Finite Group Databasebeta

← #480×D137 #31790657

Subgroups — #480×D137 #31790657

Order 17536.

512 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 135 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×C4⋊C4×D137C211yesmaximal, characteristic, complemented
3#22577718C212yesmaximal, complemented
4#22577717C211yesmaximal, characteristic, complemented
5#22577716C211yesmaximal, characteristic, complemented
6#22577715C212yesmaximal, complemented
7#22577714C211yesmaximal, characteristic, complemented
8#107×D137C212yesmaximal, complemented
9#102×D137C212yesmaximal, complemented
10C22⋊C8×D137C211yesmaximal, characteristic, complemented
11#480×C137C211yesmaximal, characteristic, complemented
12C2×D4×D137—21no—
13C2×D4×D137C2211yescharacteristic
14C22×C4×D137—21no—
15C22×C4×D137C2211yescharacteristic
16C2×C8×D137C2212yes—
17#17984063C2211yescharacteristic
18#17984062C2211yescharacteristic
19#17984061—21no—
20#17984060—21no—
21#17984059C2211yescharacteristic
22C2×C4⋊Dic137C2211yescharacteristic, complemented
23C2×#14555489—21no—
24C4⋊C4×D137—21no—
25C4⋊C4×D137—21no—
26C4⋊C4×D137C2211yescharacteristic
27C4⋊C4×D137C2212yes—
28C4⋊C4×D137C2212yescomplemented
29C22⋊C4×D137—21no—
30#17984022C2212yescomplemented
31#17984021C2212yescomplemented
32#17984021C2212yescomplemented
33#17984010C2212yescomplemented
34#17984009C2212yescomplemented
35#17984008C2212yescomplemented
36C22⋊#11764040C2211yescharacteristic, complemented
37#17984000C2212yescomplemented
38#121×C137C2211yescharacteristic
39C2×C4⋊C4×C137C2211yescharacteristic, complemented
40#107×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×D137—22no—
57C2×C4×D137—22no—
58C2×C4×D137—44no—
59C2×C4×D137D411yescharacteristic
60C2×C4×D137D411yescharacteristic
61C2×C4×D137C2311yescharacteristic
62C2×C4×D137C2311yescharacteristic
63C2×#11764040C2312yes—
64C8×D137—24no—
65C4⋊Dic137—21no—
66C4⋊Dic137C2312yes—
67C4⋊Dic137C2311yescharacteristic
68#14555489—22no—
69#14555489—22no—
70#14555489—21no—
71C22⋊Dic137—21no—
72#14555487—21no—
73#14555487—21no—
74C2×D4×C137—21no—
75C2×D4×C137C2311yescharacteristic
76C22×C4×C137—21no—
77C22×C4×C137C2311yescharacteristic
78C2×C8×C137C2312yes—
79C4⋊C4×C137—21no—
80C4⋊C4×C137C2312yes—
81C4⋊C4×C137C2311yescharacteristic
82C22⋊C4×C137—21no—
83C22×D137—22no—
84C22×D137—22no—
85C22×D137—21no—
86C22×D137—42no—
87C22×D137—42no—
88C22×D137—42no—
89C22×D137D814yes—
90C22×D137C2×D411yescharacteristic
91C22×D137C2×D411yescharacteristic
92D548—22no—
93D548—42no—
94C22⋊D137—42no—
95C22⋊D137—42no—
96C22⋊D137—42no—
97C2×Dic137—22no—
98C2×Dic137—21no—
99C2×Dic137—22no—
100C2×Dic137—42no—
101C2×Dic137C2×D411yescharacteristic
102C2×Dic137C2×D411yescharacteristic
103C4×D137—22no—
104C4×D137—22no—
105C4×D137—22no—
106C4×D137—44no—
107C4×D137—44no—
108C4×D137—42no—
109C4×D137#4114yes—
110#11764040—22no—
111C23×C137—21no—
112C23×C137C2×D411yescharacteristic
113D4×C137—22no—
114D4×C137—42no—
115C2×C4×C137—22no—
116C2×C4×C137—21no—
117C2×C4×C137—22no—
118C2×C4×C137—42no—
119C2×C4×C137C2×D411yescharacteristic
120C2×C4×C137C2411yescharacteristic
121C8×C137—22no—
122C22×C137—21no—
123C22×C137—21no—
124C22×C137—41no—
125C22×C137—41no—
126C22×C137—41no—
127C22×C137C2×D812yes—
128C22×C137C22×D411yescharacteristic
129C2×D137—21no—
130C2×D137—21no—
131C2×D137—28no—
132C2×D137—41no—
133C2×D137—41no—
134C2×D137—41no—
135C2×D137—84no—
136C2×D137#12312yes—
137C2×D137C2×D812yes—
138C2×D137C2×D812yes—
139C2×D137#13712yes—
140C4×C137—21no—
141C4×C137—42no—
142C4×C137—41no—
143C4×C137C2×#4112yes—
144Dic137—21no—
145Dic137—42no—
146Dic137—41no—
147Dic137#12312yes—
148C2×C137—22no—
149C2×C137—81no—
150C2×C137C2×#12311yescharacteristic
151C2×C137C22×D811yescharacteristic
152C2×C137C2×#13711yescharacteristic
153D137—22no—
154D137—81no—
155D137#48014yesdirect product factor
156C137C2×#48011yescharacteristic, complemented, Sylow
157C2×#480—1371nomaximal, Sylow
158C2×#121—1371no—
159C22×C4⋊C4—1371no—
160#480—1371no—
161#480—1371no—
162#480—1372no—
163#480—1372no—
164#480—1371no—
165#480D13711yescharacteristic, direct product factor
166C2×#107—1372no—
167C2×#102—1372no—
168C2×C22⋊C8—1371no—
169C22×D4—2741no—
170C22×D4—1371no—
171C23×C4—2741no—
172C23×C4—1371no—
173C22×C8—1372no—
174#121—2741no—
175#121—2741no—
176#121—1371no—
177#121—1371no—
178#121—1371no—
179#121C2×D13711yescharacteristic
180C2×C4⋊C4—2741no—
181C2×C4⋊C4—2741no—
182C2×C4⋊C4—2741no—
183C2×C4⋊C4—1371no—
184C2×C4⋊C4—1372no—
185C2×C4⋊C4—1372no—
186C2×C4⋊C4—1371no—
187C2×C4⋊C4C2×D13711yescharacteristic, complemented
188C2×C22⋊C4—2741no—
189#107—1372no—
190#107—1372no—
191#107—1372no—
192#107C2×D13712yescomplemented
193#102—1372no—
194#102—1372no—
195#102—1372no—
196#102C2×D13712yescomplemented
197C22⋊C8—1371no—
198C22⋊C8—1372no—
199C22⋊C8C2×D13711yescharacteristic, complemented
200C24—2741no—
201C24—1371no—
202C2×D4—21no—
203C2×D4—2741no—
204C2×D4—2741no—
205C2×D4—2741no—
206C2×D4—2744no—
207C2×D4—2741no—
208C2×D4—5484no—
209C2×D4—1371no—
210C2×D4C22×D13711yescharacteristic
211C22×C4—21no—
212C22×C4—2744no—
213C22×C4—2741no—
214C22×C4—2742no—
215C22×C4—2742no—
216C22×C4—2741no—
217C22×C4—5484no—
218C22×C4—1371no—
219C22×C4—1371no—
220C22×C4—1371no—
221C22×C4—1371no—
222C22×C4—1371no—
223C22×C4C22×D13711yescharacteristic
224C2×C8—2744no—
225C2×C8—1372no—
226C2×C8C22×D13712yes—
227C4⋊C4—21no—
228C4⋊C4—2742no—
229C4⋊C4—2742no—
230C4⋊C4—2741no—
231C4⋊C4—2741no—
232C4⋊C4—1372no—
233C4⋊C4—1371no—
234C4⋊C4C22×D13712yes—
235C4⋊C4C22×D13711yescharacteristic
236C22⋊C4—21no—
237C22⋊C4—2741no—
238C22⋊C4—2741no—
239C22⋊C4—2741no—
240C23—21no—
241C23—2742no—
242C23—2742no—
243C23—2741no—
244C23—5482no—
245C23—5482no—
246C23—5482no—
247C23—1374no—
248C23—1371no—
249C23—1371no—
250C23D4×D13711yescharacteristic
251D4—22no—
252D4—42no—
253D4—2742no—
254D4—5482no—
255D4—5482no—
256D4—5482no—
257D4—5482no—
258C2×C4—22no—
259C2×C4—21no—
260C2×C4—22no—
261C2×C4—42no—
262C2×C4—2742no—
263C2×C4—2742no—
264C2×C4—2742no—
265C2×C4—2742no—
266C2×C4—2741no—
267C2×C4—2742no—
268C2×C4—5484no—
269C2×C4—5484no—
270C2×C4—5482no—
271C2×C4—5482no—
272C2×C4—1371no—
273C2×C4—1371no—
274C2×C4—1374no—
275C2×C4D4×D13711yescharacteristic
276C2×C4C23×D13711yescharacteristic
277C8—22no—
278C8—2742no—
279C22—21no—
280C22—21no—
281C22—41no—
282C22—41no—
283C22—41no—
284C22—2748no—
285C22—2741no—
286C22—2741no—
287C22—5481no—
288C22—5481no—
289C22—5481no—
290C22—10964no—
291C22—1372no—
292C22—1372no—
293C22—1372no—
294C22—1372no—
295C22D8×D13712yes—
296C22C2×D4×D13711yescharacteristic
297C4—21no—
298C4—42no—
299C4—41no—
300C4—2741no—
301C4—5482no—
302C4—5481no—
303C4—1372no—
304C4#41×D13712yes—
305C2—22no—
306C2—81no—
307C2—2742no—
308C2—10961no—
309C2—1374no—
310C2#123×D13711yescharacteristic
311C2C2×D8×D13711yescharacteristic
312C2#137×D13711yescharacteristic
subgroupG/H
centerC22C2×D4×D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#480×C137C2
socleC22×C137C22×D4
prime radical#121×D137C2
hypercenter#480D137
hypocenterC137C2×#480
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C22×C4 ◁ #480

Complements

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

NH ≅ G/Ntype
2. C2×C4⋊C4×D137C2semidirect
3. #22577718C2semidirect
4. #22577717C2semidirect
5. #22577716C2semidirect
6. #22577715C2semidirect
7. #22577714C2semidirect
8. #107×D137C2semidirect
9. #102×D137C2semidirect
10. C22⋊C8×D137C2semidirect
11. #480×C137C2semidirect
22. C2×C4⋊Dic137C22semidirect
28. C4⋊C4×D137C22semidirect
30. #17984022C22semidirect
31. #17984021C22semidirect
32. #17984021C22semidirect
33. #17984010C22semidirect
34. #17984009C22semidirect
35. #17984008C22semidirect
36. C22⋊#11764040C22semidirect
37. #17984000C22semidirect
39. C2×C4⋊C4×C137C22semidirect
40. #107×C137C22semidirect
41. #102×C137C22semidirect
42. C22⋊C8×C137C22semidirect
155. D137#480direct (N × H)
156. C137C2×#480semidirect
165. #480D137direct (N × H)
187. C2×C4⋊C4C2×D137semidirect
192. #107C2×D137semidirect
196. #102C2×D137semidirect
199. 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×C4⋊C4×D1372
#225777182
#225777172
#225777162
#225777152
#225777142
#107×D1372
#102×D1372
C22⋊C8×D1372
#480×C1372
C2×#480137