Finite Group Databasebeta

← #37544201

Subgroups — #37544201

Order 23168.

596 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 596 conjugacy classes of subgroups — too many to draw. Only its 137 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×#19153840C212yesmaximal, complemented
2C2×#19153835C211yesmaximal, characteristic, complemented
3C2×D8×D181C211yesmaximal, characteristic, complemented
4#26041842C212yesmaximal, complemented
5#26041667C212yesmaximal, complemented
6#26041666C212yesmaximal, complemented
7#26041663C211yesmaximal, characteristic, complemented
8#26041662C211yesmaximal, characteristic, complemented
9#26041654C211yesmaximal, characteristic, complemented
10#26041233C211yesmaximal, characteristic, complemented
11#493×C181C211yesmaximal, characteristic, complemented
12C2×#15668448C2212yes—
13C2×D4×D181C2212yes—
14#19153840—28no—
15#19153835—24no—
16C2×Dic724C2211yescharacteristic, complemented
17C2×#15668438C2212yescomplemented
18C2×#15668437C2212yescomplemented
19C2×#15668437C2212yescomplemented
20C2×D1448C2211yescharacteristic
21C2×#15668434C2212yes—
22C2×#15668434C2212yescomplemented
23D8×D181—24no—
24C2×#15668432C2211yescharacteristic
25C2×C8×D181C2211yescharacteristic
26#19153812C2212yescomplemented
27#19153804C2212yescomplemented
28#19153803C2212yescomplemented
29#19153788—24no—
30#19153784—24no—
31#19153773C2211yescharacteristic
32#19153727C2211yescharacteristic, complemented
33#19153726C2211yescharacteristic, complemented
34C8×Dic181C2211yescharacteristic, complemented
35C2×D8×C181C2211yescharacteristic
36C2×D8×C181C2211yescharacteristic, complemented
37C2×D8×C181C2212yescomplemented
38#127×C181C2212yescomplemented
39C4×C8×C181C2211yescharacteristic, complemented
40C23×D181—22no—
41#15668448—24no—
42#15668448—24no—
43C2×Dic362C2312yes—
44C2×D724C2312yes—
45C2×C22⋊D181—22no—
46C2×C22⋊D181—22no—
47C2×C22⋊D181—24no—
48D4×D181—24no—
49D4×D181—24no—
50C22×Dic181—22no—
51C2×C4×D181C2311yescharacteristic
52Dic724—22no—
53#15668438—24no—
54#15668437—24no—
55#15668437—24no—
56D1448—22no—
57#15668434—24no—
58#15668434—24no—
59#15668432D414yescomplemented
60C2×#13089678C2311yescharacteristic
61C2×#13089678C2311yescharacteristic
62C8×D181D414yescomplemented
63#15668428D412yes—
64C22⋊Dic181—24no—
65#15668426—24no—
66#15668426D412yes—
67C4×Dic181C2311yescharacteristic
68C2×D4×C181—24no—
69C2×D4×C181C2312yes—
70C2×D4×C181C2312yes—
71D8×C181—22no—
72D8×C181—22no—
73D8×C181—24no—
74C2×C8×C181C2311yescharacteristic
75C2×C8×C181C2311yescharacteristic
76C42×C181C2311yescharacteristic
77C22×D181—22no—
78C22×D181—44no—
79C22×D181—84no—
80C22×D181C2×D411yescharacteristic
81Dic362—22no—
82Dic362—22no—
83D724—22no—
84D724—22no—
85C22⋊D181—44no—
86C22⋊D181—44no—
87C22⋊D181—84no—
88C2×Dic181—22no—
89C2×Dic181—44no—
90C2×Dic181C2×D412yes—
91C2×Dic181C2×D411yescharacteristic
92C4×D181C2×D412yes—
93C4×D181C2×D412yes—
94#13089678C2×D412yes—
95#13089678C2×D412yes—
96C23×C181—22no—
97C23×C181—22no—
98D4×C181—22no—
99D4×C181—22no—
100D4×C181—22no—
101D4×C181—22no—
102D4×C181—84no—
103C2×C4×C181C2×D412yes—
104C2×C4×C181C2411yescharacteristic
105C8×C181C2×D412yes—
106C8×C181C2×D412yes—
107C22×C181—42no—
108C22×C181—42no—
109C22×C181—44no—
110C22×C181—44no—
111C22×C181C22×D411yescharacteristic
112C2×D181—22no—
113C2×D181—42no—
114C2×D181—44no—
115C2×D181—88no—
116C2×D181#12712yes—
117C4×C181—22no—
118C4×C181C22×D411yescharacteristic
119C4×C181C22×D411yescharacteristic
120Dic181—22no—
121Dic181—42no—
122Dic181#12712yes—
123C2×C181—82no—
124C2×C181—82no—
125C2×C181C2×#12711yescharacteristic
126C2×C181#57612yes—
127D181—22no—
128D181—82no—
129C181#315511yescharacteristic, complemented, Sylow
130#3155—1811nomaximal, Sylow
131C2×AGL(1,ℤ/8)—1812no—
132C2×#135—1811no—
133C22×D8—1811no—
134#535—1812no—
135#496—1812no—
136#496—1812no—
137#495—1811no—
138#495—1811no—
139#493—1811no—
140#493D18111yescharacteristic, complemented
141#405—1811no—
142C2×#41—1812no—
143C22×D4—1812no—
144AGL(1,ℤ/8)—3628no—
145#135—3624no—
146C2×Q16—1811no—
147C2×SD16—1812no—
148C2×SD16—1812no—
149C2×SD16—1812no—
150C2×D8—3624no—
151C2×D8—1812no—
152C2×D8—1812no—
153C2×D8—1811no—
154C2×D8C2×D18111yescharacteristic
155C2×D8C2×D18111yescharacteristic, complemented
156C2×D8C2×D18112yescomplemented
157C2×M16—1811no—
158C22×C8—1811no—
159#127—1812no—
160#127C2×D18112yescomplemented
161#124—1812no—
162#124—1812no—
163#121—3624no—
164#120—3624no—
165#117—1811no—
166#97—1811no—
167#97—1811no—
168C4×C8—1811no—
169C4×C8C2×D18111yescharacteristic, complemented
170C24—3622no—
171#41—3624no—
172#41—3624no—
173C2×Q8—1812no—
174C2×D4—24no—
175C2×D4—3622no—
176C2×D4—3624no—
177C2×D4—3624no—
178C2×D4—3624no—
179C2×D4—3622no—
180C2×D4—1812no—
181C2×D4C22×D18112yes—
182C2×D4C22×D18112yes—
183C22×C4—3622no—
184C22×C4—1811no—
185Q16—3622no—
186SD16—3624no—
187SD16—3624no—
188SD16—3624no—
189D8—22no—
190D8—22no—
191D8—24no—
192D8—3624no—
193D8—3624no—
194D8—3622no—
195M16—1814no—
196C2×C8—1811no—
197C2×C8—1811no—
198C2×C8—1814no—
199C2×C8C22×D18111yescharacteristic
200C2×C8C22×D18111yescharacteristic
201C4⋊C4—1812no—
202C22⋊C4—3624no—
203C22⋊C4—3624no—
204C22⋊C4—1812no—
205C42—1811no—
206C42C22×D18111yescharacteristic
207C23—22no—
208C23—22no—
209C23—3622no—
210C23—14484no—
211C23—1811no—
212C23—7244no—
213Q8—3622no—
214Q8—3622no—
215D4—22no—
216D4—22no—
217D4—22no—
218D4—22no—
219D4—84no—
220D4—3622no—
221D4—3622no—
222D4—14484no—
223D4—7244no—
224D4—7244no—
225C2×C4—3622no—
226C2×C4—1812no—
227C2×C4—1811no—
228C2×C4—1812no—
229C2×C4—1812no—
230C2×C4—7244no—
231C2×C4D4×D18112yes—
232C2×C4C23×D18111yescharacteristic
233C8—1812no—
234C8—1812no—
235C8D4×D18112yes—
236C8D4×D18112yes—
237C22—42no—
238C22—42no—
239C22—44no—
240C22—44no—
241C22—3622no—
242C22—14488no—
243C22—1812no—
244C22—7242no—
245C22—7244no—
246C22C2×D4×D18111yescharacteristic
247C4—22no—
248C4—3622no—
249C4—1812no—
250C4—7242no—
251C4C2×D4×D18111yescharacteristic
252C4C2×D4×D18111yescharacteristic
253C2—82no—
254C2—82no—
255C2—3622no—
256C2—14482no—
257C2#127×D18111yescharacteristic
258C2#2604208012yes—
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#493×C181C2
socleC22×C181C22×D4
hypercenter#493D181
hypocenterC181#3155
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C42 ◁ #493

Complements

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

NH ≅ G/Ntype
1. C2×#19153840C2semidirect
2. C2×#19153835C2semidirect
3. C2×D8×D181C2semidirect
4. #26041842C2semidirect
5. #26041667C2semidirect
6. #26041666C2semidirect
7. #26041663C2semidirect
8. #26041662C2semidirect
9. #26041654C2semidirect
10. #26041233C2semidirect
11. #493×C181C2semidirect
16. C2×Dic724C22semidirect
17. C2×#15668438C22semidirect
18. C2×#15668437C22semidirect
19. C2×#15668437C22semidirect
22. C2×#15668434C22semidirect
26. #19153812C22semidirect
27. #19153804C22semidirect
28. #19153803C22semidirect
32. #19153727C22semidirect
33. #19153726C22semidirect
34. C8×Dic181C22semidirect
36. C2×D8×C181C22semidirect
37. C2×D8×C181C22semidirect
38. #127×C181C22semidirect
39. C4×C8×C181C22semidirect
59. #15668432D4semidirect
62. C8×D181D4semidirect
129. C181#3155semidirect
140. #493D181semidirect
155. C2×D8C2×D181semidirect
156. C2×D8C2×D181semidirect
160. #127C2×D181semidirect
169. C4×C8C2×D181semidirect

Maximal subgroups

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

subgroupindex
C2×#191538402
C2×#191538352
C2×D8×D1812
#260418422
#260416672
#260416662
#260416632
#260416622
#260416542
#260412332
#493×C1812
#3155181