Finite Group Databasebeta

← #31793929

Subgroups — #31793929

Order 17536.

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#22578205C211yesmaximal, characteristic, complemented
2#22578204C211yesmaximal, characteristic, complemented
3#22577841C211yesmaximal, characteristic, complemented
4#22577840C212yesmaximal, complemented
5#22577837C212yesmaximal, complemented
6#22577834C212yesmaximal, complemented
7#22577833C211yesmaximal, characteristic, complemented
8C2×#17984029C211yesmaximal, characteristic, complemented
9D16×D137C212yesmaximal, complemented
10#22577821C211yesmaximal, characteristic, complemented
11#509×C137C211yesmaximal, characteristic, complemented
12#17984134—21no—
13#17984133—21no—
14#17984115—22no—
15#17984114—21no—
16#17984113—21no—
17#17984112—22no—
18#17984111C2212yescomplemented
19#17984110C2211yescharacteristic, complemented
20#17984107C2212yescomplemented
21C2×D1096C2211yescharacteristic, complemented
22D8×D137C2212yescomplemented
23D8×D137C2212yescomplemented
24#17984094C2211yescharacteristic
25#17984034C2211yescharacteristic, complemented
26#17984033C2212yescomplemented
27#17984032C2211yescharacteristic, complemented
28#17984031C2211yescharacteristic, complemented
29D2192C2212yescomplemented
30#17984029C2211yescharacteristic, complemented
31#17984029C2212yescomplemented
32#17984029C2212yescomplemented
33#17984027C2212yescomplemented
34C2×#14555485C2211yescharacteristic, complemented
35C16×D137C2212yescomplemented
36#135×C137C2211yescharacteristic, complemented
37C2×D8×C137C2211yescharacteristic, complemented
38SD32×C137C2212yescomplemented
39D16×C137C2212yescomplemented
40M32×C137C2211yescharacteristic, complemented
41#14555511—22no—
42#14555510—21no—
43#14555510—41no—
44#14555510D411yescharacteristic
45#14555509—22no—
46#14555509—41no—
47C2×D548—21no—
48C2×D548—41no—
49C2×C22⋊D137—42no—
50D4×D137—22no—
51D4×D137—22no—
52D4×D137—41no—
53D4×D137—42no—
54Dic548C2311yescharacteristic
55#14555499—21no—
56#14555498—22no—
57#14555497—22no—
58D1096—21no—
59D1096C2312yes—
60D1096C2311yescharacteristic
61#14555495—22no—
62#14555495—22no—
63#14555494D411yescharacteristic
64#14555493D412yes—
65C8×D137C2312yes—
66#14555485C2311yescharacteristic
67#14555485C2311yescharacteristic
68#41×C137—21no—
69C2×D4×C137—21no—
70Q16×C137C2311yescharacteristic
71SD16×C137—21no—
72D8×C137—21no—
73D8×C137C2312yes—
74D8×C137C2311yescharacteristic
75C2×C8×C137C2311yescharacteristic
76C16×C137C2312yes—
77C22×D137—41no—
78C22×D137—42no—
79C22×D137—42no—
80Dic274—21no—
81Dic274D811yescharacteristic
82D548—21no—
83D548—22no—
84D548—42no—
85D548—41no—
86D548D811yescharacteristic
87C22⋊D137—41no—
88C22⋊D137—42no—
89C22⋊D137—42no—
90C22⋊D137—42no—
91C22⋊D137D812yes—
92C2×Dic137—42no—
93C4×D137—41no—
94C4×D137—42no—
95C4×D137C2×D412yes—
96#11764040C2×D412yes—
97C23×C137—41no—
98Q8×C137—21no—
99D4×C137—22no—
100D4×C137—21no—
101D4×C137—41no—
102D4×C137—41no—
103C2×C4×C137—41no—
104C2×C4×C137C2×D411yescharacteristic
105C8×C137C2×D411yescharacteristic
106C8×C137C2411yescharacteristic
107C22×C137—41no—
108C22×C137—42no—
109C22×C137—161no—
110C22×C137C2×D811yescharacteristic
111C2×D137—41no—
112C2×D137—42no—
113C2×D137—162no—
114C2×D137—162no—
115C2×D137—161no—
116C2×D137C2×D812yes—
117C4×C137—41no—
118C4×C137C2×D811yescharacteristic
119C4×C137C22×D411yescharacteristic
120Dic137—41no—
121Dic137C2×D812yes—
122C2×C137—21no—
123C2×C137—81no—
124C2×C137—82no—
125C2×C137C22×D811yescharacteristic
126D137—22no—
127D137—81no—
128D137—82no—
129C137#341511yescharacteristic, complemented, Sylow
130#3415—1371nomaximal, Sylow
131#576—1371no—
132#576—1371no—
133#509—1372no—
134#509—1372no—
135#509—1371no—
136#509D13711yescharacteristic, complemented
137#508—1371no—
138#508—1372no—
139C2×D16—1371no—
140C2×D16—1372no—
141#504—1371no—
142H(5,𝔽2)—2741no—
143H(5,𝔽2)—2741no—
144AGL(1,ℤ/8)—2742no—
145AGL(1,ℤ/8)—2742no—
146AGL(1,ℤ/8)—2741no—
147AGL(1,ℤ/8)—2741no—
148#135—1372no—
149#135—1372no—
150#135—1371no—
151#135C2×D13711yescharacteristic, complemented
152C2×D8—1372no—
153C2×D8—1372no—
154C2×D8—1371no—
155C2×D8C2×D13711yescharacteristic, complemented
156#131—1371no—
157Q32—1371no—
158SD32—1371no—
159SD32—1371no—
160SD32—1372no—
161SD32C2×D13712yescomplemented
162D16—1372no—
163D16—1371no—
164D16—1372no—
165D16—1372no—
166D16C2×D13712yescomplemented
167M32—1372no—
168M32C2×D13711yescharacteristic, complemented
169C2×C16—1371no—
170C2×C16—1372no—
171#41—21no—
172#41—2742no—
173#41—2742no—
174#41—2741no—
175#41—5481no—
176#41—5481no—
177#41—1371no—
178C2×D4—21no—
179C2×D4—2742no—
180C2×D4—2742no—
181C2×D4—2741no—
182C2×D4—5481no—
183C2×D4—5482no—
184C2×D4—5481no—
185C2×D4—5482no—
186Q16—1371no—
187Q16C22×D13711yescharacteristic
188SD16—21no—
189SD16—2742no—
190SD16—2742no—
191SD16—2741no—
192D8—21no—
193D8—2742no—
194D8—2742no—
195D8—2741no—
196D8—1372no—
197D8—1371no—
198D8C22×D13712yes—
199D8C22×D13711yescharacteristic
200M16—1371no—
201M16—1372no—
202C2×C8—1372no—
203C2×C8C22×D13711yescharacteristic
204C16—1371no—
205C16—1371no—
206C16C22×D13712yes—
207C23—41no—
208C23—5482no—
209C23—5482no—
210C23—5481no—
211Q8—21no—
212Q8—2741no—
213Q8—1371no—
214D4—22no—
215D4—21no—
216D4—41no—
217D4—41no—
218D4—2741no—
219D4—2742no—
220D4—5482no—
221D4—5482no—
222D4—5481no—
223D4—5482no—
224D4—5482no—
225D4—5481no—
226D4—1372no—
227D4—1371no—
228C2×C4—41no—
229C2×C4—5482no—
230C2×C4—5482no—
231C2×C4—5481no—
232C2×C4—1372no—
233C2×C4D4×D13711yescharacteristic
234C8—1372no—
235C8D4×D13711yescharacteristic
236C8C23×D13711yescharacteristic
237C22—41no—
238C22—42no—
239C22—161no—
240C22—5482no—
241C22—5481no—
242C22—1372no—
243C22—21921no—
244C22—21922no—
245C22—21922no—
246C22D8×D13711yescharacteristic
247C4—41no—
248C4—5481no—
249C4—1372no—
250C4D8×D13711yescharacteristic
251C4C2×D4×D13711yescharacteristic
252C2—21no—
253C2—82no—
254C2—81no—
255C2—2742no—
256C2—10962no—
257C2—10961no—
258C2C2×D8×D13711yescharacteristic
subgroupG/H
centerC2C2×D8×D137
commutator, cosocleC8×C137C24
FrattiniC8C23×D137
Fitting#509×C137C2
socleC2×C137C22×D8
hypercenter#509D137
hypocenterC137#3415
LCSG ▷ C8×C137 ▷ C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #509

Complements

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

NH ≅ G/Ntype
1. #22578205C2semidirect
2. #22578204C2semidirect
3. #22577841C2semidirect
4. #22577840C2semidirect
5. #22577837C2semidirect
6. #22577834C2semidirect
7. #22577833C2semidirect
8. C2×#17984029C2semidirect
9. D16×D137C2semidirect
10. #22577821C2semidirect
11. #509×C137C2semidirect
18. #17984111C22semidirect
19. #17984110C22semidirect
20. #17984107C22semidirect
21. C2×D1096C22semidirect
22. D8×D137C22semidirect
23. D8×D137C22semidirect
25. #17984034C22semidirect
26. #17984033C22semidirect
27. #17984032C22semidirect
28. #17984031C22semidirect
29. D2192C22semidirect
30. #17984029C22semidirect
31. #17984029C22semidirect
32. #17984029C22semidirect
33. #17984027C22semidirect
34. C2×#14555485C22semidirect
35. C16×D137C22semidirect
36. #135×C137C22semidirect
37. C2×D8×C137C22semidirect
38. SD32×C137C22semidirect
39. D16×C137C22semidirect
40. M32×C137C22semidirect
129. C137#3415semidirect
136. #509D137semidirect
151. #135C2×D137semidirect
155. C2×D8C2×D137semidirect
161. SD32C2×D137semidirect
166. D16C2×D137semidirect
168. M32C2×D137semidirect

Maximal subgroups

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

subgroupindex
#225782052
#225782042
#225778412
#225778402
#225778372
#225778342
#225778332
C2×#179840292
D16×D1372
#225778212
#509×C1372
#3415137