Finite Group Databasebeta

← #31819549

Subgroups — #31819549

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#22581766C211yesmaximal, characteristic, complemented
2#22581755C211yesmaximal, characteristic, complemented
3#22581395C212yesmaximal, complemented
4#22581391C211yesmaximal, characteristic, complemented
5#22581390C212yesmaximal, complemented
6#22581386C212yesmaximal, complemented
7#22581383C211yesmaximal, characteristic, complemented
8C2×#17984785C211yesmaximal, characteristic, complemented
9SD32×D139C212yesmaximal, complemented
10#22581371C211yesmaximal, characteristic, complemented
11#510×C139C211yesmaximal, characteristic, complemented
12#17984890—21no—
13#17984888—21no—
14#17984874—22no—
15#17984873—21no—
16#17984869—22no—
17#17984867—21no—
18#17984865C2212yes—
19#17984865C2212yescomplemented
20#17984864C2211yescharacteristic, complemented
21Q16×D139C2212yescomplemented
22C2×D1112C2211yescharacteristic
23D8×D139C2212yes—
24#17984848C2211yescharacteristic
25#17984788C2211yescharacteristic, complemented
26#17984787C2212yescomplemented
27#17984786C2211yescharacteristic, complemented
28#17984785C2212yescomplemented
29#17984785C2212yescomplemented
30#17984785C2211yescharacteristic, complemented
31D2224C2212yescomplemented
32#17984783C2211yescharacteristic, complemented
33#17984781C2212yescomplemented
34C2×C139⋊C16C2211yescharacteristic, complemented
35C16×D139C2212yescomplemented
36#135×C139C2211yescharacteristic
37C2×Q16×C139C2211yescharacteristic, complemented
38Q32×C139C2212yescomplemented
39SD32×C139C2212yescomplemented
40M32×C139C2211yescharacteristic, complemented
41#14555786—22no—
42#14555786—22no—
43#14555786—41no—
44#14555785—21no—
45#14555785—41no—
46#14555785—42no—
47#14555785D411yescharacteristic
48Q8×D139—22no—
49Q8×D139—41no—
50C2×D556—21no—
51C2×D556—41no—
52D4×D139—22no—
53D4×D139—42no—
54Dic556C2311yescharacteristic
55#14555775—22no—
56#14555774—21no—
57#14555772—22no—
58#14555772—22no—
59D1112—21no—
60D1112C2311yescharacteristic
61D1112C2312yes—
62#14555770—22no—
63#14555769D411yescharacteristic
64#14555768D412yes—
65C8×D139C2312yes—
66C139⋊C16C2311yescharacteristic
67C139⋊C16C2311yescharacteristic
68#41×C139—21no—
69C2×Q8×C139—21no—
70Q16×C139—21no—
71Q16×C139C2312yes—
72Q16×C139C2311yescharacteristic
73SD16×C139—21no—
74D8×C139C2311yescharacteristic
75C2×C8×C139C2311yescharacteristic
76C16×C139C2312yes—
77C22×D139—41no—
78C22×D139—42no—
79Dic278—21no—
80Dic278—42no—
81Dic278D811yescharacteristic
82D556—22no—
83D556—21no—
84D556—42no—
85D556—42no—
86D556—41no—
87D556D811yescharacteristic
88C22⋊D139—41no—
89C22⋊D139—42no—
90C22⋊D139D812yes—
91C4×D139—42no—
92C4×D139—41no—
93C4×D139—42no—
94C4×D139—42no—
95C4×D139C2×D412yes—
96C139⋊C8C2×D412yes—
97Q8×C139—21no—
98Q8×C139—22no—
99Q8×C139—41no—
100D4×C139—21no—
101D4×C139—41no—
102C2×C4×C139—41no—
103C2×C4×C139—41no—
104C2×C4×C139C2×D411yescharacteristic
105C8×C139C2×D411yescharacteristic
106C8×C139C2411yescharacteristic
107C22×C139—41no—
108C22×C139C2×D811yescharacteristic
109C2×D139—41no—
110C2×D139—42no—
111C2×D139—162no—
112C2×D139—161no—
113C2×D139C2×D812yes—
114C4×C139—41no—
115C4×C139—42no—
116C4×C139C2×D811yescharacteristic
117C4×C139C22×D411yescharacteristic
118Dic139—41no—
119Dic139C2×D812yes—
120C2×C139—21no—
121C2×C139—81no—
122C2×C139C22×D811yescharacteristic
123D139—22no—
124D139—81no—
125D139—82no—
126C139#341611yescharacteristic, complemented, Sylow
127#3416—1391nomaximal, Sylow
128#578—1391no—
129#576—1391no—
130#510—1392no—
131#510D13911yescharacteristic, complemented
132#509—1392no—
133#509—1391no—
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
149C2×Q16—1392no—
150C2×Q16C2×D13911yescharacteristic, complemented
151C2×D8—1392no—
152C2×D8—1391no—
153#131—1391no—
154Q32—1391no—
155Q32C2×D13912yescomplemented
156SD32—1392no—
157SD32—1392no—
158SD32—1391no—
159SD32—1391no—
160SD32—1392no—
161SD32C2×D13912yescomplemented
162D16—1391no—
163D16—1392no—
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—5562no—
174#41—5561no—
175#41—1391no—
176C2×Q8—21no—
177C2×Q8—2782no—
178C2×Q8—5561no—
179C2×D4—2782no—
180C2×D4—2781no—
181C2×D4—5561no—
182C2×D4—5562no—
183Q16—21no—
184Q16—2782no—
185Q16—1391no—
186Q16C22×D13912yes—
187Q16C22×D13911yescharacteristic
188SD16—21no—
189SD16—2782no—
190SD16—2782no—
191SD16—2781no—
192D8—2782no—
193D8—2781no—
194D8—1391no—
195D8—1392no—
196D8C22×D13911yescharacteristic
197M16—1391no—
198M16—1392no—
199C2×C8—1392no—
200C2×C8C22×D13911yescharacteristic
201C16—1391no—
202C16—1391no—
203C16C22×D13912yes—
204C23—5562no—
205C23—5561no—
206Q8—21no—
207Q8—22no—
208Q8—41no—
209Q8—2781no—
210Q8—5562no—
211Q8—1391no—
212D4—21no—
213D4—41no—
214D4—2782no—
215D4—2781no—
216D4—5562no—
217D4—5562no—
218D4—5561no—
219D4—5562no—
220D4—5561no—
221D4—1392no—
222D4—1391no—
223C2×C4—41no—
224C2×C4—41no—
225C2×C4—5562no—
226C2×C4—5562no—
227C2×C4—5562no—
228C2×C4—5561no—
229C2×C4—1392no—
230C2×C4D4×D13911yescharacteristic
231C8—1392no—
232C8D4×D13911yescharacteristic
233C8C23×D13911yescharacteristic
234C22—41no—
235C22—5561no—
236C22—5562no—
237C22—1392no—
238C22—22241no—
239C22—22242no—
240C22D8×D13911yescharacteristic
241C4—41no—
242C4—42no—
243C4—5561no—
244C4—1392no—
245C4D8×D13911yescharacteristic
246C4C2×D4×D13911yescharacteristic
247C2—21no—
248C2—81no—
249C2—2782no—
250C2—11122no—
251C2—11121no—
252C2C2×D8×D13911yescharacteristic
subgroupG/H
centerC2C2×D8×D139
commutator, cosocleC8×C139C24
FrattiniC8C23×D139
Fitting#510×C139C2
socleC2×C139C22×D8
hypercenter#510D139
hypocenterC139#3416
LCSG ▷ C8×C139 ▷ C4×C139 ▷ C2×C139 ▷ C139
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #510

Complements

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

NH ≅ G/Ntype
1. #22581766C2semidirect
2. #22581755C2semidirect
3. #22581395C2semidirect
4. #22581391C2semidirect
5. #22581390C2semidirect
6. #22581386C2semidirect
7. #22581383C2semidirect
8. C2×#17984785C2semidirect
9. SD32×D139C2semidirect
10. #22581371C2semidirect
11. #510×C139C2semidirect
19. #17984865C22semidirect
20. #17984864C22semidirect
21. Q16×D139C22semidirect
25. #17984788C22semidirect
26. #17984787C22semidirect
27. #17984786C22semidirect
28. #17984785C22semidirect
29. #17984785C22semidirect
30. #17984785C22semidirect
31. D2224C22semidirect
32. #17984783C22semidirect
33. #17984781C22semidirect
34. C2×C139⋊C16C22semidirect
35. C16×D139C22semidirect
37. C2×Q16×C139C22semidirect
38. Q32×C139C22semidirect
39. SD32×C139C22semidirect
40. M32×C139C22semidirect
126. C139#3416semidirect
131. #510D139semidirect
150. C2×Q16C2×D139semidirect
155. Q32C2×D139semidirect
161. SD32C2×D139semidirect
165. M32C2×D139semidirect

Maximal subgroups

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

subgroupindex
#225817662
#225817552
#225813952
#225813912
#225813902
#225813862
#225813832
C2×#179847852
SD32×D1392
#225813712
#510×C1392
#3416139