Finite Group Databasebeta

← #31792585

Subgroups — #31792585

Order 17536.

476 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 476 conjugacy classes of subgroups — too many to draw. Only its 117 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#22578207C212yesmaximal, complemented
2#22578206C212yesmaximal, complemented
3#22577779C211yesmaximal, characteristic, complemented
4#22577672C212yesmaximal, complemented
5#22577671C212yesmaximal, complemented
6#22577572C212yesmaximal, complemented
7#22577570C211yesmaximal, characteristic, complemented
8#22577517C212yesmaximal, complemented
9#453×C137C211yesmaximal, characteristic, complemented
10#17984134C2212yescomplemented
11#17984133C2211yescharacteristic, complemented
12#17984116C2212yescomplemented
13#17984115—22no—
14#17984114—22no—
15#17984114C2212yescomplemented
16#17984113—22no—
17#17984113C2212yescomplemented
18#17984112—22no—
19#17984109—22no—
20#17984108—22no—
21#17984108C2212yes—
22C2×#14555495—22no—
23C2×#14555495C2212yescomplemented
24C2×#14555495C2212yescomplemented
25D8×D137—22no—
26#17984096C2212yescomplemented
27#17984095C2212yescomplemented
28#17984085C2211yescharacteristic, complemented
29#17984023C2212yescomplemented
30#17984016C2212yescomplemented
31#17984005C2211yescharacteristic, complemented
32#17984004C2212yescomplemented
33C4×#11764040C2211yescharacteristic, complemented
34H(5,𝔽2)×C137C2211yescharacteristic, complemented
35AGL(1,ℤ/8)×C137C2212yescomplemented
36#127×C137C2211yescharacteristic, complemented
37#104×C137C2212yescomplemented
38#100×C137C2211yescharacteristic, complemented
39#14555511—22no—
40#14555510—22no—
41#14555510C2312yes—
42#14555509—22no—
43C2×D548—22no—
44C2×D548—41no—
45C2×D548C2312yes—
46C2×C22⋊D137—41no—
47D4×D137—22no—
48D4×D137—22no—
49D4×D137—42no—
50#14555500D412yescomplemented
51#14555499—22no—
52#14555498—22no—
53#14555498D412yescomplemented
54#14555497—22no—
55#14555497D412yescomplemented
56D1096—22no—
57#14555495—22no—
58#14555495—22no—
59#14555495—22no—
60#14555495—22no—
61#14555495—22no—
62#14555495—22no—
63#14555495D412yescomplemented
64#14555494—22no—
65#14555494C2312yes—
66#14555493—22no—
67C2×#11764040—22no—
68C2×#11764040C2312yes—
69C8×D137—22no—
70#41×C137—22no—
71#41×C137C2312yes—
72C2×D4×C137—22no—
73C2×D4×C137—41no—
74C2×D4×C137—41no—
75C2×D4×C137C2311yescharacteristic
76C2×D4×C137C2311yescharacteristic
77SD16×C137—22no—
78D8×C137—22no—
79M16×C137C2312yes—
80C42×C137C2311yescharacteristic
81C22×D137—22no—
82C22×D137—41no—
83C22×D137—42no—
84Dic274C2×D412yes—
85D548—22no—
86D548—22no—
87D548—42no—
88D548—82no—
89D548C2×D412yes—
90C22⋊D137—22no—
91C22⋊D137—42no—
92C22⋊D137—42no—
93C2×Dic137—41no—
94C4×D137—22no—
95C4×D137—42no—
96#11764040—22no—
97#11764040C2×D412yes—
98#11764040C2×D412yes—
99C23×C137—21no—
100C23×C137—21no—
101C23×C137—41no—
102Q8×C137C2×D412yes—
103D4×C137—22no—
104D4×C137—22no—
105D4×C137—22no—
106D4×C137—42no—
107D4×C137—82no—
108D4×C137C2×D412yes—
109C2×C4×C137—21no—
110C2×C4×C137—22no—
111C2×C4×C137—41no—
112C2×C4×C137C2411yescharacteristic
113C8×C137—22no—
114C22×C137—22no—
115C22×C137—41no—
116C22×C137—41no—
117C22×C137—81no—
118C22×C137—81no—
119C22×C137—161no—
120C22×C137H(5,𝔽2)11yescharacteristic
121C2×D137—22no—
122C2×D137—42no—
123C2×D137—82no—
124C2×D137—161no—
125C2×D137—162no—
126C4×C137—22no—
127C4×C137—22no—
128C4×C137C22×D412yes—
129Dic137—22no—
130C2×C137—21no—
131C2×C137—42no—
132C2×C137—81no—
133C2×C137—81no—
134C2×C137D4211yescharacteristic
135D137—42no—
136D137—82no—
137C137#329211yescharacteristic, complemented, Sylow
138#3292—1371nomaximal, Sylow
139#576—1372no—
140#576—1372no—
141#493—1371no—
142#472—1372no—
143#472—1372no—
144#453—1371no—
145#453—1372no—
146#453D13711yescharacteristic, complemented
147#443—1372no—
148H(5,𝔽2)—1372no—
149H(5,𝔽2)—1371no—
150H(5,𝔽2)C2×D13711yescharacteristic, complemented
151AGL(1,ℤ/8)—2742no—
152AGL(1,ℤ/8)—2742no—
153AGL(1,ℤ/8)—2742no—
154AGL(1,ℤ/8)—2742no—
155AGL(1,ℤ/8)—1372no—
156AGL(1,ℤ/8)—1372no—
157AGL(1,ℤ/8)—1372no—
158AGL(1,ℤ/8)C2×D13712yescomplemented
159#135—2742no—
160#135—2742no—
161#135—1372no—
162C2×D8—2742no—
163C2×D8—2742no—
164C2×D8—1372no—
165C2×D8—1372no—
166#131—1372no—
167#131—1372no—
168#127—1371no—
169#127C2×D13711yescharacteristic, complemented
170#108—1372no—
171#104—1372no—
172#104C2×D13712yescomplemented
173#100—1371no—
174#100—1372no—
175#100C2×D13711yescharacteristic, complemented
176C4×C8—1371no—
177#41—22no—
178#41—2742no—
179#41—2742no—
180#41—2742no—
181#41—1372no—
182#41C22×D13712yes—
183C2×D4—22no—
184C2×D4—41no—
185C2×D4—41no—
186C2×D4—2742no—
187C2×D4—2742no—
188C2×D4—2742no—
189C2×D4—5481no—
190C2×D4—5482no—
191C2×D4—5481no—
192C2×D4—1372no—
193C2×D4C22×D13711yescharacteristic
194C2×D4C22×D13711yescharacteristic
195Q16—1372no—
196SD16—22no—
197SD16—2742no—
198SD16—2742no—
199SD16—2742no—
200SD16—1372no—
201SD16—1372no—
202D8—22no—
203D8—2742no—
204D8—2742no—
205D8—2742no—
206D8—2742no—
207D8—2742no—
208D8—2742no—
209D8—2742no—
210D8—1372no—
211M16—2742no—
212M16—2742no—
213M16—1372no—
214M16C22×D13712yes—
215C2×C8—2742no—
216C2×C8—2742no—
217C2×C8—1372no—
218C42C22×D13711yescharacteristic
219C23—21no—
220C23—21no—
221C23—41no—
222C23—2742no—
223C23—5482no—
224C23—5481no—
225Q8—1372no—
226Q8D4×D13712yes—
227D4—22no—
228D4—22no—
229D4—22no—
230D4—42no—
231D4—82no—
232D4—2742no—
233D4—2742no—
234D4—2742no—
235D4—5482no—
236D4—5482no—
237D4—5482no—
238D4—10962no—
239D4—1372no—
240D4D4×D13712yes—
241C2×C4—21no—
242C2×C4—22no—
243C2×C4—41no—
244C2×C4—2742no—
245C2×C4—5482no—
246C2×C4—5481no—
247C2×C4C23×D13711yescharacteristic
248C8—22no—
249C8—2742no—
250C8—1372no—
251C8—1372no—
252C22—22no—
253C22—41no—
254C22—41no—
255C22—81no—
256C22—81no—
257C22—161no—
258C22—2742no—
259C22—5482no—
260C22—10962no—
261C22—21922no—
262C22—21921no—
263C22#1798413311yescharacteristic
264C4—22no—
265C4—22no—
266C4—2742no—
267C4C2×D4×D13712yes—
268C2—21no—
269C2—42no—
270C2—81no—
271C2—81no—
272C2—5482no—
273C2—10962no—
274C2#2257803511yescharacteristic
subgroupG/H
centerC2#22578035
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#453×C137C2
socleC2×C137D42
hypercenter#453D137
hypocenterC137#3292
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C2 ◁ C2×C4 ◁ #453

Complements

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

NH ≅ G/Ntype
1. #22578207C2semidirect
2. #22578206C2semidirect
3. #22577779C2semidirect
4. #22577672C2semidirect
5. #22577671C2semidirect
6. #22577572C2semidirect
7. #22577570C2semidirect
8. #22577517C2semidirect
9. #453×C137C2semidirect
10. #17984134C22semidirect
11. #17984133C22semidirect
12. #17984116C22semidirect
15. #17984114C22semidirect
17. #17984113C22semidirect
23. C2×#14555495C22semidirect
24. C2×#14555495C22semidirect
26. #17984096C22semidirect
27. #17984095C22semidirect
28. #17984085C22semidirect
29. #17984023C22semidirect
30. #17984016C22semidirect
31. #17984005C22semidirect
32. #17984004C22semidirect
33. C4×#11764040C22semidirect
34. H(5,𝔽2)×C137C22semidirect
35. AGL(1,ℤ/8)×C137C22semidirect
36. #127×C137C22semidirect
37. #104×C137C22semidirect
38. #100×C137C22semidirect
50. #14555500D4semidirect
53. #14555498D4semidirect
55. #14555497D4semidirect
63. #14555495D4semidirect
137. C137#3292semidirect
146. #453D137semidirect
150. H(5,𝔽2)C2×D137semidirect
158. AGL(1,ℤ/8)C2×D137semidirect
169. #127C2×D137semidirect
172. #104C2×D137semidirect
175. #100C2×D137semidirect

Maximal subgroups

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

subgroupindex
#225782072
#225782062
#225777792
#225776722
#225776712
#225775722
#225775702
#225775172
#453×C1372
#3292137