Finite Group Databasebeta

← #31793932

Subgroups — #31793932

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#22578205C211yesmaximal, characteristic, complemented
3#22577845C212yesmaximal, complemented
4#22577843C211yesmaximal, characteristic, complemented
5#22577838C211yesmaximal, characteristic, complemented
6#22577837C212yesmaximal, complemented
7#22577834C212yesmaximal, complemented
8C2×#17984033C211yesmaximal, characteristic, complemented
9SD32×D137C212yesmaximal, complemented
10#22577819C211yesmaximal, characteristic, complemented
11#508×C137C211yesmaximal, characteristic, complemented
12#17984135—21no—
13#17984134—21no—
14#17984118—21no—
15#17984117—22no—
16#17984115—22no—
17#17984113—21no—
18#17984111C2212yes—
19#17984107C2212yescomplemented
20C2×Dic548C2211yescharacteristic, complemented
21Q16×D137C2212yescomplemented
22C2×D1096C2211yescharacteristic
23D8×D137C2212yes—
24#17984094C2211yescharacteristic
25#17984034C2212yescomplemented
26#17984033C2212yescomplemented
27#17984033C2212yescomplemented
28#17984032C2212yescomplemented
29#17984031C2212yescomplemented
30#17984029C2212yescomplemented
31#17984028C2211yescharacteristic, complemented
32#17984027C2212yescomplemented
33C16×D137C2212yescomplemented
34#135×C137C2211yescharacteristic
35#135×C137C2211yescharacteristic, complemented
36Q32×C137C2211yescharacteristic, complemented
37SD32×C137C2211yescharacteristic, complemented
38SD32×C137C2211yescharacteristic, complemented
39D16×C137C2211yescharacteristic, complemented
40C2×C16×C137C2211yescharacteristic, complemented
41#14555511—22no—
42#14555510—41no—
43#14555510—41no—
44#14555510D411yescharacteristic
45#14555509—22no—
46#14555509—42no—
47C2×Dic274—21no—
48C2×Dic274—41no—
49Q8×D137—22no—
50C2×D548—21no—
51C2×D548—41no—
52D4×D137—22no—
53D4×D137—42no—
54Dic548—21no—
55Dic548C2312yes—
56#14555500—22no—
57#14555498—22no—
58#14555497—22no—
59D1096—21no—
60D1096C2312yes—
61#14555495—22no—
62#14555494D411yescharacteristic
63#14555493D412yes—
64C8×D137C2312yes—
65#14555485C2312yes—
66#41×C137—21no—
67#41×C137—21no—
68Q16×C137C2311yescharacteristic
69Q16×C137C2311yescharacteristic
70SD16×C137—21no—
71SD16×C137—21no—
72D8×C137C2311yescharacteristic
73D8×C137C2311yescharacteristic
74C2×C8×C137C2311yescharacteristic
75C16×C137C2311yescharacteristic
76C16×C137C2311yescharacteristic
77C22×D137—41no—
78C22×D137—42no—
79Dic274—22no—
80Dic274—41no—
81Dic274—42no—
82Dic274D811yescharacteristic
83D548—22no—
84D548—42no—
85D548—41no—
86D548D811yescharacteristic
87C22⋊D137—42no—
88C22⋊D137—42no—
89C22⋊D137D812yes—
90C2×Dic137—41no—
91C2×Dic137—42no—
92C4×D137—42no—
93C4×D137—42no—
94C4×D137C2×D412yes—
95#11764040C2×D412yes—
96Q8×C137—21no—
97Q8×C137—21no—
98D4×C137—21no—
99D4×C137—21no—
100D4×C137—41no—
101D4×C137—41no—
102C2×C4×C137—41no—
103C2×C4×C137—41no—
104C2×C4×C137C2×D411yescharacteristic
105C8×C137C2×D411yescharacteristic
106C8×C137C2411yescharacteristic
107C22×C137—41no—
108C22×C137—41no—
109C22×C137C2×D811yescharacteristic
110C2×D137—42no—
111C2×D137—162no—
112C2×D137—161no—
113C2×D137C2×D812yes—
114C4×C137—41no—
115C4×C137—41no—
116C4×C137C2×D811yescharacteristic
117C4×C137C22×D411yescharacteristic
118Dic137—42no—
119Dic137C2×D812yes—
120C2×C137—21no—
121C2×C137—81no—
122C2×C137—81no—
123C2×C137C22×D811yescharacteristic
124D137—22no—
125D137—82no—
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#508D13711yescharacteristic, complemented
136C2×SD32—1372no—
137C2×SD32—1371no—
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#135C2×D13711yescharacteristic
148#135C2×D13711yescharacteristic, complemented
149C2×Q16—1372no—
150C2×Q16—1371no—
151C2×D8—1372no—
152C2×D8—1371no—
153#131—1371no—
154Q32—1372no—
155Q32C2×D13711yescharacteristic, complemented
156SD32—1372no—
157SD32—1372no—
158SD32—1372no—
159SD32—1372no—
160SD32C2×D13711yescharacteristic, complemented
161SD32C2×D13711yescharacteristic, complemented
162D16—1372no—
163D16C2×D13711yescharacteristic, complemented
164M32—1371no—
165M32—1372no—
166C2×C16—1372no—
167C2×C16C2×D13711yescharacteristic, complemented
168#41—21no—
169#41—21no—
170#41—2742no—
171#41—2742no—
172#41—5481no—
173#41—5481no—
174#41—5482no—
175#41—1371no—
176C2×Q8—2742no—
177C2×Q8—2741no—
178C2×Q8—5481no—
179C2×D4—2742no—
180C2×D4—2741no—
181C2×D4—5481no—
182C2×D4—5482no—
183Q16—2742no—
184Q16—2741no—
185Q16—1372no—
186Q16C22×D13711yescharacteristic
187Q16C22×D13711yescharacteristic
188SD16—21no—
189SD16—21no—
190SD16—2742no—
191SD16—2742no—
192D8—2742no—
193D8—2741no—
194D8—1372no—
195D8C22×D13711yescharacteristic
196D8C22×D13711yescharacteristic
197M16—1371no—
198M16—1372no—
199C2×C8—1372no—
200C2×C8C22×D13711yescharacteristic
201C16—1372no—
202C16C22×D13711yescharacteristic
203C16C22×D13711yescharacteristic
204C23—5482no—
205C23—5481no—
206Q8—21no—
207Q8—21no—
208Q8—2742no—
209Q8—5482no—
210Q8—5481no—
211Q8—1371no—
212D4—21no—
213D4—21no—
214D4—41no—
215D4—41no—
216D4—2742no—
217D4—5482no—
218D4—5482no—
219D4—5482no—
220D4—5481no—
221D4—1372no—
222D4—1371no—
223C2×C4—41no—
224C2×C4—41no—
225C2×C4—5482no—
226C2×C4—5482no—
227C2×C4—5481no—
228C2×C4—5482no—
229C2×C4—1372no—
230C2×C4D4×D13711yescharacteristic
231C8—1372no—
232C8D4×D13711yescharacteristic
233C8C23×D13711yescharacteristic
234C22—41no—
235C22—41no—
236C22—5482no—
237C22—1372no—
238C22—21921no—
239C22—21922no—
240C22D8×D13711yescharacteristic
241C4—41no—
242C4—41no—
243C4—5482no—
244C4—1372no—
245C4D8×D13711yescharacteristic
246C4C2×D4×D13711yescharacteristic
247C2—21no—
248C2—81no—
249C2—81no—
250C2—2742no—
251C2—10962no—
252C2C2×D8×D13711yescharacteristic
subgroupG/H
centerC2C2×D8×D137
commutator, cosocleC8×C137C24
FrattiniC8C23×D137
Fitting#508×C137C2
socleC2×C137C22×D8
hypercenter#508D137
hypocenterC137#3416
LCSG ▷ C8×C137 ▷ C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #508

Complements

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

NH ≅ G/Ntype
1. #22578217C2semidirect
2. #22578205C2semidirect
3. #22577845C2semidirect
4. #22577843C2semidirect
5. #22577838C2semidirect
6. #22577837C2semidirect
7. #22577834C2semidirect
8. C2×#17984033C2semidirect
9. SD32×D137C2semidirect
10. #22577819C2semidirect
11. #508×C137C2semidirect
19. #17984107C22semidirect
20. C2×Dic548C22semidirect
21. Q16×D137C22semidirect
25. #17984034C22semidirect
26. #17984033C22semidirect
27. #17984033C22semidirect
28. #17984032C22semidirect
29. #17984031C22semidirect
30. #17984029C22semidirect
31. #17984028C22semidirect
32. #17984027C22semidirect
33. C16×D137C22semidirect
35. #135×C137C22semidirect
36. Q32×C137C22semidirect
37. SD32×C137C22semidirect
38. SD32×C137C22semidirect
39. D16×C137C22semidirect
40. C2×C16×C137C22semidirect
126. C137#3416semidirect
135. #508D137semidirect
148. #135C2×D137semidirect
155. Q32C2×D137semidirect
160. SD32C2×D137semidirect
161. SD32C2×D137semidirect
163. D16C2×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
#225782172
#225782052
#225778452
#225778432
#225778382
#225778372
#225778342
C2×#179840332
SD32×D1372
#225778192
#508×C1372
#3416137