Finite Group Databasebeta

← #31793935

Subgroups — #31793935

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#22578217C211yesmaximal, characteristic, complemented
2#22578204C211yesmaximal, characteristic, complemented
3#22577846C211yesmaximal, characteristic, complemented
4#22577842C212yesmaximal, complemented
5#22577837C212yesmaximal, complemented
6#22577833C211yesmaximal, characteristic, complemented
7#22577832C212yesmaximal, complemented
8C2×#17984032C211yesmaximal, characteristic, complemented
9SD32×D137C212yesmaximal, complemented
10#22577821C211yesmaximal, characteristic, complemented
11#509×C137C211yesmaximal, characteristic, complemented
12#17984135—21no—
13#17984133—21no—
14#17984118—21no—
15#17984117—22no—
16#17984114—21no—
17#17984112—22no—
18#17984110C2211yescharacteristic
19#17984107C2212yes—
20#17984107C2212yescomplemented
21C2×Dic548C2211yescharacteristic, complemented
22Q16×D137C2212yescomplemented
23D8×D137C2212yes—
24#17984094C2211yescharacteristic
25Dic1096C2212yescomplemented
26#17984034C2211yescharacteristic, complemented
27#17984033C2212yescomplemented
28#17984032C2212yescomplemented
29#17984032C2212yescomplemented
30#17984032C2211yescharacteristic, complemented
31#17984031C2211yescharacteristic, complemented
32#17984029C2211yescharacteristic, complemented
33#17984027C2212yescomplemented
34C2×#14555485C2211yescharacteristic, complemented
35C16×D137C2212yescomplemented
36#135×C137C2211yescharacteristic, complemented
37C2×D8×C137C2211yescharacteristic
38SD32×C137C2212yescomplemented
39D16×C137C2212yescomplemented
40M32×C137C2211yescharacteristic, complemented
41#14555510—21no—
42#14555510—41no—
43#14555510D411yescharacteristic
44#14555509—22no—
45#14555509—22no—
46#14555509—41no—
47#14555509—42no—
48C2×Dic274—21no—
49C2×Dic274—41no—
50Q8×D137—22no—
51C2×C22⋊D137—42no—
52D4×D137—22no—
53D4×D137—41no—
54Dic548—21no—
55Dic548C2312yes—
56Dic548C2311yescharacteristic
57#14555500—22no—
58#14555499—21no—
59#14555498—22no—
60#14555498—22no—
61D1096C2311yescharacteristic
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×C137C2311yescharacteristic
74D8×C137C2312yes—
75C2×C8×C137C2311yescharacteristic
76C16×C137C2312yes—
77C22×D137—42no—
78Dic274—21no—
79Dic274—22no—
80Dic274—41no—
81Dic274—42no—
82Dic274D811yescharacteristic
83D548—21no—
84D548D811yescharacteristic
85C22⋊D137—41no—
86C22⋊D137—42no—
87C22⋊D137—42no—
88C22⋊D137—42no—
89C22⋊D137D812yes—
90C2×Dic137—41no—
91C2×Dic137—42no—
92C2×Dic137—42no—
93C4×D137—42no—
94C4×D137—41no—
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—162no—
113C2×D137C2×D812yes—
114C4×C137—41no—
115C4×C137C2×D811yescharacteristic
116C4×C137C22×D411yescharacteristic
117Dic137—41no—
118Dic137—42no—
119Dic137C2×D812yes—
120C2×C137—21no—
121C2×C137—82no—
122C2×C137—81no—
123C2×C137C22×D811yescharacteristic
124D137—22no—
125D137—81no—
126C137#341611yescharacteristic, complemented, Sylow
127#3416—1371nomaximal, Sylow
128#578—1371no—
129#576—1371no—
130#510—1372no—
131#510—1371no—
132#509—1372no—
133#509D13711yescharacteristic, complemented
134#508—1371no—
135#508—1372no—
136C2×SD32—1371no—
137C2×SD32—1372no—
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#135C2×D13711yescharacteristic, complemented
149C2×Q16—1372no—
150C2×Q16—1371no—
151C2×D8—1372no—
152C2×D8C2×D13711yescharacteristic
153#131—1371no—
154Q32—1371no—
155Q32—1372no—
156SD32—1371no—
157SD32—1372no—
158SD32—1372no—
159SD32—1371no—
160SD32—1372no—
161SD32C2×D13712yescomplemented
162D16—1371no—
163D16C2×D13712yescomplemented
164M32—1372no—
165M32C2×D13711yescharacteristic, complemented
166C2×C16—1371no—
167C2×C16—1372no—
168#41—21no—
169#41—2742no—
170#41—2742no—
171#41—2741no—
172#41—5481no—
173#41—5481no—
174#41—5482no—
175#41—1371no—
176C2×Q8—2742no—
177C2×Q8—2741no—
178C2×Q8—5481no—
179C2×D4—21no—
180C2×D4—2742no—
181C2×D4—5481no—
182C2×D4—5482no—
183Q16—2742no—
184Q16—2741no—
185Q16—1372no—
186Q16—1371no—
187Q16C22×D13711yescharacteristic
188SD16—21no—
189SD16—2742no—
190SD16—2742no—
191SD16—2741no—
192D8—21no—
193D8—2742no—
194D8—1371no—
195D8C22×D13711yescharacteristic
196D8C22×D13712yes—
197M16—1371no—
198M16—1372no—
199C2×C8—1372no—
200C2×C8C22×D13711yescharacteristic
201C16—1371no—
202C16—1371no—
203C16C22×D13712yes—
204C23—41no—
205C23—5482no—
206Q8—21no—
207Q8—2741no—
208Q8—2742no—
209Q8—5482no—
210Q8—5481no—
211Q8—1371no—
212D4—22no—
213D4—21no—
214D4—41no—
215D4—41no—
216D4—2741no—
217D4—5482no—
218D4—5482no—
219D4—5481no—
220D4—5482no—
221D4—1372no—
222D4—1371no—
223C2×C4—41no—
224C2×C4—5482no—
225C2×C4—5481no—
226C2×C4—5482no—
227C2×C4—5482no—
228C2×C4—5481no—
229C2×C4—1372no—
230C2×C4D4×D13711yescharacteristic
231C8—1372no—
232C8D4×D13711yescharacteristic
233C8C23×D13711yescharacteristic
234C22—41no—
235C22—42no—
236C22—161no—
237C22—5481no—
238C22—1372no—
239C22—21922no—
240C22D8×D13711yescharacteristic
241C4—41no—
242C4—5481no—
243C4—5482no—
244C4—1372no—
245C4D8×D13711yescharacteristic
246C4C2×D4×D13711yescharacteristic
247C2—21no—
248C2—82no—
249C2—81no—
250C2—2742no—
251C2—10961no—
252C2C2×D8×D13711yescharacteristic
subgroupG/H
centerC2C2×D8×D137
commutator, cosocleC8×C137C24
FrattiniC8C23×D137
Fitting#509×C137C2
socleC2×C137C22×D8
hypercenter#509D137
hypocenterC137#3416
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. #22578217C2semidirect
2. #22578204C2semidirect
3. #22577846C2semidirect
4. #22577842C2semidirect
5. #22577837C2semidirect
6. #22577833C2semidirect
7. #22577832C2semidirect
8. C2×#17984032C2semidirect
9. SD32×D137C2semidirect
10. #22577821C2semidirect
11. #509×C137C2semidirect
20. #17984107C22semidirect
21. C2×Dic548C22semidirect
22. Q16×D137C22semidirect
25. Dic1096C22semidirect
26. #17984034C22semidirect
27. #17984033C22semidirect
28. #17984032C22semidirect
29. #17984032C22semidirect
30. #17984032C22semidirect
31. #17984031C22semidirect
32. #17984029C22semidirect
33. #17984027C22semidirect
34. C2×#14555485C22semidirect
35. C16×D137C22semidirect
36. #135×C137C22semidirect
38. SD32×C137C22semidirect
39. D16×C137C22semidirect
40. M32×C137C22semidirect
126. C137#3416semidirect
133. #509D137semidirect
148. #135C2×D137semidirect
161. SD32C2×D137semidirect
163. D16C2×D137semidirect
165. M32C2×D137semidirect

Maximal subgroups

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

subgroupindex
#225782172
#225782042
#225778462
#225778422
#225778372
#225778332
#225778322
C2×#179840322
SD32×D1372
#225778212
#509×C1372
#3416137