Finite Group Databasebeta

← #36365159

Subgroups — #36365159

Order 22144.

546 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 546 conjugacy classes of subgroups — too many to draw. Only its 133 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×#19149705C212yesmaximal, complemented
2#26012146C211yesmaximal, characteristic, complemented
3#26011863C211yesmaximal, characteristic, complemented
4#26011735C212yesmaximal, complemented
5#26011696C211yesmaximal, characteristic, complemented
6#26011683C211yesmaximal, characteristic, complemented
7#26011642C212yesmaximal, complemented
8#26011634C212yesmaximal, complemented
9C4×#14583811C211yesmaximal, characteristic, complemented
10#26011569C211yesmaximal, characteristic, complemented
11D42×C173C211yesmaximal, characteristic, complemented
12C2×#14583826C2212yescomplemented
13#19149705—28no—
14C2×#14583814C2212yes—
15C2×#14583814C2212yescomplemented
16C2×#14583814C2212yescomplemented
17C2×#14583811C2211yescharacteristic
18C2×#14583811C2212yescomplemented
19C2×#14583810C2212yescomplemented
20#19149679C2211yescharacteristic, complemented
21#19149661—22no—
22#19149658C2212yescomplemented
23#19149650—21no—
24C4×D692C2211yescharacteristic, complemented
25C4⋊C4×D173—21no—
26#19149612C2211yescharacteristic, complemented
27C4⋊#13063436C2211yescharacteristic, complemented
28#19149602C2212yescomplemented
29#19149600C2211yescharacteristic
30#19149600C2212yescomplemented
31#19149599C2211yescharacteristic, complemented
32C22⋊#13063436C2212yescomplemented
33C4×#13063436C2211yescharacteristic, complemented
34C22×D4×C173—21no—
35C22×D4×C173C2212yes—
36#127×C173C2211yescharacteristic
37#121×C173—21no—
38#121×C173C2212yes—
39#120×C173—22no—
40C4×D4×C173C2211yescharacteristic
41C4×D4×C173C2211yescharacteristic
42#14583826—24no—
43#14583826—42no—
44C2×Dic346C2312yes—
45C2×D692C2311yescharacteristic
46C2×C22⋊D173—22no—
47C2×C4×D173—21no—
48C2×C4×D173—22no—
49#14583814—24no—
50#14583814—24no—
51#14583814—42no—
52#14583811—24no—
53#14583811D414yescomplemented
54#14583810—24no—
55C2×#13063436C2311yescharacteristic
56C2×#13063436C2312yes—
57C2×#13063436C2311yescharacteristic
58C4⋊Dic173—21no—
59C4⋊Dic173—21no—
60C4⋊Dic173C2311yescharacteristic
61#14583805—22no—
62C22⋊Dic173—22no—
63#14583803—21no—
64C24×C173—22no—
65C2×D4×C173—24no—
66C2×D4×C173—22no—
67C2×D4×C173—21no—
68C2×D4×C173—22no—
69C2×D4×C173—21no—
70C2×D4×C173—42no—
71C2×D4×C173—44no—
72C2×D4×C173D411yescharacteristic
73C2×D4×C173C2312yes—
74C2×D4×C173C2311yescharacteristic
75C22×C4×C173—21no—
76C22×C4×C173C2312yes—
77C4⋊C4×C173D411yescharacteristic
78C4⋊C4×C173C2311yescharacteristic
79C22⋊C4×C173—21no—
80C22⋊C4×C173—22no—
81C22⋊C4×C173D412yes—
82C42×C173C2311yescharacteristic
83C22×D173—21no—
84Dic346—22no—
85Dic346—22no—
86D692—21no—
87D692C2×D412yescomplemented
88C22⋊D173—44no—
89C2×Dic173—22no—
90C2×Dic173—21no—
91C4×D173—44no—
92C4×D173—42no—
93#13063436—22no—
94#13063436—21no—
95#13063436C2×D412yescomplemented
96C23×C173—21no—
97C23×C173—22no—
98C23×C173—44no—
99C23×C173—44no—
100C23×C173—82no—
101C23×C173C2×D412yes—
102D4×C173—22no—
103D4×C173—22no—
104D4×C173—21no—
105D4×C173—22no—
106D4×C173—44no—
107D4×C173—44no—
108D4×C173—42no—
109D4×C173—81no—
110D4×C173C2×D412yes—
111C2×C4×C173—22no—
112C2×C4×C173—22no—
113C2×C4×C173—21no—
114C2×C4×C173—42no—
115C2×C4×C173C2×D412yes—
116C2×C4×C173C2×D411yescharacteristic
117C2×C4×C173C2×D411yescharacteristic
118C2×C4×C173C2411yescharacteristic
119C22×C173—22no—
120C22×C173—22no—
121C22×C173—22no—
122C22×C173—22no—
123C22×C173—41no—
124C22×C173—41no—
125C22×C173—42no—
126C22×C173—42no—
127C22×C173—42no—
128C22×C173—88no—
129C22×C173C22×D411yescharacteristic
130C2×D173—22no—
131C2×D173—41no—
132C2×D173—41no—
133C4×C173—22no—
134C4×C173—21no—
135C4×C173—41no—
136C4×C173AGL(1,ℤ/8)12yescomplemented
137C4×C173C22×D411yescharacteristic
138C4×C173H(5,𝔽2)11yescharacteristic
139Dic173—42no—
140Dic173—41no—
141C2×C173—42no—
142C2×C173—42no—
143C2×C173—82no—
144C2×C173D4211yescharacteristic
145C2×C173C2×AGL(1,ℤ/8)11yescharacteristic
146C2×C173#57711yescharacteristic
147D173—42no—
148C173#327211yescharacteristic, complemented, Sylow
149#3272—1731nomaximal, Sylow
150C2×AGL(1,ℤ/8)—1732no—
151#547—1731no—
152D42D17311yescharacteristic, complemented
153#492—1731no—
154#468—1732no—
155#461—1731no—
156#459—1731no—
157#450—1732no—
158#449—1732no—
159C4×D8—1731no—
160#436—1731no—
161C2×#41—1732no—
162C22×D4—21no—
163C22×D4C2×D17312yes—
164AGL(1,ℤ/8)—3468no—
165C2×SD16—1732no—
166C2×SD16—1732no—
167C2×SD16—1732no—
168C2×D8—1732no—
169C2×D8—1731no—
170C2×M16—1732no—
171C4⋊Q8—1731no—
172#127C2×D17311yescharacteristic
173#123—3462no—
174C22⋊Q8—1732no—
175#121—21no—
176#121—3461no—
177#121C2×D17312yes—
178#120—22no—
179C4×D4—1731no—
180C4×D4C2×D17311yescharacteristic
181C4×D4C2×D17311yescharacteristic
182C2×C4⋊C4—3461no—
183#107—1731no—
184C4⋊C8—1731no—
185#103—1732no—
186#102—1731no—
187#102—1732no—
188#102—1731no—
189C22⋊C8—1732no—
190C4×C8—1731no—
191C24—22no—
192#41—3464no—
193#41—6922no—
194C2×Q8—1732no—
195C2×D4—24no—
196C2×D4—22no—
197C2×D4—22no—
198C2×D4—21no—
199C2×D4—21no—
200C2×D4—42no—
201C2×D4—44no—
202C2×D4—3462no—
203C2×D4—1731no—
204C2×D4C22⋊D17311yescharacteristic
205C2×D4C22×D17312yes—
206C2×D4C22×D17311yescharacteristic
207C22×C4—21no—
208C22×C4—3461no—
209C22×C4—3462no—
210C22×C4C22×D17312yes—
211SD16—3464no—
212SD16—3464no—
213SD16—6922no—
214D8—3464no—
215D8—1734no—
216M16—3464no—
217C2×C8—1731no—
218C2×C8—1732no—
219C2×C8—1731no—
220C4⋊C4—3461no—
221C4⋊C4—3462no—
222C4⋊C4—3461no—
223C4⋊C4—1731no—
224C4⋊C4C22⋊D17311yescharacteristic
225C4⋊C4C22×D17311yescharacteristic
226C22⋊C4—21no—
227C22⋊C4—22no—
228C22⋊C4—3462no—
229C22⋊C4—3461no—
230C22⋊C4C22⋊D17312yes—
231C42C22×D17311yescharacteristic
232C23—21no—
233C23—22no—
234C23—44no—
235C23—44no—
236C23—82no—
237C23—3461no—
238C23C2×C22⋊D17312yes—
239Q8—3462no—
240Q8—3462no—
241D4—22no—
242D4—22no—
243D4—21no—
244D4—22no—
245D4—44no—
246D4—44no—
247D4—42no—
248D4—81no—
249D4—3461no—
250D4—1732no—
251D4—6924no—
252D4D4×D17312yes—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—21no—
256C2×C4—42no—
257C2×C4—3461no—
258C2×C4—3462no—
259C2×C4—6924no—
260C2×C4—6922no—
261C2×C4C2×C22⋊D17311yescharacteristic
262C2×C4C2×C22⋊D17312yes—
263C2×C4C2×C22⋊D17311yescharacteristic
264C2×C4C23×D17311yescharacteristic
265C8—3462no—
266C8—3461no—
267C8—1732no—
268C22—22no—
269C22—22no—
270C22—22no—
271C22—22no—
272C22—41no—
273C22—41no—
274C22—42no—
275C22—42no—
276C22—42no—
277C22—88no—
278C22—3462no—
279C22—6921no—
280C22—6921no—
281C22C22×C22⋊D17311yescharacteristic
282C4—21no—
283C4—22no—
284C4—41no—
285C4—6921no—
286C4—6922no—
287C4#1914970512yescomplemented
288C4C2×D4×D17311yescharacteristic
289C4#1914972411yescharacteristic
290C2—42no—
291C2—42no—
292C2—82no—
293C2—6922no—
294C2D4×C22⋊D17311yescharacteristic
295C2C2×#1914970511yescharacteristic
296C2#2601230211yescharacteristic
subgroupG/H
centerC22C22×C22⋊D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
FittingD42×C173C2
socleC22×C173C22×D4
hypercenterD42D173
hypocenterC173#3272
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C4×D4 ◁ D42

Complements

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

NH ≅ G/Ntype
1. C2×#19149705C2semidirect
2. #26012146C2semidirect
3. #26011863C2semidirect
4. #26011735C2semidirect
5. #26011696C2semidirect
6. #26011683C2semidirect
7. #26011642C2semidirect
8. #26011634C2semidirect
9. C4×#14583811C2semidirect
10. #26011569C2semidirect
11. D42×C173C2semidirect
12. C2×#14583826C22semidirect
15. C2×#14583814C22semidirect
16. C2×#14583814C22semidirect
18. C2×#14583811C22semidirect
19. C2×#14583810C22semidirect
20. #19149679C22semidirect
22. #19149658C22semidirect
24. C4×D692C22semidirect
26. #19149612C22semidirect
27. C4⋊#13063436C22semidirect
28. #19149602C22semidirect
30. #19149600C22semidirect
31. #19149599C22semidirect
32. C22⋊#13063436C22semidirect
33. C4×#13063436C22semidirect
53. #14583811D4semidirect
87. D692C2×D4semidirect
95. #13063436C2×D4semidirect
136. C4×C173AGL(1,ℤ/8)semidirect
148. C173#3272semidirect
152. D42D173semidirect
287. C4#19149705semidirect

Maximal subgroups

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

subgroupindex
C2×#191497052
#260121462
#260118632
#260117352
#260116962
#260116832
#260116422
#260116342
C4×#145838112
#260115692
D42×C1732
#3272173