Finite Group Databasebeta

← #31819547

Subgroups — #31819547

Order 17792.

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

normal subgroups only

This group has 360 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#22581767C211yesmaximal, characteristic, complemented
2#22581755C211yesmaximal, characteristic, complemented
3#22581395C212yesmaximal, complemented
4#22581393C211yesmaximal, characteristic, complemented
5#22581388C211yesmaximal, characteristic, complemented
6#22581387C212yesmaximal, complemented
7#22581384C212yesmaximal, complemented
8C2×#17984787C211yesmaximal, characteristic, complemented
9SD32×D139C212yesmaximal, complemented
10#22581369C211yesmaximal, characteristic, complemented
11#508×C139C211yesmaximal, characteristic, complemented
12#17984889—21no—
13#17984888—21no—
14#17984872—21no—
15#17984871—22no—
16#17984869—22no—
17#17984867—21no—
18#17984865C2212yes—
19#17984861C2212yescomplemented
20C2×Dic556C2211yescharacteristic, complemented
21Q16×D139C2212yescomplemented
22C2×D1112C2211yescharacteristic
23D8×D139C2212yes—
24#17984848C2211yescharacteristic
25#17984788C2212yescomplemented
26#17984787C2212yescomplemented
27#17984787C2212yescomplemented
28#17984786C2212yescomplemented
29#17984785C2212yescomplemented
30#17984783C2212yescomplemented
31#17984782C2211yescharacteristic, complemented
32#17984781C2212yescomplemented
33C16×D139C2212yescomplemented
34#135×C139C2211yescharacteristic
35#135×C139C2211yescharacteristic, complemented
36Q32×C139C2211yescharacteristic, complemented
37SD32×C139C2211yescharacteristic, complemented
38SD32×C139C2211yescharacteristic, complemented
39D16×C139C2211yescharacteristic, complemented
40C2×C16×C139C2211yescharacteristic, complemented
41#14555786—22no—
42#14555785—41no—
43#14555785—41no—
44#14555785D411yescharacteristic
45#14555784—22no—
46#14555784—42no—
47C2×Dic278—21no—
48C2×Dic278—41no—
49Q8×D139—22no—
50C2×D556—21no—
51C2×D556—41no—
52D4×D139—22no—
53D4×D139—42no—
54Dic556—21no—
55Dic556C2312yes—
56#14555775—22no—
57#14555773—22no—
58#14555772—22no—
59D1112—21no—
60D1112C2312yes—
61#14555770—22no—
62#14555769D411yescharacteristic
63#14555768D412yes—
64C8×D139C2312yes—
65C139⋊C16C2312yes—
66#41×C139—21no—
67#41×C139—21no—
68Q16×C139C2311yescharacteristic
69Q16×C139C2311yescharacteristic
70SD16×C139—21no—
71SD16×C139—21no—
72D8×C139C2311yescharacteristic
73D8×C139C2311yescharacteristic
74C2×C8×C139C2311yescharacteristic
75C16×C139C2311yescharacteristic
76C16×C139C2311yescharacteristic
77C22×D139—41no—
78C22×D139—42no—
79Dic278—22no—
80Dic278—41no—
81Dic278—42no—
82Dic278D811yescharacteristic
83D556—22no—
84D556—42no—
85D556—41no—
86D556D811yescharacteristic
87C22⋊D139—42no—
88C22⋊D139—42no—
89C22⋊D139D812yes—
90C2×Dic139—41no—
91C2×Dic139—42no—
92C4×D139—42no—
93C4×D139—42no—
94C4×D139C2×D412yes—
95C139⋊C8C2×D412yes—
96Q8×C139—21no—
97Q8×C139—21no—
98D4×C139—21no—
99D4×C139—21no—
100D4×C139—41no—
101D4×C139—41no—
102C2×C4×C139—41no—
103C2×C4×C139—41no—
104C2×C4×C139C2×D411yescharacteristic
105C8×C139C2×D411yescharacteristic
106C8×C139C2411yescharacteristic
107C22×C139—41no—
108C22×C139—41no—
109C22×C139C2×D811yescharacteristic
110C2×D139—42no—
111C2×D139—162no—
112C2×D139—161no—
113C2×D139C2×D812yes—
114C4×C139—41no—
115C4×C139—41no—
116C4×C139C2×D811yescharacteristic
117C4×C139C22×D411yescharacteristic
118Dic139—42no—
119Dic139C2×D812yes—
120C2×C139—21no—
121C2×C139—81no—
122C2×C139—81no—
123C2×C139C22×D811yescharacteristic
124D139—22no—
125D139—82no—
126C139#341611yescharacteristic, complemented, Sylow
127#3416—1391nomaximal, Sylow
128#578—1391no—
129#576—1391no—
130#510—1391no—
131#510—1392no—
132#509—1391no—
133#509—1392no—
134#508—1392no—
135#508D13911yescharacteristic, complemented
136C2×SD32—1392no—
137C2×SD32—1391no—
138#504—1391no—
1392−1+4—2781no—
140H(5,𝔽2)—2781no—
141#137—2782no—
142#137—2781no—
143AGL(1,ℤ/8)—2782no—
144AGL(1,ℤ/8)—2781no—
145#135—1392no—
146#135—1392no—
147#135C2×D13911yescharacteristic
148#135C2×D13911yescharacteristic, complemented
149C2×Q16—1392no—
150C2×Q16—1391no—
151C2×D8—1392no—
152C2×D8—1391no—
153#131—1391no—
154Q32—1392no—
155Q32C2×D13911yescharacteristic, complemented
156SD32—1392no—
157SD32—1392no—
158SD32—1392no—
159SD32—1392no—
160SD32C2×D13911yescharacteristic, complemented
161SD32C2×D13911yescharacteristic, complemented
162D16—1392no—
163D16C2×D13911yescharacteristic, complemented
164M32—1391no—
165M32—1392no—
166C2×C16—1392no—
167C2×C16C2×D13911yescharacteristic, complemented
168#41—21no—
169#41—21no—
170#41—2782no—
171#41—2782no—
172#41—5561no—
173#41—5561no—
174#41—5562no—
175#41—1391no—
176C2×Q8—2782no—
177C2×Q8—2781no—
178C2×Q8—5561no—
179C2×D4—2782no—
180C2×D4—2781no—
181C2×D4—5561no—
182C2×D4—5562no—
183Q16—2782no—
184Q16—2781no—
185Q16—1392no—
186Q16C22×D13911yescharacteristic
187Q16C22×D13911yescharacteristic
188SD16—21no—
189SD16—21no—
190SD16—2782no—
191SD16—2782no—
192D8—2782no—
193D8—2781no—
194D8—1392no—
195D8C22×D13911yescharacteristic
196D8C22×D13911yescharacteristic
197M16—1391no—
198M16—1392no—
199C2×C8—1392no—
200C2×C8C22×D13911yescharacteristic
201C16—1392no—
202C16C22×D13911yescharacteristic
203C16C22×D13911yescharacteristic
204C23—5562no—
205C23—5561no—
206Q8—21no—
207Q8—21no—
208Q8—2782no—
209Q8—5562no—
210Q8—5561no—
211Q8—1391no—
212D4—21no—
213D4—21no—
214D4—41no—
215D4—41no—
216D4—2782no—
217D4—5562no—
218D4—5562no—
219D4—5562no—
220D4—5561no—
221D4—1392no—
222D4—1391no—
223C2×C4—41no—
224C2×C4—41no—
225C2×C4—5562no—
226C2×C4—5562no—
227C2×C4—5561no—
228C2×C4—5562no—
229C2×C4—1392no—
230C2×C4D4×D13911yescharacteristic
231C8—1392no—
232C8D4×D13911yescharacteristic
233C8C23×D13911yescharacteristic
234C22—41no—
235C22—41no—
236C22—5562no—
237C22—1392no—
238C22—22241no—
239C22—22242no—
240C22D8×D13911yescharacteristic
241C4—41no—
242C4—41no—
243C4—5562no—
244C4—1392no—
245C4D8×D13911yescharacteristic
246C4C2×D4×D13911yescharacteristic
247C2—21no—
248C2—81no—
249C2—81no—
250C2—2782no—
251C2—11122no—
252C2C2×D8×D13911yescharacteristic
subgroupG/H
centerC2C2×D8×D139
commutator, cosocleC8×C139C24
FrattiniC8C23×D139
Fitting#508×C139C2
socleC2×C139C22×D8
hypercenter#508D139
hypocenterC139#3416
LCSG ▷ C8×C139 ▷ C4×C139 ▷ C2×C139 ▷ C139
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #508

Complements

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

NH ≅ G/Ntype
1. #22581767C2semidirect
2. #22581755C2semidirect
3. #22581395C2semidirect
4. #22581393C2semidirect
5. #22581388C2semidirect
6. #22581387C2semidirect
7. #22581384C2semidirect
8. C2×#17984787C2semidirect
9. SD32×D139C2semidirect
10. #22581369C2semidirect
11. #508×C139C2semidirect
19. #17984861C22semidirect
20. C2×Dic556C22semidirect
21. Q16×D139C22semidirect
25. #17984788C22semidirect
26. #17984787C22semidirect
27. #17984787C22semidirect
28. #17984786C22semidirect
29. #17984785C22semidirect
30. #17984783C22semidirect
31. #17984782C22semidirect
32. #17984781C22semidirect
33. C16×D139C22semidirect
35. #135×C139C22semidirect
36. Q32×C139C22semidirect
37. SD32×C139C22semidirect
38. SD32×C139C22semidirect
39. D16×C139C22semidirect
40. C2×C16×C139C22semidirect
126. C139#3416semidirect
135. #508D139semidirect
148. #135C2×D139semidirect
155. Q32C2×D139semidirect
160. SD32C2×D139semidirect
161. SD32C2×D139semidirect
163. D16C2×D139semidirect
167. C2×C16C2×D139semidirect

Maximal subgroups

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

subgroupindex
#225817672
#225817552
#225813952
#225813932
#225813882
#225813872
#225813842
C2×#179847872
SD32×D1392
#225813692
#508×C1392
#3416139