Finite Group Databasebeta

← #484×D139 #31816306

Subgroups — #484×D139 #31816306

Order 17792.

460 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 135 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
1C22⋊Q8×D139C211yesmaximal, characteristic
2C2×C4⋊C4×D139C211yesmaximal, characteristic
3#22581290C212yesmaximal, complemented
4#22581289C211yesmaximal, characteristic, complemented
5#22581288C211yesmaximal, characteristic, complemented
6#22581287C212yesmaximal, complemented
7#22581286C211yesmaximal, characteristic, complemented
8#107×D139C212yesmaximal, complemented
9#103×D139C212yesmaximal, complemented
10C22⋊C8×D139C211yesmaximal, characteristic
11#484×C139C211yesmaximal, characteristic, complemented
12C2×Q8×D139C2211yescharacteristic
13C22×C4×D139—21no—
14C22×C4×D139C2211yescharacteristic
15C2×C8×D139C2212yes—
16#17984822C2211yescharacteristic
17#17984821C2211yescharacteristic
18#17984820—21no—
19#17984819—21no—
20#17984818C2211yescharacteristic
21C2×C4⋊Dic139C2211yescharacteristic
22C2×Dic139⋊C4—21no—
23C4⋊C4×D139—21no—
24C4⋊C4×D139—21no—
25C4⋊C4×D139—21no—
26C4⋊C4×D139C2212yes—
27C4⋊C4×D139C2212yes—
28C4⋊C4×D139C2211yescharacteristic
29C22⋊C4×D139—21no—
30#17984776C2212yescomplemented
31#17984775C2212yescomplemented
32#17984775C2212yescomplemented
33#17984767C2212yescomplemented
34#17984766C2212yescomplemented
35#17984765C2212yescomplemented
36C22⋊(C139⋊C8)C2211yescharacteristic
37(C2×D139)⋊C8C2212yes—
38C22⋊Q8×C139C2211yescharacteristic
39C2×C4⋊C4×C139C2211yescharacteristic
40#107×C139C2212yescomplemented
41#103×C139C2212yescomplemented
42C22⋊C8×C139C2211yescharacteristic
43C23×D139D411yescharacteristic
44C2×Dic278—21no—
45C2×Dic278C2311yescharacteristic
46Q8×D139—24no—
47C22×Dic139—21no—
48C22×Dic139D411yescharacteristic
49C2×C4×D139—24no—
50C2×C4×D139—21no—
51C2×C4×D139—21no—
52C2×C4×D139—22no—
53C2×C4×D139—22no—
54C2×C4×D139—44no—
55C2×C4×D139D411yescharacteristic
56C2×C4×D139D411yescharacteristic
57C2×C4×D139C2311yescharacteristic
58C2×C4×D139C2311yescharacteristic
59C2×C139⋊C8C2312yes—
60C8×D139—24no—
61C4⋊Dic139—21no—
62C4⋊Dic139—21no—
63C4⋊Dic139C2311yescharacteristic
64C4⋊Dic139C2312yes—
65Dic139⋊C4—22no—
66Dic139⋊C4—22no—
67Dic139⋊C4—21no—
68Dic139⋊C4—21no—
69Dic139⋊C4—21no—
70C22⋊Dic139—21no—
71(C2×D139)⋊C4—21no—
72(C2×D139)⋊C4—21no—
73C2×Q8×C139C2311yescharacteristic
74C22×C4×C139—21no—
75C22×C4×C139C2311yescharacteristic
76C2×C8×C139C2312yes—
77C4⋊C4×C139—21no—
78C4⋊C4×C139—21no—
79C4⋊C4×C139C2311yescharacteristic
80C4⋊C4×C139C2312yes—
81C22⋊C4×C139—21no—
82C22×D139—22no—
83C22×D139—22no—
84C22×D139Q1614yes—
85C22×D139C2×D411yescharacteristic
86C22×D139C2×D411yescharacteristic
87Dic278—22no—
88Dic278—42no—
89C2×Dic139—22no—
90C2×Dic139—21no—
91C2×Dic139—21no—
92C2×Dic139—22no—
93C2×Dic139—42no—
94C2×Dic139C2×D411yescharacteristic
95C2×Dic139C2×D411yescharacteristic
96C4×D139—22no—
97C4×D139—22no—
98C4×D139—22no—
99C4×D139—44no—
100C4×D139—42no—
101C4×D139—42no—
102C4×D139—44no—
103C4×D139#4114yes—
104C139⋊C8—22no—
105C23×C139C2×D411yescharacteristic
106Q8×C139—22no—
107C2×C4×C139—22no—
108C2×C4×C139—21no—
109C2×C4×C139—21no—
110C2×C4×C139—22no—
111C2×C4×C139—42no—
112C2×C4×C139C2×D411yescharacteristic
113C2×C4×C139C2411yescharacteristic
114C8×C139—22no—
115C22×C139—21no—
116C22×C139—21no—
117C22×C139C2×Q1612yes—
118C22×C139C22×D411yescharacteristic
119C2×D139—21no—
120C2×D139—21no—
121C2×D139—28no—
122C2×D139#12312yes—
123C2×D139C2×Q1612yes—
124C2×D139C2×Q1612yes—
125C2×D139AGL(1,ℤ/8)12yes—
126C4×C139—21no—
127C4×C139—41no—
128C4×C139—42no—
129C4×C139—41no—
130C4×C139C2×#4112yes—
131Dic139—21no—
132Dic139—41no—
133Dic139—42no—
134Dic139—41no—
135Dic139#12312yes—
136C2×C139—22no—
137C2×C139C2×#12311yescharacteristic
138C2×C139C22×Q1611yescharacteristic
139C2×C139C2×AGL(1,ℤ/8)11yescharacteristic
140D139—22no—
141D139#48414yesdirect product factor
142C139C2×#48411yescharacteristic, complemented, Sylow
143C2×#484—1391nomaximal, Sylow
144C2×C22⋊Q8—1391no—
145C22×C4⋊C4—1391no—
146#484—1391no—
147#484—1391no—
148#484—1392no—
149#484—1392no—
150#484—1391no—
151#484D13911yescharacteristic, direct product factor
152C2×#107—1392no—
153C2×#103—1392no—
154C2×C22⋊C8—1391no—
155C22×Q8—1391no—
156C23×C4—2781no—
157C23×C4—1391no—
158C22×C8—1392no—
159C22⋊Q8—2781no—
160C22⋊Q8—2781no—
161C22⋊Q8—1391no—
162C22⋊Q8—1391no—
163C22⋊Q8—1391no—
164C22⋊Q8C2×D13911yescharacteristic
165C2×C4⋊C4—2781no—
166C2×C4⋊C4—2781no—
167C2×C4⋊C4—2781no—
168C2×C4⋊C4—2781no—
169C2×C4⋊C4—1391no—
170C2×C4⋊C4—1392no—
171C2×C4⋊C4—1392no—
172C2×C4⋊C4—1391no—
173C2×C4⋊C4C2×D13911yescharacteristic
174C2×C22⋊C4—2781no—
175#107—1392no—
176#107—1392no—
177#107—1392no—
178#107C2×D13912yescomplemented
179#103—1392no—
180#103—1392no—
181#103—1392no—
182#103C2×D13912yescomplemented
183C22⋊C8—1391no—
184C22⋊C8—1392no—
185C22⋊C8C2×D13911yescharacteristic
186C24—1391no—
187C2×Q8—2781no—
188C2×Q8—2784no—
189C2×Q8—1391no—
190C2×Q8C22×D13911yescharacteristic
191C22×C4—21no—
192C22×C4—2784no—
193C22×C4—2781no—
194C22×C4—2781no—
195C22×C4—2782no—
196C22×C4—2782no—
197C22×C4—2781no—
198C22×C4—5564no—
199C22×C4—1391no—
200C22×C4—1391no—
201C22×C4—1391no—
202C22×C4—1391no—
203C22×C4—1391no—
204C22×C4C22×D13911yescharacteristic
205C2×C8—2784no—
206C2×C8—1392no—
207C2×C8C22×D13912yes—
208C4⋊C4—21no—
209C4⋊C4—21no—
210C4⋊C4—2782no—
211C4⋊C4—2782no—
212C4⋊C4—2781no—
213C4⋊C4—2781no—
214C4⋊C4—2781no—
215C4⋊C4—2781no—
216C4⋊C4—2781no—
217C4⋊C4—1392no—
218C4⋊C4—1391no—
219C4⋊C4C22×D13912yes—
220C4⋊C4C22×D13911yescharacteristic
221C22⋊C4—21no—
222C22⋊C4—2781no—
223C22⋊C4—2781no—
224C22⋊C4—2781no—
225C23—2782no—
226C23—2782no—
227C23—1394no—
228C23—1391no—
229C23—1391no—
230C23D4×D13911yescharacteristic
231Q8—22no—
232Q8—2782no—
233Q8—5562no—
234C2×C4—22no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—22no—
238C2×C4—42no—
239C2×C4—2782no—
240C2×C4—2782no—
241C2×C4—2782no—
242C2×C4—2782no—
243C2×C4—2781no—
244C2×C4—2781no—
245C2×C4—2782no—
246C2×C4—5564no—
247C2×C4—5564no—
248C2×C4—5562no—
249C2×C4—5562no—
250C2×C4—5562no—
251C2×C4—1391no—
252C2×C4—1391no—
253C2×C4—1394no—
254C2×C4D4×D13911yescharacteristic
255C2×C4C23×D13911yescharacteristic
256C8—22no—
257C8—2782no—
258C22—21no—
259C22—21no—
260C22—2788no—
261C22—2781no—
262C22—2781no—
263C22—1392no—
264C22—1392no—
265C22—1392no—
266C22—1392no—
267C22Q16×D13912yes—
268C22C2×D4×D13911yescharacteristic
269C4—21no—
270C4—41no—
271C4—42no—
272C4—41no—
273C4—2781no—
274C4—5561no—
275C4—5562no—
276C4—5561no—
277C4—1392no—
278C4#41×D13912yes—
279C2—22no—
280C2—2782no—
281C2—1394no—
282C2#123×D13911yescharacteristic
283C2C2×Q16×D13911yescharacteristic
284C2AGL(1,ℤ/8)×D13911yescharacteristic
subgroupG/H
centerC22C2×D4×D139
commutator, cosocleC2×C4×C139C24
FrattiniC2×C4C23×D139
Fitting#484×C139C2
socleC22×C139C22×D4
prime radicalC23×D139D4
hypercenter#484D139
hypocenterC139C2×#484
LCSG ▷ C2×C4×C139 ▷ C2×C139 ▷ C139
UCSC1 ◁ C22 ◁ C22×C4 ◁ #484

Complements

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

NH ≅ G/Ntype
3. #22581290C2semidirect
4. #22581289C2semidirect
5. #22581288C2semidirect
6. #22581287C2semidirect
7. #22581286C2semidirect
8. #107×D139C2semidirect
9. #103×D139C2semidirect
11. #484×C139C2semidirect
30. #17984776C22semidirect
31. #17984775C22semidirect
32. #17984775C22semidirect
33. #17984767C22semidirect
34. #17984766C22semidirect
35. #17984765C22semidirect
40. #107×C139C22semidirect
41. #103×C139C22semidirect
141. D139#484direct (N × H)
142. C139C2×#484semidirect
151. #484D139direct (N × H)
178. #107C2×D139semidirect
182. #103C2×D139semidirect

Maximal subgroups

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

subgroupindex
C22⋊Q8×D1392
C2×C4⋊C4×D1392
#225812902
#225812892
#225812882
#225812872
#225812862
#107×D1392
#103×D1392
C22⋊C8×D1392
#484×C1392
C2×#484139