Finite Group Databasebeta

← #496×D139 #31816834

Subgroups — #496×D139 #31816834

Order 17792.

668 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 668 conjugacy classes of subgroups — too many to draw. Only its 145 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×SD16×D139C212yesmaximal, complemented
2C2×D8×D139C212yesmaximal, complemented
3#127×D139C211yesmaximal, characteristic, complemented
4#124×D139C211yesmaximal, characteristic, complemented
5#22581343C211yesmaximal, characteristic, complemented
6#22581342C211yesmaximal, characteristic, complemented
7#22581341C212yesmaximal, complemented
8#22581340C211yesmaximal, characteristic, complemented
9#22581339C212yesmaximal, complemented
10#97×D139C211yesmaximal, characteristic, complemented
11#496×C139C211yesmaximal, characteristic, complemented
12C2×Q8×D139C2211yescharacteristic
13C2×D4×D139—22no—
14C2×D4×D139C2211yescharacteristic
15C2×D4×D139C2212yes—
16C2×#14555774C2212yescomplemented
17C2×#14555773C2212yescomplemented
18C2×#14555772C2212yescomplemented
19SD16×D139—28no—
20C2×D1112C2212yescomplemented
21C2×#14555770C2212yescomplemented
22C2×#14555770C2212yescomplemented
23D8×D139—28no—
24C2×C8×D139C2212yes—
25#17984839C2211yescharacteristic, complemented
26#17984838—22no—
27#17984838C2212yescomplemented
28#17984831C2211yescharacteristic, complemented
29#17984830C2211yescharacteristic, complemented
30#17984829C2211yescharacteristic, complemented
31#17984828—22no—
32C22⋊C4×D139—22no—
33C42×D139C2211yescharacteristic
34#17984753C2211yescharacteristic, complemented
35#17984752C2212yescomplemented
36C2×SD16×C139C2212yescomplemented
37C2×D8×C139C2212yescomplemented
38#127×C139C2211yescharacteristic, complemented
39#124×C139C2211yescharacteristic, complemented
40#97×C139C2211yescharacteristic, complemented
41C23×D139—21no—
42C23×D139—22no—
43C2×Dic278—21no—
44C2×Dic278C2311yescharacteristic
45Q8×D139—24no—
46C2×D556—22no—
47C2×D556C2311yescharacteristic
48C2×D556C2312yes—
49C2×C22⋊D139—21no—
50C2×C22⋊D139—22no—
51C2×C22⋊D139—24no—
52D4×D139—24no—
53D4×D139—24no—
54D4×D139—24no—
55D4×D139—48no—
56C2×C4×D139—21no—
57C2×C4×D139D412yes—
58C2×C4×D139C2311yescharacteristic
59#14555774—24no—
60#14555773—24no—
61#14555772—24no—
62D1112—24no—
63#14555770—24no—
64#14555770—24no—
65C2×C139⋊C8C2312yes—
66C8×D139D418yescomplemented
67C22⋊Dic139—22no—
68(C2×D139)⋊C4—22no—
69(C2×D139)⋊C4—22no—
70C4×Dic139D412yes—
71C4×Dic139C2311yescharacteristic
72C2×Q8×C139C2311yescharacteristic
73C2×D4×C139—22no—
74C2×D4×C139C2311yescharacteristic
75C2×D4×C139C2312yes—
76SD16×C139—24no—
77D8×C139—24no—
78C2×C8×C139C2312yes—
79C22⋊C4×C139—22no—
80C42×C139C2311yescharacteristic
81C22×D139—21no—
82C22×D139—22no—
83C22×D139—42no—
84C22×D139—44no—
85C22×D139—48no—
86C22×D139—44no—
87C22×D139C2×D411yescharacteristic
88Dic278—22no—
89Dic278—42no—
90D556—22no—
91D556—22no—
92D556—22no—
93D556—44no—
94C22⋊D139—48no—
95C22⋊D139—42no—
96C22⋊D139—44no—
97C2×Dic139—21no—
98C2×Dic139C2×D412yes—
99C2×Dic139C2×D411yescharacteristic
100C4×D139—24no—
101C4×D139—42no—
102C4×D139C2×D414yes—
103C139⋊C8C2×D414yescomplemented
104C23×C139—21no—
105C23×C139—22no—
106Q8×C139—22no—
107D4×C139—22no—
108D4×C139—22no—
109D4×C139—22no—
110D4×C139—44no—
111C2×C4×C139—21no—
112C2×C4×C139C2×D412yes—
113C2×C4×C139C2411yescharacteristic
114C8×C139C2×D414yescomplemented
115C22×C139—41no—
116C22×C139—42no—
117C22×C139—44no—
118C22×C139—42no—
119C22×C139C22×D411yescharacteristic
120C2×D139—41no—
121C2×D139—42no—
122C2×D139—44no—
123C2×D139—42no—
124C2×D139—88no—
125C2×D139—84no—
126C2×D139#12712yes—
127C2×D139AGL(1,ℤ/8)14yes—
128C4×C139—22no—
129C4×C139—41no—
130C4×C139C22×D412yes—
131Dic139—22no—
132Dic139—41no—
133Dic139#12712yes—
134C2×C139—82no—
135C2×C139—81no—
136C2×C139C2×#12711yescharacteristic
137C2×C139C2×AGL(1,ℤ/8)12yes—
138D139—82no—
139D139—81no—
140D139#49614yesdirect product factor
141C139C2×#49611yescharacteristic, complemented, Sylow
142C2×#496—1391nomaximal, Sylow
143C22×SD16—1392no—
144C22×D8—1392no—
145C2×#127—1391no—
146C2×#124—1391no—
147#496—1392no—
148#496—1392no—
149#496—1391no—
150#496—1391no—
151#496—1391no—
152#496D13911yescharacteristic, direct product factor
153C2×#97—1391no—
154C22×Q8—1391no—
155C22×D4—2782no—
156C22×D4—1391no—
157C22×D4—1392no—
158C2×SD16—2788no—
159C2×SD16—1392no—
160C2×SD16—1392no—
161C2×SD16—1392no—
162C2×SD16C2×D13912yescomplemented
163C2×D8—2788no—
164C2×D8—1392no—
165C2×D8—1392no—
166C2×D8—1392no—
167C2×D8C2×D13912yescomplemented
168C22×C8—1392no—
169#127—2782no—
170#127—1392no—
171#127—1391no—
172#127C2×D13911yescharacteristic, complemented
173#124—2782no—
174#124—1391no—
175#124—1391no—
176#124—1391no—
177#124C2×D13911yescharacteristic, complemented
178C2×C22⋊C4—2782no—
179C2×C42—1391no—
180#97—1392no—
181#97—1391no—
182#97C2×D13911yescharacteristic, complemented
183C24—2781no—
184C24—2782no—
185C2×Q8—2781no—
186C2×Q8—2784no—
187C2×Q8—1391no—
188C2×Q8C22×D13911yescharacteristic
189C2×D4—22no—
190C2×D4—2784no—
191C2×D4—2784no—
192C2×D4—2784no—
193C2×D4—2781no—
194C2×D4—2782no—
195C2×D4—2784no—
196C2×D4—2782no—
197C2×D4—5568no—
198C2×D4—1391no—
199C2×D4—1392no—
200C2×D4C22×D13911yescharacteristic
201C2×D4C22×D13912yes—
202C22×C4—2781no—
203C22×C4—1392no—
204C22×C4—1391no—
205SD16—24no—
206SD16—2784no—
207SD16—2784no—
208SD16—2784no—
209D8—24no—
210D8—2784no—
211D8—2784no—
212D8—2784no—
213C2×C8—1392no—
214C2×C8—1398no—
215C2×C8C22×D13912yes—
216C22⋊C4—22no—
217C22⋊C4—2782no—
218C22⋊C4—2782no—
219C22⋊C4—2782no—
220C42—1392no—
221C42—1391no—
222C42C22×D13911yescharacteristic
223C23—21no—
224C23—22no—
225C23—2781no—
226C23—2782no—
227C23—5562no—
228C23—5564no—
229C23—5568no—
230C23—5564no—
231C23—1391no—
232Q8—22no—
233Q8—2782no—
234Q8—5562no—
235D4—22no—
236D4—22no—
237D4—22no—
238D4—44no—
239D4—2782no—
240D4—2782no—
241D4—2782no—
242D4—5564no—
243D4—5568no—
244D4—5562no—
245D4—5564no—
246C2×C4—21no—
247C2×C4—2784no—
248C2×C4—2781no—
249C2×C4—5562no—
250C2×C4—1392no—
251C2×C4—1391no—
252C2×C4—1394no—
253C2×C4D4×D13912yes—
254C2×C4C23×D13911yescharacteristic
255C8—1394no—
256C8D4×D13914yescomplemented
257C22—44no—
258C22—42no—
259C22—41no—
260C22—42no—
261C22—5564no—
262C22—5562no—
263C22—5561no—
264C22—5562no—
265C22—11128no—
266C22—11124no—
267C22—1394no—
268C22—1392no—
269C22C2×D4×D13911yescharacteristic
270C4—22no—
271C4—41no—
272C4—2782no—
273C4—5561no—
274C4—1392no—
275C4C2×D4×D13912yes—
276C2—82no—
277C2—81no—
278C2—11122no—
279C2—11121no—
280C2—1394no—
281C2#127×D13911yescharacteristic
282C2AGL(1,ℤ/8)×D13912yes—
subgroupG/H
centerC22C2×D4×D139
commutator, cosocleC2×C4×C139C24
FrattiniC2×C4C23×D139
Fitting#496×C139C2
socleC22×C139C22×D4
hypercenter#496D139
hypocenterC139C2×#496
LCSG ▷ C2×C4×C139 ▷ C2×C139 ▷ C139
UCSC1 ◁ C22 ◁ C42 ◁ #496

Complements

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

NH ≅ G/Ntype
1. C2×SD16×D139C2semidirect
2. C2×D8×D139C2semidirect
3. #127×D139C2semidirect
4. #124×D139C2semidirect
5. #22581343C2semidirect
6. #22581342C2semidirect
7. #22581341C2semidirect
8. #22581340C2semidirect
9. #22581339C2semidirect
10. #97×D139C2semidirect
11. #496×C139C2semidirect
16. C2×#14555774C22semidirect
17. C2×#14555773C22semidirect
18. C2×#14555772C22semidirect
20. C2×D1112C22semidirect
21. C2×#14555770C22semidirect
22. C2×#14555770C22semidirect
25. #17984839C22semidirect
27. #17984838C22semidirect
28. #17984831C22semidirect
29. #17984830C22semidirect
30. #17984829C22semidirect
34. #17984753C22semidirect
35. #17984752C22semidirect
36. C2×SD16×C139C22semidirect
37. C2×D8×C139C22semidirect
38. #127×C139C22semidirect
39. #124×C139C22semidirect
40. #97×C139C22semidirect
66. C8×D139D4semidirect
103. C139⋊C8C2×D4semidirect
114. C8×C139C2×D4semidirect
140. D139#496direct (N × H)
141. C139C2×#496semidirect
152. #496D139direct (N × H)
162. C2×SD16C2×D139semidirect
167. C2×D8C2×D139semidirect
172. #127C2×D139semidirect
177. #124C2×D139semidirect
182. #97C2×D139semidirect
256. C8D4×D139semidirect

Maximal subgroups

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

subgroupindex
C2×SD16×D1392
C2×D8×D1392
#127×D1392
#124×D1392
#225813432
#225813422
#225813412
#225813402
#225813392
#97×D1392
#496×C1392
C2×#496139