Finite Group Databasebeta

← #14540577

Subgroups — #14540577

Order 2176.

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

normal subgroups only

This group has 460 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#11763615C214yesmaximal, complemented
2#11763069C214yesmaximal, complemented
3#11762991C212yesmaximal, complemented
4#11762976C212yesmaximal, complemented
5#11762927C212yesmaximal, complemented
6C17×#492C211yesmaximal, characteristic, complemented
7#10587694C2212yes—
8#10587691C2212yescomplemented
9C2×SD16×C17C2214yescomplemented
10C17×C4⋊Q8C2211yescharacteristic, complemented
11C17×#127C2211yescharacteristic
12#10587628C2214yescomplemented
13#10587618C2214yes—
14#10587602—24no—
15#10587601—24no—
16#10587600—24no—
17SD16×D17—24no—
18#10587590C2212yes—
19#10587587C2214yescomplemented
20C4×C8×C17C2211yescharacteristic, complemented
21#10587507C2212yescomplemented
22#10587504C2212yescomplemented
23#10587490C2212yescomplemented
24#10587476C2214yescomplemented
25C2×Q8×C17C2312yes—
26C2×D4×C17—41no—
27C2×D4×C17C2312yes—
28C2×C22⋊D17—42no—
29#63256—24no—
30Q8×D17—24no—
31#63254—24no—
32D4×D17—24no—
33#63252—42no—
34#63252C2314yes—
35SD16×C17—24no—
36SD16×C17—24no—
37C2×C8×C17C2312yes—
38C4⋊C4×C17—41no—
39C42×C17C2311yescharacteristic
40#63231—24no—
41#63230—24no—
42#63229—24no—
43#63228—24no—
44#63223C2314yes—
45Dic68D412yescomplemented
46D136D412yescomplemented
47#63219D414yescomplemented
48#63218—24no—
49C8×D17—24no—
50C23×C17—22no—
51C22×D17—42no—
52Q8×C17—24no—
53D4×C17—24no—
54D4×C17—82no—
55C2×C4×C17—21no—
56C2×C4×C17—22no—
57C2×C4×C17C2411yescharacteristic
58C22⋊D17—24no—
59C22⋊D17—44no—
60C2×Dic17—42no—
61D68—42no—
62D68C2×D414yes—
63C4×D17—24no—
64C4×D17—44no—
65Dic34—42no—
66Dic34C2×D414yes—
67C8×C17C2×D412yes—
68C8×C17C2×D412yes—
69#3622—24no—
70#3291—171nomaximal, Sylow
71C22×C17—42no—
72C22×C17—82no—
73C22×C17H(5,𝔽2)11yescharacteristic
74C2×D17—24no—
75C2×D17—162no—
76C4×C17—22no—
77C4×C17—42no—
78C4×C17C22×D412yes—
79Dic17—24no—
80#577—174no—
81#492D1711yescharacteristic, complemented
82#471—174no—
83#456—172no—
84#453—172no—
85#443—172no—
86C2×C17—21no—
87C2×C17—82no—
88C2×C17D4211yescharacteristic
89D17—44no—
902−1+4—172no—
91H(5,𝔽2)—172no—
92#137—174no—
93#137—344no—
94AGL(1,ℤ/8)—174no—
95AGL(1,ℤ/8)—344no—
96#135—172no—
97#135—344no—
98C2×SD16—344no—
99C2×SD16C2×D1714yescomplemented
100#131—174no—
101C4⋊Q8C2×D1711yescharacteristic, complemented
102#127C2×D1711yescharacteristic
103#108—172no—
104#104—174no—
105#101—172no—
106#100—172no—
107C4×C8C2×D1711yescharacteristic, complemented
108C17#329111yescharacteristic, complemented, Sylow
109#41—174no—
110#41—344no—
111#41—344no—
112#41—682no—
113C2×Q8—344no—
114C2×Q8C22×D1712yes—
115C2×D4—41no—
116C2×D4—344no—
117C2×D4—682no—
118C2×D4C22×D1712yes—
119Q16—172no—
120Q16—344no—
121SD16—24no—
122SD16—24no—
123SD16—174no—
124SD16—344no—
125SD16—344no—
126D8—172no—
127D8—344no—
128M16—174no—
129M16—344no—
130C2×C8—344no—
131C2×C8C22×D1712yes—
132C4⋊C4—41no—
133C42C22×D1711yescharacteristic
134C23—22no—
135C23—682no—
136Q8—24no—
137Q8—174no—
138Q8—682no—
139D4—24no—
140D4—82no—
141D4—174no—
142D4—344no—
143D4—682no—
144D4—684no—
145C2×C4—21no—
146C2×C4—22no—
147C2×C4—344no—
148C2×C4—684no—
149C2×C4—682no—
150C2×C4C23×D1711yescharacteristic
151C8—344no—
152C8D4×D1712yes—
153C8D4×D1712yes—
154C22—42no—
155C22—82no—
156C22—2722no—
157C22—344no—
158C22#1058769111yescharacteristic
159C4—22no—
160C4—42no—
161C4—344no—
162C4C2×D4×D1712yes—
163C2—21no—
164C2—82no—
165C2—684no—
166C2#1176344411yescharacteristic
subgroupG/H
centerC2#11763444
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC17×#492C2
socleC2×C17D42
hypercenter#492D17
hypocenterC17#3291
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C2 ◁ C2×C4 ◁ #492

Complements

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

NH ≅ G/Ntype
1. #11763615C2semidirect
2. #11763069C2semidirect
3. #11762991C2semidirect
4. #11762976C2semidirect
5. #11762927C2semidirect
6. C17×#492C2semidirect
8. #10587691C22semidirect
9. C2×SD16×C17C22semidirect
10. C17×C4⋊Q8C22semidirect
12. #10587628C22semidirect
19. #10587587C22semidirect
20. C4×C8×C17C22semidirect
21. #10587507C22semidirect
22. #10587504C22semidirect
23. #10587490C22semidirect
24. #10587476C22semidirect
45. Dic68D4semidirect
46. D136D4semidirect
47. #63219D4semidirect
81. #492D17semidirect
99. C2×SD16C2×D17semidirect
101. C4⋊Q8C2×D17semidirect
107. C4×C8C2×D17semidirect
108. C17#3291semidirect

Maximal subgroups

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

subgroupindex
#117636152
#117630692
#117629912
#117629762
#117629272
C17×#4922
#329117