Finite Group Databasebeta

← #31789638

Subgroups — #31789638

Order 17536.

528 conjugacy classes of subgroups; only the 175 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 175 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×D8×D137C211yesmaximal, characteristic
2#22577875C212yesmaximal, complemented
3#22577516C211yesmaximal, characteristic, complemented
4#22577515C211yesmaximal, characteristic, complemented
5#22577514C211yesmaximal, characteristic, complemented
6C4×#14555495C212yesmaximal, complemented
7#22577483C212yesmaximal, complemented
8#22577438C211yesmaximal, characteristic, complemented
9#22577406C212yesmaximal, complemented
10#22577360C211yesmaximal, characteristic, complemented
11#442×C137C211yesmaximal, characteristic, complemented
12C2×D4×D137C2212yes—
13C2×D1096C2211yescharacteristic
14C2×#14555495C2212yes—
15D8×D137C418yescomplemented
16C2×#14555493C2211yescharacteristic
17C2×C8×D137C2211yescharacteristic
18C4×D548C2212yescomplemented
19D4×Dic137C2212yescomplemented
20#17984050C2212yescomplemented
21C4×C22⋊D137—22no—
22#17984048—22no—
23#17984046C2212yes—
24#17984045C2211yescharacteristic
25C2×#14555487—22no—
26C22⋊C4×D137—22no—
27#17984021C2212yescomplemented
28#17984020C2211yescharacteristic, complemented
29#17984010C2212yescomplemented
30#17984009C2212yescomplemented
31#17984008C2212yescomplemented
32#17983998C2211yescharacteristic, complemented
33C4×#11764040C2211yescharacteristic, complemented
34C8×Dic137C2211yescharacteristic, complemented
35C2×D8×C137C2211yescharacteristic
36C4×D4×C137C2212yescomplemented
37#106×C137C2211yescharacteristic, complemented
38#102×C137C2212yescomplemented
39#97×C137C2211yescharacteristic, complemented
40C23×D137—22no—
41C2×D548C2312yes—
42C2×C22⋊D137—22no—
43D4×D137—24no—
44D4×D137C2×C418yescomplemented
45C22×Dic137—22no—
46C2×C4×D137—22no—
47C2×C4×D137—22no—
48C2×C4×D137C2311yescharacteristic
49D1096C2×C414yescomplemented
50#14555495C2×C418yescomplemented
51#14555493—22no—
52C2×#11764040C2311yescharacteristic
53C2×#11764040C2311yescharacteristic
54C8×D137C2×C414yescomplemented
55C4⋊Dic137C2312yes—
56#14555489D412yes—
57C22⋊Dic137—22no—
58#14555487—22no—
59#14555487—22no—
60#14555487—22no—
61#14555487D412yes—
62C4×Dic137—22no—
63C4×Dic137C2311yescharacteristic
64C2×D4×C137C2312yes—
65C22×C4×C137—22no—
66D8×C137C2×C414yescomplemented
67C2×C8×C137C2311yescharacteristic
68C2×C8×C137C2311yescharacteristic
69C4⋊C4×C137C2312yes—
70C22⋊C4×C137—22no—
71C42×C137C2311yescharacteristic
72C22×D137—22no—
73C22×D137—28no—
74C22×D137—82no—
75C22×D137—82no—
76C22×D137C2×D411yescharacteristic
77D548—22no—
78D548C22×C414yes—
79C22⋊D137—28no—
80C2×Dic137—22no—
81C2×Dic137—82no—
82C2×Dic137C2×D412yes—
83C2×Dic137C2×D411yescharacteristic
84C4×D137—82no—
85C4×D137—82no—
86C4×D137C22×C412yes—
87C4×D137#4112yes—
88#11764040—21no—
89#11764040C22×C412yes—
90C23×C137—22no—
91D4×C137—22no—
92D4×C137C22×C414yes—
93C2×C4×C137—22no—
94C2×C4×C137—82no—
95C2×C4×C137C2×D412yes—
96C2×C4×C137C2411yescharacteristic
97C8×C137—21no—
98C8×C137C22×C412yes—
99C22×C137—24no—
100C22×C137—42no—
101C22×C137—42no—
102C22×C137C22×D411yescharacteristic
103C2×D137—22no—
104C2×D137—24no—
105C2×D137—42no—
106C2×D137—42no—
107C2×D137—88no—
108C2×D137C4×D412yes—
109C4×C137—22no—
110C4×C137—42no—
111C4×C137C23×C411yescharacteristic
112C4×C137C2×#4111yescharacteristic
113Dic137—22no—
114Dic137—42no—
115Dic137C4×D412yes—
116C2×C137—44no—
117C2×C137C2×C4×D411yescharacteristic
118C2×C137#57612yes—
119D137—22no—
120D137—44no—
121C137#294811yescharacteristic, complemented, Sylow
122#2948—1371nomaximal, Sylow
123C22×D8—1371no—
124#518—1372no—
125#442—1371no—
126#442—1371no—
127#442—1371no—
128#442D13711yescharacteristic, complemented
129C4×D8—1372no—
130C4×D8—1372no—
131#427—1371no—
132#418—1372no—
133#405—1371no—
134C22×D4—1372no—
135C2×D8—1372no—
136C2×D8—1378no—
137C2×D8—1371no—
138C2×D8C2×D13711yescharacteristic
139C2×M16—1371no—
140C22×C8—1371no—
141C4×D4—2742no—
142C4×D4—2742no—
143C4×D4—1372no—
144C4×D4—1372no—
145C4×D4—1372no—
146C4×D4C2×D13712yescomplemented
147#117—1372no—
148#117—1371no—
149C2×C22⋊C4—2742no—
150C2×C22⋊C4—2742no—
151#107—1372no—
152#106—1371no—
153#106C2×D13711yescharacteristic, complemented
154#102—1372no—
155#102—1372no—
156#102—1372no—
157#102C2×D13712yescomplemented
158#97—1371no—
159#97C2×D13711yescharacteristic, complemented
160C4×C8—1371no—
161C4×C8—1371no—
162C24—2742no—
163C2×D4—2744no—
164C2×D4—2742no—
165C2×D4—1378no—
166C2×D4—1372no—
167C2×D4C22×D13712yes—
168C22×C4—22no—
169C22×C4—2742no—
170C22×C4—2742no—
171C22×C4—2742no—
172C22×C4—1371no—
173D8—1378no—
174D8—1374no—
175D8C4×D13714yescomplemented
176M16—2742no—
177C2×C8—1371no—
178C2×C8—1371no—
179C2×C8—1374no—
180C2×C8C22×D13711yescharacteristic
181C2×C8C22×D13711yescharacteristic
182C4⋊C4—1372no—
183C4⋊C4—1372no—
184C4⋊C4C22×D13712yes—
185C22⋊C4—22no—
186C22⋊C4—2742no—
187C22⋊C4—2742no—
188C22⋊C4—2742no—
189C22⋊C4—2742no—
190C22⋊C4—1372no—
191C42—2742no—
192C42—1371no—
193C42C22×D13711yescharacteristic
194C23—22no—
195C23—2748no—
196C23—2742no—
197C23—10962no—
198C23—10962no—
199C23—1371no—
200D4—22no—
201D4—2748no—
202D4—2742no—
203D4—1374no—
204D4C2×C4×D13714yes—
205C2×C4—22no—
206C2×C4—82no—
207C2×C4—2742no—
208C2×C4—10962no—
209C2×C4—10962no—
210C2×C4—10962no—
211C2×C4—1372no—
212C2×C4—1371no—
213C2×C4—1372no—
214C2×C4—1372no—
215C2×C4D4×D13712yes—
216C2×C4C23×D13711yescharacteristic
217C8—21no—
218C8—2741no—
219C8—1372no—
220C8C2×C4×D13712yes—
221C22—24no—
222C22—42no—
223C22—42no—
224C22—2742no—
225C22—2744no—
226C22—5482no—
227C22—5482no—
228C22—10968no—
229C22—1372no—
230C22C2×D4×D13711yescharacteristic
231C4—22no—
232C4—42no—
233C4—2742no—
234C4—5482no—
235C4—1372no—
236C4C22×C4×D13711yescharacteristic
237C4#41×D13711yescharacteristic
238C2—44no—
239C2—2742no—
240C2—5484no—
241C2C4×D4×D13711yescharacteristic
242C2#2257820712yes—
subgroupG/H
centerC22C2×D4×D137
commutatorC4×C137C23×C4
FrattiniC2×C4C23×D137
Fitting#442×C137C2
socleC22×C137C22×D4
prime radicalC2×D8×D137C2
cosocleC2×C4×C137C24
hypercenter#442D137
hypocenterC137#2948
LCSG ▷ C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C42 ◁ #442

Complements

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

NH ≅ G/Ntype
2. #22577875C2semidirect
3. #22577516C2semidirect
4. #22577515C2semidirect
5. #22577514C2semidirect
6. C4×#14555495C2semidirect
7. #22577483C2semidirect
8. #22577438C2semidirect
9. #22577406C2semidirect
10. #22577360C2semidirect
11. #442×C137C2semidirect
15. D8×D137C4semidirect
18. C4×D548C22semidirect
19. D4×Dic137C22semidirect
20. #17984050C22semidirect
27. #17984021C22semidirect
28. #17984020C22semidirect
29. #17984010C22semidirect
30. #17984009C22semidirect
31. #17984008C22semidirect
32. #17983998C22semidirect
33. C4×#11764040C22semidirect
34. C8×Dic137C22semidirect
36. C4×D4×C137C22semidirect
37. #106×C137C22semidirect
38. #102×C137C22semidirect
39. #97×C137C22semidirect
44. D4×D137C2×C4semidirect
49. D1096C2×C4semidirect
50. #14555495C2×C4semidirect
54. C8×D137C2×C4semidirect
66. D8×C137C2×C4semidirect
121. C137#2948semidirect
128. #442D137semidirect
146. C4×D4C2×D137semidirect
153. #106C2×D137semidirect
157. #102C2×D137semidirect
159. #97C2×D137semidirect
175. D8C4×D137semidirect

Maximal subgroups

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

subgroupindex
C2×D8×D1372
#225778752
#225775162
#225775152
#225775142
C4×#145554952
#225774832
#225774382
#225774062
#225773602
#442×C1372
#2948137