Finite Group Databasebeta

← #489×D137 #31791115

Subgroups — #489×D137 #31791115

Order 17536.

436 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 436 conjugacy classes of subgroups — too many to draw. Only its 125 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
1C4⋊Q8×D137C211yesmaximal, characteristic, complemented
2#124×D137C211yesmaximal, characteristic
3#22577765C211yesmaximal, characteristic, complemented
4#22577764C212yesmaximal, complemented
5#22577763C211yesmaximal, characteristic, complemented
6#22577762C211yesmaximal, characteristic, complemented
7#22577761C212yesmaximal, complemented
8#103×D137C212yesmaximal, complemented
9#102×D137C212yesmaximal
10#97×D137C211yesmaximal, characteristic, complemented
11#489×C137C211yesmaximal, characteristic, complemented
12C2×Q8×D137—21no—
13C2×Q8×D137C2211yescharacteristic
14C2×D4×D137C2211yescharacteristic
15C2×C8×D137C2212yes—
16#17984088C2211yescharacteristic, complemented
17#17984087—21no—
18#17984087C2212yescomplemented
19#17984086—21no—
20#17984077C2211yescharacteristic
21#17984076C2211yescharacteristic
22#17984075C2211yescharacteristic
23#17984074—21no—
24#17984074—21no—
25C4⋊C4×D137—21no—
26C4⋊C4×D137C2212yes—
27C22⋊C4×D137—21no—
28C22⋊C4×D137—21no—
29C42×D137C2211yescharacteristic
30#17984013C2212yescomplemented
31#17984012C2212yescomplemented
32#17984011C2212yescomplemented
33#17984010C2212yes—
34#17984009C2212yes—
35#17984008C2212yes—
36#17983999C2211yescharacteristic, complemented
37#17983998C2212yescomplemented
38C4⋊Q8×C137C2211yescharacteristic, complemented
39#124×C137C2211yescharacteristic
40#103×C137C2212yescomplemented
41#102×C137C2212yes—
42#97×C137C2211yescharacteristic, complemented
43C23×D137—21no—
44C2×Dic274—21no—
45C2×Dic274—22no—
46C2×Dic274—21no—
47C2×Dic274C2311yescharacteristic
48Q8×D137—24no—
49Q8×D137—44no—
50C2×D548C2311yescharacteristic
51C2×C22⋊D137—21no—
52D4×D137—24no—
53C2×C4×D137—21no—
54C2×C4×D137—22no—
55C2×C4×D137D411yescharacteristic
56C2×C4×D137D411yescharacteristic
57C2×C4×D137C2311yescharacteristic
58C2×#11764040C2312yes—
59C8×D137—24no—
60C4⋊Dic137—21no—
61C4⋊Dic137C2312yes—
62#14555489—22no—
63#14555489—22no—
64C22⋊Dic137—21no—
65C22⋊Dic137—21no—
66#14555487—21no—
67#14555487—21no—
68#14555487—21no—
69#14555487—21no—
70C4×Dic137D411yescharacteristic
71C4×Dic137D411yescharacteristic
72C4×Dic137C2311yescharacteristic
73C2×Q8×C137—21no—
74C2×Q8×C137C2311yescharacteristic
75C2×D4×C137C2311yescharacteristic
76C2×C8×C137C2312yes—
77C4⋊C4×C137—21no—
78C4⋊C4×C137C2312yes—
79C22⋊C4×C137—21no—
80C22⋊C4×C137—21no—
81C42×C137C2311yescharacteristic
82C22×D137—21no—
83C22×D137—42no—
84C22×D137—42no—
85C22×D137—42no—
86C22×D137C2×D411yescharacteristic
87Dic274—22no—
88Dic274—44no—
89Dic274—42no—
90Dic274—42no—
91D548—22no—
92C22⋊D137—42no—
93C2×Dic137—21no—
94C2×Dic137—22no—
95C2×Dic137C2×D411yescharacteristic
96C2×Dic137C2×D411yescharacteristic
97C2×Dic137C2×D411yescharacteristic
98C4×D137—22no—
99C4×D137—22no—
100C4×D137—42no—
101C4×D137—44no—
102C4×D137#4114yes—
103#11764040—22no—
104C23×C137—21no—
105Q8×C137—22no—
106Q8×C137—42no—
107D4×C137—22no—
108C2×C4×C137—21no—
109C2×C4×C137—22no—
110C2×C4×C137C2×D411yescharacteristic
111C2×C4×C137C2×D411yescharacteristic
112C2×C4×C137C2411yescharacteristic
113C8×C137—22no—
114C22×C137—41no—
115C22×C137—41no—
116C22×C137—41no—
117C22×C137C22×D411yescharacteristic
118C2×D137—41no—
119C2×D137—41no—
120C2×D137—41no—
121C2×D137—84no—
122C2×D137#12412yes—
123C2×D137AGL(1,ℤ/8)12yes—
124C2×D137#13712yes—
125C4×C137—21no—
126C4×C137—21no—
127C4×C137—41no—
128C4×C137—42no—
129C4×C137C2×#4112yes—
130Dic137—21no—
131Dic137—21no—
132Dic137—41no—
133Dic137—42no—
134Dic137#12412yes—
135C2×C137—81no—
136C2×C137C2×#12411yescharacteristic
137C2×C137C2×AGL(1,ℤ/8)11yescharacteristic
138C2×C137C2×#13711yescharacteristic
139D137—81no—
140D137#48914yesdirect product factor
141C137C2×#48911yescharacteristic, complemented, Sylow
142C2×#489—1371nomaximal, Sylow
143C2×C4⋊Q8—1371no—
144C2×#124—1371no—
145#489—1371no—
146#489—1371no—
147#489—1372no—
148#489—1372no—
149#489—1371no—
150#489D13711yescharacteristic, direct product factor
151C2×#103—1372no—
152C2×#102—1372no—
153C2×#97—1371no—
154C22×Q8—2741no—
155C22×Q8—1371no—
156C22×D4—1371no—
157C22×C8—1372no—
158C4⋊Q8—2741no—
159C4⋊Q8—2741no—
160C4⋊Q8—1372no—
161C4⋊Q8—1371no—
162C4⋊Q8C2×D13711yescharacteristic, complemented
163#124—2741no—
164#124—2741no—
165#124—1371no—
166#124—1371no—
167#124—1371no—
168#124C2×D13711yescharacteristic
169C2×C4⋊C4—2741no—
170C2×C4⋊C4—1372no—
171C2×C22⋊C4—2741no—
172C2×C22⋊C4—2741no—
173C2×C42—1371no—
174#103—1372no—
175#103—1372no—
176#103—1372no—
177#103C2×D13712yescomplemented
178#102—1372no—
179#102—1372no—
180#102—1372no—
181#102C2×D13712yes—
182#97—1372no—
183#97—1371no—
184#97C2×D13711yescharacteristic, complemented
185C24—2741no—
186C2×Q8—21no—
187C2×Q8—2741no—
188C2×Q8—2742no—
189C2×Q8—2741no—
190C2×Q8—2744no—
191C2×Q8—5484no—
192C2×Q8—1371no—
193C2×Q8C22×D13711yescharacteristic
194C2×D4—2744no—
195C2×D4—2741no—
196C2×D4—1371no—
197C2×D4C22×D13711yescharacteristic
198C22×C4—2741no—
199C22×C4—2742no—
200C22×C4—1371no—
201C22×C4—1371no—
202C22×C4—1371no—
203C2×C8—2744no—
204C2×C8—1372no—
205C2×C8C22×D13712yes—
206C4⋊C4—21no—
207C4⋊C4—2742no—
208C4⋊C4—2742no—
209C4⋊C4—2741no—
210C4⋊C4—1372no—
211C4⋊C4C22×D13712yes—
212C22⋊C4—21no—
213C22⋊C4—21no—
214C22⋊C4—2741no—
215C22⋊C4—2741no—
216C22⋊C4—2741no—
217C22⋊C4—2741no—
218C22⋊C4—2741no—
219C22⋊C4—2741no—
220C42—1371no—
221C42—1371no—
222C42—1371no—
223C42C22×D13711yescharacteristic
224C23—21no—
225C23—2741no—
226C23—5482no—
227C23—5482no—
228C23—5482no—
229C23—1371no—
230Q8—22no—
231Q8—42no—
232Q8—2742no—
233Q8—5482no—
234Q8—5484no—
235Q8—5482no—
236D4—22no—
237D4—2742no—
238D4—5482no—
239C2×C4—21no—
240C2×C4—22no—
241C2×C4—2742no—
242C2×C4—2742no—
243C2×C4—2741no—
244C2×C4—2742no—
245C2×C4—5482no—
246C2×C4—5484no—
247C2×C4—1371no—
248C2×C4—1371no—
249C2×C4—1371no—
250C2×C4—1374no—
251C2×C4D4×D13711yescharacteristic
252C2×C4D4×D13711yescharacteristic
253C2×C4C23×D13711yescharacteristic
254C8—22no—
255C8—2742no—
256C22—41no—
257C22—41no—
258C22—41no—
259C22—5481no—
260C22—5481no—
261C22—5481no—
262C22—10964no—
263C22—1372no—
264C22—1372no—
265C22—1372no—
266C22C2×D4×D13711yescharacteristic
267C4—21no—
268C4—21no—
269C4—42no—
270C4—41no—
271C4—2741no—
272C4—2741no—
273C4—5482no—
274C4—5481no—
275C4—1372no—
276C4#41×D13712yes—
277C2—81no—
278C2—10961no—
279C2—1374no—
280C2#124×D13711yescharacteristic
281C2AGL(1,ℤ/8)×D13711yescharacteristic
282C2#137×D13711yescharacteristic
subgroupG/H
centerC22C2×D4×D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#489×C137C2
socleC22×C137C22×D4
prime radicalC2×D4×D137C22
hypercenter#489D137
hypocenterC137C2×#489
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C42 ◁ #489

Complements

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

NH ≅ G/Ntype
1. C4⋊Q8×D137C2semidirect
3. #22577765C2semidirect
4. #22577764C2semidirect
5. #22577763C2semidirect
6. #22577762C2semidirect
7. #22577761C2semidirect
8. #103×D137C2semidirect
10. #97×D137C2semidirect
11. #489×C137C2semidirect
16. #17984088C22semidirect
18. #17984087C22semidirect
30. #17984013C22semidirect
31. #17984012C22semidirect
32. #17984011C22semidirect
36. #17983999C22semidirect
37. #17983998C22semidirect
38. C4⋊Q8×C137C22semidirect
40. #103×C137C22semidirect
42. #97×C137C22semidirect
140. D137#489direct (N × H)
141. C137C2×#489semidirect
150. #489D137direct (N × H)
162. C4⋊Q8C2×D137semidirect
177. #103C2×D137semidirect
184. #97C2×D137semidirect

Maximal subgroups

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

subgroupindex
C4⋊Q8×D1372
#124×D1372
#225777652
#225777642
#225777632
#225777622
#225777612
#103×D1372
#102×D1372
#97×D1372
#489×C1372
C2×#489137