Finite Group Databasebeta

← #31790726

Subgroups — #31790726

Order 17536.

462 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 121 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#121×D137C211yesmaximal, characteristic
2C2×#17984046C211yesmaximal, characteristic, complemented
3#22577729C212yesmaximal, complemented
4#22577725C212yesmaximal, complemented
5#22577721C211yesmaximal, characteristic, complemented
6#22577717C211yesmaximal, characteristic, complemented
7#22577714C211yesmaximal, characteristic, complemented
8#22577442C212yesmaximal, complemented
9#22577406C212yesmaximal, complemented
10#22577365C211yesmaximal, characteristic, complemented
11#481×C137C211yesmaximal, characteristic, complemented
12C2×D4×D137—21no—
13C2×D4×D137C2211yescharacteristic
14C22×C4×D137C2211yescharacteristic
15C2×#14555493C2212yes—
16#17984063C2211yescharacteristic
17#17984062C2211yescharacteristic
18#17984061—21no—
19#17984060—21no—
20#17984059C2211yescharacteristic
21#17984046—21no—
22#17984046—21no—
23#17984046C2212yes—
24#17984046C2212yescomplemented
25C2×C4⋊Dic137C2211yescharacteristic, complemented
26C4⋊C4×D137C2211yescharacteristic
27C2×#14555487—21no—
28C22⋊C4×D137—21no—
29C2×C4×Dic137—21no—
30#17984022C2212yescomplemented
31#17984021C2212yescomplemented
32#17984019C2212yescomplemented
33#17984010C2212yescomplemented
34#17984009C2212yescomplemented
35#17984008C2212yescomplemented
36C22⋊#11764040C2211yescharacteristic, complemented
37#17984000C2212yescomplemented
38#121×C137C2211yescharacteristic
39C2×C4⋊C4×C137C2211yescharacteristic, complemented
40#106×C137C2212yescomplemented
41#102×C137C2212yescomplemented
42C22⋊C8×C137C2211yescharacteristic, complemented
43C23×D137—21no—
44C23×D137D411yescharacteristic
45C2×D548—21no—
46C2×D548C2311yescharacteristic
47C2×C22⋊D137—21no—
48C2×C22⋊D137—21no—
49C2×C22⋊D137—21no—
50D4×D137—24no—
51D4×D137—44no—
52C22×Dic137—21no—
53C22×Dic137D411yescharacteristic
54C2×C4×D137—24no—
55C2×C4×D137—21no—
56C2×C4×D137D411yescharacteristic
57C2×C4×D137D411yescharacteristic
58C2×C4×D137C2311yescharacteristic
59C2×C4×D137C2311yescharacteristic
60#14555493—24no—
61C2×#11764040C2312yes—
62C4⋊Dic137—21no—
63C4⋊Dic137C2312yes—
64C4⋊Dic137C2311yescharacteristic
65#14555489—21no—
66C22⋊Dic137—21no—
67#14555487—22no—
68#14555487—22no—
69#14555487—21no—
70#14555487—21no—
71C4×Dic137—22no—
72C4×Dic137—22no—
73C2×D4×C137—21no—
74C2×D4×C137C2311yescharacteristic
75C22×C4×C137—21no—
76C22×C4×C137C2311yescharacteristic
77C2×C8×C137C2312yes—
78C4⋊C4×C137—21no—
79C4⋊C4×C137C2312yes—
80C4⋊C4×C137C2311yescharacteristic
81C22⋊C4×C137—21no—
82C22×D137—22no—
83C22×D137—22no—
84C22×D137—21no—
85C22×D137—41no—
86C22×D137—42no—
87C22×D137—42no—
88C22×D137—42no—
89C22×D137C2×D411yescharacteristic
90C22×D137C2×D411yescharacteristic
91D548—22no—
92D548—42no—
93C22⋊D137—42no—
94C22⋊D137—42no—
95C22⋊D137—42no—
96C2×Dic137—22no—
97C2×Dic137—22no—
98C2×Dic137—21no—
99C2×Dic137—42no—
100C2×Dic137C2×D411yescharacteristic
101C2×Dic137C2×D411yescharacteristic
102C4×D137—22no—
103C4×D137—22no—
104C4×D137—22no—
105C4×D137—42no—
106C4×D137#4114yes—
107#11764040—22no—
108C23×C137—21no—
109C23×C137C2×D411yescharacteristic
110D4×C137—22no—
111D4×C137—42no—
112C2×C4×C137—22no—
113C2×C4×C137—21no—
114C2×C4×C137—22no—
115C2×C4×C137—42no—
116C2×C4×C137C2×D411yescharacteristic
117C2×C4×C137C2411yescharacteristic
118C8×C137—22no—
119C22×C137—21no—
120C22×C137—21no—
121C22×C137—41no—
122C22×C137—41no—
123C22×C137—41no—
124C22×C137AGL(1,ℤ/8)12yes—
125C22×C137C22×D411yescharacteristic
126C2×D137—21no—
127C2×D137—21no—
128C2×D137—21no—
129C2×D137—21no—
130C2×D137—21no—
131C2×D137—44no—
132C2×D137—41no—
133C2×D137—41no—
134C2×D137—41no—
135C2×D137—84no—
136C2×D137#12312yes—
137C4×C137—21no—
138C4×C137—41no—
139C4×C137—42no—
140C4×C137C2×#4112yes—
141Dic137—21no—
142Dic137—41no—
143Dic137—42no—
144Dic137#12312yes—
145C2×C137—22no—
146C2×C137—81no—
147C2×C137C2×#12311yescharacteristic
148C2×C137C2×AGL(1,ℤ/8)11yescharacteristic
149C2×C137#57511yescharacteristic
150D137—22no—
151D137—41no—
152D137—81no—
153C137#309411yescharacteristic, complemented, Sylow
154#3094—1371nomaximal, Sylow
155C2×#121—1371no—
156C2×#117—1371no—
157#482—1372no—
158#482—1372no—
159#481—1371no—
160#481D13711yescharacteristic, complemented
161#480—1371no—
162#480—1371no—
163#428—1372no—
164#418—1372no—
165#407—1371no—
166C22×D4—2741no—
167C22×D4—1371no—
168C23×C4—1371no—
169C2×M16—1372no—
170#121—2741no—
171#121—2741no—
172#121—1371no—
173#121—1371no—
174#121—1371no—
175#121C2×D13711yescharacteristic
176#117—2741no—
177#117—2741no—
178#117—1372no—
179#117—1372no—
180C2×C4⋊C4—1371no—
181C2×C4⋊C4—1371no—
182C2×C4⋊C4C2×D13711yescharacteristic, complemented
183C2×C22⋊C4—2741no—
184C2×C22⋊C4—2741no—
185C2×C42—2741no—
186#107—1372no—
187#107—1372no—
188#106—1372no—
189#106C2×D13712yescomplemented
190#102—1372no—
191#102—1372no—
192#102—1372no—
193#102C2×D13712yescomplemented
194C22⋊C8—1371no—
195C22⋊C8—1372no—
196C22⋊C8C2×D13711yescharacteristic, complemented
197C24—2741no—
198C24—1371no—
199C2×D4—21no—
200C2×D4—2741no—
201C2×D4—2741no—
202C2×D4—2741no—
203C2×D4—2744no—
204C2×D4—2741no—
205C2×D4—5484no—
206C2×D4—1371no—
207C2×D4C22×D13711yescharacteristic
208C22×C4—21no—
209C22×C4—2744no—
210C22×C4—2741no—
211C22×C4—2741no—
212C22×C4—1371no—
213C22×C4—1371no—
214C22×C4—1371no—
215C22×C4—1371no—
216C22×C4—1371no—
217C22×C4C22×D13711yescharacteristic
218M16—2744no—
219C2×C8—1372no—
220C2×C8C22×D13712yes—
221C4⋊C4—21no—
222C4⋊C4—2741no—
223C4⋊C4—2741no—
224C4⋊C4—1372no—
225C4⋊C4—1371no—
226C4⋊C4C22×D13712yes—
227C4⋊C4C22×D13711yescharacteristic
228C22⋊C4—21no—
229C22⋊C4—2742no—
230C22⋊C4—2742no—
231C22⋊C4—2741no—
232C22⋊C4—2741no—
233C22⋊C4—2741no—
234C42—2742no—
235C42—2742no—
236C23—21no—
237C23—2742no—
238C23—2742no—
239C23—2741no—
240C23—5481no—
241C23—5482no—
242C23—5482no—
243C23—5482no—
244C23—1371no—
245C23—1371no—
246C23D4×D13711yescharacteristic
247D4—22no—
248D4—42no—
249D4—2742no—
250D4—5482no—
251D4—5482no—
252D4—5482no—
253D4—5482no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—21no—
257C2×C4—42no—
258C2×C4—2742no—
259C2×C4—2742no—
260C2×C4—2742no—
261C2×C4—2742no—
262C2×C4—2742no—
263C2×C4—2741no—
264C2×C4—5482no—
265C2×C4—5482no—
266C2×C4—1371no—
267C2×C4—1371no—
268C2×C4—1374no—
269C2×C4D4×D13711yescharacteristic
270C2×C4C23×D13711yescharacteristic
271C8—22no—
272C8—2742no—
273C22—21no—
274C22—21no—
275C22—41no—
276C22—41no—
277C22—41no—
278C22—2741no—
279C22—2741no—
280C22—2741no—
281C22—2741no—
282C22—2741no—
283C22—5484no—
284C22—5481no—
285C22—5481no—
286C22—5481no—
287C22—10964no—
288C22—1372no—
289C22#1798411512yes—
290C22C2×D4×D13711yescharacteristic
291C4—21no—
292C4—41no—
293C4—42no—
294C4—2741no—
295C4—5481no—
296C4—5482no—
297C4—1372no—
298C4#41×D13712yes—
299C2—22no—
300C2—81no—
301C2—2742no—
302C2—5481no—
303C2—10961no—
304C2#123×D13711yescharacteristic
305C2C2×#1798411511yescharacteristic
306C2#2257819911yescharacteristic
subgroupG/H
centerC22C2×D4×D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#481×C137C2
socleC22×C137C22×D4
prime radical#121×D137C2
hypercenter#481D137
hypocenterC137#3094
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C22×C4 ◁ #481

Complements

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

NH ≅ G/Ntype
2. C2×#17984046C2semidirect
3. #22577729C2semidirect
4. #22577725C2semidirect
5. #22577721C2semidirect
6. #22577717C2semidirect
7. #22577714C2semidirect
8. #22577442C2semidirect
9. #22577406C2semidirect
10. #22577365C2semidirect
11. #481×C137C2semidirect
24. #17984046C22semidirect
25. C2×C4⋊Dic137C22semidirect
30. #17984022C22semidirect
31. #17984021C22semidirect
32. #17984019C22semidirect
33. #17984010C22semidirect
34. #17984009C22semidirect
35. #17984008C22semidirect
36. C22⋊#11764040C22semidirect
37. #17984000C22semidirect
39. C2×C4⋊C4×C137C22semidirect
40. #106×C137C22semidirect
41. #102×C137C22semidirect
42. C22⋊C8×C137C22semidirect
153. C137#3094semidirect
160. #481D137semidirect
182. C2×C4⋊C4C2×D137semidirect
189. #106C2×D137semidirect
193. #102C2×D137semidirect
196. C22⋊C8C2×D137semidirect

Maximal subgroups

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

subgroupindex
#121×D1372
C2×#179840462
#225777292
#225777252
#225777212
#225777172
#225777142
#225774422
#225774062
#225773652
#481×C1372
#3094137