Finite Group Databasebeta

← #31816853

Subgroups — #31816853

Order 17792.

656 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 656 conjugacy classes of subgroups — too many to draw. Only its 141 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×#17984869C212yesmaximal, complemented
2C2×#17984866C212yesmaximal, complemented
3#22581502C211yesmaximal, characteristic, complemented
4#127×D139C211yesmaximal, characteristic, complemented
5#22581343C211yesmaximal, characteristic, complemented
6#22581342C211yesmaximal, characteristic, complemented
7#22581340C211yesmaximal, characteristic, complemented
8#22581328C212yesmaximal, complemented
9#22581324C212yesmaximal, complemented
10#22580905C211yesmaximal, characteristic, complemented
11#496×C139C211yesmaximal, characteristic, complemented
12C2×#14555786C2211yescharacteristic
13C2×#14555784C2211yescharacteristic
14C2×D4×D139—22no—
15C2×D4×D139C2212yes—
16#17984869—28no—
17#17984866—28no—
18C2×#14555774C2212yescomplemented
19C2×#14555773C2212yescomplemented
20C2×#14555772C2212yescomplemented
21C2×D1112C2212yescomplemented
22C2×#14555770C2212yescomplemented
23C2×#14555770C2212yescomplemented
24C2×#14555768C2212yes—
25#17984840C2211yescharacteristic, complemented
26#17984839C2211yescharacteristic
27#17984838—22no—
28#17984838C2212yes—
29#17984838C2211yescharacteristic, complemented
30#17984830C2211yescharacteristic, complemented
31#17984815—22no—
32(C22⋊D139)⋊C4—22no—
33C42×D139C2211yescharacteristic
34#17984753C2211yescharacteristic, complemented
35C8×Dic139C2212yescomplemented
36C2×SD16×C139C2212yescomplemented
37C2×D8×C139C2212yescomplemented
38#127×C139C2211yescharacteristic
39#124×C139C2211yescharacteristic, complemented
40#97×C139C2211yescharacteristic, complemented
41C23×D139—22no—
42#14555786—24no—
43#14555784—24no—
44C2×Dic278C2311yescharacteristic
45C2×D556—21no—
46C2×D556—22no—
47C2×D556C2311yescharacteristic
48C2×D556C2312yes—
49C2×C22⋊D139—21no—
50C2×C22⋊D139—22no—
51C2×C22⋊D139—22no—
52C2×C22⋊D139—24no—
53D4×D139—24no—
54D4×D139—24no—
55D4×D139—48no—
56C22×Dic139—21no—
57C2×C4×D139—21no—
58C2×C4×D139D412yes—
59C2×C4×D139C2311yescharacteristic
60#14555774—24no—
61#14555773—24no—
62#14555772—24no—
63D1112—24no—
64#14555770—24no—
65#14555770—24no—
66#14555768D418yescomplemented
67C2×C139⋊C8C2312yes—
68Dic139⋊C4—22no—
69(C2×D139)⋊C4—22no—
70C4×Dic139D412yes—
71C4×Dic139C2311yescharacteristic
72C2×Q8×C139C2311yescharacteristic
73C2×D4×C139—22no—
74C2×D4×C139C2312yes—
75C2×D4×C139C2311yescharacteristic
76SD16×C139—24no—
77D8×C139—24no—
78C2×C8×C139C2312yes—
79C22⋊C4×C139—22no—
80C42×C139C2311yescharacteristic
81C22×D139—21no—
82C22×D139—22no—
83C22×D139—48no—
84C22×D139—44no—
85C22×D139C2×D411yescharacteristic
86Dic278—22no—
87D556—22no—
88D556—22no—
89D556—22no—
90D556—44no—
91D556—42no—
92C22⋊D139—42no—
93C22⋊D139—44no—
94C22⋊D139—48no—
95C22⋊D139—44no—
96C2×Dic139—21no—
97C2×Dic139—42no—
98C2×Dic139C2×D411yescharacteristic
99C2×Dic139C2×D412yes—
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—22no—
121C2×D139—41no—
122C2×D139—42no—
123C2×D139—44no—
124C2×D139—42no—
125C2×D139—88no—
126C2×D139#12712yes—
127C4×C139—22no—
128C4×C139—41no—
129C4×C139C22×D412yes—
130Dic139—41no—
131Dic139#12712yes—
132Dic139AGL(1,ℤ/8)14yescomplemented
133C2×C139—82no—
134C2×C139—81no—
135C2×C139C2×#12711yescharacteristic
136C2×C139C2×AGL(1,ℤ/8)12yes—
137D139—22no—
138D139—81no—
139D139—82no—
140C139#315111yescharacteristic, complemented, Sylow
141#3151—1391nomaximal, Sylow
142C2×AGL(1,ℤ/8)—1392no—
143C2×AGL(1,ℤ/8)—1392no—
144#532—1391no—
145C2×#127—1391no—
146#496—1391no—
147#496—1391no—
148#496—1391no—
149#496D13911yescharacteristic, complemented
150#493—1392no—
151#492—1392no—
152C4×M16—1391no—
153C2×#41—1391no—
154C2×#41—1391no—
155C22×D4—2782no—
156C22×D4—1392no—
157AGL(1,ℤ/8)—2788no—
158AGL(1,ℤ/8)—2788no—
159C2×SD16—1392no—
160C2×SD16—1392no—
161C2×SD16—1392no—
162C2×SD16C2×D13912yescomplemented
163C2×D8—1392no—
164C2×D8—1392no—
165C2×D8—1392no—
166C2×D8C2×D13912yescomplemented
167C2×M16—1392no—
168C4⋊Q8—1391no—
169#127—2782no—
170#127—1392no—
171#127—1391no—
172#127—1391no—
173#127C2×D13911yescharacteristic
174#124—1391no—
175#124C2×D13911yescharacteristic, complemented
176#121—2782no—
177C4×D4—2782no—
178C2×C42—1391no—
179#97—1391no—
180#97C2×D13911yescharacteristic, complemented
181C4×C8—1392no—
182C24—2782no—
183#41—2784no—
184#41—2784no—
185C2×Q8—1391no—
186C2×Q8C22×D13911yescharacteristic
187C2×D4—22no—
188C2×D4—2781no—
189C2×D4—2781no—
190C2×D4—2784no—
191C2×D4—2784no—
192C2×D4—2782no—
193C2×D4—2782no—
194C2×D4—2784no—
195C2×D4—2782no—
196C2×D4—5568no—
197C2×D4—1391no—
198C2×D4—1392no—
199C2×D4C22×D13911yescharacteristic
200C2×D4C22×D13912yes—
201C22×C4—2781no—
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—
213M16—1398no—
214C2×C8—1392no—
215C2×C8C22×D13912yes—
216C4⋊C4—2782no—
217C22⋊C4—22no—
218C22⋊C4—2782no—
219C42—1392no—
220C42—1391no—
221C42C22×D13911yescharacteristic
222C23—21no—
223C23—22no—
224C23—2781no—
225C23—2782no—
226C23—5568no—
227C23—5564no—
228C23—1391no—
229Q8—22no—
230Q8—2782no—
231D4—22no—
232D4—22no—
233D4—22no—
234D4—44no—
235D4—2782no—
236D4—2782no—
237D4—2782no—
238D4—5564no—
239D4—5562no—
240D4—5562no—
241D4—5564no—
242D4—5568no—
243D4—5564no—
244C2×C4—21no—
245C2×C4—2784no—
246C2×C4—2781no—
247C2×C4—5562no—
248C2×C4—5562no—
249C2×C4—1392no—
250C2×C4—1391no—
251C2×C4—1394no—
252C2×C4D4×D13912yes—
253C2×C4C23×D13911yescharacteristic
254C8—1394no—
255C8D4×D13914yescomplemented
256C22—44no—
257C22—42no—
258C22—41no—
259C22—42no—
260C22—2782no—
261C22—5564no—
262C22—5562no—
263C22—5561no—
264C22—5562no—
265C22—11128no—
266C22—1392no—
267C22C2×D4×D13911yescharacteristic
268C4—22no—
269C4—41no—
270C4—5561no—
271C4—1394no—
272C4—1392no—
273C4C2×D4×D13912yes—
274C2—82no—
275C2—81no—
276C2—2782no—
277C2—11122no—
278C2—11121no—
279C2#127×D13911yescharacteristic
280C2AGL(1,ℤ/8)×D13912yes—
subgroupG/H
centerC22C2×D4×D139
commutator, cosocleC2×C4×C139C24
FrattiniC2×C4C23×D139
Fitting#496×C139C2
socleC22×C139C22×D4
hypercenter#496D139
hypocenterC139#3151
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×#17984869C2semidirect
2. C2×#17984866C2semidirect
3. #22581502C2semidirect
4. #127×D139C2semidirect
5. #22581343C2semidirect
6. #22581342C2semidirect
7. #22581340C2semidirect
8. #22581328C2semidirect
9. #22581324C2semidirect
10. #22580905C2semidirect
11. #496×C139C2semidirect
18. C2×#14555774C22semidirect
19. C2×#14555773C22semidirect
20. C2×#14555772C22semidirect
21. C2×D1112C22semidirect
22. C2×#14555770C22semidirect
23. C2×#14555770C22semidirect
25. #17984840C22semidirect
29. #17984838C22semidirect
30. #17984830C22semidirect
34. #17984753C22semidirect
35. C8×Dic139C22semidirect
36. C2×SD16×C139C22semidirect
37. C2×D8×C139C22semidirect
39. #124×C139C22semidirect
40. #97×C139C22semidirect
66. #14555768D4semidirect
103. C139⋊C8C2×D4semidirect
114. C8×C139C2×D4semidirect
132. Dic139AGL(1,ℤ/8)semidirect
140. C139#3151semidirect
149. #496D139semidirect
162. C2×SD16C2×D139semidirect
166. C2×D8C2×D139semidirect
175. #124C2×D139semidirect
180. #97C2×D139semidirect
255. C8D4×D139semidirect

Maximal subgroups

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

subgroupindex
C2×#179848692
C2×#179848662
#225815022
#127×D1392
#225813432
#225813422
#225813402
#225813282
#225813242
#225809052
#496×C1392
#3151139