Finite Group Databasebeta

← #31819550

Subgroups — #31819550

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#22581754C211yesmaximal, characteristic, complemented
3#22581396C211yesmaximal, characteristic, complemented
4#22581392C212yesmaximal, complemented
5#22581387C212yesmaximal, complemented
6#22581383C211yesmaximal, characteristic, complemented
7#22581382C212yesmaximal, complemented
8C2×#17984786C211yesmaximal, characteristic, complemented
9SD32×D139C212yesmaximal, complemented
10#22581371C211yesmaximal, characteristic, complemented
11#509×C139C211yesmaximal, characteristic, complemented
12#17984889—21no—
13#17984887—21no—
14#17984872—21no—
15#17984871—22no—
16#17984868—21no—
17#17984866—22no—
18#17984864C2211yescharacteristic
19#17984861C2212yes—
20#17984861C2212yescomplemented
21C2×Dic556C2211yescharacteristic, complemented
22Q16×D139C2212yescomplemented
23D8×D139C2212yes—
24#17984848C2211yescharacteristic
25Dic1112C2212yescomplemented
26#17984788C2211yescharacteristic, complemented
27#17984787C2212yescomplemented
28#17984786C2212yescomplemented
29#17984786C2212yescomplemented
30#17984786C2211yescharacteristic, complemented
31#17984785C2211yescharacteristic, complemented
32#17984783C2211yescharacteristic, complemented
33#17984781C2212yescomplemented
34C2×C139⋊C16C2211yescharacteristic, complemented
35C16×D139C2212yescomplemented
36#135×C139C2211yescharacteristic, complemented
37C2×D8×C139C2211yescharacteristic
38SD32×C139C2212yescomplemented
39D16×C139C2212yescomplemented
40M32×C139C2211yescharacteristic, complemented
41#14555785—21no—
42#14555785—41no—
43#14555785D411yescharacteristic
44#14555784—22no—
45#14555784—22no—
46#14555784—41no—
47#14555784—42no—
48C2×Dic278—21no—
49C2×Dic278—41no—
50Q8×D139—22no—
51C2×C22⋊D139—42no—
52D4×D139—22no—
53D4×D139—41no—
54Dic556—21no—
55Dic556C2312yes—
56Dic556C2311yescharacteristic
57#14555775—22no—
58#14555774—21no—
59#14555773—22no—
60#14555773—22no—
61D1112C2311yescharacteristic
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×C139C2311yescharacteristic
74D8×C139C2312yes—
75C2×C8×C139C2311yescharacteristic
76C16×C139C2312yes—
77C22×D139—42no—
78Dic278—21no—
79Dic278—22no—
80Dic278—41no—
81Dic278—42no—
82Dic278D811yescharacteristic
83D556—21no—
84D556D811yescharacteristic
85C22⋊D139—41no—
86C22⋊D139—42no—
87C22⋊D139—42no—
88C22⋊D139—42no—
89C22⋊D139D812yes—
90C2×Dic139—41no—
91C2×Dic139—42no—
92C2×Dic139—42no—
93C4×D139—42no—
94C4×D139—41no—
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—162no—
113C2×D139C2×D812yes—
114C4×C139—41no—
115C4×C139C2×D811yescharacteristic
116C4×C139C22×D411yescharacteristic
117Dic139—41no—
118Dic139—42no—
119Dic139C2×D812yes—
120C2×C139—21no—
121C2×C139—82no—
122C2×C139—81no—
123C2×C139C22×D811yescharacteristic
124D139—22no—
125D139—81no—
126C139#341611yescharacteristic, complemented, Sylow
127#3416—1391nomaximal, Sylow
128#578—1391no—
129#576—1391no—
130#510—1392no—
131#510—1391no—
132#509—1392no—
133#509D13911yescharacteristic, complemented
134#508—1391no—
135#508—1392no—
136C2×SD32—1391no—
137C2×SD32—1392no—
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#135—1391no—
148#135C2×D13911yescharacteristic, complemented
149C2×Q16—1392no—
150C2×Q16—1391no—
151C2×D8—1392no—
152C2×D8C2×D13911yescharacteristic
153#131—1391no—
154Q32—1391no—
155Q32—1392no—
156SD32—1391no—
157SD32—1392no—
158SD32—1392no—
159SD32—1391no—
160SD32—1392no—
161SD32C2×D13912yescomplemented
162D16—1391no—
163D16C2×D13912yescomplemented
164M32—1392no—
165M32C2×D13911yescharacteristic, complemented
166C2×C16—1391no—
167C2×C16—1392no—
168#41—21no—
169#41—2782no—
170#41—2782no—
171#41—2781no—
172#41—5561no—
173#41—5561no—
174#41—5562no—
175#41—1391no—
176C2×Q8—2782no—
177C2×Q8—2781no—
178C2×Q8—5561no—
179C2×D4—21no—
180C2×D4—2782no—
181C2×D4—5561no—
182C2×D4—5562no—
183Q16—2782no—
184Q16—2781no—
185Q16—1392no—
186Q16—1391no—
187Q16C22×D13911yescharacteristic
188SD16—21no—
189SD16—2782no—
190SD16—2782no—
191SD16—2781no—
192D8—21no—
193D8—2782no—
194D8—1391no—
195D8C22×D13911yescharacteristic
196D8C22×D13912yes—
197M16—1391no—
198M16—1392no—
199C2×C8—1392no—
200C2×C8C22×D13911yescharacteristic
201C16—1391no—
202C16—1391no—
203C16C22×D13912yes—
204C23—41no—
205C23—5562no—
206Q8—21no—
207Q8—2781no—
208Q8—2782no—
209Q8—5562no—
210Q8—5561no—
211Q8—1391no—
212D4—22no—
213D4—21no—
214D4—41no—
215D4—41no—
216D4—2781no—
217D4—5562no—
218D4—5562no—
219D4—5561no—
220D4—5562no—
221D4—1392no—
222D4—1391no—
223C2×C4—41no—
224C2×C4—5562no—
225C2×C4—5561no—
226C2×C4—5562no—
227C2×C4—5562no—
228C2×C4—5561no—
229C2×C4—1392no—
230C2×C4D4×D13911yescharacteristic
231C8—1392no—
232C8D4×D13911yescharacteristic
233C8C23×D13911yescharacteristic
234C22—41no—
235C22—42no—
236C22—161no—
237C22—5561no—
238C22—1392no—
239C22—22242no—
240C22D8×D13911yescharacteristic
241C4—41no—
242C4—5561no—
243C4—5562no—
244C4—1392no—
245C4D8×D13911yescharacteristic
246C4C2×D4×D13911yescharacteristic
247C2—21no—
248C2—82no—
249C2—81no—
250C2—2782no—
251C2—11121no—
252C2C2×D8×D13911yescharacteristic
subgroupG/H
centerC2C2×D8×D139
commutator, cosocleC8×C139C24
FrattiniC8C23×D139
Fitting#509×C139C2
socleC2×C139C22×D8
hypercenter#509D139
hypocenterC139#3416
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. #22581767C2semidirect
2. #22581754C2semidirect
3. #22581396C2semidirect
4. #22581392C2semidirect
5. #22581387C2semidirect
6. #22581383C2semidirect
7. #22581382C2semidirect
8. C2×#17984786C2semidirect
9. SD32×D139C2semidirect
10. #22581371C2semidirect
11. #509×C139C2semidirect
20. #17984861C22semidirect
21. C2×Dic556C22semidirect
22. Q16×D139C22semidirect
25. Dic1112C22semidirect
26. #17984788C22semidirect
27. #17984787C22semidirect
28. #17984786C22semidirect
29. #17984786C22semidirect
30. #17984786C22semidirect
31. #17984785C22semidirect
32. #17984783C22semidirect
33. #17984781C22semidirect
34. C2×C139⋊C16C22semidirect
35. C16×D139C22semidirect
36. #135×C139C22semidirect
38. SD32×C139C22semidirect
39. D16×C139C22semidirect
40. M32×C139C22semidirect
126. C139#3416semidirect
133. #509D139semidirect
148. #135C2×D139semidirect
161. SD32C2×D139semidirect
163. D16C2×D139semidirect
165. M32C2×D139semidirect

Maximal subgroups

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

subgroupindex
#225817672
#225817542
#225813962
#225813922
#225813872
#225813832
#225813822
C2×#179847862
SD32×D1392
#225813712
#509×C1392
#3416139