Finite Group Databasebeta

← #31790078

Subgroups — #31790078

Order 17536.

732 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 732 conjugacy classes of subgroups — too many to draw. Only its 149 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×#17984135C211yesmaximal, characteristic, complemented
2C2×#17984113C211yesmaximal, characteristic, complemented
3C22×#14555499C211yesmaximal, characteristic, complemented
4#22577972C211yesmaximal, characteristic, complemented
5#22577549C214yesmaximal, complemented
6#22577543C214yesmaximal, complemented
7#22577411C211yesmaximal, characteristic, complemented
8#22577369C211yesmaximal, characteristic, complemented
9#416×C137C211yesmaximal, characteristic, complemented
10#17984135—24no—
11#17984135—24no—
12C2×#14555510—21no—
13C2×#14555510C2211yescharacteristic
14C2×#14555509—22no—
15C2×#14555509C2212yescomplemented
16C22×Dic274—21no—
17C22×Dic274C2211yescharacteristic
18C2×Q8×D137C2212yescomplemented
19C22×D548C2211yescharacteristic
20#17984113—24no—
21C2×#14555499—24no—
22C2×#14555499C2214yes—
23C2×#14555497C2212yes—
24C2×#14555495C2212yes—
25C2×#14555494C2211yescharacteristic
26#17984085—21no—
27#17984076—21no—
28#17984063C2214yes—
29#17984058—22no—
30#17984011C2214yescomplemented
31#17984000C2214yescomplemented
32C2×#41×C137C2211yescharacteristic
33C22×C8×C137C2211yescharacteristic
34#117×C137C2211yescharacteristic
35#103×C137C2212yescomplemented
36#102×C137C2212yescomplemented
37C23×D137—21no—
38#14555510—28no—
39#14555510—22no—
40#14555510—22no—
41#14555509—216no—
42#14555509—24no—
43#14555509—24no—
44C2×Dic274—28no—
45C2×Dic274—22no—
46C2×Dic274—21no—
47C2×Dic274—22no—
48C2×Dic274D411yescharacteristic
49C2×Dic274D414yescomplemented
50C2×Dic274C2312yes—
51Q8×D137—24no—
52Q8×D137—24no—
53C2×D548—24no—
54C2×D548—22no—
55C2×D548—22no—
56C2×D548—21no—
57C2×D548D411yescharacteristic
58C2×D548C2312yes—
59C2×C22⋊D137—22no—
60C2×C22⋊D137D412yes—
61C22×Dic137—22no—
62C22×Dic137—21no—
63C2×C4×D137—22no—
64C2×C4×D137C2312yes—
65#14555499—28no—
66#14555497—24no—
67#14555495—24no—
68#14555494—22no—
69C2×#11764040C2312yes—
70#14555487—24no—
71#41×C137—22no—
72#41×C137D414yescomplemented
73C2×Q8×C137C2311yescharacteristic
74C2×D4×C137—21no—
75C2×D4×C137C2311yescharacteristic
76C22×C4×C137—21no—
77C22×C4×C137C2311yescharacteristic
78C2×C8×C137—22no—
79C2×C8×C137C2312yes—
80C4⋊C4×C137C2312yes—
81C22⋊C4×C137—21no—
82C42×C137—21no—
83C22×D137—22no—
84C22×D137—42no—
85C22×D137—82no—
86C22×D137C2×D412yes—
87Dic274—28no—
88Dic274—22no—
89Dic274—24no—
90Dic274—22no—
91Dic274C2×D414yes—
92D548—22no—
93D548—22no—
94D548—22no—
95D548—42no—
96D548—84no—
97C22⋊D137—28no—
98C22⋊D137—24no—
99C2×Dic137—28no—
100C2×Dic137—24no—
101C2×Dic137—22no—
102C2×Dic137C2×D412yes—
103C4×D137—28no—
104C4×D137—22no—
105C4×D137—22no—
106#11764040—22no—
107C23×C137—21no—
108C23×C137C2×D411yescharacteristic
109Q8×C137—21no—
110Q8×C137C2×D412yes—
111D4×C137—24no—
112D4×C137—21no—
113D4×C137C2×D412yes—
114C2×C4×C137—21no—
115C2×C4×C137—24no—
116C2×C4×C137—22no—
117C2×C4×C137C2×D411yescharacteristic
118C2×C4×C137C2×D412yes—
119C2×C4×C137C2×D412yes—
120C2×C4×C137C2411yescharacteristic
121C8×C137—22no—
122C22×C137—22no—
123C22×C137—22no—
124C22×C137—42no—
125C22×C137#12012yes—
126C22×C137C22×D411yescharacteristic
127C2×D137—22no—
128C2×D137—24no—
129C2×D137—42no—
130C2×D137—44no—
131C2×D137—84no—
132C4×C137—22no—
133C4×C137—42no—
134C4×C137#12012yes—
135C4×C137C22×D411yescharacteristic
136C4×C137C22×D411yescharacteristic
137Dic137—24no—
138Dic137—22no—
139C2×C137—22no—
140C2×C137—42no—
141C2×C137C2×#12011yescharacteristic
142C2×C137#57712yes—
143D137—42no—
144D137—82no—
145C137#301011yescharacteristic, complemented, Sylow
146#3010—1371nomaximal, Sylow
147C2×2−1+4—1371no—
148C2×AGL(1,ℤ/8)—1371no—
149C22×SD16—1371no—
150#535—1371no—
151#449—1374no—
152#448—1374no—
153#419—1371no—
154#416D13711yescharacteristic, complemented
155#408—1371no—
1562−1+4—2744no—
1572−1+4—2744no—
158C2×#41—2741no—
159C2×#41—2742no—
160C2×#41—1372no—
161C2×#41—1371no—
162C2×#41C2×D13711yescharacteristic
163C22×Q8—2741no—
164C22×Q8—1372no—
165C22×Q8—1371no—
166C22×D4—1371no—
167AGL(1,ℤ/8)—2744no—
168C2×SD16—2744no—
169C2×SD16—1372no—
170C2×SD16—1374no—
171C2×D8—1372no—
172C2×M16—1371no—
173C22×C8C2×D13711yescharacteristic
174#127—2741no—
175#124—2741no—
176#121—1374no—
177#120—2742no—
178#117C2×D13711yescharacteristic
179#103—1374no—
180#103C2×D13712yescomplemented
181#102C2×D13712yescomplemented
182C22⋊C8—1374no—
183C24—2741no—
184#41—22no—
185#41—2748no—
186#41—27416no—
187#41—2744no—
188#41—2744no—
189#41—2742no—
190#41—2742no—
191#41D54814yescomplemented
192C2×Q8—2748no—
193C2×Q8—2742no—
194C2×Q8—2744no—
195C2×Q8—2744no—
196C2×Q8—2741no—
197C2×Q8—2742no—
198C2×Q8—1374no—
199C2×Q8—1372no—
200C2×Q8—1371no—
201C2×Q8C22×D13711yescharacteristic
202C2×D4—21no—
203C2×D4—2744no—
204C2×D4—2742no—
205C2×D4—2742no—
206C2×D4—2741no—
207C2×D4—2742no—
208C2×D4—1372no—
209C2×D4—1371no—
210C2×D4—1372no—
211C2×D4C22×D13711yescharacteristic
212C22×C4—21no—
213C22×C4—2742no—
214C22×C4—2741no—
215C22×C4—2742no—
216C22×C4—1372no—
217C22×C4C22×D13711yescharacteristic
218SD16—2748no—
219SD16—2744no—
220D8—2744no—
221M16—2742no—
222C2×C8—22no—
223C2×C8—1372no—
224C2×C8C22×D13712yes—
225C4⋊C4C22×D13712yes—
226C22⋊C4—21no—
227C22⋊C4—2744no—
228C42—21no—
229C23—21no—
230C23—2742no—
231C23—5482no—
232C23—10962no—
233C23—1372no—
234C23D4×D13711yescharacteristic
235Q8—21no—
236Q8—2748no—
237Q8—2744no—
238Q8—2742no—
239Q8—2742no—
240Q8—1374no—
241Q8C2×D54812yes—
242D4—24no—
243D4—21no—
244D4—2748no—
245D4—2744no—
246D4—2742no—
247D4—2742no—
248D4—2742no—
249D4—5482no—
250D4—10964no—
251D4C2×D54812yes—
252C2×C4—21no—
253C2×C4—24no—
254C2×C4—22no—
255C2×C4—2748no—
256C2×C4—2748no—
257C2×C4—2744no—
258C2×C4—2742no—
259C2×C4—2742no—
260C2×C4—2742no—
261C2×C4—1372no—
262C2×C4D4×D13711yescharacteristic
263C2×C4D4×D13712yes—
264C2×C4C2×D54812yes—
265C2×C4C23×D13711yescharacteristic
266C8—22no—
267C8—2742no—
268C22—22no—
269C22—22no—
270C22—42no—
271C22—2744no—
272C22—2742no—
273C22—5484no—
274C22—5482no—
275C22—10964no—
276C22#1798405812yes—
277C22C2×D4×D13711yescharacteristic
278C4—22no—
279C4—42no—
280C4—2744no—
281C4—2742no—
282C4#1798405812yes—
283C4C2×D4×D13711yescharacteristic
284C4C22×D54811yescharacteristic
285C2—22no—
286C2—42no—
287C2—5482no—
288C2—10962no—
289C2C2×#1798405811yescharacteristic
290C2#2257821012yes—
subgroupG/H
centerC22C2×D4×D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#416×C137C2
socleC22×C137C22×D4
hypercenter#416D137
hypocenterC137#3010
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C22×C4 ◁ #416

Complements

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

NH ≅ G/Ntype
1. C2×#17984135C2semidirect
2. C2×#17984113C2semidirect
3. C22×#14555499C2semidirect
4. #22577972C2semidirect
5. #22577549C2semidirect
6. #22577543C2semidirect
7. #22577411C2semidirect
8. #22577369C2semidirect
9. #416×C137C2semidirect
15. C2×#14555509C22semidirect
18. C2×Q8×D137C22semidirect
30. #17984011C22semidirect
31. #17984000C22semidirect
35. #103×C137C22semidirect
36. #102×C137C22semidirect
49. C2×Dic274D4semidirect
72. #41×C137D4semidirect
145. C137#3010semidirect
154. #416D137semidirect
180. #103C2×D137semidirect
181. #102C2×D137semidirect
191. #41D548semidirect

Maximal subgroups

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

subgroupindex
C2×#179841352
C2×#179841132
C22×#145554992
#225779722
#225775492
#225775432
#225774112
#225773692
#416×C1372
#3010137