Finite Group Databasebeta

← #31791238

Subgroups — #31791238

Order 17536.

656 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 656 conjugacy classes of subgroups — too many to draw. Only its 141 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×#17984115C212yesmaximal, complemented
2C2×#17984112C212yesmaximal, complemented
3#22577952C211yesmaximal, characteristic, complemented
4#127×D137C211yesmaximal, characteristic, complemented
5#22577793C211yesmaximal, characteristic, complemented
6#22577792C211yesmaximal, characteristic, complemented
7#22577790C211yesmaximal, characteristic, complemented
8#22577778C212yesmaximal, complemented
9#22577774C212yesmaximal, complemented
10#22577355C211yesmaximal, characteristic, complemented
11#496×C137C211yesmaximal, characteristic, complemented
12C2×#14555511C2211yescharacteristic
13C2×#14555509C2211yescharacteristic
14C2×D4×D137—22no—
15C2×D4×D137C2212yes—
16#17984115—28no—
17#17984112—28no—
18C2×#14555499C2212yescomplemented
19C2×#14555498C2212yescomplemented
20C2×#14555497C2212yescomplemented
21C2×D1096C2212yescomplemented
22C2×#14555495C2212yescomplemented
23C2×#14555495C2212yescomplemented
24C2×#14555493C2212yes—
25#17984086C2211yescharacteristic, complemented
26#17984085C2211yescharacteristic
27#17984084—22no—
28#17984084C2212yes—
29#17984084C2211yescharacteristic, complemented
30#17984076C2211yescharacteristic, complemented
31#17984061—22no—
32#17984048—22no—
33C42×D137C2211yescharacteristic
34#17983999C2211yescharacteristic, complemented
35C8×Dic137C2212yescomplemented
36C2×SD16×C137C2212yescomplemented
37C2×D8×C137C2212yescomplemented
38#127×C137C2211yescharacteristic
39#124×C137C2211yescharacteristic, complemented
40#97×C137C2211yescharacteristic, complemented
41C23×D137—22no—
42#14555511—24no—
43#14555509—24no—
44C2×Dic274C2311yescharacteristic
45C2×D548—21no—
46C2×D548—22no—
47C2×D548C2311yescharacteristic
48C2×D548C2312yes—
49C2×C22⋊D137—21no—
50C2×C22⋊D137—22no—
51C2×C22⋊D137—22no—
52C2×C22⋊D137—24no—
53D4×D137—24no—
54D4×D137—24no—
55D4×D137—48no—
56C22×Dic137—21no—
57C2×C4×D137—21no—
58C2×C4×D137D412yes—
59C2×C4×D137C2311yescharacteristic
60#14555499—24no—
61#14555498—24no—
62#14555497—24no—
63D1096—24no—
64#14555495—24no—
65#14555495—24no—
66#14555493D418yescomplemented
67C2×#11764040C2312yes—
68#14555489—22no—
69#14555487—22no—
70C4×Dic137D412yes—
71C4×Dic137C2311yescharacteristic
72C2×Q8×C137C2311yescharacteristic
73C2×D4×C137—22no—
74C2×D4×C137C2312yes—
75C2×D4×C137C2311yescharacteristic
76SD16×C137—24no—
77D8×C137—24no—
78C2×C8×C137C2312yes—
79C22⋊C4×C137—22no—
80C42×C137C2311yescharacteristic
81C22×D137—21no—
82C22×D137—22no—
83C22×D137—48no—
84C22×D137—44no—
85C22×D137C2×D411yescharacteristic
86Dic274—22no—
87D548—22no—
88D548—22no—
89D548—22no—
90D548—44no—
91D548—42no—
92C22⋊D137—42no—
93C22⋊D137—44no—
94C22⋊D137—48no—
95C22⋊D137—44no—
96C2×Dic137—21no—
97C2×Dic137—42no—
98C2×Dic137C2×D411yescharacteristic
99C2×Dic137C2×D412yes—
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—22no—
121C2×D137—41no—
122C2×D137—42no—
123C2×D137—44no—
124C2×D137—42no—
125C2×D137—88no—
126C2×D137#12712yes—
127C4×C137—22no—
128C4×C137—41no—
129C4×C137C22×D412yes—
130Dic137—41no—
131Dic137#12712yes—
132Dic137AGL(1,ℤ/8)14yescomplemented
133C2×C137—82no—
134C2×C137—81no—
135C2×C137C2×#12711yescharacteristic
136C2×C137C2×AGL(1,ℤ/8)12yes—
137D137—22no—
138D137—81no—
139D137—82no—
140C137#315111yescharacteristic, complemented, Sylow
141#3151—1371nomaximal, Sylow
142C2×AGL(1,ℤ/8)—1372no—
143C2×AGL(1,ℤ/8)—1372no—
144#532—1371no—
145C2×#127—1371no—
146#496—1371no—
147#496—1371no—
148#496—1371no—
149#496D13711yescharacteristic, complemented
150#493—1372no—
151#492—1372no—
152C4×M16—1371no—
153C2×#41—1371no—
154C2×#41—1371no—
155C22×D4—2742no—
156C22×D4—1372no—
157AGL(1,ℤ/8)—2748no—
158AGL(1,ℤ/8)—2748no—
159C2×SD16—1372no—
160C2×SD16—1372no—
161C2×SD16—1372no—
162C2×SD16C2×D13712yescomplemented
163C2×D8—1372no—
164C2×D8—1372no—
165C2×D8—1372no—
166C2×D8C2×D13712yescomplemented
167C2×M16—1372no—
168C4⋊Q8—1371no—
169#127—2742no—
170#127—1372no—
171#127—1371no—
172#127—1371no—
173#127C2×D13711yescharacteristic
174#124—1371no—
175#124C2×D13711yescharacteristic, complemented
176#121—2742no—
177C4×D4—2742no—
178C2×C42—1371no—
179#97—1371no—
180#97C2×D13711yescharacteristic, complemented
181C4×C8—1372no—
182C24—2742no—
183#41—2744no—
184#41—2744no—
185C2×Q8—1371no—
186C2×Q8C22×D13711yescharacteristic
187C2×D4—22no—
188C2×D4—2741no—
189C2×D4—2741no—
190C2×D4—2744no—
191C2×D4—2744no—
192C2×D4—2742no—
193C2×D4—2742no—
194C2×D4—2744no—
195C2×D4—2742no—
196C2×D4—5488no—
197C2×D4—1371no—
198C2×D4—1372no—
199C2×D4C22×D13711yescharacteristic
200C2×D4C22×D13712yes—
201C22×C4—2741no—
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—
213M16—1378no—
214C2×C8—1372no—
215C2×C8C22×D13712yes—
216C4⋊C4—2742no—
217C22⋊C4—22no—
218C22⋊C4—2742no—
219C42—1372no—
220C42—1371no—
221C42C22×D13711yescharacteristic
222C23—21no—
223C23—22no—
224C23—2741no—
225C23—2742no—
226C23—5488no—
227C23—5484no—
228C23—1371no—
229Q8—22no—
230Q8—2742no—
231D4—22no—
232D4—22no—
233D4—22no—
234D4—44no—
235D4—2742no—
236D4—2742no—
237D4—2742no—
238D4—5484no—
239D4—5482no—
240D4—5482no—
241D4—5484no—
242D4—5488no—
243D4—5484no—
244C2×C4—21no—
245C2×C4—2744no—
246C2×C4—2741no—
247C2×C4—5482no—
248C2×C4—5482no—
249C2×C4—1372no—
250C2×C4—1371no—
251C2×C4—1374no—
252C2×C4D4×D13712yes—
253C2×C4C23×D13711yescharacteristic
254C8—1374no—
255C8D4×D13714yescomplemented
256C22—44no—
257C22—42no—
258C22—41no—
259C22—42no—
260C22—2742no—
261C22—5484no—
262C22—5482no—
263C22—5481no—
264C22—5482no—
265C22—10968no—
266C22—1372no—
267C22C2×D4×D13711yescharacteristic
268C4—22no—
269C4—41no—
270C4—5481no—
271C4—1374no—
272C4—1372no—
273C4C2×D4×D13712yes—
274C2—82no—
275C2—81no—
276C2—2742no—
277C2—10962no—
278C2—10961no—
279C2#127×D13711yescharacteristic
280C2AGL(1,ℤ/8)×D13712yes—
subgroupG/H
centerC22C2×D4×D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#496×C137C2
socleC22×C137C22×D4
hypercenter#496D137
hypocenterC137#3151
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×#17984115C2semidirect
2. C2×#17984112C2semidirect
3. #22577952C2semidirect
4. #127×D137C2semidirect
5. #22577793C2semidirect
6. #22577792C2semidirect
7. #22577790C2semidirect
8. #22577778C2semidirect
9. #22577774C2semidirect
10. #22577355C2semidirect
11. #496×C137C2semidirect
18. C2×#14555499C22semidirect
19. C2×#14555498C22semidirect
20. C2×#14555497C22semidirect
21. C2×D1096C22semidirect
22. C2×#14555495C22semidirect
23. C2×#14555495C22semidirect
25. #17984086C22semidirect
29. #17984084C22semidirect
30. #17984076C22semidirect
34. #17983999C22semidirect
35. C8×Dic137C22semidirect
36. C2×SD16×C137C22semidirect
37. C2×D8×C137C22semidirect
39. #124×C137C22semidirect
40. #97×C137C22semidirect
66. #14555493D4semidirect
103. #11764040C2×D4semidirect
114. C8×C137C2×D4semidirect
132. Dic137AGL(1,ℤ/8)semidirect
140. C137#3151semidirect
149. #496D137semidirect
162. C2×SD16C2×D137semidirect
166. C2×D8C2×D137semidirect
175. #124C2×D137semidirect
180. #97C2×D137semidirect
255. C8D4×D137semidirect

Maximal subgroups

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

subgroupindex
C2×#179841152
C2×#179841122
#225779522
#127×D1372
#225777932
#225777922
#225777902
#225777782
#225777742
#225773552
#496×C1372
#3151137