Finite Group Databasebeta

← #36365185

Subgroups — #36365185

Order 22144.

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

normal subgroups only

This group has 556 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#26012162C211yesmaximal, characteristic, complemented
3D4×D692C211yesmaximal, characteristic, complemented
4#26011868C211yesmaximal, characteristic, complemented
5#26011744C212yesmaximal, complemented
6#26011692C211yesmaximal, characteristic, complemented
7#26011687C211yesmaximal, characteristic, complemented
8#26011638C212yesmaximal, complemented
9#26011627C212yesmaximal, complemented
10C4×#14583815C211yesmaximal, characteristic, complemented
11#436×C173C211yesmaximal, characteristic, complemented
12C2×#14583826C2212yescomplemented
13C22×D692C2212yes—
14C2×D4×D173—21no—
15#19149707—28no—
16C2×#14583815C2211yescharacteristic
17C2×#14583815C2212yescomplemented
18C2×D1384C2212yes—
19C2×D1384C2212yescomplemented
20C2×D1384C2212yescomplemented
21#19149676C2211yescharacteristic
22#19149676C2211yescharacteristic, complemented
23#19149675—21no—
24#19149654—21no—
25#19149653C2212yes—
26#19149653C2212yescomplemented
27#19149652—22no—
28#19149649—22no—
29C4×Dic346C2211yescharacteristic, complemented
30C4×D692C2211yescharacteristic
31#19149641—21no—
32#19149611C2211yescharacteristic, complemented
33#19149604C2211yescharacteristic, complemented
34#19149601C2211yescharacteristic
35#19149601C2212yescomplemented
36#19149601C2212yescomplemented
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×Dic346C2311yescharacteristic
46C2×D692—24no—
47C2×D692—21no—
48C2×D692—22no—
49C2×D692—21no—
50C2×D692—21no—
51C2×D692—21no—
52C2×D692—42no—
53C2×D692C2312yes—
54C2×D692C2312yes—
55C2×D692C2311yescharacteristic
56C2×C22⋊D173—22no—
57C2×C22⋊D173—22no—
58C2×C22⋊D173—22no—
59D4×D173—44no—
60C2×C4×D173—22no—
61C2×C4×D173—21no—
62#14583815—24no—
63#14583815D414yescomplemented
64D1384—24no—
65D1384—24no—
66D1384—42no—
67C4⋊Dic173C2311yescharacteristic
68C4⋊Dic173C2311yescharacteristic
69#14583805—21no—
70#14583803—22no—
71#14583803—21no—
72#14583803—22no—
73C4×Dic173—21no—
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—22no—
85C22×D173—22no—
86C22×D173—44no—
87C22×D173—44no—
88C22×D173—82no—
89Dic346—21no—
90Dic346C2×D412yescomplemented
91D692—22no—
92D692—22no—
93D692—21no—
94D692—22no—
95D692—22no—
96D692—44no—
97D692—42no—
98D692—42no—
99D692—81no—
100D692—81no—
101D692C2×D412yes—
102C22⋊D173—44no—
103C22⋊D173—44no—
104C22⋊D173—82no—
105C2×Dic173—21no—
106C2×Dic173—21no—
107C2×Dic173—21no—
108C4×D173—44no—
109C4×D173—42no—
110C23×C173C2×D412yes—
111D4×C173—22no—
112C2×C4×C173—22no—
113C2×C4×C173—22no—
114C2×C4×C173C2×D412yes—
115C2×C4×C173C2×D411yescharacteristic
116C2×C4×C173C2×D411yescharacteristic
117C2×C4×C173C2411yescharacteristic
118C8×C173—22no—
119C8×C173—21no—
120C8×C173C2×D412yescomplemented
121C22×C173—22no—
122C22×C173—22no—
123C22×C173—22no—
124C22×C173C22×D411yescharacteristic
125C2×D173—22no—
126C2×D173—41no—
127C2×D173—41no—
128C2×D173—42no—
129C2×D173—42no—
130C2×D173—42no—
131C2×D173—42no—
132C2×D173—42no—
133C2×D173—42no—
134C2×D173—88no—
135C4×C173—22no—
136C4×C173—21no—
137C4×C173AGL(1,ℤ/8)12yescomplemented
138C4×C173C22×D411yescharacteristic
139C4×C173H(5,𝔽2)11yescharacteristic
140Dic173—22no—
141Dic173—41no—
142Dic173—41no—
143C2×C173—42no—
144C2×C173D4211yescharacteristic
145C2×C173C2×AGL(1,ℤ/8)11yescharacteristic
146C2×C173#57611yescharacteristic
147D173—42no—
148D173—82no—
149D173—82no—
150C173#327411yescharacteristic, complemented, Sylow
151#3274—1731nomaximal, Sylow
152C2×AGL(1,ℤ/8)—1732no—
153#550—1731no—
154D42—1731no—
155#493—1731no—
156#469—1732no—
157#460—1731no—
158#459—1731no—
159#449—1732no—
160#447—1732no—
161C4×SD16—1731no—
162#436D17311yescharacteristic, complemented
163C2×#41—1732no—
164C22×D4—3461no—
165C22×D4—1732no—
166AGL(1,ℤ/8)—3468no—
167C2×SD16—1732no—
168C2×SD16—1731no—
169C2×D8—1732no—
170C2×D8—1732no—
171C2×D8—1732no—
172C2×M16C2×D17312yescomplemented
173#127—3461no—
174#127—1731no—
175#127—1731no—
176#121—3461no—
177#121—3462no—
178#121—1732no—
179#121—1732no—
180#120—3462no—
181C4×Q8—1731no—
182C4×D4—3461no—
183C4×D4—1731no—
184C4×D4C2×D17311yescharacteristic
185#106—1731no—
186C4⋊C8C2×D17311yescharacteristic, complemented
187#103—1731no—
188#102—1731no—
189#102—1732no—
190#102—1732no—
191C22⋊C8C2×D17312yescomplemented
192C4×C8C2×D17311yescharacteristic, complemented
193C24—3462no—
194#41—3464no—
195#41—6922no—
196C2×Q8—1731no—
197C2×D4—3464no—
198C2×D4—3462no—
199C2×D4—3462no—
200C2×D4—3462no—
201C2×D4—3461no—
202C2×D4—3461no—
203C2×D4—3461no—
204C2×D4—3462no—
205C2×D4—3461no—
206C2×D4—1732no—
207C2×D4—1732no—
208C2×D4—1731no—
209C2×D4—6922no—
210C2×D4—6924no—
211C2×D4D69211yescharacteristic
212C22×C4—3462no—
213C22×C4—3461no—
214C22×C4C22×D17312yes—
215SD16—3464no—
216SD16—1734no—
217D8—3464no—
218D8—3464no—
219D8—6922no—
220M16—24no—
221C2×C8C22×D17311yescharacteristic
222C2×C8C22×D17312yes—
223C2×C8C22×D17311yescharacteristic
224C4⋊C4—3461no—
225C4⋊C4—1731no—
226C4⋊C4—1731no—
227C4⋊C4D69211yescharacteristic
228C22⋊C4—3462no—
229C22⋊C4—3461no—
230C22⋊C4—3462no—
231C22⋊C4D69212yes—
232C42—3461no—
233C42C22×D17311yescharacteristic
234C23—3461no—
235C23—3462no—
236C23—3462no—
237C23—13842no—
238C23—6924no—
239C23—6924no—
240C23C2×D69212yes—
241Q8—3461no—
242Q8—1732no—
243D4—22no—
244D4—3462no—
245D4—3462no—
246D4—3461no—
247D4—3462no—
248D4—3462no—
249D4—13841no—
250D4—13841no—
251D4—13842no—
252D4—1732no—
253D4—6924no—
254D4—6924no—
255D4—6924no—
256D4—6922no—
257D4—6922no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—3461no—
261C2×C4—3461no—
262C2×C4—3461no—
263C2×C4—6924no—
264C2×C4—6922no—
265C2×C4C2×D69211yescharacteristic
266C2×C4C2×D69212yes—
267C2×C4C2×D69211yescharacteristic
268C2×C4C23×D17311yescharacteristic
269C8—22no—
270C8—21no—
271C8D4×D17312yescomplemented
272C22—22no—
273C22—22no—
274C22—22no—
275C22—3462no—
276C22—13848no—
277C22—6922no—
278C22—6922no—
279C22—6922no—
280C22—6922no—
281C22—6922no—
282C22—6922no—
283C22—6921no—
284C22—6921no—
285C22C22×D69211yescharacteristic
286C4—21no—
287C4—22no—
288C4—3462no—
289C4—6921no—
290C4—6921no—
291C4#1914970712yescomplemented
292C4C2×D4×D17311yescharacteristic
293C4#1914972511yescharacteristic
294C2—42no—
295C2—13842no—
296C2—13842no—
297C2—6922no—
298C2D4×D69211yescharacteristic
299C2C2×#1914970711yescharacteristic
300C2#2601229611yescharacteristic
subgroupG/H
centerC22C22×D692
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#436×C173C2
socleC22×C173C22×D4
hypercenter#436D173
hypocenterC173#3274
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. #26012162C2semidirect
3. D4×D692C2semidirect
4. #26011868C2semidirect
5. #26011744C2semidirect
6. #26011692C2semidirect
7. #26011687C2semidirect
8. #26011638C2semidirect
9. #26011627C2semidirect
10. C4×#14583815C2semidirect
11. #436×C173C2semidirect
12. C2×#14583826C22semidirect
17. C2×#14583815C22semidirect
19. C2×D1384C22semidirect
20. C2×D1384C22semidirect
22. #19149676C22semidirect
26. #19149653C22semidirect
29. C4×Dic346C22semidirect
32. #19149611C22semidirect
33. #19149604C22semidirect
35. #19149601C22semidirect
36. #19149601C22semidirect
37. C2×M16×C173C22semidirect
39. C4⋊C8×C173C22semidirect
40. C22⋊C8×C173C22semidirect
41. C4×C8×C173C22semidirect
63. #14583815D4semidirect
90. Dic346C2×D4semidirect
120. C8×C173C2×D4semidirect
137. C4×C173AGL(1,ℤ/8)semidirect
150. C173#3274semidirect
162. #436D173semidirect
172. C2×M16C2×D173semidirect
186. C4⋊C8C2×D173semidirect
191. C22⋊C8C2×D173semidirect
192. C4×C8C2×D173semidirect
271. C8D4×D173semidirect
291. C4#19149707semidirect

Maximal subgroups

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

subgroupindex
C2×#191497072
#260121622
D4×D6922
#260118682
#260117442
#260116922
#260116872
#260116382
#260116272
C4×#145838152
#436×C1732
#3274173