Finite Group Databasebeta

← #36365166

Subgroups — #36365166

Order 22144.

546 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 546 conjugacy classes of subgroups — too many to draw. Only its 125 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×#19149706C212yesmaximal, complemented
2#26012144C211yesmaximal, characteristic, complemented
3D4×D692C211yesmaximal, characteristic, complemented
4#26011865C211yesmaximal, characteristic, complemented
5#26011740C212yesmaximal, complemented
6#26011687C211yesmaximal, characteristic, complemented
7#26011640C212yesmaximal, complemented
8#26011639C212yesmaximal, complemented
9C4×#14583811C211yesmaximal, characteristic, complemented
10#26011568C211yesmaximal, characteristic, complemented
11#461×C173C211yesmaximal, characteristic, complemented
12C2×#14583827C2212yescomplemented
13C22×D692—21no—
14C2×D4×D173C2212yes—
15#19149706—28no—
16C2×#14583813C2212yescomplemented
17C2×#14583813C2212yescomplemented
18C2×D1384C2212yescomplemented
19C2×#14583811C2211yescharacteristic
20C2×#14583809C2212yescomplemented
21#19149676C2211yescharacteristic
22#19149664—22no—
23#19149655C2212yescomplemented
24#19149654—21no—
25#19149654C2212yes—
26#19149653—21no—
27#19149649—22no—
28C4×D692C2211yescharacteristic
29C4×D692C2211yescharacteristic, complemented
30C4⋊C4×D173—21no—
31#19149612C2211yescharacteristic, complemented
32#19149601C2212yescomplemented
33#19149600C2211yescharacteristic
34#19149599C2211yescharacteristic, complemented
35#19149591C2212yescomplemented
36C4×#13063436C2211yescharacteristic, complemented
37C2×SD16×C173C2212yes—
38C4⋊Q8×C173C2211yescharacteristic, complemented
39C4×D4×C173C2211yescharacteristic
40C4⋊C8×C173C2211yescharacteristic, complemented
41#103×C173C2212yescomplemented
42C23×D173—22no—
43#14583827—24no—
44#14583827—42no—
45C2×D692—22no—
46C2×D692—22no—
47C2×D692—21no—
48C2×D692—21no—
49C2×D692—44no—
50C2×D692D411yescharacteristic
51C2×D692C2312yes—
52C2×D692C2311yescharacteristic
53C2×C22⋊D173—22no—
54D4×D173—24no—
55D4×D173—42no—
56C2×C4×D173—22no—
57C2×C4×D173—21no—
58C2×C4×D173C2312yes—
59#14583813—24no—
60#14583813—42no—
61D1384—24no—
62#14583811D414yescomplemented
63#14583809—24no—
64C2×#13063436C2311yescharacteristic
65C2×#13063436C2311yescharacteristic
66C4⋊Dic173D411yescharacteristic
67C4⋊Dic173C2311yescharacteristic
68#14583805—22no—
69#14583803—22no—
70#14583803—22no—
71#14583803—21no—
72#14583803D412yes—
73C2×Q8×C173C2312yes—
74C2×D4×C173C2311yescharacteristic
75C22×C4×C173—21no—
76SD16×C173—24no—
77C2×C8×C173C2312yes—
78C4⋊C4×C173—21no—
79C4⋊C4×C173—21no—
80C4⋊C4×C173C2311yescharacteristic
81C22⋊C4×C173—21no—
82C42×C173C2311yescharacteristic
83C22×D173—21no—
84C22×D173—22no—
85C22×D173—44no—
86C22×D173—44no—
87C22×D173—82no—
88C22×D173C2×D412yes—
89D692—22no—
90D692—22no—
91D692—21no—
92D692—22no—
93D692—44no—
94D692—44no—
95D692—42no—
96D692—81no—
97D692C2×D412yescomplemented
98C22⋊D173—44no—
99C2×Dic173—21no—
100C2×Dic173C2×D412yes—
101C4×D173—22no—
102C4×D173—22no—
103C4×D173—44no—
104C4×D173—42no—
105#13063436—21no—
106#13063436C2×D412yescomplemented
107C23×C173—21no—
108Q8×C173—22no—
109Q8×C173—22no—
110D4×C173—21no—
111D4×C173C2×D412yes—
112C2×C4×C173—22no—
113C2×C4×C173—21no—
114C2×C4×C173—42no—
115C2×C4×C173C2×D411yescharacteristic
116C2×C4×C173C2×D411yescharacteristic
117C2×C4×C173C2411yescharacteristic
118C8×C173—22no—
119C22×C173—22no—
120C22×C173—41no—
121C22×C173—41no—
122C22×C173C22×D411yescharacteristic
123C2×D173—22no—
124C2×D173—22no—
125C2×D173—22no—
126C2×D173—22no—
127C2×D173—41no—
128C2×D173—41no—
129C2×D173—42no—
130C2×D173—42no—
131C2×D173—42no—
132C2×D173—88no—
133C4×C173—21no—
134C4×C173—42no—
135C4×C173—41no—
136C4×C173AGL(1,ℤ/8)12yescomplemented
137C4×C173C22×D411yescharacteristic
138C4×C173H(5,𝔽2)11yescharacteristic
139Dic173—22no—
140Dic173—41no—
141C2×C173—42no—
142C2×C173D4211yescharacteristic
143C2×C173C2×AGL(1,ℤ/8)11yescharacteristic
144C2×C173#57711yescharacteristic
145D173—42no—
146D173—42no—
147D173—82no—
148C173#327211yescharacteristic, complemented, Sylow
149#3272—1731nomaximal, Sylow
150C2×AGL(1,ℤ/8)—1732no—
151#547—1731no—
152D42—1731no—
153#492—1731no—
154#468—1732no—
155#461D17311yescharacteristic, complemented
156#459—1731no—
157#450—1732no—
158#449—1732no—
159C4×D8—1731no—
160#436—1731no—
161C2×#41—1732no—
162C22×D4—3461no—
163C22×D4—1732no—
164AGL(1,ℤ/8)—3468no—
165C2×SD16—1732no—
166C2×SD16—1732no—
167C2×SD16C2×D17312yes—
168C2×D8—1731no—
169C2×D8—1732no—
170C2×M16—1732no—
171C4⋊Q8C2×D17311yescharacteristic, complemented
172#127—1731no—
173#123—3462no—
174C22⋊Q8—1732no—
175#121—3461no—
176#121—3461no—
177#121—1732no—
178#120—3462no—
179C4×D4—1731no—
180C4×D4—1731no—
181C4×D4C2×D17311yescharacteristic
182C2×C4⋊C4—3461no—
183#107—1731no—
184C4⋊C8C2×D17311yescharacteristic, complemented
185#103C2×D17312yescomplemented
186#102—1731no—
187#102—1732no—
188#102—1731no—
189C22⋊C8—1732no—
190C4×C8—1731no—
191C24—3462no—
192#41—3464no—
193#41—6922no—
194C2×Q8C22×D17312yes—
195C2×D4—3462no—
196C2×D4—3462no—
197C2×D4—3464no—
198C2×D4—3462no—
199C2×D4—3461no—
200C2×D4—3461no—
201C2×D4—1731no—
202C2×D4—1732no—
203C2×D4—1731no—
204C2×D4—6924no—
205C2×D4—6922no—
206C2×D4C22×D17311yescharacteristic
207C22×C4—21no—
208C22×C4—3462no—
209C22×C4—3461no—
210C22×C4—1732no—
211SD16—24no—
212SD16—3464no—
213SD16—6922no—
214D8—3464no—
215D8—1734no—
216M16—3464no—
217C2×C8—1731no—
218C2×C8—1731no—
219C2×C8C22×D17312yes—
220C4⋊C4—21no—
221C4⋊C4—21no—
222C4⋊C4—3462no—
223C4⋊C4—1731no—
224C4⋊C4—1731no—
225C4⋊C4C22×D17311yescharacteristic
226C22⋊C4—21no—
227C22⋊C4—3462no—
228C22⋊C4—3462no—
229C22⋊C4—3461no—
230C22⋊C4—1732no—
231C42C22×D17311yescharacteristic
232C23—21no—
233C23—3461no—
234C23—3462no—
235C23—13842no—
236C23—1732no—
237C23—6924no—
238C23—6924no—
239Q8—22no—
240Q8—22no—
241D4—21no—
242D4—3462no—
243D4—3462no—
244D4—3461no—
245D4—3462no—
246D4—13841no—
247D4—1732no—
248D4—6924no—
249D4—6924no—
250D4—6924no—
251D4—6922no—
252D4D4×D17312yes—
253C2×C4—21no—
254C2×C4—22no—
255C2×C4—42no—
256C2×C4—3462no—
257C2×C4—3462no—
258C2×C4—3461no—
259C2×C4—1732no—
260C2×C4—6924no—
261C2×C4—6922no—
262C2×C4D4×D17311yescharacteristic
263C2×C4D4×D17311yescharacteristic
264C2×C4C23×D17311yescharacteristic
265C8—22no—
266C8—3461no—
267C8—1732no—
268C22—22no—
269C22—41no—
270C22—41no—
271C22—3462no—
272C22—3462no—
273C22—3462no—
274C22—3462no—
275C22—13848no—
276C22—6922no—
277C22—6922no—
278C22—6922no—
279C22—6921no—
280C22—6921no—
281C22C2×D4×D17311yescharacteristic
282C4—21no—
283C4—41no—
284C4—42no—
285C4—3462no—
286C4—6921no—
287C4#1914970612yescomplemented
288C4C2×D4×D17311yescharacteristic
289C4#1914972511yescharacteristic
290C2—42no—
291C2—13842no—
292C2—6922no—
293C2—6922no—
294C2#2601212511yescharacteristic
295C2C2×#1914970611yescharacteristic
296C2#2601230011yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#461×C173C2
socleC22×C173C22×D4
hypercenter#461D173
hypocenterC173#3272
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ #461

Complements

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

NH ≅ G/Ntype
1. C2×#19149706C2semidirect
2. #26012144C2semidirect
3. D4×D692C2semidirect
4. #26011865C2semidirect
5. #26011740C2semidirect
6. #26011687C2semidirect
7. #26011640C2semidirect
8. #26011639C2semidirect
9. C4×#14583811C2semidirect
10. #26011568C2semidirect
11. #461×C173C2semidirect
12. C2×#14583827C22semidirect
16. C2×#14583813C22semidirect
17. C2×#14583813C22semidirect
18. C2×D1384C22semidirect
20. C2×#14583809C22semidirect
23. #19149655C22semidirect
29. C4×D692C22semidirect
31. #19149612C22semidirect
32. #19149601C22semidirect
34. #19149599C22semidirect
35. #19149591C22semidirect
36. C4×#13063436C22semidirect
38. C4⋊Q8×C173C22semidirect
40. C4⋊C8×C173C22semidirect
41. #103×C173C22semidirect
62. #14583811D4semidirect
97. D692C2×D4semidirect
106. #13063436C2×D4semidirect
136. C4×C173AGL(1,ℤ/8)semidirect
148. C173#3272semidirect
155. #461D173semidirect
171. C4⋊Q8C2×D173semidirect
184. C4⋊C8C2×D173semidirect
185. #103C2×D173semidirect
287. C4#19149706semidirect

Maximal subgroups

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

subgroupindex
C2×#191497062
#260121442
D4×D6922
#260118652
#260117402
#260116872
#260116402
#260116392
C4×#145838112
#260115682
#461×C1732
#3272173