Finite Group Databasebeta

← #37544183

Subgroups — #37544183

Order 23168.

572 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 572 conjugacy classes of subgroups — too many to draw. Only its 149 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×#19153849C211yesmaximal, characteristic, complemented
2C2×#19153846C211yesmaximal, characteristic, complemented
3C2×#19153842C211yesmaximal, characteristic, complemented
4#26041825C211yesmaximal, characteristic, complemented
5C2×#19153803C211yesmaximal, characteristic, complemented
6#26041670C212yesmaximal, complemented
7#26041667C212yesmaximal, complemented
8#26041662C212yesmaximal, complemented
9#26041660C212yesmaximal, complemented
10M16×Dic181C211yesmaximal, characteristic, complemented
11C2×AGL(1,ℤ/8)×C181C211yesmaximal, characteristic, complemented
12C2×#15668449C2211yescharacteristic
13C22×Dic362C2211yescharacteristic
14#19153849—24no—
15#19153846—24no—
16#19153842—24no—
17C2×Dic724C2212yescomplemented
18C2×#15668439C2212yescomplemented
19C2×#15668438C2212yescomplemented
20C2×#15668437C2212yes—
21C2×#15668437C2212yescomplemented
22C2×#15668434C2212yes—
23C2×#15668433C2211yescharacteristic
24#19153814C2211yescharacteristic, complemented
25#19153812C2211yescharacteristic, complemented
26#19153805C2211yescharacteristic, complemented
27#19153803—22no—
28#19153803C2212yes—
29#19153803C2212yes—
30#19153803C2211yescharacteristic, complemented
31#19153788—22no—
32C4×C22⋊D181—22no—
33C2×C22⋊Dic181—22no—
34C2×C4×Dic181C2211yescharacteristic
35#19153726C2212yescomplemented
36C8×Dic181C2212yescomplemented
37C2×#41×C181C2211yescharacteristic
38C22×D4×C181C2211yescharacteristic
39AGL(1,ℤ/8)×C181—24no—
40C2×SD16×C181C2212yes—
41C2×D8×C181C2212yes—
42C2×M16×C181C2211yescharacteristic
43#15668449—22no—
44#15668449—22no—
45C2×Dic362—21no—
46C2×Dic362—22no—
47C2×Dic362—22no—
48C2×Dic362C2312yes—
49C2×Dic362C2311yescharacteristic
50C2×D724C2311yescharacteristic
51C2×C22⋊D181—22no—
52C2×C22⋊D181—21no—
53C22×Dic181—21no—
54C22×Dic181D412yes—
55C2×C4×D181—21no—
56Dic724—24no—
57#15668439—24no—
58#15668438—24no—
59#15668437—24no—
60#15668437—24no—
61#15668434—24no—
62#15668433D414yescomplemented
63C2×#13089678C2312yes—
64#15668428—22no—
65C22⋊Dic181—24no—
66C22⋊Dic181—24no—
67C22⋊Dic181—22no—
68#15668426—22no—
69C4×Dic181D412yes—
70C4×Dic181C2311yescharacteristic
71C4×Dic181C2311yescharacteristic
72C24×C181—21no—
73#41×C181—22no—
74#41×C181—22no—
75C2×Q8×C181C2311yescharacteristic
76C2×D4×C181—21no—
77C2×D4×C181—22no—
78C2×D4×C181—22no—
79C2×D4×C181—21no—
80C2×D4×C181C2311yescharacteristic
81C2×D4×C181C2312yes—
82C22×C4×C181—21no—
83C22×C4×C181C2311yescharacteristic
84SD16×C181—24no—
85D8×C181—24no—
86M16×C181D414yescomplemented
87C2×C8×C181C2312yes—
88C22×D181—21no—
89Dic362—21no—
90Dic362—22no—
91Dic362—22no—
92Dic362—21no—
93Dic362—42no—
94D724—21no—
95D724—21no—
96C22⋊D181—44no—
97C22⋊D181—42no—
98C2×Dic181—22no—
99C2×Dic181—21no—
100C2×Dic181—42no—
101C2×Dic181—42no—
102C2×Dic181C2×D412yes—
103C2×Dic181C2×D412yes—
104C4×D181—42no—
105#13089678C2×D414yes—
106C23×C181—21no—
107C23×C181—22no—
108C23×C181—44no—
109C23×C181—42no—
110C23×C181C2×D411yescharacteristic
111Q8×C181—21no—
112Q8×C181—21no—
113D4×C181—21no—
114D4×C181—22no—
115D4×C181—22no—
116D4×C181—21no—
117D4×C181—42no—
118D4×C181—42no—
119C2×C4×C181—21no—
120C2×C4×C181—42no—
121C2×C4×C181C2×D411yescharacteristic
122C2×C4×C181C2×D412yes—
123C2×C4×C181C2×D412yes—
124C2×C4×C181C2411yescharacteristic
125C8×C181C2×D414yes—
126C22×C181—22no—
127C22×C181—41no—
128C22×C181—42no—
129C22×C181—44no—
130C22×C181—42no—
131C22×C181—84no—
132C22×C181#12712yes—
133C22×C181C22×D411yescharacteristic
134C2×D181—41no—
135C2×D181—42no—
136C4×C181—41no—
137C4×C181#12712yes—
138C4×C181C22×D411yescharacteristic
139C4×C181C22×D411yescharacteristic
140Dic181—22no—
141Dic181—22no—
142Dic181—41no—
143Dic181—42no—
144C2×C181—22no—
145C2×C181—81no—
146C2×C181—82no—
147C2×C181C2×#12711yescharacteristic
148C2×C181#57512yes—
149D181—81no—
150C181#315311yescharacteristic, complemented, Sylow
151#3153—1811nomaximal, Sylow
152C2×#137—1811no—
153C2×#137—1811no—
154C2×AGL(1,ℤ/8)—1811no—
155C2×AGL(1,ℤ/8)D18111yescharacteristic, complemented
156#532—1811no—
157C2×#124—1811no—
158#497—1812no—
159#496—1812no—
160#495—1812no—
161#495—1812no—
162C4×M16—1811no—
163C2×#41—1811no—
164C2×#41C2×D18111yescharacteristic
165C22×Q8—1811no—
166C22×D4C2×D18111yescharacteristic
167#137—3624no—
168#137—3624no—
169AGL(1,ℤ/8)—24no—
170AGL(1,ℤ/8)—3624no—
171C2×Q16—1812no—
172C2×Q16—1812no—
173C2×SD16—1812no—
174C2×SD16—1812no—
175C2×SD16—1812no—
176C2×SD16C2×D18112yes—
177C2×D8—1812no—
178C2×D8C2×D18112yes—
179C2×M16—1811no—
180C2×M16C2×D18111yescharacteristic
181C4⋊Q8—1811no—
182#127—1811no—
183#124—3622no—
184#124—1811no—
185#124—1812no—
186#124—1812no—
187#124—1811no—
188#121—3622no—
189C4×D4—3622no—
190C2×C22⋊C4—3622no—
191C2×C42—1811no—
192#97—1812no—
193C4×C8—1812no—
194C24—21no—
195#41—22no—
196#41—22no—
197#41—3622no—
198#41—3622no—
199C2×Q8—3621no—
200C2×Q8—3622no—
201C2×Q8—3622no—
202C2×Q8—1812no—
203C2×Q8—1811no—
204C2×Q8C22×D18111yescharacteristic
205C2×D4—21no—
206C2×D4—22no—
207C2×D4—21no—
208C2×D4—22no—
209C2×D4—3621no—
210C2×D4—3622no—
211C2×D4—1811no—
212C2×D4C22×D18112yes—
213C2×D4C22×D18111yescharacteristic
214C22×C4—21no—
215C22×C4—3621no—
216C22×C4—3621no—
217C22×C4—1812no—
218C22×C4C22×D18111yescharacteristic
219Q16—3624no—
220Q16—3624no—
221SD16—24no—
222SD16—3624no—
223SD16—3624no—
224SD16—3624no—
225D8—24no—
226D8—3624no—
227M16—1814no—
228M16C22⋊D18114yescomplemented
229C2×C8—1812no—
230C2×C8C22×D18112yes—
231C4⋊C4—3622no—
232C22⋊C4—3622no—
233C22⋊C4—3624no—
234C22⋊C4—3624no—
235C22⋊C4—3622no—
236C42—1812no—
237C42—1811no—
238C42—1811no—
239C23—22no—
240C23—21no—
241C23—44no—
242C23—42no—
243C23—3621no—
244C23D4×D18111yescharacteristic
245Q8—21no—
246Q8—21no—
247Q8—3621no—
248Q8—3622no—
249Q8—3622no—
250Q8—3621no—
251Q8—7242no—
252D4—22no—
253D4—21no—
254D4—21no—
255D4—22no—
256D4—42no—
257D4—42no—
258D4—3621no—
259D4—3621no—
260D4—7244no—
261D4—7242no—
262C2×C4—21no—
263C2×C4—42no—
264C2×C4—3622no—
265C2×C4—3621no—
266C2×C4—1812no—
267C2×C4—1812no—
268C2×C4—7242no—
269C2×C4—7242no—
270C2×C4—7242no—
271C2×C4D4×D18111yescharacteristic
272C2×C4D4×D18112yes—
273C2×C4C2×C22⋊D18112yes—
274C2×C4C23×D18111yescharacteristic
275C8—1814no—
276C8C2×C22⋊D18114yes—
277C22—22no—
278C22—41no—
279C22—42no—
280C22—42no—
281C22—44no—
282C22—84no—
283C22—7241no—
284C22—7242no—
285C22#1915381212yes—
286C22C2×D4×D18111yescharacteristic
287C4—41no—
288C4—3622no—
289C4—3622no—
290C4—7241no—
291C4—7242no—
292C4#1915381212yes—
293C4C2×D4×D18111yescharacteristic
294C4C22×C22⋊D18111yescharacteristic
295C2—22no—
296C2—82no—
297C2—81no—
298C2—14481no—
299C2C2×#1915381211yescharacteristic
300C2#2604207512yes—
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
FittingC2×AGL(1,ℤ/8)×C181C2
socleC22×C181C22×D4
hypercenterC2×AGL(1,ℤ/8)D181
hypocenterC181#3153
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C2×#19153849C2semidirect
2. C2×#19153846C2semidirect
3. C2×#19153842C2semidirect
4. #26041825C2semidirect
5. C2×#19153803C2semidirect
6. #26041670C2semidirect
7. #26041667C2semidirect
8. #26041662C2semidirect
9. #26041660C2semidirect
10. M16×Dic181C2semidirect
11. C2×AGL(1,ℤ/8)×C181C2semidirect
17. C2×Dic724C22semidirect
18. C2×#15668439C22semidirect
19. C2×#15668438C22semidirect
21. C2×#15668437C22semidirect
24. #19153814C22semidirect
25. #19153812C22semidirect
26. #19153805C22semidirect
30. #19153803C22semidirect
35. #19153726C22semidirect
36. C8×Dic181C22semidirect
62. #15668433D4semidirect
86. M16×C181D4semidirect
150. C181#3153semidirect
155. C2×AGL(1,ℤ/8)D181semidirect
228. M16C22⋊D181semidirect

Maximal subgroups

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

subgroupindex
C2×#191538492
C2×#191538462
C2×#191538422
#260418252
C2×#191538032
#260416702
#260416672
#260416622
#260416602
M16×Dic1812
C2×AGL(1,ℤ/8)×C1812
#3153181