Finite Group Databasebeta

← #31818170

Subgroups — #31818170

Order 17792.

464 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 464 conjugacy classes of subgroups — too many to draw. Only its 117 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
1#22581759C212yesmaximal, complemented
2#22581754C212yesmaximal, complemented
3#22581220C212yesmaximal, complemented
4#22581213C212yesmaximal, complemented
5#22581130C211yesmaximal, characteristic, complemented
6#22581125C211yesmaximal, characteristic, complemented
7#22581120C212yesmaximal, complemented
8#22581076C212yesmaximal, complemented
9#496×C139C211yesmaximal, characteristic, complemented
10#17984890C2211yescharacteristic, complemented
11#17984887C2211yescharacteristic
12#17984887C2212yescomplemented
13#17984873C2212yescomplemented
14#17984871—22no—
15#17984869—22no—
16#17984868C2212yes—
17#17984868C2212yescomplemented
18#17984868C2212yescomplemented
19#17984866—22no—
20#17984866—22no—
21#17984864C2212yes—
22#17984863—22no—
23#17984861—22no—
24SD16×D139—22no—
25D8×D139—22no—
26#17984848C2212yescomplemented
27#17984848C2212yescomplemented
28#17984778C2212yescomplemented
29#17984770C2212yescomplemented
30#17984770C2212yescomplemented
31#17984761C2211yescharacteristic, complemented
32#17984759C2212yescomplemented
33#17984759C2211yescharacteristic, complemented
34C2×SD16×C139C2212yescomplemented
35C2×D8×C139C2212yescomplemented
36#127×C139C2211yescharacteristic, complemented
37#124×C139C2211yescharacteristic, complemented
38#97×C139C2211yescharacteristic, complemented
39#14555786—22no—
40#14555785—41no—
41#14555785C2312yes—
42#14555785C2312yes—
43#14555784—22no—
44#14555784—22no—
45#14555784—22no—
46Q8×D139—22no—
47C2×C22⋊D139—41no—
48C2×C22⋊D139—42no—
49D4×D139—22no—
50D4×D139—22no—
51D4×D139—22no—
52Dic556D412yescomplemented
53#14555775—22no—
54#14555774D412yescomplemented
55#14555774D412yescomplemented
56#14555773—22no—
57#14555773—22no—
58#14555773—22no—
59#14555772—22no—
60D1112D412yescomplemented
61#14555770—22no—
62#14555770—22no—
63#14555770—22no—
64#14555769C2312yes—
65#14555769C2312yes—
66#14555768—22no—
67#14555768—22no—
68C8×D139—22no—
69C8×D139—22no—
70C2×Q8×C139C2311yescharacteristic
71C2×D4×C139—41no—
72C2×D4×C139C2312yes—
73C2×D4×C139C2311yescharacteristic
74SD16×C139—22no—
75SD16×C139—22no—
76D8×C139—22no—
77D8×C139—22no—
78C2×C8×C139C2312yes—
79C22⋊C4×C139—41no—
80C42×C139C2311yescharacteristic
81C22×D139—42no—
82C22×D139—41no—
83Dic278—41no—
84Dic278C2×D412yes—
85Dic278C2×D412yes—
86D556—41no—
87D556C2×D412yes—
88D556C2×D412yes—
89C22⋊D139—22no—
90C22⋊D139—22no—
91C22⋊D139—42no—
92C22⋊D139—42no—
93C22⋊D139—42no—
94C2×Dic139—42no—
95C2×Dic139—41no—
96C4×D139—22no—
97C4×D139—22no—
98C4×D139—42no—
99C139⋊C8—22no—
100C139⋊C8—22no—
101C23×C139—21no—
102C23×C139—22no—
103Q8×C139—22no—
104D4×C139—22no—
105D4×C139—22no—
106D4×C139—22no—
107D4×C139—82no—
108C2×C4×C139—21no—
109C2×C4×C139—21no—
110C2×C4×C139C2411yescharacteristic
111C8×C139C2×D412yes—
112C8×C139C2×D412yes—
113C22×C139—41no—
114C22×C139—42no—
115C22×C139—82no—
116C22×C139—81no—
117C22×C139H(5,𝔽2)11yescharacteristic
118C2×D139—22no—
119C2×D139—22no—
120C2×D139—162no—
121C2×D139—161no—
122C4×C139—41no—
123C4×C139—41no—
124C4×C139C22×D412yes—
125Dic139—22no—
126Dic139—22no—
127C2×C139—21no—
128C2×C139—82no—
129C2×C139—81no—
130C2×C139D4211yescharacteristic
131D139—42no—
132D139—42no—
133C139#328811yescharacteristic, complemented, Sylow
134#3288—1391nomaximal, Sylow
135#577—1392no—
136#576—1392no—
137#496D13911yescharacteristic, complemented
138#472—1392no—
139#471—1392no—
140#455—1391no—
141#454—1391no—
142#453—1392no—
143#444—1392no—
1442−1+4—1391no—
145H(5,𝔽2)—1391no—
146H(5,𝔽2)—1392no—
147#137—2782no—
148#137—1392no—
149AGL(1,ℤ/8)—2782no—
150AGL(1,ℤ/8)—2782no—
151AGL(1,ℤ/8)—2782no—
152AGL(1,ℤ/8)—1392no—
153AGL(1,ℤ/8)—1392no—
154AGL(1,ℤ/8)—1392no—
155#135—2782no—
156#135—2782no—
157#135—1392no—
158C2×SD16—2782no—
159C2×SD16C2×D13912yescomplemented
160C2×D8—2782no—
161C2×D8C2×D13912yescomplemented
162#131—1392no—
163#131—1392no—
164#127C2×D13911yescharacteristic, complemented
165#124C2×D13911yescharacteristic, complemented
166#108—1392no—
167#104—1392no—
168#104—1392no—
169#101—1391no—
170#100—1392no—
171#100—1391no—
172#97C2×D13911yescharacteristic, complemented
173#41—2782no—
174#41—2782no—
175#41—2782no—
176#41—2782no—
177#41—5561no—
178#41—1392no—
179#41—1392no—
180C2×Q8—2782no—
181C2×Q8C22×D13911yescharacteristic
182C2×D4—41no—
183C2×D4—2782no—
184C2×D4—2782no—
185C2×D4—2782no—
186C2×D4—5561no—
187C2×D4—5562no—
188C2×D4C22×D13911yescharacteristic
189C2×D4C22×D13912yes—
190Q16—2782no—
191Q16—1392no—
192SD16—22no—
193SD16—22no—
194SD16—2782no—
195SD16—2782no—
196SD16—2782no—
197SD16—2782no—
198SD16—1392no—
199SD16—1392no—
200D8—22no—
201D8—22no—
202D8—2782no—
203D8—2782no—
204D8—2782no—
205D8—1392no—
206M16—2782no—
207M16—2782no—
208M16—1392no—
209M16—1392no—
210C2×C8—2782no—
211C2×C8—2782no—
212C2×C8C22×D13912yes—
213C22⋊C4—41no—
214C42C22×D13911yescharacteristic
215C23—21no—
216C23—22no—
217C23—5562no—
218C23—5561no—
219Q8—22no—
220Q8—5561no—
221Q8—1392no—
222Q8—1392no—
223D4—22no—
224D4—22no—
225D4—22no—
226D4—82no—
227D4—2782no—
228D4—2782no—
229D4—5561no—
230D4—5562no—
231D4—5562no—
232D4—5562no—
233D4—1392no—
234D4—1392no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—2782no—
238C2×C4—2782no—
239C2×C4—5562no—
240C2×C4—5561no—
241C2×C4—5562no—
242C2×C4C23×D13911yescharacteristic
243C8—2782no—
244C8—2782no—
245C8D4×D13912yes—
246C8D4×D13912yes—
247C22—41no—
248C22—42no—
249C22—82no—
250C22—81no—
251C22—2782no—
252C22—2782no—
253C22—22241no—
254C22—22242no—
255C22#1798488711yescharacteristic
256C4—41no—
257C4—41no—
258C4—2782no—
259C4—2782no—
260C4C2×D4×D13912yes—
261C2—21no—
262C2—82no—
263C2—81no—
264C2—5562no—
265C2—5562no—
266C2#2258158811yescharacteristic
subgroupG/H
centerC2#22581588
commutator, cosocleC2×C4×C139C24
FrattiniC2×C4C23×D139
Fitting#496×C139C2
socleC2×C139D42
hypercenter#496D139
hypocenterC139#3288
LCSG ▷ C2×C4×C139 ▷ C2×C139 ▷ C139
UCSC1 ◁ C2 ◁ C2×C4 ◁ #496

Complements

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

NH ≅ G/Ntype
1. #22581759C2semidirect
2. #22581754C2semidirect
3. #22581220C2semidirect
4. #22581213C2semidirect
5. #22581130C2semidirect
6. #22581125C2semidirect
7. #22581120C2semidirect
8. #22581076C2semidirect
9. #496×C139C2semidirect
10. #17984890C22semidirect
12. #17984887C22semidirect
13. #17984873C22semidirect
17. #17984868C22semidirect
18. #17984868C22semidirect
26. #17984848C22semidirect
27. #17984848C22semidirect
28. #17984778C22semidirect
29. #17984770C22semidirect
30. #17984770C22semidirect
31. #17984761C22semidirect
32. #17984759C22semidirect
33. #17984759C22semidirect
34. C2×SD16×C139C22semidirect
35. C2×D8×C139C22semidirect
36. #127×C139C22semidirect
37. #124×C139C22semidirect
38. #97×C139C22semidirect
52. Dic556D4semidirect
54. #14555774D4semidirect
55. #14555774D4semidirect
60. D1112D4semidirect
133. C139#3288semidirect
137. #496D139semidirect
159. C2×SD16C2×D139semidirect
161. C2×D8C2×D139semidirect
164. #127C2×D139semidirect
165. #124C2×D139semidirect
172. #97C2×D139semidirect

Maximal subgroups

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

subgroupindex
#225817592
#225817542
#225812202
#225812132
#225811302
#225811252
#225811202
#225810762
#496×C1392
#3288139