Finite Group Databasebeta

← #496×D173 #36363993

Subgroups — #496×D173 #36363993

Order 22144.

668 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 668 conjugacy classes of subgroups — too many to draw. Only its 145 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×SD16×D173C212yesmaximal, complemented
2C2×D8×D173C212yesmaximal, complemented
3#127×D173C211yesmaximal, characteristic, complemented
4#124×D173C211yesmaximal, characteristic, complemented
5#26011881C211yesmaximal, characteristic, complemented
6#26011880C211yesmaximal, characteristic, complemented
7#26011879C212yesmaximal, complemented
8#26011878C211yesmaximal, characteristic, complemented
9#26011877C212yesmaximal, complemented
10#97×D173C211yesmaximal, characteristic, complemented
11#496×C173C211yesmaximal, characteristic, complemented
12C2×Q8×D173C2211yescharacteristic
13C2×D4×D173—22no—
14C2×D4×D173C2211yescharacteristic
15C2×D4×D173C2212yes—
16C2×#14583815C2212yescomplemented
17C2×#14583814C2212yescomplemented
18C2×#14583813C2212yescomplemented
19SD16×D173—28no—
20C2×D1384C2212yescomplemented
21C2×#14583811C2212yescomplemented
22C2×#14583811C2212yescomplemented
23D8×D173—28no—
24C2×C8×D173C2212yes—
25#19149676C2211yescharacteristic, complemented
26#19149675—22no—
27#19149675C2212yescomplemented
28#19149668C2211yescharacteristic, complemented
29#19149667C2211yescharacteristic, complemented
30#19149666C2211yescharacteristic, complemented
31#19149665—22no—
32C22⋊C4×D173—22no—
33C42×D173C2211yescharacteristic
34#19149590C2211yescharacteristic, complemented
35#19149589C2212yescomplemented
36C2×SD16×C173C2212yescomplemented
37C2×D8×C173C2212yescomplemented
38#127×C173C2211yescharacteristic, complemented
39#124×C173C2211yescharacteristic, complemented
40#97×C173C2211yescharacteristic, complemented
41C23×D173—21no—
42C23×D173—22no—
43C2×Dic346—21no—
44C2×Dic346C2311yescharacteristic
45Q8×D173—24no—
46C2×D692—22no—
47C2×D692C2311yescharacteristic
48C2×D692C2312yes—
49C2×C22⋊D173—21no—
50C2×C22⋊D173—22no—
51C2×C22⋊D173—24no—
52D4×D173—24no—
53D4×D173—24no—
54D4×D173—24no—
55D4×D173—48no—
56C2×C4×D173—21no—
57C2×C4×D173D412yes—
58C2×C4×D173C2311yescharacteristic
59#14583815—24no—
60#14583814—24no—
61#14583813—24no—
62D1384—24no—
63#14583811—24no—
64#14583811—24no—
65C2×#13063436C2312yes—
66C8×D173D418yescomplemented
67C22⋊Dic173—22no—
68#14583803—22no—
69#14583803—22no—
70C4×Dic173D412yes—
71C4×Dic173C2311yescharacteristic
72C2×Q8×C173C2311yescharacteristic
73C2×D4×C173—22no—
74C2×D4×C173C2311yescharacteristic
75C2×D4×C173C2312yes—
76SD16×C173—24no—
77D8×C173—24no—
78C2×C8×C173C2312yes—
79C22⋊C4×C173—22no—
80C42×C173C2311yescharacteristic
81C22×D173—22no—
82C22×D173—21no—
83C22×D173—42no—
84C22×D173—44no—
85C22×D173—48no—
86C22×D173—44no—
87C22×D173C2×D411yescharacteristic
88Dic346—22no—
89Dic346—42no—
90D692—22no—
91D692—22no—
92D692—22no—
93D692—44no—
94C22⋊D173—48no—
95C22⋊D173—42no—
96C22⋊D173—44no—
97C2×Dic173—21no—
98C2×Dic173C2×D412yes—
99C2×Dic173C2×D411yescharacteristic
100C4×D173—24no—
101C4×D173—42no—
102C4×D173C2×D414yes—
103#13063436C2×D414yescomplemented
104C23×C173—22no—
105C23×C173—21no—
106Q8×C173—22no—
107D4×C173—22no—
108D4×C173—22no—
109D4×C173—22no—
110D4×C173—44no—
111C2×C4×C173—21no—
112C2×C4×C173C2×D412yes—
113C2×C4×C173C2411yescharacteristic
114C8×C173C2×D414yescomplemented
115C22×C173—41no—
116C22×C173—42no—
117C22×C173—44no—
118C22×C173—42no—
119C22×C173C22×D411yescharacteristic
120C2×D173—41no—
121C2×D173—42no—
122C2×D173—44no—
123C2×D173—42no—
124C2×D173—88no—
125C2×D173—84no—
126C2×D173#12712yes—
127C2×D173AGL(1,ℤ/8)14yes—
128C4×C173—22no—
129C4×C173—41no—
130C4×C173C22×D412yes—
131Dic173—22no—
132Dic173—41no—
133Dic173#12712yes—
134C2×C173—82no—
135C2×C173—81no—
136C2×C173C2×#12711yescharacteristic
137C2×C173C2×AGL(1,ℤ/8)12yes—
138D173—82no—
139D173—81no—
140D173#49614yesdirect product factor
141C173C2×#49611yescharacteristic, complemented, Sylow
142C2×#496—1731nomaximal, Sylow
143C22×SD16—1732no—
144C22×D8—1732no—
145C2×#127—1731no—
146C2×#124—1731no—
147#496—1732no—
148#496—1732no—
149#496—1731no—
150#496—1731no—
151#496—1731no—
152#496D17311yescharacteristic, direct product factor
153C2×#97—1731no—
154C22×Q8—1731no—
155C22×D4—3462no—
156C22×D4—1731no—
157C22×D4—1732no—
158C2×SD16—3468no—
159C2×SD16—1732no—
160C2×SD16—1732no—
161C2×SD16—1732no—
162C2×SD16C2×D17312yescomplemented
163C2×D8—3468no—
164C2×D8—1732no—
165C2×D8—1732no—
166C2×D8—1732no—
167C2×D8C2×D17312yescomplemented
168C22×C8—1732no—
169#127—3462no—
170#127—1732no—
171#127—1731no—
172#127C2×D17311yescharacteristic, complemented
173#124—3462no—
174#124—1731no—
175#124—1731no—
176#124—1731no—
177#124C2×D17311yescharacteristic, complemented
178C2×C22⋊C4—3462no—
179C2×C42—1731no—
180#97—1732no—
181#97—1731no—
182#97C2×D17311yescharacteristic, complemented
183C24—3461no—
184C24—3462no—
185C2×Q8—3461no—
186C2×Q8—3464no—
187C2×Q8—1731no—
188C2×Q8C22×D17311yescharacteristic
189C2×D4—22no—
190C2×D4—3464no—
191C2×D4—3464no—
192C2×D4—3464no—
193C2×D4—3461no—
194C2×D4—3462no—
195C2×D4—3464no—
196C2×D4—3462no—
197C2×D4—1731no—
198C2×D4—1732no—
199C2×D4—6928no—
200C2×D4C22×D17311yescharacteristic
201C2×D4C22×D17312yes—
202C22×C4—3461no—
203C22×C4—1732no—
204C22×C4—1731no—
205SD16—24no—
206SD16—3464no—
207SD16—3464no—
208SD16—3464no—
209D8—24no—
210D8—3464no—
211D8—3464no—
212D8—3464no—
213C2×C8—1732no—
214C2×C8—1738no—
215C2×C8C22×D17312yes—
216C22⋊C4—22no—
217C22⋊C4—3462no—
218C22⋊C4—3462no—
219C22⋊C4—3462no—
220C42—1732no—
221C42—1731no—
222C42C22×D17311yescharacteristic
223C23—21no—
224C23—22no—
225C23—3461no—
226C23—3462no—
227C23—1731no—
228C23—6922no—
229C23—6924no—
230C23—6928no—
231C23—6924no—
232Q8—22no—
233Q8—3462no—
234Q8—6922no—
235D4—22no—
236D4—22no—
237D4—22no—
238D4—44no—
239D4—3462no—
240D4—3462no—
241D4—3462no—
242D4—6924no—
243D4—6928no—
244D4—6922no—
245D4—6924no—
246C2×C4—21no—
247C2×C4—3464no—
248C2×C4—3461no—
249C2×C4—1732no—
250C2×C4—1731no—
251C2×C4—1734no—
252C2×C4—6922no—
253C2×C4D4×D17312yes—
254C2×C4C23×D17311yescharacteristic
255C8—1734no—
256C8D4×D17314yescomplemented
257C22—44no—
258C22—42no—
259C22—41no—
260C22—42no—
261C22—13848no—
262C22—13844no—
263C22—1734no—
264C22—1732no—
265C22—6924no—
266C22—6922no—
267C22—6921no—
268C22—6922no—
269C22C2×D4×D17311yescharacteristic
270C4—22no—
271C4—41no—
272C4—3462no—
273C4—1732no—
274C4—6921no—
275C4C2×D4×D17312yes—
276C2—82no—
277C2—81no—
278C2—13842no—
279C2—13841no—
280C2—1734no—
281C2#127×D17311yescharacteristic
282C2AGL(1,ℤ/8)×D17312yes—
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#496×C173C2
socleC22×C173C22×D4
hypercenter#496D173
hypocenterC173C2×#496
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ #496

Complements

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

NH ≅ G/Ntype
1. C2×SD16×D173C2semidirect
2. C2×D8×D173C2semidirect
3. #127×D173C2semidirect
4. #124×D173C2semidirect
5. #26011881C2semidirect
6. #26011880C2semidirect
7. #26011879C2semidirect
8. #26011878C2semidirect
9. #26011877C2semidirect
10. #97×D173C2semidirect
11. #496×C173C2semidirect
16. C2×#14583815C22semidirect
17. C2×#14583814C22semidirect
18. C2×#14583813C22semidirect
20. C2×D1384C22semidirect
21. C2×#14583811C22semidirect
22. C2×#14583811C22semidirect
25. #19149676C22semidirect
27. #19149675C22semidirect
28. #19149668C22semidirect
29. #19149667C22semidirect
30. #19149666C22semidirect
34. #19149590C22semidirect
35. #19149589C22semidirect
36. C2×SD16×C173C22semidirect
37. C2×D8×C173C22semidirect
38. #127×C173C22semidirect
39. #124×C173C22semidirect
40. #97×C173C22semidirect
66. C8×D173D4semidirect
103. #13063436C2×D4semidirect
114. C8×C173C2×D4semidirect
140. D173#496direct (N × H)
141. C173C2×#496semidirect
152. #496D173direct (N × H)
162. C2×SD16C2×D173semidirect
167. C2×D8C2×D173semidirect
172. #127C2×D173semidirect
177. #124C2×D173semidirect
182. #97C2×D173semidirect
256. C8D4×D173semidirect

Maximal subgroups

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

subgroupindex
C2×SD16×D1732
C2×D8×D1732
#127×D1732
#124×D1732
#260118812
#260118802
#260118792
#260118782
#260118772
#97×D1732
#496×C1732
C2×#496173