Finite Group Databasebeta

← #31790743

Subgroups — #31790743

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#22577970C211yesmaximal, characteristic
2#22577884C211yesmaximal, characteristic, complemented
3#22577726C212yesmaximal, complemented
4#22577725C212yesmaximal, complemented
5#22577720C212yesmaximal, complemented
6#22577715C212yesmaximal, complemented
7#22577443C211yesmaximal, characteristic, complemented
8#22577441C211yesmaximal, characteristic, complemented
9#22577408C211yesmaximal, characteristic, complemented
10#22577369C211yesmaximal, characteristic, complemented
11C2×#102×C137C211yesmaximal, characteristic, complemented
12C2×#14555510C2211yescharacteristic
13C2×#14555494C2211yescharacteristic
14#17984084—21no—
15#17984075—21no—
16#17984059C2212yes—
17#17984059C2212yes—
18#17984057—22no—
19#17984053—21no—
20#17984050—21no—
21C4×C22⋊D137—22no—
22#17984047C2211yescharacteristic
23#17984047C2211yescharacteristic
24#17984046C2212yescomplemented
25C4⋊C4×D137C2212yescomplemented
26#17984022C2211yescharacteristic, complemented
27#17984022C2211yescharacteristic, complemented
28#17984021C2212yescomplemented
29#17984020C2211yescharacteristic, complemented
30#17984020C2211yescharacteristic, complemented
31#17984019C2212yescomplemented
32#17984009C2212yes—
33#17984009C2212yes—
34#17984000C2212yescomplemented
35#17984000C2212yescomplemented
36C22×D4×C137C2211yescharacteristic
37C22×C8×C137C2211yescharacteristic
38C2×C4⋊C4×C137C2211yescharacteristic
39#102×C137C2212yes—
40#102×C137C2212yes—
41#14555510—22no—
42#14555510—22no—
43C2×Dic274D411yescharacteristic
44C2×D548D411yescharacteristic
45C2×C22⋊D137—22no—
46C2×C22⋊D137—22no—
47C2×C22⋊D137D412yes—
48C2×C4×D137—22no—
49C2×C4×D137C2312yes—
50#14555494—22no—
51C2×#11764040C2312yes—
52C4⋊Dic137C2311yescharacteristic
53C4⋊Dic137C2311yescharacteristic
54C4⋊Dic137C2311yescharacteristic
55C4⋊Dic137C2311yescharacteristic
56#14555489—22no—
57C22⋊Dic137—22no—
58C22⋊Dic137—22no—
59C22⋊Dic137—21no—
60C22⋊Dic137—21no—
61#14555487—22no—
62C4×Dic137—22no—
63C4×Dic137—21no—
64C4×Dic137—21no—
65C24×C137—21no—
66C2×D4×C137—21no—
67C2×D4×C137—22no—
68C2×D4×C137—22no—
69C2×D4×C137C2312yes—
70C22×C4×C137—21no—
71C22×C4×C137C2311yescharacteristic
72C2×C8×C137—22no—
73C2×C8×C137C2311yescharacteristic
74C2×C8×C137C2311yescharacteristic
75C4⋊C4×C137—21no—
76C4⋊C4×C137C2312yes—
77C22×D137C2×D412yes—
78Dic274—22no—
79D548—22no—
80C22⋊D137—24no—
81C22⋊D137—82no—
82C22⋊D137—82no—
83C2×Dic137—21no—
84C2×Dic137—21no—
85C2×Dic137—21no—
86C2×Dic137—21no—
87C2×Dic137C2×D412yes—
88C4×D137—22no—
89C4×D137—22no—
90C4×D137—82no—
91#11764040—22no—
92C23×C137—22no—
93C23×C137—42no—
94C23×C137—81no—
95C23×C137—81no—
96C23×C137C2×D411yescharacteristic
97D4×C137—22no—
98D4×C137—22no—
99D4×C137—42no—
100C2×C4×C137—22no—
101C2×C4×C137—81no—
102C2×C4×C137C2×D411yescharacteristic
103C2×C4×C137#4112yes—
104C2×C4×C137#4112yes—
105C2×C4×C137C2411yescharacteristic
106C8×C137—21no—
107C8×C137—21no—
108C22×C137—21no—
109C22×C137—21no—
110C22×C137—42no—
111C22×C137—42no—
112C22×C137—42no—
113C22×C137—84no—
114C22×C137#12312yes—
115C22×C137C22×D411yescharacteristic
116C2×D137—22no—
117C2×D137—22no—
118C2×D137—22no—
119C4×C137—42no—
120C4×C137#12312yes—
121C4×C137C2×#4111yescharacteristic
122C4×C137C2×#4111yescharacteristic
123Dic137—22no—
124Dic137—41no—
125Dic137—41no—
126Dic137—41no—
127Dic137—41no—
128C2×C137—22no—
129C2×C137—82no—
130C2×C137C2×#12311yescharacteristic
131C2×C137#57611yescharacteristic
132C2×C137#57711yescharacteristic
133D137—42no—
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#428—1371no—
143#427—1371no—
144#418—1371no—
145C2×#102D13711yescharacteristic, complemented
146#408—1371no—
147C2×#41—1371no—
148C22×D4C2×D13711yescharacteristic
149C2×M16—1371no—
150C22×C8C2×D13711yescharacteristic
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—1371no—
161#117—1372no—
162C2×C4⋊C4—1372no—
163C2×C4⋊C4C2×D13711yescharacteristic
164#107—1371no—
165#107—1371no—
166#107—1372no—
167#106—1371no—
168#106—1371no—
169#106—1372no—
170#102—1372no—
171#102—1372no—
172#102C2×D13712yes—
173#102C2×D13712yes—
174C22⋊C8—1372no—
175C22⋊C8—1372no—
176C24—21no—
177#41—2742no—
178#41—2742no—
179C2×Q8—1371no—
180C2×D4—21no—
181C2×D4—22no—
182C2×D4—22no—
183C2×D4—2742no—
184C2×D4—2742no—
185C2×D4—1372no—
186C2×D4—1371no—
187C2×D4C22×D13712yes—
188C22×C4—21no—
189C22×C4—2742no—
190C22×C4—1372no—
191C22×C4C22×D13711yescharacteristic
192M16—2742no—
193C2×C8—22no—
194C2×C8—1372no—
195C2×C8C22×D13711yescharacteristic
196C2×C8C22×D13711yescharacteristic
197C4⋊C4—21no—
198C4⋊C4—2742no—
199C4⋊C4—1371no—
200C4⋊C4—1371no—
201C4⋊C4—1371no—
202C4⋊C4—1371no—
203C4⋊C4C22×D13712yes—
204C22⋊C4—2742no—
205C22⋊C4—2741no—
206C22⋊C4—2741no—
207C22⋊C4—2742no—
208C22⋊C4—2742no—
209C42—2742no—
210C42—2741no—
211C42—2741no—
212C23—22no—
213C23—42no—
214C23—81no—
215C23—81no—
216C23—1372no—
217C23D4×D13711yescharacteristic
218Q8—2742no—
219D4—22no—
220D4—22no—
221D4—42no—
222D4—2744no—
223D4—2742no—
224D4—10962no—
225D4—10962no—
226C2×C4—22no—
227C2×C4—81no—
228C2×C4—2742no—
229C2×C4—2742no—
230C2×C4—2741no—
231C2×C4—2741no—
232C2×C4—2741no—
233C2×C4—2741no—
234C2×C4—10962no—
235C2×C4—1372no—
236C2×C4D4×D13711yescharacteristic
237C2×C4#1455550912yes—
238C2×C4#1455551012yes—
239C2×C4C23×D13711yescharacteristic
240C8—21no—
241C8—21no—
242C8—2742no—
243C22—21no—
244C22—21no—
245C22—42no—
246C22—42no—
247C22—42no—
248C22—84no—
249C22—2742no—
250C22—2742no—
251C22—2742no—
252C22#1798407012yes—
253C22C2×D4×D13711yescharacteristic
254C4—42no—
255C4—2742no—
256C4—5481no—
257C4—5481no—
258C4—5481no—
259C4—5481no—
260C4#1798407012yes—
261C4C2×#1455550911yescharacteristic
262C4C2×#1455551011yescharacteristic
263C2—22no—
264C2—82no—
265C2—5482no—
266C2C2×#1798407011yescharacteristic
267C2#2257820411yescharacteristic
268C2#2257820911yescharacteristic
subgroupG/H
centerC22C2×D4×D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
FittingC2×#102×C137C2
socleC22×C137C22×D4
prime radical#22577970C2
hypercenterC2×#102D137
hypocenterC137#3096
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#102

Complements

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

NH ≅ G/Ntype
2. #22577884C2semidirect
3. #22577726C2semidirect
4. #22577725C2semidirect
5. #22577720C2semidirect
6. #22577715C2semidirect
7. #22577443C2semidirect
8. #22577441C2semidirect
9. #22577408C2semidirect
10. #22577369C2semidirect
11. C2×#102×C137C2semidirect
24. #17984046C22semidirect
25. C4⋊C4×D137C22semidirect
26. #17984022C22semidirect
27. #17984022C22semidirect
28. #17984021C22semidirect
29. #17984020C22semidirect
30. #17984020C22semidirect
31. #17984019C22semidirect
34. #17984000C22semidirect
35. #17984000C22semidirect
134. C137#3096semidirect
145. C2×#102D137semidirect

Maximal subgroups

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

subgroupindex
#225779702
#225778842
#225777262
#225777252
#225777202
#225777152
#225774432
#225774412
#225774082
#225773692
C2×#102×C1372
#3096137