Finite Group Databasebeta

← #496×D137 #31791219

Subgroups — #496×D137 #31791219

Order 17536.

668 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 668 conjugacy classes of subgroups — too many to draw. Only its 145 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
1C2×SD16×D137C212yesmaximal, complemented
2C2×D8×D137C212yesmaximal, complemented
3#127×D137C211yesmaximal, characteristic, complemented
4#124×D137C211yesmaximal, characteristic, complemented
5#22577793C211yesmaximal, characteristic, complemented
6#22577792C211yesmaximal, characteristic, complemented
7#22577791C212yesmaximal, complemented
8#22577790C211yesmaximal, characteristic, complemented
9#22577789C212yesmaximal, complemented
10#97×D137C211yesmaximal, characteristic, complemented
11#496×C137C211yesmaximal, characteristic, complemented
12C2×Q8×D137C2211yescharacteristic
13C2×D4×D137—22no—
14C2×D4×D137C2212yes—
15C2×D4×D137C2211yescharacteristic
16C2×#14555499C2212yescomplemented
17C2×#14555498C2212yescomplemented
18C2×#14555497C2212yescomplemented
19SD16×D137—28no—
20C2×D1096C2212yescomplemented
21C2×#14555495C2212yescomplemented
22C2×#14555495C2212yescomplemented
23D8×D137—28no—
24C2×C8×D137C2212yes—
25#17984085C2211yescharacteristic, complemented
26#17984084—22no—
27#17984084C2212yescomplemented
28#17984077C2211yescharacteristic, complemented
29#17984076C2211yescharacteristic, complemented
30#17984075C2211yescharacteristic, complemented
31#17984074—22no—
32C22⋊C4×D137—22no—
33C42×D137C2211yescharacteristic
34#17983999C2211yescharacteristic, complemented
35#17983998C2212yescomplemented
36C2×SD16×C137C2212yescomplemented
37C2×D8×C137C2212yescomplemented
38#127×C137C2211yescharacteristic, complemented
39#124×C137C2211yescharacteristic, complemented
40#97×C137C2211yescharacteristic, complemented
41C23×D137—21no—
42C23×D137—22no—
43C2×Dic274—21no—
44C2×Dic274C2311yescharacteristic
45Q8×D137—24no—
46C2×D548—22no—
47C2×D548C2311yescharacteristic
48C2×D548C2312yes—
49C2×C22⋊D137—21no—
50C2×C22⋊D137—22no—
51C2×C22⋊D137—24no—
52D4×D137—24no—
53D4×D137—24no—
54D4×D137—24no—
55D4×D137—48no—
56C2×C4×D137—21no—
57C2×C4×D137D412yes—
58C2×C4×D137C2311yescharacteristic
59#14555499—24no—
60#14555498—24no—
61#14555497—24no—
62D1096—24no—
63#14555495—24no—
64#14555495—24no—
65C2×#11764040C2312yes—
66C8×D137D418yescomplemented
67C22⋊Dic137—22no—
68#14555487—22no—
69#14555487—22no—
70C4×Dic137D412yes—
71C4×Dic137C2311yescharacteristic
72C2×Q8×C137C2311yescharacteristic
73C2×D4×C137—22no—
74C2×D4×C137C2311yescharacteristic
75C2×D4×C137C2312yes—
76SD16×C137—24no—
77D8×C137—24no—
78C2×C8×C137C2312yes—
79C22⋊C4×C137—22no—
80C42×C137C2311yescharacteristic
81C22×D137—21no—
82C22×D137—22no—
83C22×D137—42no—
84C22×D137—44no—
85C22×D137—48no—
86C22×D137—44no—
87C22×D137C2×D411yescharacteristic
88Dic274—22no—
89Dic274—42no—
90D548—22no—
91D548—22no—
92D548—22no—
93D548—44no—
94C22⋊D137—48no—
95C22⋊D137—42no—
96C22⋊D137—44no—
97C2×Dic137—21no—
98C2×Dic137C2×D412yes—
99C2×Dic137C2×D411yescharacteristic
100C4×D137—24no—
101C4×D137—42no—
102C4×D137C2×D414yes—
103#11764040C2×D414yescomplemented
104C23×C137—21no—
105C23×C137—22no—
106Q8×C137—22no—
107D4×C137—22no—
108D4×C137—22no—
109D4×C137—22no—
110D4×C137—44no—
111C2×C4×C137—21no—
112C2×C4×C137C2×D412yes—
113C2×C4×C137C2411yescharacteristic
114C8×C137C2×D414yescomplemented
115C22×C137—41no—
116C22×C137—42no—
117C22×C137—44no—
118C22×C137—42no—
119C22×C137C22×D411yescharacteristic
120C2×D137—41no—
121C2×D137—42no—
122C2×D137—44no—
123C2×D137—42no—
124C2×D137—88no—
125C2×D137—84no—
126C2×D137#12712yes—
127C2×D137AGL(1,ℤ/8)14yes—
128C4×C137—22no—
129C4×C137—41no—
130C4×C137C22×D412yes—
131Dic137—22no—
132Dic137—41no—
133Dic137#12712yes—
134C2×C137—82no—
135C2×C137—81no—
136C2×C137C2×#12711yescharacteristic
137C2×C137C2×AGL(1,ℤ/8)12yes—
138D137—82no—
139D137—81no—
140D137#49614yesdirect product factor
141C137C2×#49611yescharacteristic, complemented, Sylow
142C2×#496—1371nomaximal, Sylow
143C22×SD16—1372no—
144C22×D8—1372no—
145C2×#127—1371no—
146C2×#124—1371no—
147#496—1372no—
148#496—1372no—
149#496—1371no—
150#496—1371no—
151#496—1371no—
152#496D13711yescharacteristic, direct product factor
153C2×#97—1371no—
154C22×Q8—1371no—
155C22×D4—2742no—
156C22×D4—1371no—
157C22×D4—1372no—
158C2×SD16—2748no—
159C2×SD16—1372no—
160C2×SD16—1372no—
161C2×SD16—1372no—
162C2×SD16C2×D13712yescomplemented
163C2×D8—2748no—
164C2×D8—1372no—
165C2×D8—1372no—
166C2×D8—1372no—
167C2×D8C2×D13712yescomplemented
168C22×C8—1372no—
169#127—2742no—
170#127—1372no—
171#127—1371no—
172#127C2×D13711yescharacteristic, complemented
173#124—2742no—
174#124—1371no—
175#124—1371no—
176#124—1371no—
177#124C2×D13711yescharacteristic, complemented
178C2×C22⋊C4—2742no—
179C2×C42—1371no—
180#97—1372no—
181#97—1371no—
182#97C2×D13711yescharacteristic, complemented
183C24—2741no—
184C24—2742no—
185C2×Q8—2741no—
186C2×Q8—2744no—
187C2×Q8—1371no—
188C2×Q8C22×D13711yescharacteristic
189C2×D4—22no—
190C2×D4—2744no—
191C2×D4—2744no—
192C2×D4—2744no—
193C2×D4—2741no—
194C2×D4—2742no—
195C2×D4—2744no—
196C2×D4—2742no—
197C2×D4—5488no—
198C2×D4—1371no—
199C2×D4—1372no—
200C2×D4C22×D13711yescharacteristic
201C2×D4C22×D13712yes—
202C22×C4—2741no—
203C22×C4—1372no—
204C22×C4—1371no—
205SD16—24no—
206SD16—2744no—
207SD16—2744no—
208SD16—2744no—
209D8—24no—
210D8—2744no—
211D8—2744no—
212D8—2744no—
213C2×C8—1372no—
214C2×C8—1378no—
215C2×C8C22×D13712yes—
216C22⋊C4—22no—
217C22⋊C4—2742no—
218C22⋊C4—2742no—
219C22⋊C4—2742no—
220C42—1372no—
221C42—1371no—
222C42C22×D13711yescharacteristic
223C23—21no—
224C23—22no—
225C23—2741no—
226C23—2742no—
227C23—5482no—
228C23—5484no—
229C23—5488no—
230C23—5484no—
231C23—1371no—
232Q8—22no—
233Q8—2742no—
234Q8—5482no—
235D4—22no—
236D4—22no—
237D4—22no—
238D4—44no—
239D4—2742no—
240D4—2742no—
241D4—2742no—
242D4—5484no—
243D4—5488no—
244D4—5482no—
245D4—5484no—
246C2×C4—21no—
247C2×C4—2744no—
248C2×C4—2741no—
249C2×C4—5482no—
250C2×C4—1372no—
251C2×C4—1371no—
252C2×C4—1374no—
253C2×C4D4×D13712yes—
254C2×C4C23×D13711yescharacteristic
255C8—1374no—
256C8D4×D13714yescomplemented
257C22—44no—
258C22—42no—
259C22—41no—
260C22—42no—
261C22—5484no—
262C22—5482no—
263C22—5481no—
264C22—5482no—
265C22—10968no—
266C22—10964no—
267C22—1374no—
268C22—1372no—
269C22C2×D4×D13711yescharacteristic
270C4—22no—
271C4—41no—
272C4—2742no—
273C4—5481no—
274C4—1372no—
275C4C2×D4×D13712yes—
276C2—82no—
277C2—81no—
278C2—10962no—
279C2—10961no—
280C2—1374no—
281C2#127×D13711yescharacteristic
282C2AGL(1,ℤ/8)×D13712yes—
subgroupG/H
centerC22C2×D4×D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#496×C137C2
socleC22×C137C22×D4
hypercenter#496D137
hypocenterC137C2×#496
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C42 ◁ #496

Complements

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

NH ≅ G/Ntype
1. C2×SD16×D137C2semidirect
2. C2×D8×D137C2semidirect
3. #127×D137C2semidirect
4. #124×D137C2semidirect
5. #22577793C2semidirect
6. #22577792C2semidirect
7. #22577791C2semidirect
8. #22577790C2semidirect
9. #22577789C2semidirect
10. #97×D137C2semidirect
11. #496×C137C2semidirect
16. C2×#14555499C22semidirect
17. C2×#14555498C22semidirect
18. C2×#14555497C22semidirect
20. C2×D1096C22semidirect
21. C2×#14555495C22semidirect
22. C2×#14555495C22semidirect
25. #17984085C22semidirect
27. #17984084C22semidirect
28. #17984077C22semidirect
29. #17984076C22semidirect
30. #17984075C22semidirect
34. #17983999C22semidirect
35. #17983998C22semidirect
36. C2×SD16×C137C22semidirect
37. C2×D8×C137C22semidirect
38. #127×C137C22semidirect
39. #124×C137C22semidirect
40. #97×C137C22semidirect
66. C8×D137D4semidirect
103. #11764040C2×D4semidirect
114. C8×C137C2×D4semidirect
140. D137#496direct (N × H)
141. C137C2×#496semidirect
152. #496D137direct (N × H)
162. C2×SD16C2×D137semidirect
167. C2×D8C2×D137semidirect
172. #127C2×D137semidirect
177. #124C2×D137semidirect
182. #97C2×D137semidirect
256. C8D4×D137semidirect

Maximal subgroups

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

subgroupindex
C2×SD16×D1372
C2×D8×D1372
#127×D1372
#124×D1372
#225777932
#225777922
#225777912
#225777902
#225777892
#97×D1372
#496×C1372
C2×#496137