Finite Group Databasebeta

← #31790400

Subgroups — #31790400

Order 17536.

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

normal subgroups only

This group has 500 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×#17984111C211yesmaximal, characteristic, complemented
2C2×M16×D137C211yesmaximal, characteristic, complemented
3#22577916C212yesmaximal, complemented
4#22577658C211yesmaximal, characteristic, complemented
5#22577657C211yesmaximal, characteristic, complemented
6#22577656C211yesmaximal, characteristic, complemented
7#22577652C212yesmaximal, complemented
8#22577651C212yesmaximal, complemented
9#22577438C211yesmaximal, characteristic, complemented
10#22577398C212yesmaximal, complemented
11#470×C137C211yesmaximal, characteristic, complemented
12C2×#14555511C2212yes—
13C22×C4×D137C2211yescharacteristic
14#17984111—24no—
15C2×#14555497C2212yescomplemented
16C2×D1096C2211yescharacteristic, complemented
17C2×#14555494C2211yescharacteristic
18C2×#14555493C2211yescharacteristic
19C2×#14555493C2211yescharacteristic
20M16×D137—24no—
21C2×C8×D137C2211yescharacteristic
22C2×C8×D137C2211yescharacteristic
23#17984073—22no—
24#17984064C2212yes—
25#17984063C2212yes—
26#17984062C2212yes—
27#17984058—22no—
28#17984050—22no—
29#17984048—22no—
30#17984046C2212yes—
31#17984021C2212yescomplemented
32#17984020C2211yescharacteristic, complemented
33#17984012C2212yescomplemented
34#17984010C2212yescomplemented
35#17984008C2212yescomplemented
36C2×Q16×C137C2211yescharacteristic, complemented
37C2×M16×C137C2211yescharacteristic
38C22⋊Q8×C137C2212yes—
39#106×C137C2211yescharacteristic, complemented
40#103×C137C2212yescomplemented
41C23×D137D411yescharacteristic
42#14555511—24no—
43#14555511—24no—
44C2×D548—22no—
45C2×D548—22no—
46C2×D548C2312yes—
47C2×C22⋊D137—22no—
48C22×Dic137D411yescharacteristic
49C2×C4×D137—22no—
50C2×C4×D137—22no—
51C2×C4×D137—22no—
52C2×C4×D137D411yescharacteristic
53C2×C4×D137D411yescharacteristic
54C2×C4×D137C2311yescharacteristic
55C2×C4×D137C2311yescharacteristic
56#14555497—24no—
57D1096—22no—
58#14555494—22no—
59#14555493—22no—
60#14555493—22no—
61C2×#11764040C2311yescharacteristic
62C2×#11764040C2311yescharacteristic
63C8×D137—22no—
64C8×D137D414yescomplemented
65C4⋊Dic137C2312yes—
66#14555489—22no—
67#14555487—22no—
68#14555487—22no—
69#14555487—22no—
70#14555487—22no—
71C4×Dic137—22no—
72C2×Q8×C137C2312yes—
73C22×C4×C137C2311yescharacteristic
74Q16×C137—22no—
75M16×C137—22no—
76C2×C8×C137C2311yescharacteristic
77C2×C8×C137C2311yescharacteristic
78C4⋊C4×C137—22no—
79C4⋊C4×C137C2312yes—
80C22⋊C4×C137—22no—
81C22×D137—22no—
82C22×D137—22no—
83C22×D137—42no—
84C22×D137C2×D411yescharacteristic
85C22×D137C2×D411yescharacteristic
86D548—22no—
87D548—22no—
88D548—44no—
89D548—44no—
90C22⋊D137—44no—
91C2×Dic137—22no—
92C2×Dic137—22no—
93C2×Dic137C2×D411yescharacteristic
94C2×Dic137C2×D411yescharacteristic
95C4×D137—22no—
96C4×D137—21no—
97C4×D137—21no—
98C4×D137—22no—
99C4×D137—44no—
100C4×D137C2×D412yes—
101C4×D137#4112yes—
102#11764040—21no—
103#11764040C2×D412yes—
104C23×C137C2×D411yescharacteristic
105Q8×C137—22no—
106Q8×C137—22no—
107C2×C4×C137—21no—
108C2×C4×C137—21no—
109C2×C4×C137—22no—
110C2×C4×C137—22no—
111C2×C4×C137C2×D411yescharacteristic
112C2×C4×C137C2411yescharacteristic
113C8×C137—21no—
114C8×C137C2×D412yes—
115C22×C137—21no—
116C22×C137—22no—
117C22×C137C22×D411yescharacteristic
118C2×D137—22no—
119C2×D137—21no—
120C2×D137—22no—
121C2×D137—44no—
122C2×D137—42no—
123C2×D137—42no—
124C2×D137—42no—
125C2×D137#12112yes—
126C4×C137—21no—
127C4×C137—42no—
128C4×C137—42no—
129C4×C137C22×D411yescharacteristic
130C4×C137C2×#4111yescharacteristic
131Dic137—21no—
132Dic137—42no—
133Dic137#12112yes—
134C2×C137—41no—
135C2×C137C2×#12111yescharacteristic
136C2×C137#57512yes—
137D137—22no—
138D137—41no—
139D137—82no—
140C137#305411yescharacteristic, complemented, Sylow
141#3054—1371nomaximal, Sylow
142C2×#135—1371no—
143C22×M16—1371no—
144#525—1372no—
145#470—1371no—
146#470D13711yescharacteristic, complemented
147#469—1371no—
148#469—1371no—
149#468—1372no—
150#468—1372no—
151#427—1371no—
152#416—1372no—
153C2×#41—1372no—
154C23×C4—1371no—
155#135—2744no—
156C2×Q16C2×D13711yescharacteristic, complemented
157C2×SD16—1372no—
158C2×D8—1371no—
159C2×M16—2744no—
160C2×M16—1371no—
161C2×M16—1371no—
162C2×M16—1371no—
163C2×M16C2×D13711yescharacteristic
164C22×C8—1371no—
165C22×C8—1371no—
166#123—2742no—
167C22⋊Q8—1372no—
168C22⋊Q8C2×D13712yes—
169#121—1372no—
170#121—1372no—
171#120—2742no—
172C4×D4—2742no—
173C4×D4—2742no—
174#117—1372no—
175#107—1372no—
176#106—1371no—
177#106C2×D13711yescharacteristic, complemented
178#103—1372no—
179#103C2×D13712yescomplemented
180#102—1372no—
181#102—1372no—
182C24—1371no—
183#41—2744no—
184#41—2744no—
185C2×Q8C22×D13712yes—
186C2×D4—2742no—
187C2×D4—2742no—
188C2×D4—2742no—
189C2×D4—1372no—
190C22×C4—2742no—
191C22×C4—2742no—
192C22×C4—2742no—
193C22×C4—1371no—
194C22×C4—1371no—
195C22×C4—1371no—
196C22×C4—1371no—
197C22×C4—1371no—
198C22×C4C22×D13711yescharacteristic
199Q16—22no—
200SD16—2744no—
201D8—2742no—
202M16—22no—
203M16—2742no—
204M16—2742no—
205M16—2742no—
206C2×C8—2742no—
207C2×C8—1371no—
208C2×C8—1371no—
209C2×C8—1374no—
210C2×C8C22×D13711yescharacteristic
211C2×C8C22×D13711yescharacteristic
212C4⋊C4—22no—
213C4⋊C4—2742no—
214C4⋊C4—1372no—
215C4⋊C4C22×D13712yes—
216C22⋊C4—22no—
217C22⋊C4—2742no—
218C22⋊C4—2742no—
219C22⋊C4—2742no—
220C22⋊C4—2742no—
221C42—2742no—
222C23—2742no—
223C23—2742no—
224C23—5482no—
225C23—1371no—
226C23—1371no—
227C23D4×D13711yescharacteristic
228Q8—22no—
229Q8—22no—
230D4—2742no—
231D4—2742no—
232D4—5484no—
233D4—5484no—
234D4—5484no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—2742no—
240C2×C4—2742no—
241C2×C4—2742no—
242C2×C4—2741no—
243C2×C4—2741no—
244C2×C4—2742no—
245C2×C4—5484no—
246C2×C4—1371no—
247C2×C4—1371no—
248C2×C4—1372no—
249C2×C4—1372no—
250C2×C4D4×D13711yescharacteristic
251C2×C4C23×D13711yescharacteristic
252C8—21no—
253C8—2741no—
254C8—1372no—
255C8D4×D13712yes—
256C22—22no—
257C22—21no—
258C22—2742no—
259C22—2742no—
260C22—2741no—
261C22—5484no—
262C22—5482no—
263C22—5482no—
264C22—5482no—
265C22—1372no—
266C22C2×D4×D13711yescharacteristic
267C4—21no—
268C4—42no—
269C4—42no—
270C4—2741no—
271C4—5482no—
272C4—1372no—
273C4C2×D4×D13711yescharacteristic
274C4#41×D13711yescharacteristic
275C2—41no—
276C2—2742no—
277C2—5481no—
278C2—10962no—
279C2#121×D13711yescharacteristic
280C2#2257820212yes—
subgroupG/H
centerC22C2×D4×D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#470×C137C2
socleC22×C137C22×D4
hypercenter#470D137
hypocenterC137#3054
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C22×C4 ◁ #470

Complements

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

NH ≅ G/Ntype
1. C2×#17984111C2semidirect
2. C2×M16×D137C2semidirect
3. #22577916C2semidirect
4. #22577658C2semidirect
5. #22577657C2semidirect
6. #22577656C2semidirect
7. #22577652C2semidirect
8. #22577651C2semidirect
9. #22577438C2semidirect
10. #22577398C2semidirect
11. #470×C137C2semidirect
15. C2×#14555497C22semidirect
16. C2×D1096C22semidirect
31. #17984021C22semidirect
32. #17984020C22semidirect
33. #17984012C22semidirect
34. #17984010C22semidirect
35. #17984008C22semidirect
36. C2×Q16×C137C22semidirect
39. #106×C137C22semidirect
40. #103×C137C22semidirect
64. C8×D137D4semidirect
140. C137#3054semidirect
146. #470D137semidirect
156. C2×Q16C2×D137semidirect
177. #106C2×D137semidirect
179. #103C2×D137semidirect

Maximal subgroups

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

subgroupindex
C2×#179841112
C2×M16×D1372
#225779162
#225776582
#225776572
#225776562
#225776522
#225776512
#225774382
#225773982
#470×C1372
#3054137