Finite Group Databasebeta

← #31792601

Subgroups — #31792601

Order 17536.

494 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 494 conjugacy classes of subgroups — too many to draw. Only its 123 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#22578056C211yesmaximal, characteristic, complemented
2#22578038C211yesmaximal, characteristic, complemented
3#22577715C212yesmaximal, complemented
4#22577680C211yesmaximal, characteristic, complemented
5#22577638C212yesmaximal, complemented
6#22577595C211yesmaximal, characteristic, complemented
7#22577538C212yesmaximal, complemented
8C4×D1096C211yesmaximal, characteristic, complemented
9C8×D548C211yesmaximal, characteristic, complemented
10#102×D137C212yesmaximal, complemented
11#486×C137C211yesmaximal, characteristic, complemented
12C2×#14555511—21no—
13C2×D4×D137—21no—
14C2×D4×D137C2212yes—
15C2×D1096C2211yescharacteristic
16C2×#14555495C2212yescomplemented
17C2×C8×D137C2212yescomplemented
18#17984073—22no—
19#17984064—21no—
20#17984063C2212yescomplemented
21#17984059—21no—
22#17984059C2212yes—
23#17984056—22no—
24C4×D548C2211yescharacteristic
25C4×D548C2211yescharacteristic
26C4×D548C2211yescharacteristic, complemented
27C4⋊C4×D137C2212yescomplemented
28#17984022C2211yescharacteristic, complemented
29#17984021C2212yescomplemented
30C4⋊#11764040C2211yescharacteristic, complemented
31#17984010C2211yescharacteristic
32#17984010C2211yescharacteristic, complemented
33#17984009C2212yescomplemented
34#17984008C2212yes—
35#17984000C2212yescomplemented
36C4⋊Q8×C137C2211yescharacteristic, complemented
37#127×C137C2211yescharacteristic
38#102×C137C2212yescomplemented
39#102×C137C2212yescomplemented
40C4×C8×C137C2211yescharacteristic, complemented
41C23×D137—22no—
42#14555511—44no—
43C2×D548—22no—
44C2×D548D411yescharacteristic
45C2×D548C2311yescharacteristic
46C2×D548C2311yescharacteristic
47C2×C22⋊D137—22no—
48C2×C22⋊D137—22no—
49D4×D137—24no—
50D4×D137—42no—
51D4×D137—44no—
52C2×C4×D137—22no—
53C2×C4×D137—21no—
54C2×C4×D137—21no—
55C2×C4×D137C2312yes—
56D1096—22no—
57#14555495—42no—
58C2×#11764040C2312yes—
59C8×D137—42no—
60C4⋊Dic137D411yescharacteristic
61C4⋊Dic137C2311yescharacteristic
62C4⋊Dic137C2311yescharacteristic
63#14555489—22no—
64C22⋊Dic137—22no—
65#14555487—22no—
66#14555487—21no—
67#14555487—21no—
68#14555487D412yes—
69C2×Q8×C137—21no—
70C2×D4×C137—21no—
71C2×D4×C137—21no—
72C2×D4×C137C2312yes—
73C2×C8×C137C2311yescharacteristic
74C2×C8×C137C2311yescharacteristic
75C4⋊C4×C137—21no—
76C4⋊C4×C137C2312yes—
77C42×C137C2311yescharacteristic
78C22×D137—21no—
79C22×D137—21no—
80C22×D137—44no—
81C22×D137—44no—
82C22×D137—82no—
83C22×D137C2×D412yes—
84D548—21no—
85D548—21no—
86D548—21no—
87D548—44no—
88D548D814yescomplemented
89D548#4112yescomplemented
90C22⋊D137—44no—
91C22⋊D137—44no—
92C2×Dic137—21no—
93C2×Dic137—21no—
94C2×Dic137C2×D412yes—
95C4×D137—22no—
96C4×D137—22no—
97C4×D137—44no—
98C4×D137—42no—
99C4×D137—42no—
100#11764040—22no—
101C23×C137—22no—
102Q8×C137—42no—
103D4×C137—22no—
104D4×C137—22no—
105D4×C137—42no—
106D4×C137—81no—
107C2×C4×C137—22no—
108C2×C4×C137C2×D411yescharacteristic
109C2×C4×C137C2×D411yescharacteristic
110C2×C4×C137C2411yescharacteristic
111C8×C137—21no—
112C8×C137—21no—
113C22×C137—42no—
114C22×C137—42no—
115C22×C137—42no—
116C22×C137C22×D411yescharacteristic
117C2×D137—22no—
118C2×D137—22no—
119C2×D137—22no—
120C2×D137—41no—
121C2×D137—41no—
122C2×D137—41no—
123C2×D137—41no—
124C2×D137—41no—
125C2×D137—88no—
126C2×D137C2×D814yes—
127C4×C137—21no—
128C4×C137—42no—
129C4×C137C2×D812yes—
130C4×C137C2×#4111yescharacteristic
131C4×C137H(5,𝔽2)11yescharacteristic
132Dic137—22no—
133Dic137—41no—
134Dic137—41no—
135C2×C137—82no—
136C2×C137#54611yescharacteristic
137C2×C137C22×D811yescharacteristic
138C2×C137#57711yescharacteristic
139D137—24no—
140D137—42no—
141D137—81no—
142C137#329411yescharacteristic, complemented, Sylow
143#3294—1371nomaximal, Sylow
144#547—1371no—
145D42—1371no—
146#486D13711yescharacteristic, complemented
147#480—1372no—
148#474—1371no—
149#466—1372no—
150#459—1371no—
151#447—1372no—
152C4×D8—1371no—
153C8×D4—1371no—
154C2×#102—1372no—
155C2×#41—2741no—
156C22×D4—2741no—
157C22×D4—1372no—
158C2×D8—1372no—
159C2×D8—1371no—
160C22×C8—1372no—
161C4⋊Q8C2×D13711yescharacteristic, complemented
162#127C2×D13711yescharacteristic
163#123—2742no—
164C22⋊Q8—2741no—
165#121—2741no—
166#121—1372no—
167#121—1372no—
168#120—2742no—
169C4×D4—1371no—
170C4×D4—1371no—
171C4×D4—1371no—
172C2×C4⋊C4—1372no—
173#107—1371no—
174#107—1372no—
175C4⋊C8—1371no—
176#102—1372no—
177#102—1371no—
178#102—1371no—
179#102—1372no—
180#102C2×D13712yescomplemented
181#102C2×D13712yescomplemented
182C22⋊C8—1372no—
183C4×C8C2×D13711yescharacteristic, complemented
184C24—2742no—
185#41—5484no—
186C2×Q8—21no—
187C2×D4—21no—
188C2×D4—21no—
189C2×D4—2742no—
190C2×D4—2744no—
191C2×D4—2742no—
192C2×D4—2742no—
193C2×D4—5482no—
194C2×D4—5484no—
195C2×D4—1371no—
196C2×D4—1371no—
197C2×D4—1371no—
198C2×D4C22×D13712yes—
199C22×C4—2742no—
200C22×C4—2741no—
201C22×C4—2741no—
202C22×C4—1372no—
203D8—2742no—
204D8—5482no—
205C2×C8—5482no—
206C2×C8—1372no—
207C2×C8C22×D13711yescharacteristic
208C2×C8C22×D13711yescharacteristic
209C4⋊C4—21no—
210C4⋊C4—2742no—
211C4⋊C4—1371no—
212C4⋊C4—1371no—
213C4⋊C4—1371no—
214C4⋊C4C22×D13712yes—
215C22⋊C4—2742no—
216C22⋊C4—2742no—
217C22⋊C4—2741no—
218C22⋊C4—2741no—
219C22⋊C4—1372no—
220C42C22×D13711yescharacteristic
221C23—22no—
222C23—2741no—
223C23—2741no—
224C23—5484no—
225C23—5484no—
226C23—10962no—
227C23—1372no—
228Q8—42no—
229D4—22no—
230D4—22no—
231D4—42no—
232D4—81no—
233D4—2741no—
234D4—2741no—
235D4—2741no—
236D4—5484no—
237D4—5484no—
238D4—5484no—
239D4—1374no—
240D4—1372no—
241C2×C4—22no—
242C2×C4—2742no—
243C2×C4—2742no—
244C2×C4—2741no—
245C2×C4—2741no—
246C2×C4—5484no—
247C2×C4—5482no—
248C2×C4—5482no—
249C2×C4—1372no—
250C2×C4D4×D13711yescharacteristic
251C2×C4D4×D13711yescharacteristic
252C2×C4C23×D13711yescharacteristic
253C8—21no—
254C8—21no—
255C8—2742no—
256C22—42no—
257C22—42no—
258C22—42no—
259C22—2742no—
260C22—2742no—
261C22—2742no—
262C22—5481no—
263C22—5481no—
264C22—5481no—
265C22—5481no—
266C22—5481no—
267C22—10968no—
268C22—1374no—
269C22C2×D4×D13711yescharacteristic
270C4—21no—
271C4—42no—
272C4—2742no—
273C4—5481no—
274C4—5481no—
275C4D8×D13712yes—
276C4#41×D13711yescharacteristic
277C4#1798413411yescharacteristic
278C2—82no—
279C2—2744no—
280C2—5482no—
281C2—10961no—
282C2#2257804611yescharacteristic
283C2C2×D8×D13711yescharacteristic
284C2#2257820911yescharacteristic
subgroupG/H
centerC22C2×D4×D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#486×C137C2
socleC22×C137C22×D4
hypercenter#486D137
hypocenterC137#3294
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C42 ◁ #486

Complements

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

NH ≅ G/Ntype
1. #22578056C2semidirect
2. #22578038C2semidirect
3. #22577715C2semidirect
4. #22577680C2semidirect
5. #22577638C2semidirect
6. #22577595C2semidirect
7. #22577538C2semidirect
8. C4×D1096C2semidirect
9. C8×D548C2semidirect
10. #102×D137C2semidirect
11. #486×C137C2semidirect
16. C2×#14555495C22semidirect
17. C2×C8×D137C22semidirect
20. #17984063C22semidirect
26. C4×D548C22semidirect
27. C4⋊C4×D137C22semidirect
28. #17984022C22semidirect
29. #17984021C22semidirect
30. C4⋊#11764040C22semidirect
32. #17984010C22semidirect
33. #17984009C22semidirect
35. #17984000C22semidirect
36. C4⋊Q8×C137C22semidirect
38. #102×C137C22semidirect
39. #102×C137C22semidirect
40. C4×C8×C137C22semidirect
88. D548D8semidirect
89. D548#41semidirect
142. C137#3294semidirect
146. #486D137semidirect
161. C4⋊Q8C2×D137semidirect
180. #102C2×D137semidirect
181. #102C2×D137semidirect
183. C4×C8C2×D137semidirect

Maximal subgroups

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

subgroupindex
#225780562
#225780382
#225777152
#225776802
#225776382
#225775952
#225775382
C4×D10962
C8×D5482
#102×D1372
#486×C1372
#3294137