Finite Group Databasebeta

← #31792616

Subgroups — #31792616

Order 17536.

462 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 123 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
1#22578113C211yesmaximal, characteristic, complemented
2#22578038C211yesmaximal, characteristic
3#22577749C211yesmaximal, characteristic, complemented
4#22577720C212yesmaximal, complemented
5#22577694C211yesmaximal, characteristic, complemented
6#22577629C212yesmaximal, complemented
7#22577552C212yesmaximal, complemented
8C4×#14555497C211yesmaximal, characteristic, complemented
9#22577468C211yesmaximal, characteristic, complemented
10#102×D137C212yesmaximal, complemented
11#460×C137C211yesmaximal, characteristic, complemented
12C2×D4×D137—21no—
13C2×D4×D137C2212yes—
14C2×#14555497C2211yescharacteristic
15C2×C8×D137C2212yescomplemented
16#17984088C2211yescharacteristic, complemented
17#17984080—21no—
18#17984066—22no—
19#17984064C2212yescomplemented
20#17984059—21no—
21#17984059C2212yes—
22#17984056—22no—
23C4×D548—21no—
24C4×D548C2211yescharacteristic
25C4×D548C2211yescharacteristic
26C4⋊C4×D137C2212yescomplemented
27#17984020C2212yescomplemented
28#17984019C2211yescharacteristic, complemented
29#17984012C2211yescharacteristic, complemented
30#17984010C2212yes—
31#17984009C2212yescomplemented
32#17984009C2212yescomplemented
33#17984008C2211yescharacteristic
34#17984000C2212yescomplemented
35C4×#11764040C2211yescharacteristic, complemented
36C2×SD16×C137C2212yescomplemented
37#127×C137C2211yescharacteristic
38C4×Q8×C137C2211yescharacteristic, complemented
39C4⋊C8×C137C2211yescharacteristic, complemented
40#102×C137C2212yescomplemented
41C23×D137—22no—
42C2×Dic274—21no—
43C2×D548D411yescharacteristic
44C2×D548C2311yescharacteristic
45C2×C22⋊D137—22no—
46C2×C22⋊D137—22no—
47D4×D137—24no—
48D4×D137—42no—
49D4×D137—44no—
50C2×C4×D137—22no—
51C2×C4×D137—21no—
52C2×C4×D137C2312yes—
53#14555497—22no—
54C2×#11764040C2311yescharacteristic
55C2×#11764040C2311yescharacteristic
56C8×D137—42no—
57C4⋊Dic137—21no—
58C4⋊Dic137D411yescharacteristic
59C4⋊Dic137C2312yes—
60C4⋊Dic137C2311yescharacteristic
61#14555489—22no—
62#14555489—22no—
63C22⋊Dic137—22no—
64#14555487—22no—
65#14555487—21no—
66#14555487D412yes—
67C2×Q8×C137C2311yescharacteristic
68C2×D4×C137—21no—
69C2×D4×C137—21no—
70C2×D4×C137C2312yes—
71SD16×C137—42no—
72C2×C8×C137C2312yes—
73C4⋊C4×C137—21no—
74C4⋊C4×C137C2311yescharacteristic
75C42×C137—21no—
76C42×C137C2311yescharacteristic
77C22×D137—21no—
78C22×D137—44no—
79C22×D137—44no—
80C22×D137—82no—
81C22×D137C2×D412yes—
82Dic274—81no—
83D548—21no—
84D548SD1614yescomplemented
85D548#4112yes—
86C22⋊D137—44no—
87C22⋊D137—44no—
88C2×Dic137—21no—
89C2×Dic137—22no—
90C2×Dic137C2×D412yes—
91C4×D137—22no—
92C4×D137—22no—
93C4×D137—42no—
94C4×D137—82no—
95#11764040—21no—
96#11764040—21no—
97C23×C137—22no—
98Q8×C137—21no—
99Q8×C137—21no—
100D4×C137—22no—
101D4×C137—22no—
102D4×C137—42no—
103D4×C137—81no—
104C2×C4×C137—21no—
105C2×C4×C137—21no—
106C2×C4×C137C2×D411yescharacteristic
107C2×C4×C137C2×D411yescharacteristic
108C2×C4×C137C2411yescharacteristic
109C8×C137—22no—
110C22×C137—42no—
111C22×C137—42no—
112C22×C137—42no—
113C22×C137C22×D411yescharacteristic
114C2×D137—22no—
115C2×D137—22no—
116C2×D137—22no—
117C2×D137—41no—
118C2×D137—41no—
119C2×D137—88no—
120C2×D137C2×SD1614yes—
121C4×C137—21no—
122C4×C137—41no—
123C4×C137—41no—
124C4×C137C2×SD1612yes—
125C4×C137C2×#4111yescharacteristic
126C4×C137H(5,𝔽2)11yescharacteristic
127Dic137—22no—
128Dic137—41no—
129Dic137—42no—
130C2×C137—82no—
131C2×C137#54611yescharacteristic
132C2×C137C22×SD1611yescharacteristic
133C2×C137#57611yescharacteristic
134D137—24no—
135D137—42no—
136C137#329511yescharacteristic, complemented, Sylow
137#3295—1371nomaximal, Sylow
138#554—1371no—
139D42—1371no—
140#486—1371no—
141#481—1372no—
142#476—1371no—
143#465—1372no—
144#460D13711yescharacteristic, complemented
145#450—1372no—
146C4×SD16—1371no—
147C8×D4—1371no—
148C2×#102—1372no—
149C22×D4—2741no—
150C22×D4—1372no—
151C2×SD16—1371no—
152C2×SD16C2×D13712yescomplemented
153C22×C8—1372no—
154C4⋊Q8—1371no—
155#127C2×D13711yescharacteristic
156#125—2741no—
157C22⋊Q8—2742no—
158C22⋊Q8—1372no—
159#121—2741no—
160#121—1372no—
161#120—2742no—
162C4×Q8C2×D13711yescharacteristic, complemented
163C4×D4—2741no—
164C4×D4—1371no—
165C4×D4—1371no—
166C2×C4⋊C4—1372no—
167#106—1372no—
168#106—1371no—
169C4⋊C8C2×D13711yescharacteristic, complemented
170#103—1371no—
171#102—1372no—
172#102—1372no—
173#102—1372no—
174#102—1371no—
175#102C2×D13712yescomplemented
176C22⋊C8—1372no—
177C4×C8—1371no—
178C24—2742no—
179C2×Q8—2741no—
180C2×Q8C22×D13711yescharacteristic
181C2×D4—21no—
182C2×D4—21no—
183C2×D4—2744no—
184C2×D4—2742no—
185C2×D4—2742no—
186C2×D4—5482no—
187C2×D4—5484no—
188C2×D4—1371no—
189C2×D4—1371no—
190C2×D4C22×D13712yes—
191C22×C4—2742no—
192C22×C4—2741no—
193C22×C4—1372no—
194SD16—42no—
195SD16—2742no—
196C2×C8—5482no—
197C2×C8—1371no—
198C2×C8—1371no—
199C2×C8C22×D13712yes—
200C4⋊C4—21no—
201C4⋊C4—2742no—
202C4⋊C4—2742no—
203C4⋊C4—2741no—
204C4⋊C4—1372no—
205C4⋊C4—1371no—
206C4⋊C4—1371no—
207C4⋊C4C22×D13711yescharacteristic
208C22⋊C4—2742no—
209C22⋊C4—2742no—
210C22⋊C4—2741no—
211C22⋊C4—1372no—
212C42—21no—
213C42C22×D13711yescharacteristic
214C23—22no—
215C23—2741no—
216C23—5484no—
217C23—5484no—
218C23—10962no—
219C23—1372no—
220Q8—21no—
221Q8—21no—
222Q8—10961no—
223D4—22no—
224D4—22no—
225D4—42no—
226D4—81no—
227D4—2741no—
228D4—5484no—
229D4—5484no—
230D4—1374no—
231D4—1372no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—2742no—
235C2×C4—2742no—
236C2×C4—2741no—
237C2×C4—2742no—
238C2×C4—5482no—
239C2×C4—10962no—
240C2×C4—1372no—
241C2×C4D4×D13711yescharacteristic
242C2×C4D4×D13711yescharacteristic
243C2×C4C23×D13711yescharacteristic
244C8—22no—
245C8—2741no—
246C8—2741no—
247C22—42no—
248C22—42no—
249C22—42no—
250C22—2742no—
251C22—2742no—
252C22—2742no—
253C22—5481no—
254C22—5481no—
255C22—10968no—
256C22—1374no—
257C22C2×D4×D13711yescharacteristic
258C4—21no—
259C4—41no—
260C4—41no—
261C4—2742no—
262C4—5482no—
263C4—5481no—
264C4SD16×D13712yes—
265C4#41×D13711yescharacteristic
266C4#1798413311yescharacteristic
267C2—82no—
268C2—2744no—
269C2—5482no—
270C2#2257805011yescharacteristic
271C2C2×SD16×D13711yescharacteristic
272C2#2257820711yescharacteristic
subgroupG/H
centerC22C2×D4×D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#460×C137C2
socleC22×C137C22×D4
prime radical#22578038C2
hypercenter#460D137
hypocenterC137#3295
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C42 ◁ #460

Complements

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

NH ≅ G/Ntype
1. #22578113C2semidirect
3. #22577749C2semidirect
4. #22577720C2semidirect
5. #22577694C2semidirect
6. #22577629C2semidirect
7. #22577552C2semidirect
8. C4×#14555497C2semidirect
9. #22577468C2semidirect
10. #102×D137C2semidirect
11. #460×C137C2semidirect
15. C2×C8×D137C22semidirect
16. #17984088C22semidirect
19. #17984064C22semidirect
26. C4⋊C4×D137C22semidirect
27. #17984020C22semidirect
28. #17984019C22semidirect
29. #17984012C22semidirect
31. #17984009C22semidirect
32. #17984009C22semidirect
34. #17984000C22semidirect
35. C4×#11764040C22semidirect
36. C2×SD16×C137C22semidirect
38. C4×Q8×C137C22semidirect
39. C4⋊C8×C137C22semidirect
40. #102×C137C22semidirect
84. D548SD16semidirect
136. C137#3295semidirect
144. #460D137semidirect
152. C2×SD16C2×D137semidirect
162. C4×Q8C2×D137semidirect
169. C4⋊C8C2×D137semidirect
175. #102C2×D137semidirect

Maximal subgroups

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

subgroupindex
#225781132
#225780382
#225777492
#225777202
#225776942
#225776292
#225775522
C4×#145554972
#225774682
#102×D1372
#460×C1372
#3295137