Finite Group Databasebeta

← #36365164

Subgroups — #36365164

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×#19149703C212yesmaximal, complemented
2#26012145C211yesmaximal, characteristic, complemented
3D4×D692C211yesmaximal, characteristic, complemented
4#26011864C211yesmaximal, characteristic, complemented
5#26011738C212yesmaximal, complemented
6#26011696C211yesmaximal, characteristic, complemented
7#26011686C211yesmaximal, characteristic, complemented
8#26011643C212yesmaximal, complemented
9#26011637C212yesmaximal, complemented
10#26011570C211yesmaximal, characteristic, complemented
11C4×D8×C173C211yesmaximal, characteristic, complemented
12C2×#14583825C2212yescomplemented
13C22×D692—21no—
14C2×D4×D173C2212yes—
15#19149703—28no—
16C2×#14583815C2212yescomplemented
17C2×#14583815C2212yescomplemented
18C2×#14583814C2212yes—
19C2×#14583811C2212yescomplemented
20C2×#14583809C2212yescomplemented
21#19149679C2211yescharacteristic, complemented
22#19149676C2211yescharacteristic
23#19149663—22no—
24#19149659C2212yescomplemented
25#19149654C2212yes—
26#19149653—21no—
27#19149653—21no—
28#19149649—22no—
29C4×D692C2211yescharacteristic
30C2×C4⋊Dic173—21no—
31C4⋊#13063436C2211yescharacteristic, complemented
32#19149602C2212yescomplemented
33#19149599C2212yescomplemented
34#19149591C2212yescomplemented
35C2×D8×C173C2211yescharacteristic
36C4×D4×C173C2211yescharacteristic
37C4×D4×C173C2211yescharacteristic, complemented
38#107×C173C2211yescharacteristic, complemented
39#102×C173C2211yescharacteristic
40#102×C173C2211yescharacteristic, complemented
41C4×C8×C173C2211yescharacteristic, complemented
42C23×D173—22no—
43#14583825—24no—
44#14583825—42no—
45C2×Dic346C2312yes—
46C2×D692—22no—
47C2×D692—21no—
48C2×D692—21no—
49C2×D692—44no—
50C2×D692D411yescharacteristic
51C2×D692C2312yes—
52C2×C22⋊D173—22no—
53C2×C22⋊D173—22no—
54D4×D173—24no—
55D4×D173—42no—
56C22×Dic173—22no—
57C2×C4×D173C2312yes—
58#14583815—24no—
59#14583815—42no—
60#14583814—24no—
61#14583811—24no—
62#14583809—24no—
63C2×#13063436C2312yes—
64C4⋊Dic173—22no—
65C4⋊Dic173—21no—
66C4⋊Dic173—21no—
67C4⋊Dic173D411yescharacteristic
68#14583803—22no—
69#14583803—22no—
70#14583803D412yes—
71C2×D4×C173C2311yescharacteristic
72C2×D4×C173C2311yescharacteristic
73C22×C4×C173—21no—
74C22×C4×C173—21no—
75D8×C173D414yescomplemented
76C2×C8×C173C2311yescharacteristic
77C2×C8×C173C2311yescharacteristic
78C4⋊C4×C173C2311yescharacteristic
79C4⋊C4×C173C2311yescharacteristic
80C22⋊C4×C173—21no—
81C22⋊C4×C173—21no—
82C42×C173C2311yescharacteristic
83C22×D173—22no—
84C22×D173—44no—
85C22×D173—44no—
86C22×D173—82no—
87C22×D173C2×D412yes—
88Dic346—22no—
89Dic346—22no—
90D692—22no—
91D692—22no—
92D692—22no—
93D692—44no—
94D692—42no—
95D692—81no—
96C22⋊D173—44no—
97C22⋊D173—44no—
98C2×Dic173—22no—
99C2×Dic173—44no—
100C2×Dic173C2×D412yes—
101C4×D173—22no—
102C4×D173—22no—
103#13063436—22no—
104C23×C173—21no—
105C23×C173—21no—
106D4×C173—21no—
107D4×C173—21no—
108D4×C173C2×D412yes—
109D4×C173C2×D412yescomplemented
110C2×C4×C173—21no—
111C2×C4×C173—21no—
112C2×C4×C173—42no—
113C2×C4×C173—42no—
114C2×C4×C173C2×D411yescharacteristic
115C2×C4×C173C2×D411yescharacteristic
116C2×C4×C173C2411yescharacteristic
117C8×C173—21no—
118C8×C173C2×D412yescomplemented
119C22×C173—22no—
120C22×C173—22no—
121C22×C173—41no—
122C22×C173—41no—
123C22×C173—41no—
124C22×C173—41no—
125C22×C173C22×D411yescharacteristic
126C2×D173—22no—
127C2×D173—22no—
128C2×D173—22no—
129C2×D173—42no—
130C2×D173—42no—
131C2×D173—42no—
132C2×D173—88no—
133C4×C173—21no—
134C4×C173—41no—
135C4×C173—41no—
136C4×C173AGL(1,ℤ/8)12yescomplemented
137C4×C173C22×D411yescharacteristic
138C4×C173H(5,𝔽2)11yescharacteristic
139Dic173—22no—
140Dic173—42no—
141C2×C173—42no—
142C2×C173—42no—
143C2×C173D4211yescharacteristic
144C2×C173C2×AGL(1,ℤ/8)11yescharacteristic
145C2×C173#57711yescharacteristic
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#461—1731no—
156#459—1731no—
157#450—1732no—
158#449—1732no—
159C4×D8D17311yescharacteristic, complemented
160#436—1731no—
161C2×#41—1732no—
162C22×D4—3461no—
163C22×D4—1732no—
164AGL(1,ℤ/8)—3468no—
165C2×SD16—1732no—
166C2×SD16—1732no—
167C2×SD16—1732no—
168C2×D8—1732no—
169C2×D8C2×D17311yescharacteristic
170C2×M16—1732no—
171C4⋊Q8—1731no—
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×D4C2×D17311yescharacteristic
181C4×D4C2×D17311yescharacteristic, complemented
182C2×C4⋊C4—3461no—
183#107C2×D17311yescharacteristic, complemented
184C4⋊C8—1731no—
185#103—1732no—
186#102—1732no—
187#102C2×D17311yescharacteristic
188#102C2×D17311yescharacteristic, complemented
189C22⋊C8—1732no—
190C4×C8C2×D17311yescharacteristic, complemented
191C24—3462no—
192#41—3464no—
193#41—6922no—
194C2×Q8—1732no—
195C2×D4—3462no—
196C2×D4—3464no—
197C2×D4—3462no—
198C2×D4—3462no—
199C2×D4—3461no—
200C2×D4—3461no—
201C2×D4—1731no—
202C2×D4—1732no—
203C2×D4—6924no—
204C2×D4—6922no—
205C2×D4C22×D17311yescharacteristic
206C2×D4C22×D17311yescharacteristic
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—3462no—
210C22×C4—1732no—
211SD16—3464no—
212SD16—3464no—
213SD16—6922no—
214D8—3464no—
215D8D69214yescomplemented
216M16—3464no—
217C2×C8—1732no—
218C2×C8C22×D17311yescharacteristic
219C2×C8C22×D17311yescharacteristic
220C4⋊C4—3462no—
221C4⋊C4—3461no—
222C4⋊C4—3461no—
223C4⋊C4—1731no—
224C4⋊C4C22×D17311yescharacteristic
225C4⋊C4C22×D17311yescharacteristic
226C22⋊C4—21no—
227C22⋊C4—21no—
228C22⋊C4—3462no—
229C22⋊C4—3462no—
230C22⋊C4—1732no—
231C42C22×D17311yescharacteristic
232C23—21no—
233C23—21no—
234C23—3462no—
235C23—13842no—
236C23—1732no—
237C23—6924no—
238C23—6924no—
239Q8—3462no—
240Q8—3462no—
241D4—21no—
242D4—21no—
243D4—3462no—
244D4—3462no—
245D4—3462no—
246D4—13841no—
247D4—6924no—
248D4—6924no—
249D4—6924no—
250D4—6922no—
251D4C2×D69212yes—
252D4C2×D69212yescomplemented
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—3462no—
258C2×C4—3462no—
259C2×C4—3462no—
260C2×C4—1732no—
261C2×C4—6924no—
262C2×C4D4×D17311yescharacteristic
263C2×C4D4×D17311yescharacteristic
264C2×C4C23×D17311yescharacteristic
265C8—21no—
266C8—3462no—
267C8C2×D69212yescomplemented
268C22—22no—
269C22—22no—
270C22—41no—
271C22—41no—
272C22—41no—
273C22—41no—
274C22—3462no—
275C22—3462no—
276C22—3462no—
277C22—13848no—
278C22—6922no—
279C22—6922no—
280C22—6922no—
281C22C2×D4×D17311yescharacteristic
282C4—21no—
283C4—41no—
284C4—41no—
285C4—3462no—
286C4—6922no—
287C4#1914970312yescomplemented
288C4C22×D69211yescharacteristic
289C4#1914972511yescharacteristic
290C2—42no—
291C2—42no—
292C2—13842no—
293C2—6922no—
294C2D4×D69211yescharacteristic
295C2C2×#1914970311yescharacteristic
296C2#2601229811yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
FittingC4×D8×C173C2
socleC22×C173C22×D4
hypercenterC4×D8D173
hypocenterC173#3272
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ C4×D8

Complements

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

NH ≅ G/Ntype
1. C2×#19149703C2semidirect
2. #26012145C2semidirect
3. D4×D692C2semidirect
4. #26011864C2semidirect
5. #26011738C2semidirect
6. #26011696C2semidirect
7. #26011686C2semidirect
8. #26011643C2semidirect
9. #26011637C2semidirect
10. #26011570C2semidirect
11. C4×D8×C173C2semidirect
12. C2×#14583825C22semidirect
16. C2×#14583815C22semidirect
17. C2×#14583815C22semidirect
19. C2×#14583811C22semidirect
20. C2×#14583809C22semidirect
21. #19149679C22semidirect
24. #19149659C22semidirect
31. C4⋊#13063436C22semidirect
32. #19149602C22semidirect
33. #19149599C22semidirect
34. #19149591C22semidirect
37. C4×D4×C173C22semidirect
38. #107×C173C22semidirect
40. #102×C173C22semidirect
41. C4×C8×C173C22semidirect
75. D8×C173D4semidirect
109. D4×C173C2×D4semidirect
118. C8×C173C2×D4semidirect
136. C4×C173AGL(1,ℤ/8)semidirect
148. C173#3272semidirect
159. C4×D8D173semidirect
181. C4×D4C2×D173semidirect
183. #107C2×D173semidirect
188. #102C2×D173semidirect
190. C4×C8C2×D173semidirect
215. D8D692semidirect
252. D4C2×D692semidirect
267. C8C2×D692semidirect
287. C4#19149703semidirect

Maximal subgroups

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

subgroupindex
C2×#191497032
#260121452
D4×D6922
#260118642
#260117382
#260116962
#260116862
#260116432
#260116372
#260115702
C4×D8×C1732
#3272173