Finite Group Databasebeta

← #14538176

Subgroups — #14538176

Order 2176.

766 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 766 conjugacy classes of subgroups — too many to draw. Only its 161 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#11763316C213yesmaximal, complemented
2#11762999C214yesmaximal, complemented
3#11762996C214yesmaximal, complemented
4#11762783C213yesmaximal, complemented
5C17×#585C211yesmaximal, characteristic, complemented
6C17×2−1+4—23no—
7C17×H(5,𝔽2)—23no—
8C17×H(5,𝔽2)C2214yes—
9C2×#41×C17—26no—
10C2×#41×C17C2213yes—
11C22×C4×D17C2211yescharacteristic, complemented
12#10587632—23no—
13#10587624—23no—
14#10587623C2216yescomplemented
15#10587621C2216yescomplemented
16C4×C22⋊D17—26no—
17#10587612C2213yescomplemented
18#10587505C22112yescomplemented
19C23×D17C2311yescharacteristic
20#41×C17—24no—
21#41×C17—212no—
22#41×C17—26no—
23#41×C17—212no—
24#41×C17—26no—
25C2×Q8×C17—26no—
26C2×Q8×C17D413yescomplemented
27C2×D4×C17—26no—
28C2×D4×C17—212no—
29C2×D4×C17D413yescomplemented
30C2×D4×C17C2316yes—
31C22×C4×C17—24no—
32C22×C4×C17D416yescomplemented
33C22×C4×C17C2311yescharacteristic
34C2×C22⋊D17—26no—
35C22×Dic17C2311yescharacteristic
36C2×C4×D17—23no—
37C2×C4×D17—23no—
38C22⋊Dic17—26no—
39C22⋊Dic17C2316yes—
40#63227—26no—
41C4⋊Dic17—23no—
42#63225—26no—
43C4×Dic17—23no—
44C23×C17—24no—
45C23×C17C2×D416yes—
46C23×C17C2411yescharacteristic
47C22×D17—23no—
48C22×D17—42no—
49Q8×C17—24no—
50Q8×C17—26no—
51D4×C17—212no—
52D4×C17—212no—
53D4×C17—26no—
54C2×C4×C17—212no—
55C2×C4×C17—26no—
56C2×C4×C17—26no—
57C2×C4×C17C2×D416yes—
58C2×C4×C17C2×D413yes—
59C2×C4×C17C2×D413yes—
60C22⋊D17—412no—
61C2×Dic17—23no—
62C2×Dic17—26no—
63C4×D17—43no—
64C4×D17—41no—
65#3025—171nomaximal, Sylow
66C22×C17—26no—
67C22×C17—41no—
68C22×C17—46no—
69C22×C17—84no—
70C22×C17C22×D413yes—
71C2×D17—26no—
72C2×D17—43no—
73C2×D17—41no—
74C4×C17—26no—
75C4×C17#12013yes—
76C4×C17#12011yescharacteristic
77Dic17—46no—
78Dic17—41no—
79#585D1711yescharacteristic, complemented
80#525—173no—
81#458—174no—
82B(4,𝔽2)—174no—
83#410—173no—
84C2×C17—23no—
85C2×C17—46no—
86C2×C17C2×#12011yescharacteristic
87D17—42no—
882−1+4—23no—
89H(5,𝔽2)—23no—
90H(5,𝔽2)C2×D1714yes—
91C2×#41—26no—
92C2×#41C2×D1713yes—
93C23×C4—171no—
94#123—176no—
95C22⋊Q8—343no—
96#121—343no—
97#120—176no—
98C4×D4—346no—
99#117—173no—
100#99—1712no—
101C17#302511yescharacteristic, complemented, Sylow
102C24—171no—
103#41—24no—
104#41—212no—
105#41—26no—
106#41—212no—
107#41—26no—
108C2×Q8—26no—
109C2×Q8C22⋊D1713yescomplemented
110C2×D4—26no—
111C2×D4—212no—
112C2×D4—346no—
113C2×D4C22⋊D1713yescomplemented
114C2×D4C22×D1716yes—
115C22×C4—24no—
116C22×C4—171no—
117C22×C4—343no—
118C22×C4—343no—
119C22×C4C22⋊D1716yescomplemented
120C22×C4C22×D1711yescharacteristic
121C4⋊C4—343no—
122C4⋊C4—346no—
123C22⋊C4—176no—
124C22⋊C4—346no—
125C22⋊C4—346no—
126C42—343no—
127C23—24no—
128C23—343no—
129C23—682no—
130C23C2×C22⋊D1716yes—
131C23C23×D1711yescharacteristic
132Q8—24no—
133Q8—26no—
134D4—212no—
135D4—212no—
136D4—26no—
137D4—6812no—
138C2×C4—212no—
139C2×C4—26no—
140C2×C4—26no—
141C2×C4—346no—
142C2×C4—343no—
143C2×C4—683no—
144C2×C4—681no—
145C2×C4C2×C22⋊D1716yes—
146C2×C4C2×C22⋊D1713yes—
147C2×C4C2×C22⋊D1713yes—
148C22—26no—
149C22—41no—
150C22—46no—
151C22—84no—
152C22—346no—
153C22—683no—
154C22—681no—
155C22C22×C22⋊D1713yes—
156C4—26no—
157C4—686no—
158C4—681no—
159C4#1058763913yes—
160C4#1058763911yescharacteristic
161C2—23no—
162C2—46no—
163C2—682no—
164C2C2×#1058763911yescharacteristic
subgroupG/H
centerC4#10587639
commutator, cosocleC23×C17C24
FrattiniC23C23×D17
FittingC17×#585C2
socleC2×C17C2×#120
hypercenter#585D17
hypocenterC17#3025
LCSG ▷ C23×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C4 ◁ C22×C4 ◁ #585

Complements

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

NH ≅ G/Ntype
1. #11763316C2semidirect
2. #11762999C2semidirect
3. #11762996C2semidirect
4. #11762783C2semidirect
5. C17×#585C2semidirect
11. C22×C4×D17C22semidirect
14. #10587623C22semidirect
15. #10587621C22semidirect
17. #10587612C22semidirect
18. #10587505C22semidirect
26. C2×Q8×C17D4semidirect
29. C2×D4×C17D4semidirect
32. C22×C4×C17D4semidirect
79. #585D17semidirect
101. C17#3025semidirect
109. C2×Q8C22⋊D17semidirect
113. C2×D4C22⋊D17semidirect
119. C22×C4C22⋊D17semidirect

Maximal subgroups

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

subgroupindex
#117633162
#117629992
#117629962
#117627832
C17×#5852
#302517