Finite Group Databasebeta

← #31819544

Subgroups — #31819544

Order 17792.

370 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 113 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#22581755C211yesmaximal, characteristic, complemented
2#22581754C211yesmaximal, characteristic, complemented
3#22581391C211yesmaximal, characteristic, complemented
4#22581390C212yesmaximal, complemented
5#22581387C212yesmaximal, complemented
6#22581384C212yesmaximal, complemented
7#22581383C211yesmaximal, characteristic, complemented
8C2×#17984783C211yesmaximal, characteristic, complemented
9D16×D139C212yesmaximal, complemented
10#22581371C211yesmaximal, characteristic, complemented
11#509×C139C211yesmaximal, characteristic, complemented
12#17984888—21no—
13#17984887—21no—
14#17984869—22no—
15#17984868—21no—
16#17984867—21no—
17#17984866—22no—
18#17984865C2212yescomplemented
19#17984864C2211yescharacteristic, complemented
20#17984861C2212yescomplemented
21C2×D1112C2211yescharacteristic, complemented
22D8×D139C2212yescomplemented
23D8×D139C2212yescomplemented
24#17984848C2211yescharacteristic
25#17984788C2211yescharacteristic, complemented
26#17984787C2212yescomplemented
27#17984786C2211yescharacteristic, complemented
28#17984785C2211yescharacteristic, complemented
29D2224C2212yescomplemented
30#17984783C2211yescharacteristic, complemented
31#17984783C2212yescomplemented
32#17984783C2212yescomplemented
33#17984781C2212yescomplemented
34C2×C139⋊C16C2211yescharacteristic, complemented
35C16×D139C2212yescomplemented
36#135×C139C2211yescharacteristic, complemented
37C2×D8×C139C2211yescharacteristic, complemented
38SD32×C139C2212yescomplemented
39D16×C139C2212yescomplemented
40M32×C139C2211yescharacteristic, complemented
41#14555786—22no—
42#14555785—21no—
43#14555785—41no—
44#14555785D411yescharacteristic
45#14555784—22no—
46#14555784—41no—
47C2×D556—21no—
48C2×D556—41no—
49C2×C22⋊D139—42no—
50D4×D139—22no—
51D4×D139—22no—
52D4×D139—41no—
53D4×D139—42no—
54Dic556C2311yescharacteristic
55#14555774—21no—
56#14555773—22no—
57#14555772—22no—
58D1112—21no—
59D1112C2312yes—
60D1112C2311yescharacteristic
61#14555770—22no—
62#14555770—22no—
63#14555769D411yescharacteristic
64#14555768D412yes—
65C8×D139C2312yes—
66C139⋊C16C2311yescharacteristic
67C139⋊C16C2311yescharacteristic
68#41×C139—21no—
69C2×D4×C139—21no—
70Q16×C139C2311yescharacteristic
71SD16×C139—21no—
72D8×C139—21no—
73D8×C139C2312yes—
74D8×C139C2311yescharacteristic
75C2×C8×C139C2311yescharacteristic
76C16×C139C2312yes—
77C22×D139—41no—
78C22×D139—42no—
79C22×D139—42no—
80Dic278—21no—
81Dic278D811yescharacteristic
82D556—21no—
83D556—22no—
84D556—42no—
85D556—41no—
86D556D811yescharacteristic
87C22⋊D139—41no—
88C22⋊D139—42no—
89C22⋊D139—42no—
90C22⋊D139—42no—
91C22⋊D139D812yes—
92C2×Dic139—42no—
93C4×D139—41no—
94C4×D139—42no—
95C4×D139C2×D412yes—
96C139⋊C8C2×D412yes—
97C23×C139—41no—
98Q8×C139—21no—
99D4×C139—22no—
100D4×C139—21no—
101D4×C139—41no—
102D4×C139—41no—
103C2×C4×C139—41no—
104C2×C4×C139C2×D411yescharacteristic
105C8×C139C2×D411yescharacteristic
106C8×C139C2411yescharacteristic
107C22×C139—41no—
108C22×C139—42no—
109C22×C139—161no—
110C22×C139C2×D811yescharacteristic
111C2×D139—41no—
112C2×D139—42no—
113C2×D139—162no—
114C2×D139—162no—
115C2×D139—161no—
116C2×D139C2×D812yes—
117C4×C139—41no—
118C4×C139C2×D811yescharacteristic
119C4×C139C22×D411yescharacteristic
120Dic139—41no—
121Dic139C2×D812yes—
122C2×C139—21no—
123C2×C139—81no—
124C2×C139—82no—
125C2×C139C22×D811yescharacteristic
126D139—22no—
127D139—81no—
128D139—82no—
129C139#341511yescharacteristic, complemented, Sylow
130#3415—1391nomaximal, Sylow
131#576—1391no—
132#576—1391no—
133#509—1392no—
134#509—1392no—
135#509—1391no—
136#509D13911yescharacteristic, complemented
137#508—1391no—
138#508—1392no—
139C2×D16—1391no—
140C2×D16—1392no—
141#504—1391no—
142H(5,𝔽2)—2781no—
143H(5,𝔽2)—2781no—
144AGL(1,ℤ/8)—2782no—
145AGL(1,ℤ/8)—2782no—
146AGL(1,ℤ/8)—2781no—
147AGL(1,ℤ/8)—2781no—
148#135—1392no—
149#135—1392no—
150#135—1391no—
151#135C2×D13911yescharacteristic, complemented
152C2×D8—1392no—
153C2×D8—1392no—
154C2×D8—1391no—
155C2×D8C2×D13911yescharacteristic, complemented
156#131—1391no—
157Q32—1391no—
158SD32—1391no—
159SD32—1391no—
160SD32—1392no—
161SD32C2×D13912yescomplemented
162D16—1392no—
163D16—1391no—
164D16—1392no—
165D16—1392no—
166D16C2×D13912yescomplemented
167M32—1392no—
168M32C2×D13911yescharacteristic, complemented
169C2×C16—1391no—
170C2×C16—1392no—
171#41—21no—
172#41—2782no—
173#41—2782no—
174#41—2781no—
175#41—5561no—
176#41—5561no—
177#41—1391no—
178C2×D4—21no—
179C2×D4—2782no—
180C2×D4—2782no—
181C2×D4—2781no—
182C2×D4—5561no—
183C2×D4—5562no—
184C2×D4—5561no—
185C2×D4—5562no—
186Q16—1391no—
187Q16C22×D13911yescharacteristic
188SD16—21no—
189SD16—2782no—
190SD16—2782no—
191SD16—2781no—
192D8—21no—
193D8—2782no—
194D8—2782no—
195D8—2781no—
196D8—1392no—
197D8—1391no—
198D8C22×D13912yes—
199D8C22×D13911yescharacteristic
200M16—1391no—
201M16—1392no—
202C2×C8—1392no—
203C2×C8C22×D13911yescharacteristic
204C16—1391no—
205C16—1391no—
206C16C22×D13912yes—
207C23—41no—
208C23—5562no—
209C23—5562no—
210C23—5561no—
211Q8—21no—
212Q8—2781no—
213Q8—1391no—
214D4—22no—
215D4—21no—
216D4—41no—
217D4—41no—
218D4—2781no—
219D4—2782no—
220D4—5562no—
221D4—5562no—
222D4—5561no—
223D4—5562no—
224D4—5562no—
225D4—5561no—
226D4—1392no—
227D4—1391no—
228C2×C4—41no—
229C2×C4—5562no—
230C2×C4—5562no—
231C2×C4—5561no—
232C2×C4—1392no—
233C2×C4D4×D13911yescharacteristic
234C8—1392no—
235C8D4×D13911yescharacteristic
236C8C23×D13911yescharacteristic
237C22—41no—
238C22—42no—
239C22—161no—
240C22—5562no—
241C22—5561no—
242C22—1392no—
243C22—22241no—
244C22—22242no—
245C22—22242no—
246C22D8×D13911yescharacteristic
247C4—41no—
248C4—5561no—
249C4—1392no—
250C4D8×D13911yescharacteristic
251C4C2×D4×D13911yescharacteristic
252C2—21no—
253C2—82no—
254C2—81no—
255C2—2782no—
256C2—11122no—
257C2—11121no—
258C2C2×D8×D13911yescharacteristic
subgroupG/H
centerC2C2×D8×D139
commutator, cosocleC8×C139C24
FrattiniC8C23×D139
Fitting#509×C139C2
socleC2×C139C22×D8
hypercenter#509D139
hypocenterC139#3415
LCSG ▷ C8×C139 ▷ C4×C139 ▷ C2×C139 ▷ C139
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #509

Complements

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

NH ≅ G/Ntype
1. #22581755C2semidirect
2. #22581754C2semidirect
3. #22581391C2semidirect
4. #22581390C2semidirect
5. #22581387C2semidirect
6. #22581384C2semidirect
7. #22581383C2semidirect
8. C2×#17984783C2semidirect
9. D16×D139C2semidirect
10. #22581371C2semidirect
11. #509×C139C2semidirect
18. #17984865C22semidirect
19. #17984864C22semidirect
20. #17984861C22semidirect
21. C2×D1112C22semidirect
22. D8×D139C22semidirect
23. D8×D139C22semidirect
25. #17984788C22semidirect
26. #17984787C22semidirect
27. #17984786C22semidirect
28. #17984785C22semidirect
29. D2224C22semidirect
30. #17984783C22semidirect
31. #17984783C22semidirect
32. #17984783C22semidirect
33. #17984781C22semidirect
34. C2×C139⋊C16C22semidirect
35. C16×D139C22semidirect
36. #135×C139C22semidirect
37. C2×D8×C139C22semidirect
38. SD32×C139C22semidirect
39. D16×C139C22semidirect
40. M32×C139C22semidirect
129. C139#3415semidirect
136. #509D139semidirect
151. #135C2×D139semidirect
155. C2×D8C2×D139semidirect
161. SD32C2×D139semidirect
166. D16C2×D139semidirect
168. M32C2×D139semidirect

Maximal subgroups

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

subgroupindex
#225817552
#225817542
#225813912
#225813902
#225813872
#225813842
#225813832
C2×#179847832
D16×D1392
#225813712
#509×C1392
#3415139