Finite Group Databasebeta

← #14540394

Subgroups — #14540394

Order 2176.

520 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 520 conjugacy classes of subgroups — too many to draw. Only its 129 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×#10587597C212yesmaximal, complemented
2#11763454C212yesmaximal, complemented
3#11763193C211yesmaximal, characteristic, complemented
4#11763056C212yesmaximal, complemented
5#11763024C212yesmaximal, complemented
6#11762972C212yesmaximal, complemented
7#11762944C212yesmaximal, complemented
8#11762888C211yesmaximal, characteristic, complemented
9C4×D8×C17C211yesmaximal, characteristic, complemented
10C2×#63254C2212yes—
11C2×D4×D17C2212yescomplemented
12C2×D8×C17C2211yescharacteristic
13C4×D4×C17C2212yescomplemented
14C2×#63229C2212yescomplemented
15C2×#63228C2212yes—
16#10587616—22no—
17C2×#63227—22no—
18#10587597—28no—
19C2×Dic68C2211yescharacteristic, complemented
20C2×D136C2211yescharacteristic
21C2×#63219C2212yescomplemented
22C2×#63218C2212yescomplemented
23#10587583C2212yescomplemented
24#10587581C2212yes—
25#10587572—22no—
26#10587568—22no—
27#10587561C2212yescomplemented
28C4×D68C2211yescharacteristic
29C17×#107C2211yescharacteristic, complemented
30C17×#102C2212yescomplemented
31C4×C8×C17C2211yescharacteristic, complemented
32#10587491C2212yescomplemented
33#10587481C2212yescomplemented
34#10587480C2212yescomplemented
35C4⋊#3622C2211yescharacteristic, complemented
36C23×D17—22no—
37C2×D4×C17C2312yes—
38C22×C4×C17—22no—
39C2×C22⋊D17—22no—
40C2×C22⋊D17—22no—
41C22×Dic17—22no—
42#63254—24no—
43#63254—42no—
44D4×D17—24no—
45D4×D17—42no—
46C2×D68—22no—
47C2×D68D411yescharacteristic
48C2×D68C2312yes—
49C2×C4×D17C2312yes—
50C2×Dic34C2312yes—
51D8×C17D414yescomplemented
52C2×C8×C17C2311yescharacteristic
53C2×C8×C17C2311yescharacteristic
54C4⋊C4×C17C2312yes—
55C22⋊C4×C17—22no—
56C42×C17C2311yescharacteristic
57#63229—24no—
58#63228—24no—
59#63227—22no—
60#63227—22no—
61#63227—22no—
62#63227—22no—
63#63227D412yes—
64C4⋊Dic17—22no—
65C4⋊Dic17D411yescharacteristic
66C2×#3622C2312yes—
67Dic68—41no—
68D136—41no—
69#63219—24no—
70#63218—24no—
71C23×C17—22no—
72C22×D17—22no—
73C22×D17—44no—
74C22×D17—44no—
75C22×D17—82no—
76C22×D17C2×D412yes—
77D4×C17—22no—
78D4×C17C2×D414yes—
79C2×C4×C17—22no—
80C2×C4×C17—82no—
81C2×C4×C17C2×D411yescharacteristic
82C2×C4×C17C2×D411yescharacteristic
83C2×C4×C17C2411yescharacteristic
84C22⋊D17—44no—
85C22⋊D17—44no—
86C2×Dic17—22no—
87C2×Dic17—44no—
88C2×Dic17C2×D412yes—
89D68—22no—
90D68—22no—
91D68—41no—
92D68—82no—
93C4×D17—22no—
94C4×D17—22no—
95Dic34—22no—
96Dic34—22no—
97C8×C17—21no—
98C8×C17C2×D412yes—
99#3622—22no—
100#3273—171nomaximal, Sylow
101C22×C17—24no—
102C22×C17—42no—
103C22×C17—42no—
104C22×C17C22×D411yescharacteristic
105C2×D17—22no—
106C2×D17—22no—
107C2×D17—22no—
108C2×D17—42no—
109C2×D17—42no—
110C2×D17—42no—
111C2×D17—88no—
112C4×C17—21no—
113C4×C17—21no—
114C4×C17—42no—
115C4×C17C22×D411yescharacteristic
116C4×C17H(5,𝔽2)11yescharacteristic
117Dic17—22no—
118Dic17—42no—
119C2×AGL(1,ℤ/8)—172no—
120#546—172no—
121#495—171no—
122#468—172no—
123#463—172no—
124#452—172no—
125#447—172no—
126C4×D8D1711yescharacteristic, complemented
127#436—171no—
128C2×C17—44no—
129C2×C17D4211yescharacteristic
130C2×C17#57511yescharacteristic
131C2×C17#57611yescharacteristic
132D17—42no—
133D17—82no—
134C2×#41—172no—
135C22×D4—172no—
136AGL(1,ℤ/8)—348no—
137C2×Q16—171no—
138C2×SD16—172no—
139C2×SD16—172no—
140C2×D8—172no—
141C2×D8—171no—
142C2×D8C2×D1711yescharacteristic
143C2×M16—172no—
144#124—172no—
145#123—342no—
146C22⋊Q8—172no—
147#121—172no—
148#121—342no—
149#120—342no—
150C4×D4—171no—
151C4×D4C2×D1712yescomplemented
152C2×C22⋊C4—342no—
153#107C2×D1711yescharacteristic, complemented
154C4⋊C8—171no—
155#103—172no—
156#102—172no—
157#102C2×D1712yescomplemented
158C22⋊C8—172no—
159C4×C8C2×D1711yescharacteristic, complemented
160C17#327311yescharacteristic, complemented, Sylow
161C24—342no—
162#41—344no—
163#41—682no—
164C2×Q8—172no—
165C2×D4—171no—
166C2×D4—172no—
167C2×D4—342no—
168C2×D4—342no—
169C2×D4—344no—
170C2×D4—342no—
171C2×D4—682no—
172C2×D4C22×D1712yes—
173C22×C4—22no—
174C22×C4—172no—
175C22×C4—342no—
176Q16—681no—
177SD16—344no—
178SD16—344no—
179D8—344no—
180D8—681no—
181D8D6814yescomplemented
182M16—344no—
183C2×C8—172no—
184C2×C8C22×D1711yescharacteristic
185C2×C8C22×D1711yescharacteristic
186C4⋊C4—171no—
187C4⋊C4—342no—
188C4⋊C4C22×D1712yes—
189C22⋊C4—22no—
190C22⋊C4—172no—
191C22⋊C4—342no—
192C22⋊C4—342no—
193C22⋊C4—342no—
194C22⋊C4—342no—
195C42C22×D1711yescharacteristic
196C23—22no—
197C23—172no—
198C23—342no—
199C23—684no—
200C23—684no—
201C23—1362no—
202Q8—342no—
203Q8—342no—
204D4—22no—
205D4—342no—
206D4—342no—
207D4—681no—
208D4—684no—
209D4—684no—
210D4—1362no—
211D4C2×D6814yes—
212C2×C4—22no—
213C2×C4—82no—
214C2×C4—172no—
215C2×C4—342no—
216C2×C4—342no—
217C2×C4—342no—
218C2×C4—684no—
219C2×C4D4×D1711yescharacteristic
220C2×C4D4×D1711yescharacteristic
221C2×C4C23×D1711yescharacteristic
222C8—21no—
223C8—342no—
224C8C2×D6812yes—
225C22—24no—
226C22—42no—
227C22—42no—
228C22—342no—
229C22—342no—
230C22—342no—
231C22—682no—
232C22—682no—
233C22—682no—
234C22—1368no—
235C22C2×D4×D1711yescharacteristic
236C4—21no—
237C4—21no—
238C4—42no—
239C4—342no—
240C4—682no—
241C4C22×D6811yescharacteristic
242C4#1058769611yescharacteristic
243C2—44no—
244C2—682no—
245C2—1362no—
246C2D4×D6811yescharacteristic
247C2#1176360611yescharacteristic
248C2#1176361011yescharacteristic
subgroupG/H
centerC22C2×D4×D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC4×D8×C17C2
socleC22×C17C22×D4
hypercenterC4×D8D17
hypocenterC17#3273
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C42 ◁ C4×D8

Complements

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

NH ≅ G/Ntype
1. C2×#10587597C2semidirect
2. #11763454C2semidirect
3. #11763193C2semidirect
4. #11763056C2semidirect
5. #11763024C2semidirect
6. #11762972C2semidirect
7. #11762944C2semidirect
8. #11762888C2semidirect
9. C4×D8×C17C2semidirect
11. C2×D4×D17C22semidirect
13. C4×D4×C17C22semidirect
14. C2×#63229C22semidirect
19. C2×Dic68C22semidirect
21. C2×#63219C22semidirect
22. C2×#63218C22semidirect
23. #10587583C22semidirect
27. #10587561C22semidirect
29. C17×#107C22semidirect
30. C17×#102C22semidirect
31. C4×C8×C17C22semidirect
32. #10587491C22semidirect
33. #10587481C22semidirect
34. #10587480C22semidirect
35. C4⋊#3622C22semidirect
51. D8×C17D4semidirect
126. C4×D8D17semidirect
151. C4×D4C2×D17semidirect
153. #107C2×D17semidirect
157. #102C2×D17semidirect
159. C4×C8C2×D17semidirect
160. C17#3273semidirect
181. D8D68semidirect

Maximal subgroups

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

subgroupindex
C2×#105875972
#117634542
#117631932
#117630562
#117630242
#117629722
#117629442
#117628882
C4×D8×C172
#327317