Finite Group Databasebeta

← #31792615

Subgroups — #31792615

Order 17536.

462 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 125 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
2D4×D548C211yesmaximal, characteristic
3#22577750C211yesmaximal, characteristic, complemented
4#22577723C212yesmaximal, complemented
5#22577693C211yesmaximal, characteristic, complemented
6#22577635C212yesmaximal, complemented
7#22577603C211yesmaximal, characteristic, complemented
8#22577552C212yesmaximal, complemented
9C8×D548C211yesmaximal, characteristic, complemented
10#102×D137C212yesmaximal, complemented
11C4×SD16×C137C211yesmaximal, characteristic, complemented
12C22×D548—21no—
13C2×D4×D137C2212yes—
14C2×#14555497C2212yescomplemented
15C2×C8×D137C2212yescomplemented
16#17984088C2211yescharacteristic, complemented
17#17984085C2211yescharacteristic
18#17984080—21no—
19#17984066—22no—
20#17984064C2212yescomplemented
21#17984063C2212yes—
22#17984062—21no—
23#17984058—22no—
24C4×D548—21no—
25C4×D548C2211yescharacteristic
26C4⋊C4×D137C2212yescomplemented
27#17984019C2212yescomplemented
28C4⋊#11764040C2211yescharacteristic, complemented
29#17984010C2212yescomplemented
30#17984010C2212yescomplemented
31#17984009C2212yes—
32#17984008C2212yescomplemented
33#17984000C2212yescomplemented
34C2×SD16×C137C2211yescharacteristic
35C4×Q8×C137C2211yescharacteristic, complemented
36C4×D4×C137C2211yescharacteristic
37#106×C137C2211yescharacteristic, complemented
38#103×C137C2211yescharacteristic, complemented
39#102×C137C2211yescharacteristic
40C4×C8×C137C2211yescharacteristic, complemented
41C23×D137—22no—
42C2×Dic274—21no—
43C2×D548—22no—
44C2×D548—21no—
45C2×D548—21no—
46C2×D548—44no—
47C2×D548D411yescharacteristic
48C2×D548C2312yes—
49C2×C22⋊D137—22no—
50D4×D137—24no—
51D4×D137—42no—
52C2×C4×D137—22no—
53C2×C4×D137C2312yes—
54#14555497—42no—
55C2×#11764040C2312yes—
56C8×D137—42no—
57C4⋊Dic137—21no—
58C4⋊Dic137D411yescharacteristic
59C4⋊Dic137C2312yes—
60#14555489—22no—
61#14555489—22no—
62#14555487—22no—
63#14555487—22no—
64#14555487D412yes—
65C2×Q8×C137C2311yescharacteristic
66C2×D4×C137C2311yescharacteristic
67C22×C4×C137—21no—
68SD16×C137—22no—
69C2×C8×C137C2311yescharacteristic
70C2×C8×C137C2311yescharacteristic
71C4⋊C4×C137—21no—
72C4⋊C4×C137C2311yescharacteristic
73C4⋊C4×C137C2311yescharacteristic
74C22⋊C4×C137—21no—
75C42×C137—21no—
76C42×C137C2311yescharacteristic
77C22×D137—22no—
78C22×D137—44no—
79C22×D137—44no—
80C22×D137—82no—
81C22×D137C2×D412yes—
82Dic274—81no—
83D548—22no—
84D548—22no—
85D548—44no—
86D548—42no—
87D548—81no—
88D548SD1614yescomplemented
89C22⋊D137—44no—
90C2×Dic137—22no—
91C2×Dic137C2×D412yes—
92C4×D137—22no—
93C4×D137—22no—
94C4×D137—82no—
95#11764040—22no—
96C23×C137—21no—
97Q8×C137—21no—
98Q8×C137—21no—
99D4×C137—21no—
100D4×C137#4112yes—
101C2×C4×C137—21no—
102C2×C4×C137—21no—
103C2×C4×C137—21no—
104C2×C4×C137—42no—
105C2×C4×C137C2×D411yescharacteristic
106C2×C4×C137C2×D411yescharacteristic
107C2×C4×C137C2411yescharacteristic
108C8×C137—21no—
109C8×C137—21no—
110C22×C137—22no—
111C22×C137—41no—
112C22×C137—41no—
113C22×C137C22×D411yescharacteristic
114C2×D137—22no—
115C2×D137—22no—
116C2×D137—42no—
117C2×D137—42no—
118C2×D137—42no—
119C2×D137—88no—
120C2×D137C2×SD1614yes—
121C4×C137—21no—
122C4×C137—41no—
123C4×C137—41no—
124C4×C137—41no—
125C4×C137C2×SD1612yes—
126C4×C137C2×#4111yescharacteristic
127C4×C137H(5,𝔽2)11yescharacteristic
128Dic137—22no—
129Dic137—42no—
130C2×C137—42no—
131C2×C137#54611yescharacteristic
132C2×C137C22×SD1611yescharacteristic
133C2×C137#57611yescharacteristic
134D137—24no—
135D137—82no—
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#460—1371no—
145#450—1372no—
146C4×SD16D13711yescharacteristic, complemented
147C8×D4—1371no—
148C2×#102—1372no—
149C22×D4—2741no—
150C22×D4—1372no—
151C2×SD16—1372no—
152C2×SD16C2×D13711yescharacteristic
153C22×C8—1372no—
154C4⋊Q8—1371no—
155#127—1371no—
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×D4C2×D13711yescharacteristic
166C2×C4⋊C4—1372no—
167#106—1372no—
168#106C2×D13711yescharacteristic, complemented
169C4⋊C8—1371no—
170#103C2×D13711yescharacteristic, complemented
171#102—1372no—
172#102—1372no—
173#102—1372no—
174#102—1372no—
175#102C2×D13711yescharacteristic
176C22⋊C8—1372no—
177C4×C8C2×D13711yescharacteristic, complemented
178C24—2742no—
179C2×Q8—2741no—
180C2×Q8C22×D13711yescharacteristic
181C2×D4—2742no—
182C2×D4—2744no—
183C2×D4—2742no—
184C2×D4—2741no—
185C2×D4—2741no—
186C2×D4—5484no—
187C2×D4—5482no—
188C2×D4—1371no—
189C2×D4—1372no—
190C2×D4C22×D13711yescharacteristic
191C22×C4—21no—
192C22×C4—2742no—
193C22×C4—1372no—
194SD16—22no—
195SD16—5482no—
196C2×C8—5482no—
197C2×C8—1372no—
198C2×C8C22×D13711yescharacteristic
199C2×C8C22×D13711yescharacteristic
200C4⋊C4—21no—
201C4⋊C4—2742no—
202C4⋊C4—2742no—
203C4⋊C4—2741no—
204C4⋊C4—1372no—
205C4⋊C4—1371no—
206C4⋊C4C22×D13711yescharacteristic
207C4⋊C4C22×D13711yescharacteristic
208C22⋊C4—21no—
209C22⋊C4—2742no—
210C22⋊C4—2742no—
211C22⋊C4—1372no—
212C42—21no—
213C42C22×D13711yescharacteristic
214C23—21no—
215C23—2742no—
216C23—5484no—
217C23—5484no—
218C23—10962no—
219C23—1372no—
220Q8—21no—
221Q8—21no—
222Q8—10961no—
223D4—21no—
224D4—2742no—
225D4—2742no—
226D4—5484no—
227D4—5484no—
228D4—5482no—
229D4—10961no—
230D4—1374no—
231D4#1455551012yes—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—42no—
236C2×C4—2742no—
237C2×C4—2742no—
238C2×C4—2742no—
239C2×C4—10962no—
240C2×C4—1372no—
241C2×C4D4×D13711yescharacteristic
242C2×C4D4×D13711yescharacteristic
243C2×C4C23×D13711yescharacteristic
244C8—21no—
245C8—21no—
246C8—2742no—
247C22—22no—
248C22—41no—
249C22—41no—
250C22—2742no—
251C22—2742no—
252C22—5482no—
253C22—5482no—
254C22—5482no—
255C22—10968no—
256C22—1374no—
257C22C2×D4×D13711yescharacteristic
258C4—21no—
259C4—41no—
260C4—41no—
261C4—41no—
262C4—2742no—
263C4—5482no—
264C4SD16×D13712yes—
265C4C2×#1455551011yescharacteristic
266C4#1798413411yescharacteristic
267C2—42no—
268C2—2744no—
269C2—10962no—
270C2#2257804911yescharacteristic
271C2C2×SD16×D13711yescharacteristic
272C2#2257820511yescharacteristic
subgroupG/H
centerC22C2×D4×D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
FittingC4×SD16×C137C2
socleC22×C137C22×D4
prime radicalD4×D548C2
hypercenterC4×SD16D137
hypocenterC137#3295
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C42 ◁ C4×SD16

Complements

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

NH ≅ G/Ntype
1. #22578113C2semidirect
3. #22577750C2semidirect
4. #22577723C2semidirect
5. #22577693C2semidirect
6. #22577635C2semidirect
7. #22577603C2semidirect
8. #22577552C2semidirect
9. C8×D548C2semidirect
10. #102×D137C2semidirect
11. C4×SD16×C137C2semidirect
14. C2×#14555497C22semidirect
15. C2×C8×D137C22semidirect
16. #17984088C22semidirect
20. #17984064C22semidirect
26. C4⋊C4×D137C22semidirect
27. #17984019C22semidirect
28. C4⋊#11764040C22semidirect
29. #17984010C22semidirect
30. #17984010C22semidirect
32. #17984008C22semidirect
33. #17984000C22semidirect
35. C4×Q8×C137C22semidirect
37. #106×C137C22semidirect
38. #103×C137C22semidirect
40. C4×C8×C137C22semidirect
88. D548SD16semidirect
136. C137#3295semidirect
146. C4×SD16D137semidirect
162. C4×Q8C2×D137semidirect
168. #106C2×D137semidirect
170. #103C2×D137semidirect
177. C4×C8C2×D137semidirect

Maximal subgroups

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

subgroupindex
#225781132
D4×D5482
#225777502
#225777232
#225776932
#225776352
#225776032
#225775522
C8×D5482
#102×D1372
C4×SD16×C1372
#3295137