Finite Group Databasebeta

← #31793931

Subgroups — #31793931

Order 17536.

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#22578216C211yesmaximal, characteristic, complemented
2#22578204C211yesmaximal, characteristic, complemented
3#22577844C211yesmaximal, characteristic, complemented
4#22577842C212yesmaximal, complemented
5#22577840C212yesmaximal, complemented
6#22577839C211yesmaximal, characteristic, complemented
7#22577835C211yesmaximal, characteristic, complemented
8#22577834C212yesmaximal, complemented
9SD32×D137C212yesmaximal, complemented
10#22577819C211yesmaximal, characteristic, complemented
11C2×SD32×C137C211yesmaximal, characteristic, complemented
12#17984136—21no—
13#17984133—21no—
14#17984120—22no—
15#17984119—21no—
16#17984114—21no—
17#17984112—22no—
18#17984111C2212yescomplemented
19#17984110C2211yescharacteristic
20#17984110C2211yescharacteristic, complemented
21#17984107C2212yes—
22Q16×D137C2212yescomplemented
23D8×D137C2212yes—
24#17984094C2211yescharacteristic
25Dic1096C2211yescharacteristic, complemented
26#17984034C2212yescomplemented
27#17984033C2211yescharacteristic, complemented
28#17984033C2211yescharacteristic, complemented
29#17984032C2212yescomplemented
30#17984031C2212yescomplemented
31D2192C2211yescharacteristic, complemented
32#17984029C2212yescomplemented
33#17984028C2211yescharacteristic, complemented
34#17984027C2212yescomplemented
35C16×D137C2212yescomplemented
36C2×Q16×C137C2211yescharacteristic, complemented
37C2×D8×C137C2211yescharacteristic
38SD32×C137C2212yescomplemented
39SD32×C137C2212yescomplemented
40C2×C16×C137C2211yescharacteristic, complemented
41#14555511—22no—
42#14555511—41no—
43#14555510—21no—
44#14555510—21no—
45#14555510—42no—
46#14555510D411yescharacteristic
47#14555509—22no—
48#14555509—41no—
49Q8×D137—22no—
50Q8×D137—41no—
51C2×C22⋊D137—42no—
52D4×D137—22no—
53D4×D137—41no—
54Dic548C2311yescharacteristic
55Dic548C2311yescharacteristic
56#14555500—22no—
57#14555499—21no—
58#14555499—21no—
59#14555498—22no—
60#14555497—22no—
61D1096C2311yescharacteristic
62D1096C2311yescharacteristic
63#14555495—22no—
64#14555494D411yescharacteristic
65#14555493D412yes—
66C8×D137C2312yes—
67#14555485C2312yes—
68C2×Q8×C137—21no—
69C2×D4×C137—21no—
70Q16×C137—21no—
71Q16×C137C2312yes—
72D8×C137—21no—
73D8×C137C2312yes—
74C2×C8×C137C2311yescharacteristic
75C16×C137C2311yescharacteristic
76C16×C137C2311yescharacteristic
77C22×D137—42no—
78Dic274—21no—
79Dic274—21no—
80Dic274—42no—
81Dic274D811yescharacteristic
82D548—21no—
83D548—21no—
84D548—42no—
85D548D811yescharacteristic
86C22⋊D137—41no—
87C22⋊D137—41no—
88C22⋊D137—42no—
89C22⋊D137—42no—
90C22⋊D137D812yes—
91C2×Dic137—42no—
92C4×D137—42no—
93C4×D137—41no—
94C4×D137—41no—
95C4×D137—42no—
96C4×D137C2×D412yes—
97#11764040C2×D412yes—
98C23×C137—41no—
99Q8×C137—22no—
100Q8×C137—41no—
101D4×C137—22no—
102D4×C137—41no—
103C2×C4×C137—41no—
104C2×C4×C137C2×D411yescharacteristic
105C8×C137C2×D411yescharacteristic
106C8×C137C2411yescharacteristic
107C22×C137—42no—
108C22×C137—161no—
109C22×C137C2×D811yescharacteristic
110C2×D137—41no—
111C2×D137—41no—
112C2×D137—162no—
113C2×D137C2×D812yes—
114C4×C137—42no—
115C4×C137C2×D811yescharacteristic
116C4×C137C22×D411yescharacteristic
117Dic137—41no—
118Dic137—41no—
119Dic137C2×D812yes—
120C2×C137—21no—
121C2×C137—82no—
122C2×C137C22×D811yescharacteristic
123D137—22no—
124D137—81no—
125D137—81no—
126C137#341611yescharacteristic, complemented, Sylow
127#3416—1371nomaximal, Sylow
128#578—1371no—
129#576—1371no—
130#510—1371no—
131#510—1372no—
132#509—1371no—
133#509—1372no—
134#508—1372no—
135#508—1371no—
136C2×SD32—1372no—
137C2×SD32D13711yescharacteristic, complemented
138#504—1371no—
1392−1+4—2741no—
140H(5,𝔽2)—2741no—
141#137—2742no—
142#137—2741no—
143AGL(1,ℤ/8)—2742no—
144AGL(1,ℤ/8)—2741no—
145#135—1372no—
146#135—1372no—
147#135—1371no—
148#135—1371no—
149C2×Q16—1372no—
150C2×Q16C2×D13711yescharacteristic, complemented
151C2×D8—1372no—
152C2×D8C2×D13711yescharacteristic
153#131—1371no—
154Q32—1372no—
155Q32—1371no—
156SD32—1372no—
157SD32—1372no—
158SD32—1371no—
159SD32—1371no—
160SD32C2×D13712yescomplemented
161SD32C2×D13712yescomplemented
162D16—1372no—
163D16—1371no—
164M32—1371no—
165M32—1372no—
166C2×C16—1372no—
167C2×C16C2×D13711yescharacteristic, complemented
168#41—2742no—
169#41—2742no—
170#41—2741no—
171#41—2741no—
172#41—5482no—
173#41—5481no—
174#41—5481no—
175#41—1371no—
176C2×Q8—21no—
177C2×Q8—2742no—
178C2×Q8—5481no—
179C2×D4—21no—
180C2×D4—2742no—
181C2×D4—5481no—
182C2×D4—5482no—
183Q16—21no—
184Q16—2742no—
185Q16—1371no—
186Q16—1371no—
187Q16C22×D13712yes—
188SD16—2742no—
189SD16—2742no—
190SD16—2741no—
191SD16—2741no—
192D8—21no—
193D8—2742no—
194D8—1371no—
195D8—1371no—
196D8C22×D13712yes—
197M16—1371no—
198M16—1372no—
199C2×C8—1372no—
200C2×C8C22×D13711yescharacteristic
201C16—1372no—
202C16C22×D13711yescharacteristic
203C16C22×D13711yescharacteristic
204C23—41no—
205C23—5482no—
206Q8—22no—
207Q8—41no—
208Q8—2741no—
209Q8—2741no—
210Q8—5482no—
211Q8—1371no—
212D4—22no—
213D4—41no—
214D4—2741no—
215D4—2741no—
216D4—5482no—
217D4—5482no—
218D4—5481no—
219D4—5481no—
220D4—5482no—
221D4—1372no—
222D4—1371no—
223C2×C4—41no—
224C2×C4—5482no—
225C2×C4—5482no—
226C2×C4—5482no—
227C2×C4—5481no—
228C2×C4—5481no—
229C2×C4—1372no—
230C2×C4D4×D13711yescharacteristic
231C8—1372no—
232C8D4×D13711yescharacteristic
233C8C23×D13711yescharacteristic
234C22—42no—
235C22—161no—
236C22—5481no—
237C22—5481no—
238C22—1372no—
239C22—21922no—
240C22D8×D13711yescharacteristic
241C4—42no—
242C4—5481no—
243C4—5481no—
244C4—1372no—
245C4D8×D13711yescharacteristic
246C4C2×D4×D13711yescharacteristic
247C2—21no—
248C2—82no—
249C2—2742no—
250C2—10961no—
251C2—10961no—
252C2C2×D8×D13711yescharacteristic
subgroupG/H
centerC2C2×D8×D137
commutator, cosocleC8×C137C24
FrattiniC8C23×D137
FittingC2×SD32×C137C2
socleC2×C137C22×D8
hypercenterC2×SD32D137
hypocenterC137#3416
LCSG ▷ C8×C137 ▷ C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ C2×SD32

Complements

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

NH ≅ G/Ntype
1. #22578216C2semidirect
2. #22578204C2semidirect
3. #22577844C2semidirect
4. #22577842C2semidirect
5. #22577840C2semidirect
6. #22577839C2semidirect
7. #22577835C2semidirect
8. #22577834C2semidirect
9. SD32×D137C2semidirect
10. #22577819C2semidirect
11. C2×SD32×C137C2semidirect
18. #17984111C22semidirect
20. #17984110C22semidirect
22. Q16×D137C22semidirect
25. Dic1096C22semidirect
26. #17984034C22semidirect
27. #17984033C22semidirect
28. #17984033C22semidirect
29. #17984032C22semidirect
30. #17984031C22semidirect
31. D2192C22semidirect
32. #17984029C22semidirect
33. #17984028C22semidirect
34. #17984027C22semidirect
35. C16×D137C22semidirect
36. C2×Q16×C137C22semidirect
38. SD32×C137C22semidirect
39. SD32×C137C22semidirect
40. C2×C16×C137C22semidirect
126. C137#3416semidirect
137. C2×SD32D137semidirect
150. C2×Q16C2×D137semidirect
160. SD32C2×D137semidirect
161. SD32C2×D137semidirect
167. C2×C16C2×D137semidirect

Maximal subgroups

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

subgroupindex
#225782162
#225782042
#225778442
#225778422
#225778402
#225778392
#225778352
#225778342
SD32×D1372
#225778192
C2×SD32×C1372
#3416137