Finite Group Databasebeta

← #31790749

Subgroups — #31790749

Order 17536.

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

normal subgroups only

This group has 418 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#22577972C211yesmaximal, characteristic
2#22577884C211yesmaximal, characteristic, complemented
3#22577730C212yesmaximal, complemented
4#22577729C212yesmaximal, complemented
5#22577723C212yesmaximal, complemented
6#22577718C212yesmaximal, complemented
7#22577439C211yesmaximal, characteristic, complemented
8#22577409C211yesmaximal, characteristic, complemented
9C2×#17984008C211yesmaximal, characteristic, complemented
10#22577371C211yesmaximal, characteristic, complemented
11#428×C137C211yesmaximal, characteristic, complemented
12C2×#14555510C2211yescharacteristic
13C22×D548C2211yescharacteristic
14C22×#11764040C2211yescharacteristic
15#17984085—21no—
16#17984076—21no—
17#17984063C2212yes—
18#17984063C2212yes—
19#17984058—22no—
20#17984053—21no—
21#17984050—21no—
22C4×C22⋊D137—22no—
23#17984047C2211yescharacteristic
24#17984046C2212yescomplemented
25C4⋊C4×D137C2212yescomplemented
26#17984021C2211yescharacteristic, complemented
27#17984021C2211yescharacteristic, complemented
28#17984019C2211yescharacteristic, complemented
29#17984019C2211yescharacteristic, complemented
30#17984010C2212yes—
31#17984010C2212yes—
32#17984008C2212yes—
33#17984008C2212yes—
34#17984000C2212yescomplemented
35#17984000C2212yescomplemented
36C2×M16×C137C2211yescharacteristic
37#117×C137C2211yescharacteristic
38C2×C4⋊C4×C137C2211yescharacteristic
39#107×C137C2212yescomplemented
40#106×C137C2212yescomplemented
41C23×D137—21no—
42#14555510—22no—
43#14555510—22no—
44C2×Dic274D411yescharacteristic
45C2×D548—22no—
46C2×D548—22no—
47C2×D548—21no—
48C2×D548—22no—
49C2×D548—22no—
50C2×D548D411yescharacteristic
51C2×D548C2312yes—
52C2×C22⋊D137D412yes—
53C2×C4×D137—22no—
54C2×C4×D137C2312yes—
55C2×#11764040—22no—
56C2×#11764040C2311yescharacteristic
57C2×#11764040C2311yescharacteristic
58C4⋊Dic137C2311yescharacteristic
59C4⋊Dic137C2311yescharacteristic
60#14555489—22no—
61C22⋊Dic137—21no—
62#14555487—22no—
63#14555487—22no—
64#14555487—22no—
65C4×Dic137—22no—
66C4×Dic137—21no—
67C22×C4×C137—21no—
68C22×C4×C137C2311yescharacteristic
69M16×C137—22no—
70C2×C8×C137C2312yes—
71C4⋊C4×C137—21no—
72C4⋊C4×C137C2312yes—
73C4⋊C4×C137C2311yescharacteristic
74C4⋊C4×C137C2311yescharacteristic
75C22⋊C4×C137—21no—
76C42×C137—21no—
77C22×D137—22no—
78C22×D137—42no—
79C22×D137—81no—
80C22×D137—81no—
81C22×D137C2×D412yes—
82Dic274—22no—
83D548—22no—
84D548—22no—
85D548—22no—
86D548—42no—
87D548—82no—
88D548—82no—
89C22⋊D137—24no—
90C2×Dic137—21no—
91C2×Dic137—21no—
92C2×Dic137C2×D412yes—
93C4×D137—22no—
94C4×D137—22no—
95C4×D137—82no—
96#11764040—21no—
97#11764040—21no—
98C23×C137C2×D411yescharacteristic
99C2×C4×C137—21no—
100C2×C4×C137—21no—
101C2×C4×C137—22no—
102C2×C4×C137—81no—
103C2×C4×C137C2×D411yescharacteristic
104C2×C4×C137#4112yes—
105C2×C4×C137#4112yes—
106C2×C4×C137C2411yescharacteristic
107C8×C137—22no—
108C22×C137—21no—
109C22×C137—21no—
110C22×C137#12312yes—
111C22×C137C22×D411yescharacteristic
112C2×D137—22no—
113C2×D137—22no—
114C2×D137—22no—
115C2×D137—42no—
116C2×D137—42no—
117C2×D137—42no—
118C2×D137—84no—
119C4×C137—42no—
120C4×C137—41no—
121C4×C137—41no—
122C4×C137#12312yes—
123C4×C137C2×#4111yescharacteristic
124C4×C137C2×#4111yescharacteristic
125Dic137—22no—
126Dic137—41no—
127Dic137—41no—
128C2×C137—22no—
129C2×C137C2×#12311yescharacteristic
130C2×C137#57611yescharacteristic
131C2×C137#57711yescharacteristic
132D137—42no—
133D137—82no—
134C137#309611yescharacteristic, complemented, Sylow
135#3096—1371nomaximal, Sylow
136#535—1371no—
137#520—1371no—
138#482—1372no—
139#482—1372no—
140#481—1372no—
141#480—1372no—
142#428D13711yescharacteristic, complemented
143#427—1371no—
144#418—1371no—
145C2×#102—1371no—
146#408—1371no—
147C2×#41—1371no—
148C22×D4—1371no—
149C2×M16C2×D13711yescharacteristic
150C22×C8—1371no—
151#127—2741no—
152#124—2741no—
153#121—1372no—
154#121—1372no—
155#120—2742no—
156C4×Q8—2741no—
157C4×D4—2741no—
158C4×D4—2742no—
159#117—1371no—
160#117—1372no—
161#117C2×D13711yescharacteristic
162C2×C4⋊C4—1372no—
163C2×C4⋊C4C2×D13711yescharacteristic
164#107—1371no—
165#107—1371no—
166#107C2×D13712yescomplemented
167#106—1371no—
168#106—1371no—
169#106C2×D13712yescomplemented
170#102—1372no—
171#102—1372no—
172#102—1372no—
173#102—1372no—
174C22⋊C8—1372no—
175C22⋊C8—1372no—
176C24—2741no—
177#41—2742no—
178#41—2742no—
179C2×Q8—1371no—
180C2×D4—2742no—
181C2×D4—2742no—
182C2×D4—2741no—
183C2×D4—2742no—
184C2×D4—2742no—
185C2×D4—1372no—
186C2×D4—1371no—
187C2×D4—1372no—
188C22×C4—21no—
189C22×C4—2742no—
190C22×C4—1372no—
191C22×C4C22×D13711yescharacteristic
192M16—22no—
193C2×C8—2742no—
194C2×C8—1371no—
195C2×C8—1371no—
196C2×C8C22×D13712yes—
197C4⋊C4—21no—
198C4⋊C4—2742no—
199C4⋊C4—1371no—
200C4⋊C4—1371no—
201C4⋊C4C22×D13712yes—
202C4⋊C4C22×D13711yescharacteristic
203C4⋊C4C22×D13711yescharacteristic
204C22⋊C4—21no—
205C22⋊C4—2742no—
206C22⋊C4—2742no—
207C22⋊C4—2742no—
208C22⋊C4—2741no—
209C42—21no—
210C42—2742no—
211C42—2741no—
212C23—2742no—
213C23—5482no—
214C23—10961no—
215C23—10961no—
216C23—1372no—
217C23D4×D13711yescharacteristic
218Q8—2742no—
219D4—2744no—
220D4—2742no—
221D4—2742no—
222D4—2742no—
223D4—5482no—
224D4—10962no—
225D4—10962no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—22no—
229C2×C4—81no—
230C2×C4—2742no—
231C2×C4—2742no—
232C2×C4—2741no—
233C2×C4—2741no—
234C2×C4—10962no—
235C2×C4—1372no—
236C2×C4D4×D13711yescharacteristic
237C2×C4#1455551012yes—
238C2×C4#1455551112yes—
239C2×C4C23×D13711yescharacteristic
240C8—22no—
241C8—2741no—
242C8—2741no—
243C22—21no—
244C22—21no—
245C22—2742no—
246C22—2742no—
247C22—2742no—
248C22—5482no—
249C22—5482no—
250C22—5482no—
251C22—10964no—
252C22#1798407312yes—
253C22C2×D4×D13711yescharacteristic
254C4—42no—
255C4—41no—
256C4—41no—
257C4—2742no—
258C4—5481no—
259C4—5481no—
260C4#1798407312yes—
261C4C2×#1455551011yescharacteristic
262C4C2×#1455551111yescharacteristic
263C2—22no—
264C2—5482no—
265C2—10962no—
266C2C2×#1798407311yescharacteristic
267C2#2257820711yescharacteristic
268C2#2257821211yescharacteristic
subgroupG/H
centerC22C2×D4×D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#428×C137C2
socleC22×C137C22×D4
prime radical#22577972C2
hypercenter#428D137
hypocenterC137#3096
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C22×C4 ◁ #428

Complements

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

NH ≅ G/Ntype
2. #22577884C2semidirect
3. #22577730C2semidirect
4. #22577729C2semidirect
5. #22577723C2semidirect
6. #22577718C2semidirect
7. #22577439C2semidirect
8. #22577409C2semidirect
9. C2×#17984008C2semidirect
10. #22577371C2semidirect
11. #428×C137C2semidirect
24. #17984046C22semidirect
25. C4⋊C4×D137C22semidirect
26. #17984021C22semidirect
27. #17984021C22semidirect
28. #17984019C22semidirect
29. #17984019C22semidirect
34. #17984000C22semidirect
35. #17984000C22semidirect
39. #107×C137C22semidirect
40. #106×C137C22semidirect
134. C137#3096semidirect
142. #428D137semidirect
166. #107C2×D137semidirect
169. #106C2×D137semidirect

Maximal subgroups

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

subgroupindex
#225779722
#225778842
#225777302
#225777292
#225777232
#225777182
#225774392
#225774092
C2×#179840082
#225773712
#428×C1372
#3096137