Finite Group Databasebeta

← #36365165

Subgroups — #36365165

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×#19149707C212yesmaximal, complemented
2#26012146C211yesmaximal, characteristic, complemented
3D4×D692C211yesmaximal, characteristic, complemented
4#26011864C211yesmaximal, characteristic, complemented
5#26011737C212yesmaximal, complemented
6#26011697C211yesmaximal, characteristic, complemented
7#26011687C211yesmaximal, characteristic, complemented
8#26011644C212yesmaximal, complemented
9#26011638C212yesmaximal, complemented
10C4×D1384C211yesmaximal, characteristic, complemented
11#436×C173C211yesmaximal, characteristic, complemented
12C2×#14583826C2212yescomplemented
13C22×D692C2212yes—
14C2×D4×D173—21no—
15#19149707—28no—
16C2×#14583815C2212yes—
17C2×#14583815C2212yescomplemented
18C2×#14583815C2212yescomplemented
19C2×D1384C2211yescharacteristic
20C2×D1384C2212yescomplemented
21#19149679C2211yescharacteristic, complemented
22#19149676C2211yescharacteristic
23#19149661—22no—
24#19149658C2212yescomplemented
25#19149654—21no—
26#19149653C2212yes—
27#19149650—21no—
28#19149649—22no—
29C4×D692C2211yescharacteristic
30C4×D692C2211yescharacteristic, complemented
31C4⋊C4×D173—21no—
32#19149613C2211yescharacteristic, complemented
33#19149604C2212yescomplemented
34#19149601C2211yescharacteristic
35#19149601C2212yescomplemented
36#19149601C2211yescharacteristic, complemented
37C2×M16×C173C2212yescomplemented
38C4×D4×C173C2211yescharacteristic
39C4⋊C8×C173C2211yescharacteristic, complemented
40C22⋊C8×C173C2212yescomplemented
41C4×C8×C173C2211yescharacteristic, complemented
42C23×D173—22no—
43#14583826—24no—
44#14583826—42no—
45C2×Dic346C2312yes—
46C2×D692—24no—
47C2×D692—22no—
48C2×D692—21no—
49C2×D692—21no—
50C2×D692—42no—
51C2×D692C2312yes—
52C2×D692C2311yescharacteristic
53C2×D692C2311yescharacteristic
54C2×C22⋊D173—22no—
55C2×C22⋊D173—22no—
56D4×D173—44no—
57C2×C4×D173—21no—
58C2×C4×D173—22no—
59C2×C4×D173—21no—
60#14583815—24no—
61#14583815—24no—
62#14583815—42no—
63D1384—24no—
64D1384D414yescomplemented
65C4⋊Dic173—21no—
66C4⋊Dic173—21no—
67C4⋊Dic173C2311yescharacteristic
68C4⋊Dic173C2311yescharacteristic
69#14583805—22no—
70C22⋊Dic173—22no—
71#14583803—21no—
72#14583803—21no—
73#14583803—22no—
74C2×D4×C173D411yescharacteristic
75C22×C4×C173C2312yes—
76M16×C173—24no—
77C2×C8×C173C2312yes—
78C2×C8×C173C2311yescharacteristic
79C2×C8×C173C2311yescharacteristic
80C4⋊C4×C173D411yescharacteristic
81C22⋊C4×C173D412yes—
82C42×C173C2311yescharacteristic
83C22×D173—21no—
84C22×D173—21no—
85C22×D173—22no—
86C22×D173—44no—
87C22×D173—44no—
88C22×D173—82no—
89Dic346—22no—
90Dic346—22no—
91D692—22no—
92D692—21no—
93D692—21no—
94D692—22no—
95D692—44no—
96D692—42no—
97D692—81no—
98D692C2×D412yes—
99D692C2×D412yescomplemented
100C22⋊D173—44no—
101C22⋊D173—44no—
102C2×Dic173—21no—
103C2×Dic173—22no—
104C2×Dic173—21no—
105C4×D173—44no—
106C4×D173—42no—
107C4×D173—42no—
108C23×C173C2×D412yes—
109D4×C173—22no—
110C2×C4×C173—22no—
111C2×C4×C173—22no—
112C2×C4×C173C2×D412yes—
113C2×C4×C173C2×D411yescharacteristic
114C2×C4×C173C2×D411yescharacteristic
115C2×C4×C173C2411yescharacteristic
116C8×C173—22no—
117C8×C173—21no—
118C8×C173C2×D412yescomplemented
119C22×C173—22no—
120C22×C173—22no—
121C22×C173—22no—
122C22×C173C22×D411yescharacteristic
123C2×D173—22no—
124C2×D173—22no—
125C2×D173—41no—
126C2×D173—41no—
127C2×D173—41no—
128C2×D173—41no—
129C2×D173—42no—
130C2×D173—42no—
131C2×D173—42no—
132C2×D173—88no—
133C4×C173—22no—
134C4×C173—21no—
135C4×C173AGL(1,ℤ/8)12yescomplemented
136C4×C173C22×D411yescharacteristic
137C4×C173H(5,𝔽2)11yescharacteristic
138Dic173—42no—
139Dic173—41no—
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#461—1731no—
156#459—1731no—
157#450—1732no—
158#449—1732no—
159C4×D8—1731no—
160#436D17311yescharacteristic, complemented
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×D8—1731no—
170C2×M16C2×D17312yescomplemented
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×D4—1731no—
181C4×D4C2×D17311yescharacteristic
182C2×C4⋊C4—3461no—
183#107—1731no—
184C4⋊C8C2×D17311yescharacteristic, complemented
185#103—1732no—
186#102—1731no—
187#102—1731no—
188#102—1732no—
189C22⋊C8C2×D17312yescomplemented
190C4×C8C2×D17311yescharacteristic, complemented
191C24—3462no—
192#41—3464no—
193#41—6922no—
194C2×Q8—1732no—
195C2×D4—3464no—
196C2×D4—3462no—
197C2×D4—3462no—
198C2×D4—3461no—
199C2×D4—3462no—
200C2×D4—3461no—
201C2×D4—1732no—
202C2×D4—1731no—
203C2×D4—1731no—
204C2×D4—6922no—
205C2×D4—6924no—
206C2×D4D69211yescharacteristic
207C22×C4—3461no—
208C22×C4—3462no—
209C22×C4—3461no—
210C22×C4C22×D17312yes—
211SD16—3464no—
212SD16—3464no—
213SD16—6922no—
214D8—3464no—
215D8—1734no—
216M16—24no—
217C2×C8C22×D17311yescharacteristic
218C2×C8C22×D17312yes—
219C2×C8C22×D17311yescharacteristic
220C4⋊C4—3461no—
221C4⋊C4—3462no—
222C4⋊C4—3461no—
223C4⋊C4—1731no—
224C4⋊C4—1731no—
225C4⋊C4D69211yescharacteristic
226C22⋊C4—3462no—
227C22⋊C4—3461no—
228C22⋊C4—3461no—
229C22⋊C4—3462no—
230C22⋊C4D69212yes—
231C42C22×D17311yescharacteristic
232C23—3461no—
233C23—3461no—
234C23—3462no—
235C23—13842no—
236C23—6924no—
237C23—6924no—
238C23C2×D69212yes—
239Q8—3462no—
240Q8—3462no—
241D4—22no—
242D4—3462no—
243D4—3461no—
244D4—3461no—
245D4—3462no—
246D4—13841no—
247D4—1732no—
248D4—1732no—
249D4—6924no—
250D4—6924no—
251D4—6924no—
252D4—6922no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—3461no—
256C2×C4—3461no—
257C2×C4—3462no—
258C2×C4—6924no—
259C2×C4—6922no—
260C2×C4—6922no—
261C2×C4C2×D69211yescharacteristic
262C2×C4C2×D69212yes—
263C2×C4C2×D69211yescharacteristic
264C2×C4C23×D17311yescharacteristic
265C8—22no—
266C8—21no—
267C8D4×D17312yescomplemented
268C22—22no—
269C22—22no—
270C22—22no—
271C22—3462no—
272C22—3462no—
273C22—13848no—
274C22—6922no—
275C22—6922no—
276C22—6922no—
277C22—6921no—
278C22—6921no—
279C22—6921no—
280C22—6921no—
281C22C22×D69211yescharacteristic
282C4—21no—
283C4—22no—
284C4—6921no—
285C4—6921no—
286C4—6922no—
287C4#1914970712yescomplemented
288C4C2×D4×D17311yescharacteristic
289C4#1914972511yescharacteristic
290C2—42no—
291C2—13842no—
292C2—6922no—
293C2—6922no—
294C2D4×D69211yescharacteristic
295C2C2×#1914970711yescharacteristic
296C2#2601230311yescharacteristic
subgroupG/H
centerC22C22×D692
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#436×C173C2
socleC22×C173C22×D4
hypercenter#436D173
hypocenterC173#3272
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C4×D4 ◁ #436

Complements

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

NH ≅ G/Ntype
1. C2×#19149707C2semidirect
2. #26012146C2semidirect
3. D4×D692C2semidirect
4. #26011864C2semidirect
5. #26011737C2semidirect
6. #26011697C2semidirect
7. #26011687C2semidirect
8. #26011644C2semidirect
9. #26011638C2semidirect
10. C4×D1384C2semidirect
11. #436×C173C2semidirect
12. C2×#14583826C22semidirect
17. C2×#14583815C22semidirect
18. C2×#14583815C22semidirect
20. C2×D1384C22semidirect
21. #19149679C22semidirect
24. #19149658C22semidirect
30. C4×D692C22semidirect
32. #19149613C22semidirect
33. #19149604C22semidirect
35. #19149601C22semidirect
36. #19149601C22semidirect
37. C2×M16×C173C22semidirect
39. C4⋊C8×C173C22semidirect
40. C22⋊C8×C173C22semidirect
41. C4×C8×C173C22semidirect
64. D1384D4semidirect
99. D692C2×D4semidirect
118. C8×C173C2×D4semidirect
135. C4×C173AGL(1,ℤ/8)semidirect
148. C173#3272semidirect
160. #436D173semidirect
170. C2×M16C2×D173semidirect
184. C4⋊C8C2×D173semidirect
189. C22⋊C8C2×D173semidirect
190. C4×C8C2×D173semidirect
267. C8D4×D173semidirect
287. C4#19149707semidirect

Maximal subgroups

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

subgroupindex
C2×#191497072
#260121462
D4×D6922
#260118642
#260117372
#260116972
#260116872
#260116442
#260116382
C4×D13842
#436×C1732
#3272173