Finite Group Databasebeta

← #31790060

Subgroups — #31790060

Order 17536.

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

normal subgroups only

This group has 784 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×#17984134C211yesmaximal, characteristic, complemented
2C2×#17984113C211yesmaximal, characteristic, complemented
3C22×D1096C211yesmaximal, characteristic, complemented
4#22577972C211yesmaximal, characteristic, complemented
5#22577551C214yesmaximal, complemented
6#22577538C214yesmaximal, complemented
7#22577407C211yesmaximal, characteristic, complemented
8#22577369C211yesmaximal, characteristic, complemented
9#416×C137C211yesmaximal, characteristic, complemented
10#17984134—24no—
11#17984134—24no—
12C2×#14555511C2212yescomplemented
13C2×#14555510—21no—
14C2×#14555510C2211yescharacteristic
15C22×D548—21no—
16C22×D548C2211yescharacteristic
17C22×D548C2211yescharacteristic, complemented
18C2×D4×D137—22no—
19C2×D4×D137C2212yescomplemented
20#17984113—24no—
21C2×#14555497C2212yescomplemented
22C2×D1096—24no—
23C2×D1096C2214yescomplemented
24C2×#14555495C2212yescomplemented
25C2×#14555494C2211yescharacteristic, complemented
26#17984085—21no—
27#17984076—21no—
28#17984063C2214yescomplemented
29#17984058—22no—
30#17984008C2214yescomplemented
31#17984000C2214yescomplemented
32C2×#41×C137C2211yescharacteristic
33C22×C8×C137C2211yescharacteristic, complemented
34#117×C137C2211yescharacteristic, complemented
35#103×C137C2212yescomplemented
36#102×C137C2212yescomplemented
37C23×D137—22no—
38C23×D137—21no—
39C23×D137—21no—
40#14555511—24no—
41#14555511—24no—
42#14555510—28no—
43#14555510—22no—
44#14555510—22no—
45C2×Dic274D411yescharacteristic
46C2×D548—28no—
47C2×D548—22no—
48C2×D548—24no—
49C2×D548—21no—
50C2×D548—22no—
51C2×D548—22no—
52C2×D548—22no—
53C2×D548—21no—
54C2×D548D411yescharacteristic
55C2×D548D414yescomplemented
56C2×D548C2312yes—
57C2×D548C2312yes—
58C2×C22⋊D137—22no—
59C2×C22⋊D137D412yes—
60D4×D137—216no—
61D4×D137—24no—
62D4×D137—24no—
63C2×C4×D137—22no—
64C2×C4×D137C2312yes—
65#14555497—24no—
66D1096—28no—
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—28no—
85C22×D137—24no—
86C22×D137—22no—
87C22×D137—42no—
88C22×D137—84no—
89C22×D137—82no—
90C22×D137—82no—
91C22×D137C2×D412yes—
92Dic274—22no—
93D548—28no—
94D548—22no—
95D548—24no—
96D548—22no—
97D548—22no—
98D548—22no—
99D548—42no—
100D548—84no—
101D548C2×D414yes—
102C22⋊D137—28no—
103C22⋊D137—24no—
104C2×Dic137C2×D412yes—
105C4×D137—28no—
106C4×D137—22no—
107C4×D137—22no—
108#11764040—22no—
109C23×C137—21no—
110C23×C137C2×D411yescharacteristic
111Q8×C137—21no—
112Q8×C137C2×D412yes—
113D4×C137—24no—
114D4×C137—21no—
115D4×C137C2×D412yes—
116C2×C4×C137—21no—
117C2×C4×C137—24no—
118C2×C4×C137—22no—
119C2×C4×C137C2×D411yescharacteristic
120C2×C4×C137C2×D412yes—
121C2×C4×C137C2×D412yes—
122C2×C4×C137C2411yescharacteristic
123C8×C137—22no—
124C22×C137—22no—
125C22×C137—22no—
126C22×C137—42no—
127C22×C137#12012yes—
128C22×C137C22×D411yescharacteristic
129C2×D137—22no—
130C2×D137—24no—
131C2×D137—24no—
132C2×D137—44no—
133C2×D137—42no—
134C2×D137—44no—
135C2×D137—84no—
136C2×D137—88no—
137C2×D137—84no—
138C4×C137—22no—
139C4×C137—42no—
140C4×C137#12012yes—
141C4×C137C22×D411yescharacteristic
142C4×C137C22×D411yescharacteristic
143Dic137—22no—
144C2×C137—22no—
145C2×C137—42no—
146C2×C137C2×#12011yescharacteristic
147C2×C137#57612yes—
148D137—42no—
149D137—44no—
150D137—82no—
151C137#300811yescharacteristic, complemented, Sylow
152#3008—1371nomaximal, Sylow
153C2×H(5,𝔽2)—1371no—
154C2×AGL(1,ℤ/8)—1371no—
155C22×D8—1371no—
156#535—1371no—
157#449—1374no—
158#447—1374no—
159#418—1371no—
160#416D13711yescharacteristic, complemented
161#408—1371no—
162H(5,𝔽2)—2744no—
163H(5,𝔽2)—2744no—
164C2×#41—2741no—
165C2×#41—1372no—
166C2×#41—1371no—
167C2×#41C2×D13711yescharacteristic
168C22×D4—2741no—
169C22×D4—2742no—
170C22×D4—1372no—
171C22×D4—1371no—
172C22×D4—1371no—
173AGL(1,ℤ/8)—2744no—
174C2×SD16—1372no—
175C2×D8—2744no—
176C2×D8—1372no—
177C2×D8—1374no—
178C2×M16—1371no—
179C22×C8C2×D13711yescharacteristic, complemented
180#127—2741no—
181#124—2741no—
182#121—1374no—
183#120—2742no—
184#117C2×D13711yescharacteristic, complemented
185#103C2×D13712yescomplemented
186#102—1374no—
187#102C2×D13712yescomplemented
188C22⋊C8—1374no—
189C24—2742no—
190C24—2741no—
191C24—2741no—
192#41—22no—
193#41—2748no—
194#41—2744no—
195#41—2744no—
196#41—2742no—
197#41—2742no—
198#41D54814yescomplemented
199C2×Q8—1371no—
200C2×Q8C22×D13711yescharacteristic
201C2×D4—21no—
202C2×D4—2748no—
203C2×D4—2742no—
204C2×D4—2744no—
205C2×D4—27416no—
206C2×D4—2744no—
207C2×D4—2744no—
208C2×D4—2742no—
209C2×D4—2741no—
210C2×D4—2742no—
211C2×D4—2741no—
212C2×D4—2742no—
213C2×D4—2742no—
214C2×D4—1372no—
215C2×D4—1371no—
216C2×D4—1374no—
217C2×D4—1372no—
218C2×D4—1372no—
219C2×D4C22×D13711yescharacteristic
220C22×C4—21no—
221C22×C4—2742no—
222C22×C4—1372no—
223C22×C4C22×D13711yescharacteristic
224SD16—2744no—
225D8—2748no—
226D8—2744no—
227M16—2742no—
228C2×C8—22no—
229C2×C8—1372no—
230C2×C8C22×D13712yes—
231C4⋊C4C22×D13712yes—
232C22⋊C4—21no—
233C22⋊C4—2744no—
234C42—21no—
235C23—21no—
236C23—2748no—
237C23—2742no—
238C23—2744no—
239C23—2742no—
240C23—5482no—
241C23—10962no—
242C23—10962no—
243C23—10964no—
244C23—1372no—
245C23D4×D13711yescharacteristic
246Q8—21no—
247Q8—2742no—
248Q8C2×D54812yes—
249D4—24no—
250D4—21no—
251D4—2748no—
252D4—2748no—
253D4—2744no—
254D4—2742no—
255D4—2744no—
256D4—2742no—
257D4—2742no—
258D4—2742no—
259D4—5482no—
260D4—10964no—
261D4—1374no—
262D4C2×D54812yes—
263C2×C4—21no—
264C2×C4—24no—
265C2×C4—22no—
266C2×C4—2748no—
267C2×C4—2742no—
268C2×C4—2742no—
269C2×C4—1372no—
270C2×C4D4×D13711yescharacteristic
271C2×C4D4×D13712yes—
272C2×C4C2×D54812yes—
273C2×C4C23×D13711yescharacteristic
274C8—22no—
275C8—2742no—
276C22—22no—
277C22—22no—
278C22—42no—
279C22—2744no—
280C22—2742no—
281C22—2744no—
282C22—5484no—
283C22—5484no—
284C22—5482no—
285C22—10964no—
286C22—10964no—
287C22—10968no—
288C22#1798405812yes—
289C22C2×D4×D13711yescharacteristic
290C4—22no—
291C4—42no—
292C4—2742no—
293C4#1798405812yes—
294C4C2×D4×D13711yescharacteristic
295C4C22×D54811yescharacteristic
296C2—22no—
297C2—42no—
298C2—5482no—
299C2—5484no—
300C2—10962no—
301C2C2×#1798405811yescharacteristic
302C2#2257820512yes—
subgroupG/H
centerC22C2×D4×D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#416×C137C2
socleC22×C137C22×D4
hypercenter#416D137
hypocenterC137#3008
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×#17984134C2semidirect
2. C2×#17984113C2semidirect
3. C22×D1096C2semidirect
4. #22577972C2semidirect
5. #22577551C2semidirect
6. #22577538C2semidirect
7. #22577407C2semidirect
8. #22577369C2semidirect
9. #416×C137C2semidirect
12. C2×#14555511C22semidirect
17. C22×D548C22semidirect
19. C2×D4×D137C22semidirect
21. C2×#14555497C22semidirect
23. C2×D1096C22semidirect
24. C2×#14555495C22semidirect
25. C2×#14555494C22semidirect
28. #17984063C22semidirect
30. #17984008C22semidirect
31. #17984000C22semidirect
33. C22×C8×C137C22semidirect
34. #117×C137C22semidirect
35. #103×C137C22semidirect
36. #102×C137C22semidirect
55. C2×D548D4semidirect
72. #41×C137D4semidirect
151. C137#3008semidirect
160. #416D137semidirect
179. C22×C8C2×D137semidirect
184. #117C2×D137semidirect
185. #103C2×D137semidirect
187. #102C2×D137semidirect
198. #41D548semidirect

Maximal subgroups

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

subgroupindex
C2×#179841342
C2×#179841132
C22×D10962
#225779722
#225775512
#225775382
#225774072
#225773692
#416×C1372
#3008137