C2×Q8—24no—
192C2×Q8—3624no—
193C2×Q8—3621no—
194C2×Q8—1812no—
195C2×Q8C22×D18112yes—
196C2×D4—3624no—
197C2×D4—3628no—
198C2×D4—3622no—
199C2×D4—3622no—
200C2×D4—3621no—
201C2×D4—1812no—
202C22×C4—24no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—3628no—
206C22×C4—3621no—
207C22×C4—3622no—
208C22×C4—3622no—
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—22no—
220C4⋊C4—22no—
221C4⋊C4—21no—
222C4⋊C4C22×D18112yes—
223C22⋊C4—22no—
224C22⋊C4—22no—
225C23—22no—
226C23—22no—
227C23—41no—
228C23—3624no—
229C23—3622no—
230C23—7242no—
231C23—7242no—
232C23C2×C22⋊D18111yescharacteristic
233C23C2×C22⋊D18111yescharacteristic
234C23C2×C22⋊D18111yescharacteristic
235Q8—24no—
236Q8—42no—
237Q8—3624no—
238Q8—3622no—
239Q8—1814no—
240D4—3624no—
241D4—3628no—
242D4—3628no—
243D4—3622no—
244D4—1814no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—3628no—
255C2×C4—3624no—
256C2×C4—3622no—
257C2×C4—3628no—
258C2×C4D4×D18114yes—
259C2×C4C2×C22⋊D18111yescharacteristic
260C2×C4C2×C22⋊D18111yescharacteristic
261C2×C4C2×C22⋊D18111yescharacteristic
262C2×C4C23×D18111yescharacteristic
263C8—3624no—
264C22—21no—
265C22—21no—
266C22—21no—
267C22—21no—
268C22—21no—
269C22—21no—
270C22—21no—
271C22—44no—
272C22—3624no—
273C22—7248no—
274C22—7242no—
275C22—7242no—
276C22#1915378412yes—
277C22#1915384712yescomplemented
278C22C22×C22⋊D18111yescharacteristic
279C4—21no—
280C4—21no—
281C4—42no—
282C4—42no—
283C4—3624no—
284C4#1915378412yes—
285C4C2×D4×D18112yes—
286C2—22no—
287C2—22no—
288C2—41no—
289C2—7244no—
290C2C2×#1915378411yescharacteristic
291C2C2×#1915384711yescharacteristic
292C2#2604207711yescharacteristic
subgroupG/H
centerC22C22×C22⋊D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
FittingC2×C22⋊Q8×C181C2
socleC22×C181C22×D4
prime radicalC22×#15668449C2
hypercenterC2×C22⋊Q8D181
hypocenterC181#3007
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. C2×#19153847C2semidirect
3. #26041439C2semidirect
4. #26041434C2semidirect
5. #26041429C2semidirect
6. #26041278C2semidirect
7. #26041243C2semidirect
8. C2×C22⋊Q8×C181C2semidirect
19. C2×#15668439C22semidirect
20. C2×#15668436C22semidirect
21. C2×#15668433C22semidirect
22. #19153739C22semidirect
23. #19153736C22semidirect
25. C22⋊#13089678C22semidirect
26. C22×Q8×C181C22semidirect
31. C22⋊Q8×C181C22semidirect
33. C2×C4⋊C4×C181C22semidirect
126. C22×C181#137semidirect
146. C181#3007semidirect
150. C2×C22⋊Q8D181semidirect
162. C22×Q8C2×D181semidirect
174. C22⋊Q8C2×D181semidirect
176. C2×C4⋊C4C2×D181semidirect
277. C22#19153847semidirect

Maximal subgroups

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

subgroupindex
C22×#156684492
C2×#191538472
#260414392
#260414342
#260414292
#260412782
#260412432
C2×C22⋊Q8×C1812
#3007181