Finite Group Databasebeta

← #36365184

Subgroups — #36365184

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×#19149706C212yesmaximal, complemented
2#26012163C211yesmaximal, characteristic, complemented
3D4×D692C211yesmaximal, characteristic, complemented
4#26011868C211yesmaximal, characteristic, complemented
5#26011745C212yesmaximal, complemented
6#26011691C211yesmaximal, characteristic, complemented
7#26011686C211yesmaximal, characteristic, complemented
8#26011639C212yesmaximal, complemented
9#26011626C212yesmaximal, complemented
10#26011570C211yesmaximal, characteristic, complemented
11C4×SD16×C173C211yesmaximal, characteristic, complemented
12C2×#14583827C2212yescomplemented
13C22×D692—21no—
14C2×D4×D173C2212yes—
15#19149706—28no—
16C2×#14583813C2212yescomplemented
17C2×D1384C2212yescomplemented
18C2×D1384C2212yescomplemented
19C2×#14583811C2212yes—
20C2×#14583809C2212yescomplemented
21#19149676—21no—
22#19149676C2211yescharacteristic
23#19149676C2211yescharacteristic, complemented
24#19149654—22no—
25#19149654C2212yes—
26#19149654C2212yescomplemented
27#19149653—21no—
28#19149649—22no—
29C4×D692—21no—
30C4×D692C2211yescharacteristic
31C4⋊#13063436C2211yescharacteristic, complemented
32#19149599C2212yescomplemented
33#19149599C2212yescomplemented
34#19149591C2212yescomplemented
35C2×SD16×C173C2211yescharacteristic
36C4×Q8×C173C2211yescharacteristic, complemented
37C4×D4×C173C2211yescharacteristic
38#106×C173C2211yescharacteristic, complemented
39#103×C173C2211yescharacteristic, complemented
40#102×C173C2211yescharacteristic
41C4×C8×C173C2211yescharacteristic, complemented
42C23×D173—22no—
43#14583827—24no—
44#14583827—42no—
45C2×D692—22no—
46C2×D692—22no—
47C2×D692—22no—
48C2×D692—21no—
49C2×D692—21no—
50C2×D692—21no—
51C2×D692—21no—
52C2×D692—44no—
53C2×D692D411yescharacteristic
54C2×D692C2312yes—
55C2×D692C2312yes—
56C2×C22⋊D173—22no—
57D4×D173—24no—
58D4×D173—42no—
59C2×C4×D173—22no—
60C2×C4×D173C2312yes—
61#14583813—24no—
62D1384—24no—
63D1384—42no—
64#14583811—24no—
65#14583809—24no—
66C2×#13063436C2312yes—
67C4⋊Dic173D411yescharacteristic
68#14583803—22no—
69#14583803—22no—
70#14583803D412yes—
71C2×Q8×C173C2311yescharacteristic
72C2×D4×C173C2311yescharacteristic
73C22×C4×C173—21no—
74SD16×C173D414yescomplemented
75C2×C8×C173C2311yescharacteristic
76C2×C8×C173C2311yescharacteristic
77C4⋊C4×C173—21no—
78C4⋊C4×C173C2311yescharacteristic
79C4⋊C4×C173C2311yescharacteristic
80C22⋊C4×C173—21no—
81C42×C173—21no—
82C42×C173C2311yescharacteristic
83C22×D173—22no—
84C22×D173—22no—
85C22×D173—44no—
86C22×D173—44no—
87C22×D173—82no—
88C22×D173C2×D412yes—
89D692—22no—
90D692—22no—
91D692—22no—
92D692—22no—
93D692—22no—
94D692—44no—
95D692—44no—
96D692—42no—
97D692—42no—
98D692—81no—
99D692—81no—
100D692—82no—
101C22⋊D173—44no—
102C2×Dic173C2×D412yes—
103C4×D173—22no—
104C4×D173—22no—
105C4×D173—44no—
106#13063436—22no—
107C23×C173—21no—
108Q8×C173—21no—
109Q8×C173C2×D412yescomplemented
110D4×C173—21no—
111D4×C173C2×D412yes—
112C2×C4×C173—21no—
113C2×C4×C173—21no—
114C2×C4×C173—21no—
115C2×C4×C173—42no—
116C2×C4×C173C2×D411yescharacteristic
117C2×C4×C173C2×D411yescharacteristic
118C2×C4×C173C2411yescharacteristic
119C8×C173—21no—
120C8×C173C2×D412yescomplemented
121C22×C173—22no—
122C22×C173—41no—
123C22×C173—41no—
124C22×C173C22×D411yescharacteristic
125C2×D173—22no—
126C2×D173—22no—
127C2×D173—22no—
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×C173—41no—
138C4×C173—41no—
139C4×C173AGL(1,ℤ/8)12yescomplemented
140C4×C173C22×D411yescharacteristic
141C4×C173H(5,𝔽2)11yescharacteristic
142Dic173—22no—
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×SD16D17311yescharacteristic, complemented
162#436—1731no—
163C2×#41—1732no—
164C22×D4—3461no—
165C22×D4—1732no—
166AGL(1,ℤ/8)—3468no—
167C2×SD16—1732no—
168C2×SD16C2×D17311yescharacteristic
169C2×D8—1732no—
170C2×D8—1732no—
171C2×D8—1732no—
172C2×M16—1732no—
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×Q8C2×D17311yescharacteristic, complemented
182C4×D4—3461no—
183C4×D4—1731no—
184C4×D4C2×D17311yescharacteristic
185#106C2×D17311yescharacteristic, complemented
186C4⋊C8—1731no—
187#103C2×D17311yescharacteristic, complemented
188#102—1732no—
189#102—1732no—
190#102C2×D17311yescharacteristic
191C22⋊C8—1732no—
192C4×C8C2×D17311yescharacteristic, complemented
193C24—3462no—
194#41—3464no—
195#41—6922no—
196C2×Q8C22×D17311yescharacteristic
197C2×D4—3462no—
198C2×D4—3462no—
199C2×D4—3462no—
200C2×D4—3464no—
201C2×D4—3462no—
202C2×D4—3461no—
203C2×D4—3461no—
204C2×D4—3461no—
205C2×D4—3461no—
206C2×D4—1731no—
207C2×D4—1732no—
208C2×D4—1732no—
209C2×D4—6924no—
210C2×D4—6922no—
211C2×D4C22×D17311yescharacteristic
212C22×C4—21no—
213C22×C4—3462no—
214C22×C4—1732no—
215SD16—3464no—
216SD16D69214yescomplemented
217D8—3464no—
218D8—3464no—
219D8—6922no—
220M16—3464no—
221C2×C8—1732no—
222C2×C8C22×D17311yescharacteristic
223C2×C8C22×D17311yescharacteristic
224C4⋊C4—21no—
225C4⋊C4—1731no—
226C4⋊C4C22×D17311yescharacteristic
227C4⋊C4C22×D17311yescharacteristic
228C22⋊C4—21no—
229C22⋊C4—3462no—
230C22⋊C4—3462no—
231C22⋊C4—1732no—
232C42—21no—
233C42C22×D17311yescharacteristic
234C23—21no—
235C23—3462no—
236C23—3462no—
237C23—13842no—
238C23—1732no—
239C23—6924no—
240C23—6924no—
241Q8—21no—
242Q8C2×D69212yescomplemented
243D4—21no—
244D4—3462no—
245D4—3462no—
246D4—3462no—
247D4—3462no—
248D4—3462no—
249D4—13841no—
250D4—13841no—
251D4—13842no—
252D4—6924no—
253D4—6924no—
254D4—6924no—
255D4—6922no—
256D4—6922no—
257D4C2×D69212yes—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—42no—
262C2×C4—3462no—
263C2×C4—3462no—
264C2×C4—1732no—
265C2×C4—6924no—
266C2×C4D4×D17311yescharacteristic
267C2×C4D4×D17311yescharacteristic
268C2×C4C23×D17311yescharacteristic
269C8—21no—
270C8—3462no—
271C8C2×D69212yescomplemented
272C22—22no—
273C22—41no—
274C22—41no—
275C22—3462no—
276C22—3462no—
277C22—3462no—
278C22—13848no—
279C22—6922no—
280C22—6922no—
281C22—6922no—
282C22—6922no—
283C22—6922no—
284C22—6922no—
285C22C2×D4×D17311yescharacteristic
286C4—21no—
287C4—22no—
288C4—41no—
289C4—41no—
290C4—3462no—
291C4#1914970612yescomplemented
292C4C22×D69211yescharacteristic
293C4#1914972511yescharacteristic
294C2—42no—
295C2—13842no—
296C2—13842no—
297C2—6922no—
298C2D4×D69211yescharacteristic
299C2C2×#1914970611yescharacteristic
300C2#2601229311yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
FittingC4×SD16×C173C2
socleC22×C173C22×D4
hypercenterC4×SD16D173
hypocenterC173#3274
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ C4×SD16

Complements

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

NH ≅ G/Ntype
1. C2×#19149706C2semidirect
2. #26012163C2semidirect
3. D4×D692C2semidirect
4. #26011868C2semidirect
5. #26011745C2semidirect
6. #26011691C2semidirect
7. #26011686C2semidirect
8. #26011639C2semidirect
9. #26011626C2semidirect
10. #26011570C2semidirect
11. C4×SD16×C173C2semidirect
12. C2×#14583827C22semidirect
16. C2×#14583813C22semidirect
17. C2×D1384C22semidirect
18. C2×D1384C22semidirect
20. C2×#14583809C22semidirect
23. #19149676C22semidirect
26. #19149654C22semidirect
31. C4⋊#13063436C22semidirect
32. #19149599C22semidirect
33. #19149599C22semidirect
34. #19149591C22semidirect
36. C4×Q8×C173C22semidirect
38. #106×C173C22semidirect
39. #103×C173C22semidirect
41. C4×C8×C173C22semidirect
74. SD16×C173D4semidirect
109. Q8×C173C2×D4semidirect
120. C8×C173C2×D4semidirect
139. C4×C173AGL(1,ℤ/8)semidirect
150. C173#3274semidirect
161. C4×SD16D173semidirect
181. C4×Q8C2×D173semidirect
185. #106C2×D173semidirect
187. #103C2×D173semidirect
192. C4×C8C2×D173semidirect
216. SD16D692semidirect
242. Q8C2×D692semidirect
271. C8C2×D692semidirect
291. C4#19149706semidirect

Maximal subgroups

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

subgroupindex
C2×#191497062
#260121632
D4×D6922
#260118682
#260117452
#260116912
#260116862
#260116392
#260116262
#260115702
C4×SD16×C1732
#3274173