Finite Group Databasebeta

← #31790073

Subgroups — #31790073

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×#17984114C211yesmaximal, characteristic, complemented
3C22×#14555498C211yesmaximal, characteristic, complemented
4#22577546C214yesmaximal, complemented
5#22577541C214yesmaximal, complemented
6#22577411C211yesmaximal, characteristic, complemented
7#22577397C211yesmaximal, characteristic, complemented
8#22577370C211yesmaximal, characteristic, complemented
9#535×C137C211yesmaximal, characteristic, complemented
10#17984135—24no—
11#17984135—24no—
12C2×#14555510C2211yescharacteristic
13C2×#14555510C2212yescomplemented
14C2×#14555509—22no—
15C2×#14555509—21no—
16C22×Dic274C2211yescharacteristic
17C22×Dic274C2212yescomplemented
18C2×Q8×D137—21no—
19#17984114—24no—
20C2×#14555498—24no—
21C2×#14555498C2214yes—
22C2×#14555498C2212yes—
23C2×#14555495C2212yes—
24C2×#14555494C2211yescharacteristic
25C22×#11764040C2211yescharacteristic
26#17984011C2214yescomplemented
27#17984011C2212yescomplemented
28#17984008C2212yescomplemented
29C22⋊#11764040C2214yescomplemented
30C2×#41×C137C2211yescharacteristic
31C22×D4×C137C2211yescharacteristic
32#127×C137—21no—
33#124×C137—21no—
34#121×C137C2214yes—
35#120×C137—22no—
36#117×C137C2211yescharacteristic
37#14555510—24no—
38#14555510—24no—
39#14555510—22no—
40#14555510D414yescomplemented
41#14555509—216no—
42#14555509—28no—
43C2×Dic274—24no—
44C2×Dic274—24no—
45C2×Dic274—21no—
46C2×Dic274—22no—
47C2×Dic274—22no—
48C2×Dic274D414yescomplemented
49C2×Dic274C2312yes—
50C2×Dic274C2311yescharacteristic
51Q8×D137—28no—
52C2×D548C2311yescharacteristic
53C2×C22⋊D137—21no—
54C2×C22⋊D137—22no—
55C22×Dic137—21no—
56C22×Dic137—22no—
57C2×C4×D137—21no—
58C2×C4×D137—22no—
59#14555498—28no—
60#14555498—24no—
61#14555495—24no—
62#14555494—22no—
63C2×#11764040—22no—
64C2×#11764040C2312yes—
65C2×#11764040C2312yes—
66C24×C137—21no—
67#41×C137—22no—
68#41×C137—22no—
69C2×Q8×C137D411yescharacteristic
70C2×D4×C137—24no—
71C2×D4×C137—22no—
72C2×D4×C137—22no—
73C2×D4×C137—21no—
74C2×D4×C137D412yes—
75C2×D4×C137D411yescharacteristic
76C2×D4×C137C2312yes—
77C22×C4×C137C2312yes—
78C22×C4×C137C2311yescharacteristic
79C4⋊C4×C137C2312yes—
80C22⋊C4×C137—21no—
81C22⋊C4×C137—24no—
82C42×C137—21no—
83C22×D137—21no—
84Dic274—28no—
85Dic274—24no—
86Dic274—21no—
87Dic274—22no—
88Dic274C2×D412yes—
89Dic274C2×D414yes—
90D548—21no—
91D548C2×D412yes—
92C22⋊D137—28no—
93C22⋊D137—24no—
94C2×Dic137—28no—
95C2×Dic137—24no—
96C2×Dic137—21no—
97C2×Dic137—22no—
98C4×D137—28no—
99C4×D137—24no—
100#11764040—22no—
101#11764040—22no—
102C23×C137—22no—
103C23×C137—42no—
104C23×C137—82no—
105C23×C137C2×D412yes—
106C23×C137C2×D411yescharacteristic
107Q8×C137—22no—
108D4×C137—24no—
109D4×C137—22no—
110D4×C137—22no—
111D4×C137—22no—
112D4×C137—42no—
113D4×C137—84no—
114C2×C4×C137—22no—
115C2×C4×C137—22no—
116C2×C4×C137—22no—
117C2×C4×C137C2×D412yes—
118C2×C4×C137C2×D412yes—
119C2×C4×C137C2×D412yes—
120C2×C4×C137C2×D411yescharacteristic
121C2×C4×C137C2411yescharacteristic
122C22×C137—22no—
123C22×C137—22no—
124C22×C137—24no—
125C22×C137—42no—
126C22×C137—44no—
127C22×C137—84no—
128C22×C137#12012yes—
129C22×C137C22×D411yescharacteristic
130C2×D137—22no—
131C2×D137—42no—
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×C137—82no—
142C2×C137C2×#12011yescharacteristic
143C2×C137#57712yes—
144D137—42no—
145C137#301011yescharacteristic, complemented, Sylow
146#3010—1371nomaximal, Sylow
147C2×2−1+4—1371no—
148C2×AGL(1,ℤ/8)—1371no—
149C22×SD16—1371no—
150#535D13711yescharacteristic, complemented
151#449—1374no—
152#448—1374no—
153#419—1371no—
154#416—1371no—
155#408—1371no—
1562−1+4—2744no—
1572−1+4—2744no—
158C2×#41—2742no—
159C2×#41—2741no—
160C2×#41—1372no—
161C2×#41—1371no—
162C2×#41C2×D13711yescharacteristic
163C22×Q8—2741no—
164C22×Q8—1372no—
165C22×Q8—1371no—
166C22×D4C2×D13711yescharacteristic
167AGL(1,ℤ/8)—2744no—
168C2×SD16—2744no—
169C2×SD16—1374no—
170C2×SD16—1372no—
171C2×D8—1372no—
172C2×M16—1371no—
173C22×C8—1371no—
174#127—21no—
175#124—21no—
176#121C2×D13714yes—
177#120—22no—
178#117C2×D13711yescharacteristic
179#103—1372no—
180#103—1374no—
181#102—1372no—
182C22⋊C8—1374no—
183C24—21no—
184#41—22no—
185#41—22no—
186#41—2744no—
187#41—2744no—
188#41—27416no—
189#41—2748no—
190#41—2742no—
191#41—1374no—
192C2×Q8—2744no—
193C2×Q8—2744no—
194C2×Q8—2748no—
195C2×Q8—2742no—
196C2×Q8—2742no—
197C2×Q8—2741no—
198C2×Q8—1374no—
199C2×Q8—1371no—
200C2×Q8—1372no—
201C2×Q8C22⋊D13711yescharacteristic
202C2×D4—24no—
203C2×D4—22no—
204C2×D4—22no—
205C2×D4—21no—
206C2×D4—2742no—
207C2×D4—2741no—
208C2×D4—1371no—
209C2×D4C22⋊D13712yes—
210C2×D4C22⋊D13711yescharacteristic
211C2×D4C22×D13712yes—
212C22×C4—2742no—
213C22×C4—2741no—
214C22×C4—2742no—
215C22×C4—2741no—
216C22×C4C22×D13712yes—
217C22×C4C22×D13711yescharacteristic
218SD16—2748no—
219SD16—2744no—
220D8—2744no—
221M16—2742no—
222C2×C8—2742no—
223C2×C8—1372no—
224C2×C8—1372no—
225C4⋊C4C22×D13712yes—
226C22⋊C4—21no—
227C22⋊C4—24no—
228C42—21no—
229C23—22no—
230C23—42no—
231C23—82no—
232C23—2741no—
233C23C2×C22⋊D13712yes—
234C23C2×C22⋊D13711yescharacteristic
235Q8—22no—
236Q8—2748no—
237Q8—2744no—
238Q8—2741no—
239Q8—2742no—
240Q8—1372no—
241Q8—1374no—
242D4—24no—
243D4—22no—
244D4—22no—
245D4—22no—
246D4—42no—
247D4—84no—
248D4—2748no—
249D4—2744no—
250D4—2741no—
251D4—1372no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—2748no—
256C2×C4—2748no—
257C2×C4—2744no—
258C2×C4—2742no—
259C2×C4—2741no—
260C2×C4—2744no—
261C2×C4D4×D13712yes—
262C2×C4D4×D13712yes—
263C2×C4C2×C22⋊D13712yes—
264C2×C4C2×C22⋊D13711yescharacteristic
265C2×C4C23×D13711yescharacteristic
266C8—2742no—
267C8—2742no—
268C22—22no—
269C22—24no—
270C22—22no—
271C22—44no—
272C22—42no—
273C22—84no—
274C22—2742no—
275C22—5482no—
276C22#1798405612yes—
277C22C22×C22⋊D13711yescharacteristic
278C4—22no—
279C4—42no—
280C4—2744no—
281C4—2742no—
282C4#1798405612yes—
283C4C2×D4×D13711yescharacteristic
284C4C2×D4×D13711yescharacteristic
285C2—22no—
286C2—42no—
287C2—82no—
288C2—5482no—
289C2C2×#1798405611yescharacteristic
290C2#2257821412yes—
subgroupG/H
centerC22C22×C22⋊D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#535×C137C2
socleC22×C137C22×D4
hypercenter#535D137
hypocenterC137#3010
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C2×#41 ◁ #535

Complements

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

NH ≅ G/Ntype
1. C2×#17984135C2semidirect
2. C2×#17984114C2semidirect
3. C22×#14555498C2semidirect
4. #22577546C2semidirect
5. #22577541C2semidirect
6. #22577411C2semidirect
7. #22577397C2semidirect
8. #22577370C2semidirect
9. #535×C137C2semidirect
13. C2×#14555510C22semidirect
17. C22×Dic274C22semidirect
26. #17984011C22semidirect
27. #17984011C22semidirect
28. #17984008C22semidirect
29. C22⋊#11764040C22semidirect
40. #14555510D4semidirect
48. C2×Dic274D4semidirect
145. C137#3010semidirect
150. #535D137semidirect

Maximal subgroups

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

subgroupindex
C2×#179841352
C2×#179841142
C22×#145554982
#225775462
#225775412
#225774112
#225773972
#225773702
#535×C1372
#3010137