Finite Group Databasebeta

← #31790731

Subgroups — #31790731

Order 17536.

462 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 127 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×#17984063C211yesmaximal, characteristic
2#117×D137C211yesmaximal, characteristic, complemented
3#22577730C212yesmaximal, complemented
4#22577729C212yesmaximal, complemented
5#22577723C212yesmaximal, complemented
6#22577718C212yesmaximal, complemented
7#22577443C211yesmaximal, characteristic, complemented
8#22577409C211yesmaximal, characteristic, complemented
9#22577407C211yesmaximal, characteristic, complemented
10#22577366C211yesmaximal, characteristic, complemented
11#428×C137C211yesmaximal, characteristic, complemented
12C22×D548—21no—
13C22×D548C2211yescharacteristic
14C22×C4×D137C2211yescharacteristic
15C2×#14555494C2211yescharacteristic
16#17984063—21no—
17#17984063—21no—
18#17984063C2212yes—
19#17984063C2212yes—
20#17984047C2211yescharacteristic
21#17984046C2211yescharacteristic, complemented
22#17984046C2211yescharacteristic, complemented
23#17984045—21no—
24#17984044—21no—
25C4⋊C4×D137C2211yescharacteristic, complemented
26C4⋊C4×D137C2211yescharacteristic, complemented
27C2×#14555487—21no—
28C22⋊C4×D137—21no—
29C42×D137—21no—
30#17984021C2212yescomplemented
31#17984019C2212yescomplemented
32#17984010C2212yescomplemented
33#17984010C2212yescomplemented
34#17984008C2212yescomplemented
35#17984008C2212yescomplemented
36#17984000C2212yescomplemented
37#17984000C2212yescomplemented
38C2×M16×C137C2211yescharacteristic
39#117×C137C2211yescharacteristic
40C2×C4⋊C4×C137C2211yescharacteristic
41#107×C137C2212yescomplemented
42#106×C137C2212yescomplemented
43C23×D137—21no—
44C23×D137D411yescharacteristic
45C2×D548—22no—
46C2×D548—22no—
47C2×D548—21no—
48C2×D548—22no—
49C2×D548—22no—
50C2×D548—44no—
51C2×D548C2312yes—
52C22×Dic137D411yescharacteristic
53C2×C4×D137—21no—
54C2×C4×D137—21no—
55C2×C4×D137—22no—
56C2×C4×D137—22no—
57C2×C4×D137D411yescharacteristic
58C2×C4×D137D411yescharacteristic
59C2×C4×D137C2311yescharacteristic
60C2×C4×D137C2311yescharacteristic
61#14555494—22no—
62C2×#11764040C2312yes—
63C4⋊Dic137C2311yescharacteristic
64C4⋊Dic137C2311yescharacteristic
65#14555489—21no—
66#14555489—21no—
67C22⋊Dic137—21no—
68#14555487—21no—
69#14555487—21no—
70#14555487—22no—
71#14555487—22no—
72C4×Dic137—21no—
73C4×Dic137—21no—
74C4×Dic137—21no—
75C22×C4×C137—21no—
76C22×C4×C137C2311yescharacteristic
77M16×C137—22no—
78C2×C8×C137C2312yes—
79C4⋊C4×C137—21no—
80C4⋊C4×C137C2311yescharacteristic
81C4⋊C4×C137C2311yescharacteristic
82C4⋊C4×C137C2312yes—
83C22⋊C4×C137—21no—
84C42×C137—21no—
85C22×D137—22no—
86C22×D137—22no—
87C22×D137—22no—
88C22×D137—41no—
89C22×D137—42no—
90C22×D137—42no—
91C22×D137—42no—
92C22×D137C2×D411yescharacteristic
93C22×D137C2×D411yescharacteristic
94D548—22no—
95D548—22no—
96D548—44no—
97D548—44no—
98D548—42no—
99C2×Dic137—22no—
100C2×Dic137—21no—
101C2×Dic137—21no—
102C2×Dic137C2×D411yescharacteristic
103C2×Dic137C2×D411yescharacteristic
104C4×D137—22no—
105C4×D137—22no—
106C4×D137—21no—
107C4×D137—21no—
108C4×D137—21no—
109C4×D137—21no—
110C4×D137—42no—
111C4×D137—42no—
112#11764040—22no—
113C23×C137C2×D411yescharacteristic
114C2×C4×C137—21no—
115C2×C4×C137—21no—
116C2×C4×C137—22no—
117C2×C4×C137—42no—
118C2×C4×C137C2×D411yescharacteristic
119C2×C4×C137#4112yes—
120C2×C4×C137#4112yes—
121C2×C4×C137C2411yescharacteristic
122C8×C137—22no—
123C22×C137—21no—
124C22×C137—21no—
125C22×C137#12312yes—
126C22×C137C22×D411yescharacteristic
127C2×D137—21no—
128C2×D137—21no—
129C2×D137—21no—
130C2×D137—21no—
131C2×D137—21no—
132C2×D137—44no—
133C2×D137—42no—
134C2×D137—42no—
135C2×D137—42no—
136C2×D137—84no—
137C2×D137AGL(1,ℤ/8)12yes—
138C4×C137—42no—
139C4×C137—41no—
140C4×C137—41no—
141C4×C137#12312yes—
142C4×C137C2×#4111yescharacteristic
143C4×C137C2×#4111yescharacteristic
144Dic137—21no—
145Dic137—21no—
146Dic137—41no—
147Dic137—41no—
148C2×C137—22no—
149C2×C137C2×#12311yescharacteristic
150C2×C137C2×AGL(1,ℤ/8)11yescharacteristic
151C2×C137#57511yescharacteristic
152D137—22no—
153D137—41no—
154D137—82no—
155C137#309411yescharacteristic, complemented, Sylow
156#3094—1371nomaximal, Sylow
157C2×#121—1371no—
158C2×#117—1371no—
159#482—1372no—
160#482—1372no—
161#481—1372no—
162#480—1372no—
163#428—1371no—
164#428D13711yescharacteristic, complemented
165#418—1371no—
166#418—1371no—
167#407—1371no—
168C22×D4—2741no—
169C22×D4—1371no—
170C23×C4—1371no—
171C2×M16—1371no—
172C2×M16C2×D13711yescharacteristic
173#121—2741no—
174#121—2741no—
175#121—1372no—
176#121—1372no—
177#117—2741no—
178#117—2741no—
179#117—1371no—
180#117—1371no—
181#117—1371no—
182#117C2×D13711yescharacteristic
183C2×C4⋊C4—1371no—
184C2×C4⋊C4—1371no—
185C2×C4⋊C4C2×D13711yescharacteristic
186C2×C22⋊C4—2741no—
187C2×C22⋊C4—2741no—
188C2×C42—2741no—
189#107—1372no—
190#107C2×D13712yescomplemented
191#106—1372no—
192#106C2×D13712yescomplemented
193#102—1372no—
194#102—1372no—
195#102—1372no—
196#102—1372no—
197C22⋊C8—1372no—
198C22⋊C8—1372no—
199C24—2741no—
200C24—1371no—
201C2×D4—2742no—
202C2×D4—2742no—
203C2×D4—2741no—
204C2×D4—2742no—
205C2×D4—2742no—
206C2×D4—5484no—
207C2×D4—1372no—
208C22×C4—21no—
209C22×C4—2741no—
210C22×C4—2741no—
211C22×C4—2742no—
212C22×C4—2742no—
213C22×C4—1371no—
214C22×C4—1371no—
215C22×C4—1371no—
216C22×C4—1371no—
217C22×C4—1371no—
218C22×C4C22×D13711yescharacteristic
219M16—22no—
220M16—2742no—
221C2×C8—1372no—
222C2×C8C22×D13712yes—
223C4⋊C4—21no—
224C4⋊C4—2741no—
225C4⋊C4—2741no—
226C4⋊C4—1371no—
227C4⋊C4—1371no—
228C4⋊C4C22×D13711yescharacteristic
229C4⋊C4C22×D13711yescharacteristic
230C4⋊C4C22×D13712yes—
231C22⋊C4—21no—
232C22⋊C4—2741no—
233C22⋊C4—2741no—
234C22⋊C4—2742no—
235C22⋊C4—2742no—
236C22⋊C4—2741no—
237C42—21no—
238C42—2741no—
239C42—2741no—
240C42—2741no—
241C23—2742no—
242C23—2742no—
243C23—2742no—
244C23—5481no—
245C23—5482no—
246C23—5482no—
247C23—5482no—
248C23—1371no—
249C23—1371no—
250C23D4×D13711yescharacteristic
251D4—2742no—
252D4—2742no—
253D4—5484no—
254D4—5484no—
255D4—5482no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—22no—
259C2×C4—42no—
260C2×C4—2742no—
261C2×C4—2742no—
262C2×C4—2742no—
263C2×C4—2741no—
264C2×C4—2741no—
265C2×C4—2741no—
266C2×C4—2741no—
267C2×C4—2741no—
268C2×C4—2741no—
269C2×C4—5482no—
270C2×C4—5482no—
271C2×C4—1371no—
272C2×C4—1371no—
273C2×C4D4×D13711yescharacteristic
274C2×C4#1455551012yes—
275C2×C4#1455551112yes—
276C2×C4C23×D13711yescharacteristic
277C8—22no—
278C8—2742no—
279C22—21no—
280C22—21no—
281C22—2741no—
282C22—2741no—
283C22—2741no—
284C22—2741no—
285C22—2741no—
286C22—5484no—
287C22—5482no—
288C22—5482no—
289C22—5482no—
290C22—10964no—
291C22—1372no—
292C22#1798407312yes—
293C22C2×D4×D13711yescharacteristic
294C4—42no—
295C4—41no—
296C4—41no—
297C4—2741no—
298C4—2741no—
299C4—5481no—
300C4—5481no—
301C4#1798407312yes—
302C4C2×#1455551011yescharacteristic
303C4C2×#1455551111yescharacteristic
304C2—22no—
305C2—2742no—
306C2—5481no—
307C2—10962no—
308C2C2×#1798407311yescharacteristic
309C2AGL(1,ℤ/8)×D13711yescharacteristic
310C2#2257820211yescharacteristic
subgroupG/H
centerC22C2×D4×D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#428×C137C2
socleC22×C137C22×D4
prime radicalC2×#17984063C2
hypercenter#428D137
hypocenterC137#3094
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C22×C4 ◁ #428

Complements

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

NH ≅ G/Ntype
2. #117×D137C2semidirect
3. #22577730C2semidirect
4. #22577729C2semidirect
5. #22577723C2semidirect
6. #22577718C2semidirect
7. #22577443C2semidirect
8. #22577409C2semidirect
9. #22577407C2semidirect
10. #22577366C2semidirect
11. #428×C137C2semidirect
21. #17984046C22semidirect
22. #17984046C22semidirect
25. C4⋊C4×D137C22semidirect
26. C4⋊C4×D137C22semidirect
30. #17984021C22semidirect
31. #17984019C22semidirect
32. #17984010C22semidirect
33. #17984010C22semidirect
34. #17984008C22semidirect
35. #17984008C22semidirect
36. #17984000C22semidirect
37. #17984000C22semidirect
41. #107×C137C22semidirect
42. #106×C137C22semidirect
155. C137#3094semidirect
164. #428D137semidirect
190. #107C2×D137semidirect
192. #106C2×D137semidirect

Maximal subgroups

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

subgroupindex
C2×#179840632
#117×D1372
#225777302
#225777292
#225777232
#225777182
#225774432
#225774092
#225774072
#225773662
#428×C1372
#3094137